IN MEMORIAM 
 FLORIAN CAJORl 
 
HIGH SCHOOL ALGEBRA 
 
 BY 
 
 J. H. TANNER, Ph.D. 
 
 PROFESSOR OF MATHEMATICS IN CORNELL UNIVERSITY 
 
 NEW YORK .:• CINCINNATI •:• CHICAGO 
 
 AMERICAN BOOK COMPANY 
 
The Modern Mathematical Series, 
 lucien augustus wait, 
 
 (^Senior Professor of Mathematics in Cornell University,) 
 GENERAL EDITOR. 
 
 This series includes the following works : 
 ANALYTIC GEOMETRY. By J. H. Tanner and Joseph Allen. 
 DIFFERENTIAL CALCULUS. By James McMahon and Virgil Snyder. 
 INTEGRAL CALCULUS. By D. A. Murray. 
 
 DIFFERENTIAL AND INTEGRAL CALCULUS. By Virgil Snyder and J. 
 Hutchinson. 
 
 HIGH SCHOOL ALGEBRA. By J. H. Tanner. 
 ELEMENTARY ALGEBRA. By J. H. Tanner. 
 ELEMENTARY GEOMETRY. By James McMahon. 
 
 The High School Algebra and the Elementary Algebra cover substantially the 
 same ground : each of them is designed to meet college entrance requirements in 
 elementary algebra ; the High School Algebra, however, presents the briefer and 
 simpler treatment of the two. 
 
 COi'VKlGHT, 190T, BY 
 
 J. H. TANNER. 
 
 TANNBB'8 HIGH 8CH. ALG. 
 W. P. I. 
 
-3+ 
 
 PREFACE 
 
 In the preparation of this book the author's aim has been : 
 
 (1) To make the transition from arithmetic to algebra as easy 
 and natural as possible, and to arouse the pupil's interest by 
 showing him early some of the advantages of algebra over 
 arithmetic. 
 
 (2) To present the several topics in the order of their sim- 
 plicity, giving definitions only where they are needed, and insur- 
 ing clearness of comprehension by an abundance of concrete 
 illustrations and inductive questions. 
 
 (3) To provide a large, well-chosen, and carefully graded set of 
 exercises, the solution of which will help not only to fix in the 
 pupil's mind the principles involved, but also further to unfold 
 those principles. 
 
 (4) To omit non-essentials, and yet provide a book that fully 
 meets the entrance requirements in elementary algebra of any 
 college or university in this country. 
 
 Among other features of this book to which attention is in- 
 vited are: (1) the careful statement of definitions and principles; 
 (2) the emphasis laid upon translating formulas and equations 
 into verbal language, and vice versa; (3) the inclusion of many 
 formulas from physics which the pupils are asked to solve for 
 the various letters which they contain; and (4) the extensive 
 cross-references, as well as the many " hints " and " suggestions " 
 found among the exercises and problems, all calculated to throw 
 sidelights upon the work. 
 
 On the request of several prominent mathematics teachers the 
 author has put an elementary chapter on quadratic equations 
 (Chap. XII) before the chapters on radicals, imaginaries, and the 
 theory of exponents. This arrangement is made possible by the 
 treatment of radicals of the second order given in § 121 (only such 
 
iv PREFACE 
 
 radicals are met with, in Chap. XII), and it has important peda- 
 gogical as well as practical advantages over the more usual 
 arrangement. Those teachers, however, who prefer the usual 
 order, may omit § 121 altogether, and take Chaps. XIV and XV, 
 except §§ 162, 163, and 170, before taking Chap. XII. 
 
 In order to avoid unnecessary repetition, the work on graphs 
 has nearly all been collected into a single chapter. This arrange- 
 ment has made it possible to give this topic a somewhat more 
 adequate treatment than is usual in a book of this kind, and to 
 do so without giving it more than its rightful amount of space. 
 By this arrangement also those schools w^hich do not take graphs 
 in their first year's work will find their algebra work unin- 
 terrupted, while appropriately .placed footnotes indicate the 
 connection in which the parts of this chapter may be most advan- 
 tageously read by those who wish to include graphs. 
 
 Scattered through the book are a few articles (marked with 
 a *) which, should be taken in connection with the review work 
 when time permits ; the omission of these articles, however, does 
 not anywhere break the continuity of the work. For the benefit 
 of the brightest pupils in a class there have been inserted here 
 and there references to the author's Elementary Algebra, where 
 the topics concerned are discussed somewhat more fully ; a few 
 copies of the Elementary Algebra placed in the school library can 
 in this way be made to serve a very useful purpose. 
 
 It is with great pleasure that the author acknowledges his in- 
 debtedness to the many experienced teachers of Algebra in High 
 Schools and Academies in all quarters of our country whose sug- 
 gestions to him have added so much of value to this book. Special 
 acknowledgments are due to Prof. J. M. McPherron of the Los 
 Angeles High Schools for reviewing the manuscript before it 
 went to press, and to Miss Cora Strong of the State Normal 
 School of Greensboro, N.C., who secured a leave of absence from 
 her school so that she might give her entire time to assisting in 
 the work on this book ; to her mainly belongs the credit for the 
 excellent exercises which are found in the following pages. 
 
CONTENTS 
 
 [See Index in back of book for particular topics.] 
 
 CHAPTER PAGE 
 
 I. Introduction 
 
 I. Literal Numbers . 1 
 
 II. Elementary Operations 8 
 
 11. Positive and Negative Numbers 16 
 
 III. Addition and Subtraction — Parentheses' 
 
 I. Addition 27 
 
 II. Subtraction 32 
 
 III. Parentheses 35 
 
 IV. Multiplication and Division 
 
 I. Multiplication . . . . . . . .38 
 
 11. Division 45 
 
 V. Equations and Problems 
 
 Review Exercise — Chapters I-V 67 
 
 VI. Type Forms in Multiplication — Factoring 
 
 I. Some Type Forms in Multiplication ... 71 
 11. Factoring 77 
 
 VII. Highest Common Factor — Lowest Common Multiple 
 
 I. Highest Common Factor 98 
 
 II. Lowest Common Multiple 105 
 
 VIII. Algebraic Fractions 109 
 
 IX. Simple Equations 125 
 
 X. Simultaneous Simple Equations 
 
 I. Two Unknown Numbers 144 
 
 11. Three or More Unknown Numbers .... 156 
 Review Exercise — Chapters VI-X . . . . .166 
 
 V 
 
VI 
 
 CONTENTS 
 
 C^KVT^^ PAGE 
 
 XI. Involution and Evolution 
 
 I. Involution 170 
 
 II. Evolution 176 
 
 XII. Quadratic Equations (Elementary) 
 
 I. Equations in One Unknown Number . . ' . 190 
 
 II. Simultaneous Equations involving Quadratics . 206 
 
 XIII. Graphic Representation of Equations . . . 219 
 
 XIV. Irrational Numbers — Radicals 234 
 
 XV. Imaginary Numbers 257 
 
 XVI. Theory of Exponents (Zero, Negative, and Fractional 
 
 Exponents) 265 
 
 XVII. Quadratic Equations (Supplementary to Chapter XII) 277 
 
 Review Exercise — Chapters XI-XVII .... 284 
 
 XVIII. Inequalities 287 
 
 XIX. Ratio, Proportion, and Variation 
 
 I. Ratio 293 
 
 II. Proportion 295 
 
 III. Variation . 301 
 
 XX. Series — The Progressions 
 
 I. Arithmetical Progression 307 
 
 II. Geometric Progression 313 
 
 XXI. Mathematical Induction — Binomial Theorem . 321 
 
 XXII. Logarithms 329 
 
 Table of Common Logarithms 341 
 
 INDEX 343 
 
HIGH SCHOOL ALGEBRA 
 
 • CHAPTER I 
 INTRODUCTION 
 I. LITERAL NUMBERS 
 
 1. Algebra. In the following pages, we shall continue to 
 use the symbols 0, 1, 2, 3, etc. to represent numbers, and 
 the signs +, — , x, -^, and =, to denote addition, subtrac- 
 tion, multiplication, etc.; that is, we shall use all of these 
 characters just as we used them in arithmetic. We shall 
 presently see, however, that algebra greatly simplifies the 
 solution of certain kinds of problems (§ 3), that it introduces 
 new kinds of number (§§ 13, 146, 164), and that it makes 
 extensive use of letters to represent numbers. 
 
 2. Numbers represented by letters. In arithmetic, num- 
 bers are almost always expressed by means of the symbols 
 0, 1, 2, 3, etc., but letters also are sometimes used. For 
 example, in interest problems p often stands for principal^ 
 r for rate^ t for time^ i for interest^ and a for amount. 
 
 In algebra, on the other hand, the use of letters to repre- 
 sent numbers is very common ; thus, just as in arithmetic 
 we speak of 4 books, 7 bicycles, 85 pounds, 3 men, etc., so 
 in algebra we use, not only these expressions, but also such 
 expressions as a books, n bicycles, x pounds, y men, etc. 
 Numbers represented by letters are often called literal 
 numbers. 
 
 In the case of literal numbers the operations of addition, 
 subtraction, etc. may be indicated just as these operations 
 are indicated with arithmetical numbers. Thus, for example, 
 
 1 
 
2 HIGH SCHOOL ALGEBRA [Ch. I 
 
 if n represents one number and k another, then n-{-k stands 
 for their sum, n — k for their difference, n x k for their 
 
 product, n-i-k or - for their quotient. 
 k 
 
 EXERCISE I 
 
 If a stands for 3, b for 2, and x for 12, find the value of each 
 of the following expressions : 
 
 - 2 a -\- X o 3 abx — ah 
 3. • o. • 
 
 h ab-{-hx 
 
 ^ a-\-hx ^ X 4:a x 
 
 1. 
 
 a-\-h. 
 
 2. 
 
 x — a. 
 
 3. 
 
 x-i-a. 
 
 4. 
 
 5b--* 
 
 3a a 2b 2 
 
 7 ah -{-2 X —10 ^ 10 ^ 4- ^~^^ 
 
 4a-f 46 a b 
 
 11. If s represents 16, what number is represented by 2s? 
 
 s 3 s 
 
 by Js, or (as usually written) -? by — ? 
 
 4 8 
 
 12. If a suit of clothes costs 8 times as much as a hat, and 
 if h stands for the cost of the hat, how may the cost of the suit 
 be represented ? 
 
 13. Does h-\-Sh (i.e., 9h) represent the combined cost of the 
 hat and suit in Ex, 12 ? Explain your answer. 
 
 14. The side of a square is 5 feet long. How long is the 
 bounding line of this square? How long is the bounding line 
 if the side is x feet long ? 
 
 15. A boy's present age is 15 years ; indicate, without perform- 
 ing the subtraction, his age 4 years ago. What was his age n 
 years ago ? What will it be y years hence ? 
 
 16. At 5 cents each how many erasers can be bought for 15 
 cents ? for x cents ? for n dollars ? 
 
 17. What number multiplied by 8 gives the product 40? If 
 8 a; = 40, what is the value of x? If 5y -\-2y = 21, what is the 
 value of y ? 
 
 * 6b means 5 times b ; so too ab means a times b ; and 3 ax means the 
 product of 3, a, and x. 
 
2-^ INTliODUCTION 3 
 
 3. One advantage of literal numbers. The following ex- 
 amples show how the solution of problems may often be 
 simplified by using letters to represent numbers. 
 
 Prob. 1. A gentleman paid $45 for a suit of clothes and a 
 hat. If the clothes cost 8 times as much as the hat, what was 
 the cost of each ? 
 
 ARITHMETICAL SOLUTION 
 
 The hat cost "a certain sum/' and since the clothes cost 8 times 
 as much as the hat, therefore the cost of the clothes was 8 times 
 " that sum," and the cost of the two together was 9 times " that 
 sum." Hence 9 times "that sum" is $45, and therefore 'Hhat 
 sum" is $5, and 8 times "that sum" is $40; i.e., the hat cost 
 $5, and the clothes $40. 
 
 This solution may be put into the following more systematic 
 form, still retaining its arithmetical character. 
 
 A certain sum = the cost of the hat ; 
 then •.• * 8 tim,es that sum = the cost of the clothes, 
 
 9 times that sum = the cost of both, 
 i.e.j 9 times that sum = $45. 
 
 that sum = $ 5, the cost of the hat, 
 and 8 times that sum = $ 40, the cost of the clothes. 
 
 ALGEBRAIC SOLUTION 
 
 The solution just given becomes much simpler if we let a 
 single letter, say x, stand for the number of dollars in " a certain 
 sum " and " that sum " as used above, thus : 
 
 Let X = the number of dollars the hat cost. 
 
 Then Sx = the number of dollars the clothes cost, 
 
 and x + S x = the number of dollars both cost ; 
 i.e., 9 a; = 45. 
 
 a; = 5, and 8 a? = 40 ; 
 i.e., the hat cost $5, and the clothes cost $40. 
 
 N.B. The letter x, above, stands for a number, not for the cost of the hat. 
 
 * The symbols •.• and .-. stand for the words "shice" and "therefore," 
 respectively. 
 
4 HIGH SCHOOL ALGEBRA [Ch. I 
 
 Prob. 2. If a locomotive weighs 3 times as much as a car, and 
 the difference between their weights is 50 tons, what is the weight 
 of the locomotive ? 
 
 SOLUTION 
 
 Let w = the number of tons in the weight of the car. 
 Then Sw = the number of tons in the weight of the locomotive, 
 and, since the difference between their weights is 50 tons, 
 
 3w — w = 50, 
 i.e.f 2w = 50, 
 
 whence iv = 25, 
 
 and 3w = 75; 
 
 i.e., the locomotive weighs 75 tons. 
 
 Prob. 3. Of three numbers the second is 5 times the first, and 
 the third 2 times the first ; if the sum of these numbers equals 
 the third number increased by 42, what are the numbers ? 
 
 SOLUTION 
 
 Let n = the first of the three numbers. 
 
 Then 5n = the second number, 
 
 and 2 71= the third number ; 
 
 now since the sum of the three equals the third number increased 
 by 42, 
 
 .-. n-{-5n-j-2n = 2n + 4:2, 
 i.e., Sn = 2n-\- 42, 
 
 hence 6 n = 42. [Subtracting 2 n from each of the equal 
 
 sums above.] 
 n = 7, 5n = 35, and 2 n = 14 ; 
 i.e., the numbers are 7, 35, and 14, respectively. 
 
 Remark. Observe that the steps in each of the foregoing solu- 
 tions are : 
 
 1. To let some letter, say x, stand for one of the unknown 
 numbers (preferably the smallest). 
 
 2. To express the other unknown numbers in terms of x. 
 
 3. To translate into algebraic language those relations between 
 the unknown numbers which the problem states in words ; this 
 
3] INTRODUCTION 5 
 
 translation gives an equation, and from it the required numbers 
 are easily found. 
 
 Observe also that while the above problems can be solved by 
 arithmetic, the algebraic solution is much simpler. 
 
 EXERCISE 11 
 
 Solve the following problems : 
 
 4. In a room containing 45 pupils there are twice as many 
 boys as girls. How many boys are there in the room ? 
 
 5. If a horse costs 7 times as much as a saddle, and if the 
 difference in the cost of the two is $ 90, find the cost of each. 
 
 6. A house is worth 5 times as much as the lot on which it 
 stands, and the two together are valued at $4200. Find the 
 value of each. 
 
 7. If the house and lot of Ex. 6. differ in value by $4200, how 
 much is each worth ? 
 
 8. The double of a certain number taken from 10 times the 
 same number leaves 72. What is the number ? 
 
 9. If n represents a certain number, how may we represent: 
 
 (1) the number, plus 4 times itself, plus 5 times itself ? 
 
 (2) the sum of the number, its double, and its half ? 
 What does 5 -a? -f- 7 n — 3 n represent ? 
 
 10. A number, plus twice itself, plus 4 times itself, is equal to 
 56. What is the number ? 
 
 11. Divide 98 into three parts such that the second is twice 
 the first and the third is twice the second. 
 
 12. Divide 160 into three parts such that two of them are equal, 
 while the third is twice either of the others. 
 
 13. In a yachting party consisting of 36 persons, the number 
 of children is 3 times the number of men, and the number of 
 women is one half that of the men and children combined. How 
 many women are there in the party ? 
 
 14. If I have s nickels, how many cents have I ? How many 
 cents in s dimes? in s quarters? in the sum of s nickels, s 
 dimes, and s quarters ? 
 
6 TUGn SCHOOL ALGEUnA [Ch. I 
 
 15. A boy found that he had the same number of 5, 10, and 25 
 cent pieces, and that the total amount of' his money was ^ 3.20. 
 How many coins of each kind had he ? 
 
 16. Alice buys Christmas gifts at 25 cents, 15 cents, and 10 
 cents — the same number at each price. If she spends $2 in 
 all, how many gifts does she buy ? 
 
 17. If a; stands for a certain number, what would stand for 
 
 (1) the double of the number, increased by 7 ? 
 
 (2) the difference between 3 times the number and 8 ? 
 
 18. How would you represent two numbers whose difference is 
 4 ? two numbers whose sum is 13 ? 
 
 19. Find the number whose double, with 4 added, equals 46. 
 
 20. Find two numbers, differing by 7, whose sum is 35. Also 
 find two numbers whose sum is Q^ and whose difference is 15. 
 
 21. William has 8 cents more than his sister Harriet, and 
 together the two have 80 cents. How much money has each? 
 If Harriet's money is made up of an equal number of nickels and 
 one-cent pieces, how many nickels has she ? 
 
 22. In a family of seven children each child is 2 years older 
 than the next younger. If the sum of their ages is 84 years, how 
 old is the youngest child ? 
 
 23. A father's age is now 3 times that of his son ; 5 years hence, 
 the sum of their ages will be 62 years. Find the present age of 
 each. (Cf. Ex. 15, p. 2.) 
 
 24. Four years ago Isabel was twice as old as Mabel ; the sum 
 of their present ages is 32 years. How old is each ? • 
 
 25. In a business enterprise, the combined capital of A, B, and 
 C is f 21,000. A's capital is twice B's, and B's is twice C's. 
 What is the capital of each ? 
 
 26. In the triangle MNP, NP'is 2 inches longer 
 than MN, while PM and JfJVare of equal length. 
 If the sum of the three sides is S6 inches, find 
 the length of each. 
 
 27. An east-bound and a west-bound train 
 leave Chicago at the same hour, the first running twice as fast 
 
 N P 
 
3] INTRODUCTION 7 
 
 as the second ; after one hour they are 90 miles apart. Find the 
 speed of each. 
 
 28. In a fishing party consisting of four boys, two of the boys 
 caught each the same number of fish, another caught 2 more than 
 this number, and the fourth, 1 less. If the total number of fish 
 caught was 29, how many did each catch ? 
 
 29. An estate valued at $24,780 is to be divided among a family 
 consisting of a mother, two sons, and three daughters. If the 
 daughters are to receive equal shares, each son twice as much as a 
 daughter, and the mother twice as much as all the children to- 
 gether, what will be the share of each ? ^ ^ 
 
 30. ABCD represents the floor of a room. 
 Find the dimensions of the floor if its bound- 
 ing line is 48 feet long. 
 
 31. A gallon of cream is poured into two 
 
 (a;+4) feet 
 
 pitchers, one of which holds 7 times as -D G 
 
 much as the other. How many gills does each pitcher hold? 
 
 32. If i of a number is added to the number, the sum is 120. 
 What is the number ? 
 
 Suggestion. Let 3 x = the number. 
 
 33. If ^ of a number is added to twice the number, the sum is 
 35. What is the number ? 
 
 34. Of two numbers, twice the first is 7 times the second, and 
 their difference is 75. Find the numbers. 
 
 Suggestion, Let 1 x = the first number, then 2 x = the second. 
 
 35. An estate of f 19,600 was so divided between two heirs 
 that 5 times what one received was equal to 9 times what the 
 other received. What was the share of each ? 
 
 36. A tree whose height was 150 feet was broken off by the 
 wind, and it is found that 3 times the length of the part left 
 standing is the same as 7 times that of the part broken off. How 
 long is each part ? 
 
 37. If two boys together solved 65 problems, and if 8 times the 
 number solved by the first boy equals 5 times the number solved 
 by the second boy, how many did each boy solve ? 
 
8 HIGH SCHOOL ALGEBRA [Ch. I 
 
 II. ELEMENTARY OPERATIONS 
 
 4. Addition. In algebra, as in arithmetic, such an ex- 
 pression as 7 + 3 is read " 7 plus 3," and means that 3 is to 
 be added to 7. 
 
 To perform this addition we begin at 7 and count 3 for- 
 ward, obtaining the result 10, which is called the sum of 
 these two numbers.* 
 
 So also if a and b stand for any two numbers whatever, 
 the expression a-\-b is read "a plus 5," and means that h is 
 to be added to a. 
 
 The result obtained by adding two or more numbers is 
 called their sum, and the numbers that are to be added are 
 called the summands. 
 
 5. Subtraction. To what number must 3 be added to 
 obtain the sum 8? If 8 was obtained by adding 3 to some 
 number (i.e.^ by counting 3 forward}, how may we, starting 
 with 8, find the number at which the counting began ? 
 
 Here, as in arithmetic, the operation of finding this num- 
 ber is indicated by the expression 8 — 3, which is read " 8 
 minus 3." We may say that 8 — 3=5 because 5 + 3 = 8. 
 
 The process of finding one of two numbers when their 
 sum and the other number are given, is called subtraction. 
 It consists, as we have just seen, in counting backward, i.e., 
 in undoing the work of addition, which consists in counting 
 forward. 
 
 If a and b stand for any two numbers whatever, the ex- 
 pression a — b is read " a minus 5," and means that b is to be 
 subtracted from a. 
 
 The result obtained by subtracting one number from an- 
 other is called their difference (also the remainder). The 
 number which is to be subtracted is called the subtrahend, and 
 
 * If fractions are to be added, we first reduce tliera to a common denomi- 
 nator and tlien add their numerators ; it is still a counting process. 
 
4-C] INTRODUCTION 9 
 
 the one from which the subtraction is to be made is called 
 the minuend. 
 
 6. Inverse operations. Of two operations which neutralize 
 each other when performed in succession, each is called the 
 inverse of the other. Thus the operations of addition and 
 subtraction are each the inverse of the other (cf. Exs. 
 8-10, below). 
 
 EXERCISE- III 
 
 Read each of the following expressions, then name its parts : 
 1. 8 + 12 = 20. 2. 9-7 = 2. 3. 12y-9y = 3y, 
 
 4. Since 4 + 9 = 13, therefore 13 - 9= ? 13 -4 = ? 
 
 5. In subtraction, what name is used to denote the given sum ? 
 the given summand? the required summand ? Illustrate, using 
 Ex. 2 above. 
 
 6. Add 4 to 7 by counting. Where do you begin to count ? 
 In what direction do you count? 
 
 7. By counting, subtract 4 from 11. Do you count in the 
 same direction as in Ex. 6 ? 
 
 8. How may you combine the subtrahend* and remainder to 
 get the minuend ? Why ? 
 
 9. How would you test the correctness of an answer in sub- 
 traction? Illustrate. Could you use subtraction to test the 
 correctness of a sum ? 
 
 10. When is one operation said to be the inverse of another ? 
 Using the numbers 8 and 6, illustrate the fact that subtraction is 
 the inverse of addition. 
 
 11. If m and n stand for any two given integers whatever, can 
 you, by counting, find the value of ?7i + n ? of m — n ? 
 
 12. What is the value of 5 - 3 ? of5-4? of5-5? of5-6? 
 of 5 — 8 ? In order that subtraction be possible, how must the 
 subtrahend compare in size with the minuend ?"* 
 
 * With our present (arithmetical) meaning of number such a subtraction 
 as 5 — 8 is, of course, impossible ; in Chapter II, however, we shall so extend 
 the meaning of number as to make the subtraction a — b possible even when 
 b is greater than a. 
 
10 niGH SCHOOL ALGEBRA [Cit. I 
 
 7. Multiplication, (i) In arithmetic, multiplication is 
 usually defined as the process of taking (additively) one 
 of two numbers, called the multiplicand, as many times as 
 there are units in the other, called the multiplier. In this 
 sense, 6x4 (read "6 multiplied by 4") means 6 + 64-6 + 6; 
 i.e.^ this multiplication may be regarded as an abbreviated 
 addition. 
 
 Strictly speaking, however, the above definition of multi- 
 plication applies only when the multiplier is an arithmetical 
 integer : under this definition, for instance, we could not find 
 such a product as 8 x 51, because we could not take the mul- 
 tiplicand two thirds of a time any more than we could fire a gun 
 two thirds of a time. 
 
 (ii) A broader definition of multiplication, and one bet- 
 ter suited to our present purpose, may be stated thus : 
 
 Multiplication is the process of performing upon one of 
 two given numbers (the multiplicand) the same operation 
 as that which is performed upon unity to get the other given 
 number (the multiplier) ; the result thus obtained is called 
 the product of these numbers. The multiplicand and multi- 
 plier are called factors of the product. 
 
 To illustrate, consider again the question of multiplying 
 8 by 5|-. The multiplier, 5|-, is obtained from unity by tak- 
 ing the unit 5 times, and J of the unit twice, as summands, 
 i.e., 5f = 1 + 1 + 1 + 1 + 1 + 1+ J; 
 
 and, therefore, by this new definition of multiplication, 
 8x5f = 8 + 8 + 8 + 8 + 8 + | + f = 40 + J^ = 45f 
 
 (iii) Just as 6 X 5 means that 6 is to be multiplied by 5, 
 so 5 X 3 means that b is to be multiplied by 3. Similarly, 
 kxn xy means that h is to be multiplied by n^ and that 
 their product is then to be multiplied by y. 
 
 Instead of the oblique cross ( x ), a center point (•) placed 
 between two numbers (a little above the line to distinguish 
 it from the decimal point) is frequently used as a sign of 
 
7-8] INTttOhVCTlON . 11 
 
 multiplication. And even the center point is usually omitted 
 if doing so causes no confusion. Thus, 8xri = o-'/i=3n; 
 so, too, p xr xt=p • r ' t=prt, and 3 x 7 = o • 7. But the 
 sign (cross or center point) must not be omitted between two 
 arithmetical numbers. (Why not?) 
 
 EXERCISE IV 
 Kead each of the following expressions, then name its parts : 
 1. 8x3 = 24. 2. f. 15 = 10. 3. 5«.4 = 20a. 
 
 4. What is the value of 5 • 3 ? How is this product obtained 
 under the old definition of multiplication [§ 7 (i)] ? 
 
 5. Using the new definition [§ 7 (ii)], show that 5 • 3 means 
 5 + 5 + 5. Similarly, explain the meaning of 9 • 4; of 4 • 9. 
 
 6. Show that 2| • 8 has the same meaning under the old defi- 
 nition of multiplication as under the new. 
 
 7. To get f from 1, we divide 1 into how many equal parts ? 
 How many of these parts do we take '? What, then, should be 
 done to 10 in multiplying it by J ? 
 
 As in Ex. 7, find the following products : 
 
 8. 16. f. 9. 12. 2f 10. 7.5f. 
 
 If a = 2, 6 = 5, and a? = |, find the value of each of the follow- 
 ing expressions : 
 
 11. Sabx. 13. 2bx — ax + 3b. 15. Saax + 4:bx. 
 
 12. 5b + 6x-ab. 14. 7abx + 3a-2b. 16. aab — 10 x. 
 
 8. Division. Division is the inverse (i.e.^ihe "undoing") 
 of multiplication. Thus, since 4 x 9 = 36, therefore 36 -=- 9 = 4, 
 and 36 -4 =9. 
 
 The expression 36 -^ 9 = 4 is read " 36 divided by 9 equals 
 4." Here 36 is called the dividend, 9 the divisor, and 4 the 
 quotient. 
 
 In multiplication we have given two numbers, and are 
 asked to find their product ; in division we have given the 
 product (now called the dividend} and one of the factors 
 
 HIGH SCH. ALG. — 2 
 
1^ tilGH SCHOOL ALGEBRA [Ch. 1 
 
 (now called the divisor')^ and are asked to find the other 
 factor (now called the quotient). 
 
 Hence we may say : division is the process of finding from 
 two given numbers, called dividend and divisor, respectively, 
 a third number (called the quotient) such that the divisor 
 multiplied by the quotient equals the dividend. 
 
 E.g., 36 -- 9 = 4, because 4 x 9 = 36. 
 
 If 8 and t represent any two numbers whatever, then each 
 
 of the expressions, s -?- f , -, s/U and s : t indicates that s is to 
 be divided by t. 
 
 If the divisor is not exactly contained in the dividend, 
 then, as in arithmetic, the indicated division is called a 
 fraction. 
 
 E.g.^ ^, -— , — , and — ^t_ are called fractions. 
 6 D n y . 
 
 It is to be remarked, however, that literal numbers 
 may be fractional in form but integral in value, and 
 
 vice versa. Thus, -, though fractional in form, has the inte- 
 gral value 3 if a = 12 and 5 = 4. 
 
 9. Powers, exponents, etc. (i) In algebra, as in arithmetic, 
 such a product as 5 • 5 • 5 is usually written in the abbrevi- 
 ated form 5^, the small 3 showing the number of times that 
 5 is used as a factor. 
 
 Similarly, 23 z= 2 • 2 • 2, a^^a-a-a, 2^ . 52= 2 • 2 • 2 • 5 • 5, 
 7i2jt>4 z= n^ ' p^ = n ' 71 • p ' p ' p • p, etc. 
 
 The expression k^ is usually called the fourth power of h. 
 In this expression, 4 is called the exponent and h the hase of 
 the power. 
 
 (ii) Hence the following definitions : A power of a number 
 is the product arising from using the given number one or 
 more times as a factor. 
 
 An exponent is a number placed (in small symbols) at the 
 
8-9] INTRODUCTION " 13 
 
 right and slightly above a given number, to show how many 
 times the latter is to be used as a factor. 
 
 Thus, if X represents any number whatever, and n any 
 arithmetical integer,* then the expression x^ is called the nth 
 power of 07, and means the product arising from using x as 
 a factor n times ; n is the exponent of the power. 
 
 Note. Observe that under the above definitions a^ has the same mean- 
 ing as a ; the exponent 1, therefore, need not be written. 
 
 The second and third powers of numbers are, for geometrical reasons, 
 often called by the special names of square and cube respectively. Thus a^ 
 is called " the second power of a," " the square of a," and also " a squared." 
 
 EXERCISE V 
 
 Eead each of the following expressions, name its parts, and 
 test the correctness of the results : 
 
 1. 18-6 = 3. 4. ?5^ = 4a. 7. ^ = Z2. 
 
 2. 28^14 = 2. 5. 9.^81. k^^4a^ 
 
 3. 6_|o = 70. 6. 2^.32 = 288. * 32 3 * 
 Read the following expressions and tell what operations are 
 
 indicated in each case; then find the numerical value of each 
 expression when a = 5, 6 = 2, 7i = l, and ic = 4. 
 
 .5 ,^ 3 6^ „ a^-lOW 
 
 9. a\ 12. ^^1^. 15. 
 
 16 3 a 
 
 10- «' + ^'- 13. 8W-10a^. ^^ 75 -aW 
 
 11. 7wV. 14. 7i^ + bax\ ' y?-a ' 
 
 17. Write 7«7-7«7 by means of the exponent notation. 
 Also a ' a'a\ 5 • 5 • a; • a; • a; ; and 9 • 9 • 9 • 9 • a • a • ?/ • ?/ • 2/. 
 
 18. How may we use multiplication to test the correctness of 
 an example in division ? Why ? 
 
 19. The sum of any two integers is integral. Is this true of 
 their difference ? of their product ? of their quotient ? Illus- 
 trate your answers. 
 
 * We shall later (Chapter XVI) enlarge the scope of such a symbol as x" by 
 giving it a meaning even when n does not represent an arithmetical integer. 
 
14 HIGH SCHOOL ALGEBRA [Ch. I 
 
 20. When the dividend is not exactly divisible by the divisor, 
 what name is given to the indicated quotient ? 
 
 5 
 
 21. How are fractions defined in arithmetic ? Is — a frac- 
 
 tion under the arithmetical definition ? If not, why not ? 
 
 10. The order in which arithmetical operations are to be 
 performed. What is the value of 2 + 6 • 5 — 8 ^ 2 ? Is it 
 28, 16, 'or 12? In order that such an expression shall have 
 the same meaning for all of us, mathematicians have agreed 
 that, when there is no express statement to the contrary : 
 
 (1) A succession of multiplications and divisions shall 
 mean that these operations are to be performed in the order 
 in which they occur from left to right. 
 
 (2) A succession of additions and subtractions shall mean 
 that they are to be performed in the order in which they 
 occur. 
 
 E.g., 9 . 8 -- 6 . 2 = 72 -- 6 . 2 = 12 . 2 = 24, 
 
 but 9 • 8 -J- 6 • 2 is 710^ equal to 72 -h 12, i.e., to 6. 
 
 So, too, 7 + 9-6 + 3 = 16 -6 + 3 = 10 + 3 = 13, 
 but 7 + 9 — 6 + 3 is 710^ equal to 16 — 9, i.e., to 7. 
 
 (3) A succession of the operations of addition, subtrac- 
 tion, multiplication, and division shall mean that all the 
 operations of multiplication and division are to be performed 
 before ani/ of those of addition and subtraction, and in accord 
 with (1) above. The additions and subtractions are then 
 to be performed in accord with (2) above. 
 
 E.g., 2 + 6- 5-8--2 = 2 + 30 -4 = 28. 
 
 Note. While such an expression as 3 • a -^ 2 • a; • ?/ means [(8 a) ^ 2] • a; • y, 
 the expression Sa -^2xy is usually understood to mean (3 a) ~ (2 xy) ; 
 i.e., 3 a and 2 xy are here understood to vei>resent products rather than unper- 
 formed multiplications. 
 
 11. Signs of aggregation, (i) Any desired departure from 
 the order of operations given in § 10 may be indicated by 
 employing one or more of the so-called signs of aggregation ; 
 
1-11] tNTROTWCrtON 15 
 
 among these are the parenthesis ( ), the brace \\, the bracket 
 [ ], and the vinculum . 
 
 (ii) An expression within a parentliesis, brace, or bracket, 
 or under a vinculum, is to bo regarded as a whole, and is to 
 be treated as though it were represented by a single symbol. 
 
 U.g., (2 + 6) . 5 - 3 - (7 + 8 -^ 2) = 8 • 5 ^ 3 - 11, i.e,, 
 '2^. So, too, (4 + 6) -^ 2 = 5, while without the parenthesis 
 its value would be 7. 
 
 It may sometimes be useful even to employ one sign of 
 aggregation within another, 
 
 Mg., 72- J47-7(15-10)S =72- ^47-35? =72-12 = 6. 
 
 EXERCISE VI 
 Find the value of each of the following expressions: 
 
 1. 20 + 5-3. 4. 12-2x4. 7. 16-2-1-6. 
 
 2. 20-5 + 3. 5. 12 -(2x4). 8. 16 - (2 + 6). 
 
 3. 20 -(5 + 3). 6. 9.(6-2). 9. 11 • 4 - 6 • 3-2. 
 
 10. 28 -(6 + 13) -(10 -2). 13. 42-7x5-5 + 6x2. 
 
 11. 32-9 + 6--2 + 1. 14. 12 + 9-3-30-2 + 8. 
 
 12. 32- (9 + 6) --(2 + 1). 15. (12-f 9).3-(30-2 + 8). 
 
 16. J25 - (10 + 13)S -2 + 31-5 + 4. 
 
 17. 16- 9 -4(36 -3-2) +54 -(17 -12-5). 
 Read each of the following expressions, and tell in what order 
 the indicated operations are to be performed : 
 
 18. ac-^b. 21. (c-by. 24. 6a H-2c-2d ^ 
 
 19. a(c-^b\ 22. c'-^b'-2d. 
 
 20. c-b\ 23. c----(b^-2d). 25. ^ -— . 
 
 ^ ^ c<i^+[2(c — a)]- 
 
 26. If a = 8, b — 3, c = 12, and 2d= 1, find the numerical value 
 of each of the expressions in Exs. 18-25. 
 
CHAPTER II 
 
 POSITIVE AND NEGATIVE NUMBERS 
 
 12. Introductory. * Suppose the present reading of a ther- 
 mometer is 5° above zero, and the temperature is falling ; 
 what will be the reading when it has fallen 1°? 2°? 3°? 
 4°? 5°? 6°? 7°? 8°? 
 
 * Note to the Teacher. It will stimulate interest in the work, as well 
 as enlarge the pupil's view, if the teacher will amplify and present to the 
 class the following considerations (cf. also El. Alg. pp. 18, 19) : 
 
 1. Man's earliest idea of number came from counting., and in this way 
 there arose the number system consisting of the integers 1, 2, 3, 4, •••. 
 Presently he advanced a step and counted backward as well as forward, and 
 by groups of things as well as by single things ; this led the way to the opera- 
 tions of addition, subtraction, multiplication, and division. 
 
 2. Having devised this number system and these operations to meet the 
 needs of his daily life (just as later he invented the clock, the steam engine, 
 the telephone, etc.), he soon found the system inadequate: 7 h- 3, for ex- 
 ample, represents no number in the above system. His increasing desire for 
 exactness, however, as civilization advanced, demanded that division should 
 be always possible; for this and other reasons he invented fractions, and 
 included them in his number system . 
 
 3. Many of the things with which we are concerned bear a relation of 
 opposition to each other (assets and liabilities, thermometer readings above 
 zero and below zero, etc.), and the change from one of these opposites to 
 the other may be regarded as a subtraction (cf. § 12) ; for this and other 
 reasons it was found advantageous again to extend the number system, and 
 thus to make subtraction always possible. 
 
 4. The use of literal notation, too, quite apart from the foregoing con- 
 siderations, demands a number system in which addition, subtraction, etc., 
 are always possible. If subtraction, for example, is not always possible, 
 such an expression as a — 6 may or may not represent a number ; hence, 
 should a — h occur in the solution of a problem before the relative values of 
 a and b were known, our work would come to a standstill. It is wiser, there- 
 fore, to include negatives in- our number system, and let the solution proceed, 
 giving the necessary interpretation to our results later. 
 
 16 
 
12-18] POSITIVE AND NEGATIVE NUMBERS 17 
 
 Wliicli one of the arithmetical operations (addition, sub- 
 traction, etc.) did you use in answering these questions? 
 
 Again, if a business man's financial losses are large enough, 
 they will decrease his capital, not only to zero but through 
 zero, and bring him into debt. 
 
 So, too, if a traveler now in north latitude goes south far 
 enough, he will pass through zero latitude, and into south 
 latitude. 
 
 Observe that, in each of the statements just made, the 
 change from a given condition to its opjjosite is essentially 
 a process of subtraction; it is a subtraction, moreover, in 
 which we can subtract not only to, but through zero. 
 
 In our present number system, we can subtract only to 
 (not through^ zero ; in order, therefore, to express m the 
 simplest possible wag the numerical relations between such 
 opposite things as those given above (gains and losses, lati- 
 tude north and south, etc.), we must extend our number 
 system so as to make subtraction through zero possible. 
 
 13. The number system extended. The arithmetical inte- 
 gers arranged in a series increasing by one from left to right, 
 and therefore decreasing by one from right to left, are 
 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ... 
 
 Addition is performed by counting toward the right (cf. 
 § 4), and subtraction by counting toward the left, in this 
 series. Moreover, addition is always possible because this 
 series extends without end toward the right, and subtraction 
 is arithmetically possible only when the subtrahend is not 
 greater than the minuend because this series is limited at 
 the left. 
 
 Hence, to make subtraction with arithmetical integers 
 always possible, it is only necessary to continue the above 
 series indefinitely toward the left. 
 
 Let the result of subtracting 1 from 1 be designated by ; 
 of subtracting 1 from 0, by — 1 ; of subtracting 1 from — 1, 
 
18 HIGH SCHOOL ALGEBRA [Ch. II 
 
 by — 2 ; of subtracting 1 from — 2, by — 3, etc. ; with these 
 new numbers included, the series becomes 
 
 ..., -6,-5,-4,-3,-2,-1, 0, 1, 2, 3, 4, 5, 6, 7, ..., 
 
 whicli extends without end toward the left as well as toward 
 the right. 
 
 Since in this enlarged series each number is less by one 
 than the next number at its right (and hence greater by 
 one than the next number at its left), therefore addition and 
 subtraction with arithmetical integers may, as before, be 
 performed by counting toward the right and left respectively. 
 
 E.g.^ to subtract 8 from 5, ^.e., to find the number which 
 is 8 less than 5, we begin at 5 and count 8 toward the left, 
 arriving at — 3 ; hence, 5 — 8 = — 3. 
 
 Similarly, 4-6=— 2, 4-9=- 5, 11 -16 =-5, etc. ; 
 hence, besides indicating a particular place in the enlarged 
 number series, — 5 also indicates that the subtrahend is 5 
 greater than the minuend. 
 
 Again, to add 7 to —4, i.e.^ to find the number which is 
 7 greater than — 4, we begin at — 4 and count 7 toward the 
 right, arriving at 3 ; hence —4 + 7 = 3. 
 
 Note. Such an expression as 4 — 9 = — 5 is, of course, not to be under- 
 stood to mean that 9 actual units of any kind can be subtracted from 4 such 
 units ; 4 of the 9 units may be immediately subtracted, leaving the other 5 
 units to be subtracted later if there is anything from which to subtract ; in 
 this sense the number —5 may be said to indicate a postponed subtraction, 
 and thus to have a subtractive quality ; hence the appropriateness of attaching 
 the minus sign to such numbers. 
 
 14. Positive and negative numbers. Numbers which are 
 less than zero * are called negative numbers ; while numbers 
 greater than zero are, for distinction, called positive numbers. 
 
 In writing positive and negative numbers, the latter are 
 always preceded by the minus sign, while positive numbers 
 may be written either with or without the plus sign. Thus, 
 
 * "Less than zero" in the sense suggested in § 13, 
 
13-15] POSITIVE AND NEGATIVE NUMBEES 19 
 
 — 5, —2.3, —48, and — | are negative numbers, while 3, 
 + 84, and 1.7 are positive numbers. 
 
 Such a number as — 2 is read either as negative two or 
 as minus two, while + 6 is read as positive six or plus six. 
 
 15. Algebraic numbers, etc. Positive and negative num- 
 bers together (including, of course, fractions as well as inte- 
 gers) are often called algebraic numbers, wliile positive 
 numbers alone are called arithmetical numbers. 
 
 By the absolute value of a number is meant merely its size 
 without regard to its quality/ ; thus — 2 and + 2 have the 
 same absolute value; so also have — 17.25 and + 17.25. 
 
 Two numbers which have the same absolute value but 
 which are of opposite quality, are called opposite numbers ; 
 such, for example, are 5 and — 5. 
 
 Note. The signs + and — as used above are called signs of quality. 
 We shall continue, however, to use them as signs of operation also ; hence 
 it is sometimes necessary, in order to avoid possible confusion arising from 
 this double use, to inclose a number with its quality sign in a parenthesis 
 (cf. Ex. 20, p. 21). 
 
 EXERCISE VII 
 By counting along the algebraic number series given in § 13 : 
 
 1. Add 3 to 8. 5. Subtract 3 from 8. 
 
 2. Add 7 to — 3. 6. Subtract 7 from 7. 
 
 3. Add 4 to — 4. 7. Subtract 7 from 4. 
 
 4. Add 5 to — 12. 8. Subtract 5 from — 3. 
 
 9. If temperature above zero is indicated by positive numbers, 
 how may temperature below zero be indicated ? 
 
 10. Interpret the following temperature record taken from a 
 U. S. Weather Bureau report : Albany, -f- 8° ; Bismarck (S.D.), 
 - 11° ; Buffalo, - 2° ; Denver, - 5° ; and Galveston, + 34°. . 
 
 11. In the above record how much warmer is it at Albany than 
 at Bismarck ? at Buffalo than at Denver ? at Buffalo than at 
 Galveston ? Illustrate your answers by diagrams. 
 
20 HIGH SCHOOL ALGEBRA [Ch. II 
 
 12. The total value of a man's property (his assets) is $ 15,860, 
 and his total indebtedness (liabilities) is $1420. What is the net 
 value of his estate? What is the net value of an estate of which 
 the assets are a dollars and the liabilities h dollars ? 
 
 13. If h exceeds a in Ex. 12, is a — 6 positive or negative ? 
 How should a — 6 be interpreted in this case ? What is meant 
 by saying that the net value of an estate is — $ 750 ? 
 
 14. If assets are indicated by positive numbers, interpret the 
 financial conditions indicated by the following numbers : $ 783 ; 
 - $ 2568 ; - $ 374.20 ; and $ (856-1232). 
 
 15. Kegarding longitude west of Greenwich as positive, indi- 
 cate by a number and sign that a place is : (1) in 24° east longi- 
 tude; (2) on the meridian of Greenwich; (3) in 10° west longitude; 
 (4) in 15° east longitude. 
 
 16. At 6 A.M. a thermometer records 15° below zero; by noon 
 it has risen 32°. Indicate by a number and sign the reading at 6 
 A.M. and at noon. Illustrate by a diagram. 
 
 17. If on the line X'OX distances to the ^^ ? ^ 
 
 left of are called negative, locate the point 
 
 on this line whose distance from is : + .3 in. ; — .5 in. ; + .1 in. 
 
 18. If positive numbers are used to denote assets, gains in busi- 
 ness, increase of any kind, temperature above zero, easterly motion, 
 south latitude, west longitude, distance down stream, etc., what are 
 the corresponding meanings to be attached to negative numbers ? 
 
 19. An ocean steamer is in 12° east longitude. If east longitude 
 is indicated by positive numbers, and if the vessel moves west- 
 ward through 7° of longitude per day, indicate by a number and 
 sign the longitude of the vessel 4 days hence ; 1|- days hence ; 2 
 days ago. Illustrate by a drawing. 
 
 20. What number must be added to — 12 to make the sum 4 ? 
 Which, then, is the greater, —12 or 4 ? How much greater? 
 
 "Which of these numbers has the greater absolute value ? 
 
 • 
 
 16. Addition of algebraic numbers. As we have already 
 seen (§§ 4, 13), to add a positive number to any given num- 
 ber, we begin at the given number and aoxmi forivard. 
 
la-lC] POSITIVE AND NEGATIVE NUMBERS 21 
 
 Moreover, since a negative number represents an unper- 
 formed subtraction (§ 13, Note), therefore adding a negative 
 number means performing this subtraction. Hence, to add 
 a negative number to any given number, we begin at the given 
 number and count hachvard. 
 
 E.g.^ to add — 6 to 54 we begin at 54 and count 6 backward; 
 i.e., 54 + (- 6) = 54 - 6 = 48 (cf. Exs. 19, 20, below). 
 
 Such an expression as 11—4—8 + 3, which, by what is 
 said above, equals 11 + (— 4) + (— 8) + 3, is usually spoken 
 of as an algebraic sum. 
 
 EXERCISE VIII 
 Perform the following additions, and explain your work : 
 1. 2. 3. 4. 5. 6. 
 
 9 
 
 12 -8 9 14 -7 
 
 8 8-3 -6 -3 
 
 7. 
 
 8. 
 
 9. 
 
 10. 
 
 11. 
 
 12. 
 
 -4 
 
 23 
 
 -11 
 
 -31 
 
 7 
 
 -9 
 
 -12 
 
 -15 
 
 -26 
 
 45 
 
 5 
 
 — 5 
 
 13. 
 
 14. 
 
 15.' 
 
 16. 
 
 17. 
 
 18. 
 
 -3 
 
 7 
 
 -6 
 
 11 
 
 -22 
 
 -72 
 
 4 
 
 5 
 
 -8 
 
 -4 
 
 31 
 
 ^b 
 
 -8 
 
 -9 
 
 — 2 
 
 -9 
 
 15 
 
 -21 
 
 19. If a boy weighing 54 lb. were weighed while holding a 
 toy balloon which pulls upward with a force of 6 lb., what 
 would be the combined weight? If +54 lb. represents the 
 weight of the boy, what would represent the iveight of the 
 balloon ? 
 
 20. In Ex. 19 the combined weight of the boy and the balloon 
 is (+54) + (— 6) lb. ; hence adding the negative number destroys 
 part of the positive number ; is this true in general for additions 
 of positive and negative numbers ? Illustrate your answer. 
 
22 HIGH SCHOOL ALGEBRA [Ch. II 
 
 21. When does the addition of a negative number to a positive 
 number destroy the latter wholly ? When only in part ? Illus- 
 trate your answers by means of assets and liabilities (cf. Ex. 12, 
 p. 20). 
 
 22. Ex. 21 suggests a useful rule for algebraic addition, viz. : 
 
 (1) To add two numbers with like signs, find the sum of their 
 absolute (arithmetical) values, and to this prefix their common 
 sign. 
 
 (2) To add two numbers with unlike signs, find the difference 
 of their absolute values, and to this prefix the sign of the larger. 
 
 Test this rule in the examples on p. 21. 
 
 23. A wheelman after riding 37 miles westward from a certain 
 point rides back 12 miles. If distances to the westward are indi- 
 cated by positive numbers, show that 37 + (— 12) miles indicates 
 both his direction and his distance from the starting point. 
 
 24. Indicate by a sum of positive and negative numbers what 
 temperature is now registered by a thermometer which stood at 
 54° above zero, then rose 2°, later fell 9°, and then rose 2i°. 
 
 25. Eind the following indicated algebraic sums : 18 + (— 3) + 
 (_10) + 2; 42+(-27)4-(-64); _5 + 18 + (-11) + 23. 
 
 26. Is algebraic addition sometimes performed by arithmetical 
 subtraction ? Is it so when the two summands have like signs ? 
 when they have unlike signs ? Illustrate your answers. 
 
 17. Subtraction of algebraic numbers. We already know 
 how to subtract positive numbers (of. §§ 5, 13); therefore 
 we now need to consider only the question of subtracting 
 negative numbers. 
 
 Now, since subtraction is the inverse of addition (§ 6) 
 and since 7 + (-8) = 4 (§ 16), therefore 4 -(-3)= 7; i.e., 
 4-(-3) = 4 + 3. Similarly: 11 - (- 5) = 11 + 5; -3- 
 (-12)= -3 + 12; etc. 
 
 Hence, subtracting a negative number from any given 
 number gives the same result as adding its opposite to the 
 given number. 
 
10-17] POSITIVE AND NEGATIVE I^ UMBERS 23 
 
 EXERCISE IX 
 
 Subtract the numbers written below, each from the one above 
 it, giving the necessary explanation in each case: 
 
 1. 2. 3. 4. 5. 
 
 12 4 -2 8 -4 
 
 8 7 5 -3 -2 
 
 6. 
 
 7. 
 
 8. 
 
 9. 
 
 10. 
 
 -3 
 
 11 
 
 6 
 
 -4 
 
 -31 
 
 -10 
 
 -8 
 
 -15 
 
 -12 
 
 -25 
 
 11. The weight of a boy while holding a toy balloon, which 
 pulls upward with a force of 6 lb., is 48 lb. If we take away 
 (subtract) the balloon, how much will the boy weigh ? Show that 
 in this case 48 — (— 6) = 54. 
 
 Supply the missing numbers in the following equations : 
 
 12. •.•-5h-? = -2, .-. _2-(-5) = ? 
 
 13. ... _54-? = -9, ... -9-(-5) = ? 
 
 14. |-(_|) = ? 16. 10-3-(-5) = ? 
 
 15. _l|-(-5f) = ? 17. 23-(-a) + (-3) = ? 
 
 If a = 5, b = — 6, c = — 3, and d = 2, find the value of each of 
 the following algebraic sums : 
 
 18. a-b + c + d. 21. a^^b-(d^-c). 
 
 19. a-^c — (b — d). 22. ad — b-{-(a^ — b). 
 
 20. a—(b + G-d). 23. 2 a-^ - c — 3d^-\-b. 
 
 24. Using positive numbers to represent assets, illustrate the 
 fact that subtracting a negative from a positive number increases 
 the latter (cf. Ex. 11). 
 
 25. Make up concrete examples (like Ex. 11 or Ex. 24) to illus- 
 trate Exs. 4, 8, and 3, above. 
 
 26. Solve Exs. 1, 2, 5, and 9, above, by counting ; and explain 
 in each case why you count in one direction rather than in the 
 opposite direction. 
 
24 HIGH SCHOOL ALGEBRA [Ch. II 
 
 27. A rule for subtraction is often stated thus : " Reverse the 
 sign of the subtrahend and proceed as in addition." Show that 
 this rule is correct when the subtrahend is a negative number. 
 
 28. Mt. Washington is 6290 feet above sea level, Pikes Peak 
 is 14,083 feet above sea level, and a place near Haarlem, in Holland, 
 is 16|^ feet below sea level. By subtraction find how much higher 
 Pikes Peak is than Mt. Washington ; and also how much higher 
 Mt. Washington is than the place near Haarlem. 
 
 29. When is algebraic subtraction equivalent to arithmetical 
 addition (cf. Ex. 2%, p. 22) ? Illustrate your answer. 
 
 30. Write the following algebraic sums so that they shall not 
 contain minus as a sign of operation, then find the value of each : 
 36-19-13+2; 14a-26i«+9}a-15a; -6a;-47a;-3a;. 
 
 18. Product of algebraic numbers. Rule of signs.* The 
 product of any two algebraic numbers is tlie result obtained 
 by performing upon the multiplicand the same operation as 
 that which is performed upon positive unity to obtain the 
 multiplier [§ 7 (ii)]. 
 
 E.g,, since 3 = 1 + 1 + 1, 
 
 therefore 8 • 3 = 8 + 8 + 8 = 24, 
 
 the product 24 being obtained from 8 just as 3 is obtained from 
 positive unity. 
 
 Similarly, - 8 • 3 = (-8) + (- 8) + (- 8) = - 24. [§ 16 
 
 Again, since — 3 = — 1 — 1 — 1, i.e., since — 3 is obtained by 
 subtracting positive unity three times, 
 therefore 8 • (- 3) = - 8 - 8 - 8 = - 24, 
 
 and _8.(-3) = -(-8)-(-8)-(-8) = 8 + 8 + 8 = 24. 
 
 Observe that two of the above products are positive and 
 two are negative. How do the signs of the factors compare 
 when the product is positive? when it is negative? How 
 does the absolute value of the product (cf. § 15) compare 
 with the absolute values of the factors ? 
 
 * Teachers who prefer to give more drill on addition and subtraction at 
 this point may omit §§ 18 and 19 until after Chapter III has been read. 
 
17-19] POSITIVE AND NEGAriVE NUMBERS 25 
 
 Applying the tlefinitioii of a product to any two numbers 
 whatever, just as we did to 8 and 3,-8 and 3, etc., above, 
 we see that : (1) if two factors have like signs, their product 
 is positive; (2) if they have unlike signs, their product is 
 negative; and (3) the absolute value of the product equals 
 the product of the absolute values of the factors. 
 
 EXERCISE X 
 
 Find the following products and explain your work : 
 1.-5-2. 4. 9 . - 3. 7. 3 . - 4 . - 5 . 2. 
 
 2.-5.-2. 5. -7|.-6. 8. 2'-'^x,i.e.,2'-^-x. 
 
 3.-12.8. 6.-^.-5. 9. _4«.3.-5.-7. 
 
 10. Show that ( - 2)\ i.e., _ 2 . - 2 • - 2 • - 2, is 16. What is 
 
 the value of (- 2)' ? of (- 3)^ • (- 4)^ ? 
 
 11. What is the sign of (- 7)^ ? Why ? What is its absolute 
 value ? What is the sign of (- 7)'^ - 2 . 5 ? 
 
 12. A succession of multiplications (as in Ex. 7, for instance) 
 is called a continued product. Can the sign of a continued product 
 be obtained without actually performing the multiplication ? 
 How ? What is the sign if there are 5 negative factors ? 
 
 13. An odd power of a negative number (i.e., a power whose 
 exponent is odd) has what sign ? An even power ? Is a power 
 of a. positive number ever negative ? Explain. 
 
 If a = — 4, 6 = — 2, c = 3, rf = — 1, and e = 2, find the value of : 
 
 14. abode. 16. ah^d'. 18. If — c^—d^. 
 
 15. cHh\ 17. {a + hf. 19. 4.c'-cd^d\ 
 Find the value of (a-\-h) - (x — y) : 
 
 20. when a =2, b = —3, x =8, and 2/ = — 5. 
 
 21. when a= — 4, b = 6, x = 3, and ?/ = — 1. 
 
 22. when a = ^, b = —2a,x= — \, and y = \. 
 
 19. Division of algebraic numbers. Division is the inverse 
 of multiplication (cf. § 8); i.e., it consists in finding one of 
 
26 HIGH SCHOOL ALGEBRA [Ch. 11 
 
 two numbers when their product and the other number are 
 given. Hence the results of § 18 may be used to show how 
 to divide algebraic numbers. 
 
 Thus, since 8-3 = 24, 8.(-3) = -24, _8.3 = -24, and 
 (_8).(-3)=24, 
 
 therefore 24 -- 3 = 8, 
 
 _24-(-3)=8, 
 _24-3=3_8, 
 24 - (- 3) = - 8. 
 
 So, too, whatever the given numbers. Therefore : (1) the 
 absolute value of the quotient of two algebraic numbers is 
 the quotient of their absolute values, (2) this quotient is 
 positive if the dividend and the divisor have like signs, and 
 (3) it is negative if they have uiilike signs. 
 
 EXERCISE XI 
 Find the value of each of the foljowing indicated quotients : 
 
 1. 14-2. 4. -31- (-If). 7. 15 -(-1). 
 
 2. 14-- (-2). 5. -24-9. a -365--(-9i). 
 
 3. _18^41 6. (-6)2-(-2)3. 9. _63a2^(-9). 
 
 10. Of what operation is division the inverse? How, then, 
 may the correctness of a quotient be tested ? Illustrate. 
 
 11. If the dividend is positive, and the divisor negative, what 
 is the sign of the quotient ? Compare the signs of divisor and 
 quotient when the dividend is positive ; when it is negative. 
 
 Find the value of each of the following expressions : 
 
 12. 24-28-(-7)+(-16)-(-4).(-3). 
 
 13. _8.(-6)--24-27--(-6)^3. 
 
 14. j28-(-7)-2.(-4-2) + 24j-(-2)3. 
 
 Verify that = —. -„ : 
 
 •^ x-i-y X — y x^ — y^ 
 
 15. when a = Q, b = 2, a? = 10, and y = 6. 
 
 16. when a= — 8, b = 12, x= — 9, and y = 7. 
 
CHAPTER III 
 
 ADDITION AND SUBTRACTION OF ALGEBRAIC EXPRESSIONS 
 — PARENTHESES 
 
 I. ADDITION 
 
 20. Algebraic expressions, monomials, etc. Any combina- 
 tion of letters, or of letters . and numerals, representing a 
 number is called an algebraic expression. 
 
 The terms of an algebraic expression are the parts into 
 which it is separated by the signs + and — (or, rather, these 
 parts together with the signs preceding them). Thus, 3 a, 
 + m^, and — 5cx are the terms of the expression 3 a + w^ — 5 ex. 
 
 An algebraic expression which is not separated into parts 
 by + or — signs is said to consist of a single term, and is 
 called a monomial. Thus, 3 a, 5mx^, and —llcV are 
 monomials. 
 
 An expression consisting of two or more terms is called 
 a polynomial. Thus, S a -{-7 m^— 5 ex is a polynomial. 
 
 A polynomial consisting of two terms is usually called a 
 binomial, and one of three terms, a trinomial. Thus, 2 s—5 xy 
 is a binomial, while 4 a; — a + 7 ^^^ is a trinomial. 
 
 21. Coefficients. Any one of the factors of a term, or the 
 product of two or more of them, is called the coefficient 
 (co-factor) of the product of the remaining factors. Thus, 
 in the term 5 axy^., the coefficient of axy^ is 5, the coefficient 
 of x'lp' is 5 a, the coefficient of 5 xy^ is a, 
 
 A coefficient consisting of numerals only is called a numer- 
 ical coefficient, while one that contains one or more letters 
 is called a literal coefficient. Thus, in the term — 3 a^m^ the 
 
 HIGH SCH. ALG. — 3 27 
 
28 HIGH SCHOOL ALGEBRA [Ch. Ill 
 
 numerical coefficient of ax^m is — 3 ; but — 3 a and — '6 am 
 are literal coefficients of xhii and x?^ respectively. 
 
 Eemakk. The word " coefficient " is usually understood to 
 mean '' numerical coefficient " and to include the sign preceding 
 the term. Observe also that ax means the same as lax (cf. § 9, 
 
 Note). 
 
 22. Positive and negative terms. Like and unlike terms. 
 A term preceded by the sign + is called a positive term, and 
 one preceded by the sign — is called a negative term. If 
 the first term of an algebraic expression is positive, its sign 
 is usually omitted, but the sigli of a negative term is never 
 omitted. 
 
 Terms which either do not differ at all, or which differ 
 only in their coefficients, are called like terms, also similar 
 terms ; terms which differ in other respects are called unlike 
 terms, also dissimilar terms. Thus, 3 x^y^ — 5 a^y, and | x^y 
 are called similar terms. 
 
 EXERCISE XII 
 
 1. Name the coefficient of d^x in each of the following terms : 
 
 3 a^x, — 5 d^x, dx. 4 dhx. — | a^x. —^ , — 9 a^x. 
 
 2. In Ex. 1 which coefficients are literal and which numerical? 
 Which terms are positive and which negative ? 
 
 3. Do the positive terms in Ex. 1 necessarily represent positive 
 numbers for all values that may be assigned to the letters in- 
 volved ? Try a = 3 and a; = - 2. 
 
 4. What is the coefficient of x — ym each of the follow- 
 ing expressions : 13 (a? — y), — a (x — y), ^ m (x — y), and 
 (4 — a^) (x — y) ? Which of these coefficients are numerical ? 
 Which literal ? 
 
 5. Consult a good dictionary for the derivation of the words 
 "monomial," "binomial," "trinomial," and " polynomial." Write 
 three monomials, three binomials, three trinomials, and three 
 polynomials. 
 
21-23] ADDITION OF ALGEBRAIC EXPRESSIONS 29 
 
 6. Distinguish between the meanings of 5 in the expressions 
 5 X and oc^. What name is given to the 5 in each case ? 
 
 7. What are like terms ? By what other name are they known ? 
 In what respects may they differ and still be like terms ? 
 
 8. Are 3 x^y, — 2 x-y, and | x^y similar ? Are 4 ax^ and — 6 bx^ 
 similar ? Are these last two terms similar if 4 a and —6b are 
 regarded as their coefficients ? 
 
 9. Write three sets of like terms, some terms being positive 
 and some negative, and each set containing at least four terms. 
 
 23. Addition of monomials. Since 5 times any given num- 
 ber plus 2 times that number is 7 (^^e., 5 + 2) times the 
 given number, therefore 5a + 2a=(5-f2)a=7a, whatever 
 the number represented by a. Similarly, 8 mx^i/ + 8 mx^i/ — 
 (3 + S)mx^i/ = 11 mx^i/. Hence, 
 
 To add two or more similar monomials, add their coeffieie7it>i 
 and to this result annex the common literal factors, each with 
 its proper exponent. 
 
 It is usually more convenient to write the terms to be 
 added under one another, as in arithmetic, thus : 
 3 XT/ 153 ahnx 18 aks 
 
 8 xi/ 74 a!^mx — 7 aks 
 
 Wxy ' ? ? (cf. § 16) 
 
 If the monomials to be added are dissimilar, they cannot 
 be united into a single term, but their sum may be indicated 
 in the usual way ; thus, the sum of 5 a and 2 a:^ is 5 a H- 2 a;^. 
 Similarly, the sum of 3m and — 6a is 3m + (— 6 a), which 
 equals 3m — 6 a (cf. § 16). 
 
 EXERCISE XIII 
 
 Add the following sets of similar termsf and explain your work: 
 1. 2. 3. 4. 
 
 6 n 18 a^ —9 mx 31 abvr 
 
 3 n - 10 a^ 5mx - 22 abx" 
 
 — 2n — 3 a^ — 6 mx — 6 aba^ 
 
30 HIGH SCHOOL ALGEBRA [Ch. Ill 
 
 5. State a convenient rule for adding any number of like 
 terms. Does your rule apply to cases in which some of the terms 
 are negative ? 
 
 Find the algebraic sum of: 
 
 6. 4 a?^, — 2 x^y — 5 x^. 10. 12 a^n, a^n, — 4 a% — 9 ahi, 
 
 7. 11 ax, ax, — 9 ax. 11. S xz, — 8 xz, — xz, 2 ics;. 
 
 8.-4 cs^, — cs^, 8 cs^. 12. — a6^, — 7 a6^, a5% — 5 ah*. 
 
 9. — 3 aa;"2/j ''^ cl^Vj ctx^y. 13. — ic^/? ~ ^ ^I/) 12 a;?/, — 3 a;y. 
 
 Simplify the following expressions; i e., unite like terms, and 
 indicate the results where the terms are unlike : 
 
 14. 3 bxy^ + (- 4 bxy^) + (- 12 bxij^ + 5 bxy^ + bxy^ + (- bxy^. 
 
 15. — 4 mp^ + 13 a'^x + 7 mp^ -f 3 mp^ -f ( — 5 ax^ — 2 acc^ + mp^ 
 
 16. 25 c^s^^ - 10 6^^ - bH - c^s^ -\-3bH-{- bH + cV - 8 (fs\ 
 
 17. 7.5a; + fx — a; — i£C + iic + i^x — 3.45 x + li a;. 
 
 18. 3d*-5 c/ + 2ic^^-11.5d^-7ic^'^-5d^ + d*-c^. 
 
 19. - 6 (a- 62) 4-3 (a- 62) _ («_ ^,2) _ 5 (ct_ 52^ j^^a-b"), 
 
 20. 23 a^ + 5 6" - 8 a^^** - 13 6« + 24 a^^" _ 19 a^ + a^ - 6" - a^ft". 
 
 21. How many a^'s in 5 a; + 3 a; ? in 10 a? — 2 a; ? in 8 a; — aa; ? 
 in4aj — aj? ina; — Sa^ + lla;? in ma? -f na; — 3 a; ? 
 
 22. How many s^^'s in 8 s^^ + 2 sH ? in 3 sH + as't ? in 3^ s^^ 
 + 2 ms^^ - s2^ ? in 3 asH + (- 2 fts^^) ? 
 
 23. Add 15 a;2, - 2 aa;2, - 7 a^^^ 4 fta^^^ and - 2 Za;^. 
 [In Exs. 23-25 let «, 6, and ^ belong to the coefficients.] 
 
 24. 2 aa;?/) ~ 8 ^2/> ^^^ ^ ^^2/- 25. 5 xz^, — axz^, and 2 bxz^. 
 
 24. Addition of polynomials. Any two polynomials, e.g.^ 
 
 S a^ — 7 xi/ -}- 12 7/^ ai^d 5 a^ 4- 6 2:3/ — 3 «/2, may be added 
 
 thus: 
 
 3a2_7^y + 12^2 
 
 5 a^ _|_ (5 2;^ _ 3 ^2 
 
 8a2_ a;y+ 9?/2 
 
28-26] ADbtriON OF ALGEBRAIC EXPRESSIONS 31 
 
 This procedure may be stated tliiis : To add two or more 
 polynomials^ write them under 07ie another so that similar terms 
 shall stand in the same column, and then add each columii sep- 
 arately as in § 28. 
 
 EXERCISE XIV 
 
 Add the following sets of polynomials : 
 
 1. 
 
 
 2. 
 
 3. 
 
 
 4. 
 
 4a-26 
 
 6m 
 
 4-3^2 
 
 ax — 5 
 
 y 
 
 8|) + 2.s-3^2 
 
 2a4-56 
 
 4m 
 
 -7 7l' 
 
 2ax- 
 
 1 
 
 Wp-bs-^lt^ 
 
 5. 
 
 
 
 6. 
 
 
 7. 
 
 6 m — 4 ^i 
 
 + 7p 
 
 Sa 
 
 -40^ + 5?/ 
 
 
 - a^a; - 8 6 -h 2 / 
 
 - 2m+ n 
 
 -5p 
 
 6a 
 
 -i-5x'- y 
 
 
 — 2 a?x —^'f 
 
 -Sm-{-2n 
 
 -4.P 
 
 -4.a 
 
 ^2^ 
 
 
 8 a-x + 2 6 - 9 .v*" 
 
 8. 12 ace — 5 a?^ — 9 2/, — 3 ao? — 6 a^^ + 2 ?/, and ax + x'^' — y. 
 
 9. 3 m — 7 71 + 2p, m H- 4 ?? — 6p, and n — 2m +p. 
 
 10. a^ — 2 .T + 1, a; — 3 + 8 a^, and 4 a; — 3. 
 
 11. 2a-76 + 3(a^-l), 4.b- a-6{x' -I), and 6+ (a;--l). 
 
 12. 3(a + &) - 2 c' - 5, 7 - 6{a + &) + 4 c', and c' - (a + h). 
 
 13. 2(?7i - n) + 4(|> + 1), 3 a — 5(m — w), and — 7{p + 1) — 9 a. 
 
 14. 2f A:-6.5? + 3im, 5fc-6im, and4m + 2Z. 
 
 15. s''-^t-^v,2t-^s\^.\i^i^v-^t+2s\ 
 
 Supply the missing coefficients in the following equations : 
 
 16. 12a; — 4a?/ — 5aj-h7a2/= ^x + '^ay. 
 
 17. aoi? — 2 xy -\- dxy — CO? = ? x^ -\- ? xy. 
 
 18. 6rs-s^ + 5cs3+(2-3c)ns=?rs+?s3. 
 
 19. (a2-c)p + (3a4-5c-5)i?=?/>. 
 
 20. Add 3 a?2 + 4 a;/ — 2/"* — 7 a;?/ + 2 a:^?/, 10 a^y -f- (5 c - 10) xy-, 
 (c + 1) a:^ — 3 2/% and xy -\- ^ xy"^ -{■ {2 d — l)x^y. 
 
 25. Checking results. If a result (iu addition, for example) 
 is correct, then it must, of course, remain correct when we 
 assign any arbitrary values to its letters. This is the basis 
 
32 BIGH SCHOOL ALGEBRA [^'h. Ill 
 
 of a very useful test of the correctness of algebraic work 
 (usually called a '' check "). 
 
 Thus, find the sum of 10 a^ - 3 7/ and - 2 a;^ _^ 9 2/. 
 
 SOLUTION CHECK 
 
 . 10.^-3., = 10- 6=. 4 1 j^^^^^^^ 
 
 -^^ + ^^ =-2 + 18=26 L,d ,= 2 
 80^ + 62/ = 8 + 12 = 20 J 
 
 And since the sum of 4 and 16, the values of the summands, is 
 20, which is also the value of the sum, therefore the work is prob- 
 ably correct. 
 
 EXERCISE XV 
 
 In Exs. 4-11, p. 31, check your results as above by putting any 
 convenient arbitrary values for the letters. 
 
 II. SUBTRACTION 
 
 26. Subtraction of monomials. Since subtraction is the 
 inverse of addition, therefore (cf. § 23) : 
 
 To suhtraet one of tivo similar monomials from the other, sub- 
 tract the coefficient of the subtrahend from that of the minuend^ 
 and to this remainder annex the common literal factors. 
 
 The work may be arranged thus : 
 
 73 a2 27n^k mx^f Uax 
 
 24 a2 -Sn^k 19 2;^ - 9 ax 
 
 49rt2 ~J5M ~1 ~~f~ (cf. § IT) 
 
 If the given monomials are dissimilar, the subtraction can, 
 of course, only be indicated. 
 
 A good practical rule for subtraction is : To subtract one 
 of tivo similar monomials from the other^ reverse the sign of the 
 subtrahend and proceed as in addition. In order to avoid 
 confusion in reviewing one's work, it is best, however, not 
 actually to reverse the written sign but only to conceive it 
 to be reversed. 
 
25-27] SUBTRACTION OF ALGEBRAIC EXPRESSIONS 38 
 
 EXERCISE XVI 
 
 Perform the following indicated subtractions : 
 1. 18-5; -18-5; _18-(-5); 18-(-5); 9-(-9). 
 2. 3. 4. 5. 6. 
 
 7a 16ba^ - 18 m^ -ISr^a^ 26 ^V 
 
 4 a -Sbx^ Im^ - Tr^cc^ _ 9 ^y 
 
 7. 8. 9. 10. 
 
 3 ex" - 6 7nY 6. 8 k'a^y'- - 21 a^m* 
 
 11. Show that " changing the sign of the subtrahend and pro- 
 ceeding as in addition" will give the remainder in each of the 
 above exercises. 
 
 12. From 7 aoc^y take 3 ax-y ; from 5 njf take — 8 7q)^ ; from 
 4 (a - 2 b^) take - 11 (a - 2 ¥) ; from the sum of 13 2/"V and - 5 y-^'^ 
 take 4 i/^a;^. 
 
 13. Indicate the subtraction of b^ from 3 a-; of 4(0^ + ?/^) from 
 — 6 (c + 2/) ; of 2 a;?/ from the sum of x^ and 2/" ; of — a^ft" from the 
 sum of 3 a* and — 6^**. 
 
 14. How many x^y's in 8 aic-^/ — 2oify? in mxry-{-nx-y—2 cx^y ? 
 in 7 cx'y-(- 3 x'y)? 
 
 15. Supply the missing coefficient in : 125 7nz — 97 mz= ? Z', 
 
 c2a2-(-9a2)=?a2; 5 ax^ + ? ao;^ = 2 aa;=^ ; 4:C8-?s = 2bs. 
 
 27. Subtraction of polynomials. One polynomial may be 
 subtracted from another by writing the subtrahend under 
 the minuend, similar terms under one another, and subtract- 
 ing term by term, thus : 
 
 55_3^2+ Q^l 32:4-5 
 
 85 + 5:^2 _ 9^^ _7a;4 4- 4 -2:?: 
 
 2 6 - 8 ^2 _^ 15 ab 10ai^'^)-\-2x 
 
34 HIGH SCHOOL ALGEBRA [Ch. Ill 
 
 EXERCISE XVII 
 
 In the following pairs of expressions subtract the second from 
 the first, and check your results as in § 25 : 
 
 1. 8a-562, 2a + h\ 3. ^x' + x, Zx-2x', 
 
 2. 3m2-7, m^-lO. 4. s' + ^t, 2t-5s', 
 
 5. a^-2ab + b', -3 a' + 12 ab-12b\ 
 
 6. x^-\-5oi^y + 7 xif — 2y^, 3 ic/ — ar' — 2 / — 5 a?y. 
 
 7. Check your answer to Ex. 6 by letting x = 2 and y = l. 
 
 8. From the definition of subtraction show that the minuend 
 equals the sum of the subtrahend and remainder. What means 
 of checking the result does this suggest (cf . § 17) ? 
 
 9. In each of Exs. 1-6, p. 31, subtract the second expression 
 from the first. 
 
 10. From c^ -\-d subtract c^ — d — 4 A;. Check result in two ways. 
 
 11. From 5 x^ + 4 a'b take 8 a^6 — 2 x^ + 5 abx, and check result. 
 
 12. Subtract 15 y^ + 10 aV + 4 mV from 34 aV - 10 mV, and 
 check result. 
 
 13. Subtract 15 — 3 a? + 10 aj^ from 12 a^ + 5 j also from — 2x; 
 also from 0. 
 
 14. Subtract 5^ a^-2i + a;-4J a;^ from 7 a;^~2i- a; + a^-4; 
 also from 0. 
 
 15. Subtract the sum of 5 a — 31 6^ + 2 a^ and 26 6^ — 4 a; from 
 ix^-2a' + 7b\ 
 
 16. From what must a^b"" + 3 ca; + cZ*" be subtracted if the result 
 is to be 4 a^b'' — d'-^2cx? 
 
 In Exs. 17-20 let a, b, and m belong to the coefficients : 
 
 17. From 2ax — Sby-\- mxy take x — 2y-\-^^xy. 
 
 18. From (m - 2 6) ?/ + 3\z take 2z+(a- b)y\ 
 
 19. From Qr -l)xy -\- bY - 3 mx^ take 2 V'xy - 5 mx^ + af. 
 
 20. From (a^ - 3 a6 + m^)^ -f (4 a^ - 5 «?> + 2 6*' + 7 m^) ?/ + 
 2 amz^ subtract {cir - 5 o6 + &^ - m')^ - 3|- a^62^ + (a^ - 2 a6 + m") y. 
 
27-28] PARENTHESES 35 
 
 III. PARENTHESES* 
 
 28. Parentheses removed and inserted. Such an expres- 
 sion as 2 a; — (?/ — 3 0) means that i/ — 3 2 is to be subtracted 
 from 2 X ; hence (§ 27) 
 
 2x-(iy-Zz) = 2x-y + ^z, 
 Similarly, a — (^—h + c— d — e) = a-\-h — c + d-{- e; etc. 
 
 These equations (read from left to right) show that a 
 parenthesis inclosing any number of terms, and preceded by 
 the minus sign, may be removed provided that the sign of each 
 term within the parenthesis is reversed. 
 
 Again, reading the above equations from right to left 
 shows that any number of the terms of an expression may 
 be inclosed within a parenthesis preceded by the minus sign, 
 provided that the sign of each term so inclosed is reversed. 
 
 Remark. A parenthesis preceded by the plus sign may, of 
 course, be removed or inserted without changing the signs of the 
 terms inclosed. (Why ?) 
 
 EXERCISE XVIII 
 
 By means of § 27 show that : 
 
 1. 5a — (3a + ^)=5a — 3a — & = 2a — &. 
 [What is the sign of 3 a in (3 a + 6) ?] 
 
 2. ^x — 2y—{—^x-\-y)=^x — 2y-{-^x — y=zlx — Sy. 
 
 3. m^ — 3 np 4-p^ = m^ — (3 np —p^). 
 
 4. a — 2b-^c — 4.x = c — 4:X-{—a-\-2b). 
 
 5. Using § 11, show that 8 - (10 - 7) = 8 - 3 = 5 ; and then 
 show that the same result may be obtained by using § 27, i.e., 
 show that 8-(10-7)=8-10 + 7 = -24-7 = 5. 
 
 Simplify each of the following expressions by two methods, 
 , as in Ex. 5, and compare results : 
 
 * " ^Parenthesis " here means any sign of aggregation whatever (cf. § 11). 
 
36 HIGH SCHOOL ALGEBRA [Ch. Ill 
 
 6. ll_(3 + 6). 9. 27 + (-5-3)-13-7. 
 
 7. ii_(_3 4.6). 10. 27 -(-5 -3) + 13 -7. 
 
 a _(8-5) + 10. 11. -(6-4 + 9)+3-(-2 + 7). 
 
 12. In Ex. 9 what is the quality sign of 13 ? What does the 
 minus sign preceding 13 indicate ? 
 
 In Exs. 13-19 remove parentheses and unite similar terms : 
 
 13. 7x — Sac-\-(x — 2 ac). [Compare § 28, Remark.] 
 
 14. 3a — 46 + (6 — 2a). 17. x — y + {x-\-y) — {^x — y). 
 
 15. 2f-{-x' + f-xy). 18. a-2/2-(a-3)-(-3/-l). 
 
 16. 5a2 + 3?>-(-2a). 19. -(2m-5)-(-6+a^-3m). 
 
 In each of the following examples inclose the last two terms 
 in a parenthesis preceded by — : 
 
 20. 2s-3^ + w. 23. aa^ — 4 5x — 3 + 2^. 
 
 21. 6 4-5 X- — 3?/. 24. 2 /i — 3 A; — 7 a? — 5. 
 
 22. a^-f2 6 + c*. 25. 3 m* — 2 m"ic — 5 ma^" + a;*. 
 
 29. Parentheses within parentheses. It often happens 
 that one parenthesis incloses one or more others. In such 
 cases the expression within an inner parenthesis forms a 
 single term of the next outer parenthesis [cf. § 11 (ii)]. 
 
 These parentheses, too, may be removed as in § 28, 
 thus : 
 3 a^ - J 9 m - [ - a^ - (4 s3 _ 5 ^^^) ^ ,,3-| j 
 
 = 3 a^ — 9 m -f- [ — a^ — (4 s^ — 5 m) + s^] [Removing brace 
 = 3 a^ — 9 m — a^ — (4 s^ — 5 m) -+- s^ [Removing bracket 
 
 = 3a^ — 9m— a^ — 4s* + 5m + s^ [Removing parenthesis 
 = 2 a^ — 4 m — 3 s'. [Collecting terms 
 
 Let the pupil simplify the above expression by first 
 removing the innermost parenthesis, then the next inner- 
 most, and so on, and compare his work with what is here 
 
28-29] PAIiENTHESES 37 
 
 EXERCISE XIX 
 
 Simplify the following expressions; that is, remove the paren- 
 theses and combine like terms : 
 
 1. s - [t^ -\- (u'^ ~ s)']. 5. (m — 4p) - (a —p + m). 
 
 2. s-[e-(ir-s)]. 6. Sx'-\2a-(-x'-{-a)l, 
 
 3. 6a-[6-(-2a-|-36)]. 7. _ J2a - (-ic^.^, ^^ j^ 
 
 4. a^^l-f-(2f-3x')l 8. -J_(-a^ + a)J. 
 9. mx^— [8?/— (6a; — W.1') — 2aj. 
 
 10. _ (60 - 25) -[92 -(18 + 27) I . 
 
 11. 3p + 4g + [7i)-2g-(5i)-3-5g)]. 
 
 12. a — y— \a — (—y — a — 2)1. 
 
 13. a;-S3a.--[-(-3a; + 22/)+o?/]-32/: 
 
 14. 8a;2^2a;?/-[3x2-?/2-(2a;.?/-.T- + /)]-2/2. 
 
 15. 2 a' - 5 [3 6" - (a' - 2 c - 5 a')] _ [7 6" + 5 c - 6« - 2 a'] \ . 
 
 16. 8a-25-{-(3c-f?)-[4c-d-(-8aH-2&)]-2d;. 
 
 17. 4.-[5y-l3-(2x-2)-4.xl'}- Ja; + 52/-^+3J. 
 
 18. In Ex. 2, how many minus signs affect u^? How often, 
 then, will its sign be reversed by removing the parentheses ? 
 What will be its sign finally ? Answer the above questions for 
 - «2 in Ex. 7. 
 
 19. By considering the number of minus signs affecting the 
 respective terms, remove together all parentheses in Ex. 9. Also 
 in Exs. 11, 12, and 14. 
 
 20. In the expression 3 m — 4 a -f- 10 a;^ — 5 ?/ + 3 a^^ — 8 aa;, in- 
 close the 4th and 5th terms in a parenthesis preceded by the 
 minus sign ; then inclose this parenthesis, together with the two 
 preceding terms, in a bracket preceded by the minus sign.* 
 
 21. Make the changes asked for in Ex. 20, in the expressions 
 3m + 4a-10a;2_5^_^3^^2_g^^^ 3m-4a -2a?2-h52/-3a6-, 
 and — 5 a;'' + 3 ?/'" - 4 a - 14 6c + 8 A;2. 
 
 * The value of the expression is, of course, to be left unchanged. 
 
CHAPTER IV 
 
 MULTIPLICATION AND DIVISION OF ALGEBRAIC EXPRES- 
 SIONS 
 
 L MULTIPLICATION 
 
 30. Law of exponents in multiplication. What is the 
 
 meaning of 5^ (cf . § 9) ? of o^ ? of a;" ? 
 
 How many times is s used as a factor in the product 
 8^-8^? Is 8^ ' 8^ equal to 8^ ? Explain why. 
 
 How may the exponent of the product ^ - 8"^ he obtained 
 from the exponents of the factors s^ and 8^ ? Would your 
 answer remain true if we were to put other exponents in 
 place of 3 and 2 ? 
 
 Is cc^x^a^ equal to x^'^, i.e., to 2^^+2+5? W^hy ? Is x"x^ equal 
 toaj«+*? Why? 
 
 The results of these considerations may be expressed in 
 symbols thus : 
 
 wherein a may stand for any number whatever, but m, w, 
 and p are positive integers. 
 
 Translated into common language, this law of exponents is : 
 The product of two or more power8 of any number is that 
 power of the given number who8e exponent i8 the 8um of the 
 exponents of the factors. 
 
 31. Product of two or more monomials. The product of 
 any two or more monomials may be obtained by a simple 
 extension of § 30. 
 
 J7.^., in the product of 2 ax^ and 3 ab'^x, how many numeri- 
 cal factors ? What is their prodilct ? How many a\s in the 
 
 38 
 
30-31] MULTIPLICATION OF ALGEBRAIC EXPRESSIONS 39 
 
 entire product? How many 5's ? How many a;'s ? ^\^rite 
 down this product, using the exponent notation. What is 
 its sign ? Why ? 
 
 What is the product of 3 aV^ — 2 abaP'^ and 5 «5 ? Is it 
 — 30 a'^h'^x^ ? Explain in detail, mentioning the sign, the 
 coefficient, the letters, and their exponents. 
 
 These considerations lead to the following rule for obtain- 
 ing the product of two or more monomials: To the product 
 of the numerical coefficients of the several monomials, annex the 
 different letters which these monomials contain, giving to each 
 letter an exponent equal to the sum of the exponents of that 
 letter in the several monomials. 
 
 EXERCISE XX 
 
 Find the following indicated products, and explain your re- 
 sults, especially the signs and exponents: 
 
 1. 2. 3. 4. 5. 
 
 3a2 3a2 6-3^ -Smh -7aV 
 
 5a^ -5a^ -2.3^ -3ms^ -2 2^ 
 
 6. 7. 8. 9. 10. 
 
 -5-23 mV 2 a;" -Sa^^" _4a;2y 
 
 8.2^ -ba^x -QxP -Sab^ -9x^y^ 
 
 11. 7 s2 . _ 3 as . - 2 a\ 14. - 2 a"6 • 6f ab^'- 1 2| a'^bK 
 
 12. 3 mx^ • 2 m^ • — 7 am. 15. 7.5 mv'^ - — 4| am^ • —3 a"^. 
 
 13. _ 2i s? . - s^ . 3f a^^. 16. - 12 A;^/^ . _ aA:Z" • 2^ a"A:^ 
 
 17. 2 (a -6)2. - 5 (a -b) ' 3 x'y ' -2^ x (a-b) .2.5(a-6)Y. 
 
 18. Define and illustrate the meaning of exponent, power, 
 base (of. § 9). May the base be negative ? May it be a frac- 
 tion ? May the exponent be fractional or negative ? 
 
 19. If 7z represents a negative number, is n^ positive or nega- 
 tive (cf. Ex. 13, p. 25)? How does 3^ compare with (-3)^? 
 2^ with (-2)^? 
 
 20. What is the meaning of 2/"~^ ? In this expression may n 
 be less than 2 ? What is the product of 4 a^ and —Ta""-^? 
 
40 HIGH SCHOOL ALGEBRA [Ch. IV 
 
 21. Determine, by inspection, the sign of the result in each of 
 the following products when a = — 2, 6 = 3 : (a — 6)^ ; (a — 6)^ ; 
 (a + hy ; {att^Y ; (a^b^y ; (a^ — by. State your reason in each case. 
 
 32. Product of a monomial and a polynomial. From the 
 
 definition of a product [cf. § 7 (ii), § 18], 
 
 5 . (2 + 9) = 5 . 2 + 5 . 9, 
 since 2 + 9 is obtained by taking positive unity 2 times, 
 then 9 times, and adding the two results. 
 
 Similarly, a{m -\- x— y) = q^i -\- ax — ay^ 
 
 whatever the numbers represented by a, m, x, and y. 
 
 Hence we may say : To find the product of a polynomial and 
 a monomial multiply each term of the polynomial by the mono- 
 mial and add the partial products. 
 
 The actual work may conveniently be arranged thus : 
 
 Check 
 3 a%- 4x2 + 11 2/2 = 10 
 
 -2xy = - 2 
 
 - 6 aH^y + 8 x^y - 22 xy^ = - 20 
 
 EXERCISE XXI 
 
 when a = 1, X = 1, 
 and y = 1 
 
 1. How is 2 -\-a — x obtained from + 1 ? How, then, may 
 2/(2 + a — ic) be obtained from y ? 
 
 
 2. 
 
 3. 
 
 4. 
 
 Multiply 
 
 3x-5y 
 
 a-4a6 + 362 
 
 2 m — Sn^ — mn 
 
 by 
 
 2x 
 
 -56 
 
 — 4 mn 
 
 Find the following products, and check results (cf. § 25) : 
 
 5. (2a^-4:y')'7xy. 9. -Sx^y^(2 x"" -4:X^y). 
 
 6. (4:ax-5xy)(-2x^). 10. 5 a\a^-~4 a^ -2 a-\-7), 
 
 7. (~2s^-Sst)(-9st% 11. -7ax'(Sax^-3a'x-5ax). 
 
 8. {-6u'-\-nv)(-4:v'x). 12. - 8 a^^" (3 - 4 a"* + 12 6"). 
 
 13. -12xy{2}x^-5lx-4:). 
 
 14. 2iabc(1.5a'-{-7.5ab'-6b'). 
 
31-33] MULTIPLICATION OF ALGEBEAIC EXPRESSIONS 41 
 
 15. {^x'z-^x^^-4.xz''-^xz + ll){-^xz). 
 
 16. [a^+(a + iy4-(a-l)ic + l].2aa:2_ 
 
 17. («/ - 2 xy -.15 a;y + 4 ary - 7 a?y (- cc^^/^-s). 
 
 18. 2 (%2-3 i;) [(^2 - 3 i;)^ - 13 a; (?^2_3 '^^+2 a;2(^2_ 3 ^,)_l]. 
 
 19. 3a[7ar^-4(2a;2 + aaj) + «(2a-3a; + l)]. 
 
 20. Multiply da—bh-\-c—x — y by — 1, and show that the 
 result agrees with § 28. 
 
 21. By what nust x — ly — 2az be multiplied, to obtain the 
 product 12 ah: — 14 a^y — 4 ah ? 
 
 22. Find a monomial and a polynomial whose product is 
 6 ax" -10 a'x - 14 aV 4- 8 aV. 
 
 23. Are the values of m and n in Exs. 9, 12, and 18, limited in 
 any way ? If so, how ? 
 
 33. Product of two polynomials. Since m + ^ is obtained 
 by taking positive unity m times, then n times, and adding 
 the two results, therefore (of. § 32) 
 
 (a -\- h ■{- c) ' (m -\- n) = (a -\- h -\- c)m + (a + 5 + c)n 
 = am -\-hm+ cm -\- an -{- hn -\- en. 
 
 Similarly for any polynomials whatever ; i.e.^ the 'product of 
 two polynomials is obtained by multiplying each term of the 
 multiplicand by each term of the multiplier^ and adding the 
 partial products. 
 
 If any two or more terms of a product are similar, they 
 should, of course, be united. 
 
 Such a multiplication and its check may be arranged thus : 
 
 Check 
 
 a2 + 2 a& - 62 = + 2, 
 
 a+ h =+2. 
 
 (a2+2a&-&2).a= a^ + 2a%- aW' 
 
 (a'^ + 2a&-62).6=: ggft + 2 a&2 _ &8 
 
 a3 + 3a-26 _f. aW- - 63 = + 4, 
 
 when a — \ 
 and 6 = 1 
 
 Remark. The product of three or more polynomials may be 
 obtained by multiplying the product of the first two by the third, 
 this product by the fourth, and so on [cf. § 10 (1)]. 
 
42 BIGB SCHOOL ALGEBRA [Ch. IV 
 
 EXERCISE XXII 
 Perform the following indicated multiplications : 
 
 1. {x+2d)'(x-a). 9. (x^-xy + y^)'(x-y). 
 
 .2. {Sa^-5x)'(2a''-{-3x). 10. (a^ -xy -i-y^) - (x + y). 
 
 3. (7-2m3).(3-5m=^). 11. (0^-2 ay -\-y^) • {a- y). 
 
 4. (a'A-ab-\-b')'{a-b). 12. (5 s^ - 2 i^^^ . (5 ^3 _^ 2 ^^^^ 
 
 3. (2s-Sf)-(Ss-4.t). 13. (7ey2-2c/«).(7ey2 + 2^). 
 
 6. {x-{-a)'(x-{-a), 14. (-4/g- 6 r) • (3^)-^?^)^ 
 I.e., (ic + ay. 15. (m^ — w^ + p^) • (5 m — 2 np). 
 
 7. (3m2-10)l 16. (2a;-32/ + 0-4)2. 
 
 8. {Gx'-Sayy. 17. (a^- 3 a6 + 62_2 c)^. 
 
 18. (3m2 — 2mn) • (5m + 3w^) = ? Check your result by 
 letting m = 1 and n — 1. If, in the product, the exponent of m 
 should be wrong, would this check reveal the error ? Explain. 
 Would the error be revealed if m were taken equal to 2 ? 
 
 Multiply (and check your work) : 
 
 19. m* — 2 m^ — 6 m^ + m — 1 by 3 m^ + m — 2. 
 
 20. 2ii? — 1xy + Sx^ — 4.x-\-2y-\-lhjxy — Zx — 2y. 
 
 21. a2-62_c2_2a6-25c+2acby a-26 + c. 
 
 22. I.^x^-2xy-2.3y^-^x-2.^yhy3ly-l^x. 
 
 23. x"" + y"" hy X — y '^ hj y? -\- y'^ ', by a?" — i/**. 
 
 24. ic"" — 3 a;'"~^y + 3/"* — 3 x?/'"-^ by a?^ — 2 a;?/ + /. 
 
 25. 2x+(3n — l)?/by (n 4-l)a^ — (3n + l)y. 
 
 26. r2-(2ri-«2) by4?2_4 (^2r^-r2)-l. 
 
 34. Degree and arrangement of integral expressions. In 
 
 multiplications with polynomials, and elsewhere, it is often 
 advantageous to arrange the terms of a polynomial in a par- 
 ticular order; such arrangements will now be explained. 
 -^ A term is said to be integral if it contains no letters in its 
 denominator ; it is integral in a particular one of its letters if 
 that letter does not appear in its denominator. A polynomial 
 
83-35] MULTIPLICATION OF ALGEBRAIC EXPRESSIONS 43 
 
 is integral, or integral in a particular letter, if each of its 
 terms is so. 
 
 U.g., 3 ax^ + ^^ — ^f ^ is integral in 6, m, x, and v ; 
 a 3 
 
 it is fractional in a; its first and last terms are altogether 
 
 integral, while its second term is integral only in 5, m, and ^. 
 
 e^ By the degree of an integral term in any letter (or letters) 
 
 is meant the number of times this term contains the given 
 
 letter (or letters) as a factor. Thus, 7 a^x^ is of the second 
 
 degree in x, and of the fifth degree in a and x together. 
 
 ^The degree of an integral polynomial is the same as the 
 
 degree of its highest term. Thus, Sa^ — 5 a^jp^y — 2 bx^y^ is 
 
 of degree 4 in a;, 3 in ?/, and 5 in a; and 7/ together. 
 
 A polynomial is said to be arranged according to ascending 
 powers of some one of its letters if the exponents of that 
 letter, in going from term to term toward the right, increase ; 
 and that letter is then called the letter of arrangement. If 
 the exponents of the letter of arrangement decrease from 
 term to term toward the right, the expression is said to be 
 arranged according to descending powers of that letter. 
 
 Thus, 2x^ — 5 ax^y — 7 Pxy^ 4- 3 m^y^ is arranged according 
 to descending powers of x, and ascending powers of y, 
 
 35. Multiplication in which the polynomials are arranged. 
 
 If each of two polynomials is arranged according to powers 
 of some letter which is contained in each, then their product 
 will arrange itself according to powers of that letter, and the 
 actual multiplication will take on an orderly appearance. 
 
 E. g. , Ito 6nd the1>roduct«l 7 ic - 2 ic^ + 5 + x^ by 3 ic + 4 a;2 - 2, arrange the 
 tft)rk<to«B: 
 
 Check 
 
 a-3- 2x2+ 7x + 5 =11/ 
 
 4 x2 + 3 x - 2 =5, 
 
 4 a;5 - 8 x* + 28 x3 + 20 x2 
 
 3x* - 6x3 + 21x2 + 15 X 
 
 - 2 x3 + 4 x2 - 14 X - 10 
 
 4x6 _5a4 + 20x3 + 45x2 + X- 10 =55, 
 
 HIGH SCH. ALG. — 4 
 
 when X = 1 
 
44 niGIl SCHOOL ATMEBJIA [Ch. TV 
 
 EXERCISE XXIII 
 
 1. Is 1^ a^oc^ integral or fractional ? In what letters is ' :;^ 
 
 integral ? ' In what letters is it fractional ? 
 
 2. May an integral term have a fractional coefficient ? Illus- 
 trate. Write a term integral in m and n and fractional in d. 
 
 3. When is a polynomial fractional in a particular letter^? 
 Write a binomial fractional in a and b ; a trinomial integral in a 
 and b, but fractional in c. 
 
 4. In Ex. 11 below, 
 
 (1) Of what degree is each term of the multiplicand ? each 
 term of the multiplier ? 
 
 (2) Of what degree in a? is the first term of the multiplicand ? 
 Of what degree in y ? Answer the same questions for the other 
 terms of the multiplicand. 
 
 5. How is the degree of an integral polynomial determined ? 
 Give three illustrations from the exercises on p. 42. 
 
 6. Name the degree as regards x of each polynomial in Exs. 
 11 and 12 below. 
 
 7. Arrange the expression 
 
 3ay^f + xf>-S xY — 6 T^y^ + x^y — Sf-{- 5 y^ 
 according to ascending powers of y. How is it then arranged 
 with reference to aj? Of what degree is this expression? What 
 is the degree of 3 x^y^ ? 
 
 Multiply : 
 
 8. 6x^-2 + 5x + Si^hj x^-\-5-x. 
 
 [In Exs. 8-16 arrange both multipUer and multiplicand according to some 
 letter contained in each, and observe that the product has then a correspond- 
 ing arrangement.] 
 
 9. 2a + a3-a2_iby 4-a2-fa. 
 
 10. Sa-x-4.ax^-{-x''-a^hj a^-ax-^a^. 
 
 11. 3xy'^-f-Sx^y-^x''hy-2xy-\-a^-^y\ 
 
 12. x-y'^ — xy^ -f y^ — x^y-\-x^ by x^-\-xy — y^. 
 
 13. 4.hh-hr^-h^-\-27^\)yh-2r. 
 
35-37] DIVISION OF ALGEBRAIC EXPRESISIONS 45 
 
 14. Q>y^ + ^ xY + 2x'-3o?y-xfhy y'' + Zx'-2xy. 
 
 15. af — 5 aj^2/^ —^f — &xy^-\- 15 ^y^ + 2 x^y hj 'd y"^ -{- x^ — 2 xy. 
 
 16. 4.f-lQ>s^t^-^&^-\-^sH + ^st^hj^s'-f-^sH. 
 
 17. a"+i - 3 a"+- + a "+^ - 2 «"+'* by 2 a"-i + 3 a"-^ - 4 a'*-=^. 
 
 18. x^'y^ + i»"+^?/ + 0^**+^ — £c"+^?/2 by or^ + 2/^ — ic?/^ — x^y. 
 
 19. In Ex. 8, of what degree is the multiplicand ? the multi- 
 plier? the product? The term of highest degree in the product 
 is the product of what two terms ? 
 
 20. Of what degree is the multiplicand in Ex. 11 ? the multi- 
 plier? What, then, should be the degree of the product? Should 
 all the terms of the product be of the same degree ? Why ? 
 
 II. DIVISION 
 
 36. Law of exponents in division. Since division is the 
 inverse of multiplication (§ 19), therefore the results of § 30 
 may be employed to find the law of exponents in division. 
 
 Thus : since a^ - a^— a^ therefore a^ -j- a^ = ? Is o^ -r- oc^ 
 equal to x^^ i.e., to a:;^^? Why? 
 
 . Write the following indicated quotients and explain your 
 answer in each case : aJ -^a^\ 8^ -i- s^; 2^" -^ 2^ ; x^ -i- x. 
 
 How is the exponent of the product of two powers of any 
 given number obtained (cf . § 30) ? How, then, should the 
 exponent of the quotient be obtained ? 
 
 If m and n are positive integers, m greater than /t, and x 
 any number whatever, then (cf. § 9, also Exs. 18, 20, p. 39) 
 the above results may be expressed in symbols thus : 
 
 j,m _j^ w-re __ j,m — n 
 
 This equation states the law of exponents in division ; 
 translate this law into common language (cf. § 30). 
 
 37. Division of monomials. Since the quotient multiplied 
 by the divisor always equals the dividend (§8), therefore 
 12 a^-f- 3 a:^ = ? that is, what is the number which, when 
 multiplied by 3 x'^., gives 12 a^ as product (gL § 31) ? 
 
46 HIGH SCHOOL ALGEBliA [Ch. IV 
 
 Similarly: 8 a^rr^ ^ 4 aV = ? Why? 24mV-^8mY = ? 
 Why? -18 a^^"-^ 6^362=? Why? Sml^xy^ -i- (^-^.m^xy) ^^^ 
 
 How is the sign of the quotient determined ? the coeffi- 
 cient? the exponents? How may § 8 be used 'to test the 
 correctness of the quotient ? From the above write a rule 
 for dividing one monomial by another, mentioning the sign, 
 coefficient, letters, and exponents of the quotient (cf. § 31). 
 
 EXERCISE XXIV 
 
 Perform the following divisions ; check results by § 8 : 
 
 1. Q>a^^2a. 4. - 1^ a^h' ^ (S ah\ 
 
 2. 15aV-f;3ax2. 5. 10 c«dV -- 5 c^de. 
 
 3. 12 mV -T-4:X^. 6. — 45 m^n^ -^ ( — ^ m^n). 
 
 ^ -48aV ^^ -lmh\ ^^ 3.1 (xyz)' 
 
 12 aV ' ' --i-mV' ' -.Sa^yV' 
 
 g * 15 7ty ^2 -<33mny ^^ - 12 g^^c^^ 
 
 -mfgY\ 13 _J/^. ,„ 2^ 
 
 ,»»+3 
 
 25 //p2 -T\fh' 6x^ 
 
 10. — :r—rr' 14. 18. . 
 
 — 7 a^6 4^ m"?^'^ 2 a;" 
 
 19. If two monomials have like signs, what is the sign of their 
 product ? of their quotient ? How do we find the exponent of 
 any given letter in the quotient of two monomials ? 
 
 In Exs. 20-25, multiply the first monomial by the second; 
 also divide the second monomial by the first : 
 
 20. - 16 «^ 51 m''. 23. I c (m + n), - 10 c' (771 + 71)1 
 
 21. 42 (p + qy, — 14 (p + qy. 24. 8 ax^Y'"+% — 2 ^^^+2^2 (m+»)^ 
 
 22. 5afy% 15 a^^+y. 25. 13 (.t-:^)^, -26(x-zy. 
 
 38. Division of a polynomial by a monomial. Since (§ 32) 
 a (m + 2; — ^) = «m + ax — ay, 
 therefore (am + t«:r — ay} -i- a = m + x — y, 
 whatever the numbers represented by a, w, a: and y. Hence, 
 
;}7-;]8] niVTSlON OF ALGKliUAIC EXPRESSIONS 47 
 
 To divide a polynomial hy a monomial^ divide each term of 
 the polynomial hy the monomial^ and add the quotients so 
 obtained. 
 E.g., (15 aV - 10 hx^y + cV) -^ 5 x^ =^ S a^x - 2 bx^y + i c\ 
 
 EXERCISE XXV 
 
 Perform the following indicated divisions : 
 , 4a»-12a^ ^ 9 mhi^ + 12 mn^ - 30 m^n* 
 
 D. 
 
 4 a^ — 3 mn^ 
 
 -2Axy + lSiK^y\ ^ -18a^-81a; + 9a^ 
 
 xy 
 
 c2 
 
 14 7-2- 
 
 -3^0^ 
 2l7-2s^ + 8 
 
 -39> 
 
 
 26 a^m' 
 
 7 r 
 ^-52 aV- 
 
 «%® 
 
 1 a'W- 
 
 13 a-m^ 
 
 -4a«6^ + 6 
 
 a^«Z>^ 
 
 
 ^ 3 m - 2 n + 11 a; ^ 
 
 9mV + 12myi2-30mV ^ 
 
 3m7i2 ' • -2a26« 
 
 11. How may any polynomial whatever be divided by a mono- 
 mial ? How are the signs of the several quotient terms deter- 
 mined ? their coefficients ? their letters ? their exponents ? 
 
 Divide [and check the work in each case (cf . Ex. 10, p. 2Q>)'] : 
 
 12. 07-1-4 ax^ — 3 m^x — 6 a7nx by — x. 
 
 13. a^6%i3 - 4 a36V« + 12 ci'ft'ic by 4 rt^^s^ 
 
 14. 1 7-^5 + J cr^^s^ — I r^s^ by 2 rs ; also by f rs. 
 
 15. a"* — 2 a^'^^ - 5 a"*+2 + 9 a"»+^ by a"* ; also by a\ 
 
 16. 2«+* — 3 z""^ -h 4 aV — 2;^ by — 1 2:1 
 
 17. - 10 (h - 1)*-' - 6 (/i - 1)^A: + 15 (7i - 1)^A:2 by - 5 (h - 1). 
 
 18. x(x-\- yy —^(x-\- yf + x'^ (x +yy by — a? (aj + ?/)l 
 
 19. 2 (s - ^)- - s^ (s - O'^+i - 5 (s - ^)-+3 by i (s - O'"-^ 
 Separate each of the following expressions into two factors, 
 
 one of which is a^ : 
 
 20. c^a^-hd^^. 22. -Sx^y-{-5x^z-7x^. 
 
 21. aV — a^x^ + aj2. 23. —x^ + Q eV -f — - • 
 
 4 
 
48 HIGH SCHOOL ALGEBRA [Ch. IV 
 
 In Exs. 24-26, group the like powers of y (cf. Exs. 20-23) : 
 
 24. ty^ -f c?/' - ry' -~3sy^-{- y\ 
 
 25. ay* --2by — S cy^ — my* + dy^ — 9 y. 
 
 26. (a + l)/-(«-l).v' + /-3 2/^-(3a4-4)2/3 + a/. 
 
 39. Division of a polynomial by a polynomial. Since, by 
 § 35, the product of (4:a^ + 2>x-2^ and (jx^ -2x^+1 x^b) 
 is 4 a:^ — 5 a^ + 20 a;^ + 45 a:2 + a; — 10, therefore, with this last 
 expression as dividend, and a^— 2a:^+7a7 + 5as divisor, the 
 quotient must be 4:x^-\- S x — 2; i.e., 
 ( 4 rcs _ 5 ^4 + 2 :z^ + 4 5 2^ + a; - 1 ) -f- (2^3 _ 2 2^ + 7 rr + 5) 
 
 = ^x^ + Sx-2, 
 
 The process of obtaining this quotient from the given 
 dividend and divisor will now be explained. 
 
 Since the dividend is the product of the divisor by the 
 quotient, therefore the highest term in the dividend is the 
 product of the highest term in the divisor multiplied by 
 the highest term in the quotient (cf . Ex. 19, p. 45) ; and 
 therefore if 4 x^, the highest term in the dividend, is divided 
 by a^, the highest term in the divisor, the result, 4a^, will be 
 the highest term in the quotient. 
 
 Moreover, since the dividend is the algebraic sum of the 
 several products obtained by multiplying the divisor by each 
 term of the quotient, therefore if 4 a;^ — 8 ic* + 28 a^ 4- 20 a^, 
 the product of the divisor by the highest term of the 
 quotient, is subtracted from the dividend, the remainder, 
 viz., 3a::* — 8a^ + 25a:2^^_;1^0^ ^[n }yQ ^\^q g^^ Qf ^\^q prod- 
 ucts obtained when the divisor is multiplied by each of the 
 other terms of the quotient except this one. 
 
 For the reason given above, if 3 a;*, the highest term of this 
 remainder, is divided by a^, the highest term of the divisor, 
 the result, 3 x, is the next highest term of the quotient. 
 
 By continuing this process all the terms of the quotient 
 may be found. The work may be arranged as follows : 
 
J9] DIVISION OF ALGEBRAIC EXPRESSIONS 49 
 
 DIVIDEND 
 
 x3-2a:2+7x+5 
 
 4x2 + 3x-2 
 
 (x3-2x2 + 7x+6).4x2= 4 x5-8a;H 28x3+20x2 
 
 3x4- 8x3 + 25x2+ x-10 — -' 
 
 (a;3_2a;2+7x + 5).3x = Sx^- 6x3+21x2+ 15 x Quotient 
 
 - 2x3+ 4x2-14 X- 10 
 (x3-2x2+7x+5) .(-2)= - 2x3+ 4x2-14 x- 10 
 
 
 Check 
 
 When X = 1, dividend = 55, divisor = 11, and quotient = 5, as it should. 
 
 Even if it is not known beforehand that the dividend is 
 the product of two polynomials, the process of division may 
 still be applied as above. This process may be formulated 
 thus : 
 
 (1) Arrange both dividend and divisor according to the 
 descending powers of some one of the letters involved iii each, 
 and place the divisor at the right of the dividend. 
 
 (2) Divide the first term of the dividend by the first term of 
 the divisor y and write the result as the first term of the quotient. 
 
 (3) Multiply the entire divisor by this first quotient term^ 
 and subtract the result from the dividend, 
 
 (4) Treat the remainder as a new dividend^ arranging as 
 before^ and repeat this process until a zero remainder is reached^ 
 or until the remainder is of lower degree in the letter of arrange- 
 ment than the divisor. 
 
 EXERCISE XXVI 
 
 Divide (and check your results by § 25) : 
 
 1. ar^+7a; + 12by ic + 3. 5. IS x +6x^-i-6 by 3x-{-2. 
 
 2. ay^-x-20hj x-5. 6. 8 + 3 a^- 14 a; by 2-3a;. 
 
 3. b'-6b-16hjb + 2. 7. 10a^+lla2_8byl-2al 
 
 4. s2-14s + 49by s-7. 8. Sx^ -4.a^-7 hj - sc^-l. 
 
 9. c3 + 6c24.12c+8by c+2. 
 10. 2a^-{-llx''-\-19x-\-10hy2x''-i-7xi-5. 
 U. 75 m- + m^ - 15 m' -125 by 25-\-m'- 10 ml 
 
50 HIGB SCHOOL ALGEBRA [Ch. IV 
 
 12. p*-\-4:p^-\-6p'' + 5p + 2hjp'+p + l. 
 
 13. 2ic^-f6a^ — 4a; — 5a^ + lbya^ — a; + l. 
 
 14. 3a^+3a^ + 3 + 3a + a^ + 5a3by 1 + a. 
 
 [Here, as in arithmetical "long division," labor may be saved by "bring- 
 ing down " at any stage of the vs^ork only so much of the remainder as is 
 needed for the next step.] 
 
 15. Divide 6 a^o^ - 4 a^x-4:ac(f + a"^ -\- x'^ hj a^ -\- x^ - 2 ax. 
 
 SOLUTION 
 
 X* -iax^ + 6 a^x^ - 4 a% + a* bc^ - 2 ax + a^ 
 
 x^ — 2 ax^ + (i^x^ ^•■^ — 2 ax -{■ a^ 
 
 — 2 ax^ + 5 a'^ic-^ — 4 a^x 
 -2ax^ + 4 a^x^ - 2 a^x 
 
 a-x- — 2 a^x + a* 
 a-x^ -2a^x-\- g^ 
 
 Note. To make the explanation of § 39 apply when two or more letters 
 are involved, replace "highest term" by "term of highest degree in the 
 letter of arrangement." 
 
 16. In Ex. 15 perforin the division when both dividend and 
 divisor are arranged according to the descending powers of a. 
 
 17. Divide 4 a^i/^ + 8 a^ + 2/^ + 8 ic^2/ ^J ?/ + ^ x. 
 
 18. Divide 2 a^ + A;^ - 5 o^k - 4 ak^ + 6 a'k' by ¥ + a^- ak. 
 
 19. (10 x^y^ -^ ii^ -10 xhf -\-5 xy' - 5 x'y -f)-^(a^+y^-2 xy) = ? 
 
 20. If the partial quotient, at any stage of the process of divi- 
 sion, is multiplied by the divisor, and the corresponding remain- 
 der added, how must the result compare with the dividend ? 
 
 21. What check for division is suggested by Ex. 20 ? Is this 
 check more or less complete than that given in § 25 ? Explain. 
 
 22. Divide 2a^ + a;^ + 49a^-13aj-12by a^-2«2 4.7aj4_3. 
 
 [Since there is no term in x^ in the dividend, care must be used to keep 
 the remainders properly arranged.] 
 
 Divide (and check the results as the teacher directs) : 
 
 23. v^ — 'v* — l-}-2v-\-v^ — v-hjv — l-\- v\ 
 
 24. 0(^^-41 a-120 by a2 + 4a-|-5. 
 
39] DIVISION OF ALGEBRAIC EXPliEbSIONS 51 
 
 25. m* + 16 + 4 m^ by 2 m + m^ + 4. 
 
 26. T\x*-ia^'y + lixY + ixfhy^x-\-iy. 
 
 27. 1.2 aa;^ -\-a^x^-2a'- 3.4 a-V + 6 aa; by 6 aa; -2 al 
 
 28. {2x-{-3a^-l+2x^){l-\-a^-x)hjl + x + x^. 
 
 29. a^ - 6^ by (a^ + 63^) (a + 6) + a'b\ 
 
 30. a^ 4- 6^ + c^ — 3 a6c by a^ -{- b^ -{- c^ — ab — ac — be. 
 
 31. a;*-3a^ + a^ + 2a;-l byaj2-aj-2. 
 
 fin Ex. 31 the complete quotient is ic'^ - 2 x + 1 + ~ ^ '^ .~| 
 
 32. v^ — v-{-7hjv-\-A. 34. 2 s^ — 3 s + 8 by s^ — 4. 
 
 33. a^ — 1 by a + 1. 35. a^ + a; — 25 by a; — 3. 
 
 36. a*-7a2-9a-6a3-6by3+a2-2a. 
 
 37. 3 a;^ 4- 11 a^ + 11 a;^ + 9 a; + 10 by 4 x + 5 4- a^. 
 
 38. Divide p^ -{- q^ by p -^ q until 4 quotient terms are obtained ; 
 divide 1 by 1 — r to 8 quotient terms ; 1 by 1 — mx to 4 quotient 
 terms ; and a by a — a; to 5 quotient terms. 
 
 Divide : 
 
 39. cd-d2 + 2c2by c + c?. 42. h^ - 7i^ by h^ + Ic^. 
 
 40. oc^ — i/ by x — y. 43. a-" — a;-" by a" — a;". 
 
 41. a*-166^by a-26. 44. u^'' -^ 11 u"" -{- 30 by u"" -\- 6. 
 
 45. a5'"+'* — a?**?/""^ — a:*"?/** +2/^""^ t>y ^"^ ~ 2/**"^- 
 
 46. af"+"-^ - 3 xY" "^ — 5 a;"*" V + 1^ /''-^ by a;" - 5 y. 
 
 47. Divide a6c + aoif + a^ + a6a; + 6a;^ + ca;^ + aca; + bcx by 
 a^ -\- ax -\- ab -\- bx. 
 
 Solution. Since x occurs in more terms than any other letter, it will 
 be best to arrange the work thus (cf. Exs. 24-26, p. 48) : 
 
 or^ + (a + ?> +c) x2 + (ab -\- ac ■{- bc)x + abc \c^ + (a + b)x + ab 
 x^ + (a + h)x/^ + ahx 
 
 cx^ + {ac + bc)x + ahc 
 
 cx/^ + {ac + hc)x + cthc 
 
 
 I 
 
62 HIGH SCHOOL ALGEBRA [Ch. IV 
 
 Divide : 
 
 48. 1/ -\- cy -\- cd -\- dy by 2/ + c. 
 
 49. -ab-\-ay + y^-byhy y-b. 
 
 50. f-\-2 dy' + 2 / + ^^2/ + 4 d?/ + 2 d- by / + 2 ^2/ + d^- 
 
 51. a-f{a-\-&)+^y'-y^-2,ay' + aY-2ayhySf-y + a. 
 
 52. Divide a^ — 21 by x — a) note that the remainder is what 
 the dividend would become if a were substituted for x. 
 
 53. Divide a^ + Sa^ + l by x — a] note that the remainder 
 differs from the dividend only in that a replaces x. 
 
 54. Divide m^4-7 by m — c and compare the remainder with 
 the dividend. Similarly, divide v^ — 1 by v — 2 ; 5 m^ — 8 m + 3 
 by m-3; 2/^-4/+ 32/-1 by y-h-, 2?-* -7^ + 10 by r-1; 
 by r — 2 ; by r — 3. 
 
 55. Divide 2 xy^ + Sx* — 4: x^y^ — 7a^y-\-y^ by a^-\-y^ — xy,' ar- 
 ranging first according to powers of x, then according to powers 
 of y, and compare the results. 
 
 56. As has just been seen in Ex. 55, the form of the quotient 
 depends upon the choice of the letter of arrangement when the 
 division is not exact ; is this the case when the division is exact ? 
 
 40. Finite numbers. As we pass from left to right the 
 numbers of the series 2, 2^, 2^ 2*, etc., increase without end ; 
 and the numbers of the series 1, J, |^, etc., decrease without 
 end. Hence we see that, in mathematical operations, there 
 may arise numbers which are greater, and others which are 
 smaller, than any fixed number that we can name or even 
 conceive of; such numbers are called infinitely large and in- 
 finitely small numbers, respectively. All other numbers are 
 called finite numbers. An infinitely large number is repre- 
 sented by the symbol 00. 
 
 41. Zero. Operations involving zero, (i) The result of 
 subtracting any given finite number from itself is called 
 zero (cf. § 13). Thus if a represents any finite number, 
 then I a — a == 0. 
 
39-41] DIVISION OF ALGEBBAIC EXPRESSIONS 53 
 
 (ii) From this definition of zero and the definitions of 
 addition, subtraction, etc., already given, it follows that, if k 
 is any finite number whatever, 
 then k-\-0 = k-0 = k, 
 
 and A; . = . ^ = 0. 
 
 J£^.^., ^ . = because k • = k • {a — a) — ka — ka = 0. 
 
 (iii) If k is any finite number whatever, then 
 
 A; -7- = no finite number whatever. 
 For, if A; -7-0=/, a finite number, then/- would equal k 
 (§ 19), but this is impossible (ii). 
 
 (iv) 0-^0 =/ [i^iiy finite number 
 
 for /•0 = 0. 
 
 (v) From (iii) and (iv) above it follows that we must 
 not divide hy zero^ since doing so leads, at best, to an inde- 
 terminate result. 
 
 EXERCISE XXVII 
 
 1. When the values 1, \, \, \, y^g, • • • are assigned to x, how 
 do the successive values of the fraction 5/a; compare ? Can you 
 name a number so large that none of these values will exceed it? 
 Can you name a number so near that none of the series of num- 
 bers 1, -^, \, ^, yi^, • • • will be still nearer to ? 
 
 2. What is meant by an infinitely small number ? by an in- 
 finitely large number ? 
 
 3. Define zero. How does it follow from your definition that 
 
 (1) 3-0 = 3? (2) 0-5 = 0? 
 
 4. Can the equation ax = be' true if neither a nor x is zero? 
 Does it require that both a and x should be zero ? 
 
 5. What is the value of f ? Why ? What is the value of 0/a, 
 wh^re a is any number except zero ? 
 
 6. May ^ = 5? 1000? -72? ^? Explain. What is 
 meant by saying that ^ gives an indeterminate quotient ? 
 
 7. The quotient ^ cannot be a finite number. W^hy ? Will 
 it be an infinitely large or an infinitely small number (cf. Ex. 1)? 
 
54 HIGH SCHOOL ALGEBllA [Ch. IV 
 
 42 * Some elementary laws. What is the meaning of the expres- 
 sion 5 + 2 + 8 (cf . § § 4 and 10) ? of 2 +5 + 8 ? Wherein do these 
 expressions differ? 
 
 (i) Although a change in the order in which operations are 
 performed may, in general, change the result (cf. § 10), yet 
 some such changes of order do not affect the result. Thus : 
 5 + 2 + 8=2 + 5 + 8 = 8 + 5 + 2, 
 
 5.2.8 = 2.5.8 = 8.5.2, 
 5 + 2 + 8 = 5 + (2 + 8) = 5 + 10, 
 5.2.8 = 5. (2. 8) = 5. 16, 
 and 5.(2 + 8) = 5.2 + 5.8. 
 
 (ii) Moreover, based upon our experience with particular sets 
 of numbers, we have silently assumed, in the preceding pages, 
 that the above changes may be made with any numbers whatever 
 without affecting the result. Thus, if a, 6, and c represent any 
 numbers whatever (positive, negative, integral, etc.), we have as- 
 sumed (without proof) that : 
 
 a + & + c = 6+a + c = c + a + 6, etc., (1) 
 
 a 'h ' c = h ' a • c = c • a 'h, etc., (2) 
 
 a^h + c = a-\-(h + c), etc., (3) 
 
 a 'b ' c = a ' (b ' c), etc., (4) 
 
 and a- {b + c) = ab-^ ac. (5) 
 
 Of these equations, (1) states what is known as the commutative 
 law of addition; (2), the commutative law of multiplication; (3), 
 the associative law of addition; (4), the associative laiv of multipli- 
 cation; and (5), the distributive law of multiplication as to addi- 
 tion: all of them taken together are often spoken of as the 
 combinatory laws of algebra. 
 
 These laws are easily verified in any particular cases : through- 
 out this book we shall continue to assume their correctness. We 
 wish, however, to point out to the pupil that mere verifications, 
 however numerous, do not establish a general law. 
 
 * This article may, if the teacher prefers, be omitted till the subject is 
 reviewed. For a full discussion of these laws see El. Alg. Chap. IV. 
 
CHAPTER V 
 EQUATIONS AND PROBLEMS 
 
 43. Equation. Members of an equation. A statement 
 that each of two expressions has the same value (i.e., repre- 
 sents the same number) as the other, is called an equation. 
 These two expressions are called the members of the equa- 
 tion, the expression preceding the sign of equality being the 
 first member, and the other the second member. 
 
 Thus, Sx—16 = Sx-i-4: is an equation ; 8 a; — 16 is its 
 first member, and S x -\- 4: its second member. 
 
 Remark. In algebraic work, the equation is a most important 
 instrument ; to it is due the chief advantage of algebra over arith- 
 metic. We have already seen some evidence of this in § 3, but 
 much more is to follow. In a recent book Sir Oliver Lodge says : 
 " An equation is the most serious and important thing in mathe- 
 matics." 
 
 44. Conditional equations. Identical equations. Is the 
 
 statement Sx — lQ=Sx-\-4:, true when x = l? when x=2? 
 when a: = 3 ? when a^ = 4 ? when x = 5? Answer the same 
 questions with regard to 2x= (x -{- 1)^ — Qa^ + 1). 
 
 The equation Sm-\- ^n = 22 is true if m = 4 and n = 2, 
 but is not true for any other positive integral values of m 
 and n ; while the equation Sx^ + k — a^ = k-\-2x^is true for 
 all values that may be assigned to x and k. 
 
 An equation which is true for all values that may be as- 
 signed to its letters is called an identical equation, and also 
 an identity; while one which is true only on condition that 
 certain particular values be assigned to its letters, is called 
 a conditional equation. In the following pages we shall use 
 the word equation to mean conditional equation unless the 
 contrary is expressly stated. 
 
 55 
 
56 HIGH SCHOOL ALGEBRA [Ch. V 
 
 As we shall see later, by performing the indicated opera- 
 tions the two members of an identity may be reduced to 
 exactly the same form; hence the name "identical equation." 
 
 45. Roots of an equation. Checking. The roots of an 
 
 equation are those values which satisfy the equation; ^.e., 
 they are those values which, when substituted for the letters 
 the equation contains, make the two members identical. 
 
 Any process by which the roots of an equation are found 
 is called solving the equation. 
 
 The final test of the correctness of supposed roots is to 
 substitute them for the letters in the equation ; if they satisfy 
 the equation they are roots, otherwise not. This process is 
 called checking the roots. Thus, 4 is a root of the equation 
 
 8a;-16 = 3a:H-4, 
 because 8-4 — 16 = 3-4+4. [each member being 16 
 
 46. Some axioms and their uses. The following principles, 
 usually called axioms^ are useful in solving equations : 
 
 1. If equals (i.e., equal numbers^ are added to equals^ the 
 sums are equal. 
 
 2. If equals are subtracted from equals, the remainders are 
 equal. 
 
 3. If equals are multiplied hi/ equals^ the products are equal. 
 
 4. If equals are divided hy equals, the quotients are equal. 
 Here, however, as elsewhere, it is not permissible to divide 
 by zero [cf. § 41 (v)]. 
 
 The correctness of these axioms rests upon the fact that 
 equal numbers are in reality the same number, differing at 
 most in form. Thus, 24 + 11, 7 • 5, and 6^ — 1 are merely 
 different forms of writing 35. 
 
 Suggestion to the Teacher. It is strongly recommended that the teacher 
 illustrate the physical meaning of an equation, and also the meaning of the 
 above axioms, by means of a pair of balances (easily made, if not provided 
 by the school). 
 
44-47] EQUATIONS AND PROBLEMS 57 
 
 47. Solution of equations. To show how the above axioms 
 may be used in solving equations, let it be required to 
 solve the equation H x—lQ = '^ x -\- 4, ^.e., to find the value of 
 X which satisfies it. 
 
 
 
 SOLUTION 
 
 
 Since 
 
 
 Sx-16 = 3x-\-4r, 
 
 
 therefore 
 
 Sx 
 
 -16 + 16 = 3x + 4+16, 
 
 [Axiom 1 
 
 i.e., 
 
 
 8 a^ = 3 a; + 20, 
 
 
 and therefore 
 
 
 8x-3^ = 3a; + 20-3a-, 
 
 [Ax. 2 
 
 i.e., 
 
 
 5x = 20, 
 
 
 whence 
 
 
 x = 4.. 
 
 [Ax. 4 
 
 CHECK 
 
 On substituting 4 for x in the original equation, that equation 
 becomes 
 
 8. 4- 16 =3. 4 + 4, i.e., 16 = 16 ; 
 
 hence the equation is satisfied, and 4 is a root (cf. § 45). 
 
 EXERCISE XXVIII 
 
 1. Is 2 a root of a^ — 5 07 + 6 = ? Is 3 also a root ? Explain. 
 How may we check a supposed root of an equation ? 
 
 Solve the following equations, give the reasons for each step in 
 the work, and check the roots : 
 
 2. 10a; = 40. 10. 3m + 2 = m4-30. 
 
 3. Sy = -32. 11. 7a;-10 = 5x + 18. 
 
 4. k + l = 7. 12.-4^ = 3 + ^ — 15. 
 
 5. m-9 = 4. 13. 20-12w + 5 = 0. 
 
 6. 2i; + 7 = 63. ' 14. 13s-9-2s = 24. 
 
 7. 2v-7 = 63. 15. 7x-55 = lS-2x-l. 
 
 8. 46 = 5s-4. 16. 6v-(v-3)-12 = 0. 
 
 9. -13 = 3a; + 8. 17. 2/'-(/+2/ + 8) = -6. 
 
 18. In Exs. 7-13, point out the members of each equation. 
 Which is the first member of the equation in Ex. 14 ? What is 
 the other member called ? 
 
58 BIGII SCHOOL ALGEBRA [Ch. V 
 
 19. What is meant by solving an equation ? Describe briefly 
 the process used in solving an equation. 
 
 20. Are the equations in Exs. 2-17, above, conditional equations 
 or identities ? Why ? In which class of equations would you place 
 
 2x + S = 2(4:X-{-3)-(6x + 3)? Why ? 
 
 21. If 2 a is subtracted from each member of the equation 
 5x-\-2a = 3x-\-4:b, what is the resulting equation ? What does 
 this show with reference to removing a term from the first to the 
 second member of an equation ? Is the same thing true when a 
 term is removed from the second member to the first? Show 
 this by adding —3 a; to each member of the given equation. 
 
 48. Transposition. Directions for solving equations. Re- 
 moving a term from one member of an equation to the other 
 is called transposing that term. 
 
 If 
 
 x-}-a = b, 
 
 
 
 then 
 
 x-\-a — a = b- 
 
 -a. 
 
 [Ax. 2 
 
 i.e., 
 
 x=h- 
 
 -a. 
 
 [•.•a-a = 
 
 Again, if 
 
 x = b- 
 
 -a, 
 
 
 then 
 
 x-\-a = b- 
 
 -a-\-a, 
 
 [Ax. 1 
 
 i.e., 
 
 x-^a = b. 
 
 
 [... -a-i-a = 
 
 Hence, since a msij represent any term whatever, a term 
 may he transposed from one member of an equation to the other 
 by merely reversing its sign (cf. also Ex. 21, above). 
 
 For solving equations such as those considered in § 47 the 
 following directions may now be given:. 
 
 1. Transpose all the terms containing the unknown number 
 to the first member of the equation., and all other terms to the 
 second member. 
 
 2. Unite the terms of each member., and then divide both 
 members by the coefficient of the unknown number. 
 
 3. Check the root thus found by substituting it in the given 
 equation. 
 
47-48] EQUATIONS AND PROBLEMS 69 
 
 Ex. 1. Solve the equation 4 s — 15 = 2 s + 11. 
 
 SOLUTION 
 
 Transposing, we have 4 s — 2 .s = 11 + 15; 
 uniting like terms, 2 s = 26; 
 
 dividing by 2, s = 13. 
 
 Check: 4-13-15 = 
 
 : 2 • 13 + 11. f each mem 
 
 
 
 Lber is 37 
 
 EXERCISE XXIX 
 
 Solve (and check) the following equations : 
 
 2. 32/-5 = 22. 
 
 3. -10 = 6 a + 8. 
 
 13. 
 
 ^-7 = 12. 
 4 
 
 4. 3(aj-5) = 48. 
 
 14. 
 
 2^ + ^ = ??. 
 
 5. iz-\-2-z^n. 
 
 
 3 6 
 
 6. A.-x = -ll-\-2x. 
 
 15. 
 
 ^ ^^^ = m + 10. 
 3y-7 = A-2y-5. 
 
 7. {d-\-lf-d'' = -ll, 
 a 20-5fc = 3A; + 3. 
 9. -8a; = 4(aj-2) + 10. 
 
 16. 
 17. 
 
 10. 4(-3 + /i') = (2/i-3)2. 
 
 XX. 1^-1 = 10. 
 
 18. 
 19. 
 
 i(2/-6) = K2/-2). 
 x-{-l x + 6 o 
 - 7 + 2 - ^* 
 
 [Multiply both members of the 
 
 20. 
 
 3ia = 5a-9ia-16. 
 
 equation by 12 (see Ax. 3).] 
 
 21 
 
 12-3a; + 20 = 44 + 3a;. 
 
 • ^'- f'-ro-''- 
 
 22. 
 
 x-9 x-5 ^ 
 3 ~ 12 ' 
 
 23. 14A;-(20-7A:-2)=:6A:+68. 
 
 24. (c + 5)(2c-l)-(2c-3)(c + 7)=0. 
 
 25. What is meant by transposing a term from one member of 
 an equation to the other ? What change must be made in a term 
 thus transposed? 
 
 26. State in order the axioms thus far used in solving equa- 
 tions. Illustrate the use of each. Why does the division axiom 
 not apply when the divisor is zero? [Cf. § 41 (v).] 
 
 HIGH SCH. ALG. — 5 
 
00 HIGH SCHOOL ALGEBRA [Ch. V 
 
 27. Point out the fallacy in the following reasoning : 
 
 If X = a, 
 
 then x^ = ax, 
 
 and xF — a^ = ax — a', [subtracting a^ from 
 
 each member 
 i.e., (x + a)(x — a) = a (x — a) ; 
 
 therefore 2a(x — a)=a(x — a), [since x = a 
 
 and, therefore, 2 = 1. [dividing by a (x — a) 
 
 49. Translation of common language into algebraic lan- 
 guage, and vice versa. Tlie equation a^ — 8 = 3 is an algebraic 
 sentence ; it may be translated into common (verbal) lan- 
 guage thus : " X exceeds 8 by 3 " or ^^x is 3 greater than 8." 
 
 Similarly, the verbal statement " the excess of a over the 
 product of 8 and t is 9," when expressed algebraically, be- 
 comes a — st=9. 
 
 In order to use equations easily in the solution of prob- 
 lems we must learn to translate freely from either of these 
 two languages into the other. 
 
 EXERCISE XXX 
 
 Write as algebraic sentences : 
 
 1. Nine is 2 greater than x. 
 
 2. 2/ is 8 less than 3 x. 
 
 . 3. a^ exceeds 2 a by 1. • 
 
 4. The excess oi Sx over 6 a; is 2 a?. 
 
 5. The difference of two given numbers is five less than three 
 times their sum. 
 
 [Hint. Let a and b be the given numbers.] 
 
 6. The product of two given numbers exceeds half the larger 
 number by 17. 
 
 7. Twenty-one is divided into two parts, the smaller of which 
 is p. What is the larger part? Express by an equation that 
 the larger part exceeds the smaller by 3. 
 
48-49] EQUATIONS AND PROBLEMS 61 
 
 8. Translate into verbal language the equations in Exs. 5-11, 
 p. 57. In how many different ways may we translate the equa- 
 tion in Ex. 8 ? 
 
 9. A father is now 4 times as old as his son. Eepresent 
 the age of each 5 years ago ; 5 years hence. Also express by an 
 equation the fact that 5 years ago the father's age was 7 times 
 that of his son. 
 
 [Hint. Let the son's present age be s years.] 
 
 10. Translate into algebraic language the following statement : 
 a rectangular flower bed whose length is y feet, and whose 
 width is 6 feet less than its length, contains 40 square feet. 
 
 11. If butter costs m cents a pound, eggs n cents a dozen, and 
 milk r cents a quart, express in algebraic language that 
 
 (1) the combined cost of 8 qt. of milk and 6 doz. eggs is $ 1.90. 
 
 (2) the cost of 9 qt. of milk is 30 cents less than the cost of 
 2i lb. of butter. 
 
 12. Express as common fractions : 50 % of n dollars ; 26 % of 
 k bushels ; m % of $ 525. Show that the amount of x dollars at 
 5 % simple interest for 3 years is cc -f- -^^ x. 
 
 13. If the units' digit of a number is 2, the tens' digit 4, the 
 number itself is 4 • 10 + 2, i.e., 42. What is the number whose 
 units' digit is 8 and whose tens' digit is 3 ? the number whose 
 tens' digit is x and whose units' digit is a; 4- 7 ? 
 
 14. The smallest of three consecutive integers is a ; what are 
 the other two ? If n is any integer, 2 n is an even integer ; write 
 the even integer next higher than «2 n ; next lower than 2 n. 
 Write the odd integer next lower than 2 n ; next higher than 2 n. 
 
 15. A walks 2i miles an hour ; B, 3 miles an hour. How far 
 does each walk in 3 hours ? in f hours ? How much farther 
 than A does B walk in 1 hour ? Express by an equation that in 
 t -\-2 hours B walks 3 miles farther than A. 
 
 16. At the rate given in Ex. 15, in how many hours will A 
 walk 10 miles ? 15 miles ? s miles ? Answer the same ques- 
 tions for B. 
 
62 HIGH SCHOOL ALGEBRA [Ch. V 
 
 17. If I can do a certain piece of work in 6 days, what part of 
 it can I do in 1 day ? in 5 days ? in a? days ? If I can finish 
 a job in d days, what part can I finish in 1 day ? in 3 days ? 
 
 50. Problems leading to equations. A problem is a ques- 
 tion proposed for solution; it always asks to find one or 
 more numbers which at the beginning are unknown, and it 
 states certain relations (conditions) between these numbers, 
 by means of which their values may be determined. 
 
 In solving a problem the important steps are : 
 
 1. To represent one of the unknown numbers involved in the 
 problem by some letter^ as x. 
 
 2. To translate the common language of the problem into 
 algebraic language. 
 
 3. To solve the equation thus founds — called the equation of 
 the problem. 
 
 4. To check the result. 
 
 These steps are illustrated in the solutions of the follow- 
 ing problems. 
 
 Prob. 1. The sum of the ages of a father and son is 54 years, 
 and the father is 24 years older than the son. How old is each ? 
 
 Solution. Stated in verbal language, the given conditions are : 
 
 (1) The number of years in the father's age plus the number of 
 years in the son's age is 54. 
 
 (2) The number of years in the son's age plus 24 equals the 
 number of years in the father's age. 
 
 To translate these conditions into algebraic language, 
 let X stand for the number of years in the son's age ; 
 then, by the second condition, 
 
 0? + 24 stands for the number of years in the father's age, 
 and, by the first condition, 
 
 a? + cc + 24 = 54, 
 which is the equation of the problem. 
 
 Solving this equation, we find x — 15, whence a; + 24 = 39. 
 On substitution in the problem, these numbers are found to satisfy 
 
49-50] EQUATIONS AND PliOBLEMS 63 
 
 its conditions (i. e., to check) ; therefore the father's and son's 
 ages are, respectively, 39 years and 15 years. 
 
 Prob. 2. A boy was given 39 cents with which to buy 3-cent 
 and 5-cent postage stamps, and was told to purchase 5 more of 
 the former than of the latter. How many of each kind should 
 he purchase ? 
 
 Solution. Stated in verbal language, the given conditions are: 
 
 (1) The total expenditure is 39 cents. 
 
 (2) There are to be 5 more 3-cent stamps than 5-cent stamps. 
 To translate these conditions into algebraic language, 
 
 let X stand for the number of 5-cent stamps purchased, 
 then 6x stands for the number of cents in their cost; 
 and, by the second condition, 
 
 a? -f 5 stands for the number of 3-cent stamps purchased, 
 and 3 07 4-15 stands for the number of cents in their cost; 
 hence, by the first condition, 
 
 5aj + 3a;-f 15 = 39, 
 which is the equation of the problem. 
 
 Solving this equation, we have cc = 3, whence a; + 5 = 8. Sub- 
 stitution in the problem shows that .these values check. Hence 
 the number of 5-cent stamps is 3, and the number of 3-cent 
 stamps is 8. 
 
 Prob. 3. If a certain number is diminished by 6, and twice 
 this diiference is added to 5 times the number, the result will 
 equal 88 minus 3 times the number. What is the number ? 
 
 Solution. To form the equation of the problem, 
 let n represent the number sought, 
 
 then 6 n = 5 times the number, 
 
 and 2 {n — 6) = twice the difference of this number and 6, 
 
 and 88 — 3 71 = 88 minus 3 times the number. 
 
 Hence the given condition becomes 
 
 5n-f 2(n-6)=88-3w. 
 
 The solution of the equation gives n = 10, which checks; there- 
 fore 10 is the required number. 
 
64 nWU SCHOOL ALGEBRA [Ch. V 
 
 Prob. 4. A number consists of two digits whose sum is 5 ; if 
 the digits are interchanged, the number is diminished by 9. 
 What is the number ? 
 
 Solution. Let x represent the digit in the units' place ; 
 then, by the first condition, 
 
 5 — a? = the digit in the tens' place, 
 and 10 (p — x) -\- X =^ the number, [cf . Ex. 13, p. 61. 
 
 and 10 X + (5 — ic) = the number with its digits interchanged. 
 
 Hence, by the second condition, 
 
 1 a; 4- T) - a; = 1 (5 - a?) + a; - 9, 
 whence x — 2, and 5 — a? = 3. 
 
 These two digits are found to satisfy both conditions of the 
 problem ; therefore the number sought is 32. 
 
 EXERCISE XXXI 
 
 5. John has 14 cents less than Henry ; together they have 
 60 cents. How much money has each ? 
 
 6. Divide 28 into two parts whose difference is 4. 
 
 7. The sum of two numbers is 63, and the larger exceeds the 
 smaller by 17. What are the numbers ? 
 
 8. If 16 is added to a certain number, the result is the same 
 as it would be if 7 times the number were subtracted from 56. 
 What is the number ? 
 
 9. Of four given numbers each exceeds the one below it by 
 3, and the sum of these numbers is 58. Find the numbers. 
 
 10. Divide $2200 among A, B, and C in such away that B 
 shall have twice as much as A, and C $ 200 more than B. 
 
 11. I take a trip of 90 miles, partly by train, partly by trolley. 
 If I go 42 miles farther by train than by trolley, how far do I 
 go by each ? 
 
 12. Three boys together have 140 marbles. If the second has 
 twice as many as the first, but only half as many as the third, 
 how many marbles has each boy ? 
 
60] EQUATIONS AND PROBLEMS 65 
 
 13. After taking 3 times a certain number from 11 times that 
 number, and then adding 12 to the remainder, the result is less 
 than 117 by 7 times the number. What is the number ? 
 
 14. I spend $ 2.50 for 3-cent and 4-cent stamps, getting 25 more of 
 the former than of the latter. How many of each kind do I buy ? 
 
 15. A man who is 32 years old has a son who is 8 years old. 
 How many years hence will the father be 3 times as old as his 
 son (cf . Ex. 9, p. 61) ? 
 
 16. The sum of two consecutive integers is 73. What are the 
 integers (cf. Ex. 14, p. 61) ? 
 
 17. Find three consecutive integers whose sum is 51. Show 
 that the sum of any three consecutive integers is 3 times the 
 second of these integers. 
 
 18. The difference between the squares of two consecutive 
 integers is 19. Find the integers. 
 
 19. Find two consecutive even integers whose sum is 98. • 
 
 20. Find the even integer whose square subtracted from that 
 of the next higher even integer leaves 52. 
 
 21. The accompanying diagram represents 
 the floor of a room. If the perimeter (the dis- 
 tance around it) is 5 times the width, how wide 
 is the floor ? how long ? How many square 
 yards in its area ? 
 
 22. The length and breadth of a rectangular floor differ by 
 5 ft. ; the perimeter is 60 ft. Find the dimensions and area of 
 this floor ; also make an accurate diagram of the floor. 
 
 23. If each side of a square lot were increased by 2 yd., the 
 area of the lot would be increased by 96 sq. yd. Find the side 
 of the given lot. Draw an appropriate diagram. 
 
 24. A certain rectangle is 5 ft. longer than it is wide ; if each 
 dimension were increased by 2 ft., the area would be increased by 
 38 sq. ft. Find the length, the breadth, and the area of this 
 rectangle. 
 
 (aH-6)feet 
 
66 HIGH SCHOOL ALGEBRA [Ch. V 
 
 25. Five boys agreed to purchase a pleasure boat, but one of 
 them withdrew, and it was then found that each of the remaining 
 boys had to pay $ 2 more than would have been necessary under 
 the original plan. How much did the boat cost ? 
 
 26. A laborer was engaged to do a certain piece of work on 
 condition that he was to receive $2 for every day he worked, and 
 to forfeit 50 cents for every day he was idle ; at the end of 18 
 days he received $28.50. How many days did he work? 
 
 27. A number consists of two digits whose sum is 8; and if 
 36 is subtracted from this number, the order of its digits is 
 reversed. What is the number? 
 
 28. In a certain two-digit number the tens' digit is twice the 
 units' digit, and the number formed by interchanging the digits 
 equals the given number diminished by 18. What is the number ? 
 
 29. A two-digit number equals 7 times the sum of its digits ; 
 the tens' digit exceeds the units' digit by 3. Find the number. 
 
 30. What principal at 5 % interest yields an annual income of 
 $250 ? What principal at 4 % simple interest amounts in 5 years 
 to $2400 (cf. Ex. 12, p. 61)? 
 
 31. I make two equal investments, one at 6%, one at 4%. If 
 the difference in the annual income from the two is $80, find the 
 total sum invested. 
 
 32. Two trains which travel, respectively, 30 and 50 miles an 
 hour, start toward each other at the same time from two cities 
 240 miles apart. How long before they meet ? 
 
 Suggestion. Let x = the number of hours before they meet ; then 
 30x + 50x = 240 (cf. Ex. 15, p. 61). 
 
 33. Two bicyclists ride toward each other from towns 104 miles 
 apart, the first at the rate of 12 miles an hour, the second at the 
 rate of 14 miles an hour. If they start at the same time, how 
 long before they meet (cf. Ex. 15, p. 61) ? 
 
 34. Two bicyclists, A and B, whose rates are, respectively, 12 
 and 15 miles an hour, start from the same town and rid^ in the 
 same direction. If A starts 1| hours before B, how long before 
 B overtakes him ? 
 
60] EQUATIONS AND PROBLEMS 67 
 
 35. A walks m miles at the rate of 3 miles an hour, returning 
 at the rate of 2i miles an hour. If the entire walk is made in 
 5^ hours, what is the value of m ? 
 
 Hint. The first half of the walk can be made in — hours, the second in 
 
 ^ hours (cf. Ex. 16, p. 61). 
 5 
 
 36. A tourist climbs a certain mountain at an average rate of 
 2 miles an hour, and descends at an average rate of 3 miles an 
 hour. If the round trip takes 6 hours, how long is the path ? 
 
 37. The diiference of the radii of two circles is 4 inches ; the 
 sum of their circumferences is 88 inches. Find the radius of 
 each. (The circumference of a circle equals 27r times its radius ; 
 and 7r = 3|, approximately.) 
 
 38. Divide $351 among three persons in such a way that for 
 every dime the first receives, the second shall receive 25 cents, 
 and the third a dollar. 
 
 39. Divide 48 into two parts such that twice the larger part 
 equals 10 times the smaller part (cf. Ex. 34, p. 7). 
 
 40. Three times Harry's age equals 5 times the age of his 
 sister ; the sum of their ages is 24 years. How old is each ? 
 
 41. A, working alone, can do a certain piece of work in 3 days; 
 B, in 6 days. In how many days can they complete it, working 
 together (cf. Ex. 17, p. 62)? 
 
 Hint. Let x = the required number of days ; then - = -+-. 
 
 aj 3 o 
 
 42. Solve Prob. 41, if A can do the work in 8 days, B in 6 days ; 
 also, if A can do the work in 4|^ days, B in 4 days. 
 
 REVIEW EXERCISE-CHAPTERS J-V 
 
 1. Define : negative number, absolute value of a number, coef- 
 ficient, exponent, term, polynomial, degree of a term, finite num- 
 ber, equation, root of an equation, identity. Illustrate each of 
 your definitions; 
 
 2. Define and illustrate : inverse operations, multiplication, 
 division. Point out at least one advantage which the definition 
 of multiplication given in § 7 (ii) has over that in § 7 (i). 
 
68 HIGH SCHOOL ALGEBRA [Ch. V 
 
 3. If distances above sea level are called positive, what would 
 — 25 feet mean? +35 feet? What is the difference between 
 these elevations ? 
 
 4. A walks east at the rate of 3 miles an hour, B at the rate 
 of —2 miles an hour. How long before the two are 15 miles 
 apart? Illustrate by a drawing. 
 
 5. Translate into algebraic language : 
 
 (1) The number formed by interchanging the digits of a cer- 
 tain two-place number exceeds the number itself by 18. 
 
 (2) A certain number diminished by 5 % of itself equals 76. 
 
 (3) The sum of two consecutive even integers equals half the 
 difference of their squares. 
 
 6. Point out at least one advantage in using letters to repre- 
 sent numbers. 
 
 7. How are two or more similar monomials added? State a 
 rule for subtracting one polynomial from another. 
 
 8. How may a parenthesis which incloses several terms, and 
 which is preceded by the minus sign, be removed without affect- 
 ing the value of the expression ? Why ? 
 
 9. State the law of signs for multiplication ; for division. 
 What powers of —3 between the 1st and 12th are negative in 
 sign ? Why ? Find the value of ( - 1)3 . ( - 10)^ -- ( - 5)4. 
 
 10. State the exponent law for multiplication ; for division. 
 By reference to these laws find the value of ^^"^'^' ' ^^"'^^^" . 
 
 If c = 9, d = -5, e = -2,/=-2,^ = l^ find the value of; 
 
 11. 5c + 3d-e4-/-c--/+2c.3^--e + 8/. 
 
 12. 4 c'f -i-6eg^-\-d^-\-5[e- 3/+ c'-3 ed]. 
 
 13. -^i^llc-(12g'-6efg)^-\-(c'-^d^^{c + dy. 
 Perform the following indicated operations : 
 
 14. (x—5y){4:y — x). 17. (a^ — 1)1 
 
 15. (am — en) (3 a -f 5 c). 18. (m^ ~mn-{-7')(m^-{-mn — r). 
 
 16. (k-2qf. 19. (62- + c^)(&2-_c-). 
 
50] REVIEW EXERCISE 69 
 
 20. (a^« + 2/^-i)(a:-l). 22. (x"^+* -5x''+'+6)--r(x'^+^-3). 
 
 21. (r2"' — r"*H-l) (3 + ?•'»-'). 23. {x^' -y^'^) ^ (y^ — x"). 
 
 24. (1x2+ ia;-i) + (f a.'- 3 x^-r"')- (-2x^ + 1^-1).' 
 
 25. 4p - [p2 4. 22)r - (2 q +p') j + (1 -j^r + g). 
 
 26. 5m — Sn—\ — 7n-\-m — 5n — 3ml. 
 
 27. 2a;-a;-2/ + 22-[-S-(y + 42;)|]. 
 
 28. a(6 + c — cZ) — 6(a — 2 c + d) — 3 c(— a — d). 
 
 If itf=7a6-3Z>2_4a2, ^=36^-4 a^- «?>, P=a-b, and 
 Q = ?,* _ 4 a^ft - 4 a6^ -h 6 a262 + a^ find the value of : 
 
 29. M-\-N. 33. J»f2. 37. Qh-P. 
 
 30. Jf-iV: 34. itfP. 38. Q^M. 
 
 31. |(4jlf+iV^). 35. MN. 39. 2Q-NP. 
 
 32. P^-5N. 36. ^-J-iJf. 40. Q--P'^ + 3P2. 
 
 41. Check your work in Exs. 32-37. What two checks may 
 be used for Ex. 30 ? for Ex. 36 ? 
 
 42. In Ex. 44 below, insert the second and third terms in a 
 parenthesis preceded by + ; place the fourth and fifth terms 
 under a vinculum preceded by — (see footnote, p. 37). 
 
 43. In Ex. 45 below, inclose the first three terms in a paren- 
 thesis preceded by — ; place the last two under a vinculum pre- 
 ceded by + . 
 
 In each of Exs. 44-49, collect the coefficients of r, s, and t 
 (cf. Exs. 24-26, p. 47). 
 
 44. 2 a^s — 3 ar — 6s — cs — 4 r. 
 
 45. -r-\-5fh + 2s+fh-lft. 
 
 46. -nH-{-5<^ds-2hH-'ieft-\0<?cls, 
 
 47. t -f aV + 6=^s H- r -i- s — 2 c? + 2 hs -{-cH — 2 ar. 
 
 48. (a H- 2 c)s -f (c + d)r + 2 cs — 5dr — 7 ds — er. 
 
 49. (m^ - l)s - 2(1 - ny + nV -3s- (m* - 2 mhi^)r + 91^^. 
 
 50. Multiply x^ -\- {a -{- c)x hj X — c -, hj x^ -j- ex -\- k. 
 
70 HIGH SCHOOL ALGEBRA [Ch. V 
 
 51. Divide 2 a^ -f- 2 wV — n^x — mn^ — a^(2 m -f- 2 n) — wa^ — 
 2 mnic + mna;^ by —x^ + {m-{- n)x + mn (cf . Ex. 47, p. 51). Check 
 your result by multiplication. 
 
 Solve, and check the roots: 
 
 52. x{2 x-B)-51 = 2 x{x-\-l). 54. |a;-(|a; + i) = -a; + f 
 
 53. ^±2 = ?^±3^_2^. 55. |(.-2)(.-l)=^^-^^ 
 
 5 3 2 4 
 
 56. Four times the number of seniors in a certain school 
 exceeds the number of freshmen by 15. If the total enrollment 
 in the two classes is 130, find the size of each class. 
 
 57. A certain number is subtracted from 50 and 42 in turn. If 
 \ of the first remainder equals ^ of the second, find the number. 
 
 58. If the population of a town has increased 30 % in the last 
 10 years, and is now 5200, find the population 10 years ago. 
 
 59. A man spends \ of his income for living expenses and 
 insurance, ^^ for books, -^^ for travel, -^^ for charities. If he 
 saves $425, what is his income ? 
 
 60. A certain rectangle is 8 ft. longer and 5 ft. narrower than 
 a given square, and its area exceeds that of the square by 5 sq. ft. 
 Find the side of the square. Draw an appropriate figure. 
 
 61. A and B together can do a certain piece of work in 6 days. 
 If A can do it alone in 10 days, in how many days can B do it ? 
 
 62. Silk marked to sell at a gain of 33|^ % has its marked price 
 reduced 20 %, and then sells for 80 cents a yard. Find its cost. 
 
CHAPTER VI 
 
 TYPE FORMS IN MULTIPLICATION — FACTORING 
 I. SOME TYPE FORMS IN MULTIPLICATION 
 
 51. Type forms. Although all exercises in multiplication 
 and division of integral polynomials can be readily solved 
 by § 38 and § 39, yet there are a few special cases of these 
 operations which occur so frequently in practice that it is 
 well worth one's while to memorize them ; they are often 
 spoken of as type forms. Some of these type forms are con- 
 sidered in the next few paragraphs. 
 
 52. Square of a binomial. Let a and h represent any two 
 numbers whatever; then by actual multiplication (§ 33), 
 
 (« + ^») (a + 6) = ^2 4- 2 a6 + ^2, 
 and (a - 6) (a - 6) = a2 - 2 a6 + 52 . 
 
 i.e., (a+6)2 = a2 + 2a6 + 62, I 
 
 and (a_6)2 = a2-2a6 + 62, II 
 
 whatever the numbers represented by a and h. 
 
 Translated into common language, I becomes : 
 
 The square of the sum of any two numbers equals the square 
 of the first number^ plus twice the product of the two numbers^ 
 plus the square of the second number. 
 
 The student may translate II into common language. 
 
 By means of I and II, we can now write down (without 
 actually performing the multiplication) the expanded form 
 of the square of any binomial whatever. Thus : 
 
 Ex. 1. (m + 3)2 = m^ + 6 m + 9. 
 
 Ex. 2. (x — yy = x^~2xy-\- ?/. 
 
 Ex. 3. (2 s - 3 ty = (2 sy - 2(2 s)(3 t) + (3 ty = 4 s2 - 12 st + 9 tK 
 
 71 
 
72 HIGH SCHOOL ALGEBRA [Ch. VI 
 
 EXERCISE XXXII 
 
 Expand the following (check your work as teacher directs) : 
 
 4. {x^yf. 12. (1+2^)'. 20. (la -2)2. 
 
 5. {m + n)\ 13. {l-vf, 21. {\yJ^Qf. 
 
 6. Qi-^kf. 14. (2 a; -6)2. 22. (5a^-f)2. 
 
 7. (w + it;)l 15. (a. + 7ic)2. 23. (5m + |w)2. 
 
 8. (a-p)l 16. (3m^-2)2. 24. (2 a^x + 3 62/^)2. 
 
 9. (c-/i)l 17. (2g^-5hy. 25. (5 rs - 3 r^s^)^ 
 
 10. (x + Sy. 18. (11-7A;)2. 26. (ic'» + r)2. 
 
 11. (a - 5)2. 19. (4 b^ -h 1)2. 27. (3 a" - 2 s^f. 
 
 28. Expand: (a -\- b - 5)% i.e., ](a -^ b) -5\^', (c + 2H-d)2; 
 (_2c-(Z + e«)2; (7-ha2-c)2; (3a.-«-p-5)2. 
 
 29. Since a — b = a^( — b), show that II, § 52, is included 
 under I. 
 
 30. What must be added to x^-^6x to make it the square of a;+3 ? 
 
 31. What must be added to a'* + a262 -f- b* to make it the square 
 oia'-i-b'? 
 
 32. What must be added to 25 — 10 a^ to make it the square of 
 
 33. What must be added to ic^ + 2 x*y^ + 4 / to make it the 
 square of a;'' + 2/? 
 
 By the method of § 52 write down the squares of the following 
 numbers : 
 
 34. 16, i.e., 10 + 6. 36. 28. 38. 71. 
 
 35. 19, i.e., 20 - 1. 37. 43. 39. 83. 
 
 40. Expand (a + 1)2 ; also (—a — 1)2. Compare and explain. 
 
 53. Product of sum and difference. If a and b represent 
 any two numbers whatever, then, by actual multiplication, 
 
 whatever the numbers represented by a and h. 
 
 The student may translate this formula into common lan- 
 guage (cf. § 52). 
 
52-54] TYPE FORMS IN MULTIPLWAriON 73 
 
 EXERCISE XXXIII 
 
 Write the following products by inspection and check results : 
 
 1. ix^y){x-y). 12. (4+a«)(4_,,3), 
 
 2. (7n-\-n){m — 7i). 13. (2/"» — 11)(^'" + 11). 
 
 3. {^x + y){^x-y). 14. {ax'' + W){ax''-W). 
 
 4. {x-2y)(x + 2y). 15. \{x-y) + z\\{x-y)-z\. 
 
 5. (4a + 15Z/)(4a-15?>). 16. K« + &) + cn(« + &) -cj. 
 
 6. (G /) — 5 q) (6 ji) + 5 q). * 17. (m + n +p)(m + n — p). 
 
 7. (2a;2/— 7)(2a;?/ + 7). 18. (c - d + 5)(c — d — 5). 
 
 8. (4m2-3n)(4m2 + 3w). 19. j2-(a;4-2/)n2 +(^4-2/)S. 
 
 9. (9 + 5pV)(9 — 5pV). 20. (7 + m + rz)(7 — m — w). 
 
 10. (a^ + i/)(a^-i/). 21. {a-h^x){a-\-h-x). 
 
 11. (10mn-6)(10mn + 6). 22. (2 A: - Z + 3)(2 A: + Z-3). 
 
 23. (9lx''-4.f)-^(^x-2y) = ? Why? 
 
 24. (16a2-2562)-f-(4a + 56) = ? Why? 
 
 25. (a;^-2/')^(a;^-2/0 = ? Why? 
 
 26. (a;«-2/*)-(ar^-/)=? 27. (a;i« - ^/S) -^ (a^^ + 2/') = ^ 
 
 28. Find, by the above method, the product of 22 by 18, i.e., 
 of 20 + 2 by 20-2; of 17 by 23; of 42 by 38; of 56 by 44. 
 
 54. Product of binomials having a common term. By ac- 
 tual multiplication, 
 
 (2^+3)(:r+5) = a;2 + 8:r4-15 = rr2^(3 + 5)2:4-15; 
 and (a:+3)(a:-5) = a:2-2a;-15 = a;2 + (3-5>-15. 
 And, in general, 
 
 (jr + a)(jr + 6)= jr2 + (a + 6)jr + ab, 
 
 whatever the numbers represented by a, h, and x. 
 Translating this formula into words, it becomes: 
 The product of two binomials having a term in common equals 
 
 the square of the common term, plus the algebraic sum of the 
 
 unlike terms multiplied by the cornmon term, plus the product 
 
 of the unlike terms. 
 
74 HIGH SCHOOL ALGEBRA [Ch. VI 
 
 EXERCISE XXXIV 
 Write down the following products (check as teacher directs) : 
 
 1. (a + 5)(a + 7). 16. {xy - ^)(xy + U). 
 
 2. (a-5)(a-7). 17. (-8-fmV)(2 + mV). 
 
 3. (a + 5)(a-7). 18. (s4- 7'2)(3s + r^). 
 
 4. (a_5)(a + 7). 19. \{l + m) -2\\{l^m) -^, 
 
 5. (2/-c)(2/ + 2c). 20. 5(Z + m) + 8n(^ + m)-15i. 
 
 6. (a^-f.4)(ar' + 5). 21. (m -n- 5)(m-n- 9). 
 
 7. (a^ + 4)(^2_5>)^ 22. (s-« + 4)(s-^-4). 
 
 8. (a^_4)(aj2_5)^ 23^ (,.p_ 10)(r^ + 15). 
 
 9. (a^_4)(aj2-|.5). 24. (a;^- + 3)(a^2- -7). 
 
 10. (3 4-m)(5 + m). 25. (3 a^-a;)(2.T + 3 a^). 
 
 11. (6 + a)(c + a). 26. (&^ + 2/)( -c + 2/). 
 
 12. (2a; + l)(-5 + 2aj). 27. \^{axY + 2\\^{a.xf -\-l\. 
 
 13. (a-&)(a-c). 28. (5 - 4 i«y ) (5 + a^y ) . 
 
 14. (4 + 3a)(-6+3a). 29. .(m2-c)(- 7 m^- c). 
 
 15. (4s2-5)(4s2-|-i). 30. (3j9-g-7)(3p-g + 7). 
 31. When 6 = a, what does the formula of § 54 become ? What 
 
 does it become when b =— a? Are the formulas of §§52 and 
 53 only special cases of (x -{- a) (x -\- b) = a^ + (a -\- b)x + a&? 
 
 55. Product of two binomials whose corresponding terms are 
 similar. By actual multiplication we obtain 
 
 ^x -1y 
 15 x^ + 20 xy 
 
 — ^ xy —%y^ 
 15x^-{-Uxy-Sy^ 
 
 Here the term 14 xy is the algebraic sum of the " cross 
 products " 5x'4:y and — 2 y • S x. 
 
 With a little practice the final product of two such bi- 
 nomials may be written down by inspection, i.e.^ without 
 first writing the partial products. 
 
54-66] TYPE FORMS IN MULTIPLICATION 76 
 
 EXERCISE XXXV 
 Write the following products by inspection : 
 
 1. (3a; + 2)(4a;-3). 4. (o- 11)(3 a- 1). 
 
 2. (5m-l)(2m-3). 5. (3x-^2 y){Ax-\-3y), 
 
 3. (2r + 5)(r-5). 6. (x-S y)(5 x-h 6y). 
 
 7. In each of the above products, how is the first term ob- 
 tained ? the third ? the second ? 
 
 8. What is meant by the expression " cross products " as used 
 in § 55 ? Illustrate from Ex. 3, above. 
 
 Write down the following products by inspection, and check 
 results as the teacher directs : 
 
 9. (7-2m)(7-m). 18. (ia-2)(|a+4). 
 
 10. (3 - 4 a) (4 -f 3 a). 19. {x-{- a){x-^b). 
 
 11. (9x — 2y)(x-\-y). 20. (3x+ c){x-^d). 
 
 12. (2a-4 62)(5a-6 62). 21. (3x-c)(x-d). 
 
 13. (7(y'-{-d'){3c^-\-Sd^. 22. (3 x-{- c){5 x + d). 
 
 14. (a"'-2e)(a'"-e). 23. (3 x-c){5 x-d). 
 
 15. (6ic^-f4)(3a;^-2). 24. (ky -\-l)(ny-l). 
 
 16. (11-7 cd^){6-{- 3 cd^). 25. (Jcy -^a){ny + b). 
 
 17. (ic + 2)(ic + l). 26. Qcy-a){l-cy). 
 
 56. The square of any polynomial. By actual multiplica- 
 tion it is found that 
 
 (a 4. 5 + c)2 = ^2 4- 52 + c2 + 2 a6 + 2 a^ + 2 he, 
 (^a-^b-{-c-\-dy = a'^-{-P + c'^-{-d^ + 2ah-\-2ac + 2ad 
 + 2bc-{-2bd-h2cd, 
 (^a-]-b-^c+d+ey=a^-^h^-{-(P + d^-{-e^ + 2ab-^2ac-{-2ad 
 + 2ae-{-2bc-{-2bd+2be-\-2cd-\-2ce-{-2de, 
 
 and so on for any polynomials whatever; that is, The square 
 of any 'polynomial whatever equals the sum of the squares of 
 all the terms of the poly)fiomial, plus twice the product of each 
 term by all the terms that follow it (for proof see § 209). 
 
 HIGH SCH. ALG. — 6 
 
76 HIGH SCHOOL ALGEBRA [Ch. VI 
 
 EXERCISE XXXVI 
 
 Expand by inspection (check as teacher directs) : 
 
 1. 
 
 {c + d + ey. 
 
 12. 
 
 (4a3-&_5)2. 
 
 2. 
 
 (m + n — sf. 
 
 13. 
 
 (5-^x-fy, 
 
 3. 
 
 (a-b-cf. 
 
 14. 
 
 (-5-x + yy. 
 
 4. 
 
 (m + r + l)2. 
 
 15. 
 
 (a — b-\-c — dy. 
 
 5. 
 
 (^m-r-Sy. 
 
 16. 
 
 (ax-\-by-\-czy. 
 
 6. 
 
 (2x-\-y-^zy. 
 
 17. 
 
 (ia-ic + iey. 
 
 7. 
 
 (2x-j-3y-zy, 
 
 18. 
 
 (mn — np—pqy. 
 
 8. 
 
 {2x-3y-\-zy. 
 
 19. 
 
 (abx — acy — bczy. 
 
 9. 
 
 l-(2x + 3y^l)y. 
 
 20. 
 
 (2x-3y + 4.z-ay. 
 
 10. 
 
 (3c2 + d-4)l 
 
 21. 
 
 (x^-\-x + iy. 
 
 11. 
 
 (4.a'-\-b' + Sc'y. 
 
 22. 
 
 (/+m+w+p+^4-r+s)2. 
 
 23. Could any of the above products have been found by 
 means of formulas alrfea^dy used (cf. § 52, also Ex. 28, p. 72) ? 
 
 24. Give a rule for writing down the square of any polynomial 
 whatever. What does this rule become when the polynomial is 
 a binomial (cf . § 52) ? 
 
 57. Cube of a binomial. The cube of a binomial is another 
 product which, because of its frequent occurrence, should be 
 memorized. By actual multiplication we obtain 
 
 and (a-by = a^-3aH-{-Zab^-b^ 
 
 whatever the numbers represented by a and b. 
 
 By means of these formulas (which the pupil should trans- 
 late into words) we may write by inspection the cube of 
 any binomial whatever. 
 
 Note. § 52 and § 57 are particular cases of what is known as the 
 binomial theorem; this theorem is considered in § 112. 
 
 Ex.1. (x-\-2y = a^-\-3 x''2-\-3x-2^-\-2^=x'-{-6x^-{-12x-\-S. 
 Ex. 2. (2 a-5 by = (2 a)«- 3 (2 ay • (5 b) +3 (2 a) • (5 by- (5 by 
 = 8 a«- 60 a'b + 150 ab' - 125 b'. 
 

 EXERCISE XXXVII 
 
 
 Expand the following 
 
 : expressions : 
 
 
 3. (x-^yf. 
 
 7. (c + l)«. 
 
 11. 
 
 4. (m-ty. 
 
 8. (a -3)3. 
 
 12. 
 
 5. (2x-yy. 
 
 9. (d' + c^y. 
 
 13. 
 
 6. (z-Syy. 
 
 10. (2yz-5y. 
 
 14. 
 
 56-68] FACTORING 77 
 
 (l + 2m)3. 
 (3a2-2 67^ 
 (_5_v)3. 
 
 15. What is the difference in meaning between "the cube 
 of the sum of two numbers" and "the sum of the cubes of two 
 numbers " ? Illustrate, using 3 and 4 as the two numbers. 
 
 16. Give a rule for finding the cube of the difference of two 
 numbers. 
 
 17. Expand (c-\-dy, also (—c—dy. If we change the signs of 
 an expression, do we change the signs in its cube ? Why ? 
 
 II. FACTORING 
 
 58. Definitions. In a broad sense, any two or more num- 
 bers whose product is a given number are factors of that 
 number. Thus, since J -1^. 15=6, therefore ^, |^, and 15 
 are factors of 6; so also are ^, 18, and ^; the important 
 factors of 6 are, however, 2 and 3. 
 
 In order to exclude fractional and other unimportant fac- 
 tors, we shall (as it is customary to do) define factors thus: 
 
 The factors of a number or algebraic expression are its 
 rational* integral exact divisors. 
 
 JEJ.g.^ the factors of 3 a; (a^ — b^} are S, x, a-\- 6, and a — b, 
 as well as the product of any two or more of these. 
 
 Observe, too that if 3, a + 6, etc., are factors of any given 
 expression, then — 3, — (a-f-J), etc., also are factors of this 
 expression. 
 
 A factor (or expression) is said to be prime if it contains 
 no factors except itself and 1 ; otherwise, it is composite. 
 
 * An expression \\raUonal with regard to a particular letter if it contains 
 no indicated root of that letter (see § 113). 
 
78 HIGH SCHOOL ALGEBRA [Ch. VI 
 
 By factoring a number (or expression) is usually meant 
 the procetis of separating it into its prime factors. 
 
 Factoring an expression, as will appear later, often greatly 
 simplifies algebraic work ; it is therefore important that the 
 pupil should early master those cases of factoring which 
 present themselves most frequently. 
 
 59. Factors of a monomial. The literal factors of a mono- 
 mial are evident by inspection, and the factors of the numer- 
 ical coefficient are found as in arithmetic. 
 
 E.g.y the factors of 30 a^x^y^ are 2, 3, 5, a, oi?, and t/^. (The ic-and 
 ^/-factors are as evident in the forms ni? and y^ as from x-x- x and 
 
 yy-) 
 
 60. Monomial and polynomial factors of a polynomial. If a 
 
 polynomial contains a monomial factor, the latter is readily 
 discovered by mere inspection. 
 
 E.g., in 12aVH-4a6a^?/ — 8aicy, it is seen that each term contains 
 the factor 4 ax^, hence (see § 38) the other factor is Sax-\-by—2y^; 
 i. e., 12 a V + 4 abx^y — 8 ax^y^ = 4 aar^ (3 ax -{-by— 2 y^). 
 
 To factor a polynomial completely/ requires (1) the removal 
 of all monomial factors, and (2) the factoring of the poly- 
 nomial thus freed from its monomial factors. The simpler 
 cases of (2) are considered in the next few articles; (1) 
 may always be accomplished as above. 
 
 EXERCISE XXXVIII 
 
 Factor : 
 
 1. Qa'x^. 3. 42 s¥. 5. 408 mVt/l 
 
 2. WmjA^ 4. 210 2/V. 6. 572 a'(fwv\ 
 
 7. The expression 5 a — 10 6 + 30 a^ has what monomial fac- 
 tor? what polynomial factor? How do you find the former? 
 the latter ? 
 
 Separate the following expressions into their monomial and 
 polynomial factors, and check your results : 
 
58-61] FACTORING 79 
 
 8. 17a;2-51^. 14. 3 a' - 6 a'b + a'b\ 
 
 9. 4 ar' — 6 x-y. 15. 7nIn'^-\- m^w^ + m^/i^. 
 
 10. 4a-^62-26a-6l 16. 3 r^ - 12 r^s^ + 6 rs^ 
 
 11. 10 mhi^ — 15 m%\ 17. ac — bc — cd — abed. 
 
 • 12. -16x--2abx. 18. 32 .^ry - 28 a^y + 12 .ti/. 
 
 13. 15a;^-10ar^ + 25a^. 19. Uxyz^-2 afyh''-{-Sxyh\ 
 
 20. 60 mhih^ — 45 m^nh'^ -\- 90 m*n^r^. 
 
 21. 12a;267/-18a.7y«6 + 24a;y?>*. 
 
 22. 14 a^mii^ — 21 aSnhi^ — 49 a*mn^. 
 
 23. 35 c^cZar^ + ^ c\? V - 55 c^d V. 
 
 24. 51i«?/V-68ar^/22_^85a;y^l 
 
 25. 52 a%V- 65 ^6=^6^ + 91 a2?>V. 
 
 26. Write (m + 7?.)^ — 3(m + 7i)^ + (m + n) as the product of two 
 factors, one of which is m + 71. 
 
 27. Write 2(3 0- - 1)2 - 5(3 a; -1) + 4(3 0^-1)3 as the product 
 of two factors; also 6 (2 -a)^ - 8 (2 - a)^ - 12 (2 - a)«; also 
 x\a - c)-(l-3x') (a-c)-(a- c). 
 
 28. If — 5 mhi^ is one factor of 10 m'^71^ — 15 mhi^, what is the 
 other (cf. Ex. 11) ? Factor again the expressions in Exs. 12-16, 
 in each case taking the monomial factor as negative. 
 
 61. Factoring by means of type forms. Expressions of the 
 type a^ + 2 a6 + 6^. Factoring being the inverse of multipli- 
 cation, it follows that to every case of multiplication there 
 corresponds a case of factoring. Ease in factoring, as in 
 every inverse process, depends upon a ready knowledge of 
 the corresponding direct process. 
 
 Thus, if we promptly recognize the form 
 
 a2 + 2 a5 + 52, [see § 52 
 
 then we can as promptly write down its factors, viz. : 
 a-\-h and a-{-h. 
 So, too, the factors of 
 
 ^2 _ 2 a5 + 52 
 are a — h and a — h. [see § 52 
 
80 HIGH SCHOOL ALGEBRA [Ch. VI 
 
 The expressions 6 mn + m^ -f 9 w^ and 4 a;^ + 25 — 20 rr be- 
 long to this type form, for, in each case, two terms of the 
 trinomial are the squares of certain numbers, and the third 
 term is twice the product of these numbers. These expres- 
 sions may, therefore, be written as (m + 3 9i)(m + 3 w) and 
 (2 2: — 6) (2 a? — 5), or as (m + 3 ^)2 and (2x — 5)^, respectively. 
 
 EXERCISE XXXIX 
 
 Factor the following expressions : 
 
 1. x'-'Zhx-^W. 8. a'V'-2ah-\-l. 
 
 2. u^ + 2uw + 'u^. 9. l-12?/ + 36i/2. 
 
 3. a^-6a; + 9. 10. aJ^-4a^ + 4. 
 
 4. 2/'-42/H-4. 11. 30a^ + 225 + a;i». * 
 
 5. l-\-2a + o?. 12. 9 a;2 - 12 a;2/2 + 4 2/V. 
 
 6. m^- 10 m + 25. 13. 6 a5cd + 9 c^d^ _^ a^ftl 
 
 7. 49 - 14 s 4- s'. 14. 4-36a^&2^81a^6l 
 
 15. What first suggests to you that a;^ + 9 ?/^ + 6 xy may be the 
 square of a binomial ? How do you test the correctness of this 
 supposition ? When is a trinomial the square of a binomial ? 
 
 16. Write out a carefully worded rule for factoring expressions 
 of the types ar + 2ab-\-W and a^-2ab + h'^'? How do we 
 find the terms of the binomial ? How do we determine the 
 sign by which they are to be connected ? 
 
 17. Is a* + 2 c^Jf — W the square of a binomial ? Explain. 
 Factor the following expressions, and check your work. 
 
 18. 5 a7? — 80 aa? + 320 a. [Remove monomial factor first.] 
 
 19. 7 n^ + l^ahri'^l o?h''n. 25. a^P-6cM" + 9 6^ 
 
 20. 18a3t/-60a% + 50a6Y 26. ic^-ic + f 
 
 21. 27 cV- 36 0^^^7134-12 c^c^V. 27. l + f^s^ + fs. 
 
 22. — a^ + 2 0^2/ — 2/^. 28. m2p+2 + 2 mP+V+3 + 7i2'+« . 
 
 23. — m* H- 2 m^n^ — m^n^ 29. (a -f x)^ + 2(a + cc) + 1. 
 
 24. a:2n4.4ajy4.42/6. 30. i6_8(ic + 2/) + (» + 2//. 
 
 31. 9(m — n)2 — 6a7(m — n) + a;^ 
 
1. 
 
 /-^^. 
 
 2. 
 
 y'-^z\ 
 
 3. 
 
 4 2/2 _ 49 62. 
 
 4. 
 
 2^a?W-lQ>. 
 
 5. 
 
 9 /-I. 
 
 6. 
 
 225 a;^- 9/. 
 
 (Jl-62] FACTORING 81 
 
 62. Expressions of the type a^ — 6^. From § 53 it follows 
 that the factors of aP' — b^ are a -\- b and a —b. 
 
 Again, the expression 25 n^ — 9 t^ is of the above type, 
 and its factors are 5^4-3^ and 5 w — 3 ^. 
 
 EXERCISE XL 
 
 Factor the following expressions : 
 
 7. a^x-h^x. 13. 49-36a^/. 
 
 a 36aV-81dl 14. m^^-n^-. 
 
 9. i»2n_4 3^5 49a^/-16 
 
 10. 121a^-36 6^ 16. 64a^/-81. 
 
 11. 64 0^2/'" -144 21 17. 2S^o^z^-f^z. 
 
 12. {x + yf-\. 18. 4.d?-{x-yy. 
 
 19, In factoring the difference of two squares {e.g., a^—b^), 
 how are the terms of each factor found ? How are these terms 
 connected in the first factor ? in the second ? 
 
 20. Write a rule for factoring the difference of two squares. 
 
 By rearranging and grouping terms factor the following ex- 
 pressions, and check your work : 
 
 21. b^-2bc-d^-{-c\ 28. - a^+ 6*-2 a^-l. 
 [i.e.,62_26c+c2-d!2,i.e.,(6-c)2-d2]. 29. - 18 A: + 81 + A;^ - 25 ^^^^ 
 
 22. c^ + 2cd-{-d'-e'. 30. -9 ^^ + 49-12 wv -4^^^ 
 
 23. x^-b'-2xy-^y\ 31. S cH^ - 4: -{- c* -\- 16 d\ 
 
 24. x^ + 4:xy-4:z' + 4.yK^^^ 32. s^ -4.7^ -^f-2 st. 
 
 25. m2-6m + 9-p2. / 33. -524- ^2_4^_4 ^^^ 
 
 26. l-s'-2st-t\ 34. 36c2-aV-36 + 12aa;. 
 
 27. 25-m24-2m7i-n2. 35. - 22 a;?/ + 121 - 2^ + aj^^/^. 
 36. Supply the required factor in each of the following: 
 
 a'-b' = (-a-{-b)'(?)', 16x'-9y' = (-4rX-Sy).( ? ). 
 
 May the factors in Ex. 4 be written ( — 5a6— 4)(— 5a6H-4)? 
 Explain (cf. §§ 18, 58). 
 
82 HIGH SCHOOL ALGEBRA [Ch. VI 
 
 63. Expressions of the type x^+ mx-\- n. From § 54 it fol- 
 lows that the factors oi aP' -{- (^a -\- b) x -\- ab are 
 x-\- a and x -\- b. 
 Again, since the expression A;^ + 7 A; + 12 may be written 
 in the form A;^ + (3 + 4)A: + 3-4, therefore its factors are 
 A: + 3 and k + 4:. 
 So, too, j92 4-2jt?-15=jt?2+(5-3)jt? + 5.(-3) 
 
 = (^ + 5)(p-3). 
 From these illustrations we see that we can separate the 
 trinomial x^ + mx -\- n into two binomial factors whenever 
 we can separate n into two factors whose sum is m. Hence, 
 whenever such an expression as x'^ + mx + n can be factored, 
 its factors may be found by a few trials — the number of 
 trials never exceeding the number of pairs of factors of n. 
 
 EXERCISE XLI 
 
 I. If the expression cc^ + 5 aj — 36 is the product of two bino- 
 mial factors, what is the product of the unlike terms in these 
 two binomials ? Have these terms like or unlike signs ? Why? 
 What is the sum of these unlike terms ? Is the larger of them 
 positive or negative ? Why ? 
 
 Factor the following expressions and check the work : 
 
 ax — 90 al 
 
 nx^^i2x 
 
 [i.e., x(x2- 17 a; + 72)]. 
 2x'-Qx + 4.. 
 Uv^-S2v'\ 
 
 8. r- 122? + 35. 18- ax'-i-Ta^x + ea^ 
 
 9. m2 + 15m + 50. 19- 66 + 39/ + 3/. 
 10. Jc'-Sk-AO. 20. -a2-27 + 12a. 
 
 II. 'y2-7v-18. 21. a2&2_7a& + 10 
 12. f + 13 ^ - 30. • ^ U-e-, (aby - l{ah) + 10]. 
 
 2. 
 
 x--^x + 2. 
 
 13. 
 
 0? 
 
 3. 
 
 x'-\-x-Q. 
 
 14. 
 
 v' 
 
 4. 
 
 :^-x-2. 
 
 15. 
 
 :^ 
 
 5. 
 
 s2+_i2s4-36. 
 
 
 \i. 
 
 6. 
 
 f^^y^h. 
 
 16. 
 
 2i 
 
 7. 
 
 o2 + 7a-30. 
 
 17. 
 
 v' 
 
63-64] FACTORING 83 
 
 22. 4ic2 + 4a;-3 30. ^-'"-24r + 63. 
 [i.e., (2a:)2 + 2(2 x) - 3]. 31. ^^2 _ ^^ ^^^ _^ 28 n\ 
 
 23. 4a;2-8a;-21. 32. s'^-st-4t2f. 
 
 24. Oo.'^ + eaj-S. 33. -12a;2/2 + ar/ + 32 22. 
 
 25. 9x2-21a;-8. 34. (m + ri)^^- 7(m + n) + 6. 
 
 26. 9x' + 21x + U. 35. (s-A;)2-26(s-A:) + 69. 
 
 27. 16ar^-56ic + 33. 36. Q-y-y\ 
 
 28. 15 + 32a;H-16a;2^ 37. r^ - {b - f)r - bf. 
 
 29. 25a:2_8_;l^()^ 33^ ic2 + (3a- 2 5)a;-6a&. 
 
 64. Expressions of the type kx^ + mx + n. Every trinomial 
 of this type which is the product of two binomials, may be 
 readily factored by an extension of the method of § 63. 
 For example, to factor 6 x^ — 11 x — Sb^ we proceed thus: 
 62:2_ii^_35^ 1 (36 2;2- 66 a; -210) 
 
 = i[(6^)2-ll(6.'c)-210] 
 = 1 (6 ^ - 2 1) (6 2; + 10) [§ 63 
 
 = (2x-l}(Sx+5). 
 The given expression is first multiplied by 6 so as to make 
 the first term an exact square, and the factor ^ is then 
 inserted so as to keep the value unchanged. 
 
 Note to the Teacher. The above method may, if the teacher prefers, 
 be replaced by the following ; in that case § 64 should follow § 67. 
 
 Let 6 x^ — 11 a; — 35 = (ax + b) (ex + d), wherein a, &, c and d are to 
 
 be determined ; 
 
 then 6 aj2 -11 X - 35 = acx^ + (ad + bc)x + bd, 
 
 whence — 11 = ad + be and 6 ( — 35) = ac . &d (i. e. , ad • be). 
 
 If, therefore, we separate 6 (—35), le., — 210, into two factors whose 
 sum is — 11, we shall then have found ad and be ; these factors are — 21 and 
 10, hence we have 
 
 6 a;2 - 11 X - 35 = 6 x2 + ( - 21 + 10) x -35 
 = 6x2-21x+10x-35 
 = 3x(2x-7)+5(2x-7) 
 = (2x-7)(3x + 5). 
 
 When this method is used with young pupils, special care will be needed to 
 keep the work from becoming merely mechanical. 
 
84 HIGH SCHOOL ALGEBRA [Ch. VI 
 
 EXERCISE XLII 
 
 1. In factoring 3a^ + 13a;-fl4 by the method of § 64, what 
 
 multiplier should be used ? Why ? What divisor must then 
 be used ? Why ? 
 
 Factor the following expressions and check your work : 
 
 2. 3a^ + 13aj + 14. 19. 10-19a:4-6a^. 
 
 3. 6a2-lla + 4. 20. 56 x-\- 15 + 20 a^, 
 
 4. 322 + ^-10. 21. Sc^-10cd-3(^. 
 
 5. 4a^ + 16a; + 15. 22. -28 + 39s-8s2. 
 
 6. lOy^-lSy-S. ^ 23. - 30 i2_;i^9 ^.^5 
 
 7. 9a^ + 7x-2. 24. 12j92_ 28^ + 11. 
 
 8. 10ar^ + a;-2. 25. 16 ar' + 4 a^^^^ _ 3Q ^^4^ 
 
 9. 12i»2_j_43._5^ 26. 4: at)'- 73 abc-\- IS ac\ 
 [Multiply and divide by 3.] 27. — 14: y — 16+15 /. 
 
 10. lSs^-9s-5. 28. 15a^'* + 16a;"2/ + 4/. 
 [Multiply and divide by 2.] 29. 14 k^ — 27 A;^'' — 20. 
 
 11. 12m2 + 7m-10. 30. 3(a + &)' + 10(a + &) - 8. 
 
 12. 20m2-7m-6. 31. 5(c-dy -7(c-d) -6. 
 
 13. 2a^ + a-55. 32. 15 a.-^^ - a;^ - 28. 
 
 14. 8^2_^7^-18. 33. sx'+(7 s-t)x-7t. 
 
 15. 10/ + 7 2/ -12. 34. cz^+(fc-d)z-fd. 
 
 16. 8^2 + 14 71 — 15. 35. locP — Ikx — mx + km. 
 
 17. 6i)2-29p + 35. 36. 6ay^ + 2aby — Scy -be. 
 
 18. 10 62 + 37 6-12. 37. 90 a;^/^^ - 98 a^a??/^ + 8 a^a^y. 
 
 38. Is a product altered when two of its factors are changed 
 in sign? Explain (cf. § 18, also Ex. 36, p. 81). Change the 
 signs in each factor found for Ex. 2 above, and thus write the 
 factors of 3 x' + 13 a? + 14 in a new form. Similarly, in each of 
 Exs. 3-8 write the factors in a new form. 
 
 65. Squares of polynomials. Cubes of binomials. These 
 types may be recognized by comparing them with the for- 
 mulas of § 56 and § 57. 
 
04-65] FACTORING 85 
 
 Thus, since the expression a^ + z^ — 4:2/z + 2xz-\-4^^ — 4ixy 
 consists of three square terms and three double products, 
 it may be the square of a trinomial. On rearranging its 
 terms thus : x^ -\- ^y'^-\-z'^ — 4:xy -{-^xz — -^yz, and compar- 
 ing with § 56, we see that the given expression is the square 
 oi x — 2y -\- z. 
 
 Again, the expression 12 am^ — 6 a^m — 8 m^ + «^, consisting, 
 as it does, of four terms, two of which are cubes, may be the 
 cube of a binomial ; further examination shows that it is the 
 cube oi a — 2 m. 
 
 EXERCISE XLIII 
 
 1. Is a^ — 2a6 + c^ + 26c — 2ac -\- h^ the square of a trino- 
 mial ? What suggests to you that it may be ? How do you find 
 the terms of this trinomial ? Which of them are alike in sign ? 
 Which unlike in sign ? Why ? 
 
 Factor, and check your results as the teacher directs : 
 
 2. m^ + w^ -h s^ + 2 mn — 2 ms — 2 ns. 
 
 3. 4: x^ -\- y^ -\- 2 yz -{- 4: xy + 2- + 4 xz. 
 
 4. ^v^-{-2kx + x'-Q>kv-&vx-^'k?, 
 
 5. 6ac + 8 6c + 9a- + c2 + 24a6 + 1662. 
 
 6. 4c2-f 9a2-12ac-fl6ftc-24a6-fl662. 
 
 7. l-\-2r — 2m + m^ — 2rm + r^, 
 
 8. 2lm — 2ln +/ — 2lp-^m- + l--2mn — 2mp + n^-\-2np. 
 
 9. If an expression {e.g., 3 pq- — (f -\- 2)^ — 3 p^q) is the cube of 
 a binomial, how do you find the terms of this binomial ? By 
 what sign do you connect them ? Illustrate. 
 
 Factor, and check by § 25 : 
 
 10. a^-Sa''y-{-3ay'--y\ 13. - y^ - 12 x'y -^ 6 xy^ -\- S x\ 
 
 11. m^-f-{-3mf-37nH. 14. - 27 yh -\- 27 y^-z^ + 9 yz\ 
 
 12. 3a*-f 1-f 3a2 + a«. 15. S - c^ -12 c' -{-6 c\ 
 
 16. a;«-2a^ + 10a;2 + cc*-10aj3 + 25. 
 
 17. 216 - 108 s¥ -f 18 s¥ - sH\ 
 
 18. 25-^6m^n-10n-\-9m*-30m^-\-nK 
 
86 
 
 HIGH SCHOOL ALGEBRA 
 
 [Ch. VI 
 
 66. Factoring the type forms jr^— /" and x" +/". By actual 
 division we obtain the following results : 
 
 (^x^ — a^^ -^ (^x — a') = X -{■ a, 
 
 (ar^ — a") -7- (x — a^ = x"^ -^ ax -{- a^^ 
 
 (^x^ — a^ ) ^ (x — (i) = x^ 4- ax^ + oP'x + a^, 
 
 (a^ — a^) ^{x — a) = x^-\-ax^ + a?x^ + a^x + aS etc. 
 
 I. 
 
 11. 
 
 III. 
 
 (^x^ — a^) -^ (x -{- a) = X — a, 
 (a;^ — a^) -^ {x + «) not exactly divisible, 
 {x"^ — rt^) -h (x -]- a) = x^ — ax^ + a^x — a^ 
 (x^ — a^ ) -V- (^x -\- a) not exactly divisible, etc. 
 
 (^x^ + «2) _^ (2; — a) not exactly divisible, 
 (^x^ 4- rt!^) -^ (a; — a) not exactly divisible, 
 (a^* + a^) -r- (x— a) not exactly divisible, etc. 
 
 (x^ + a^) -7- (2: + «) not exactly divisible, 
 
 T V J (^^ + «^ ) ^ (a: + «) = a;2 — a:r 4- ^^ 
 
 (2:* + a*) -f- (2; + a) not exactly divisible, 
 
 (^x* 4- a^) -=- (2: 4- «) = ^* — «a;3 4- oP'x^ — (^x 4- a*, etc. 
 
 These quotients illustrate the following principles (for 
 proofs see Exs. 17-19, p. 94): 
 
 (i) From I, x^ — a^ is always exactly/ divisible hy x — a; 
 the quotient terms are all positive. 
 
 (ii) From II, x^ — a^ is exactly divisible by x -\- a only ivhen 
 n is even; the quotient terms are alternately positive and 
 negative. 
 
 (iii) From III, x^-\-a^ is never exactly divisible by x— a. 
 
 (iv) From lY, x"-\-a^ is exactly divisible by x-\-a only 
 ivhen n is odd ; the quotient terms are alternately positive and 
 negative. 
 
 (v) The order of the letters and exponents is the same in all 
 the quotients ; the exponent of the first letter decreasing., and 
 that of the second increasing^ in passing toward the right. 
 
66] FACTORING 87 
 
 EXERCISE XLIV 
 
 Write the following quotients by inspection and then verify 
 them by actual division : 
 
 1. :^— ^. 7. •" '^•'^ . 13. ■ 
 
 x-\-y 
 
 2. 
 
 x^- 
 
 -r 
 
 X 
 
 -y 
 
 T^- 
 
 -f 
 
 X- 
 
 -y 
 
 a'- 
 
 -b' 
 
 a 
 
 -b 
 
 m^ 
 
 -n« 
 
 m 
 
 + rt 
 
 u'- 
 
 -v' 
 
 u- 
 
 — V 
 
 u'- 
 
 -v' 
 
 ^jj-jf^ T, x^y 
 
 x + y 
 x^ 4- y^ 
 
 x + y 
 
 [i.e., W^-OTh . 
 L x2 - y-i J 
 
 x' 
 
 -f 
 
 (x^r - (y^r 
 
 
 X2 - y^ 
 
 ^10 
 
 + P 
 
 s^ 
 
 + ^^ 
 
 ^ao 
 
 -?/" 
 
 s' 
 
 -/ 
 
 x'' 
 
 _y2 
 
 x' 
 
 -t 
 
 x'' 
 
 -f' 
 
 9. '£±^. 14. 
 
 m-\-s 
 
 4. •.^^. 10. ^^!±A'. 
 
 a-\-b 
 
 5. :^^. 11. (■^7' + 0/y. 16. 
 
 i^ + f 
 
 u-\-'\} (J? — & x? — y^ 
 
 18. In Exs. 5-11, above, express the dividend as the product of 
 the quotient and the divisor. 
 
 19. Of which of the following binomials is r — s a factor : 
 ,.8 _|_ ^8 . ^10 _ ^10 . ^.7 _ ^7 . ^.11 _|_ gU 9 Answer the same question for 
 the factor r + s. 
 
 Write each of the following as the product of two factors : 
 
 20. m^ — n^. 26. x^-^y*^, 32. l^p'^ — q'^, 
 
 21. d^' + e^ 27. r^ — s^. 33. 32.^'^4-l. 
 
 22. x'^-y'^. • 28. 2/^ + 8 34. 8-27r^. 
 
 23. F-Z^. [Le.,?/3 + (2)3]. 35^ Ho x' -'^l. 
 
 24. 2/3 +z3. 29. 0^3 + 27. 36_ 27v^-64m;«. 
 
 25. aio + &''. ^^' 8^'-l- 37. / + 32a.'io. 
 [Cf.Ex. 14.] 31.^^-32. 38. 64r-r^ 
 
 39. Factor a^ — & in two ways : (1) by taking out tlie factor a — c, 
 (2) by using § 53 (cf. Ex. 12, above) and then refactoring the two 
 factors thus found. Which is the better plan to use when the 
 prime factors of a^— c^ are sought ? Show that this plan is advis- 
 able in general, e.g.^ with a^ — y^ and p^ — q-^. 
 
88 HIGH SCHOOL ALGEBRA [Ch. VI 
 
 Resolve the following expressions into their prime factors : 
 
 40. x*-y\ 45. aiV^-2/i«. 50. a^-\-y^ 
 
 41. a^-b^ 46. 64a^-l. 51. x^^-xy^. 
 
 42. a^-b^ 47. a«-81. 52. Sas^'-Saf^ 
 
 43. m«-l. 48. SI a*b*- 16 xy. 53. 64ic« + 2/^ 
 
 44. r^^—n^, 49. a;^ — 2/^ 54. /4-1- 
 
 67. Factoring by rearranging and grouping terms. A re- 
 arrangement and grouping of the terms of an expression will 
 often reveal a factor which could not be easily seen before. 
 
 E.g., ax— 3 by-{-bx — Say = ax-^bx — 3by — Say 
 
 = x(a-hb)-Sy(a-^b) 
 = (a-\-b)(x-Sy), 
 
 i.e., ax — 3by + bx — S ay = (a-\-b) (x — 3y) . 
 
 Again, a;(a; + 4) — y(j/ -f- 4) = a;^ + 4a; — / — 4?/ 
 
 = xF - y'^ -\- 4:(x — y) 
 = (x-y)(x-{-y)-{-4:{x^y) 
 = (x-y){x + y-\-4:). 
 
 I.e., x(x + 4:)-y(y + 4.) = (a;-?/) (a; -h2/ + 4). 
 
 EXERCISE XLV 
 - Factor the following expressions and check your work : 
 
 1. cx — cy-{-Sx — Sy, 12. m^ — n^ — (m — ny. 
 
 2. ay-^1cx-\-ax-i-ky. 13. 'Sxy(x^-{-y)-^16(x^ + f). 
 
 3. p3_p2_^7^_7^ 14_ x^ _ xy^ — ax^ -{- ay\ 
 
 4. p3_y_7^_^7^ 15, ab-\-bx'' — x''y'^ — ay'^. 
 
 5. ac + bd — ad— be. 16. a^ — 9 a;^ + 4 c^ — 4 ac 
 
 6. 9cy — 6cx-12mx-\-lSmy. [i.e., (a2-4ac+ 4c2)- 9 a;^]. 
 
 7. aV-acd-a6c + 6d. 17. - Uk^ - A9 b^ -\- A k' -\- 121. 
 
 8. 7mr-3rs + 21ms-9s2. 18. ac^ + M^ - ad^ - fec^. 
 
 9. 5a^-«2 + 2-10a;. 19. 1 + ds - (c^ + c(^y. 
 
 10. 5a^-hl-x^-5x. 20. (a + 1)2 -(4 a + 3)2. 
 
 11. aa^ + l + a + ar\ 21. (p^ - q^ - (p^ - pqy. 
 
00-68] FACTORING 89 
 
 22. a^x + ahx + ac 4- b^y + a6?/ + &c. 
 
 23. (ar' + 6a; + 9)2-(aj2 + 5a; + 6)^ 
 
 24. a^-a2 + 2/=^-62 4.2a;2/-2a6. 
 
 25. h'-m^-\-10m + k^-25-2hk. 
 
 26. (a; + 2/)^ + 12(a; + 2/)-85 (cf- Exs. 34-35, p. 83). 
 
 27. x2 + 4a;^ + 4i/2-f.3a;H-62/ + 2 (cf. Ex. 26 above). 
 
 28. 4: x"^ + 10 X — 6 — 5 a — A ax -\- a\ 
 
 29. Show that by changing the signs of two of them at a time 
 the factors in Ex. 10 may be written in three different forms 
 (cf. Ex. 38, p. 84). Is the same true in Ex. 18 ? 
 
 68. Factoring by means of other devices. It often happens 
 that the factors of an expression will become apparent by 
 adding a certain number to, and subtracting the same num- 
 ber from, the given expression ; this, of course, leaves the 
 value of the expression unchanged. 
 
 Ex. 1. Find the factors ofx^ + x^ + l. 
 
 Solution. If the second term in this expression were 2x^ 
 instead of x^, then (§ 61) the expression could be written (x^ + 1)" ; 
 this suggests that x^ be both added and subtracted, which gives 
 x*-{-x^-{-l=x' + 2a^-{-l-x' 
 
 ==(^^^l+x)(x^ + l-x), [§62 
 i.e., x* + x'i-l=y' + x + l)(a^-x + l). 
 
 EXERCISE XLVI 
 
 2. Find the factors of a* + a^b^ + b\ 
 Suggestion. By the method of Ex. 1, 
 
 a4 _,. ^2^2 + 54 = (^4 _^ 2 a-^62 + 54 _ ^252 
 
 = (a2 + &2)2_(a5)2. 
 
 3. Find the factors of a^ - 4 a; - 32. 
 
 Suggestion. Here the first two terms, plus 4, form an exact square ; this 
 suggests the following arrangement : 
 
 x2_4x-32 = ic2-4x + 4-32-4 
 = (x-2)2-36. 
 
90 EIGH SCHOOL ALGEBUA [Ch. VI 
 
 4. What must be added to ic^H-3a^+4 to make it an exact 
 square? What must then be subtracted to leave the value un- 
 changed ? Factor the given expression. 
 
 5. Can the sum of two squares be factored (cf . § 66)? Is a;^ -f 4 
 the sum of two squares ? Can it be factored ? 
 
 6. What must be added to a;^ -f- 4 to make it (a^ + 2)^? Is the 
 added term a square ? Factor x"^ + 4. 
 
 Factor : 
 
 7: p^^q^\ 16. 4a8-21a46^ + 96^ 
 
 a x^-f-64 2/^ * 17. ^x^-lOxY + ^y^' 
 
 9. m^ + mhi^^-n\ 18. ^ a"" ^2Q> a'h'' + 25 h\ 
 
 10. x''^a'x^ + a\ 19. o? + 2 ab -d''-2hd, 
 
 11. a^ + a;y-f/. 20. 4a^ + 81. 
 
 12. a^ + 6a;4-5. 21. a^/-h4aj/. 
 
 13. 9 s^ -f 30 si 4- 16 ^^ 22. m^-\-4:m7i\ 
 
 14. a^6^ + «'&Vd2 + c4d^. 23. a' + Sa'-12S. 
 
 15. 9 a;^ + 8 a^2/^ + 4 2/^ 24. 5 na;^ — 70 wa;^ + 200 n. 
 25. Find the four factors oi x* i-y' + z^-2 a^/ - 2 a^2;2 _ 2 ^^V. 
 
 69. General plan for factoring a polynomial. 
 
 1. By inspection, find and remove all monomial factors. 
 
 2. By comparison with type forms, by rearrangement and 
 grouping of terms, or by some other device, separate the 
 resulting polynomial factor into two factors. 
 
 3. Then, if possible, separate each of these factors into 
 two others, and so continue until all factors are prime. 
 
 Note. By the above plan the simpler expressions can usually be factored. 
 For determining the binomial factors of longer polynomials, see § 71. 
 
 EXERCISE XLVII 
 Factor : 
 
 1. 4aa^-4a/. 4. s^ + 16 s^ -]- 15 s. 
 
 2. 49A;-fc3^ 5. x*-Sx^-^15x\ 
 
 3. fp—p\ 6. mV + my. 
 
68-69] FACTORING 91 
 
 7. tv - nw - nv -\- tw. 14. k* — 17 k^ -\-16. 
 
 . 8. v^ -7?;--2i; + 14. 15. Sx-^ — lOxy + 37f, 
 
 9. 1 - 24 .s -h 144 si 16. 49mV + 42wms-h9n2. 
 
 10. a''b'-4.abhj-^4:by. 17. c- + a;2-2 ex- 1. 
 
 11. p^ — Sr\ 18. &* + 6y + 2/^ 
 
 12. 1-2/'. 19. 2u^-Uu^ + 70-107L 
 
 13. c2-5c-14. 20. 4.c'-25a^-\-b^-4:bc. 
 
 21. Give two methods for checking an exercise in factoring. 
 Illustrate, using Ex. 15 above. 
 
 Factoi: and check as the teacher directs : 
 
 22. 71^ — 1. 38. a^x^ — oc^, 
 
 23. q*p — t^*. 39. s^^ — t^^. 
 
 24. 216 + 2/3. 40. Tr'-.QOT. 
 
 25. a« + 4. 41. a^^ + l. 
 
 26. 3(a;-2/)^-27. 42. (a -6)^-03. 
 
 27. rsv^ — ar^st — 4: cr^ . 43. /c^^ + 4p^ 
 
 28. x^-\-ax-ay-yx. 44. 12 4-s(^2_4) _3 ^2^ 
 
 29. m(d2-3)+d2_3^ 45; x^''^^-{-2x''-'by + by, 
 
 30. m^" — 4: m^b -\- 4. b\ 46. m^ — 1 — 3 m(m — 1). 
 
 31. fc(Z2-4)-/2 + 4. 47. 3a2p-28rg-21ri94-4aV 
 
 32. mV + 4-5mV. 48. (5a + 2/)2-7(5 a +?/) + 10. 
 
 33. 7(a; + a)-ll(a;2-tt2). 49. 8 -12 mn + 6 mV-m%l 
 
 34. y^-y^L 50. m2 + 6m/i-16.T2/ + 9nl 
 
 35. -a;y-;2(3 2/-a;-3 2;). 51. (a? -1/)^- 2 2/ + 2 aj+l. 
 
 36. x^ — y^ — 3xy + 3a^y\ 52. m^n^ + 2 mVrV + mV W. 
 
 37. 6x'' + 12a^~lS. 53. (a2 + 5a4-4)2-(a2-5a-6)l 
 
 54. a;2-2x2/ + l+/ + 2(a;-2/). 
 
 55. (c-3)«-l-3(c-3)2 + 3(c-3). 
 
 56. m^~2mn-[-n^-s--{-2st-f. 
 
 57. {c'-2cd-\-dy-(Sc'-cd-2d'y. 
 
 58. «2 + 9/ + 2522_6iC2/-10^^ + 302/«. 
 
 59. 2(a262_aV-6V) + a^ + 6* + c*. 
 
 HIGH SCH. ALG. — 7 
 
92 HIGH SCHOOL ALGEBRA [Cii. VI 
 
 60. a¥ ^ yiy. _ ^2g _^ ^2^ _|_ 52^2 _ ^2g^ 
 
 61. a2_2a6 + 6'-2ac + 26c + c2-2ad + 26d + 2cd + d2. 
 
 62. iB2-9a;-|-14 = (aj-7).( ? ) = (7-a;).( ? ). 
 
 63. c3-r3=(c-r).( ? ) = (r-c).( ? ). 
 
 64. 7'3_36r = r(r+6).( ? ) = -r(r + 6).( ? ). 
 
 65. Write the four factors of a;* — 10 a^ + 9 in seven different 
 ways (cf. Ex. 29, p. 89). 
 
 70* Remainder theorem. In Ex. 53, p. 52, it was seen that 
 if oc^ -^ 3 X -\- 1 is divided by a; — a the remainder is a^ + 3 a 4- 1 ; 
 i.e., the remainder is what the dividend would become if a were sub- 
 stituted for X. (Cf. also Ex. 52, p. 52.) 
 
 And this relation between dividend and remainder is not acci- 
 dental ; it is true for all such expressions. For, let 
 
 Ax"" + Bx""-^ + Cx""-^ H [-Hx -\-K • 
 
 be any polynomial in x, let it be divided by x — a, and let Q and 
 Rj respectively, represent the quotient and remainder; then 
 
 Ax^ + Baf-^ + Cx^-'-^ ...j^Hx + K= Q(x-a) + R. 
 Moreover, since the second member of this equation, when 
 multiplied out, must be exactly like the first, therefore this equar 
 tion is true for all values that may be assigned to x; but if the 
 value a be given to x, the equation becomes 
 
 Aa'^ + 5a"-i + Ca^-^ + - -- + Ha -\- K=: R,f 
 hence, in every such division, the remainder may be obtained by 
 simply substituting a for x in the dividend. 
 
 71.* Application of the remainder theorem to factoring. By 
 
 means of the remainder theorem (S 70), and without actually 
 performing the division, write down the remainder resulting 
 from dividing a^— 3a^ + 3a;-f2 by a; — a. Also write the re- 
 mainder when aj^ — 3a^-f-3a; — 2 is divided by a; — 2. What is 
 
 * Articles 70 and 71, with Exercise XLVIIJ, may, if the teacher prefers, 
 be omitted till the subject is reviewed. 
 
 t Since, in that case, Q(x — a) becomes Q'(a — a), i.e., zero; and M is 
 the same as before substituting, since it does not contain x. 
 
60-71] FACTORING 93 
 
 the value of this last remainder? Does this show that a;— 2 is 
 a factor of y? -'d x^ + Sx-2'! 
 
 Binomial factors of many polynomials may be found in this 
 way, for, from § 70, it follows that if 
 
 ^a" + BoT-"^ + (7a"-2 -\ +Ha + K^ 0, 
 
 then, and then only, is Ax"" + Bx""-^ + Ca;"-^ -\ h ^a; + /f ex- 
 actly divisible by ic — a ; for in that case, and in that case only, 
 is the remainder zero. 
 
 Thus, we know that a? — 3 is a factor of y? — 2y? — ^:X -\-Z 
 because 3^ — 2-3-— 4.34-3 = 0; and a;-f-l, ?.e., a?— (— 1), is a 
 factor of a;2 + 7 a; -f 6 because (- 1)2 ^_ 7(_ i) + 6 = 0. 
 
 Again, if a; — a is a factor of a^ — a;^ — 2 a; + 8, then a is a factor 
 of 8 ; hence, in seeking such factors of a?^ — a;^ — 2 a; -f 8 we need 
 try only 1, — 1, 2, — 2, 4, — 4, 8, and — 8 in place of a. 
 
 When, by any process whatever, any factor of an expression 
 has been discovered, this factor may be divided out; the remain- 
 ing factors may then be more easily found. 
 
 EXERCISE XLVIII 
 
 1. If a;^-f 6a^— 12 a; + 5 is divided by x — a, what is the 
 remainder ? Without performing the division, find the re- 
 mainder when the divisor is x — 2; also when it is a; — 1 and 
 when it is x-\-l. Which of these divisors is a factor of the 
 given expression ? 
 
 2. If the expression x^ — 3a^ — a;-f-3 has a factor of the form 
 X— a, what are the four possible values of a ? Find all the 
 binomial factors ofa;^ — 3ar^ — a;-f3. 
 
 By the above method, factor the following expressions : 
 
 3. a^_7a;-f6. 7. T<? -\-4.k^ -Ilk -30. 
 
 4. a;3-9a^4-23a;-15. 8. w^ - 15 w;^ + 10 lo -f 24. 
 
 5. a.-3 + 14ar-f-35a;-f-22. 9. a^-\-l a" -\-2 a- ^0. 
 
 6. a.-3-lla;^-f-31a^-21. 10. c^-S c^- 29 c + 105. 
 
 11. .^•* - a.-*^ - 7 a^ + a; 4- 6. 
 
 12. /-102/' + 402/^-80/-f 80y-32. 
 
94 HIGH SCHOOL ALGEBRA [Ch. VI 
 
 13. If X — A: is a factor of any given expression, what does the 
 value of that expression become when x = k ^ Why ? If any 
 given expression becomes zero when x = k, is x — k a factor of 
 the expression ? Why ? 
 
 14. By means of the remainder theorem show that a—b, b — c, 
 and G — a are factors of a(b^ — c'-^) + b(c^ — a^ + c{a^ — b^). 
 
 15. Write the remainder when (2x — Saf + (Sx — ay is divided 
 by ic — a ; also the remainder when (x— y -\-zy — 'if-\-x^ is divided 
 by a; — 2/ ; and hj x-{-y, that is, hj x—{ — y). 
 
 16. What is the remainder when or"*— ctMs divided hjx — a? 
 Why? Write the remainder when x'' -\- a^ is divided by a; — a; 
 when ic^ + a^ is divided by x -\- a. 
 
 17. By means of the remainder theorem, show that x^ — a" is 
 exactly divisible by a; — a ; also that x'' + a" is not exactly divis- 
 ible by £c — a (cf. § Q^Q). 
 
 18. By means of the remainder theorem, show that x"" — a"* is 
 exactly divisible by x + a only when n is an even positive integer. 
 
 19. By means of the remainder theorem, show that x"^ + a"* is 
 exactly divisible hy x-\-a only when n is an odd positive integer. 
 
 72. Solving equations by factoring. Factoring greatly 
 simplifies the solution of certain kinds of equations. The 
 following examples illustrate the procedure. 
 
 Ex. 1. Given ic^ — 5a; + 6 = 0; to find its roots, i.e., to find 
 those values of x for which this equation is satisfied (cf. § 45). 
 
 Solution. By § 63 the first member of this equation is the 
 product of ic — 3 and a; — 2 ; hence the equation may be written 
 thus: 
 
 (a;-2)(a;-3)=0. 
 
 Now this equation is satisfied if either 
 
 x-2 = or a?-3 = 0, [§ 41 
 
 i.e., if either a? = 2 or a; = 3. 
 
 On substitution these values are found to check; they are, 
 therefore, the roots of the given equation. 
 
71-72] FACTORING 95 
 
 Ex. 2. Given oc^ = 8 x -\- 4: ; to find its roots. 
 
 Solution. On transposing, tliis equation becomes 
 x^-3x~4:={), 
 i.e., (x_4)(a; + l)=0; [§63 
 
 hence either ic — 4 = or a^ + 1 = 0, 
 
 i.e., x = 4: or£c = — 1; 
 
 and these numbers check, therefore the roots are 4 and — 1. 
 
 Ex. 3. Solve the equation 6 a?^ — 11 cc = 35. 
 
 Solution. On transposing and factoring (§ 64), this equation 
 becomes 
 
 (3x+5)(;2x-7) = 0', 
 hence 305 + 5 = or 2a; — 7 = 0; . 
 
 therefore the roots are — | and J. 
 
 Remark. Since the roots of the equation (x — a)(x—b)=:0 
 are a and b, therefore an equation which shall have any given 
 numbers as roots may be immediately written down; thus the 
 equation whose roots are 3 and 8 is 
 
 (x-3){x-S) = 0, I.e., a^- 11 a; + 24 = 0. 
 Similarly, the equation whose roots are 2, — 1, and 5 is 
 (x-2)(x-\-l)(x-5) = 0, i.e.,a^-(jx'-{-Sx-\-10 = 0. 
 
 EXERCISE XLIX 
 
 4. What is meant by a root of an equation (cf . § 45) ? May 
 an equation have more than one root ? 
 
 5. What values of x satisfy the equation (x—2)(x—3')=0? 
 Can any values of x other than 2 or 3 sat^* sfy this equation ? 
 Explain. How many roots, then, has this equation ? 
 
 Solve the following equations by factoring, and check the roots : 
 
 6. 2/--62/ + 5 = 0. 11. ^2-4 = 0. 
 
 7. x^-4:x-21 = 0. 12. m2-36 = 0. 
 
 8. ^-13s-\-A0 = 0. 13. /-7?/ = 0. 
 
 9. a;2-2a; = 15. 14. c2 + 22c = -121. 
 10. A;2 + 47(; = 45. 15. '^^-3^-50 = 38. 
 
96 HIGH SCHOOL ALGEBRA [Cii. VI 
 
 16. 32/- + ?/-, 10 = 0. 22. a;--3aa;-54a2 = 0. 
 
 17. 6ar^ — a;=l. 23. s^ — (c + d)s -f cd = 0. 
 
 18. 4y2_27 = 12'y. 24. 8a^ + 10.T = 3. 
 
 19. 82/2 + 15 = -26y. 25. 36 = -x^+13a:2^ 
 
 20. 5if2— 7a; = 0. 26. ^-\-x~ — x = l. 
 
 21. 1222 = _42. 27. 2a^ + 5.^ = 2a; + 5. 
 
 28. What are the roots of {x -l){x -2){x -^2) = 0? Explain. 
 Determine by inspection the roots of (a;+l)(3 a;— 2) = 0. 
 
 29. Determine by inspection the roots of : 
 
 (1) (5a;-3)(a;-l)=0. 
 
 (2) (2/-f)(22/ + 9) = 0. 
 
 (3) m(3 m + 1)(4 m - 3) = 0. 
 
 (4) («-a)(2aj-lla)(4a; + 5a) = 0. 
 
 30. Write an equation whose roots are 5 and 2. Also one 
 whose roots are 3, 1, and 7. 
 
 31. Write the equations whose roots are : 1 and — 5 ; f and 6 ; 
 a and 65 3, — 1, and o ; a, —a, and 2a; 1, 2, 3, and 4. 
 
 The following problems lead to equations whose roots may be 
 found by factoring. Solve and check each problem. 
 
 32. Find a number such that if 3 and 5 are subtracted from it 
 in turn, the product of the two remainders is 24. How many 
 solutions has this problem ? Explain. 
 
 33. The sum of two numbers is 12, and the square of the 
 larger is 1 less than 10 times the smaller. Find the numbers (cf. 
 Ex. 18, p. 6). • 
 
 34. Tlie difference between two numbers is 2, and the sum of 
 their squares is 130. What are these numbers ? 
 
 35. One side of a rectangle is 3 feet longer than the other. If 
 the longer side be diminished by 1 foot and the shorter side in- 
 creased by 1 foot, the area of the rectangle will then be 30 square 
 feet. How long is this rectangle ? 
 
 Note. The equation of this problem has two roots, one positive and one 
 negative ; but only the positive root will satisfy the problem itself, for it is 
 implied that the dimensions of the rectangle are positive. 
 
72] . FACTORING 97 
 
 36. How may $ 128 be divided equally among a certain num- 
 ber of persons so that the number of dollars received by each 
 person shall exceed the number of persons by 8 ? 
 
 37. The senior class of a certain school present the school with 
 a picture whose cost is $12. If each senior contributes 3 times 
 as many cents as there are members in the class, how large is the 
 class ? How much does each member pay ? 
 
 38. A rectangular orchard contains 2800 trees, and the number 
 of trees in a row is 10 less than twice the number of rows. How 
 many trees are there in a row ? 
 
 39. If the dimensions of a certain rectangular box which con- 
 tains 120 cubic inches were increased by 2, 3, and 4 inches, 
 respectively, the new box would be cubical in form. Find the 
 dimensions of this box (cf. § 71). 
 
CHAPTER VII 
 
 HIGHEST COMMON FACTOR — LOWEST COMMON MULTIPLE 
 
 I. HIGHEST COMMON FACTOR 
 
 73. Definitions. A factor of each of two or more algebraic 
 expressions (or numbers) is called a common factor of these 
 expressions. The highest common factor (H. C. F.) of two 
 or more expressions is the product of all the prime factors 
 (§ 58) that are common to these expressions; it is, therefore, 
 the factor of highest degree common to the given expressions. 
 
 Thus, the H. C. F. of 9 a%V and 6 abh"^ is 3 ah\ because 
 when this factor is removed from the given expressions they 
 have no common factor left. 
 
 So, too, 2x(^a- 1)2 is the H. C. F. of 6 aV (a - 1)* and 
 Sx(a-iy(is-ty, 
 
 Two or more algebraic expressions which have no common 
 factor except unity are said to be prime to each other. 
 
 74. Highest common factor of two or more monomials. The 
 
 H. C. F. of two or more monomials may, obviously, always 
 be found by inspection. 
 
 E.g., to find the H. C. F. of 12 a^b^xy, 6 a6V, and 9 ab^x\ 
 Inspection shows that 3, a, b^, and x are the only factors com- 
 mon to the given monomials; hence the H. C. F. of these mo- 
 nomials is 3 ab^x. 
 
 A rule for writing down the H .C. F. of several monomial 
 expressions may be formulated thus : To the H. C. F. of the 
 numerical coefficients annex those letters that are common to 
 the given monomials, and give to each of these letters the lowest 
 exponent which it has in any of the mono^nials. 
 
 98 
 
73-75] HIGHEST COMMON FACTOR 99 
 
 75. H. C. F. of polynomials whose factors are known. By 
 
 first writing any given polynomials in their factored forms 
 their H. C. F. may be found by inspection. 
 
 For example, to find the H. C. F. of 4 ax^ — 20 ax -\- 24: a and 
 6 abx^ + 24 abx — 126 ab, we write : 
 
 4:ax'-20ax-\-24:a = 4.a{x-2)(x-3), [§ 63 
 
 and 6 aba^ + 24:abx-126 ab = ^ ab(x-\-7) (x-S)-, 
 
 hence their H. C. F. is 2 a (x—3). 
 
 EXERCISE L 
 
 Find the H. C. F. of each of the following sets of expressions : 
 
 1. 3 a'b' Sind 6 ab\ 
 
 2. 15 afy% 24 xY, and 18 xSj. 
 
 3. 16 x'fz', 52 yh% and'39 x'f- 
 
 4. 195 a*6V and 260 a'bc\ 
 
 5. 96 ?/V, 100 ?/V, and 56 t/V. 
 
 6. 104 x'^y^^'z^' and 364 x^'^f^'z^'. 
 
 7. (c + d)\c - d) and (c + d)(c - d)\ 
 
 a 6(c + d)2(c-d)2and 15(c-(^)2(c+c?). 
 9. 24 a^x (y - zf and 56 a'^b^ (y - z)*. 
 
 10. a^ — b^, a(a — b), and a^ — 2ab-\- 6^ 
 
 11. a;2_^7a; + 10and a.-2 + 12x + 20. 
 
 12. 'm? —m— 12 and di? — 4 m — 21. 
 
 13. 15 (2/2; — z) and 35 (/s; — yz). 
 
 14. Is — (tt — b), I.e., 6 — a, a common factor of the expressions 
 in Ex. 10 (cf. Ex. 36, p. 81)? May we then call the H. C. F. 
 of these expressions either a — 6 or 6 — a? 
 
 15. Show that the H. C. F. of rn? — mn and n^ — mn is either 
 m — 71 or 71 — m. 
 
 In each of Exs. 16-25 find two forms of the H. C. F. : 
 
 16. 1^ — s^ and ^ — rl 
 
 17. 5 a — as and 3 s' — 75. 
 
 18. p^ - 125 and p^ - 10 p^ + 25 p. 
 
100 HIGH SCHOOL ALGEBRA [Ch. VII 
 
 19. a^ 4- a^ and 3 a^ + 3 a^x — 5 aoiy^ — 5 a.*^. 
 
 20. 28^2-17?! -3 and 4n2 + 5/i-6. 
 
 21. 5-19 A;-4A;2 and A;2^2A;-15. 
 
 22. a^a; — x — y-\- a^y and a^a? + 4 a^a; — 5x. 
 
 23. 12 a6^a; + 4 a6^a;^ — 40 aft^, 18 a^mx^— 54 a%a; + 36 a^m, and 
 6 aV?/ — 6 a^xy — 12 a^y. 
 
 24. ?^v — w^, u^— 5v -\-5u — uVj and 3 w^ — 10 iii; + 7 v^. 
 
 25. 15 aV 4- 15 a'6V + 15 ft^aj^ and 3 {a' - ab^ + ^>')- 
 
 Find the H. C. F. of each of the following sets of expressions: 
 
 26. 2a^-a;-3 and2a^ + llar^-a;-30. 
 
 Suggestion. Find the factors of 2 x^ — x — 3 and determine by trial which 
 of these are factors of 2 cc^ + 11 a;^ _ ^ — 30 also. This plan may be used 
 whenever any one of a given set of expressions is easily factored. 
 
 27. (a;+3)(aj2-4) anda;^-h4a^ + 2a.'2-aj + 6. 
 
 28. a^ + 1, Sa^-4.a^-\-4.a-l, and 2 a^ + a^- ^ + 3. 
 
 29. b^-S, b'^b' + 2b-4., ^ndb'-\-2b'-b'-10b-20. 
 
 30. Of what is the H. C. F. of two or more expressions com- 
 posed ? State a rule for finding the H. C. F. of two or more ex- 
 pressions which may easily be separated into their prime factors. 
 
 31. Is the H. C. F. as above defined the same as the greatest 
 common divisor (G. C. D.) in the arithmetical sense ? What is 
 the H. C. F. of a^(a; - 1)^ and xia^-l)? Is this also the G. C. D. 
 of these expressions for all values of x? Try a; = 3, also a; = 4. 
 
 76.^ H. C. F. of polynomials neither of which is easily factored. 
 The H.C.F. of two or more polynomials can always be found by 
 what is known as the Euclidean (division) process. This process 
 is essentially the same as that used in arithmetic to find the 
 G. C. D. of two numbers. 
 
 The steps in the arithmetical process are : (1) Divide the larger 
 number by the smaller ; (2) if there is a remainder, divide the 
 smaller number [I'.e., the divisor in step (1)] by this remainder; 
 
 * Articles 76, 77, and 78, with Exercises LI and LIT, may, if the teacher 
 prefers, be omitted till' the subject is reviewed. 
 
:(J] 
 
 HIGHEST COMMON FACTOR 
 
 101 
 
 (3) divide the remainder in (1) by the' remainder in (2); (4) so 
 continue, dividing each remainder by the one following, until 
 there is no remainder; (5) the last divisor is the G.C.D. sought. 
 
 This work may be more compactly 
 arranged thus : 
 
 2639 
 2866 
 
 Thus, to find the G. C. D. 
 of 1183 and 2639. 
 1183)2639(2 
 2366 
 
 273)1183(4 
 1092 
 91)273(3 
 273 
 
 
 The last divisor, 91, is the G. C. D. of the given numbers. 
 Similarly, the H. C. F. of a:* + 3 a;^ + 2 ic-^ - x - 5 and x^ + x^ - 2 may be 
 found thus : 
 
 1183 
 
 1092 
 91 
 
 QUOTIENTS 
 
 273 
 
 273 
 
 QUOTIENTS 
 
 x* + 3 x3 + 2 x"^ 
 x*+ x3 
 
 X 
 
 2x 
 
 2 x3 + 2 x2 + X 
 2 x3 + 2 x2 
 
 X- 1 
 
 x + 2 
 
 a;3 4x2-2 
 
 
 X3-X2 
 
 
 2x2-2 
 
 
 2 x2 - 2 X 
 
 x2 + 2 X + 2 
 
 2x -2 
 
 
 2x -2 
 
 
 
 
 Hence x — 1, the last divisor, is the H. C. F. of the given polynomials. 
 
 EXERCISE LI 
 
 By the above method, find the H. C. F. of each of the following 
 pairs of expressions : 
 
 1. a^ + 5i»4-6and4a^ + 21a;2 + 30a; + 8. 
 
 2. 6a2-13a-5andl8a3-51a2 + 13a + 5. 
 
 3. 5 m''' — 2 m — 3 and 15 m^ — 6 m^ — 4 m -f- 3. 
 
 4. c3_2c2-2c-3 andc^-c3-3c2^4c-2. 
 
 5. 12x'-Sa^-55a^-2x-^5and6a^-x'-29x-15. 
 
 6. lSx^-\-75a^-{-17x'-2Sx-lSand6a^-}-2Sx^-3x-10. 
 
 7. S0y' + 16y* + 16f-Sy^-3y-2a,nd20f + 4:y'-y-3. 
 
 8. 4.k'-\-201(^-10k'-^SJc-\-S5sind21i^ + llk'-25, 
 
 9. 5n*-10n3 + lln2-67i-hl and 
 
 10 
 
 5n*-7n^ + ldn'-Un-{'2. 
 
102 
 
 HIGH SCHOOL ALGEBHA 
 
 [Ch. VII 
 
 11* Proof of principle involved in § 76 (see footnote, p. 100). 
 The success of the method employed in § 70 is due to the follow- 
 ing considerations : 
 
 Let A and B represent any two polynomials in ic, the degree of 
 A being at least as high as that of B, and let q and R represent 
 the quotient and remainder respectively, when A is divided by B\ 
 then A = qB^R. [Ex. 20, p. 50 
 
 This equation shows that: (1) every divisor common to B and 
 II is a divisor of A also (why ?), and (2) every divisor common to 
 A and ^ is a divisor of H also (why?); hence the H. C.F. of B 
 and R is the same as that of A and B. 
 
 If now B is divided by R, giving p and M as quotient and 
 remainder respectively, then, by reasoning as above, we see that 
 the H. C. F. of M and 7^ is the same as that of B and R, and 
 therefore the same as that of A and B. 
 
 Suppose now that this series of divisions is continued ; then, by 
 the above reasoning, the H. C. F of ^ and B is the same as that of 
 any ttvo successive remainders. 
 
 If now the last one of this series of divisions is exact, i.e., if the 
 final remainder is zero, then the H. C. F. of the two preceding 
 remainders is the last divisor itself ; hence the last divisor is the 
 H.C.F. of A and B, which was to be found. 
 
 Remark, The H. C. F. of two expressions is evidently not 
 altered by multiplying (or dividing) either of them by any num- 
 ber which is not a factor of the other ; this fact enables us to 
 avoid fractional coefficients in the division process. 
 
 Thus, to find the H. C. F. of 3 x^ -|- 8 x'^ -h 3 x - 2 and a;^ - 2 a;^ + x + 4 : 
 
 3x3 + 8x2 + 3x- 2 
 3x3-6x2 4- 3x + 12 
 
 14)14x2-14 
 
 x2-l 
 
 x2-f-X 
 
 -x-1 
 -x-1 
 
 x-2 
 
 x=^ - 2 x2 4- X -f- 4 
 
 - 2 x2 -}- 2 X + 4 
 
 - 2 x2 -H 2 
 
 2)2x4- 
 
 X4-1 
 
 Before beginning 
 the second division 
 the factor 14 is sup- 
 pressed (see Remark 
 above), and later 2 
 is suppressed also ; 
 fractional coeffi- 
 cients are thus 
 avoided. 
 
 Hence x 4- 1, the last divisor, is the H. C. F. of the given expressions. 
 
77-78] 
 
 HIGHEST COMMON FACTOR 
 
 103 
 
 As a further illustration, let us find the 11. C. F. of 
 
 x4 + 4 x^ + 2 x2 - X 4- 6 and 2 x^ + 9 x^ + 7 x 
 
 6. 
 
 xH4x3 + 2x2-x+6 
 
 2 
 
 X, +1 
 2x-l 
 
 2x3+ 9^2_^ 7a^_6 i 
 2x3+10x2+12x 
 
 2xH8xa + 4x2-2x+12 
 2x4+9x3 + 7 x2-6x 
 
 -x-i- 5x-6 
 -x2- 5x-6 
 
 - x3-3x2+4x+12 
 
 -2 
 
 
 
 2x3+6x2- 8x-24 
 2x3+9x2+ 7x- 6 
 
 _3)_3x2-15x-18 
 
 x2+ 5x+ 6 
 
 Before beginning 
 the division the fac- 
 tor 2 is introduced 
 so as to avoid frac- 
 tional coefficients in 
 the quotient ; later 
 
 — 2 is introduced 
 for the same pur- 
 pose ; and finally 
 
 — 3 is rejected. 
 Hence x2 + 5 x + 0, the last divisor, is the H. C. F. of the given expressions. 
 
 78.* Supplementary to § 77 (see footnote, p. 100). (i) If the 
 polynomials whose H. C. F. is sought contain monomial factors, 
 these should be set aside before the division process is begun. 
 Monomial factors that are common to the given polynomials must, 
 of course, be reserved ; all others may be rejected. 
 
 (ii) The H. C. F. of three or more polynomials is found by first 
 finding the H. C. F. of any two of them, then the H. C. F. of that 
 result and the third polynomial, and so on until all the poly- 
 nomials have been used. 
 
 (iii) To find whether polynomials which involve more than one 
 letter have a common factor containing any particular one of these 
 letters, they need only be arranged according to powers of that 
 letter, and divided as already described. By a repetition of this 
 process all the common factors of such polynomials, and hence, 
 their H. C. F., may be found. 
 
 [For fuller discussion of H. C. F. see El. Alg. pp. 116-121.] 
 
 EXERCISE Lll 
 Find, by the Euclidean method, the H. C. F. of : 
 
 1. ar^-3a;2^3^_j^ andaj4-2ar^+-2a;2-2ic + l. 
 
 2. c^ + 4c3-12c2-f c + Gaud c^- 0^-2 C2+-C + 1. 
 
 3. 5z^ + lSz^-3Sz-\-10sind2z^-{-5z^-22z-\-W. 
 
 4. 2ii^-{-Sx'-\-2x-2sindx' + 2a^-{-x'-2x-2. 
 
 5. r^-2r^ + 2r2-4andr^+-2r^-7'3-2. 
 
104 HIGH SCHOOL ALGEBRA [Ch. Vll 
 
 6. 1 — 4 m^ 4- 3 m'' and 1 — 5 m^ + 4 m'* + m — m^. 
 
 7. 63 + A:^ - 9 A; - 7 A;2 and 40 A; + A:^ - 5 Ar^ + 111 - 23 fe2^ 
 
 8. ar^- 4 a;3_ 2 a^ _ 8 H- a;4 and 2 a^+ 9 ic^-a^- 4 ar^ + 14 a; -16. 
 
 9. 8a^-22a^ + 17a;-3and6x=^-17a^ + 14aj-3. 
 
 10. 2a;2-3a;-35anda;4 + 14a!--9cc3 + 35a;-25. 
 
 11. What is meant by the H. C. F. of two expressions A and B ? 
 If a is not a factor of A, how does the H. C. F. of A and a • B 
 compare with the H. C. F. of ^ and B ? Explain. 
 
 12. If a is a factor of A, but not of B, how does the H. C. F. 
 of A and a • B compare with the H. C. F. of ^ and B? In intro- 
 ducing and suppressing factors during the process of division 
 (§ 77), what precaution must be exercised, and why ? 
 
 Find the H. C. F. of the following expressions : 
 
 13. m* — 3 m^ + 1 and m^ — 2 m - 2 — m*^ — m^ + 2 m^. 
 
 14. a' + 2a'-5a'-10anda'-{-a^-a'-2a-2. 
 
 15. a^-4x'-2-\-3x-3i^-{-Ba^Siudx-\-2s(^-{-2-5x'. 
 
 16. s^-2s^-2s3-lls=^-s-15 and 
 
 2s^-7s^ + 4s3-15s2 + s-10. 
 
 17. a;^ + 3 ar^ - 2 «2 _ 6 a; and 4 iK^ - a^ + a^ + 4 a;^ - 12 + 4 a^. 
 
 18. 21 ax — 17 ax^ — 5 aar^ 4- ax^ and 5 aa^ — 34 aa^ — 7 ax. 
 
 19. 7 mV— 49 m-^x + 42 m^ and 
 
 14 a^mx^ + 14 a^mx^ — 56 a^mx — 56 a^m. 
 
 20. 48 s^to* - 162 ^tx^ + 54 s^^ and 
 
 18 ^fu -9 s fux - 48 ^fux" -f 24 sH'uo^. 
 
 21. 4a;*-12a^2/ + 5a^2/^-fl2a;i/3-92/*and 
 
 12 a;^ - 36 a:^^^ + 11 a;y + 48 aJ2/3 _ 36 2/4. 
 
 22. x^-x^y-llxy^-4:f and a^^+a^y - 12 a^/ - 30 a;^ - 8 /. 
 
 23. The H. C. F. of any number of expressions must be a 
 factor of the H. C. F. of any two of these expressions. Why ? 
 Must it be the H. C. F. itself of any two of these expressions ? 
 Explain. 
 
78-79] LOWEST COMMON MULTIPLE 105 
 
 FindtheH.C.F. of: 
 
 24. a^ + 4a3 + 4a2, a^b-4:ab, and a*6 4- 5 a^fe + 6 a%. 
 
 25. Sx*-9a^+6a^, a^-9x^-\-26 x-24:, and a:^- 8 0^24. 19 a; -12. 
 
 26. a + a^a; — 2 a^, a + 3 a^a; + 4 ax^ + 2 a;^, and 
 
 2a3 4-3a2a;4-2aa^-2a^. 
 
 II. LOWEST COMMON MULTIPLE 
 
 79. Multiples of algebraic expressions. A multiple of an 
 
 algebraic expression is another algebraic expression that is 
 exactly divisible by the given one ; hence it contains all the 
 prime factors of the given expression. A common multiple 
 of two or more algebraic expressions is a multiple of each of 
 these expressions. 
 
 j5^.^., 12 a^Q^{]f'— 1) is a common multiple of 3 aVQy + 1) 
 and 2a%(^ — 1). 
 
 The lowest common multiple (L. C. M.) of two or more 
 algebraic expressions is the algebraic expression of lowest 
 degree which is exactly divisible by each of the given expres- 
 sions ; hence it contains all the prime factors of each of the 
 given expressions, but no superfluous factors. 
 
 E.g.^ a common multiple of 2 a%'^o[^ and 3 a^a^y^ must con- 
 tain the factors 2, 3, a^, 5^, x^^ and ?/* ; it may contain other 
 factors also, but it need not do so. Therefore 6 a%^j[^y^ is the 
 lowest common multiple (L. C. M.) of 2 a^h^a:^ and 3 a^j(^y^. 
 
 So, too, the L.C.M'. of 12 m\x^-lc^) and 8 JV.2(s+0(^-^)^ 
 is 24 m%\s -\-t')(x — k')^(^x -|- ^), — show that this last expres- 
 sion contains all the necessary^ but no superfluous^ factors. 
 
 The procedure for finding the L. C. M. of two or more 
 expressions whose prime factors are known (or easily found) 
 may be formulated thus : 
 
 To the L. C. M. of the numerical coefficients annex all the 
 different prime factors that occur in the given expressions^ and 
 give to each of these factors the highest exponent which that 
 factor has in any of the given expressions. 
 
106 HIGH SCHOOL ALGEBRA [Ch. VII 
 
 EXERCISE LIII 
 
 Find the H. C. F. and the L. C. M. of : 
 
 1. 8 a'h'', 24 a^6V, and 18 ahc\ 5. x" - ?/' and aj^ + 2 ic?/ + if. 
 
 2. 15a^6^ -20a26V, and 30 ac^ 6. 21 a;^ and 7 ^^^(x -h 1). 
 
 3. lQ>o?h\ 24:aMc, and 36 a^ft^dl 7. ic^-l and a^ + a;. 
 
 4. 18a^6r^, 12 pV^, and— 54a6y. 8. 4:X^y — y Siud 2 x^ -\-x. 
 
 9. Is 12a^6''(a^ — 1/^) a common multiple oi 2 a^b {x — y) Siud 
 3 ab\x -y)? Is it their L. C. M. ? 
 
 10. What factors must an expression contain in order that it 
 may be a common multiple of two or more other expressions ? 
 that it may be their L. CM.? 
 
 11. Are both 6 ax^ and — 6 ax"^ multiples of 3 a; ? Explain. 
 If a multiple of an expression has its sign reversed, does it 
 remain a multiple of the given expression ? 
 
 12. Does a change in the sign of an expression affect the de- 
 gree of the expression ? If the L. C. M. of several expressions 
 has its sign reversed, it may still be regarded as their L. C. M. 
 Why ? (Cf. Exs. 14-15, p. 99.) 
 
 FindtheL. C. M. of: 
 
 13. a + &, a — b, o? + W, and a* + b\ 
 
 14. 3 + a, 9 - a^ 3 — a, and 5 a + 15. 
 
 15. a^ — if, :ii? -\-xy + 2/^, and a^ — xy. 
 
 16. 4a + 4&, Q>o?-24.b\ and a--3 a& + 2 ftl 
 
 17. a? -f 2/^, ^y — y^, and a;^ — 2/^. 
 
 18. y'^-^y + Q and i/^ - 7 2/ + 10. 
 
 19. a^— (a+6)a;4-aft and a;-— (a— 6)a; — a&. 
 
 20. 3s2-7s + 2and6-s-s2. 
 Hint. 6 - s - s^ = - 1 (s2 + ^ _ 6). 
 
 21. c2-4c4-4, 4-c2, andc^-16. 
 
 22. 3p«-13p4-14 andl3p-5i)2-6. 
 
 23. r^" — s^" and (.s« — r^'f. 
 
 24. (m + ny—p^ and (m + w +i))^. 
 
79-81 j LOWEST COMMON MULTIPLE 107 
 
 25. 63^262-46-8, 86-12 + 62-63, and 6^ + 4 6^-3 6-18. 
 
 26. Find the L. C. M. of each of the sets of expressions in 
 Exs. 19-25, p. 100. 
 
 80.* The L. C. M. of two algebraic expressions found by means of 
 their H. C. F. The use of the H. C. F. in finding the L. C. M. may 
 
 be shown as follows : 
 
 Let it be required to find the L. C. M. of 3x* — a:r^ — x'^-\-x — 2 
 and2ar^-3a;2-2a; + 3. 
 
 By § 76 it is found that the H. C. F. of these expressions is 
 x^ —1', they may, therefore, be written thus : 
 
 Sx*-x^-x^-\-x-2={x''-l)(3x'~x + 2), 
 and 2aj3-3a;2-2aj-f-3 = (x2-l)(2x-3), 
 
 wlierein Sx- — x-\-2 and 2 x — 3 have no common factor. Hence 
 the L. C. M. of the given expressions is 
 
 (x'-l)(3x^-x + 2)(2 X - 3). 
 
 This shows that the L. C. M. of the given expressions may he 
 found by dividing their product by their H. C. F. 
 
 Obviously, the L. C. M. of any other pair of expressions may 
 be found in the same way ; hence. 
 
 To find the L. C. M. of two algebraic expressions, divide either 
 of the given expressions by their H. C. F. and multiply the other 
 expression by the resulting quotient. 
 
 81.* The L. C. M. of three or more expressions. The L. C. M. of 
 
 three or more algebraic expressions whose factors are not easily 
 found, may be obtained by first finding the L. C. M. of two of the 
 given expressions, then the L. C. M. of that result and another of 
 the given expressions, and so on. 
 
 EXERCISE LIV 
 Find the L. C. M. of : 
 
 1. a;3_6a^_^lla;_6anda.'3-9a^ + 26a;-24. 
 
 2. a^- 5x--4i» + 20anda^ + 2ar^-25a;-50. 
 
 *Articles 80 and 81, with Exercise LIV, may, if the teacher prefers, be 
 omitted till the subject is reviewed. 
 
 HIGH SCH. ALG. — 8 
 
108 HIGH SCHOOL ALGEBRA [Ch. VIl 
 
 3. 2 2/3 - 11 / -h 18 ?/ - 14 and 2 2/^ + 3 / - 10 2/ -h 14. 
 
 4. 6a^x-o a?x - 18 aa; - 8 a; and 6 o?h - 13 c^h - 6 a6 + 8 5. 
 
 5. 4 a;^ - 17 ^f 4- 4 / and 2y^-Qi?y-?, xY -^xf-2y^. 
 
 6. 2a;^-9a^ + 18i»2-18a; + 9and3a;*-llar'H-17a^-12a;+6. 
 
 7. If ^, B, and stand for any three given expressions, and 
 if i»f is the L. C. M. of A and B, while iV^is the L. C. M. of M 
 and (7, show that iVis the L. C. M. of A, B, and (7; that is, show 
 that N contains all the factors necessary in such a multiple, and 
 no superfluous factors. 
 
 Find the L. CM. of: 
 
 a s*-2s^-i-s^-l, s'-s^^2s-l, ands*-3.s2-fl. 
 9. c3 + 3c2-6c-8, c^-2c--c + 2, andc^ + c-G. 
 
 10. a52-4a2, ar5 + 2aa^^-4A-}-8a^ anda^-2aa^ + 4a2a;_8a3. 
 
 11. a3-j-7a2 + 14a + 8, a^ + Sa'-e a-S, and a3 + a'-10a+8. 
 
 12. Ar^-9 A:24.23 k^W, k'+k'-17 k+W, and ]i^+7 k'+7 k-15. 
 
CHAPTER VIII 
 ALGEBRAIC FRACTIONS 
 
 82. Definitions. An algebraic fraction is an indicated divi- 
 sion in which the divisor is an algebraic expression : the 
 dividend may be either an algebraic or a numerical expres- 
 sion. (Cf. § 8.) 
 
 Here, as in arithmetic, the fraction A-i-B is usually written 
 
 A 
 
 in the form — or A/B ; A and B are called the terms of the 
 B 
 
 fraction, A being the numerator and B the denominator. 
 
 If A is exactly divisible by B, then A/B is, in reality, an 
 
 integral expression, but is written in the form of a fraction. 
 
 E.g.^—- , — -•,and — are algebraic fractions; while 
 
 ab—oj^ m—2n -.a^— a;^ . , . 
 
 , , and are integral expressions written 
 
 a 1 a— X 
 
 in fractional form. 
 
 If both terms of a fraction involve the same letter, and if 
 
 the numerator is not of lower degree than the denominator 
 
 (in this letter), then the fraction is said to be improper ; 
 
 otherwise it is proper. An expression that is partly integral 
 
 and partly fractional is called a mixed expression. 
 
 E.g.^ ^ — and — — — are improper fractions, and 
 
 X — 1 a 
 
 4 2: — 3 H — ^^—— is a mixed expression. 
 x — \ 
 
 83. Operations with fractions. The reduction of fractions, 
 and the various operations with fractions (addition, subtrac- 
 tion, etc.), are essentially the same in algebra as in arith- 
 luetic. 
 
 109 
 
110 BIGH SCHOOL ALGEBRA [Ch. Vlll 
 
 A 
 
 Thus, if — and ~ are any two fractions whatever, then 
 
 ^ ^ B D~BI)' ^ ^ B^I)~b'~C' 
 
 These formulas state the rules for finding the product and 
 quotient, respectively, of two fractions ; the pupil may trans- 
 late each formula into verbal language. 
 
 (i) The proof of (1) follows directly from the definition 
 of a fraction (cf. §§ 82, 8). 
 
 Thus, let — = a;and — =?/, 
 
 then A = X'B •Awdi 0=y'D, [§§82,8 
 
 hence A- C= xByB = xy'BD, [Ax. 3 
 
 AO 
 and therefore — — = xy [Ax. 4 
 
 BB 
 
 ^4 O fsince A/B = x 
 
 ~ B' B' L and C/B = y 
 
 A C^AO 
 B' D BB" 
 which was to be proved. 
 
 (ii) To prove (2) above, let - - - = t, 
 
 B B 
 
 then | = ^'|' [§§82,8 
 
 hence A.^=t.^.R [Ax. 3 
 
 B Q BO •- 
 
 n 
 
 i.e.^ 
 
 I.e., 
 
 ^'{b'o)^*' ^^^^ ^^^^^ 
 
 and therefore A^^=4.^ 
 
 B B B O' 
 
 which was to be proved. 
 
 Kemark. The reciprocal of any given number is 1 divided by 
 that number; e.g., the reciprocal of 3 is ^. 
 
 Hence it follows from (ii) that the reciprocal of a fraction is 
 that fraction inverted. 
 
 Note. Observe that the validity of § 42 is assumed in the proofs of (i) 
 and (ii) above. 
 
83-85] ALGEBRAIC FRACTIONS 111 
 
 84. Reducing an improper fraction to a mixed expression. 
 
 This change in form is made in algebra precisely as it is 
 made in arithmetic. 
 
 E.g.f just as -ijQ=3|^, i.e., 3 + J, so, too, since a fraction is an 
 
 .,.",,... x^-^-'Jaf + B ' ,-,, 4r-2x 
 indicated division, — -^ -— = a? + 1 + — -. 
 
 EXERCISE LV 
 
 Reduce each of the following improper fractions to an equal 
 integral or mixed expression, and explain your work : 
 a^ — 2ab-\-ac 
 
 8. 
 
 2. 
 
 a 
 
 3ar^+9a; + 2 
 
 3a; 
 
 ^ 2a;'^ + 4aa; + 2a^ __ 
 
 x-{-a 
 
 2s-l. 
 
 11. 
 
 12. 
 
 ar^-ar^-2ar^- 
 
 -2a;-l 
 
 a^-x- 
 
 1 
 
 6-hQc-oc^- 
 
 -2c3 + c* 
 
 c'-3 
 
 
 8a;''-10a;2_3a; + 5 
 
 4x^^-3 
 
 
 3a,6_^2a;-5 
 
 
 a;» + 2a;4-l 
 
 
 15^^-13 i;2- 
 
 8i;-l 
 
 A: + 2 Sv'-^ + S'y + l 
 
 g^ + g^ + l 7fe^-l 
 
 a + 1 * Jc' + k + l 
 
 a^4-7a;2_5 I8a;^-a^-2a;^- 7 
 
 a;2_i • ■»•*• ^3_:3^_^1 
 
 ^3 2 a 4- 1 
 
 15. Is ^^^-^ a proper or an improper fraction ? Why ? 
 
 16. Write the reciprccal of 11 ; of —a; of — |; of each frac- 
 tion in Exs. 1-5 (cf. Remark, § 83). 
 
 85. Reducing fractions to lowest terms. In algebra, as in 
 arithmetic, a fraction is said to be in its lowest terms when 
 its numerator and denominator have no common factor. 
 
 Hence, to reduce a fraction to its lowest terms, divide 
 both numerator and denominator by their H. Q. F. Instead of 
 dividing at once by the H. C. F., we may, of course, divide 
 
112 HIGH SCHOOL ALGEBUA [Ch. Vlll 
 
 by any common factor, then by another, etc., until all com- 
 mon factors are divided oat. Multiplying or dividing both 
 terms of a fraction by any given number leaves the value 
 of the fraction unchanged; for, whatever the algebraic ex- 
 pressions represented by A, B, and m, 
 
 ^ = i.^ [§83 (i) 
 
 Bm B m l^ \ J 
 
 = — , [since m/m = 1 
 
 which was to be proved. -" 
 
 3 ax^ 3 ax 
 
 E.g., ~ = , which is in its lowest terms ; so, too, 
 
 4 bxy 4 by 
 
 , ^~^ — = (a? + l)(a;- l) ^ x±l^ ^^.gj^ .g .^ .^g lowest terms. 
 a^-2a; + l {x-l){x-l) x-l' 
 
 EXERCISE LVI 
 
 Beduce each of the following fractions to its lowest terms : 
 a^ — ah m^ -\-2 mn -f- w^ „ m? -\-if 
 
 a^ — h^ m® + n^ x^ + x^y"^ + y^ 
 
 34«^6V 2a^ + 3a; + l ^ 3a^-2a-l 
 
 Sla^ft^c* ' a^ + 5x + 4 * ' l + a-a^-d"' 
 
 c'-d^ {r-qY-s^ a*-a^-20 
 
 {c-df' ' (r-q-sf' ' a'-9a'-{-20' 
 
 10. May equal factors be canceled from the numerator and 
 denominator of a fraction ? May equal parts (or factors of parts) 
 
 be thus canceled ? Is 1^L±^ equal to -^? Is P^^^^, 
 obc-^x obc ox — on^ 
 
 equal to ^ + ^y ? Explain fully. 
 
 Eeduce the following fractions to their lowest terms, and 
 check your work by § 25: 
 
 n ^-t\ T« xy-zy-x + z 
 
 ^^' '^r~t' ^®- f^^ 
 
 c3-17c^ + 72c 50-40m + 8m^ 
 
 c2(cd4-16-2c-8d)* ' 125-8m« 
 
85-86] ALGEBRAIC FRACTIONS ' 113 
 
 16. ^^:^ 21. ^"-^ 
 
 (a -«)(«- 6) l + Ai^"* 
 
 (s-2)(8-3) . (a^-5)(a.- + 2) , 
 
 (2-s)(3-s)(s-4) • aj3-7a^ + 2a; + 40 
 
 18 p^-llp + 24 3a^ + 8a -3 
 
 56+i)-i)^ 3a3 + 17a2 + 21a-9 
 
 ^!±l!. 24 10^ + 20 a:'-a?- 2 
 
 * y'-x^' ' 3a^ + 6a^ + 21a;+42' 
 
 86. Reducing fractions to equal fractions having given de- 
 nominators. Since multiplying both terms of a fraction 
 by the same number does not change its value (§ 85), there- 
 fore any given fraction may be reduced to an equal fraction 
 whose denominator is any desired multiple of the given 
 denominator. 
 
 E.g., to reduce - — - to an equal fraction whose denominator 
 
 shall be 12 cx^y, multiply both terms of the given fraction by 
 12 cx^y -7- 4:X^, i.e., by 3 cy, 
 
 EXERCISE LVII 
 Find the required part in each of the following equations : 
 1 ^ = L. 1 m-2 _ (m-2y 
 
 3. 
 
 4 12 
 Sab^ ? 
 
 4 12^' 
 2 cd 16 cH^ 
 
 8. 
 
 6t 
 
 9 
 
 3c- 
 
 2d' 
 
 1 ? 
 
 Scd* 
 
 4:X 
 
 9 
 
 1 
 
 7x^-5 
 
 a-b 
 
 a'-b' 
 
 10. 
 
 5. ^ = , ; ^ ' 11. 
 
 a-\-b ? "■ 2(2a;-5) -6x{5-2x) 
 
 9m 
 
 9 
 
 — r 
 
 9 
 
 r2 + 3 r^ + 10r2+21 
 
 2u-v 
 
 ? 
 
 ^f +2 V Su 
 
 ^j^5uv-2v'' 
 
 u^ — uv-\-v^ 
 
 _2{u' + if)^ 
 
 lu^-1 
 
 9 
 
 3m-8 
 
 9 
 
 2a;-5 - 
 
 -2.TH-5 
 
 3m-8 
 
 9 
 
114 ' HIGH SCHOOL ALGEBRA [Ch. VIII 
 
 13. If the denominator of a fraction is multiplied by any given 
 expression, what must be done to the numerator in order to pre- 
 serve the value of the fraction ? 
 
 14. Change { to an equal fraction whose numerator is 
 
 J9^ — 9 
 
 — 3j9 — 2; to one whose numerator is 3jp^''H-5p^4-2j9; to one 
 whose denominator is 9p— p^; to one whose denominator is 
 3y4_p2_27p_9. 
 
 87. Reduction of fractions to common denominators. To 
 
 reduce any given fractions to equal fractions having a com- 
 mon denominator it is necessary only (1) to choose some 
 common multiple (§ 79) of the denominators of the given 
 fractions as the new denominator, (2) to divide this com- 
 mon multiple by the denominators of the given fractions in 
 turn, and (3) to multiply both terms of the given fractions 
 by the respective quotients (cf. § 86). 
 
 3 h hn 
 
 E.g., to reduce -^^ — and — - to equal fractions having a com- 
 2 ax 3 x^ 
 
 mon denominator, we choose 6 ax^ as the new denominator, and 
 
 find, by § 86, that 
 
 3h _ 9 hx T hn _ 2 dbn 
 
 2 ax 6 ax? 3 x- 6 ax^ 
 
 The lowest possible common denominator is, of course, the 
 
 L. C. M. of the given denominators. 
 
 EXERCISE LVIII 
 
 Reduce the following to equal fractions having the lowest pos- 
 sible common denominator: 
 
 1. -, — T' '^"^^ -r—^' 
 
 111 inr 5 m"* 
 
 ^ 3a + l T 3i»-f4 
 
 2. — and ■ 
 
 4 6 
 
 ^ 9-3 a ,34-5a^ 
 
 3. and — ^ „ - 
 
 16 h 20 h^ 
 
 4. 
 
 « + ^and^-^ 
 a — h a-\-h 
 
 5. 
 
 ^, ¥, - 1^ 
 
 6. 
 
 rfyl|."0|^ 
 
86-88] 
 
 ALGEBRAIC FRACTIONS 
 
 115 
 
 7. -^:z^and ^ + ^ 
 
 ar + aic 2 aa^ — 2 a^a; 
 
 and 
 
 9. — !-^^ and ' -^ 
 
 10. 
 
 
 a;2 + a-2/ + t/'^ 
 and 
 
 11. 
 
 12. 
 13. 
 14. 
 15. 
 
 16. 
 
 (m — l)(m — 2) m — 3 * x + y'x — y^ x^ — xi/ 
 
 Hint. First multiply both terms 
 of by — 1, so as to arrange 
 
 and 
 
 2 — m' m + 2' m- — 4 
 
 2 - m 
 the denominators in the same order. 
 
 m 
 
 and -^ 
 
 m — n rr — m' 
 
 m 4-^*' 
 
 6x 
 
 2 J 3 
 
 . and 
 
 ar^-1 
 3 a 
 
 and 
 
 3 a — 6 X 
 
 ^ ^-2 and ^ + ^ , 
 
 5 — a' a^ — 8 a + 15' a^ — 6 o + 5 
 
 and 
 
 (aj-l)(a;-3) (x-8)(3-a^) 
 
 17. Show that 
 
 Hint. 
 
 equal 
 
 (x-S)iS-x) 
 
 -7 
 
 (x-S)(x-S) 
 Cf. Hint, Ex. 11. 
 
 (c-l)(2-c) (c-l)(c-2) 
 2-a; x-2 
 
 18. Show that _ ^^ ox — / Kx/ ox- 
 
 (5 — x-)(aj — 3) (a;-5)(a;— 3) 
 
 Reduce to equal fractions with the lowest common denominator ; 
 
 ? , ? , and — (cf.Ex.l6). 
 
 (a; _ 7) (a? - 2)' (2 - x)(x - 4)' (4 - aj)(a; - 7) ^ ^ 
 
 1 2.-3 
 
 19. 
 20. 
 
 and 
 
 (.y _ 2)(v - 5) ' (v - 5)(3 - ?;)' (v - 3)(7 - v) ' 
 
 21. 
 
 a+ 5 
 
 a-2 
 
 and 
 
 a + 1 
 
 a2_ 4 a + 3' 8 a- a2_ 15' 6a -5- a^ 
 
 88. Addition and subtraction of fractions. From § 38 it 
 follows that 
 
 a h a + h -.a h a — h 
 - -4- - = and = ; 
 
 c c 
 
 c c 
 
116 HIGH SCHOOL ALGEBBA [Ch. VIIl 
 
 that is, ill algebra, as in arithmetic, the sum (or difference^ of 
 two given fractions which have a common denominator is a 
 fraction whose numerator is the sum (or difference^ of the given 
 numerators, and whose denominator is the common denominator 
 of the given fractions. 
 
 ^ m^ 2h ^ m^^2h . 2 a b^c ^ 2a-l?c 
 
 '^'"^ax Sax Sax ' ^^ 6(x-l) 5(a^-l) 5(a:-l)* 
 
 If the given fractions have unlike denominators, they must, 
 
 of course, be reduced to equal fractions having a common 
 
 denominator (§ 87) before they can be added or subtracted. 
 
 3 7 
 
 Ex. 1. Find the sum of and 
 
 x — 2 x-\-l 
 
 Solution. The L. C. M. of the denominators is (x — 2) (a; -j- 1) 
 
 and, by § 87, 
 
 and 
 
 3 ^ 3(a;+l) ^ 3a; + 3 
 x-2 (x-~2)(x + l) (aj-2)(a! + iy 
 
 7 ^ 7(a;-2) ^ 7a;-14 
 x + 1 (x + l)(x-2) (a;-fl)(a;-2)' 
 7 Sx + 3-{-7x-U 10a!-ll 
 
 + 
 
 x-2 aj + 1 (x-{-l){x^2) (a; + l)(a;-2) 
 
 7 3 
 
 Ex. 2. Subtract from 
 
 x-^1 x — 2 
 
 Solution. Proceeding as in Ex. 1, we obtain 
 
 _3 7 ^ 3a; + 3- (7a;-14) ^ -4a; + 17 
 
 x-2 x-\-l (^x-\-l)(x-2) (^x + l)(x-2) 
 
 Note. The minus sign before the second fraction means, of course, that all 
 of this fraction is to be subtracted, hence the need of the parenthesis in the 
 numerator of the next fraction. 
 
 EXERCISE LIX 
 
 Simplify the following expressions and check your results : 
 
 3 ^4.^ 5 a?-l x-^3 x-\-7 
 
 ■ 3 "^6 • ' 2 5 "^ 10 * 
 
 . a + S a-{-5 c-^-d b_ 
 
 5 7 * ' d 2d' 
 
88] ALGEBRAIC FRACTIONS 117 
 
 - 1 +^. 16. 1 3 
 
 x+y x — y 2s^—s — l 6 s- — s — 2 
 
 - X a; # _ _ ct^ — aa.' H- 0^ a + a; 
 
 1 — a^ 1 + a^ a^ + ax + a;^ a — x 
 
 10. ^Lz^H-^^I^ + ^r:i^. [Suggestion. x = ^.] 
 
 ah he ac L 1 J 
 
 11. r-j_^r+^__r^--^> ^^ _a c^ ^ s. 
 
 2rs s — as^ — w 
 
 (a5-2/)^ a;^ + 4a;?/-5/ a^-7a; + 12 a^-5a;+6 
 
 13. ft + ^-c a-6 + c ^ 21. 1 I ^-?/ a:^-a;y ^ 
 a? — ih — cf {a—hy—c^ ' x-\-y x^—xy-\-y^ 3^-\-'if 
 
 14. J~-^ 22. -i- + -^ 1. 
 
 a;2_i aj2_^_2 s(s — t) t{s + t) st 
 
 15 a? 4- 7 a; + 2 ^^ 2a;-3c 2a;-c ^^ 
 
 a^_3a;-10 a^4-2a;-35' * a;-2c x-c 
 
 Remark. Since a fraction is a quotient, its sign (the sign 
 before the fraction) is governed by the law of signs in 
 division; hence, whatever the expressions represented by a 
 and hy 
 
 __a_ — g _ a 
 h~ b ~-h' 
 Exercises in subtraction may therefore be changed into exercises 
 in addition ; the results of such exercises are often called algebraic 
 sums (cf. § 16). 
 
 E.g., Ex. 23 may be written ^^~^^ + ~^^"^^ + 3 a;. 
 
 X— 2 c x~c 
 
 24. Write Exs. 6, 8, 11, 13, 14, 15, 19-22, above, as exercises 
 in addition. 
 
 * Cf . Ex. 2, Note. 
 
118 HIGH SCHOOL ALGEBRA [Ch. VIH 
 
 Write the following as exercises in addition, and find the alge- 
 braic sum in each case : 
 
 a a + 1 a + 2 s—1 2(s + 1) 
 26. h « — :i 30. 1 
 
 ic — 1 1 — X a — 1 a(a — 1) 
 
 27. -i L.-.!^. 31. 1 1 
 
 a + 6 a-6 5'^ - a^ 2a^-a;-l 3-aj-2a^ 
 
 d cd cc?2 ^^ 2h-a 3x(a-b) , ?^-2a 
 
 G + d {c-\-df (c + df x-b b'^-x" x + b 
 
 Simplify : 
 
 {a-b){a-c) {b-c){b-a) (c-a){c-b) 
 [Hint. The given expression, witli the letters in alphabetical order, is 
 
 ^i + -I + J .1 
 
 (a-6)(a-c) (,b-c)(a-b) (o-c)(6-c)-l 
 34. "" + * + <' 
 
 (a — b)(a — c) (b — a)(b — c) {c — a){c — b) 
 
 35 a;-l 2(a^-2) x-S 
 
 (x-2)(x-3) {3-x){x-l) (x-l)(2-x) 
 
 36. ., . ^ .. „ - „ . V „ „ + ' 
 
 x^ — 5 xy -\- 6 y^ x^ — A xy -{- 3 y- x' — 3xy-^2y' 
 37. ~ H f- ^ 
 
 x^ — 5x-{-6 3a7 — 2 — ic- 4a; — 3 — a;'-' 
 38. ^^ [ ^^ I «^ 
 
 (a-c)(a-6) (6_c)(6-a) (c-a)(c-6) 
 
 89. Reducing mixed expressions to improper fractions. 
 
 Since an integral expression may be written in the frac- 
 tional form with the denominator 1, therefore reducing mixed 
 expressions to improper fractions is merely a special case of 
 addition. 
 
 E.g.,x + 1+ 1 ~»' + l . 1 
 
 a; — 1 1 a; — 1 
 
 x—1 x — 1 x — 1 
 
88-90] ALGEBRAIC FRACTIONS 119 
 
 EXERCISE LX 
 
 By the method of § 89 simplify the following expressions : 
 
 af — l a + 2b 
 
 , -i 2x „ 9 oX'^-j-oi^ — x-\-l 
 
 2. x-\-l • 7. X — or — XT ■ — 
 
 x — 1 l-\-x-{-x~ 
 
 c^— 2c + l 1 — 2a; + a;- 
 
 a4-& 2aH-36 
 
 _ -, 2 1— V^ -.r* -1 I, ax-\-bx-\-ab 
 5. 1 — y — if — ^ • 10. 1 — aa; — 6ir :; — 
 
 1 — 2/ 1 — ax 
 
 11. May the numerator in the answer to Ex. 1 above be found 
 by multiplying x~l by a^ — 1 and adding x'^ to the product ? 
 Explain this method fully (of. § 84, also Ex. 20, p. 50). 
 
 12. By the method of Ex. 11, solve Exs. 2-5, and 8-10, above. 
 
 90. Product of two or more fractions. In § 83 it was 
 shown that, whatever the expressions represented by A^ B, 
 (7, and i>, 
 
 A O^AO 
 B ' J) BD' 
 This principle is easily extended to finding the product of 
 any number of fractions ; 
 
 A ^ ^ ^ = A^ ^ Oi^ACE a^ACEa 
 
 ^*^*' B' d' f' H BB' f' H BBF ' H BBFH 
 
 Hence, the product of two or more fractions is a fraction ivhose 
 numerator is the product of the numerators of the given fractions^ 
 and whose denominator is the product of their denominators, 
 
 Ex. 1. Find the product of ^^^ ~ ^ and , ^^ ^^ • 
 
 '6xy a (or — 1) 
 
 SOLUTION 
 
 a(x — l) 6x _ Gax{x — \) _ 
 
 3xy a(ar — 1) Saxy{x^ — 1) y{x + l) 
 
 [§85 
 
120 HIGH aCHOOL ALGEBRA [Ch. VllI 
 
 Remark. Observe that the factors 3, a, x, and a^ — 1 might 
 have been " canceled " even before the multiplication was actu- 
 ally performed. Pupils should cancel wherever possible, and thus 
 simplify their work. 
 
 EXERCISE LXI 
 
 Find the following products, and simplify your results : 
 
 2. ^ . ^. 9. 
 
 xyz ac^ m- — mn ef—f^ 
 
 g Qxy l^yh\ ^^ x-1 . a; + l . 
 
 * 82 * 9a^ * ' a^ + 2a; + l x-1 
 
 ^ - 5 iH^ Sr^t ^^ x-1 x2 + 4a; + 3 
 
 12iu^ lOs^v^ 3aj2-|.8i» + 5 ar^-1 
 
 — Ix^yz f 6 fey^ Y /-^ — ?-8 + s^ j? — <f 
 
 IS xyz'- '\ 11 x'zj ' {p-qf *r^ + ?V + «^* 
 
 4myi^ —am (a — xY x^-\-xy + y^ 
 
 1 m^n^ x^ — y^ d^ — 2 ax-\-x^ 
 
 7. Isx^y^ . t . 4-. 14. i—^r-r . a±b+l, 
 
 y 9xY (a + 6)2_l a-b-1 
 
 ^m+2 * ^m •52* * 2s^ * si + i^ 's2_g^* 
 
 16 ^ + 5 a4-3 gg-Ta + lO . 
 a-2 * a-5 ' 2a- + 5a-3* 
 
 17 a.'^ + y' a* + a'b' + b* x^-f 
 
 ' a^-^b^' a^-b^ 'x*-2afy''-\-y*' 
 
 1^ _ f 3 
 18- -^ — r • r^r-: = ? May i^ be canceled in this example ? Ex- 
 
 plain (cf. Ex. 10, p. 112). 
 
 19. Simplify [x-\-2y ) • — ^ by finding the product of each 
 
 V y) a-\-x 
 
 term of the multiplicand by the multiplier, and then adding the 
 partial products (cf. § 32). 
 
 20. Simplify [ a; + 2 ?/ — - ) • — ^ by first reducing the multipli- 
 
 V • 2/7 a + a; 
 
 cand to an improjjer fraction. 
 
9(V01] 
 
 ALGEBRAIC FRACTIONS 
 
 121 
 
 Simplify, and check your results : 
 22. (5s2-36s+7) 
 
 25. 
 
 26. 
 
 27. 
 
 28. 
 
 29. 
 
 + 
 
 (f - ?> 
 
 1-6- 
 
 3s-21* 
 
 23. 
 
 24. 
 
 a + 25 
 2b-a 
 
 1 ^'" 
 
 V 
 
 
 x' + fj x'-f 
 
 Sa(a-b) 
 
 V af-9x-^20j\'jf-6x + 9 J\ x J 
 
 b(x^ 
 
 a 
 
 9x-^ 
 -hx + y 
 
 3 a' + 3 a'bx + 3 a*by 
 ax — 3by—Sbx-\-ay 
 
 1 ^a + b^(a-\-byi\ (a-\-by\a 
 
 a_b__c 2\A 2c V 5a 
 
 6c ac a6 « A a + 6 + cy l^ — a — b 
 
 (a + 6)2fa + 6 + c 
 
 30. What doesr-Y mean [cf. § 9 (ii)!? Show that (^-Y 
 
 Ve/ \/-i \o/ 
 
 and that ("^ = i^?!^' = ^^ . 
 
 fey f rs' Y / m - 2 n V / -Sc V / a + 6 
 
 31. Raise the following fractions to the powers indicated : 
 
 b-c\\ r 2.<i/V Y 
 / ' \—'Smyip) 
 
 32. Write the following fractions as powers of other fractions : 
 
 6^ + 26 + 1 . _t_. s^-^sH-^-^sf-f 
 a? ' 16aj8' 64 + 48i/ + 122/' + 2/'* 
 
 91. Division of fractions. In algebra, as in arithmetic, to 
 divide by any fraction gives the same result as to multiply hy 
 the reciprocal of that fraction [of. 83 (ii)]. 
 
 a^x 
 
 E.g., -^1^ - 
 
 a'^x by 
 
 by- by b-y^ 
 
 ex 
 
 af_ 
 
 bey 
 
 Note. If the divisor is an integral (or mixed) expression, it should be 
 written in fractional form (that is, as an improper fraction) before proceed- 
 ing as above. 
 
122 HIGH SCHOOL ALGEBRA [Ch. VllI 
 
 EXERCISE LXII 
 
 Simplify the following : 
 
 Ud'b' 2a'b' /^H-8^ + 7 ^ ^ 
 
 a" -121 . a + 11 ,^ / _^^^^V «' i\ . '^ 
 
 2- ^ — r -^ — ^ • 11. « + /i .> '> - i - 2 
 
 • ^V ' -5q^' ' (a+6)2-9 ' a + 6 + 3 
 
 4. l^!i£±fU-^. , 13. ^l±^^(t^-^2tv + 2v^y 
 
 7'.s — .9^ r — s —or 
 
 ^ oc^ — a" (x — ay , ^ -y^ — 25 5 — 'i? 
 
 5^ J.- N z_ , 2,4. r- • 
 
 a^ -j-a^ x^ — a^ v^-\-v-\-l v^—1 
 
 ^ Ux^-7x 2aj-l „ x^-1 x''-12x+?>n 
 
 15. 
 
 12x^-\-24:X^ x'^ + 2x lO-^'Sx-a^ x^ + 3x + 
 
 „ p(^ Q„4 5 r)V ^ /h 2cd \ d — c 
 7. OO^rrs^-? '- — 16. 1 ' • 
 
 
 a. (.^-3.-10).-^;. 1. ^14-.-^^ 
 
 a^-b* , (g - by r'-l , 1 - r^ 
 
 • a^ + a2^2_^6* • a'-b' ' t^-dr ' 3-r' 
 
 0^^-5^-36 V ^'-y 
 
 20. 2r^-21r-ll ^/^_r-3 
 ^^-^-ISr-T • V r-7 
 
 ^. 
 
 22. 
 
 i»2_49 a;2_^_42 
 
 5 m^n — 5 ri^ 
 
 urn + 2 mrr + n^ V wi 
 
 
 23 2a^ + 13.T + 15 . 2a?^ + lla; + 5 
 
 4x^-9 ■ 4x2-1 
 
 2^ x^-nx- + l^ . / a^2_3^_4 
 
 9rt^-34a2 + 2o V 3a2 + 8a + 5 
 
91-92] ALGEBUAIC FRACTIONS 123 
 
 {p — qY P — Q p^-\-q^ p^ — q'^ 
 
 25- f:^=^,^^~^^ + z^ ■ P-'P\^1 (cf. § 10). 
 
 I'oTj—WjfY ^ (^ — ^y^ P^^ — 4 py—^p^y+px 
 
 ' \p-q J 
 
 p^ — pqj \p—qj ^ -\- ^ xy -\- ^ if 
 
 92. Complex fractions. In algebra, as in arithmetic, a frac- 
 tion whose numerator or denominator, or both, are them- 
 selves fractional expressions, is called a complex fraction. 
 
 1 1 
 
 a X 
 
 ^'Q--> ^ and r- are complex fractions. 
 
 X 
 
 Since a complex fraction is merely an indicated quotient, 
 it may be simplified by means of §§89 and 91. 
 
 ^g ^ _ ^ _aP — 1 X _ ^ — 1 
 
 '.,. .1" x^-\-2x-\-l~ X ' x^-\-2x-{-l~ x + 1 
 
 X X 
 
 In many cases, however, a complex fraction is most easily 
 simplified by first multiplying both its terms by the L. C. M. 
 of their own denominators. 
 
 Thus, in the above example, multiplying both terms by x 
 
 gives o^ ^ T") which (by § 85) equals ~ , as before. 
 
 X^ -}- ij X -\~ J~ X -J- JL 
 
 EXERCISE LXIII 
 
 Simplify the following expressions : 
 
 ^ (m — nf -4-i 
 
 s—p _m- g. s s 
 
 1 =— . 3. . 9. • 
 
 ■ s +p m^ — n^ 1 _ I 
 
 p^ mn V s 
 
 c-^d (x-S)(x-5) 1_1 
 
 ^ c— d x — 7 6 ^ -^ 
 
 (^-d^ x-n i^_j. 
 
 c^-fd^ (a!-3)(.T-7) e" f 
 
 HIGH SCH. ALG. — 9 
 
124 
 
 7. 
 8. 
 
 c 
 
 HIGH SCHOOL AH 
 
 1 + 7 
 
 9 ^ 
 
 a 6 c 
 
 9EBRA [Ch. VIII 
 
 3 a-c 
 11. «-f 
 
 . ah' 
 
 x — 6 
 x-2 ' 
 
 X 
 
 a 
 
 (a - 6)^ 
 2a-3&+c 
 
 ^ ,.4 
 
 oca 
 a 
 
 1+- + J 
 
 
 a — 
 3 
 
 18. "^^ 
 
 6 a-\-h ^^ 
 1 ^ 
 
 a^-b' a^ + W 
 a-{-h a—h 
 
 15 
 
 & ' a + 6 
 
 a — b a + b 
 
 1 
 
 
 1 I ^ 
 
 a «-l 
 
 
 ^ + 1 + 0. 
 a; 
 
 a+1 
 
 
 " 1 / 1 ^ 
 
 
 
 * To simplify such an expression we begin at the end and work backward. 
 The first step here is to add ^ to 1 , then divide 4 by this sum, then add this 
 quotient to 1, and finally divide 3 by this sum, obtaining the result j\. 
 
CHAPTER IX 
 SIMPLE EQUATIONS 
 
 93 Introductory remarks and definitions. As we have 
 already seen in Chapter V, an algebraic problem states a re- 
 lation between numbers whose values are known (called 
 known numbers), and others whose values are at first un- 
 known (called unknown numbers). It is by means of this 
 relation, translated into an equation, that we can find the 
 values of the unknown numbers. 
 
 Besides the numerals 1, 2, 3, ••• the letters a, 5, <?, •••are 
 often used to represent known numbers ; unknown numbers 
 are usually (though not necessarily) represented by the later 
 letters of the alphabet, such as ic, ^, and z. 
 
 A literal equation is one in which one or more of the 
 known numbers are represented by letters; while in a nu- 
 merical equation all known numbers are represented by the 
 numerals 1, 2, 3, etc. An integral equation is one whose 
 members are integral in the unknown numbers (cf. § 34); 
 known numbers may appear as divisors and the equation 
 still be integral. 
 
 By the degree of an integral algebraic equation is meant 
 the highest number of unknown factors which it contains in 
 any one term. 
 
 E.g., of the equations (1) 4a;-5y=:10, (2) —-8 = -, (3) 4ta'^ = 2+x, 
 
 a h 
 
 (4) 3 x2 - 9 a3 = 4 y2^ and (5) |- -3 a;y = 5 y, all are integral, (1), (2), and (3) 
 
 are of the first degree, (4) and (5) are of the second degree, (2), (3), and 
 
 (4) are literal, and (1) and (5) are numerical. 
 
 Equations of the first degree are usually called simple 
 equations, and often also linear equations (cf. § 140, Note) ; 
 
 125 
 
126 HIGH SCHOOL ALGEBRA [Ch. IX 
 
 while equations of the second and third degrees are called 
 quadratic and cubic equations, respectively. 
 
 94. Equations having fractional coefficients. Equations 
 having fractional coefficients may be solved as follows: 
 
 Ex. 1. Given -— — ^ = « 5 *^ ^^^ ^• 
 
 Solution. On multiplying both members of this equation by 
 6 (Ax. 3), it becomes 5 oj — 48 = 3 a;, 
 
 whence 2 a; = 48, [Ax. 1 
 
 and therefore a; = 24. [Ax. 4 
 
 On substitution in the given equation, 24 is found to check; it 
 is therefore a root of that equation. 
 
 Multiplying both members of an equation by a common 
 multiple of its denominators is usually spoken of as clearing 
 the equation of fractions. 
 
 EXERCISE LXIV 
 Solve the following equations, checking the root in each case : 
 
 _ 2x — 4 3a; — 7 o 
 
 O. — = Zi. 
 
 2. 
 
 X 4:X f, 
 
 2==Y"^' 
 
 3. 
 
 2a; 5a; „ 
 3 4 ~ 
 
 4. 
 
 3a; = | + 25. 
 
 5. 
 
 7a; 5 w 2a; 
 10 6 15" 
 
 6. 
 
 4. ._23. 
 9 6 36 
 
 7. 
 
 7?i 4- 8 m -f 4 ^ 
 6 11 ~ • 
 
 
 [Cf. §88, Ex. 2, Note.] 
 
 
 5 
 
 
 7 
 
 9. 
 
 V+s= 
 
 :91 
 
 -lOv. 
 
 10. 
 
 T-'i- 
 
 = - 
 
 -i + 2ia, 
 
 ,, X 2x , 2x-3 ^ 
 
 12. l„-~+-^ — . 
 
 ,, a; — 3 , 7x 4x — S 
 3 18 5 
 
 14. Are the above equations integral or fractional ? Why ? 
 
93-95] SIMPLE EQUATIONS 127 
 
 15. How is an equation cleared of fractions ? Upon what 
 axiom does the process depend ? 
 
 16. In each of Exs. 5-7 name three factors, any one of which 
 might be used to clear the equation of fractions. How may we 
 in each case find the least factor which can be used ? 
 
 17. Name the degree of each equation in Exs. 22-28 below. 
 Which of these equations are simple ? quadratic ? cubic ? 
 
 Solve the following equations, and check as the teacher directs: 
 
 ^3 7,a;-4^^^ 4.x + 7^ 25. a:^-x=6. (Cf . § 72.) 
 
 5 " 7 * 
 
 . /o ox X 26. x'-15x' + mx = 0. 
 
 19. -2 a; + 4 -(3 a; 4- 2)=--. 
 
 20 Mzz^ = ^±2_3^-2 27. v'-^2v' = v + 2. 
 
 5 G 7 ' 
 
 ^, Ax 21X-25 4.(3x-2) 28. ^±i ^ _A_ = ^"zi^S^ 
 
 21. -3- = — 12 2~ ^ -^-^ 1^ 
 
 22. 3s-15_2g-^^3 -^^ 29. 
 
 8 4 2 
 
 23. 
 
 1.25+ .5 a ^ .25 g- 2.375 
 .25 1.125 
 
 7 
 
 ' - 
 
 k-i 
 
 5 
 
 ;22- 
 
 -2^- 
 
 15 
 
 :2| = 
 
 
 ^-3 
 
 
 
 3 
 
 
 
 2a: 
 3 
 
 + 4 
 
 161 
 
 : — 
 
 — X 
 
 a-32 2a-3 _15-3a _3 _ lb^-x x 
 
 2 4 ~ 8 ' ^^- 2 32' 
 
 31. J[c2- i(c2_3) + 6c-7] = 9fi. 
 
 - (i-0(i-^^)-(i-'^)(3-^^)=-^- 
 
 95. Equivalent equations, (i) Two equations are said to 
 be equivalent if every root of either is a root of the other 
 also. Thus, the several equations in the solution of Ex. 1, 
 § 94, are equivalent ; each has the root 24, and that only. 
 
 The method of solution used in Ex. 1, § 94, consists (1) in 
 deducing from the given equation, by means of the axioms, 
 a succession of new and simpler equations, and (2) in finding 
 the root of the last and simplest of tliem all. 
 
128 Jliail SCHOOL A LG Eli HA [Ch. IX 
 
 That the root of this last equation (24, in this case) 
 happens to be a root of the given equation also is due to the 
 fact that applying the axioms to equations usually produces 
 equivalent equations. 
 
 (ii) Although the axioms are correct, their application to 
 equations does not always result in equivalent equations. 
 
 E.g., given the equation 3 a; — 4 = 1. 
 
 Multiplying both members (Ax. 3) by x — 2, we obtain 
 
 3a---10aj+ 8 = a;-2. 
 
 Simplifying, 3a^- 11 a; + 10 = 0, » 
 
 i.e., (a;-2)(3aj-5) = 0; 
 
 whence (§ 72), aj = 2 or a; = f ; 
 
 but 2 is not a root of the given equation. 
 
 The axioms must, tlierefore, be used with caution, and 
 results should always be checked. 
 
 (iii) The following changes in equations will, however, 
 always produce equivalent equations (cf. El. Alg. p. 143). 
 
 (1) Transposing and uniting terms (Axioms 1 and 2). 
 
 (2) Multiplying and dividing by any expression whicli is 
 not zero, and which does not contain the unknown number 
 (Axioms 3 and 4). 
 
 96. Literal equations. Literal equations in one unknown 
 number, and of the first degree, may evidently be solved by 
 the method already employed for numerical equations. 
 
 E.g., given the equation — — = 3; to find x. 
 
 hah 
 
 On multiplying through by ah, to clear of fractions, the given 
 equation becomes 
 
 ax-hx-2W = a^-Zah. [Ax. 3 
 
 Hence ax — hx = a- — 'S ah -\-2 h^, [Ax. 1 
 
 i.e., {a — h)x = or — 3 a6 + 2 ft^j 
 
 and, therefore, x = ^'-3a6 + 2 h^ r^^ ^ 
 
 a — b 
 = a-26. 
 
96-96] SIMPLE EQUATIONS 129 
 
 Check. On substituting a — 2 b for x in the given equation, 
 we obtain 
 
 a — 2b a — 2b-\-2b ^a g 
 b a b ' 
 
 in which the first member readily simplifies, and becomes - — 3; 
 hence a — 2 6 is a root of the given equation. 
 
 EXERCISE LXV 
 
 Solve, and check as the teacher directs : 
 1.3cx — cl = 2d. 11 i_l_i 
 
 2. (a-b)x = 3b-Sa. * e / 
 
 3. 2 ax = 0? — ex. 12_ _1 i_ _- J 
 
 4. «6- + c.s = a2-c2. ^-/ ^+/ 
 
 c 7 ^ 
 
 a 
 
 ^ -|^ 14. (6 — c)aj — (a4-&)a; = c? — 20. 
 
 ^' d^^~ ■^^d' 15. ^^^lA^4.^ = 16 + i^. 
 7. z-Saz=(l-Say. b a a 
 
 a cx-c' + id' = 2dx. 16- 3d(aj + 3cd)=c(c2-a^). 
 
 9. 
 
 cz + d z + 9cd 17. ^-2a&_-^^^-3c 
 
 3(^ 
 
 ab 
 
 10. n^a;-a; = n-a;-e. la ^nl^ + ^ ^ ^^!±iA\ 
 
 2 6 a a6 
 
 19. Solve Ex. 1 for c ; for d. Similarly, solve Exs. 2, 5, 9, and 
 11 for each letter in turn, and check your results. 
 
 Solve the following equations : 
 
 20. b(c — x)-\-a(b — x) — b(b — x) = 0. 
 
 21. ah + bh = 3ab-3 s{a'b + ab""). 
 
 22. (a —b)(x — c) — (b — c)(x — a) = (c — a)(a; — &), 
 
 ^^. — 1 1 = 7H 1 r-i- 
 
 b a c b c a 
 
 24. What is a literal equation ? a numerical equation ? To 
 which class does 2 u; — 13 + ax = 14 a? belong ? 
 
130 HIGH SCHOOL ALGEBRA [Ch. IX 
 
 25. Each member of the equation m- — 5 m — 24 = — 3 (m — 8) 
 is divided by m — 8 ; are the quotients equal ? Why ? Show- 
 that the new equation is not equivalent to the given equation. 
 
 97. Fractional equations. Equations containing expres- 
 sions that are fractional in the unknown number (§ 34) are 
 called fractional equations. The methods already employed 
 apply to such equations also. 
 
 3 15 1 
 
 Ex. 1. Given the equation - — - = [■-'•> to find the value of x. 
 
 X 2 3x 6 
 
 Solution. Clearing of fractions by multiplying each member 
 by 6 Xj the L. C. M. of the denominators, we obtain 
 
 18-3x = 10 + », 
 whence x=2; 
 
 moreover, this value of x checks, hence 2 is a root of the given 
 equation. 
 
 Ex.2. Given = — ? — ; to find a;. 
 
 2(a;-l) 7(x + l) x + l 7(0^2-1)' 
 
 Solution. On multiplying each member by 2 • 7 (a;-|-l) • {x—1), 
 
 to clear the equation of fractions, we obtain 
 
 3-7(a; + l)-2(a;-l) = 8-2.7 (x-l)-20, 
 
 le., 21a; + 21-2a; + 2 = 112a;-112-20, 
 
 whence ^ = |> 
 
 which checks, and is, therefore, a root of the given equation. 
 
 7 af — 1 
 
 Ex. 3. Given - + — = x: to find x. 
 
 6 ar — 1 
 
 Solution. On multiplying this equation by 6 (a^ — 1), to clear 
 of fractions, we obtain 
 
 7 (x"" -1)+ 6 (x^ -1) = 6 x(a^ -1), 
 i.e., 7x^-7-\-6x^-6 = 6a^-6x, 
 
 whence 7 a^ + 6a;-13 =0, 
 
 i.e., (a;-l)(7a;H-13) = 0, 
 
 and the roots of this equation are 1 and — J/. [§ 72 
 
 On substituting these values of x in the given equation, it is 
 found that — -y- checks, but that 1 does not check ; hence — 4^ is 
 (and 1 is not) a root of the given equation. 
 
96-97] SIMPLE EQUATIONS 131 
 
 Note. This shows that clearing an equation of fractions may introduce 
 extraneous roots, i.e., roots which do not belong to the given equation. 
 
 ~3 _ 1 
 
 In this example the fraction might have been reduced to its lowest 
 
 x'^ — 1 
 
 terms before clearing the equation of fractions. In that case the multiplier 
 
 6(x + 1), instead of 6(x^ — 1), would have sufficed to clear of fractions; 
 
 the unnecessary factor x — 1 brought in the extraneous root 1. 
 
 No extraneous roots are brought into a fractional equation unless an 
 
 unnecessary factor is used in clearing of fractions (cf. El. Alg. § 99). Such 
 
 roots, if introduced, are always discovered in checking. 
 
 EXERCISE LXVI 
 
 Solve and check the roots : 
 
 4 0^-3 a;-f >^ ^ a; + 26 ^ _3 ?_ =0 
 
 73 6 ' '.T + 1 x-l~ ' 
 
 ic-l,ir-2 11-13a; o^/-l i 1 
 
 23 12 y + 1 y 
 
 X 16 8a; 10 ^y by 
 
 10. ^(2-a;)-|(3-2a:) = ^±i^. 
 
 11. -M_H--^_=-A_ + 3. 
 x^ —1 x—1 x-\-l 
 
 12. Clearing Ex. 11 of fractions, we obtain a;^ — 2a;— 3 = 0; 
 are the roots of this equation, viz., 3 and — 1, roots of the given 
 equation also (cf. Ex. 3, Note) ? 
 
 13. Define a fractional equation. Which of the equations in 
 Exs. 4-11 are fractional ? Explain. 
 
 Solve the following, and check as the teacher directs: 
 10a;4-17 5a;-2 12a;-l 
 
 14. 
 
 18 9 11 a; -8 
 
 Hint. Multiply both members by 18, and combine similar terms ; then 
 multiply both members of the resulting equation by 11 aj — 18. 
 
 5 (^-4-4) 7 a;- 3 ^ 3 (3 a; + 1) 
 11 3a;H-2 22 
 
132 HIGH SCHOOL ALGEBRA [Ch. I.\ 
 
 3-2/ 8^ 2/ + 3 8(2/ + 3) 
 
 17 g-5 g-lQ ^g-4 g-9 
 
 '^ + 5 2; + 10 2 + 4 ;2 + 9' 
 
 Hint. Simplify each member before clearing of fractions. 
 lo> — — — ——^— • 
 
 x + 2 x + 'd x-\-Q> x-^7 
 
 19 ^~^ I -'^ — '^ _ ^ — 5 , g; — 3 
 03 — 2 0?- 8 x — 6 X— 4: 
 
 0^ + 2 a^-2 ^ 10-2a^ 2a; + l 8 ^ 2a;-l 
 
 ^°* oj + l ic-1 x'-l ' ' 2x-l 4.0^-1 2x4-1' 
 
 21 1 + ^_ = ^±^. 25 ^-^II^-?: = 0. 
 
 • 2 2(07 + 1) cc + 6 X bx a 
 
 2 s 5 s- 3 1 ^^ 2c , 6 c(a-2a;) 
 
 22 — • =-• 26. \--=— -• 
 
 3 10s2_l 3 a x a{2-x) 
 
 23. 1 ^ 6 (1 - a?) ^ - g; ^7. ^ + ^a^ =^-^. 
 
 1 — x X x — 1 ' x^ — cx-^ax — ac c — x 
 
 3v 15 ^ 10 g 
 
 v + l 3v2 + v-2 3v-2 
 
 2 5 m ^ m + 29 _ ^ 
 
 • ^_5 3m + 2 (m-5)(3m+2) 
 
 ^^ 07 + 7 a, x — a x-^7 a a — x 
 30. — --: h — 
 
 aj + 6a x — 3a x-\-a 2a-\-x 
 
 a(b — x) o{c — x) a {c — x) 
 
 17+3 ,^18 21 _^ 100^5 
 
 „ « . X X X 3 
 
 ^^- -§- + — 5- -9- + -T5— 
 
 33. If C represents the circumference of a circle whose radius 
 n 
 is Rf then - — = 7r(cf. Ex. 37, p. 67) ; solve this equation for O; 
 2 7? 
 
 for i? . Taking tt = 3|, find the value of it when 0—56. 
 
34. v = -- 
 
 t 
 
 37. II'=-ii'.s'. 
 
 35. at = v. 
 
 38 F^^". 
 
 36. Z> = f 
 
 39. V =u — gt. 
 
 
 PROBLEMS 
 
 07] SIMPLE EQrATlONS 183 
 
 Solve each of the following equations for each letter it contains: 
 
 40. i(F-32) = a 
 
 41. s = ig(2t-l). 
 
 42. - + -=-• 
 ^ J?' / 
 
 1. Three fourths of a certain number exceeds | of it by 25. 
 What is the number ? 
 
 2. The sum of a certain number, its half, and its third is 
 .36. Find the number. 
 
 3. If f of a certain number diminished by J of that number 
 equals 3 more than i of the number, what is the number ? 
 
 4. The sum of two numbers is 18, and the quotient of the less 
 divided by the greater is \. What are the numbers ? 
 
 5. Divide the number 32 into two parts such that -^ of the 
 larger shall equal ^ of the smaller. 
 
 6. Divide the number 80 into two parts such that -| of the 
 smaller shall exceed ^ of the greater by 2. 
 
 7. Divide the number 25 into two parts such that the square 
 of the greater shall exceed the square of the smaller by 75. 
 
 8. Wliat number must be added to each term of the fraction 
 YY so that the resulting fraction shall be equal to J? 
 
 9. If a certain number is added to, and also subtracted from, 
 each term of the fraction |, the first result exceeds the second 
 by i; find the number. How many solutions has this problem ? 
 
 10. B's present age is 18 years, which is | of A's age ; after 
 how many years will B's age be | of A's age ? 
 
 11. The combined cost of a table and a chair is $ 11, of the 
 table and a picture, $ 14, and the chair and the picture together 
 cost 3 times as much as the table. What is the cost of each ? 
 
 12. Divide a line 28 inches long into two parts such that the 
 length of one part shall be | that of the other. 
 
184 HIGH SCHOOL ALGEBRA [Ch. IX 
 
 13. A field is twice as long as it is wide, and increasing its 
 length by 20 rods and its width by 30 rods would increase its 
 area by 2200 square rods. What are the dimensions of this field 
 (cf. Exs. 23-24, p. 65) ? 
 
 14. An orchard has twice as many trees in a row as it has 
 rows. By increasing the number of trees in a row by 2, and the 
 number of rows by 3, the whole number of trees will be increased 
 by 126. How many trees are there in the orchard ? 
 
 15. An officer in forming his soldiers into a solid square, with 
 a certain number on a side, finds that he has 49 men left over ; 
 and if he puts one more man on a side, he lacks 50 men of com- 
 pleting the square. How many men has he? 
 
 16. A boy was engaged at 15 cents a day to deliver a daily 
 paper, with the added condition, however, that he was to forfeit 
 5 cents for every day he failed to perform this service ; at the 
 end of 60 days he received $ 7. How many days did he serve ? 
 
 17. A man was hired for 30 days on the following terms : for 
 every day he worked he was to receive $ 2.50 and board ; for 
 every day he was idle he was to receive nothing, and was to pay 
 75 cents for board. If his total earnings were $49, how many 
 days did he work ? 
 
 18. The square of a certain number is diminished by 9, and the 
 remainder is divided by 10, giving a quotient w^hich is 3 greater 
 than the number itself. Find the number (two solutions). 
 
 19. If a certain number is subtracted from each of the four 
 numbers 20, 24, 16, and 27, the product of the first two remain- 
 ders equals the product of the second two. What is the number ? 
 
 20. Find a fraction whose numerator is greater by 3 than one 
 half of its denominator, and whose value is f . 
 
 21. The numerator of a certain fraction is less by 8 than its 
 denominator, and if each of its terms is decreased by 5, its value 
 will be i ; what is the fraction ? 
 
 22. What principal at 4% interest for 3 years amounts to 
 f 784 (cf. Ex. 12, p. 61) ? Solve the same problem if the amount 
 is $ 10,140. 
 
97] SIMPLE EQUATIONS 135 
 
 23. I invest $6000, part at 6 %, part at 5%, thus securing 
 a total yearly income of $ 325 ; how large is each investment? 
 
 24. A gentleman made two investments amounting together 
 to $ 4330 ; on one he lost 5 % , on the other he gained 12 % . If 
 his net gain was $ 251, how large was each investment ? 
 
 25. In a certain quantity of gunpowder, made up of saltpeter, 
 sulphur, and charcoal, the saltpeter weighs 6 lb. more than i of 
 the whole, the sulphur 5 lb. less than ^ of the whole, and the 
 charcoal 3 lb. less than ^ of the whole. How many pounds of 
 each constituent does this gunpowder contain ? 
 
 26. A boy bought some apples for 24 cents; had he received 
 4 more for the same sum, the cost of each would have been 1 
 cent less. How many did he buy ? 
 
 27. Knowing the time consumed by an automobile in making 
 a run of a given number of miles, how can you find the average 
 speed ? How, from the distance and the rate, can you find the 
 time ? How, from the rate and the time, can you find the dis- 
 tance ? Illustrate your answers (cf . Exs. 15-16, p. 61). 
 
 28. A tourist ascends a certain mountain at an average rate 
 of 1^ miles an hour, and descends by the same path at an aver- 
 age rate of 4^ miles an hour. If it takes him 6J hours to make 
 the round trip, how long is the path (cf . Exs. 35-36, p. 67) ? 
 
 29. A north-bound and a south-bound train leave Chicago at 
 the same time, the former running 2 miles an hour faster than 
 the latter. If at the end of 1|- hours the trains are 141 miles 
 apart, find the rate of each. 
 
 30. In running 180 miles, a freight train whose rate is f that 
 of an express train takes 2 hours and 24 minutes longer than the 
 express train. Find the rate of each. 
 
 31. If the freight train of Ex. 30 requires 6 hours longer than 
 the express train to make the run between Buffalo and New York, 
 how far apart are these two cities ? 
 
 32. An express train whose rate is 40 miles an hour starts 
 1 hour and 4 minutes after a freight train and overtakes it in 
 1 hour and 36 minutes. Find the rate of the freight train. 
 
136 HIGH SCHOOL ALGEBRA [Ch. IX 
 
 33. An automobile runs 10 miles an hour faster than a bicycle, 
 and it takes the automobile 6 hours longer to run 255 miles than 
 it does the bicycle to run 63 miles. Find the rate of each. 
 
 How many solutions has the equation of this problem ? Is 
 each of these also a solution of the problem itself ? 
 
 34. A steamer now goes 5 miles downstream in the same time 
 that it takes to go 3 miles upstream, but if its rate each way is 
 diminished by 4 miles an hour, its downstream rate will be twice 
 its upstream rate. What is its present rate in each direction ? 
 
 35. A steamer can go 20 miles an hour in still water. If it 
 can go 72 miles with the current in the same time that it can 
 go 48 miles against the current, how swift is the current? 
 
 Hint. Let x = the rate of the current (in miles per hour) ; then 20 — jc = 
 the steamer's rate upstream, and 20 + x its rate downstream. (Why ?) 
 
 36. A man rows downstream at the rate of 6 miles an hour, 
 and returns at the rate of 3 miles an hour. How far downstream 
 can he go and return if he has 2^ hours at his disposal ? At 
 what rate does the stream flow ? 
 
 37. At what time between 2 and 3 o'clock are the hands of a 
 clock together ? 
 
 Hint. Make drawing, or use model of clock face. Let x = the number 
 of minute spaces over which the minute hand passes after 2 o'clock before 
 the two hands come together ; then — = the number of minute spaces over 
 
 which the hour hand passes in the same time (why?); and ic = -^ + 10. 
 (Why?) ^^ 
 
 38. At what time are the hands of a clock together between 
 8 and 9 ? between 5 and 6 ? 6 and 7 ? 11 and 12 ? 
 
 39. At what time between 3 and 4 o'clock is the minute hand 
 15 minute spaces ahead of the hour hand ? 
 
 40. At what time do the hands of a clock extend in opposite 
 directions between 4 and 5 ? between 2 and 3 ? 7 and 8 ? 
 
 41. The tens' digit of a certain two-digit number is ^ the units' 
 digit, and if this number, increased by 27, is divided by the sum 
 of its digits, the quotient will be 6^1^. What is the number 
 (cf . Prob. 4, p. 64) ? 
 
07] SIMPLE EQUATIONS 137 
 
 42. Divide 72 into four parts, such that if the first is divided 
 by 2, the second multiplied by 2, the third increased by 2, and the 
 fourth diminished by 2, the results will all be equal. 
 
 43. M can do a certain piece of work in 8 days, and N can do 
 it in 12 days ; in how many days can the two do it when working 
 together (cf. Ex. 41, p. 67)? 
 
 44. Two plasterers, A and B, working together, can plaster a 
 house of a certain size in 12 days, while A, working alone, can 
 plaster such a house in 18 days. In how many days can B alone 
 do the work ? 
 
 45. A reservoir is fitted with three pipes, one of which can 
 empty it in 4 hours, another in 3 hours, and the third in 1 J hours. 
 If the reservoir is half full, and the three pipes are opened, in 
 what time will it be emptied ? 
 
 46. The first of three outlet pipes can empty a certain cistern 
 in 2 hr. and 40 min., the second in 1 hr. and 15 min., and the 
 third in 2 hr. and 30 min. If the cistern is f full, and all three 
 pipes are opened, in what time will it be emptied ? 
 
 47. A can do a piece of work in 6 days, and B can do it in 14 
 days. A, having begun this work, had later to abandon it ; B took 
 his place and finished the work in 10 days from the time it was 
 begun by A. How many days did B work? 
 
 48. A certain number is increased by 1, and also diminished 
 by 1 ; it is then found that twice the reciprocal of the second 
 result minus 3 times the reciprocal of the first result equals \. 
 What is this number ? How many solutions has this problem ? 
 
 49. A picture whose length lacks 2 inches of being twice its 
 width is inclosed in a frame 4 inches wide. If the length of the 
 frame divided by its width, plus the length of the picture divided 
 by its width, is 3^, what are the dimensions of the picture ? How 
 many solutions has the equation of this problem? Is each of 
 these a solution of the problem also ? 
 
 50. A gentleman invested ^ of his capital in 4% bonds 
 (i.e., bonds yielding 4 % interest per annum), f of it in S^ % 
 bonds, and the remainder in 6 % bonds, purchasing all these bonds 
 at par. If his total annual income is $ 3412.50, find his capital. 
 
138 HIGH SCHOOL ALGEBRA [Ch. IX 
 
 51. At what time between 9 and 10 o'clock is the hour hand 
 20 minute spaces in advance of the minute hand? 
 
 52. A pedestrian finds that his uphill rate of walking is 3 miles 
 an hour, and his downhill rate 4 miles an hour. If he walked 60 
 miles in 17 hours, how much of this distance was uphill ? 
 
 53. A wheelman and a pedestrian start at the same time for a 
 place 54 miles distant, the former going 3 times as fast as the 
 latter ; the wheelman, after reaching the given place, returns and 
 meets the pedestrian 6| hours from the time they started. At 
 what rate does each travel ? 
 
 54. In a mixture of water and listerine containing 21 ounces 
 there are 7 ounces of listerine. How much listerine must be 
 added to make the new mixture J pure listerine ? 
 
 Hint. Let x = the number of ounces of listerine to be added. Then 
 
 7 +x 
 
 (Wliy?) 
 
 21 +rc 4 
 
 55. In an alloy of silver and copper weighing 90 oz. there are 
 6 oz. of copper ; find how much silver must be added in order 
 that 10 oz. of the new alloy shall contain but f oz. of copper. 
 
 56. If 80 lb. of sea water contains 4 lb. of salt, how much 
 fresh water must be added in order that 45 lb. of the new solu- 
 tion may contain 1 J lb. of salt ? 
 
 57. If a mixture of water and alcohol is y% pure alcohol, how 
 much water must be added to one gallon of the mixture to make 
 a new mixture ^ pure alcohol ? 
 
 58. Solve Prob. 57 if the given mixture is 80 % pure alcohol 
 and the required mixture 50 % pure alcohol. 
 
 59. How much alcohol must be added to one gallon of a mixture 
 40 % pure to make a new mixture 75 % pure ? 
 
 60. What fractional part of a 6 % solution of salt and water 
 (salt water of which 6 % by weight is salt) must be allowed to 
 evaporate in order that the remaining portion of the solution may 
 contain 12 % of salt ? that it may contain 8 % of salt ? 10 % ? 
 
07-98] SIMPLE EQUATIONS 139 
 
 61. A physician having a 6% solution of a certain kind of 
 medicine wishes to dilute it to a 3| % solution. What percent- 
 age of water must he add to the present mixture ? 
 
 62. If the specific gravity of brass is 8^,* while that of iron is 7|-, 
 and if, when immersed in water, 57 lb. of an alloy of brass and iron 
 displaces 7 lb. of water, find the weight of each metal in the alloy. 
 
 63. If, on being irtimersed in water, 97 oz. of gold displaces 
 5 oz. of water, and 21 oz. of silver displaces 2 oz. of water, how 
 many ounces of gold and of silver are there in an alloy of these 
 metals which weighs 320 oz. and which displaces 22 oz. of 
 water ? Find the specific gravity of the alloy ; also of gold. 
 
 98. General problems. Formulas. Interpretation of results. 
 
 A problem in which the known numbers are represented by 
 letters, instead of by arithmetical numerals, is often called 
 a general problem ; it includes all those particular problems 
 which may be obtained by giving particular values to these 
 letters. Some problems of this kind are given below. 
 
 Prob. 1. A yacht was chartered for a pleasure party of 12, the 
 expense to be shared equally ; 3 members of the proposed party 
 being unable to go, the share of each of the others had to be 
 increased by $ 2. How much was paid for the yacht ? How 
 much was each to pay under the original arrangement ? 
 
 SOLUTION 
 
 Let X = the number of dollars each member was to have paid, 
 then x-\-2 = the number of dollars each participant did pay ; hence 
 12 X and 9 (a? -f 2) each represent the number of dollars charged 
 for the yacht ; 
 
 therefore 12 a; == 9 (a; + 2), 
 
 i.e.y 12 a; = 9 a; + 18, 
 
 and therefore • x = 6j and 1 2 a; = 72 ; 
 
 hence the amount each was to have paid is $ 6, and the rental price 
 of the yacht is $ 72. 
 
 * This means that a given volume of brass weighs 8f times as much as an 
 equal volume of water. 
 
 HIGH SCH. ALG. — 10 
 
140 BIGH SCHOOL ALGEBRA [Cii. IX 
 
 Prob. 2. Substitute p, q, and d, for 12, 3, and 2, respectively, 
 in Prob. 1, and solve the problem thus formed. 
 
 SOLUTION 
 
 Let X = the number of dollars each member was to have paid, 
 then x-\-d = the number of dollars each participant did pay; hence 
 2}x and (/) — q) - {x-\-d) each represent the number of dollars 
 charged for the yacht ; 
 therefore px =z {p — q) {x -\- d)= px -\- pd — qx — qd ; 
 
 whence x = ^^^ ~ -^^ , the amount each was to pay, 
 
 and px=:p • ^^~^\ the rental price of the yacht. 
 
 Remark. The solutions of Probs. 1 and 2 are alike except 
 in this: In the solution of Prob. 1 the numbers given in that 
 problem (12, 3, and 2) have, by combining, completely lost their 
 identity before the result is reached ; but in the solution of Prob. 
 2 the given numbers {p, q, and d) preserve their identity to the end. 
 
 For this reason the result in Prob. 2 may be used as a formula, 
 by means of which the answer to Prob. 1, or to any like problem, 
 may be immediatel}^ written down. 
 
 E.g., substituting 12, 3, and 2 for p, q, and d respectively, in 
 the solution of Prob. 2, gives the answer to Prob. 1. 
 
 The solution of Prob. 2, therefore, includes that of Prob. 1. 
 The first problem, and all like numerical problems, are merely 
 particular cases of the second, which is called a general problem. 
 
 Prob. 3. Divide m golf balls into two groups, in such a way 
 that the first group shall contain n balls more than the second. 
 
 Solution. Let a? = the number of balls in the first group. 
 Then m — x = the number of balls in the second group, 
 
 and, therefore, by the condition of the. problem, 
 x = m — x-\-n] 
 
 whence x = -, the number in the first group, 
 
 2 
 
 m — x = m 
 second group, 
 
 and m — x = m ^~- = — — ^, the number in the 
 
98] SIMPLE EQUATIONS 141 
 
 As in Prob. 2, so here, the general solution may be employed to solve any 
 particular problem of the same kind. For example, if m = 30 and n = 4, 
 
 then the two groups contain, respectively, — ^^ and — ^^^— balls, i.e., 17 and 
 
 18 ; while, if w = 10 and w = 2, then the two groups contain 6 and 4 balls, 
 respectively. 
 
 If, however, m = 10 and n = 14, then the number of balls in the two groups, 
 
 as given by the above solution, is — ^ — and - — ^^- — , respectively, i.e., 12 
 
 and — 2 ; but since there cannot be an actual group containing — 2 golf 
 balls, therefore this last problem is impossible, and the impossibility is indi- 
 cated by the negative result. 
 
 Eemark. Some problems admit of negative results, and some 
 do not, just as some problems admit of fractional results, while 
 others do not. The nature of the things with which any particu- 
 lar problem is concerned will always make it evident whether or 
 not fractional or negative solutions are admissible. 
 
 Prob. 4. Two boys, A and B, are running along the same 
 road, A at the rate of a, and B at the rate of h, yd. per minute ; 
 if B is m yd. in advance of A, and if they continue running at the 
 same rates, in how many minutes will A overtake B ? 
 
 Solution. Let i)j = the number of minutes that must elapse 
 before A overtakes B. Then by the conditions of the problem, 
 ax = hx 4- m, 
 
 whence x = , the number of minutes before 
 
 A overtakes B. ^ ~ 
 
 As in the two previous problems, so here, the general solution may be 
 employed to solve any particular problem of the same kind. 
 
 QO 
 
 E.g.,\ia = 280, h = 270, and m = 90, then x = — = 9; i.e., A will 
 
 '' ' ' ' ' 280 270 
 
 overtake B in 9 minutes. 
 
 Again, if a = 280, b = 280, and m = 90, then x = = '— ; i.e., an 
 
 '280 - 280 
 
 infinite number of minutes will elapse before A overtakes B ; in other words, 
 
 A will never overtake B. Compare § 41 (iii), also Ex. 7, p. 53. 
 
 QO 
 
 But if a =280, b = 290, and m = 90, then x = — = -9; i.e., the 
 
 280 - 290 
 
 two boys are together — 9 minutes from the moment they were observed, 
 
 i.e., the two boys loere together 9 minutes ago. 
 
 Let the pupil show that this interpretation of the negative result accords 
 
 fully with the physical cuuditions of the problem. 
 
142 HIGH SCHOOL ALGEBRA [Ch. IX 
 
 Prob. 5. The present ages of a father and son are respectively 
 50 and 20 years ; after how many years will the father bfe 4 times 
 as old as the son ? 
 
 Solution. Let x = the number of years from now to the time 
 when the father's age shall be 4 times that of the son. Then, by 
 the conditions of the problem, 
 
 50 + a; = 4(20 + a;), 
 whence a? = — 10. 
 
 This means that 10 years ago the father's age was 4 times the 
 son's. 
 
 N.B. The general problem of which Prob. 5 is a particular case, may be 
 stated thus : The present ages of a father and son are, respectively, m and n 
 years ; after how many years will the father be p times as old as the son ? 
 
 EXERCISE LXVII 
 
 6. The sum of two numbers is a, and the larger exceeds the 
 smaller by b. What are the two numbers ? 
 
 7. By substituting in the formula obtained from the solution 
 of Prob. 6 above, solve Probs. 6' and 7, p. 64. Could Prob. 16, 
 p. 65, be solved by means of the same formula ? 
 
 8. Is Prob. 9, p. 64, a particular or a general problem ? Why ? 
 Make a general problem which shall include this one as a par- 
 ticular case. Solve the new problem and thus find a formula by 
 which Prob. 9, p. 64, may be solved. 
 
 9. Answer the questions in Ex. 8 above, supposing them to 
 have been asked with regard to Probs. 4 and 12, p. 133. 
 
 10. Which of the following admit of fractional results : Probs. 
 14, 15, 18, p. 134; Probs. 24-26, p. 135? 
 
 11. Do any of the problems mentioned in Prob. 10 above admit 
 of negative results ? Explain. 
 
 12. By a slight change in the wording of Prob. 5 above^ make 
 an equivalent problem whose answer shall be positive. This an- 
 swer should agree with the interpretation of the negative result 
 oriven in Prob. 5. 
 
98] SIMPLE EQUATIONS 148 
 
 13. By slightly changing the wording in the last particular 
 case under Prob. 4 above, make an equivalent problem whose 
 answer shall be positive. 
 
 14. What principal at c % for t years will earn i dollars simple 
 interest ? By substituting in your answer, find the principal 
 when c = 5, I = 270, i = 3 ; also, when c = 3i, i = 224, t = 8. 
 
 15. A father is now m times as old as his son; in p years, the 
 father's age will be n times that of the son. Find the present 
 age of each. Also interpret your result when m is less than n. 
 Is p positive or negative in this case ? 
 
 16. Solve the equation of Prob. 2 above for d, and then find 
 the value of d corresponding to p = 12, g = 2, ic = 4. May d be 
 fractional in value ? negative ? Explain. 
 
 17. M can do in a days a piece of work which N can do in 6 
 days. In how many days can they do it when working together ? 
 Use this answer to solve Prob. 43, p. 137. 
 
 18. A merchant has two kinds of sugar worth, respectively, a 
 and h cents a pound. How many pounds of each kind must he 
 take to make a mixture of n pounds worth c cents a pound ? 
 
 19. How many solutions has Prob. 18 if a = 5 = c? ifa = 6 
 while c differs from a ? Does the answer to Prob. 18 show these 
 facts [cf. § 41 (iii) and (iv)] ? 
 
 20. An alloy of two metals is composed of m parts (by weight) 
 of one to n parts of the other. How many pounds of each of the 
 metals are there in a pounds of the alloy ? 
 
 21. A bell made from an alloy of 5 parts (by weight) of tin to 
 16 of copper, weighs 4200 lb. ; how many pounds of tin and of 
 copper in the bell ? How is Ex. 22 related to Ex. 21 ? 
 
 22. At what time between n and n + 1 o'clock will the hands 
 of a clock be together ? By means of your answer write down 
 the answers to Prob. 38, p. 136. 
 
 23. At what time between n and w + l o'clock will the hands 
 of a clock be pointing in opposite directions if n is less than 6 ? 
 if n is greater than 6 ? if n equals 6 ? By means of your 
 answer write down the answers to Prob. 40, p. 136. 
 
CHAPTER X 
 
 SIMULTANEOUS SIMPLE EQUATIONS 
 
 I. TWO UNKNOWN NUMBERS 
 
 99. Indeterminate equations. A simple equation in one 
 unknown number has but one solution (i.e., one root, cf. 
 Chapter IX), but an equation that contains two or more un- 
 known numbers has many solutions. 
 
 E.g., in the equation 3x+2y = 6, which, when solved 
 
 for y, becomes ^ o 
 
 ^' b — ox 
 
 we see that if the values 1, 2, 3, — 1, etc., are assigned to x, then 
 // will take the corresponding values 1, — ^, — 2, 4, etc. That is, 
 this equation is satisfied by the pairs of numbers : 
 
 a; = 11. x = 2 1. » = 3 1. ^ = -11. ^^^ 
 
 An equation, such as the one just now considered, which 
 has an infinite number of solutions, is for that reason called 
 an indeterminate equation. 
 
 EXERCISE LXVm 
 
 By the method of § 99 find five solutions of each of the fol- 
 lowing equations : 
 
 1. a; + 3 2/ = 7. 3. 5 a; + 3?/ = 11. 5. 2^y = 5 + 32;. 
 
 2. a; 4- 2/ = 5. 4. 5 m + 2 ti = 15. 6. v — vj — 1. 
 
 7. How many solutions has each of the above equations ? 
 Why ? What are such equations called ? 
 
 8. How many positive integral solutions (i.e., solutions in 
 which both x and y are positive integers) has the equation 
 3a; + 22/=ll? 
 
 Hint. Solve the equation for y, and thus show that x cannot exceed 3. 
 
 144 
 
99-100] SIMULTANEOUS SIMPLE EQUATIONS 145 
 
 9. By the method of Ex. 8 find four positive integral sohi- 
 tions of the equation 2x-\-y = 9. How many such solutions has 
 this equation ? 
 
 10. If possible, find positive integral solutions of the equations 
 in Exs. 1-6 above. 
 
 Show that the following have no positive integral solutions : 
 
 11. 2x-4:y = l. 12. 3x + 6y = 5. 13. 9x-{-3y = 17. 
 
 14. Find three solutions of the equation 2x — 5y + Sz = 6', 
 also, three solutions of the equation 2 x + 3y -\-4:Z = 20. 
 
 15. A farmer spent $22 in purchasing two kinds of lambs, the 
 first kind costing him $3 each, and the second kind $5 each. 
 How many of each kind did he buy ? 
 
 Hint. Let X = the number of the first kind, and y = the number of the 
 second kind ; then Sx -\- 5y = 22, where x and y are positive integers. 
 
 16. A man spends $300 for cows and sheep costing, respec- 
 tively, $4:5 and $6 a head; how many of each does he buy? 
 
 17. In how many ways may a 19-pound package be weighed 
 with 5-pound and 2-pound weights ? 
 
 18. How many pineapples at 25 cents each, and watermelons 
 at 15 cents each, can be purchased for $2.15? 
 
 19. Divide a line which is 100 feet long into two parts, one of 
 which shall be a multiple of 11 feet, the other of 6 feet. 
 
 20. Find the least number which when divided by 4 gives a 
 remainder of 3, but when divided by 5 gives a remainder of 4. 
 
 100. Simultaneous equations. Independent equations.* 
 
 The equations Sx-\-2i/ = 5 
 
 and x—2y = l^ 
 
 have, individually, an infinite number of solutions (cf . § 99) ; 
 
 they also have 07ie solution, viz., x=3 and «/= —2, in common ; 
 
 I.e., these values of x and y satisfy each of the given equations. 
 
 xA set of equations, like those above, having one or more 
 
 * If time permits, read §§ 137-140, also § 142, in connection with §§ 100-101. 
 This plan will make the definitions, and also the operations, more concrete. 
 
146 HIGH SCHOOL ALGEBRA [Ch. X 
 
 solutions in common, is usually called a system of simul- 
 taneous equations. 
 
 Simultaneous equations are often called consistent equa- 
 tions, while two equations which have no solution in com- 
 mon are called inconsistent equations. Thus, x-\-^ = 4: and 
 
 2 a; -f 2 ^ = 9 are inconsistent equations. 
 
 Two or more equations, no one of which can be derived 
 from the others, are called independent equations. Thus, 
 
 3 a?+^ = 11 and 7 x — i/=9 are independent ; but 3 a;-|-^ = ll 
 and 6 a; H- 2 ?/ = 22 are not independent, the second being 
 obtained by multiplying each member of the first by 2. 
 
 101. Solving simultaneous equations. The solving of a 
 system of simultaneous equations is the process of finding 
 the solutions which these equations have in common. 
 
 x-\-y = 4., (1) 
 
 Ex. 1. Solve the equations , 
 
 ^ 'x-y = 2. (2) 
 
 Solution. Adding these two equations, member to member 
 (Ax. 1), gives 2i. = 6, 
 
 whence x = 3. 
 
 Substituting this value of x in Eq, (1) gives 
 
 whence 2/ = 1- 
 
 Moreover, these numbers, viz., x = S and y = l, when substi- 
 tuted in the given equations, check; therefore they constitute a 
 solution of these equations. 
 
 3x-\-2y = 26, (1) 
 
 Ex. 2. Solve the equations , 
 
 ^ '5x + 9y = SS. (2) 
 
 Solution. On multiplying both members of Eq. (1) by 5, and 
 of Eq. (2) by 3, these equations become, respectively, 
 
 15aj-f-10i/ = 130, (3) 
 
 15ic + 27?/ = 249; (4) 
 
 and (Ax. 2) subtracting Eq. (3) from Ex. (4) gives 
 
 17 2/ = 119, 
 whence y=T, 
 
100-102] SIMULTANEOUS SIMPLE EQUATIONS l-iT 
 
 Substituting this value of y in any one of the equations con- 
 taining both X and y gives 
 
 a; = 4; 
 and since these numbers, viz., x = 4 and y = 7, check, therefore 
 they constitute a solution of the given system of equations. 
 
 102. Elimination. Any process of deducing from two or 
 more simultaneous equations other equations which contain 
 fewer unknown numbers is called elimination. Such a process 
 eliminates (i.e., gets rid of) one or more of the unknown 
 numbers, and thus makes the finding of a solution easier. 
 
 That particular plan of elimination which was followed in 
 the examples given in § 101 is known as elimination by 
 addition and subtraction. It is evident, moreover, that this 
 method is applicable to any pair of such equations. The 
 procedure may be formulated thus : 
 
 (1) Multiply the given equations hy such numbers as will 
 make the coefficient of the letter to he eliminated the same (in 
 absolute value) in both equations. 
 
 (2) Subtract or add these last two equations (according as 
 the terms to be eliminated have like or unlike signs). 
 
 (3) Solve the resulting equation for the unknown number 
 which it contains. 
 
 (4) Substitute that value in any one of the earlier equations 
 and thus find the other unknown number o 
 
 (5) Check the results. 
 
 Note. Number (2) above is permissible only because the letters have the 
 same value in both equations (of. § 101). 
 
 EXERCISE LXIX 
 Solve each of the following systems of equations and check 
 the results : 
 
 {2x 
 
 4. 
 
 -2/ = 5, g ja; + 3y=ll, 
 
 H-32/ = 17. ' |3a;-4.y = 7. 
 
 = 15, 
 = 33, 
 
 |a;4-2?/ = 9, {2v-\-du 
 
 1 2 a; + 2/ = 15, ' [4^ + 9?^ 
 
148 HIGH SCHOOL ALGEBRA [Ch. X 
 
 3a; + 7?/ = 6. [12x-9y = 0. 
 
 2x-\-5y=:S, f5s + 6f = 17, 
 
 8. ^ _ _ . _ 11. 
 
 7 a: + 10 2/ =-17. [6.9 + 5^ = 16. 
 
 15 oj + 77 2/ = 92, f 4 m - 15 7i = 32, 
 
 9. i 12. ' 
 
 5aj-32/=2. [10m-9w = -34. 
 
 13. What is meant by saying that two equations are simul- 
 taneous ? consistent? inconsistent? independent? Show the 
 appropriateness of these terms. 
 
 14. If in two simultaneous equations the coefficients of the 
 letter to be eliminated are prime to each other (cf. Ex. 11), what 
 is the simplest multiplier for the first equation ? for the second ? 
 Answer the same questions when the coefficients under considera- 
 tion are 7iot prime to each other (cf. Ex. 12). 
 
 Solve the following systems of equations and check the results : 
 
 5p-{-3q=6S, {3m-2n = 7, 
 
 15. r. . ' 19. 
 
 2^ + 5^ = 69. [4 m — 7w=— 47. 
 
 22 .T- 8 ^ = 50, (4r-f 5s=-19, 
 
 26x-\-6y = 175. ' [2 r-\-3 s= -10^-^. 
 
 15 it- -f 14 ?/ = - 45, f 35 a; - 27 ?/ = - 19, 
 
 25 a^ - 21 2/ = - 75. [ 21 2/ -f- 40 a; = 82 
 
 18 t^ + 10 V = 59, f 28 a; - 23 y = 33, 
 
 18. <! _ _ _ 22. ' -^ 
 
 12w-15v = 28i. [63 a; -25 2/ = 199. 
 
 103. Other methods of elimination. Besides the method of 
 elimination described in § 102, there are several other methods 
 that serve the same purpose ; two of these, which are often 
 useful, will now be explained. 
 
 (i) Elimination by substitution. 
 
 3x-Ay = 7, (1) 
 
 Ex. 1. Solve the system of equations , ^ ^ ^ 
 
 ^ ^ '2x + 3y = 16. (2) 
 
102-108] SIMULTANEOUS SIMPLE EQUATIONS 149 
 
 SOLUTION 
 
 From Eq. (1) x = ^^^; [§ 99 
 
 o 
 
 on substituting this expression for cc, Eq. (2) becomes 
 
 2(^) + 32,= 16, (3) 
 
 whence (§ 94) 2/ = 2; 
 
 on substituting this vahie in either Eq. (1) or Eq. (2), we obtain 
 
 X = r). 
 Moreover, these values, viz., x = 5 and y = 2, check ; therefore, 
 they constitute a solution of the given system of equations. 
 
 This method of elimination is known as elimination by 
 substitution ; it is manifestly applicable to any such system 
 of equations as the above. 
 
 The student may solve, by this method, the system 
 
 I 8 M - 4 u = 19, 
 
 1 5 w + 2 V = 10, 
 being careful to check the result, and then write out a "rule" for applying 
 this method to all such exercises. 
 
 (ii) Elimination by comparison. 
 
 r. . . n . f3a;-42/ = 7, (1) 
 
 Ex. 2. Solve the system of equations < 
 
 -^ ^ \2x-{-3y = 16. (2) 
 
 SOLUTION 
 
 From Eq. (1), x = I^til, and from Eq. (2), x= l^Llzll . Now, 
 
 since a? is to have the same value in each of these equations, 
 therefore Tj^^lfi-l^. ^3^ 
 
 Solving Eq. (3) gives y = 2, 
 whence, substituting this value in either of the given equations, 
 
 X = 5. 
 
 Moreover, these values, viz., x = 5 and y = 2, check ; therefore, 
 they constitute a solution of the given system of equations. 
 
 This method of elimination is called elimination by com- 
 parison ; it is applicable to all such systems of equations. 
 
150 niGll SCHOOL ALGEBBA [Ch. X 
 
 The student may solve, by this method, the system 
 I 8r + 5s = 3, 
 ll2r-7s=48, 
 and then write out a " rule " for applying this method to all such exercises. 
 
 EXERCISE LXX 
 
 Solve the following systems of equations, using first elimina- 
 tion by substitution, and then that by comparison ; observe which 
 method is easier in the different exercises : 
 
 • 5a;-2^ = 10. 
 
 4. ^ - 7. 
 
 llx-lOy 
 
 = 14, 
 
 ox+1 y= 
 
 41. 
 
 21 2/ H- 20 a; 
 
 = 165, 
 
 772/-30«^ 
 
 = 295. 
 
 8 ^ - 10 y = 
 
 14, 
 
 6^ + 35 'y = 
 
 :41. 
 
 4x + 2/ = 34, 
 42/ + a^ = 16. 
 
 ^ |2a^ + 72/ = 34, 
 ■ l5a; + 92/ = 51. 
 
 9. Using the method of elimination by comparison, solve each 
 of the systems of equations in Exs. 3-6, p. 147. 
 
 10. Using the method of elimination by substitution, solve each 
 of the systems of equations in Exs. 7-10, p. 148. 
 
 11. Show that elimination by comparison is merely a special 
 case of elimination by substitution. 
 
 Solve Exs. 12-21 below ; use the simplest method of elimination 
 in each case, giving reasons for your choice of method ; and check 
 your results as the teacher directs : 
 
 7ic + 42/ = l, r8?/-21v = 5, 
 
 ±2. { ^ , ' 15. i ' 
 
 9a; + 42/ = 3. [6ii + 14'y = -26. 
 
 ^^ 3a; + 52/ = 19, ^^ |34a^-152/ = 4, 
 
 5a;-42/ = 7. * [ 51 a; + 25 2/ = 101. • 
 
 14. 
 
 f aj - 11 ?/ = 1, I 39 .'c - 1 5 2/ = 93, 
 
 jlll7/-9a? = 99. ^^' [6535 + 17 2/ = 113. 
 
103-104] SIMULTANEOUS SIMPLE EQUATIONS 
 
 151 
 
 18. 
 
 19. 
 
 19s + 85^ = 350, 
 17 s + 119 ^ = 442. 
 
 j8.9-llw = 0, 
 [258-17 w; = 139. 
 
 20. 
 
 21 
 
 I 
 
 3a;-ll^ = 0, 
 19a;-19 2/ = 8. 
 
 aj + 9 2/+42 = 0, 
 25 2/ + a; 4- 20 = 0. 
 
 104. Equations containing fractions. The solution of si- 
 multaneous equations containing fractions is illustrated by 
 the following examples : 
 
 -2 
 
 Ex. 1. Given 
 
 3 
 
 13__2/ 
 
 4-¥ = 4i 
 
 ; to find a? and y. 
 
 Solution. On multiplying these equations by 12 and 6, re- 
 spectively, and collecting terms, we obtain 
 
 4 a; + 3 2/ = 29, 
 and 3 ic + 4 2/ = 27 ; 
 
 whence (§ 101) x = 5 and y = S. 
 
 Moreover, when substituted in the given equations, these values 
 check ; they are, therefore, the solution of those equations. 
 
 Ex. 2. Given 
 
 3 s r r (? 
 
 1-5 = 0. 
 
 16 
 
 3s) 
 
 > ; to find r and s. 
 
 Solution. On multiplying these equations by r (r — 3 s) and 
 3, respectively, they become r + 4 (r — 3 s) == 16, 
 and r_3_3s = 0; 
 
 whence (§ 101) r = 4 and s — \. 
 
 When substituted in the given equations, these values check ; 
 they are, therefore, the solution sought. 
 
 Ex. 3. Given 
 
 1 + 1 = 3 
 ?-? = ! 
 
 to find u and v. 
 
162 
 
 HIGH SCHOOL ALGEBRA 
 
 [Cn. X 
 
 Solution. Instead of clearing of fractions here, it is better 
 
 to treat - and - as the unknown numbers ; we may even substi- 
 
 tute a single letter for each of these unknown fractions. 
 
 Thus, on substituting x for - and y for -, the given equations 
 
 u V 
 
 become 
 
 
 x + y = 3, 
 
 
 and 
 
 
 2x-3y=l, 
 
 
 respectively. 
 
 Whence 
 
 x = 2 and y = lf 
 
 [§103 
 
 i.e., 
 whence 
 
 
 i = 2and ^ = 1; 
 
 U V 
 
 u = ^ and v = l', 
 
 [§97 
 
 and these values are found to check. 
 
 EXERCISE LXXI 
 
 Solve the following systems of equations^ and check the results ; 
 eliminate before clearing of fractions when practicable : 
 
 4. 
 
 7. 
 
 M=i^' 
 
 4 2 
 
 3 + 3 ' 
 
 2. 
 
 X y 
 6 2 
 
 6i. 
 
 -4-^-7 
 2 + 3" ' 
 
 3^4 
 
 + 5;2 = -4, 
 
 10. 
 
 11. 
 
 ^^ + 5 = 3, 
 X y 
 
 ?-? = l 
 
 X y 
 
 s r 
 
 ^-? = 7. 
 
 s r 
 
 2r + 3t , ^+6 
 
 5 7 
 
 2r — bt , r + 7 
 
 3 
 
 h-2 
 3 
 
 2/^-7 
 
 4 
 
 = 2, 
 = 1. 
 
 3 
 
 13-J 
 
 
 = 0, 
 
 10. 
 
104j 
 
 BIMULTANEOUS SIMPLE EQUATIONS 
 
 153 
 
 12. 
 
 13. 
 
 14. 
 
 15. 
 
 16. 
 
 17. 
 
 18. 
 
 19. 
 
 3x-{-2y-\-6_. 
 4:x-2y ' 
 
 3-7.v_2 
 
 2ic + l • 
 
 8 15 
 
 y 
 
 20. . 
 
 21. . 
 
 llr 5^ ^o 
 |l2 8=^^- 
 
 r3 6 _17 
 2.T"^5.y 40' 
 7 4 11 
 2x 5 ;y 120 
 
 5x + 16y 3x—4: 
 
 Sy-2x = 7. 
 
 n 
 
 0, 
 
 2w-5 3m-7 
 
 2n-3 3m + l 
 r 3a;-2y + | _16 
 
 a; -2/ 
 15 + y — 2a; _g 
 4a7— 5?/— 2 
 
 2a;H-i/-50 = 0, 
 
 ? = ?l4-3. 
 
 4 3 
 
 5 + ? = 20, 
 a; y 
 
 ^ + ^ = 10. 
 
 a; i/ 
 
 3' 
 
 2v IV 
 
 = -3, 
 
 :|- + - = 23. 
 
 2^ w 
 
 22. 
 
 23 
 
 24. 
 
 25. 
 
 26. 
 
 -4s + i^-? = 0. 
 
 7« 7 
 
 |-i(2/-2)=K^-3), 
 ^-i(2/-l)=i(^-2). 
 
 0, 
 
 ^ + ' 
 
 x-2 3-y 
 x-1 _2y + 11 
 
 6 5.5 
 
 .2y + .5 ^ .49a;-.7 
 
 1.5 4.2 ' 
 .5 a;-. 2 ^41 1.5.^ 
 
 1.6 16 
 
 11 
 
 27. 
 
 '5« + 6: 
 
 / + 13_ 
 
 3 
 
 
 42/-2 
 
 .'K + e 
 
 2' 
 
 
 13 
 
 1 x—y 
 
 _ 2i 
 
 
 
 x-y- 
 
 ■3- . 
 
 
 r 1 
 
 2 
 
 1 
 
 
 4 ?,i + V 
 
 u — u (4 u 
 
 -f-v)' 
 
 -8 
 
 v 2 
 
 1^ 1 
 
 5 
 
 
 1^— V I 
 
 ^(2^^- 
 
 -V) 
 
154 
 
 HIGH SCHOOL ALGEBRA 
 
 [Cii. X 
 
 28. 
 
 29. 
 
 30. 
 
 31. 
 
 32. 
 
 7; + i (3 V - w - 1) = J- + f (to - 1), 
 ^(4^ + 3^(;) = 3-V(7^o+24). 
 
 2^-x 2 ' 
 
 ^ - |f^ = ^^ + 3^-^ ^(.^ -g^ ^^^ p ^3^^ 
 
 s o_5i + 2s,s — 3 
 
 2~^-~t::7'"^~2~^ 
 
 2t-^s 
 
 «+l ' 
 
 
 22,. 
 
 A or A- 
 
 a; 
 
 2 
 
 ' 17- 
 
 -3a; 
 
 50 2'- 
 
 -1 
 
 + ^ = 12--^ 
 
 -2^ 
 
 2 
 16 a; -f 19 
 
 3(^-2) 
 
 = 8?/ + 
 
 147 - 24 y 
 
 U2x I 3^ I ^^ + ^y -31 I 3a; + 4 
 
 8t/ + 7 6a;-3.i/ ^^ 4y-9 
 10 "^2(2/-4) "^ 5 
 
 J^J.gf., given 
 
 ; to find X and y. 
 
 105. Literal equations. Literal equations may be solved 
 by the methods already employed in solving numerical 
 
 equations. 
 
 ax-\-by — c 
 
 hx -^ky= I 
 
 Solution. On multiplying the first of these equations by k 
 and the second by b, they become 
 
 aJcx + hky = cJc, 
 and bhx 4- bky = bl. 
 
 Subtracting member from member, we obtain 
 akx — bhx = ch — bl^ 
 i.e. , (oik — bh)x = ck — bl\ 
 
 ck — bl 
 
 whence 
 
 ak — bh 
 
104-105] SIMULTANEOUS SIMPLE EQUATIONS 
 
 ir,5 
 
 If we multiply the first of the given equations by h, and the 
 second by a, and subtract, we eliminate x, and find 
 
 ch — al 
 
 y = • 
 
 hh — dk 
 
 Moreover, these values of x and y check, and are, therefore, a 
 solution of the given equations. 
 
 EXERCISE LXXII 
 
 Solve the following systems of equations and check the re- 
 sults ; eliminate without clearing of fractions where practicable : 
 
 I ax -f hy = m, 
 
 1. 
 
 2. 
 
 3. 
 
 4. 
 
 x + y = c, 
 x — y = d. 
 ax = by, 
 x-{-y = ab. 
 'y-\-az = 0, 
 by + z=^l. 
 
 X y 
 
 + 1-2 = 0, 
 a b 
 
 ' — ay = 0. 
 
 a b 
 
 ^ + JL = a + b. 
 ab ab 
 
 [Hint. Let s = - and« = ^; 
 a 
 
 cf. Ex. 3, p. 152]. 
 
 7. 
 
 X y 
 
 ^4-^ = 1. 
 
 X y 
 
 HIGH 8CH. ALG. 
 
 10. 
 
 11. 
 
 12. 
 
 13. 
 
 14. 
 
 15. 
 
 I bx + ay = n. 
 x — y=a — b, 
 ax-\-by = a^ — b^. 
 
 0, 
 
 x-{-y x — y 
 
 a b 
 
 X -\-y _ x — y 
 a 
 1 
 c 
 1 
 b 
 
 b 
 a b 
 X y 
 c _a 
 X y 
 
 1. 
 
 ax by & 
 
 
 bx cy 0? 
 
 
 (a + &)aj4-(«+c)2/: 
 
 = a+6. 
 
 (a+c)a;4-(a+% = 
 
 = a-\-c. 
 
 a; + l a + & + l 
 
 
 ^z + l^a-ft + l' 
 
 
 x-y=^2b. 
 
 
 'hx + lcy = 4.h\ 
 
 
 1 . 1 _ 
 
 h 
 
 x—k y — h k(y — h) 
 
 11 
 
150 HIGH SCHOOL ALGEBllA [Cii. X 
 
 16. Under what circumstances has Ex. 8 above no finite solu- 
 tion ? Explain [cf. § 41 (iii)]. Answer this question with regard 
 to Ex. 9 also ; and with regard to Ex. 3. 
 
 II. THREE OR MORE UNKNOWN NUMBERS 
 
 106. Equations containing more than two unknown num- 
 bers. The methods already emploj^ed in the solution of 
 systems of equations containing two unknown numbers 
 (§§ 101-105) are easily extended to systems containing three 
 or more iinknoAvn numbers. 
 Thus, to solve the system of equations 
 
 r a^ + 32/-« = 5, (1) 
 
 3a^-j-62/ + 2^ = 3, (2) 
 
 [2x-3y-2z=^e>, (3) 
 
 we first eliminate some one of the unknown numbers, say 2;, 
 
 between (1) and (2), then eliminate the same unknown number 
 
 between (1.) and (3); in this way we obtain two new equations, 
 
 each containing the two unknown numbers x and y. On solving 
 
 these two equations we find x and y, and substituting their values 
 
 in (1) we find z, which completes the solution of the given system. 
 
 r x + ?.y- z = r,, (1) 
 
 Ex. 1. Given \Zx + (Sy -{-2z = ^, (2) 
 
 [2x-^y-3z=:Q', (3) 
 
 to find X, 2/, and z. 
 
 SOLUTION 
 
 Adding 2 times Eq. (1) to Eq. (2), member to member, gives 
 
 5x + 12y = l^, (4) 
 
 and subtracting Eq. (3) from 3 times Eq. (1) gives 
 
 a^-fl2y = 9. (5) 
 
 Now subtracting Eq. (5) from Eq. (4) gives 
 
 4 a! = 4, 
 whence x — 1. 
 
 On substituting this value of x in Eq. (5), we obtain 2/ = |; and, 
 with these values of x and y in Eq. (1), we obtain z = — 2. 
 
 Moreover, these values of x, y, and z check ; therefore they 
 constitute a solution of the given system of equations. 
 
lOo-luej SIMULTANEOUS SIMPLE EQUATIONS 157 
 
 Note. Had the given system coiisisttd of four equations, containing four 
 unknown numbers, the same method of solution would still have sufficed. 
 For, by eliminating- some one of the unknown numbers, say x, between (1) 
 and (2), (1) and (3), and (1) and (4) in turn, we should have obtained a 
 system of three equations containing the remaining three unknown numbers, 
 which could then have been solved as in Ex. 1. And the vakies of these three 
 unknown numbers, being substituted in any one of the given equations, would 
 have determined the value of the remaining unknown number. 
 
 Similarly, a system consisting of five equations containing five unknown 
 numbers can, by eliminating some one of these, be made to depend upon a 
 system of four equations in four unknown numbers ; and so in general (see 
 also § 107). 
 
 (2x-3y-2z==-l, (1) 
 
 Ex. 2. Given 1 3 .« + 2; = 6, (2) 
 
 l-c-\-y + z = S; (3) 
 
 to find the values of x, y, and z. 
 
 SoLUTiox. Since the second of these equations is already free 
 from the unknown number y, therefore it is best to combine Eqs. 
 (1) and (3) so as to eliminate y, and thus obtain another equation 
 involving only x and z. On adding Eq. (1) to three times Eq. (3) 
 we obtain 
 
 6x + z='6, (4) 
 
 and on subtracting Eq. (2) from Eq. (4), we obtain 
 
 2a? = 2, 
 whence a; = l. (5) 
 
 On substituting this value of x in Eq. (2), we obtain 
 
 2^=3; 
 and on substituting these two values in Eq. (3), we obtain 
 
 Moreover, these values of x, y, and z, viz., 1,-1, and 3, check, 
 and therefore constitute a solution of the given equations. 
 
 EXERCISE LXXIH 
 
 Solve each of the following systems of equations: 
 
 (2x-\- 3 // + 4 :^ = 20, Ux-y-z = 5, 
 
 3. J3.r + 4//-f-r);^ = 26, 4. hx-Ay-\-16 = 6z, 
 
 [3x + oy-}-(Jz = 'Sl. [3y + 2(z — l) = x. 
 
158 
 
 HIGH SCHOOL ALGEBRA 
 
 [Cii. X 
 
 5. 
 
 6. 
 
 7. 
 
 8. 
 
 9. 
 
 10. 
 
 11. 
 
 12. 
 
 13. 
 
 7x + 3y-2z = W, 
 2x-^5y-\-Sz = 3d, 
 5x — y-\-5z = 31. 
 
 5x — 6y-\-4:Z = lo, 
 7x-\-Ay-3z = 19, 
 2x-\-y-^6z = 4.6. 
 2x-{-Ay-\-5z = 19, 
 -3x-\-5y-{-7z = S, 
 Sx-3y-\-5z = 23. 
 
 5x-{-6y-12z = 5, 
 
 2x-2y-6z = -l, 
 4:X — 5y + 3z = 7^. 
 
 y^z-S6 = 72-5x, 
 93-ix-ly = ^^y-2z, 
 ix-^ly-\-lz = 5S. 
 
 ix-{-ly = 12-iz, 
 iy + lz==8-\-lx, 
 
 2x-5y-\-19 = 0, 
 3y-4:Z + 7 = 0, 
 2z-5x-2 = 0. 
 
 5 4"^5~ ' 
 4 3^2 
 
 X y 
 
 1 + 1 = 8. 
 
 2 a; 
 
 10, 
 
 14. 
 
 15. 
 
 16. 
 
 17. 
 
 18. 
 
 .S 2 1 
 
 X y z 
 
 1+^=1. 
 
 x + z = 3a-\-h, 
 
 x-\-3y = ^c, 
 
 y-{-2z = x, 
 y-{-z=^x-2d. 
 
 s + e/i+l'j^o, 
 
 X yj 
 
 0, 
 
 u-3(!.l)=.. 
 
 a^y ^1 
 
 x-\-y a 
 
 yz 1 
 
 5-2(1 + 1 
 
 y ^ 
 
 19. 
 
 2/ + 2; 5' 
 xz _\ 
 
 x-{-z c 
 
 [Hint. If ^^ = 1, then 
 
 x + y a 
 
 xy y X 
 
 (2v-{-Sx + y-z = 0, 
 3y-2x-\-z-4:V = 21, 
 2z — 3v — y-\-x = 6, 
 v+4:X-^2y-3z = 12. 
 
IOC] 
 
 SIMULTANEOUS SIMPLE EQUATIONS 
 
 159 
 
 20. 
 
 v-}-y + z = 17, 
 
 V -\-x + z = 16. 
 
 Hint. Adding these equations 
 and dividing the sum by 3 gives 
 
 'y-\-z-\-v — x = 22, 
 z-\-v-^x — y = lS, 
 V -\- X -{- y — z =14:, 
 x + y-\-z — v = 10. 
 
 21. 
 
 22. 
 
 23. 
 
 y-\-z — Zx = 2a, 
 
 z-v-3y = 2b, 
 
 y-\-x — 3z = 2cy 
 
 l2v-{-2y =a-b. 
 
 (3u-\-5v-2x + Sz = 2, 
 2u-{-4:X-3y — z = 3, 
 
 u-{-v-\-z = 2, 
 (Sy-\-4iV + u = 2y 
 5 2; H- 4 a; — 7 v = 0. 
 
 PROBLEMS 
 
 [Leading to simultaneous equations in two or more unknown numbers.] 
 
 1. Find two numbers whose difference is 3^5 of their sum, and 
 such that 5 times the smaller minus 4 times the larger is 39. 
 
 SOLUTION 
 
 Let 
 and 
 
 Then, by the conditions of the problem 
 
 X = the larger number, 
 ?/ = the smaller number. 
 
 x-y = 
 
 ~35"' 
 
 and 5 y — 4 aj = 39. 
 
 Solving these equations, we obtain 
 
 X = 54: and y = 51 ; 
 and these numbers, which constitute a solution of the equations of the prob- 
 lem, also satisfy the problem itself, and are, therefore, the numbers sought. 
 
 2. Find two numbers snch that 3 times the greater exceeds 
 twice the less by 29, and twice the greater exceeds 3 times the 
 less by 1. 
 
 3. A lady purchased 20 yd. of gingham, and 50 yd. of linen, 
 for $ 29 ; she could have purchased 30 yd. of gingham, and 20 of 
 linen, for $ 16. What was the price of each material ? 
 
 4. If A's money were increased by $ 4000, he would have twice 
 as much as B. If B's money were increased by $5500, he would 
 have 3 times as much as A. How much money has each ? 
 
ir>o 
 
 UIGIl SCHOOL ALGEBRA 
 
 [Ch. X 
 
 5. One eleventh of A's age is greater by 2 years than 1- of B's, 
 and twice B's age equals what A's age was 13 years ago. Find 
 the present age of each. 
 
 6. ABC represents a triangle whose perimeter is 
 82 inches. If AB = BC and 7 5(7= 17 AC, find the 
 length of each side of the triangle. 
 
 7. A man having $45 to distribute among a group 
 of children, finds that he lacks $1 of being able to 
 give $ 3 to each girl and $ 1 to each boy, but that he 
 has just enough to give $2.50 to each girl and $1.50 
 to each boy. How many boys and how many girls are 
 
 there in this group ? 
 
 8. John said to James, " Give me 8 cents and I shall have as 
 much as you have left." James said to John, " Give me 16 cents 
 and I shall have 4 times as much as you have left." How much 
 money had each ? 
 
 C 9. ABCD represents a flower bed in which 
 
 BC = ^ AB. If the perimeter of the bed is 
 40 feet, find the length of each of its sides. 
 
 10. A pound of tea and 6 lb. of sugar 
 together cost $ .96 ; if sugar were to advance 
 50%, and tea 10%, then 2 lb. of tea and 
 
 12 lb. of sugar would cost $2.28. Find the present price of tea, 
 
 and also of sugar. 
 
 11. A grain dealer sold to one customer 5 bushels of wheat, 
 2 of corn, and 3 of rye, for $6.60; to another, 2 of wheat, 3 of 
 corn, and 5 of rye, for $5.80; and to another, 3 of 
 wheat, 5 of corn, and 2 of rye, for $5.60. What was 
 the price per bushel of each kind of grain ? 
 
 12. The perimeter of the triangle CDE is 68 in. ; 
 four times CE equals CD increased by four times DE, 
 while twice CE equals DE increased by twice CD. 
 How long is each side of the triangle? 
 
106] SIMULTANEOUS SIMPLE EQUATIONS 161 
 
 13. Divide 800 into three parts such that the first, plus -J- of 
 the second, plus |- of the third, shall equal the second, plus f of 
 the first, plus J of the third : each of these sums being 400. 
 
 14. Divide 90 into three parts such that ^ of the first, plus -| 
 of the second, plus i of the third, shall be 30 ; while the first part 
 increased by twice the second shall equal twice the third. 
 
 15. A boy spent $ 4.10 for oranges, buying some at the rate of 
 
 2 for 5 cents, some at 3 for 10 cents. Later he sold all at 4 cents 
 apiece, thereby clearing $ 1.58. How many of each kind did he 
 buy ? 
 
 16. If a certain rectangular floor were 2 ft. broader and 3 ft. 
 longer, its area would be increased by 64 sq. ft. ; but if it were 
 
 3 ft. broader and 2 ft. longer, its area would be increased by 68 
 sq. ft. Find its length and breadth. 
 
 17. Three rectangles are equal in area ; the second is 6 meters 
 longer and 4 meters narrower than the first, and the third is 2 
 meters longer and 1 meter narrower than the second. What are 
 the dimensions of each ? 
 
 18. The sum of the ages of a father and son will be doubled in 
 25 years ; the difference of their ages 20 years hence will just equal 
 ^ of their sum at that time. Find the present age of each. 
 
 19. A merchant sold to Mrs. A. 2 yd. of cambric, 4 of silk, and 
 3 of flannel, for $5.05, and to Mrs. B., 4 yd. of cambric, 5 of 
 flannel, and 2 of silk, for $4.30. If 2 yd. of flannel cost 10 cents 
 more than 2 yd. of cambric and -J yd. of silk combined, find the 
 price of each per yard. 
 
 20. The tickets to a concert were 50 cents for adults and 35 
 cents for children. If the proceeds from the sale of 100 tickets 
 were $39.50, how many tickets of each kind were sold ? 
 
 Solve this problem also by using but one letter to represent an 
 unknown number. 
 
 21. Find three numbers such that the sum of the reciprocals of 
 the first and second is y\, the sum of the reciprocals of the first 
 and third is -f-g, and the sum of the reciprocals of the second 
 and third is ||. 
 
162 HIGH SCHOOL ALGEBRA [Cii. X 
 
 22. The sum of the reciprocals of three numbers is 34 ; the re- 
 ciprocal of the second minus that of the third equals 4 ; the sum 
 of 3 times the reciprocal of the first and twice the reciprocal of 
 the second is less by 1 than 5 times the reciprocal of the third. 
 Find the three numbers. 
 
 23. In a certain two-digit number which equals 8 times the sum 
 of its digits, the tens' digit exceeds 3 times the units' digit by 1. 
 Find the number. 
 
 24. The sum of the digits of a two-digit number is 12, and if 
 the digits are interchanged, the number thus formed will lack 12 
 of being twice the original number. What is the number ? 
 
 25. The sum of the digits of a 3-digit number is 11 ; the double 
 of the second digit exceeds the sum of the first and third by 1, 
 and if the first and second digits are interchanged, the number 
 will be diminished by 90. What is the number ? 
 
 26. The third digit of a 3-digit number is as much larger than 
 the second as the second is larger than the first ; if the number 
 is divided by the sum of its digits, the quotient is 15 ; and the 
 number will be increased by 396 if the order of its digits is 
 reversed. What is the number ? 
 
 27. A capitalist invested ^4000, part at 5%, part at 4%, and 
 found that his annual income from this investment was ^175. 
 How much was invested at 5 %, and how much at 4 % ? 
 
 Solve this problem also by using only one unknown letter. 
 
 28. A capitalist invested A dollars, part at p %, part at q%, 
 and found that his annual income from this investment was B 
 dollars. How much was invested at p % ? at g % ? 
 
 Show that this problem includes Prob. 27 as a special case. 
 
 29. Divide the number N into two such parts that 1/m of the 
 first part, plus l/^i of the second, shall exceed the first part by M. 
 
 Specialize this problem, and find the solution of the special 
 problem by substituting in the general solution. 
 
 30. Three cities. A, B, and C, are situa,ted at the vertices of a 
 triangle ; the distance from A to C by way of B is 50 miles, from 
 A to B by way of C is 70 miles, and from B to C by way of A is 
 60 miles. How far apart are these cities ? (Make diagram.) 
 
106] SIMULTANEOUS SIMPLE EQUATIONS 163 
 
 31. In the triangle ABC, AB = 12 inches, BC = 10 
 inches, BE^BF, FC=GG, AG = 4.iBF. If the 
 perimeter of the triangle is 42, find AG, AE, BE, FC. 
 
 32. A quantity of water which is just sufficient to 
 fill three jars of different sizes, will fill the smallest 
 jar exactly 4 times ; or the largest jar twice, with 4 
 gallons to spare; or the second jar 3 times, with 2 
 gallons to spare. Find the capacity of each jar. 
 
 33. Two men, A and B, rowed a certain distance, 
 alternating in the work ; A rowed at a rate sufficient to cover the 
 entire distance in 10 hours, while B's rate would require 14. If 
 the journey was completed in 12 hours, how long did each row ? 
 
 34. Two boys, A and B, run a race of 400 yards, A giving B a 
 start of 20 seconds and winning by 50 yards. On running this 
 race again. A, giving B a start of 125 yards, wins by 5 seconds. 
 What is the speed of each ? Generalize this problem. 
 
 35. If A and B can do a certain piece of work in 10 days, A 
 and C in 8 days, and B and C in 12 days, how long will it take 
 each to do the work alone ? 
 
 36. A and B together can build a wall in 5^^ days; being 
 unable to work at the same time, A works 5 days, then B takes up 
 the work, finishing it in 6 days more. In how many days could 
 each have built the wall alone ? Generalize this problem. 
 
 37. A man can row m miles downstream in c hours and m 
 miles upstream in d hours; what is his rate of rowing in still 
 water, and what is the rate of the current ? 
 
 38. From the solution of Prob. 37 find the solution of the 
 special problem in which m = 6, c = 1^, d = 4. 
 
 39. Two trains whose respective lengths are 1200 feet and 960 
 feet run on parallel tracks; when moving in opposite directions, 
 the trains pass each other in 24 seconds ; when moving in the 
 same direction, each at the same rate as before, the faster passes 
 the slower in 1^ minutes. Find the rate of each train. 
 
164 HIGH SCHOOL ALGEBRA [Ch. X 
 
 40. Two trains are scheduled to leave the cities A and B, m 
 miles apart, at the same time, and to meet in h hours ; but, the 
 train from A being a hours late in starting, and running at its 
 regular rate, the trains met k hours later than the scheduled time. 
 What is the rate at which each train runs ? 
 
 41. From the solution of Prob. 40 iind the sohition of the 
 special problem in which m = 800, /i = 10, a = If , and k = Jq. 
 
 42. A train was scheduled to make a certain run at a uniform 
 speed. After traveling 2 hours it was delayed 1 hour by an 
 accident, after which it proceeded at -y- its usual rate and arrived 
 ^ hour late. Had the accident occurred 36 miles farther on, the 
 train would have been 36 minutes late. Find the usual rate of 
 the train and the entire distance traveled. 
 
 43. Two boats which are d miles apart will meet in a hours if 
 they sail toward each other, and the second will overtake the first 
 in b hours if they sail in the same direction. Find the respective 
 rates at which these boats sail. Also discuss fully your solution, 
 i.e., interpret the results (cf. Prob. 4, p. 141). 
 
 44. Find an expression of the form ax^ -\-bx-\-c whose value is 
 
 6 when x = 2, 3 when x= —1, and 10 when a? = 4. 
 
 Hint. 4 a + 2b + c is the value of ax^ + bx -\-c when x = 2 ; therefore, 
 4a+26 + c = 6, etc. 
 
 45. Find an expression of the form ax^ + &x + c whose value is 
 
 7 when x = S,9 when x= —1, and 17 when x = — 5. 
 
 46. Find an expression of the form ax^ + bx^ -\- ex -\-d which 
 equals — 16 when x= —1,-4 when x = l, — 43 when x = — 2, and 
 — 100 when x= — 3. 
 
 47. Of three alloys, the first contains 35 parts of silver, to 5 
 of copper, to 4 of tin ; the second, 28 parts of silver, to 2 of 
 copper, to 3 of tin ; and the third, 25 parts of silver, to 4 of copper, 
 to 4 of tin. How many ounces of each of these alloys melted 
 together will form 600 oz. of an alloy consisting of 8 parts of 
 silver, to 1 of copper, to 1 of tin ? 
 
10(5-107] SIMULTANEOUS SIMPLE EQUATIONS 1()5 
 
 48. If Prob. 47 demanded merely that the alloy should contain 
 8 parts of silver to 1 of copper (without specifying the amount of 
 tin), how many ounces of each of the given alloys would then be 
 required ? Why is this problem indeterminate (of. § 107) ? 
 
 107.* Determinate and indeterminate systems of equations. 
 
 As we have already seen, a system containing as many inde- 
 pendent equations as unknown numbers, can always be solved, i.e., 
 the unknown numbers can be determined (§§ 101-106). Such a 
 system is, therefore, a determinate system. 
 
 On the other hand, a system in which there are fewer inde- 
 ])eudent equations than unknown numbers is an indeterminate 
 system. It is easy to show that this statement — already seen 
 to be true in the case of a single equation containing two un- 
 known numbers (§ 99) — is true generally. 
 
 Thus, suppose we have three equations containing four unknown 
 numbers. By regarding one of these numbers temporarily as 
 kuown, we can solve the given equations for the other three ; i.e., 
 we can express any three of the four unknown numbers in terms of 
 the fourth. To every assigned value, therefore, of this fourth un- 
 known number, there corresponds a set of values of the other 
 three (cf. § 99) ; hence the system is indeterminate. 
 
 Again, there can never be in a system more independent equa- 
 tions than there are unknown numbers. 
 
 For, if that were possible, suppose there are three independent 
 equations, viz., 
 
 ax-{-hy = c, (1) 
 
 hx+jy^k, (2) 
 
 and Zfl? -f my = n. (3) 
 
 containing but two unknown numbers, x and y. 
 
 On solving (1) and (2) we obtain 
 
 cj — bk -, ch — ak 
 
 X = -^ and y = , 
 
 aj — bh ' bh — aj 
 
 and on substituting these values for x and y in (3), we obtain 
 
 \aj — bh) \bh — qjj 
 
 * This article may be omitted till the subject is reviewed. 
 
166 HIGH SCHOOL ALGEBRA ICu. X 
 
 i.e., the known numbers of these equations are not independent 
 (n, for example, is expressed in terms of a, b, c, Z, etc.), hence the 
 given equations are themselves not independent. 
 
 REVIEW EXERCISE-CHAPTERS VI-X 
 
 Find the H. C. F. and also the L. C. M. of: 
 
 1. 6a^ + 13a;-5 and3a^ + 2a.'2-f 2a;-l. 
 
 2. 12a^-29a; + 14and8a.-2-30 + 6a;*-f-lla^ + 33a;. 
 
 3. Chanere '^ ^^~ to an equal fraction whose denomi- 
 
 nator is 24 a? — 6 a^x^ ; also to an equal fraction whose numerator 
 is 1 — 10 ay — 5 a + 2 y. 
 
 Simplify : 
 
 go;'" — bx'^"^''- 1 + x l + o^ 
 
 ** a^ftic-dV * „ 1 + aJ- 1+a^ 
 
 l-hx^ 1 + a^ 
 
 5. 
 
 4a^ 
 6xy-\-9y^ . Sa^-27f 
 
 r^«I-^)_yAp-l) ^ 4^ 
 
 ^4 _ ^ _ J ,^.2 _|_ ^, _j_ g 4 ic^ — 6 ict/ 
 
 1^1 1 
 
 10. 
 
 (a-6)(6-c) (c-6)(c-d) {d-c)(b-a) 
 k-1 1-k 
 
 (]c - l)(k -m)(n- Tc) (I -k){n- k){k -p) 
 
 11. Why may a term be transposed from one member of an 
 equation to the other by merely changing its sign ? 
 
 12. When are equations conditional ? identical ? integral ? 
 fractional ? literal ? numerical ? indeterminate ? Illustrate each 
 of your answers. 
 
 Solve, and check as the teacher directs : 
 
 13. 3a;2_5i»-12 = 0. 2 -\- x ^ 19 
 
 14. 6m2-13m = -6. ' 2-a;~ 21 
 
107] 
 
 SIMULTANEOUS SIMPLE EQUATIONS 
 
 167 
 
 16. 1---^ 
 
 3^ 
 
 2s 
 
 3s 
 
 X a 
 
 17 
 
 = a-\-b. 
 
 19. {x-2y+(x+oy=(x + 7y. 
 
 20. 7x + 5fl-^^ = a(x-a). 
 
 3 2 
 ic a 
 18. 6x'-\-7x^-20a^ = 0. 
 
 y — 4: ?/ — 6 _ y — 5 _ y_ 
 
 21. 
 
 + 
 
 8v 
 
 2vH-5 2v-5 25-4^2 
 
 22. 
 
 2/-5 2/-7 2/-6 2/-8 
 
 23. (a? — a)(a — & + c) = (icH-a)(6 — a + c). 
 
 24. Show that while 2 is a root of the integral equation which 
 
 results from clearing — ^ + — rr- = 8 H of fractions, 
 
 x-^o {x + 5)(x-2) x-2 
 
 it is not a root of the given fractional equation. How could we 
 
 avoid introducing this extraneous root ? 
 
 25. Form the equations whose roots are: 2,-9; — 3|-, 4; 
 I, f ; -2a, -6a; 1,3,-7; l-c,c-l. 
 
 26. When are two equations equivalent ? inconsistent ? simul- 
 taneous ? independent ? Illustrate each of your answers. 
 
 27. Explain the term '^ elimination" as applied to simultaneous 
 equations, and outline three methods of elimination. 
 
 Solve the following systems ; check as the teacher directs : 
 
 28. 
 
 29. 
 
 7-2a; ^3 
 5-37/ 2' 
 y — x = 4:. 
 
 2x- 
 
 y 
 
 = 4, 
 
 31. 
 
 32. 
 
 30. 
 
 32, + ^ = 9. 
 
 1 + 1-1 
 X y 4 
 
 33. 
 
 1_ 
 
 12 
 
 5 ahx -\-2y = 
 
 166, 
 
 3 a5a.' + 4 ?/ = 
 
 18 6. 
 
 a 6 
 
 
 3a 66 3 
 
 
 y — z x-\-z 
 
 1 
 
 2 4 
 
 '2' 
 
 x—y x—z 
 
 — 
 
 5 6 
 
 '-'> 
 
 ?/ 4- 2; _ a) 4- ?/ 
 4 2 
 
 -4. 
 
168 HIGH SCHOOL ALGEBRA [Ch. X 
 
 34. If asi^ -\-hx-\-c becomes 8, 22, 42, respectively, when a; 
 becomes 2, 3, 4, what will it become when x becomes — \ ? 
 
 35. The sum of two numbers is 5760, and their difference is \ 
 of the greater. Find the numbers. 
 
 36. What number added to its reciprocal gives 5.2 ? 
 
 37. It takes 2000 square tiles of a certain size to pave a hall, 
 or 3125 square tiles whose dimensions are 1 inch less. Find the 
 area of the hall floor. How many solutions has the equation of 
 this problem ? How many has the problem itself ? 
 
 38. Divide the number a into two parts such that the second 
 part shall equal n increased by m times the first part. 
 
 39. What number must be added to m and to n in order that 
 the first sum divided by the second shall equal p/q ? What does 
 your answer become when p = q"} What does this indicate 
 (1) when m = n, (2) when m and n are unequal ? 
 
 40. In order to build a new clubhouse, a country club assessed 
 each of its 200 members a certain sum ; later an increase of 50 
 in the membership reduced the individual assessments by ^10. 
 Find the cost of the proposed house. 
 
 41. At what time between 3 and 4 o'clock is the minute-hand 
 25 minute spaces ahead of the hour-hand ? 
 
 42. The freezing point of Avater is marked 0° on a Centigrade 
 thermometer, and 32° above zero on a Fahrenheit thermometer. 
 If 100° Centigrade = 180° Fahrenheit, find the reading on a Centi- 
 grade thermometer corresponding to 68° Fahrenheit. (Make a 
 diagram of each scale.) 
 
 43. State and solve the general problem of which Prob. 42 is 
 a particular case. By substitution in the formula thus obtained 
 express in the Centigrade scale the following Fahrenheit readings : 
 44°; 212°; -10°; 0°. 
 
 44. A man rows a boat with the tide 8 miles in If hr. and 
 returns against a tide 1 as strong in 4 hr. What is the rate of 
 the stronger tide ? At what rate does the man row in still water ? 
 
107] REVIE]r EXEliCISE 169 
 
 45. A man selling eggs to a grocer counted them out of his 
 basket 4 at a time and had 1 e^^ left over ; the grocer counted 
 them into his box 5 at a time and there were 3 left over. If the 
 man had between 6 and 7 dozen eggs, how many must there have 
 been (cf. § 99) ? 
 
 46. Of two wheelmen, A and B, A starts c hours in advance 
 of B, and travels at the rate of a miles in h hours, while B follows 
 at the rate of p miles in q hours. How far will A travel before he 
 is overtaken by B ? 
 
 Under what conditions is this solution positive ? negative ? 
 zero ? infinite ? Interpret the result in each case. 
 
CHAPTER XI 
 
 INVOLUTION AND EVOLUTION 
 
 I. INVOLUTION 
 
 108. Introductory. For the meaning of the words hase^ 
 exponent^ and power ^ as used in algebra, see §§9, 30, and 36. 
 The process of raising a number or expression to any given 
 power is called involution. 
 
 In this chapter, as in the earlier treatment of powers, 
 we shall use only positive integers as exponents. Later on 
 (Chapter XVI), however, we shall find it advantageous to 
 employ such symbols as aP^ a~^^ and or also, and we shall 
 then assign suitable meanings to such symbols. 
 
 109. Even powers, odd powers, powers of fractions, etc. A 
 
 power of any given number is called even or odd according 
 as its exponent is even or odd. 
 
 From the law of signs given in § 18 it follows that : 
 
 (1) All integral powers of a positive number are positive. 
 
 (2) All even integral powers of a negative number are 
 positive. 
 
 (3) All odd integral powers of a negative number are 
 negative. 
 
 And from § 83 (i) [cf. also Ex. 30, p. 121], it follows that 
 
 a^ fmY w* . 
 J5' [nJ^V''- 
 
 Let pupils fully explain each of the above statements: 
 
 170 
 
108-110] INVOLUTION AND EVOLUTION 171 
 
 EXERCISE LXXIV 
 
 1. Answer again questions 18-20 on p. 39. 
 
 2. Write that power whose base is k and whose exponent is 
 m — 3. Are there any limitations here on the value of A;? on 
 the value of m? 
 
 3. From the definition of an exponent show that a^ • ar^ = a^. 
 Also that 2^. 2. 22 = 2^ 
 
 4. For what values of n between 1 and 10 is (—3)" • (— S)^" 
 positive ? Explain. 
 
 5. Show that an even power of a negative number is positive. 
 
 6. How is a fraction raised to a power (of. Ex. 30, p. 121) ? 
 Illustrate your answer. 
 
 Simplify each of the following expressions : 
 
 8. 
 
 (-!!)■■ - (-drj 
 
 110. Exponent laws. The following formulas state what 
 are known as the exponent laws. The bases (<2, 5, and c) stand 
 for any numbers or algebraic expressions whatever, but the 
 exponents are positive integers. 
 
 (i) First exponent law. a"" - a"" = a"*+«. [§ 30 
 
 For, just as a^ • a^ = (a • a • a) • (a • a) 
 
 = a^ i.e., «^+2. 
 so, too, 
 
 a^ ' a" =: (^a ' a ' a '" to m factors)(a - a • a ••• to n factors) 
 = a ' a • a '•• to (m + n) factors 
 
 Similarly, a"^ - a"" - a^ = o^^^'-^p. 
 
 HIGH SCU. ALG. — 12 
 
172 BIGH SCHOOL ALGKhRA [Cn. XI 
 
 (ii) Second exponent law. (a'"y — a"***. 
 For, just as {aF)'^ = {a - a ■ a^ 
 
 = (^a ' a ' a^ ' (^a ' a ' a) 
 
 = a^, i.e.^ a^'2; 
 so, too, (^a'^y = (a - a ' a ••' to m factors)** 
 
 = a • a • a • • • to mn factors 
 
 (iii) Third exponent law. a^ - b^ = (aby. 
 For, just as a^ • b^ = a • a ■ a - b - b -h 
 = ab • ab ' ab 
 ^(aby; 
 so, too, 
 
 a"b^ = (^a • a • a ••' to n factors) - (b - b • b -" to n factors) 
 — ab ' ab • ab "• to n factors 
 
 = iaby. 
 
 Similarly, a^^c"" = (obey. 
 
 (iv) Fourth exponent law. a'" -r- a" = a""-". [§ 80 
 
 This law is an immediate consequence of (i) above, and 
 of the definition of division (§ 8), for since 
 
 therefore a"» ^ a" = «"*"". 
 
 EXERCISE LXXV 
 
 Simplify, and explain your work in each case : 
 7. (-5a)2. 
 
 9. (IT^S^)^ 
 
 ^ \2 
 
 11. ' ^ 
 
 1. 
 
 a'h^ ' ah\ 
 
 
 2. 
 
 3A-(- 
 
 2a?f). 
 
 3. 
 
 a ' 0? ' gC 
 
 ■a'. 
 
 4. 
 
 
 
 5. 
 
 ccPe 
 
 
 6. 
 
 {x'zy. 
 
 
 m- 
 
 12. 
 
 X'^ ' oc^. 
 
 13. 
 
 X"" -T-Olf. 
 
 14. 
 
 (x-y. 
 
 15. 
 
 (2x'"'y, 
 
 16. 
 
 s« . s^. 
 
 17. 
 
 V^ ' V^ ' v^. 
 
 18. 
 
 c^r 
 
110-111] 
 
 INVOLUTION AND EVOLUTION 
 
 
 - (-ir' - m. 
 
 - C-^J- 
 
 - (5T- - c^; 
 
 21. (H^)*. 
 
 22. (-c)2- 
 
 25. (,,,•.->)'. ^^ r.„_ 
 
 26. (5 »•")». • H^a + 
 
 173 
 
 Write the following as powers of products [cf. law (iii) above] : 
 
 30. /i%2. 33. o?if. 36. a' • (2 6)»^. 
 
 31. r^s¥. 34. a^2/*- 37. 3* • ( - m)^ • (?i/. 
 
 32. c^d^ 35. — 2» . 3^ 38. ir-" • /« • 2^". 
 
 39. What does a represent in the proofs of § 110 ? May it rep- 
 resent a polynomial as well as a number ? 
 
 40. Translate the first exponent law (§ 110) into verbal lan- 
 guage (cf. § 30). 
 
 41. Translate the second, third, and fourth exponent laws into 
 verbal language. 
 
 42. Is (a • 6 • cf equal to a^ • 6^ • c^ ? Is (a + 6 -f c -h df equal to 
 a'- -f ^^ + c^ + c^^ ? Explain your answers. 
 
 43. Is [(-2)3]2 equal to [(-2)^^? Why? Is (a^)* equal to 
 {xy^ Why? 
 
 111. Powers of binomials. We have already seen (§§ 52 
 and 57) that 
 
 and (a + hy = a^-\-^a%^2>ah'^ + h^. 
 
 These powers (expansions) were obtained by direct multi- 
 plication, and the higher powers may, of course, be obtained 
 in the same way. Thus, 
 (a + 6)4 = ^4 4. 4 ^35 ^ 6 ^252 + 4 aJ3 4. 54^ 
 (a -f 6)5 = ^5 + 5 a^i + 10 ^352 4. 10 aW 4- 5 aft* _|_ js^ 
 (a 4- 6)6 = ^6 4. (3 ^56 4. 15 ^4^2+ 20 a%^+ 15 ^254 4. 5 ^554. je, etc. 
 
174 HIGH SCHOOL ALGEBRA [Ch. XI 
 
 The following questions may serve to bring out the strik- 
 ing similarity of these expansions : 
 
 1. How does the exponent of the first term in each ex- 
 pansion compare with that of the corresponding binomial ? 
 
 2. How, in each expansion, does the exponent of a change 
 as we pass from term to term toward the right ? 
 
 3. In which term of each expansion does b first appear ? 
 How does the exponent of b change from term to term ? 
 
 4. How many terms in each expansion? What is the 
 sign of each term ? 
 
 5. What coefficient has the first term of each expansion ? 
 the second term ? 
 
 6. Multiply the coefficient of any term in any of the ex- 
 pansions by the exponent of a in that term, and divide this 
 product by the number of the term ; how does this quotient 
 compare with the coefficient of the next term ? 
 
 7. Assuming that the expansion of (a + by is similar in 
 form to the expansion of (a + ^)^ (a+i)^ etc., complete 
 the statement : 
 
 (a + 5)8 = a8 + 8 a^b + 28 a%^ + .... 
 
 112. Binomial theorem, (i) The answers to the first six 
 questioHS in § 111, when combined, may be expressed sym- 
 bolically thus : 
 
 (a + by = a--\-'^ a^-'b + ^(f -^) a"-2^,2 
 ^ "^ 1 1.2 
 
 1 . ^ . o 
 
 This formula, which was discovered by the celebrated 
 English mathematician Sir Isaac Newton (1642-1727), is 
 called the binomial theorem; its correctness is proved in 
 §§ 206-207. 
 
111-112] INVOLUTION AND EVOLUTION 175 
 
 (ii) Since a — 5 = aH-(— 5), 
 therefore 
 (a_5)3=[a+(-*)]3=«H3a2(-5) + 3a(-6)2 + (-6)3' 
 
 ?'.e., (a — by differs from (a + b^ only in having the signs of 
 its even terms negative. So also for other powers of a — b, ^.e., 
 
 (a -by=a^-^ a^-'b + ^i(!L:^^«-2j2 
 
 1.2.3 « ^ + • 
 
 EXERCISE LXXVI 
 
 Write down the expansions of the following binomials : 
 
 1. (a + xy. 5. {a-\-cy. 9. (m2 + 6)3. 
 
 2. (mH-f)^ 6. (i» + 2/)*. 10. {m-\-hy. 
 
 3. (1* + ?;)^ 7. (2/ + ;^)'. 11. (m^+fty. 
 
 4. (p + g)'. 8. {k-\-iy. 12. (m2 + 63)6^ 
 
 13. Expand each of the following expressions : (x-\-yy, (^+2/)^ 
 and (x + yy ; then multiply the first two expanded forms together 
 and thus verify that (x -\- yY • {x -\- yy = (a; + yy. 
 
 14. What terms in the expansion of {c — dy are negative? 
 Why? 
 
 15. Write the first five terms of (s — 2 ty and simplify your 
 result (cf. Ex. 3, p. 71). 
 
 16. How many terms are there in the expansion of (m-f-?iy? 
 How many in (a — by ? How many in (3 s — 2 ty ? 
 
 Write each of the following expressions in its expanded form : 
 
 17. (k-cy. 23. (4c + ^')'. 29. {v'-2y. 
 
 18. (r — sy. 24. (mii. — rsy. 30. (2 xy — ly. 
 
 19. (m — ny. 25. (ab + cdy. 31. (c + a)». 
 
 20. (c4-dy«. 26. (3a2 + c«(^)^ 32. (2 771+3)1 
 
 21. (x' + yy. 27. (A;4-l)'. 33. (2 -3 0-2^)5. 
 
 22. (2r-ay. 28. (^^-2)^ 34. (2-a'by. 
 
176 HiGn SCHOOL algebea [Ch. xi 
 
 35. (.r-.9vy. gg_ fr-'lW 40- [.ci + (b + c)J. 
 
 36. (2xy^-\-x'yy. V ^^ 41. (c + d + e)^ 
 
 37. fa-j-fl] . 39. - + - • ^ ^ ^ 
 
 V a?; V« V 43. (2a^-m-l)^ 
 
 44. Write the first four terms of (a + x)'-^ ; the first three terms 
 of (a; — y)^ ; the first three terms of (2 ax — 3 k^y. 
 
 II. EVOLUTION 
 
 113. Definitions. Here, as in arithmetic, by the square 
 root of any given number we mean a number whose square 
 equals the given number. 
 
 Thus, since T^ = 49, therefore 7 is a square root of 49. 
 
 Similarly, the third or cube root of a number is a number 
 whose third power equals tlie given number. 
 
 Roots are usually indicated by the radical sign ( V)i which 
 is a modification of the letter r, the initial letter of the Latin 
 word radix, meaning root. A small figure, called the index 
 of the root, is written in the opening of the radical sign to 
 indicate the particular root to be extracted. When no index 
 is written, the index is understood to be 2. 
 
 U.g., -yja indicates the second or square root of a, (1) 
 
 -^a indicates the third or cube root of a, (2) 
 
 and -y/a indicates the seventh root of a, (3) 
 
 and, in general, ya means the rith root of a, 
 
 t.e., i^ay = a. (4) 
 
 An indicated root is said to be an even root or an odd root 
 according as its index is an even or an odd number. 
 
 The process of finding a root of any given number is called 
 evolution ; it is the inverse of involution (cf. § 108). 
 
 Note. In practice the radical sign is usually combined with a vinculum 
 (§ 11) to indicate clearly just how much of the expression following the radi- 
 cal sign is to be affected by that sign ; thus \/9 + 16 means tlie square root 
 of the sum of 9 and 16, while VO + 16 indicates that 16 is to be added to the 
 square root of 9. 
 
112-115] INVOLUTION AND EVOLUTION 177 
 
 114. Law of signs of roots. From the definition of root 
 (§ 113), and from § 109, it follows that : 
 
 1. An odd root of any number has the same sign as 
 the number itself. Thus, V8 = 2, and V— 8 = — 2, because 
 23=8 and (-2)3= -8. 
 
 2. An even root of a positive number has two opposite 
 values, i.e.^ one positive, the other negative. Th us, VSl = + 3 
 or— 3, since (+ 3)* = (— 3)*= 81. Instead of writing 
 V81 = -f 3 or — 3, we usually write V81 = ± 3 ; this expres- 
 sion is read, ''The fourth root of 81 equals plus or minus 3." 
 
 3. An even root of a negative number is neither a positive 
 nor a negative number. Thus, V— 9 is neither + 3 nor —3, 
 since ( -f 3)2 = ( - 3 )2 = + 9, and not - 9. 
 
 Note. Such indicated roots as V— 9 are called imaginary numbers 
 (cf. §§ 146, 164) ; all other numbers are, for distinction, called real numbers. 
 To provide for such roots as V— 9 we must again extend the number system, 
 just as we did when subtractions like 3 — 8 first presented themselves (cf . 
 Chap. II). 
 
 115. Roots of monomials. If a monomial is an exact 
 power, the corresponding root can usually be written down 
 by inspection. 
 
 E.g.y </S aV = 2 a% because (2 a^xf = 8 aV (§ 110) ; 
 
 ■V'9^\>f=±3xy, because (+3 ay'y'y=(-Sxyy=9xY', 
 i/-32x'^ = - 2 x% because (-2x'y = -32 x''; 
 
 ^ = 2^^ because f2^Y = 8™\ 
 
 EXERCISE LXXVII 
 
 1. What is meant by the square root of a number ? Are both 
 5 and — 5 square roots of 25 ? Why ? 
 
 2. What are the square roots of 64 ? the fourth roots of 16 ? 
 Why ? If a is any ecen root of a number, then — a also is a root 
 (with the same index) of that number, — explain, 
 
178 HIGH SCHOOL ALGEBRA [Ch. XI 
 
 3. What is the cube root of 27 ? of - 27 ? of 64 ? of -64 ? 
 Explain. How does -^32 compare with V— 32 ? 
 
 4. How does the sign of an odd root of a number compare with 
 the sign of the number itself ? Why ? Answer these questions 
 for an even root also. 
 
 5. Give the cube of each integer between 1 and 7. Name the 
 cube root of : - 8 ; 1000 ; - 1728 ; - f 7 . ii5 . _ 216 ; 8000. 
 
 6. What is the sign of any even power of a positive or negative 
 number? Can, then, an even root of a negative number be 
 positive ? negative ? Illustrate your answer. 
 
 7. Is — 13 a square root of 169 ? Why ? Is 5 as^ the cube 
 root of 125 aV ? Why ? How can you tell whether one given 
 number is a square root of another given number? a fifth root? 
 
 8. How do we find the exponents in the cube root of 8 o}^y?'}f ? 
 in the 4th root of a%^h' ? in the 6th root of m^^^^ ? in the nth 
 root of a^^n^in 9 Explain. 
 
 Find the following indicated roots, and check your answers. 
 Also, tell which are even and which are odd roots, and name the 
 index in each case : 
 
 9. Va'6V^ 16. V128a^^6i^. 3/ 21^ &d}'' 
 
 22. 
 
 343 (c-df 
 
 10. Vl6aV2/^ 17 3 125<i/« , 
 
 • A/ 112% ah^ 23 xl^m^hK 
 
 11. ^32^«. r^ -i, ^225 
 
 12. V-243aiV. ■ ^ 128 0^^^ ' 24. 
 
 1000 c^ 
 125 d'' 
 
 13. A/a^a^Y". 19. V J^ ^''^'"- 25 W ^''""'"^'" 
 
 14. ^'/-64^. 20 A^/ ^^^y . ^, x|aF5^^ 
 
 2V 
 
 15. 
 
 5/--32_aV^ 3/ .O27 gV 4/ 8I (g-c)^^"^ 
 
 \ 2432/2^ ' ' \. 064 6V' * \ 625 a*'"" » 
 
 28. Write a rule for the extraction of such roots as the above, 
 emphasizing particularly the matter of exponents and signs. 
 Does your rule apply to roots of polynomials also ? 
 
116-116] INVOLUTION AND EVOLUTION 179 
 
 29. Is V9T16 equal to V9 • Vi6 ? Why ? Is V9Tl6 equal 
 to V9 + vTe ? Is VoM^ equal to Vo^^- Vfe' (cf. Ex. 42, p. 173)? 
 Is Va^ • h'^ equal to Va^ • V6^ ? State in words your conclusion 
 as to the square roots of sums and products. 
 
 116. Roots of polynomials extracted by inspection. If a 
 
 polynomial is an exact power of a binomial, or the square of 
 a polynomial, a little study usually reveals the corresponding 
 root. 
 
 Ex. 1. Find the square root of m^ + 4 m^n + 4 n^. 
 
 Solution. This expression is easily seen to be (m^H-2n)^; 
 therefore Vm*^ + 4 m^n -\- 4 n^ = ± (m^ + 2 n). 
 
 Ex. 2. Find the cube root of 8 a^ - 36 a^b -27b^ + 54 ab^. 
 
 Solution. This polynomial consisting of four terms, two of 
 which, viz., 8 a^ and — 27 b^, are exact cubes, may be the cube of 
 a binomial (§ 57) ; if so, that binomial must be 2 a — 3 6. (Why ?) 
 
 On cubing 2 a — 3 b, we see that 
 
 ■\/Sa^-36a'b-27b^ + 54:ab^ = 2a-Sb. 
 
 Ex. 3. Find the square root of a^ + 6^ — 2 a6 — 4 6c + 4 c^ + 4 ac. 
 
 Solution. This polynomial consisting of six terms, three of 
 which are exact squares, and three of which are double products, 
 7nay be the square of a trinomial whose terms are the square roots 
 of the square terms (§ 56). A little further examination shows 
 that Va^ + 62 _ 2 a6 - 4 6c + 4 c^ + 4 ac = ± (a - 6 -f 2 c). 
 
 EXERCISE LXXVIII 
 
 By inspection find the following roots, and check results : 
 
 4. V4 aj- + 12 a; 4- 9. 6. V(m+^)^ — 4 (m + ?i) +4. 
 
 5. V25 2/^ -40 2/ + 16. 7. -Va^ -{-2xy + y^- 2 xz~2yz-\-z\ 
 
 a V 8 u^ — 12 u^v — v^-\-6 uv^. 
 9. Va;'* — 4: oc^y -^ y^ — 4: xy^ + 6 icy. 
 10. v^8 /r - 84 h^k + 294 hk^ - 343 Ar^. 
 
180 HIGH SCHOOL ALGEBRA [Ch. XI 
 
 11. VciP- b'-5a'b-^5 ab* + 10 a^b' - 10 a%\ 
 
 12. Va' + 962_6a6 + 6(a;-2i/)(a-36)4-9(a^-4a;y + 4?/2). 
 
 13. ^x^ -6aba:^-\- 15 a^^x* - 20 a'b^x^-{- 15 a'b'x^- 6 a^6'x + a'6^ 
 
 117. Square roots of polynomials.* 
 
 Sincev (A; + w)^ = /i:^ -f- 2 A:2* + w^^ 
 
 therefore Vk^ -^ 2ku-tu^= k -\- u, 
 
 and we shall now try to find a method by which the root, 
 k -\- u, may be found from the power, k^ + 2ku + u^. 
 
 Manifestly the first term of the root (viz., k) is the square 
 root of the first term of the power. 
 
 And having subtracted k'^ (the square of this root term) 
 from the power, the next term of the root (viz., u} is found 
 by dividing the first term of the remainder (viz., 2ku-\-u^) 
 by twice the root term previously found (viz., 2 k^. 
 
 The actual work may be arranged thus : 
 
 k'^ + 2 ^•M + u^ \k -\-u 
 
 T^ 
 
 Trial divisor f =2k 
 
 Complete divisor = 2 A; + 
 
 2 ku + w2 
 
 'Iku + u'^ = (2 k + u) u 
 
 
 The same method may be applied to other polynomials. ^ 
 
 Ex. 1. Find the square root of 4 s- - 28 .s^ + 49 f. 
 
 Solution. Let k represent the first term of this root, and u 
 the next term ; then 
 
 4 s^ - 28 st^ + 49 t' contains (k + uf, i.e., k- + 2 few + u\ 
 
 Now the first term of the square root of 4 s^ — 28 st^ + 49 f 
 is, manifestly, 2s, i.e., k = 2s; and the next root term may be 
 found as above, thus : 
 
 * For a more detailed discussion of this topic, see El. Alg. § 125. 
 
 t Twice the root already found at any stage of the work is usually called 
 the trial divisor (T. D.) and the trial divisor plus the next root term is called 
 the complete divisor (C. D.). % See Note 1, p. 181. 
 
10-117] INVOLUTION AND EVOLUTION 181 
 
 4 .s2 - 28 .<?«3 + 49 «« I 2 s - 7 i-"^ 
 
 k^ = (2 s) 
 
 •2-4 .V ' 
 
 T.T>.=2k = 4s 
 
 C. D. =2^ + w=4s-7<5 
 
 - 2«.s'«' + 49 «6 
 
 - 28 st^ + 49 ^ = (2 A: + m) w 
 
 Checks : (1) Square 2 s — 7 f, or (2) substitute special values 
 for s and t (gL § 25). 
 
 Ex. 2. Find the square root of 9 x^ -\- 6 x^ — 11 x^ — 4: x -{- 4. 
 
 Solution. At any stage of the process of finding this root, let 
 k represent the term or terms already known, and let ii represent 
 the next term ; then 
 
 9x^-{-6x^ — lla^ — 4:X-\-4: contains k^-{-2ku-\- \C\ [§ h^ 
 
 Here the first term of the root is 3 a^, i.e., k = 3x^, and the next 
 term (u) may be found as in Ex. 1, thus : 
 
 9x*+6a;3-llx2-4x+4 I 3 y.2 + X - 2 
 k^ = (3 a;2)2 = 9x4 
 
 T. D. = 2 ^• = 6 a;2 
 
 C.I).= 2k-{- u = 6x^ + x 
 
 T.B. = 2k* = 6x^ + 2x 
 
 C. D. = 2 A; + u = 6x^ + 2X-2 
 
 (5 x3 - 1 1 x2 - 4 X + 4 
 
 y3 + .7:2 = (2 A: + it) u 
 
 12x2 -4x + 4 
 
 12 x2 - 4 X + 4 = (2 k*-h n) u 
 
 Checks : (1) Square Sx'-{-x-2', or (2) use § 25. 
 
 Note 1. Before applying the process of Exs, 1 and 2 a polynomial should 
 be arranged according to ascending or descending powers of one of its 
 letters. 
 
 Note 2. Exs. 1 and 2 show how to find the square root of a polynomial 
 which is an exact square ; i.e., if the above process is continued until a zero 
 remainder is reached, then the square of the root thus found will be the given 
 polynomial. If, however, the same process is applied to a polynomial 
 which is not an exact square, then as many root terras as desired may be 
 found, and the square of this root, at any stage of the work, equals the 
 result of subtracting the corresponding remainder from the given polynomial ; 
 such a root is called an approximate root, and also the root to n terms. 
 
 * Here k represents 3 x2 + x, and u represents — 2. Observe also that the 
 first and second subtractions in this solution are together equivalent to the 
 subtraction of (3 x2 + x)2 from the given expression. 
 
182 IIIGB SCHOOL ALGEBRA [Ch. XI 
 
 EXERCISE LXXIX 
 
 Find the square ropt of each of the following expressions, and 
 check your work : 
 
 3. 9 m^* - 66 m2 + 121. 5. 4: -\- S x - 4: a^ -\- x\ 
 
 4. 16/ +104 7-^ + 169. 6. l + 2m-3m2-4m3 + 4m^ 
 
 7. l-6y-{-5y^-\-12f-\-4:y\' 
 
 8. 9a* + 30a^a; + aV-40aa^ + 16a;*. 
 
 9. 4a^ + 17a^-22.r^ + 13a;*-24a;-4a^ + 16. 
 
 10. 4 a^ + 64 6* - 20 a^b + 57 a^b^ - 80 a¥. 
 
 11. 6a^2/ + 2a;«i/^-28aj.v^ + 9.T« + 42/« + 45a^/ + 43icy. 
 
 12. 3x''-2:f^-af-i-2x + l-\-x\ 
 
 13. 48a* + 12a2+l-4a-32a3 + 64a«-64a^ 
 
 14. 46x' + 25x'-Ua^-^0x + 4:xr'-\-25-12xi', 
 
 15. x'^ - 2 x'^y + 2 afz^ -2 yz^ + y^ -hz\ 
 
 16. ^ + 16ay + 8a^v'. 18. 9 a;2_24a; + 28-^^ + i. 
 
 17. aj2_j.2a;_i_?+l :* 19. 4^2- 20« + 21 + — + i. 
 
 X x^ a a^ 
 
 20. n^ + 4 71^ + -i + 2 7i + 4 + 4 n^. 
 
 21. a)'' + i + 4aj3 + i + 6a^ + -i- + 5 + 5a; + 5. 
 
 cc* ar 4 ar x 
 
 22. ^4_^_L..ii_. 
 
 r / ^ 4e^ 
 
 23. (a; - 2/)2 -2{xy -i-xz- y^ - yz) + (y + zf. 
 
 25. 1 + », to three terms (cf. Note 2, p. 181). 
 
 * Observe that this expression is already arranged according to descending 
 powers of x. 
 
117-118] INVOLUTION AND EVOLUTION 183 
 
 26. 1+2 m-, to four terms. 
 
 '27. a^ -f 1, to three terms. 
 
 28. 1 -f-.aj — a^, to four terms. 
 
 29. X* -\- 2 afy -{- y* -\- xy^ + ar^2/^ ^o four terms. 
 
 30. In Ex. 1 is not — 2 s, as well as -f 2 s, a square root of the 
 first term ? Solve Ex. 1, using — 2 s. Does your result check ? 
 
 31. Solve Ex. 2, using —Sx^ as the square root of the first 
 term, and compare your answer with that found in the text. 
 
 32. By extracting the square root until a numerical remainder is 
 reached, show that x*-\-4:a^-{-SQif-\-Sx—5 equals (ic-+2a; + 2)^—9, 
 and thus find the factors ofic'* + 4a^ + 8a:^ + 8ic — 5. 
 
 33. As in Ex. 31, find the factors of x'*'-\-6af-\-llx^-\-6x — S; 
 also of a« - 6 ci^ + 10 a^ + 9 a^ _ 30 a + 9. 
 
 118. Square roots of arithmetical numbers.* In order 
 to proceed systematically, and find the successive digits of 
 the root in their order from left to right, we first separate 
 the given number into periods of two figures each, toward the 
 right and left from the decimal point. The root may then 
 be extracted by virtually the same process as that used in 
 §117. 
 
 Note. The reason for the separation into periods lies in this : the square of 
 any number of tens ends in two ciphers, and hence the first two digits at the 
 left of the decimal point are useless when finding the tens' digit of the root ; 
 they are, therefore, set aside until needed to find the units' digit of the root. 
 So, too, the square of any number of hundreds ends in four ciphers, and hence, 
 for a like reason, two periods are set aside when the hundreds' digit of the 
 root is being found, and so on. Similarly for the periods at the right of the 
 decimal point. 
 
 Ex. 1. Eind the square root of 1156. 
 
 Solution. This number consists of two periods, hence its 
 square root consists of two digits. Again, since 9 is the greatest 
 
 * For a more complete discussion of this topic see El. Alg. § 126. 
 
184 
 
 HIGH SCHOOL ALGEBRA 
 
 [Ch. XI 
 
 square in the left-hand period, therefore 3 is the first figure in the 
 root. Now, let k represent the known part of the root at any stage 
 of the work, and u the next root figure, then 
 
 1156 contains Tc^ + 2ku-\- u^, 
 
 and the work may be arranged as follows : 
 
 k^ = (30)'-^ = 
 T. D. = 2 A: = 60 
 C. D. = 2 ^' + w = 60 + 4 
 
 11'56 I 30 + 4 : 
 900 
 
 34 
 
 256 
 256 
 
 {2 k + u)u 
 
 Check: (34)2 = 1156. 
 Ex. 2. Find the square root of 315844. 
 Solution. Using k and u as in Ex. 1, we have 
 315844 contains A;^ + 2 ku + u% 
 and the work may be arranged as follows : 
 
 
 fc-^ =(500)2 = 
 
 31'58'44 1 500 + 60 + 2 = 
 250000 
 
 562 
 
 T.D. 
 CD. 
 
 = 2 A = 1000 
 = 2k + u = 1060 
 = 2k* =1120 
 = 2k + u = 1122 
 
 65844 
 
 63600 =(2 A; + u)u 
 
 
 T.D. 
 CD. 
 
 2244 
 
 22U =(2 k-\-u)u 
 
 
 Check: (562)=-' = 315844. 
 
 Note. When some familiarity with the above process has been gained, 
 the work may be abridged by omitting unnecessary ciphers, as shown below 
 in finding the square root of 315844 and of 10.5625. 
 
 31'58'44 
 25 
 
 10.'66'25 
 9 
 
 13.26 
 
 106 
 1122 
 
 658 
 636 
 
 2244 
 2244 
 
 62 
 645 
 
 156 
 
 124 
 3225 
 3225 
 
 * Here k = 560 ; compare footnote, p. 181. 
 
118-119] INVOLUTION AND ^VOLUTION 185 
 
 EXERCISE LXXX 
 
 Extract the square root of each of the following numbers, and 
 check your results : 
 
 3. 1296. 6. 9216. 9. 667489. 12. 17424. 
 
 4. 841. 7. 12.96. 10. 26.2144. 13. 36.8449. 
 
 5. 2209. 8. 62.41. 11. 1664.64. 14. 101.0025. 
 
 15. How may the square root of a fraction be found? 
 Illustrate, using the fractions ^\ and f||. Is — 14 also a square 
 root of the latter fraction ? Why ? 
 
 16. A number contains one decimal place ; how many decimal 
 places in its square? How many, if the number contains two 
 decimal places ? if it contains three ? if it contains n ? Explain. 
 
 17. Show from Ex. 16 that if the right-hand period of a decimal 
 is incomplete, we must annex a cipher to complete it. Is this 
 true of the left-hand period of an integral number also ? 
 
 18. Extract the square root of 2 to two decimal places (cf. 
 p. 181, Note 2). How many periods of ciphers must be annexed to 
 2 for this purpose ? Why ? 
 
 Find the square root of each of the following numbers, correct 
 to two decimal places : 
 
 19. 13.5. 21. .017. 23. |. 25. 4|. 
 
 20. |. 22. 1.1105. 24. ■^. 26. .049. 
 
 27. Is V36 equal to V9- Vi? Is V27 (i.e., V9T3) equal to 
 3V3 where the roots are extracted to two decimal places? to 
 three decimal places ? 
 
 28. Is V450 (i.e., •\/225 • 2) equal to 15 V2 where the roots are 
 correct to two decimal places ? Show that V96 and 4 V6 are 
 equal, to at least two decimal places. 
 
 119.* Cube root of polynomials. The procedure here is like 
 that in § 117. 
 
 * Articles 119, 120, with Exercises LXXXI and LXXXII, may, if the 
 teacher prefers, be omitted till the subject is reviewed. 
 
186 HIGir SCHOOL algebra [Ch. XI 
 
 Since Qc^uy = 7c^-{-^k^u-h3ku^-hu\ 
 
 therefore VA;^ + 3 k'^u -\- 3 ku^ -{-u^ = k-{- u. 
 
 And this equation shows : 
 
 (1) that the j^rs^ tei^m of the cube root (viz., k) is the cube root 
 of the first term of the polynomial ; 
 
 (2) that the trial divisor for finding the next term of the root 
 is3A^; 
 
 (3) that the complete divisor is 3 A;^ + 3 ku -{-u^. 
 
 The actual work may be arranged thus : 
 
 F + 3 khi + P> ku^ + v^ \k-\-u 
 
 J^ 
 
 T. D. = 3 k^ 
 
 C. D. = 3 A:2 + 3 ku + u 
 
 3 A:2m + 3 kv!^ + u^ 
 
 3 k'^u + ^ku^ + m3 = (3 k'^ + 3 A;?( + ifi) u 
 
 
 
 If now we let k represent the part of the root already known 
 at any stage of the work, and let u represent the next term, then 
 the above method will serve to extract the cube root of any 
 "arranged" polynomial (cf. § 117, Exs. 1 and 2). 
 
 Thus, the cube root of ofi -9x^ + 30 x^ - i^x^ + SOx'^ - 9x + 1 may be 
 found as follows : 
 
 I a;2 - 3 a; + 1 
 5c6 _ 9 a:^ + 30 0^4 - 45 a:3 + 30 ic2 _ 9 X + 1 
 
 (x^y = x6 
 
 T. D. =3(^2)2 = 3 X* 
 
 CD. =3x4-9x3 + 9^2 
 T.D. =3(x2-3x)2 = 3x*- 18x3 + 27x2 
 C. D. = 3 (x2 - 3 x)2 + 3 (x2 - 3 X) + 1 
 = 3x*- 18x3 + 30x2 -9x + 1 
 
 - 9x5 + 30x4 -45x3 + 30x2 -9x + l 
 -9x5 + 27x4-27x3 
 
 3x4- 18x3 + 30x2 -9x + 1 
 3x4 - 18x3 + 30x2 -9x + l 
 
 
 
 EXERCISE LXXXI 
 
 Find the cube root in Exs. 1-14, and check your results ; 
 
 1. Sa^-12x^ + 6x-l. 
 
 2. 27x^-189x'y-hUlxy^-34.Sf/ 
 
 3. 125 n"" - 150 mn^-Sm^-\- 60 m'n. 
 
 4. 225uh + lS5uv^ + 125u''-\-27'i^. 
 
liy-120] INVOLUTION AND EVOLUTION 187 
 
 5. a;^-20a^-6aj-f 15a;4-6x-^4-15ic2 + l. 
 
 6. 3aj^ + 9a;^-|-ic'' + 8 + 12a; + 13a^ + 18a;l 
 
 7. 342 x^ - 108 a; - 109 a^ + 216 + 171 x"" -21x^ + 27 x\ 
 
 8. 156a;*-144aj«-99a^ + 64aj6 4-39aj2-9a; + l. 
 
 9. u X ■\- - -112 -^ + — + 0^-12 xUai. Ex. 17, p. 182). 
 
 10. 20 + i^ + 15c2 + c« + | + i + 6o^ 
 
 11. 25 + ^ + 82/« + 302/-12 2/-^-25-i|. 
 
 2/ 2/ y 
 
 12. 6 aV-4aV -2 aV + 6 aV + 3 a^a; + «' + a^^ - 3 a«8. 
 
 13. 108 'ifz -21 f- 90 2/V + 8 2« - 80 y V ^ 50 yh^ + 48 2/2^ 
 
 14. 'y3» -f- 9 ^Sn-S ^ 21 i;3"-2 — 42 'y3«-4_ 35 -y3n-5_9 ^3«-l_g ,y3«-6^ 
 
 15. Find the first three terms of v 1 + x. 
 
 16. Find the first four terms of V 1 — 3 a? + aj^. 
 
 120* Cube root of numbers. The cube root of a number may be 
 found by virtually the same process as that used in § 119 for 
 finding the cube root of a polynomial (cf. §§ 118 and 117). The 
 number should be separated into periods of 3 figures each, begin- 
 ning at the decimal point (why ?), and the right-hand period, if 
 incomplete, should be completed by annexing ciphers. 
 
 Ex. 1. Find the cube root of 42875. 
 
 SOLUTION 
 
 k -\- u 
 42'875 1 30 + 5 = 35 
 27000 
 
 15875 
 
 T^ = (30)8 = 
 T.D. = 3 A:2 = 3 . (30)2 ^ 2700 
 CD. =Sk^ + 'Sku + u^ , 
 
 = 3 (30)2 + 3 (30)5 + 52 = 3175 1 15875 
 
 
 
 * See footnote, p. 185. 
 
 HIGH SCH. ALG. — 13 • 
 
188 HIGH SCHOOL ALGEliRA [Cii. Xi 
 
 Ex 2. Find the cube root of 9825.17, correct to tenths. 
 
 SOLUTION 
 
 F = (20) 
 
 9'825.'170 |20+ 1 + .4 = 21.4 
 8000 
 
 564.170 
 539.344 
 
 T.D. = 3 k-^ = 3 (20)2 ^ 1200 1825 
 CD. =3^•2 + 3^•« + M2 
 = 3 (20)2 + 3 (20) . 1 + 12 = 1201 1261 
 T.D. = 3A;2=: 3(21)2 =1323 
 CD. = 3 (21)-^ +3 (21) (.4) + (.4)2 = 1348.36 
 
 24.826 
 Check. (21.4)=^ = 9825.17 - 24.826 (cf. Note 2, p. 181). 
 
 EXERCISE LXXXII 
 Extract the cube root of each of the following numbers : 
 
 1. 1728. 3. 31855.013. 5. 39304. 
 
 2. 571787. 4. 148877. 6. 426.957777. 
 
 7. 75.686967. 9. .04, to two decimal places. 
 
 8. 34.7, to two decimal places. 10. 3^, to two decimal places. 
 
 121. Transformation of indicated roots.* From the defini- 
 tion of the symbol -Va (§ 113) it follows that, whatever the 
 values of h and k, 
 
 ■Vm = hVk: (1) 
 
 for (h-Vky^JiVk-h^k 
 
 = hh--Vk^ 
 
 = h^k, [since Vk Vk = k~\ 
 
 i.e., (hVky=h%, 
 
 and h VJ is, therefore, a square root of hj^k. 
 
 Equation (1), read forward, tells how to simplify an indi- 
 cated square root of a number which contains a square factor ; 
 and read backward, it tells how to insert a coefficient under 
 
 * Omit § 121 if radicals are to be studied before quadratics : see Preface. 
 
120-121] 
 
 INVOLUTION AND EVOLUTION 
 
 189 
 
 a square root 8i(/n. Let the pupil translate this eqiuttion 
 into verbal language, reading it both ways. 
 
 It also follows from the definition of -Va (§ 113) that 
 
 (V^^0^=-^- (2) 
 
 Equations (1) and (2) will be useful in Chapter XII. 
 
 EXERCISE LXXXIII 
 
 Simplify the following expressions [cf. Eq. (1), § 121]: 
 1. Vl2. 
 
 2. V50. 
 
 3. V48. 
 
 4. V63. 
 
 5. V4a26. 
 
 '■4- ■ 
 
 "■ x/f- 
 
 
 20. 
 
 |V9. 
 
 23. —5cVb. 
 
 6. V54ajy. 
 Insert the following coefficients under the radical signs 
 
 15. 3V2. 19. -i-VS. 22. aV2^. 
 
 16. 5V7. 
 
 17. — 4V5. 
 
 18. 7ViO. 21. f Vii. 24. |aV6'-4ac. 
 Expand the following expressions, and unite like terms : 
 
 25. (3+V5)l 28. (6-V^2)l 31. [^(3-2V5)?. 
 
 26. (3 + V^^)'. 29. (&-V6'-4ac)l 
 
 27. (-H-V3)'. 30. (6-V4ac-62)^ 
 
 32. 
 
 D 
 
 & + V62 - 4 
 
 ac 
 
 2a 
 
CHAPTER XII* 
 
 QUADRATIC EQUATIONS (Elementary) 
 
 I. EQUATIONS IN ONE UNKNOWN NUMBER 
 
 122. Definitions. A quadratic equation has already been 
 defined (§ 93) as an equation wliicli, when simplified, is of 
 the second degree in the unknown number or numbers. 
 Thus, 3 §2 _ 4 = 7 s^ aoiP' + 5a; + c = 0, and dm = 4:m^ are 
 quadratic equations in s, x^ and 7n, respectively. 
 
 By transposing and uniting terms every quadratic in x^ 
 say, may evidently be reduced to the standard form 
 
 ax^ -{- bx -^ c = 0, 
 wherein a, 6, and c represent known numbers, and are usu- 
 ally called the coefficients of the equation. The term free 
 from X, viz., c, is called the absolute (also constant) term. 
 
 U.g.^ by transposing, etc., 2x'^-^b — Sx = lx— S becomes 
 2:^2 — 10a;-|-13 = 0: hence, for this particular equation a= 2, 
 5 = -10, and c = U. Similarly, 6x^- 2x=Sx^- 4:-2x 
 becomes 3 a:^ -}- 4 = ; here a = 3, 5 = 0, and c = 4:. 
 
 An equation of the form ax^ + c = is often called a pure 
 quadratic, while one containing both the first and second 
 powers of the unknown number 'is called an affected quadratic. 
 
 123. Solution of pure quadratics. All pure quadratic equa- 
 tions may be solved like Exs. 1 and 2 below. 
 
 Ex. 1. Given 3 a;^ _ 12 = ; to find x. 
 
 * Chapter XII may, if the teacher prefers, be omitted until Chapters XIV 
 and XV have been studied: see Preface. 
 
 190 
 
122-123] QUADBATIC EQUATIONS 191 
 
 Solution. On dividing through by 3, and transposing, the 
 
 given equation becomes 
 
 a;2 = 4, 
 
 whence x= ±2,* 
 
 i.e., a; = 2ora;=— 2; 
 
 and each of these values is found to check. 
 
 45 s^ 
 
 Ex. 2. Solve the equation 5 -— - = 0. 
 
 16 
 
 Solution. On dividing the given equation through by 5, clear- 
 ing of fractions, and transposing, we obtain 
 
 whence 3 s = ± 4, 
 
 I.e., s=4ors = — I; 
 
 and each of these values is found to check. 
 
 EXERCISE LXXXIV 
 
 Solve and check : 
 
 3. 4a^ = 36. 11. 3cx'-10Sc^ = 0. 
 
 4. x' = ^. 12. (a + l)V = 4a2. 
 
 5. i_a^=_48. 13. (k-6y = 72-^12k. 
 
 6 — = 27 ■*■*• <^-^) = 2-^^' 
 
 4 * 15. a;(a;4-l)+3a^ = x + |. 
 
 7. ^-Sy = 0. 16. (r-Sf = 25. 
 
 ^y 17. (u+iy-^%=o. 
 
 8. 4('y2 + 3)=32. 18. (a?-a)2 = 9&l 
 
 9. a^ + 3a; = 3(a;+ll)-ll. 19. 4a^-l = a2 + 2a. 
 
 ^ (, + 3)(r-3), 20. ^(x-^)=(l-f 
 
 9 8 16^ ^ V 8. 
 
 * Using the double sign in each member here gives ± x= ±2, 
 i.e., either x = 2, (1) or - a; = 2, (3) 
 
 or a; = - 2, (2) or - ic = -2. (4) 
 
 But (3) and (4) give the same values of x as (2) and (1), respectively; 
 hence in solving such an equation as x^ = 4, the double sign need be used in 
 one member only. 
 
192 HIGH SCHOOL ALGEBRA [Ch. XII 
 
 21. If X and y stand for unknown numbers, tell which of the 
 following equations are simple, and which quadratic (cf. § 93) : 
 
 aV -\-a^x-{-a = 0\ ~ =-; 6x—ly = ll; b x + xy — 1 y — 11', 
 
 Z X 
 
 y 2/ + 2 
 
 22. Reduce 5a:^H-2 — 8a; = 4(8 — x) to the " standard form." 
 What are its coefficients ? What is its absolute term ? 
 
 23. Are the equations in Exs. 3-14 pure or affected? Explain. 
 What is the absolute term in Ex. 9? 
 
 24. Show that the equation of Ex. 16 is a pure quadratic in 
 r — 3 but an affected quadratic in r. 
 
 Solve each of the following equations for each letter it contains : 
 
 25. S = \qt\ 26. 1 = ^. 27. E=^^ 28. R = K-- 
 
 s V- 2 d- 
 
 A 
 
 29. The square on the hypotenuse (longest side) of a 
 right-angled triangle equals the sum of the squares on 
 the other two sides. In the right-angled triangle ABC, 
 the hypotenuse AO = 5, while -B(7=| AB; find AB 
 and BC. 
 
 30. The area of a square is 169 square inches; find the perime- 
 ter and the diagonal of the square. 
 
 31. How many rods of fence will inclose a square garden 
 whose area is 2^ acres ? 
 
 E B 32. From a rectangular field, ABCD, whose width 
 is f of its length, there is cut off a square field, 
 AEFD, whose area is 10 acres. Find the area of 
 the rectangular field. 
 
 33. The surface area of a cube is 150 square 
 inches. Find one edge and also the volume of the cube. 
 
 124. Quadratics solved by factoring. A quadratic equation 
 ma}^ often be easily solved by reducing it to standard form 
 (§ 122), and then factoring its first member (§ 72). 
 
123-124] QUADRATIC EQUATIONS 198 
 
 Ex. 1. Solve the equation 3 ic^ + 4 = ar' — li a; + 16. 
 
 Solution. On transposing, uniting, and dividing through by 
 2, the given equation becomes 
 
 ar -h a; — 6 = 0, 
 i.e., (x-2){x-{-3) = 0. 
 
 Now, as in § 72, this last equation is satisfied when 
 a^-2 = or ic-|-3 = 0, 
 
 I.e., when x = 2 or when a; = — 3 ; 
 
 and each of these vahies of x, when substituted in the given 
 equation, is found to check. 
 
 Ex. 2. Solve the equation x (a; — 3) + a; + 2 = 2 (1 — .-c^). 
 Solution. On transposing, etc., the given equation becomes 
 3ar'-2a; = 0, 
 
 i.e., a;(3a;-2) = 0, 
 
 wlience x = or -; 
 
 3' 
 
 and these values of x are found to check. 
 
 EXERCISE LXXXV 
 
 Solve the following equations by factoring, and check the roots 
 in each case : 
 
 3. /-5?/-24 = 0. 
 
 4. m--16 = 0. 
 
 5. a^ + 5x = 21 + x. 
 
 6. 5 a;-=a^- 14. 
 
 7. 5s- = Ss. 
 
 8. 2v--30 = 9v-v- 
 
 9. 2ar'-x=3. 
 
 10. 4. 0^ = 9. 
 
 11. 2c^ = x(x-^r). 
 
 12. (4 x)- = 14 (4 x) — 
 
 13. 22x + Sx' = 4:x'- 
 
 14. ^-^"- = 38. 
 
 
 15. 
 
 .t'--4a; = 117. 
 
 
 16. 
 
 13y + 2f=.5y-^4.y\ 
 
 
 17. 
 
 2iK2-20x = x- -51. 
 
 
 18. 
 
 3a(3a-l)=3a + 24. 
 
 
 19. 
 
 2/-72/ 4-3 = 0. 
 
 
 20. 
 
 ic (a; + 7n) =n{—x—m) 
 
 
 21. 
 
 ~U0 + x^=^-23x. 
 
 
 22. 
 
 7-2_5r = 5r-25. 
 
 
 23. 
 
 Zaj^ — Ikx -}- A:w = ma;. 
 
 )•). 
 
 24. 
 
 5^2_3 =10^-3^1 
 
 48. 
 
 25. 
 
 ao(? -\-bx — ex. 
 
 
 26. 
 
 XT _X i» , -j 
 
 bo be 
 
194 HIGH SCHOOL ALGEBRA [Ch. XII 
 
 27. If a quadratic equation in one unknown number has no 
 absolute term, show that one root of the equation must be zero. 
 
 125. Completing the square. What must be added to 
 x^+Qx to make it the square oi x+ S? What must be added 
 to m^—14:m to make it the square of m — 7 ? 
 
 Since Qx ±k')^ = x^ ±2kx -\- k^, therefore the expression 
 x^±2 kx, whatever the value of k, lacks only the term k^ of 
 being the square of x±k; hence, if the square of half the 
 coefficient of the first power of x he added to an expression of 
 the form x^ + bx, the result will be an exact square. 
 
 Such an addition is usually spoken of as completing the 
 square. 
 
 j^.^., if (f )2 is added to «/2 + 5 ?/ it becomes (?/ + 1)2. 
 
 126. Solution of quadratics by completing the square.* 
 
 There are many quadratic equations which cannot easily be 
 solved by the method of factoring given in § 124. All quad- 
 ratic equations, however, may be solved by the method of 
 completing the square, which is illustrated below. 
 
 Ex. 1. Solve the equation 2a;^ — 3— 5a; = 7a;+ll. 
 
 Solution. On transposing, etc., this equation becomes 
 
 Now, adding 9 to each member (§ 125, and Ax. 1), we obtain 
 a^-6x + 9 = 16, 
 i.e., (a; -3)2 = 16, 
 
 whence (§ 123) a; - 3 = ± 4, 
 
 i.e., a; — 3 = 4 or a; — 3 = — 4, 
 
 and therefore a; = 7 or — 1. 
 
 Moreover, these values check, and are, therefore, the required 
 roots. 
 
 *For the solution of quadratic equations by means of a formula, see § 178. 
 
124-126] QUADRATIC EQUATIONS 195 
 
 Ex. 2. Solve the equation x^ + 11 a; + 1 = 8 a:. 
 Solution. On transposing, the given equation becomes 
 
 a^-|-3ic = — 1, 
 whence, adding (f )2, x'-\-Sx + (^f = - 1 + (f )', [§ 125 
 
 i.e., (^-\-iy = h 
 
 and hence . a; + f = ± V|= ± iV5, [§121 
 
 3^1 /- -3±V5 
 
 ^ = -2^2^^ = ^ 
 
 Moreover, these values otx, viz., ~'^^ ^ and ~^~ ^ , check (cf. Ex. 
 25, p. 189), and are, therefore, the required roots. 
 
 EXERCISE LXXXVI 
 
 3. Solve the equation ax^ -\-bx+ c =0. 
 
 Solution. On transposing and dividing by a, this equation becomes 
 
 whence 
 
 a a 
 
 5 \2 12 c b^-4ac 
 
 a \2al ^a^ a 4 a^ ' '■* 
 
 \ 2a/ 4 a^ 
 
 therefore ^ + A = ± J^!^ ^ ± ^Z"' - 4 ac ^ [5121 
 
 2 a ^ 4 a^ 2 a 
 
 ig ^^ b ±Vb'^-4ac ^ -b±y/b'^-4:ac _ 
 
 2a 2a 2a 
 
 Moreover, thesevaluesof a;, viz., -h + ^b^-4.ac ^^^ -b-Vb^-4a c^ 
 
 2a 2a 
 
 check (cf. Ex. 29, p. 189) and are, therefore, roots of the given equation. 
 
 4. What must be added to each of the following expressions 
 in order to complete the square (cf. § 125): x^-\-Sx', P^ — 5P; 
 (x + yy-4.(x + y)? 
 
 5. How do we find the number which added to r^ + ar com- 
 pletes the square ? Explain. 
 
6. 
 
 m^ -6 m =40. 
 
 7. 
 
 y'-10y = 75. 
 
 8. 
 
 W = 2x-hx'. 
 
 9. 
 
 -S = 2x^-h'^0x. 
 
 10. 
 
 a^ = x-\-l. 
 
 11. 
 
 Sx'-2x = l. 
 
 196 HIGH SCHOOL ALGEBRA [Ch. XII 
 
 Solve the following equations by the method of completing the 
 square, and check the roots in each case : 
 
 20. a!2-3a;-2 = 0. 
 
 21. ar^- 3 a; H- 4 = 0. 
 
 22. |c^— 7c = c(cH-l). 
 
 23. 6 + ot = et\ 
 
 24. 12a^- x = 6, 
 
 25. «- = -j6' + 2. 
 
 12. 2a;2-f3 = 7a;. 26. r'-er=f. 
 
 13. 3a^-10 = 7a;. 27. 3 x^ -\-5x-7 =x^-2 x. 
 
 14. (2 2/-3)2 = 6 2/ + l. 28. 8m-10 = 3m^. 
 
 15. m(m + 4) = 7. 29. A-a;-|ar^+2=0. 
 
 16. a2-6a + 10 = 0. 30. (y -Sf -4.(y-3) = 117. 
 
 17. 3 ('y^-'?;) =2^2.^ 5 V + 4. 31. (2m-3)2-6(mH-l)4-8=0. 
 
 18. y^ — 2cy = l. 32. cV + 2(Za;=— e. 
 
 19. r2 + 2ar = d 33. (n + 1)2- 8 (n + 1) = 16. 
 
 34. Write a carefully worded rule for solving such quadratic 
 equations as those in Exs. 6-33 above. 
 
 35. Show that the rule asked for in Ex. 34 will serve to solve 
 such an equation asa^ + 6i« = 0. Is this equation more easily 
 solved by completing the square or by factoring ? 
 
 127. Avoiding fractions in completing the square. The 
 
 method employed in § 126 for completing the square often 
 introduces fractions into the work, and these sometimes be- 
 come troublesome (cf. Exs. 2 and 3, p. 195). A method 
 which avoids fractions is illustrated below. 
 
 Ex. 1. Solve the equation 5 x^ —6x= — 1. 
 
 Solution. On multiplying through by 5, we obtain 
 25aj2-30a;=-5, 
 i.e., (5aj)2-6(5a;) = -5, 
 
 whence, adding 9, (5 a;)^ — 6 (5 a;) + 9 = 4, 
 i.e., (5 X- 3)2 = 4; 
 
12(i-127J QUADRATIC EQlATIOlSfS 197 
 
 therefore 5 ic — 3 = ± 2, 
 
 from which 5a; = 3±2=:5orl, 
 
 i.e.f x = \ or I ; 
 
 and each of these values is, on substitution, found to check. 
 
 Ex. 2. Solve the equation as? -{-hxz^ — c. 
 
 Solution. On multiplying through by 4 a, we obtain 
 4 a^ic- + 4 abx = — 4 «c, 
 i.e., (2 axf + 2 6 (2 ax) = - 4 ae, 
 
 whence, adding 6^, (2 aa;)^ + 2 6 (2 aa;) + 6^ = 6^ — 4 ac, 
 t.e., (2 ax -j- 6)- = 6^ _ 4 ^f^ . 
 
 therefore 2ax-\-h = ± V 6^ — 4 ac, 
 
 from which x = -l>±^h^-^ ^^^^ Ex. 3, p. 195. 
 
 • 2 a 
 
 Note. From the above solutions we see that, if an equation of the form 
 aofi + &a; + c = is multiplied through by a or 4 a, according as b is even or 
 odd, fractions can be avoided in the solution. 
 
 EXERCISE LXXXVII 
 
 3. What must be added to each of the following expres- 
 sions in order to complete the square : 4 x- -f 8 x; 9 m* + 12 m^; 
 25 cH"" - 10 cd; and 4 cV - 4 cm ? 
 
 In each of Exs. 4-11 belo^, (1) name the factor 'by which both 
 members of the equation must be multiplied if fractions are to be 
 avoided in completing the square; (2) solve the equation by the 
 method of § 127. 
 
 4. 5 0^4-6 x = 8. 8. 322 = 2 + 52. 
 
 5. 3/ + 4?/ = 95. 9. 7 = 2x + 3.^;^'. 
 
 6. 2m2-f3m = 27. 10. 6x^-x-^ = 0. 
 
 7. 2^2^7^ + 6 = 0. 11. 15a;2-7a!-2 = 0. 
 
 12. By the method of § 127, solve Exs. 12-14 and 20-25, p. 196. 
 
198 HIGH SCHOOL ALGEBRA • [Ch. XII 
 
 Solve the following equations by the method of § 127 : 
 
 f^3t-{-5 ^ t + 1 u'-^u '^^ + ^ ^0 
 
 2 3 * 3 "^ 4 
 
 14. ma^ — 6 a; + 3 = 0. 18. 3/ — 4% + 2 = 0. 
 
 15. x^-\-px + q=:0. 19. mx^ -^ nx -\- p — 0. 
 
 16. mx^ = 2nx — k, 20. ax^ — 5ax = a — 11. 
 
 128. Fractional equations which lead to quadratics. As in 
 
 § 97, so here, we first clear the given equation of fractions, 
 then solve the resulting integral equation, and finally check 
 the results so as to guard against the introduction of extra- 
 neous roots (§ 97, note). 
 
 Ex. 1. Solve the equation —^ — |- 1 = 3 a;. 
 
 ^ x + 2 
 
 Solution. On clearing of fractions, etc., this equation becomes 
 3 a;2 + 4 .T - 7 = 0, 
 whence, solving as in § 127, we obtain 
 
 aj = l or — 1^; 
 and each of these values, when substituted in the given equation, 
 is found to check. 
 
 Ex. 2. Solve the equation -^— _^4a; + 3_ 2 x" 
 
 1 — x x-\-l x^ — 1 
 Solution. On clearing of fractions, etc., we obtain 
 a^_2x-3 = 0, 
 whence a; = 3 or - 1, [§ 126 
 
 of which 3 checks, but — 1 is extraneous (§ 97, note). 
 
 EXERCISE LXXXVIII 
 
 Solve the following fractional equations, being careful to ex- 
 clude all extraneous roots : 
 
 3. 15 a; + - = 11. 5. - = -• 
 
 X bx-{-b x-\-± 
 
 4. l-2 + .. = 2. 6. -1- + _!- = §. 
 X X 1— sl+s3 
 
127-129] qUADRATlG EQUATIONS 199 
 
 7 __3 L_ = l. 11 2y + l 5^y-8 
 
 ' 2(a^-l) 4(aj + l) 8 ■ 1-22/ 7 2 
 
 8 ^"^ I ^ + ^ = 2/^ ^"^^ Y 12 ^^ + ^ I ct-2a; ^22. 
 
 07 + 2 a;— 2 V^~^/ 2a — x a-{-2x 
 
 9. _l_ + ^_= J_. 13. -*^ + 6 = «J£+26). 
 a; — 1 a;— 2 3 — a; a — a; a-{-b 
 
 10. ■20.+ 40 ^_3M:7. ^^_ . __ ._.__,. 
 
 :+3 s'+is-\-3 S + 1 X— 1 2 + 1 
 15.^^ + ^ -=8+ ^ 
 
 x + 5 (a; + 5)(a; — 2) a; — 2 
 
 129. Problems which lead to quadratics. As in § 50, so 
 
 here, the important steps in the solution of a problem are : 
 
 1. To translate the verbal language of the problem into 
 algebraic language, i.e., into equations. 
 
 2. To solve these equations. 
 
 3. To check, and interpret, the results. 
 
 Special emphasis should be laid upon testing and interpret- 
 ing results: a problem often contains restrictions upon its 
 numbers, expressed or implied, which are not translated into 
 the equations, hence the solutions of the equations may or 
 may not be solutions of the problem itself (cf. § 98). 
 
 Prob. 1. A farmer purchased some sheep for $ 168, and later 
 sold all but four of them for the same sum. If his profit on 
 each sheep sold was $ 1, how many sheep did he buy ? 
 
 SOLUTION 
 
 Let X = the number of sheep purchased. 
 
 Then — - = the number of dollars each sheep cost, 
 
 X 
 
 1f>8 
 and = the number of dollars received for each sheep. 
 
 A I. 168 168 ^ 
 and hence = 1 , 
 
 X — 4: X 
 
 therefore (§ 128) a; = 28 or -24. 
 
 "Profit on each sheep 
 being $1 
 
200 BIGH SCHOOL ALGEBRA [Ch. XII 
 
 The first of these values, viz., 28, is found to be a solution of 
 the problem as well as of the equation, but while the second satis- 
 fies the equation it cannot satisfy the problem, since the number 
 of sheep purchased is necessarily a positive integer. 
 
 Prob. 2. At a certain dinner party it is found that 6 times the 
 number of guests exceeds the square of | their number by 8; 
 how many guests are there ? 
 
 SOLUTION . ^ 
 
 Let X = the number of guests. 
 
 Then the expressed condition of the problem is 
 
 «-(¥)'=»■ 
 
 i.e., 2x^-27 x + 36 = 0, 
 
 whence x = 12 or |. 
 
 Here, too, an implied condition of the problem is that the 
 answer must be a positive integer ; hence f , although it satisfies 
 the equation, it is not a solution of the problem. 
 
 Prob. 3. The sum of the ages of a father and his son is 100 
 years, and one tenth of the product of the numbers of years in 
 their ages, minus 180, equals the number of years in the father's 
 age ; what is the age of each ? 
 
 SOLUTION 
 
 Let X = the number of years in the father's age. 
 
 Then 100 — x = the number of years in the son's age, 
 and the condition of the problem states that 
 
 whence cc = 60 or 30. 
 
 Although both 60 and 30 are positive integers, yet 30 is not a 
 solution of the problem : it would make the son older than the 
 father. Hence the father is 60 years old, and the son 40. 
 
 If, in the above problem, " two persons " be substituted for " a 
 father and his son," etc., then both solutions are admissible, and 
 the ages are either 60 and 40 years, or 30 and 70 years. 
 
1210 QUADliATlC EQUATIONS 201 
 
 EXERCISE LXXXIX 
 
 4. Divide 10 into two parts whose product is 22f . 
 
 5. Find two numbers whose difference is 11, and whose sum 
 multiplied by the greater is 513. 
 
 6. A man bought a flock of sheep for $ 75. If he had paid 
 the same sum for a flock containing 3 sheep more they would 
 have cost him $ 1.25 less per head. How many did he buy ? 
 
 Is each solution of the equation of this problem a solution of 
 the problem itself? Explain. 
 
 7. A clothier having bought some cloth for $30 found that if 
 he had received 3 yards more for the same money, the cloth would 
 have cost him 50 cents less per yard. How many yards did he 
 buy ? Has this problem more than one solution ? 
 
 8. Find two numbers whose sum is 10 and whose product 
 is 42. Can these numbers be real (see Note, § 114) ? 
 
 9. Find two consecutive integers the sum of whose squares is 
 61. How many solutions has the equation of this problem ? 
 Show that each of these is a solution of the problem also. 
 
 10. Are there two consecutive integers the sum of whose squares 
 is 118 ? Are there two numbers whose difference is 1, and the 
 sum of whose squares is 118 ? What are they? How does the 
 second of these questions differ from the first ? 
 
 11. Find three consecutive integers whose sum is equal to the 
 product of the first two. 
 
 12. Is it possible to find three consecutive integers whose sum 
 equals the product of the first and last ? How is the impossibility 
 of such a set of numbers shown ? 
 
 13. In selling a yard of silk at 75 cents, a merchant gains as 
 many per cent as there are cents in its cost. Find the cost. 
 
 14. A cow staked out to graze can graze over a circle 616 square 
 feet in area ; how long is the rope by which she is tied ? [The 
 area of a circle of radius r is tt • ?'^ ; 7r = 3|, approximately.] 
 
202 HIGH SCHOOL ALGEBRA [Ch. XII 
 
 15. Two circles are such that the difference of their radii is 
 3 inches, and the sum of their areas 279|- sq. in. Find the radius 
 of each circle. 
 
 16. Find two numbers whose sum is |, and whose difference is 
 equal to their product. How many solutions has this problem ? 
 
 17. The product of three consecutive integers is divided by 
 each of them in turn, and the sum of the three quotients is 74. 
 What are these integers ? How many solutions has this prob- 
 lem ? Explain. 
 
 18. If the product of two numbers is 6, and the sum of their 
 reciprocals is ff, what are the numbers ? How many solutions 
 has the equation of this problem ? How many solutions has the 
 problem itself ? Explain. 
 
 19. A merchant who had purchased a quantity of flour for 
 found that if he had obtained 8 barrels more for the same money, 
 the price per barrel would have been f 2 less. How many barrels 
 did he buy ? How many solutions has this problem ? Explain. 
 
 20. Why is it that the solutions of the equation of a problem 
 are not always solutions of the problem itself ? (Cf. § 129.) 
 
 21. In a rectangle whose area is 55i sq. in., the sum of the 
 length and breadth is 15 in. ; find the length. 
 
 22. Find the length of a rectangle whose area is 464 sq. in., 
 and the sum of whose length and breadth is 16 in. 
 
 Interpret the imaginary result in this problem (cf. § 98). Does 
 an imaginary result always show that the conditions of the problem 
 are impossible of fulfillment (cf. Prob. 8, p. 201) ? 
 
 23. The number of square inches in the surface area of a cube 
 exceeds the number of cubic inches in its volume by 8 times the 
 number of inches in one edge. Find the edge of the cube. How 
 
 many solutions has this problem? Explain. 
 
 24. In the trapezoid ABCD, whose area is 
 75 sq. in., the altitude BE equals BCy and AD 
 is 5 in. longer than BC \ find BC and AD. 
 
 A E 
 
 ^ [The area of ABCD = \{BC-\- AD) • BE.-] 
 
120 j QUADRATIC EQUATIONS 208 
 
 25. A triangle whose base is 2 in. longer than its altitude has 
 an area equal to that of a rectangle 10 in. by 4 in. Find the 
 base and altitude of the triangle. [The area of a triangle equals 
 half the product of its base and altitude.] 
 
 26. A boating club on returning from a short cruise found that 
 its expenses had been $90, and that the number of dollars each 
 member had to pay was less by 4i than the number of members in 
 the club. How many members were there in the club ? 
 
 27. If in Prob. 26 the expense of the cruise had been $145 
 the other conditions remaining unchanged, how many members 
 would the club contain ? 
 
 What is the significance of the fractional and negative results 
 in this problem ? Do such results always indicate that the con- 
 ditions of a problem are impossible of fulfillment ? 
 
 28. The number of miles in the distance between two cities is 
 such that its square root, plus its half, equals 12. What is this 
 distance ? Has this problem more than one solution ? Explain. 
 
 29. When a certain train has traveled 5 hours it is still 60 
 miles from its destination. If by traveling 5 miles faster per 
 hour, it could make the entire trip in 1 hour less than the sched- 
 uled time, find the entire distance ; also the actual speed. 
 
 30. The hypotenuse of a right-angled triangle is 10 inches, 
 and one of the sides is 2 inches longer than the other ; required 
 the length of the sides (cf. Ex. 29, p. 192). 
 
 31. It took a number of men as many days to dig a trench as 
 there were men. If there had been 6 more men, the work would 
 have been done in 8 days. How many men were there ? 
 
 32. A crew can row 5|- miles downstream and back again in 2 
 hours and 23 minutes ; if the rate of the current is 3-^ miles an 
 hour, find the rate at which the crew can row in still water. 
 
 33. From a thread whose length is equal to the perimeter of a 
 square, one yard is cut off, and the remainder is equal to the 
 perimeter of another square whose area is -| that of the first. 
 What was the length of the thread at first ? 
 
 HIGH SCH. ALG. — 14 
 
204 HIGH SCHOOL ALGEBRA [Ch. XII 
 
 34. The diagonal and the longer side of a rectangle are to- 
 gether equal to five times the shorter side, and the longer side 
 exceeds the shorter by 35 yards. Find the area of the rectangle. 
 ■ 35. A ladder 13 ft. long leans against a vertical wall. When 
 the distance from the base of the >vall to the foot of the ladder is 
 7 ft. less than the height of the wall, the ladder just reaches to 
 the top of the wall. How high is the wall (cf. Frob. 30) ? 
 
 36. If one train, by going 15 miles an hour faster than another, 
 requires 12 minutes less than the other to run 36 miles, what is 
 the speed of each train ? 
 
 37. A tank can be filled by one of its two feed-pipes in 2 hours 
 less time than by the other, and by both pipes together in 1|- 
 hours. In what time can each pipe separately fill the tank ? 
 
 38. The owner of a lot 56 rods long and 28 rods wide divided 
 it into 4 equal rectangular lots, by constructing through it two 
 streets of uniform width. If these streets decrease the available 
 area of the lot by 2 acres, what is their width ? 
 
 39. One of two casks contains twice as many gallons of water 
 as the other does of wine ; 6 gallons are drawn from each cask, 
 exchanged, and emptied into the other; it is then found that the 
 percentage of wine in each cask is the same. How many gallons 
 of water did the first cask originally contain ? 
 
 40. A and B together can do a given piece of work in a certain 
 time ; but if they each do one half of this work separately, A works 
 one day less, and B two days more, than when they work 
 together. In how many days can they do the work together ? 
 
 41. In going a mile, the hind wheel of a carriage makes 145 
 revolutions less than the front wheel, but if the hind wheel were 
 
 16 inches greater in circumference, it would then 
 make 200 revolutions less than the front wheel. 
 What is the circumference of the front wheel ? 
 
 42. In the figure, AB = BC= CD = DA = 10 
 inches, the diagonals AC and DB bisect each 
 other at right angles, and DB is 4 inches longer 
 than AC. Find the lengths of AC and DB, and 
 the area of the figure. 
 
l-2i1-K30] (QUADRATIC KQrATIONS 205 
 
 130. Equations in quadratic form. Equations which con- 
 tain onl}^ two different powers of the unknown number, one 
 of these powers being the square of the other, are said to be 
 in quadratic form. Thus, a^+1 x^=S, av?"" + 6m" + c = 0, 
 and (2 s^ + 1)^ — 5 (2 s^ 4. 1) = 4 are in quadratic form. 
 
 Such equations may be solved as follows : 
 
 Ex. 1. Solve the equation 2y?{y? -\-X)=h — oi?. 
 
 Solution. When simplified, the given equation becomes 
 
 or, putting y for a^, 2 2/^ + 3 ?/ — = 0, 
 
 whence y=\ oy -\, [§126 
 
 Li 
 
 i.e. (since y — oc^), a^ = 1 or — -f , 
 
 whence a; = ± 1 or ± V — |. 
 
 Moreover, each of these values (1, —1, V— f, and — V— f)* 
 checks (§ 121), and is therefore a root of the given equation. 
 
 Ex. 2. Solve the equation Vx' — 5 x + 10 = 2 .'e^ — 10 aj + 14. 
 Solution. This equation may be written thus : 
 Va;2-5a^ + 10 = 2(a.-2-5a; + 10)-6; 
 and, on putting y for Vaf — 5 ;r + 10, the given equation becomes 
 
 2/ = 2/-6, 
 whence y = 2 ov—l, [§ 126 
 
 i.e., Vaj^ — 5 a; + 10 = 2 or — -|, 
 
 and therefore aj'^ — 5 a; + 10 = 4 or f , [Squaring 
 
 whence a; = 2, 3, ^ ^ ~- ^ ^ or ^-V^ ^ ^^ ^26 
 
 all of which values check (cf. Ex. 28, p. 189), and are therefore the 
 required roots. 
 
 EXERCISE XC 
 
 Solve, and check as the teacher directs : 
 
 3. wt^ -16 = 0. 5. ?/' - 25 ?/2 + 144 = 0. 
 
 4. a;*_8x2 + 12 = 0. 6. n* = 18n2-32. 
 
206 BIGH SCHOOL ALGEBRA CCh. XII 
 
 o r» /- Hint. Write equation thus : 
 
 8. x = 3 — 2-Vx. ^ , 
 
 (2 k^ _ 1) - 6 \/2 A;-^ - 1 = 7. 
 
 9. 20x'-23x' = -6. , 
 
 19. y?-x-\-^x^—x-?, = ^. 
 
 10. 4 — ?ii^ = 18 m^. 
 
 20. 5a;2 + 2V5a;2_a; = 8 + iK. 
 
 11. 13 V^! — 5 = 62;. 
 
 . 1 21. ?2^±2 + --A_=2. 
 
 13. (x' + iy + 4(a^ + l) = 45. Hmx. Let»: = <±2 j^^^i^y 
 
 15. x-2=Vi^+6. 23. '^±5 + ^^ = 7. 
 
 r r' + b 
 
 16. (m + l)-5Vm + l = 6. y + 2 2(/ + 4) ^51 
 
 17. 2s-3 = 7V2i^^-12. ^*' / + 4 2/ + 2 5* 
 
 25. 5Vm2-10m + 42 = m2-10m + 6. 
 
 26. ^2_7i_pVi2_7^^18^24. 
 
 27. a;4 + 4i»3-8x + 3 = 0. 
 
 Hint. By extracting the square root of the first member this equation may 
 be written in the form (x^ + 2 a; - 2)^ = 1. (Cf. Ex. 32, p. 183.) 
 
 28. 2/' + 22/^ + 52/' + 42/ = 60. 
 
 29. 16a;^-8aj3-31i»2 + 8a; + 15 = 0. 
 
 30. 9a;* + 6a^-83a;2_28x + 147 = 0. 
 
 II. SIMULTANEOUS EQUATIONS INVOLVING QUADRATICS 
 
 [Two Unknown Numbers] 
 
 131. One equation simple and the other quadratic. Wlien 
 one equation is simple and the other quadratic, we may 
 alwa.ys eliminate by substitution [§ 103 (i)]. 
 
 Ex. 1. Solve the following system of simultaneous equations: 
 
 (1) 
 
 (2) 
 
 I 3 a; - 2 2/ = 3, 1 
 
l;jo-l3l] 
 
 QUADRATIC EQUATIONS 
 
 3 + 2j 
 »- 3 : 
 
 Solution. From Eq. (1), 
 whence, by substituting this value of x, Eq. (2) becomes 
 
 and, on expanding and simplifying, Eq. (4) becomes 
 
 102/^ + 32/-27 = 0, 
 whence (§ 126) ^ = f or - 1. 
 
 Substituting these values of y in Eq. (3) shows that 
 
 207 
 (3) 
 
 (4) 
 
 2/ = |or -f. 
 
 according as 
 
 ia? = 2, ix = — h 
 
 and \ g 
 
 satisfy the given system, and are therefore the solutions sought. 
 
 EXERCISE XCI 
 
 Solve the following systems of equations and check your results: 
 
 6. 
 
 'I 
 
 8. 
 
 x'-f = SO, 
 
 9. 
 
 x-\-y = 10. 
 
 
 x^-\.xy = 12, 
 
 
 x-y = 2. 
 
 10. 
 
 3 uv — v—lOuj 
 
 
 u-^2 = V. 
 
 
 4=x + Sy = 9, 
 
 11. 
 
 2x'-{-5xy=:3. 
 
 
 (a^ + 3)(2/-7) = 48, 
 
 12. 
 
 x-{-y = lS. 
 
 
 ■st=A2, 
 
 13. 
 
 s-t = 19. 
 
 
 1 + ^ = 10. 
 
 14. 
 
 f-f=^- 
 
 
 2s + 3^=10, 
 8. 
 
 f2s + 3^ = 
 \t(s + t) = 
 
 x^y 15' 
 x—y = — 10. 
 faj + f 2/ = 15, 
 3ir- — = 24. 
 
 y 
 
 1.5 ic— .52/ = 6, 
 .1 x^ -\- .1 xy = 16. 
 
 16 + 4'y + 2w2 = 5i^v, 
 11 v — 5 w = 4. 
 
 -H-l^ = J + % 
 xy ar ^/^ 
 
 2-1 = 7. 
 o; y 
 
208 HIGH SCHOOL ALGEBRA [Ch. XII 
 
 132. Both equations quadratic : one homogeneous. An 
 
 equation is said to be homogeneous if all of its terms are of 
 the same degree in the unknown numbers (of. §§ 34, 93). 
 
 If one of two given quadratic equations is homogeneous, 
 the system may always be solved as follows : 
 
 Ex. 1. Solve the following system of equations : 
 
 6£c24-5a;2/-^6/ = 0,l (1) 
 
 2x2-2/2 + 5 0^ = 9. J (2) 
 
 Solution. On dividing Eq. (1) by y^, it becomes 
 
 whence (§126) ^^|or^^-|, (4) 
 
 i.e., x = lyov x = -^y. (5) 
 
 On substituting the first of these two values of x, viz., ^y, in 
 Eq. (2), we obtain 
 
 2(t2/)^-/ + 5(f.v)=9, • 
 i.e., /- 30 2/4-81 = 0, 
 
 whence (§ 124) 2/ = ^^ or' 2/ = 3, 
 
 and, since x = ^y, the coiTesponding values of x are 18 and 2. 
 
 'x = 2 
 
 Moreover, these pairs of values, viz., <j ' and 
 
 .y = *^' J 
 
 J are 
 
 ^ = 3, 
 
 found to check, and are therefore solutions of the system. 
 
 Again, by substituting in Eq. (2) the second of the two values 
 of X in Eq. (5), we find two other solutions of the given system 
 
 VIZ. \ ^ ^ and \ ^' 
 
 Note. The above method may be somewhat simplified by substituting a 
 single letter, say v, for the fraction x/y in Eq. (3), i.e., by putting x — vym 
 the homogeneous equation. Thus, putting vy for x, Eq. (1) becomes 
 
 6 i?2y2 4. 5 W2/2 - 6 2/2 = 0^ 
 and hence, dividing by ?/2^ 6 v^ + 5 u — 6 = 0, 
 
 whence (§ 126) w = f or t7= -f ; 
 
 and, since x = vy, therefore x = ^ y or x = — ^ y. From here on the work 
 is the same as that already given, 
 
132-133] 
 
 QUADRATIC EQUATIONS 
 
 209 
 
 EXERCISE XCII 
 
 2. Which of the equations in Ex. 3-12 below are homogene- 
 ous ? Why ? Write a homogeneous cubic equation involving 
 the unknown numbers r and s. 
 
 By the method of Ex. 1 (or of the note) solve the following 
 systems of equations ; check your results as the teacher directs : 
 
 6. 
 
 xy-\-f=2S, 
 
 4:x'-27 xy + lSy^ = 0. 
 
 x^ -\-xy — 14: = y — Xj 
 
 2 a^ — 3y^ = xy. 
 
 ^(s + Q=36, 
 
 b s" -1^ St + 12 f = 0. 
 
 ^ 
 
 i3y?-5uv=:2v\ 
 \u{u — v) = 8. 
 
 10. 
 
 11. 
 
 12. 
 
 5a^-7a;2/-24 2/2 = 0, 
 
 xy-ir2y'^z=5. 
 
 aJ2/ + 3/-20 = 0, 
 
 5x^ = 13xy —^y^. 
 
 8aj2 + 2/2 = 36, 
 4a^-9a;2/ + 52/2 = 0. 
 2{:»? + f) = nxy, 
 x'-y' = 75. 
 5af + 4:xy = y% 
 x^-\-3x = 5-{-y. 
 
 Ex. 1. Solve 
 
 133. Both equations homogeneous except for the absolute 
 term. A system consisting of two quadratic equations each 
 of which is homogeneous in the terms containing the unknown 
 numbers can be solved by the method of § 132, Note. 
 
 x^ + 3xy + 2f = S, (1) 
 
 xy-4:f = 2. (2) 
 
 Solution. By substituting vy for x in Eqs. (1) and (2) we 
 obtain 3 vY + 3 vy^ -{- 2 y^ = S, (3) 
 
 and vY — vy^ — 4:y^ = 2, (4) 
 
 whence, from (3) and (4), respectively, 
 
 2 
 
 \x' 
 
 y 
 
 therefore 
 whence 
 
 3 -y^ + 3 v + 2 
 8 
 
 and 2/^ 
 
 V' 
 
 4' 
 
 3v^ + 3v + 2 'y2_-y_4' 
 
 v = - 2 or 9. 
 
 (5) 
 (6) 
 
210 
 
 HIGH SCHOOL ALGEBRA 
 
 [Ch. XII 
 
 Hence, from (5), 
 
 I.e. 
 
 if = l or/=3\, ^ 
 2/ = 1 or — 1 or 2/ = + V 3^; 
 
 34 
 
 or 
 
 V,>j, 
 
 and substituting these values of y and v in 
 
 x=vy, 
 we obtain, as corresponding values, 
 
 a? = — 2 or 4-2, and also + 9 V^^^ or — 9 V^; 
 and, checking, we find the solutions of the given system to be : 
 
 Note. The success of the '■'•vy method" employed above is due to the 
 fact that by eliminating the absolute term from the given system of equations 
 we obtain a homogeneous equation (cf. § 132). 
 
 EXERCISE XCIII 
 
 Solve the following systems, and check your results: 
 
 2. 
 
 3. 
 
 6. 
 
 7. 
 
 ar^ + 2/2 = 29, 
 xy = 10. 
 m^ — mn = 8, 
 mn -\-n^ = 12. 
 af — Sxy = — 14, 
 xy-\-2f = S9. 
 f + 2xy = -% 
 x^ — xy=z70. 
 2x^-xy = 2S, 
 a^ + 22/' = 18. 
 ?/2 4. 15 = 2 xy, 
 x' + y' = 21 + xy. 
 
 8. 
 
 10. 
 
 11. 
 
 12. 
 
 x^ + xy — 40, 
 
 27 + 2y'-3xy = 0. 
 
 2a?- + 3a!?/ + / = 20, 
 5a;2 + 42/' = 41. 
 
 xF — xy — y^= 20, 
 a;2_3(»2/ + 22/2 = 8. 
 i*^ H- 3 wv + 'y^ = 61, 
 u^-v^^Sl-2uv. 
 3f_5y2x_ ^ 
 
 4 iC 
 
 ~3" 
 
 '^ _ y^ + 2xy 
 l~2{l-x)' 
 
 13. Solve Ex. 1 by eliminating the absolute term and then 
 applying § 132. 
 
 By the method suggested in Ex. 13, solve : 
 
 x'-3xy + 4:f=:Sy, ^^ i3x'-5 xy -4.f = Sx, 
 
 x^-4:xy-^2y^ = 2y. ' [9 a^ + xy- 2 y^=6 x. 
 
 x'-4.xy-^3f = -3y, ^^ Ux' -^6xy -f = ^,y, 
 3x^ — 5xy = 6y. \6x-—9xy + 2y- = 2y. 
 
 14. 
 
 15. 
 
138-134] QUADRATIC EQUATIONS 211 
 
 134. Special devices. The kinds of systems of equations 
 specified in §§ 131-133 occur frequently, and, although they 
 present themselves in a great variety of forms, they may 
 always be solved by the methods there given. 
 
 Special devices for elimination, however, often give sim- 
 pler and more elegant solutions; some of these devices are 
 illustrated below. 
 
 (i) Solving hy first finding x-\- y and x — y, 
 
 x-y = 5, (1) 
 
 Ex. 1. Solve the equations , 
 
 [xy = -6. (2) 
 
 Solution. From (1), x" - 2xy -^y^ = 25, (3) 
 
 and from (2), 4 a;?/ = - 24 ; (4) 
 
 adding (4) to (3), x^-\-2xy-{- y' = 1, . (5) 
 
 whence x-\-y = ±lj (6) 
 
 and from (1) and (6), « = 3 or 2. 
 
 The corresponding values of y are y = —2ov — 3 ; 
 
 . • P 1 . (x = 3, (x = 2, 
 
 i.e., the solutions of the ffiven system are: J ^ and J 
 
 ^ ^ \y = -2, l.y = -3. 
 
 (a;2 + 2/' = 5, (1) 
 
 Ex. 2. Solve the equations i „ , „ 
 
 ^ [x'-xy + f^^- (2) 
 
 Solution. Subtracting (2) from (1), xy = 2, (3) 
 
 whence, adding 2 • (3) to (1), x'' + 2 xy + 2/^ = 9, (4) 
 
 and subtracting 2 • (3) from (1), x^ — 2xy -{-y'^ = l\, (5) 
 
 therefore, from (4), a; + ?/ = ± 3, (6) 
 
 and from (5), x — y = ±l] (7) 
 
 from (6) and (7), we now easily find the following solutions : 
 
 \y = l, \y = 2, U = -l, U = -2, 
 
 all of which check. 
 
212 HIGH SCHOOL ALGEBRA [Ch. XII 
 
 (ii) Solving hy dividing one equation hy the other. 
 
 a^^-/ = 3, (1) 
 
 Ex. 3. Solve the equations . 
 
 '^-2/=l. (2) 
 
 Solution. On dividing (1) by (2), member by member, we 
 obtain x-\-y = d, (3) 
 
 whence, from (2) and (3), a; = 2, and y = l. 
 
 ar^ + ^« = 26, • (1) 
 
 Ex. 4. Solve the equations , 
 
 ^ [x +y= 2. (2) 
 
 Solution. On dividing (1) by (2), member by member, we 
 obtain x^ — xy-^y^ = 13, (3) 
 
 and (2) and (3) may now be solved either like Exs. 1 and 2 above, 
 or by the method of § 131. 
 
 (iii) Solving by considerations of symmetry. An equation 
 is symmetric with regard to two of its letters if it is not 
 changed by interchanging those letters. Thus : x + y = S^ 
 and §2 ^ st -\- 1^ = 5 (^s -{- f) are symmetric equations. 
 
 Two equations which are symmetric (or symmetric except 
 for the signs of one or more terms) may often be solved by 
 substituting u + v and u — v, respectively, for their unknown 
 numbers. 
 
 ^ + 2/^=6, (1) 
 
 Ex. 5. Solve the equations 
 
 ^xy = 2(x^y)-5. (2) 
 
 Solution. On putting x = it-{-v and y = u — v, the given equa- 
 tions become, respectively, 
 
 2u^ + 2v^ = 6, Sindu^-v^ = 4:U-5', (3) 
 
 therefore, eliminating v^ and simplifying, 
 u'-2u + l = 0, 
 whence u = l. 
 
 Substituting this value of u in either one of Eqs. (3), gives 
 
 v=±V2, 
 whence (since x = u-^ v, and y = u — v), 
 
 x=l± V2, and y~lT V2. 
 
134] QUADRATIC EQUAIIONS 213 
 
 ( oc^ -^ Tf = xy — 5f (1) 
 
 Ex. 6. Solve the equations { ^ ^ 
 
 \x-^y + l = 0. (2) 
 
 Solution. On putting x = u-\-v and y =u — v, (1) and (2) be- 
 come, respectively, 
 
 2u^ + Quv'-u^ + v^ + b = 0, (3) 
 
 and 2 2^ + 1 = 0. (4) 
 
 From (4) u = — |, 
 
 and substituting this value in (3) gives 
 
 whence a; = 1 or — 2, and 2/ = — 2 or 1. 
 
 EXERCISE XCIV 
 
 By the method of 134 (i) solve the following systems : 
 ja^ + / = 13, |a^ + 2/' = l, 
 
 ^' \xy = ^. ^^' |25a^y + 12 = 0. 
 
 1^2 + ^2^61, 
 'm4-n = 24. 
 
 [5mn-2 = 0. 
 
 mn Q |a;2 + 2/2 = a, 
 
 — y. 12. 
 
 9 [a;-f-y = 6. 
 
 By the method of 134 (ii) solve the following systems : 
 
 13- „ 15. ^ 
 
 \r — s = l. [r— p = 7. 
 
 14. \ 16 
 
 r4-s= 7. 
 
 aj^ + 2/3 = a, 
 a: + 2/ = 6. 
 By the method of 134 (iii) solve the following systems : 
 
 17. |-^ + -"-* = 26. f2(. + ,) = -§f^, 
 I a; + 2/ = 6. 20. ] 5 
 
 a: -2/ = 7. 
 
 18. \ ■ -2/) = -^^2/' f2x2_ _^22/^ = 62, 
 2. 21. { _ ^. 
 
 [ a; — 2/ — o 0^2/ = ~~ "1- 
 
 ^^- [ar^ + 2r'' = 37. ^^' 1^ + 2.9/4-^-22. 
 
 ^ |3(a;-2/ 
 [a; + 2/ = 
 
214 
 
 HIGH SCHOOL ALGEBBA 
 
 [Ch. XII 
 
 Solve the following systems of equations, choosing for each 
 the method (§§ 131-134) which seems to you best: 
 
 23. 
 
 24. 
 
 25. 
 
 26. 
 
 27. 
 
 28. 
 
 29. 
 
 30. 
 
 31. 
 
 ^_?/_16 
 y X lo' 
 a;-2/ = 2. 
 
 xF — xy = 6, 
 a;2-f2/2 = 61. 
 
 ^+^. = 74, 
 a;- 2/ . 
 
 1-1 = 2. 
 
 rcf. Ex."! 
 13, §104. J 
 
 ;i-^ y^ 
 
 i+l= 
 
 = 91, 
 
 7. 
 
 •1 + 1-1 
 
 X y Z 
 
 .^2/ 18 
 
 a; + ?/ . X —y _ 10 
 a; — ?/ x-{-y 3' 
 
 a;^ + 2/^ = 45. 
 
 1 1 
 
 = 119, 
 
 -7 = i. 
 
 aJ^ + 2/^ + i» = 2/ + 14:, 
 a;?/ = 6. 
 
 = 96 — 4mn , 
 6. 
 
 m -\-n 
 
 33. 
 
 34. 
 
 35. 
 
 36. 
 
 37. 
 
 38. 
 
 x-^y = 25, 
 
 V^+ ■Vy = 7. 
 
 a^ + 2/ + 2Va; + 2/ = 24, 
 a; — ^ -f- 3Va/' — 2/ = 10. 
 
 2(a^2 + /)=o.r2/, 
 
 x^-2xy = S y% 
 y{x + y)=4.. 
 
 (2+^(2/ + l)=4, 
 V2 + aj-V// + l = -J-. 
 
 {a' + h' = n, 
 
 |a262 = 4. 
 
 39. I 
 
 40. 
 
 41. 
 
 42. 
 
 43. 
 
 44. 
 
 45. 
 
 a^ 4- 2/^ + 6 Va;^ + 2/^ = So^ 
 a;2_2/2 = 7. 
 
 tv=- 26. 
 
 ^' + sV + ^^ = 9, 
 
 s2 + 6-^ + 2/' = 3. 
 
 s^-t^ = 37, 
 st(s-t) =12. 
 x^ + 2/^ = 97, 
 a; + 2/ = — 1- 
 
 o , 3 771 f. 
 
 3 mn H = o, 
 
 n 
 
 3^n + — = 2.5. 
 m 
 
 2/ a; ' 
 
i;U] qUADHATIC EQUATIONS 215 
 
 PROBLEMS 
 
 1. The sum of two numbers is 14, and the difference of their 
 squares is 28. What are the numbers ? 
 
 2. Find two numbers whose difference is 15, and such that if 
 the greater is diminished by 12, and the smaller increased by 12, 
 the sum of the squares of the results will be 261, 
 
 3. Find two numbers whose difference is 80, and the sum of 
 whose square roots is 10. 
 
 4. Given that one root of a quadratic equation is 4 times the 
 other and that their product is ^, find the roots ; then form the 
 equation which has these roots (cf. § 72). 
 
 5. The sum of the roots of a quadratic equation is 12, and 
 their product is — 189. What is the equation ? 
 
 6. The sura of two numbers, their product, and also the dif- 
 ference of their squares are all equal ; find the numbers. 
 
 7. If the length of the diagonal of a rectangular field, contain- 
 ing 30 acres, is 100 rods, how many rods of fence will be required 
 to inclose the field ? 
 
 8. Find the dimensions of a rectangular field whose perimeter 
 is 188 rods and whose area "will remain unchanged if the length 
 is diminished by 4 rods and the width increased by 2 rods. 
 
 9. The sum of the circumferences of two circular flower beds 
 is 56|^ feet, and the sum of their areas is 141^ square feet. Find 
 the radius of each. (Cf. Ex. 14, p. 201.) 
 
 10. A circular table whose radius is 3J- feet has the same area 
 as a rectangular table whose length is 5 inches more than its 
 breadth. Find the dimensions of the rectangular table. 
 
 11. A sum of money lent at a certain rate of interest gives an 
 annual income of $ 450 ; if the sum were ^ 500 more and the 
 rate 1 % less, the annual income would be $ 50 less. Find the 
 principal and the rate. 
 
 12. A sum of money at interest for one year at a certain rate 
 amounted to $11,130. If the rate had been 1 % less and the 
 principal $100 more, the amount would have been the same. 
 What was the principal, and what the rate? 
 
216 IHGU SCHOOL ALdKliRA [Ch. XII 
 
 13. A formal rectangular flower garden is to be enlarged by a 
 border whose uniform width is 10 % of the length of the garden. 
 If the area of the border is 900 sq. ft., and the width of the old 
 garden is 75 % of the width of the new one, find the dimensions 
 of the garden and the width of the border. 
 
 14. A certain kind of cloth loses 2 % in width and 5 % in 
 length by shrinking. Find the dimensions of a rectangular piece 
 of this cloth whose shrinkage in perimeter is 38 in., and in area 
 8.625 sq. ft. 
 
 15. The perimeter of a right-angled triangle is 24 ft., and its 
 area is 24 sq. ft. Find the length of each side in the triangle. 
 
 16. In the right-angled triangle ABC, BD is 
 drawn perpendicular to AC. If BC =12, AC =11, 
 and BD=^AD'DC, find BD, AD, and DC 
 
 17. The combined capacity of two cubical coal 
 bins is 2728 cu. ft., and the sum of their lengths 
 is 22 ft.; find the length of the diagonal of the 
 smaller bin. 
 
 18. Find two numbers whose product is 8 greater than twice 
 their sum, and 48 less than the sum of their squares. 
 
 19. Find two numbers such that the sum of their fourth powers 
 is 881 while the sum of their squares is 41. 
 
 20. The total area of the walls and ceiling in a room 9 ft. 
 high is 575 sq. ft. Find the length and breadth of the room 
 if their sum is 24 ft. 
 
 21. A farmer found that he could buy 16 more sheep than 
 cows for $ 100, and that the cost of 3 cows was $ 15 greater than 
 the cost of 12 sheep. What was the price of each ? 
 
 22. If 5 times the sum of the digits of a certain two-digit 
 number is subtracted from the number, its digits will be inter- 
 changed; and if the num.ber is multiplied by the sum of its digits, 
 the product will be 648. What is the number ? 
 
 23. Find two numbers such that the square of either of them 
 equals 112 diminished by 12 times the other. 
 
1;)4-135J QUADRATIC EQUATIONS 217 
 
 24. If 5 is added to the numerator and subtracted from the 
 denominator of a given fraction, the result equals the reciprocal 
 of the fraction ; and if 2 is subtracted from the numerator, the 
 result equals J of the original fraction. Find the fraction. 
 
 25. The distance (s) in meters, through which a body falling 
 from a position of rest passes in the ^th second of its fall is 
 given by the formula s = \g (2t—l) ; and the total distance (S) 
 fallen in t seconds is S=^gt^. How long has a body been falling 
 when s = 44.1 meters and S = 122.5 meters? If g is less than 10, 
 what is its value? 
 
 26. Solve the problem of Ex. 25 if s and S are each expressed 
 in feet, and s = 112^7^ and S = 2571 
 
 27. In going 40 yards more than ^ of a mile the fore wheel of 
 a carriage revolves 24 times more than the hind wheel ; but if the 
 circumference of each wheel were 3 ft. greater, the fore wheel 
 would revolve 16 times more than the hind wheel. What is the 
 circumference of the hind wheel ? 
 
 28. A merchant paid $125 for an invoice of two grades of 
 sugar. By selling the first grade for $91, and the second for 
 $36, he gained as many per cent on the first grade as he lost on 
 the second. How much did he pay for each grade ? 
 
 29. Two trains start at the same time from stations A and 
 B, 320 miles apart, and travel toward each other. If it re- 
 quires 6 hr. and 40 min., from the time the trains meet, for the 
 first train to reach B, and 2 hr. and 24 min. for the second to 
 reach A, find the rate at which each train runs. 
 
 30. After traveling 2 hr., a train is detained 1 hr. by an acci- 
 dent ; it then proceeds at 60 % of its former rate, and arrives 
 7 hr. 40 min. late. Had the accident occurred 50 miles farther 
 on, the train would have been 6 hr. 20 min. late. Find the 
 distance traveled by the train. (Cf. Ex. 42, p. 164.) 
 
 135.* Simultaneous quadratics not always solvable by methods 
 already given. While many systems containing quadratics (and 
 
 * § 135 may, if the teacher prefers, be omitted till the subject is reviewed. 
 
'21S HIGH SCHOOL ALGEBRA [Ch. XIl 
 
 some containing still higher equations) may be solved by the 
 methods of §§ 131-134, these methods do not always suffice for 
 the solution of such systems. 
 
 Thus, inspection shows that the system 
 a^-3a; + 82/ = 4, 
 3a;^-16/ + 20 2/ = 9, 
 cannot be solved by the methods of §§ 132-134; and elimination 
 by substitution (as in § 131) leads to an equation of the fourth 
 degree in one unknown number, viz., to 
 
 a,>4 _ 6 ar^-i»2- 6 a; +12 = 0, 
 which cannot be solved by the elementary methods already studied. 
 Such equations are discussed in higher algebra. 
 
 For all systems like the above, however, approximate solutions 
 may be obtained by means of graphs (cf. § 143). 
 
 136.* Systems containing three or more unknown numbers. Some 
 systems containing three or more simultaneous equations, some 
 of which are quadratic, may be solved by elementary methods. 
 
 E.g., if one equation of a given system is quadratic, and all the 
 others are of the first degree, then a slight modification of the 
 method of § 131 will provide a solution (cf. El. Alg. § 180). 
 
 The solution of such systems in general is, however, beyond 
 the limits of this book. 
 
 * This article may, if the teacher prefers, be omitted till the subject is 
 reviewed. 
 
CHAPTER XIII 
 
 GRAPHIC REPRESENTATION OF EQUATIONS* 
 
 137. Introductory. Although an equation in two unknown 
 numbers has (§ 99) an infinitely large number of solutions, 
 and is in that sense indeterminate, yet by a beautiful device, 
 due to the celebrated mathematician and philosopher Des- 
 cartes (pronounced da-kart', born 1596, died 1650), a perfectly 
 definite picture of such an equation may be made (cf. § 139). 
 
 138. Axes. Coordinates. Let us draw (as Descartes did) 
 two perpendicular straight lines X'X 
 
 and Y' Z", cutting each other in the 
 point 0, and call these lines the 
 coordinate axes. If we now agree 
 that distances measured toward the 
 right from Y'Y^ or upward from 
 X'X^ shall be positive, while dis- 
 tances toward the left, or downward, 
 shall be negative, then any point in 
 the plane of this page can be located 
 as soon as we know its distances from the axes X' X and Y' Y. 
 
 Thus, to locate the point P, 3 inches from F'^and 2 
 inches from X' X^ we measure off 3 inches (represented in the 
 diagram by 3 spaces) toward the right from 0, to the point 
 M^ say, and then 2 inches upward from M. 
 
 The numbers which serve to locate a point (in this case 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ( 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 p 
 
 
 
 
 — -1 
 
 , 
 
 
 
 
 
 
 
 
 
 
 
 
 r 
 
 
 
 
 
 
 
 
 
 
 
 
 
 J 
 
 
 
 
 
 
 
 
 
 
 
 M 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 - 
 
 - 
 
 - 
 
 
 
 
 
 
 
 
 
 
 
 
 
 p? 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 *This chapter should he included in the course whenever possible 
 omission, however, will not break the continuity of the work. 
 
 HIGH SCH. ALG. — 15 219 
 
 its 
 
220 
 
 HIGH SCHOOL ALGEBRA 
 
 [Ch. XIII 
 
 3 and 2) are called the coordinates of the point. The point 
 P may be represented by the symbol (3, 2). 
 
 Similarly the point ^ (— 3, 4) is located by measuring 3 
 spaces toward the left from 0, and then four spaces upward. 
 The point i2 (— 2, — 3), also, is represented in the figure. 
 
 Note. This plan of locating points in the figure somewhat resembles that 
 used to locate places on the earth's surface by their latitude and longitude. 
 The coordinate axes correspond to the equator and the prime meridian. 
 
 
 
 
 
 
 
 
 y 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 p 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 X 
 
 , 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 X 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Q 
 
 
 
 
 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 , 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 EXERCISE XCV 
 
 1. Name the ic-coordinate (i.e., the distance from the axis 
 Y'Y) of each point located in the figure below. Also name the 
 ^/-coordinates of these points. 
 
 Draw a pair of axes as in § 138 and 
 locate the following points : 
 
 2. (5,4); (3,7); (4,-2); (-3,1); 
 and (-4, -6). 
 
 3. (-i,l); (i,|); (H, -3); (4,1); 
 and ( — I, —5). 
 
 4. (3, 0); (-5, 0); (0, 8); (0, 0) ; 
 and (0, -2). 
 
 5. Where are the points whose ^/-coordinate is ? Where are 
 those whose ^-coordinate is ? those whose ^/-coordinate is 3^ ? 
 
 6. Locate five points each of which has its a7-coordinate equal 
 to its ^/-coordinate, and draw a line through these points. Does this 
 line contain any other points whose two coordinates are equal ? 
 
 7. Where are the points which have their ^/-coordinates oppo- 
 site in value to their respective ic-coordinates ? 
 
 8. Verify that the equation 2 x — y==3 is satisfied by each of 
 the following number-pairs : (0, — 3) ; (1, — 1) ; (2, 1) ; (3, 3) ; 
 (4, 5) ; then locate the point corresponding to each pair. On 
 what kind of a line do these points lie ? 
 
 9. Measure the coordinates of several other points on the line 
 mentioned in Ex. 8. Are these coordinates solutions of 2 x—y=3? 
 
i;W-140] (lliAPlIJC liEPIiESKNTATION OF EQUATIONS 221 
 
 10. Locate the following points : (0, 5) ; (0, — 5) ; (5, 0) ; 
 (-5,0); (4,3); (-4,3); (4,-3); (-4,-3); (3,4); (-3,4); 
 (3, —4); ( — 3, —4). On what kind of a line do they lie? 
 
 139. The picture (graph) of an equation. Consider the 
 equation 2 a; — y = 3. 
 
 This equation is, manifestly, satisfied by the following 
 pairs of values of x and 7/ (§ 99) : 
 (-1, -5); (0, -3); (1, -1); (2, 1); (3, 3); (4,5); etc. 
 
 If we now locate (as in § 138) the points A, B, (7, etc., 
 corresponding to these number-pairs, we find that they are 
 not scattered at random over the 
 page, but that the^ all lie upon the 
 straight line US in the figure (cf. 
 Ex. 8, p. 220). Moreover, the co- 
 ordinates of every point in the line 
 RS, and those of no other points 
 whatever, satisfy the given equa- 
 tion.* 
 
 For these reasons, the line MS may 
 be regarded as the picture of the 
 equation ; it is usually called the 
 graph, also the locus of the equation. That is, the graph 
 (or locus) of an equation is the line (or lines) containing 
 all the points (and no others) whose coordinates satisfy the 
 given equation. 
 
 140. Drawing of graphs. The method illustrated in § 139, 
 for finding the graph of an equation in x and «/, may be 
 stated thus : 
 
 (1) Solve the given equation for ?/, in terms of x. 
 
 V /a 
 
 7 
 
 t 
 
 7 
 
 T 
 
 
 X o~r^ zE. 
 
 / 
 
 M 
 
 t 
 
 i 
 
 f 
 
 ^ 
 
 t. 
 
 -hi v^ 
 
 ^ r 
 
 * Let the pupil test this statement by careful measurement on a large and 
 well-drawn figure. The proof of its correctness follows easily from the 
 theory of similar triangles in geometiy. 
 
•>•^:> 
 
 Hid 1 1 S(1I(K)L ALGEBRA 
 
 [Ch. XIII 
 
 (2) Assign to rr a succession of values, such as 0, 1, 2, 
 3, ••• (also —1, —2, — 3, •••), and find the corresponding 
 values of y ; ^.e., find a succession of solutions of the given 
 equation. 
 
 (3) By means of a pair of axes locate the points corre- 
 sponding to these solutions, — use cross-section paper. 
 
 (4) Draw a line connecting these points in regular order ; 
 this line is (approximately) the graph of the given equation. 
 
 E.g., to find the graph of the equation 3y—x^ = 0, we solve 
 the equation for y in terms of x, and tabulate the corresponding 
 values of x and y, thus : 
 
 Locating the points 0, A, B, C, 
 etc., and connecting them in order, 
 we obtain the line NML •••£/, which 
 
 X-. 
 
 X 
 
 y 
 
 Points 
 
 
 
 
 
 
 
 1 
 
 i 
 
 A 
 
 2 
 
 1 
 
 B 
 
 3 
 
 3 
 
 C 
 
 -1 
 
 i 
 
 H 
 
 — 2 
 
 i 
 
 K 
 
 -3 
 
 3 
 
 L 
 
 • 
 
 : 
 
 ' 
 
 II ITT 
 
 ~E ji T 
 
 j-i^ 4 
 
 t " 7 
 
 \ t 
 
 \ I 
 
 -M^ J? 
 
 A i^ t 
 
 -> f- 
 
 X ~K- 
 
 % t 
 
 \ / 
 
 T ' k\ /(^ t. 
 
 ^ Tt- -^ ^ 
 
 ± :£ ^ 
 
 
 vr' 
 
 
 is a good approximation to the graph 
 of the given equation. 
 By assigning to x values between and 1, 1 and 2, etc., and 
 finding the corresponding values of y, we can locate points 
 between and A, A and B, etc., and thus draw a closer approxima- 
 tion to the required graph. 
 
 Note. The graph of a first decree equation in x and y is (cf. § l.SO) a 
 straight line (hence a first degree equation is often called a linear equation). 
 In this case, of course, only two points (i.e., two solutions of the equa- 
 tion) need be found in order to draw the complete graph. 
 
UO-Ulj GRAPHIC REPRESENTATION OF EqUATlONti 228 
 
 EXERCISE XCVI 
 
 1. Find six solutions of 2x-{-y = 12, locate the points 
 determined by these solutions, and draw the graph of the equation. 
 
 Using the plan of Ex. 1, draw the graph of : 
 
 2. a; + 2^ = 8. 5. 2a;-3?/ = 0. 
 
 3. x-2y = l. 6, 'Sx + 2y = 12. 
 4c. 3x = y. 7. 2 2/ - ar = 0. 
 
 8. Draw the graph of 3 a; = 2 [i.e., 3x-\-0'y = 2, cf . Ex. 5, 
 p. 220]; of 22/ = 5; of x' = -l; of ar = 9. 
 
 9. What is the graph of x = ? of ?/ = ? Without making a 
 drawing, show that the graph of 2 x — 7 y = must pass through 
 the point in which the axes intersect. Is this true for the graph 
 of every equation of the form ax -\-by = 0? 
 
 10. In the equation Ax — 5 y = 10, when x = 0, y=? When 
 y = 0, x=? From these two solutions of the equation draw its 
 graph (cf. § 140, Note). 
 
 Draw the graph of : 
 
 11. x-y = 0. 17. 5x-2y = 20. 
 
 12. x-\-y = 0. 18. Sx + 5y = 7^. 
 
 13. 3 a; = — 11. 19. 7x — y = S^. 
 
 14. ?/2-16 = 0. 20. Dx-\-2y=21. 
 
 15. 2x-\-Sy = 6. 21. 4:X = y\ 
 
 16. 2x-3y = 6. 22. 3a^-4i/ = 0. 
 
 Calling the coordinate axes S'S and TT instead of X'X and 
 Y' Y, draw the graph of : 
 
 23. 4:t = Ss. 24. s-t = 5. 25. 2t-3s^ = 0. 
 
 141. Drawing of graphs (continued). Thus far we have 
 considered only the simplest kind of graphs ; the method 
 employed will serve, however, for any equations whatever 
 in two unknown numbers. 
 
224 
 
 HIGH SCHOOL ALGEBRA 
 
 [Ch. XIII 
 
 Ex. 1. Construct the graph of 2y — x^ = 0. 
 
 CoxsTRucTiON. Solving this equation for y in terms of x, and 
 tabulating the corresponding values of x and y, we obtain : 
 
 X 
 
 y 
 
 Points 
 
 
 
 1 
 
 2 
 
 
 4 
 
 
 
 A 
 B 
 
 -1 
 
 -2 
 
 -4 
 
 H 
 K 
 
 : 
 
 : 
 
 ' 
 
 Yi- 
 
 
 T 
 
 L 
 
 it 
 
 -4^ 
 
 jT 
 
 ^ t V- 
 
 ^^ - ,-i^ i 
 
 v^ 
 
 iy 
 
 if 
 
 ^ 
 
 J 
 
 j 
 
 I 
 
 I 
 
 ' ^ 
 
 • X 
 
 On locating these points and connecting them in order, we ob- 
 tain the required graph, viz., : KHOA •••. 
 
 Ex. 2. Construct the graph of 4 a^ + 9 2/^ = 144. 
 
 Construction. Proceeding as in Ex. 1, we obtain : 
 
 
 a; = |Vl6- 
 
 2/^. 
 
 y 
 
 
 
 X 
 
 Points 
 
 6 or -6 
 
 ^or^' 
 
 1 
 
 1 Vis or - 1 \/l5 
 
 BovB' 
 
 -1 
 
 1 Vis or - f Vl5 
 
 Hov H' 
 
 -2 
 
 3 V8 or - 3 \/3 
 
 Kov K' 
 
 
 .IjL 
 
 
 
 
 AJ^"--^ 
 
 ^^ 
 
 ^. 
 
 V ^/ 
 
 S 
 
 ^44- 
 
 -2.]"-^ 
 
 4^ 
 
 XaT 
 
 n\ 
 
 t 
 
 #s.± 
 
 n' ^^ 
 
 
 
 
 \7 
 
 
 I 
 
 On locating these points, using approximate values of the 
 square roots, and connecting them by a smooth curve, we obtain 
 the graph ABNAN'A. 
 
141-142 ] GliAPHIC liEPRESENTA TION OF EQ UA TIONS 225 
 
 Note. The limitations of the graph in Ex. 2 are interesting. Thus, since 
 X = |\/16 — 2/2, X must be imaginary when y is greater than 4 ; hence, as our 
 graphic representation admits real values only, there are no points on the 
 curve whose ^/-coordinate is greater than 4. Similarly, it may be shown that 
 there are no points on the graph below y = — 4. And solving the given equa- 
 tion for y in terms of x shows that there are no points on the graph at the 
 right of ic = 6, or at the left of aj = — 6. 
 
 Ex. 3. Construct the graph of xy = 4. 
 Construction. Proceeding as in Exs. 1 and 2, we obtain: 
 4 
 
 X 
 
 X 
 
 y 
 
 Points 
 
 
 
 00 
 
 
 1 
 
 4 
 
 A 
 
 2 
 
 2 
 
 B 
 
 3 
 
 1 
 
 C 
 
 . 
 
 • 
 
 • 
 
 • 
 
 • 
 
 • 
 
 . 
 
 • 
 
 • 
 
 -1 
 
 -4 
 
 H 
 
 -2 
 
 -2 
 
 K 
 
 
 
 
 
 
 
 
 
 Y 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \a 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 V 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 B 
 
 
 
 
 
 
 
 X 
 
 ^, 
 
 
 
 
 
 
 
 
 
 N 
 
 '< 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 
 
 - 
 
 
 — . 
 
 
 ^ 
 
 
 
 
 o 
 
 
 
 
 
 
 
 X 
 
 
 
 
 
 
 
 
 N 
 
 K 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 fl\ 
 
 
 
 
 
 
 
 
 
 - 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 On locating these points and connecting them by a smooth 
 curve, we obtain the graph AB • • • HK. 
 
 142. Intersection of graphs. Since (0, — 3) and (4, 0) are 
 
 solutions of the equation Z x — ^y 
 = 12, therefore its graph is the line 
 AB in the figure (of. § 140, Note). 
 
 If we now draw the graph of 
 3 rr + «/ = 2, using the same axes as 
 before, we obtain the line HK. 
 
 Moreover, since P, the point in 
 which AB and HK intersect (i.e, 
 cut) each other, lies on each of these 
 
 -L4-F- 
 
 kY^ 
 
 r 
 
 X 
 
 ^ B 
 
 , >^ 
 
 -_U ^^ - 
 
 xT o^H 7^ -x 
 
 W- 
 
 ^^ 
 
 ^'^ -t 
 
 ^^ t 
 
 ^^ \h 
 
 v' 
 
 X 
 
226 IIIGa SCHOOL algebra [Ch. XIII 
 
 graphs, therefore its coordinates (§ 138) must satisfy each 
 of the given equations (cf. § 139). 
 
 Hence, we may find the coordinates of P by merely solving 
 the given equations as in § 101, and without even drawing 
 their graphs. 
 
 Approximate values of the coordinates of P may, of course, 
 be found by direct measurement of OM and MP ; this 
 measurement constitutes a graphical solution of the given 
 equations. Let pupils use both methods for finding these 
 coordinates, and compare results. 
 
 Remark. From what has just been said, and from the defini- 
 tions in § 100, it follows that (let pupils explain why) : 
 
 (1) The graphs of consistent equations intersect each other. 
 
 (2) The graphs of inconsistent linear equations are parallel 
 lines. 
 
 
 
 EXERCISE XCVII 
 
 Construct the graph 
 
 of 
 
 : 
 
 4. y' = Sx. 
 
 
 8. xy = 5. 
 
 5. y={x-iy. 
 
 
 9. 3x' + 4.f- = 12. 
 
 6. x' + y' = 25. 
 
 
 10. 3x'-4.y' = 12. 
 
 7. 16a^+/ = 64. 
 
 
 11. 4.f- = a^. 
 
 12. Show from the equation of Ex. 4 that no part of its graph 
 lies to the left of the ^/-axis (the line Y' Y) . 
 
 13. Show from the equation of Ex. 6 that no part of its 
 graph lies outside a certain square whose side is 5 ; similarly, 
 show that the graph of Ex. 7 is contained within a certain rec- 
 tangle whose dimensions are 16 and 4. 
 
 14. Show from the equation of Ex. 8 that its graph consists of 
 two infinitely long branches, one in the quarter XOY and one in 
 the quarter X'OY'. 
 
 15. If the graph of xy = — 5 were drawn, how would it differ 
 from that of xy = 5? Why ? 
 
142-143J GEAPHW REPRESENTATION OF EQUATIONS 227 
 
 16. Draw the graph of 2x-\-y = dr -{-S and show that this 
 graph differs from that of Ex. 5 only in being moved two divi- 
 sions upward. Explain why this should be so. 
 
 17. Find, both by solving the equations, and by measurement, 
 the coordinates of the point in which the graphs of x-{-y = ^ and 
 2 a; — 2/ = 4 intersect ; compare your results. 
 
 In each of Exs. 18-23 below find, as in Ex. 17, the coordinates 
 of the point in which the graphs of the two equations intersect : 
 
 18. i ' 21. ' ^^ ' 
 
 \2y-x= 
 
 6. 2a; + 2?/ = 8. 
 
 ^^ 1 2.^^72/, . 22^ [^•+2/=3, 
 
 U-4- 
 \\x 
 
 y-x=5, ^ [^x + ^y=li. 
 
 2^ {Sy + 2x = 17, ^^ \2x-y = S, (cf. §§ 139, 
 
 2x-y = 5. [3y-x' = 0. 140.) 
 
 24. How are the graphs of two first degree equations in a; and 
 y related when the equations are inconsistent (cf. Ex. 21) ? 
 when they are simultaneous and independent (cf . Ex. 20) ? simul- 
 taneous and not independent (cf. Ex. 22) ? 
 
 143. Graphic solution of simultaneous equations. If the 
 
 graph of one of two simultaneous equations is drawn across 
 the graph of the other (^i.e., if the same axes are used in 
 both drawings), then the measured coordinates of each point 
 in which these graphs intersect constitute an approximate 
 solution of the given system (cf. § 142). The following 
 examples will illustrate this procedure. 
 
 Ex. 1. Solve graphically the simultaneous equations 
 Sx-4:y = 12, 
 3x-^y=2. 
 
 Solution. The graphs of these equations are the lines AB 
 and HK (figure, § 142) ; and the (measured) coordinates of P, 
 their point of intersection, are approximately .x- = | and y =—2, 
 which, by trial, are found to be a solution of the given system. 
 
228 
 
 HIGH SCHOOL ALGEBRA 
 
 [Ch. XIII 
 
 Ex. 2. Solve graphically the system J ^ ^' 
 
 \y + 3 = 2x. 
 
 Solution. The graphs of 
 these equations are, respec- 
 tively, SPOQT and AB. 
 The coordinates of P, one of 
 their points of intersection, 
 are approximately a: =3.4 and 
 y = 3.7, which constitute an 
 approximate solution of the 
 given equations. 
 
 So, too, the coordinates of 
 Q (viz., X = .65 and y = —1.6) 
 constitute an approximate 
 solution of the system. 
 
 -y^ y 
 
 Ml ^- 
 
 
 l! ^-^"^ 
 
 2^^ 
 
 ^^ 
 
 VI 
 
 4^ ZJ- ~t 
 
 X "--jt X- 
 
 ^^t 
 
 M 
 
 f\ 
 
 / ^s. 
 
 ^ ^^ 
 
 ^ >- ~ 
 
 
 
 ^r 
 
 Ex. 3. Solve graphically the system 
 
 xy = 4:. 
 
 Solution. The graphs 
 of these equations are, re- 
 spectively, AB'A'B and 
 CPQC'P'Q'-, and the co- 
 ordinates of P, one of their 
 points of intersection, are 
 approximately x = 1.4 and 
 y= 2.9, which constitute an 
 approximate solution of 
 these equations. 
 
 So, too, the coordinates 
 of Q (x= 5.75 and y = .7), 
 P'(x = -1.4 and y= -2.9), 
 and Q' (x = — 5.75 and y = 
 — .7) are approximate solutions of the given equations. 
 
 Note to the Teacher. In the case of a pair of simple equations the 
 solution by the method of § 101 is usually easier than the graphic method, 
 and its results are exact instead of approximate. There are, however, many 
 other cases in which the graphic method is advantageous ; hence some prac- 
 tice with it, even on simple equations, is recommended. 
 
 
 
 Y J 
 
 1 
 
 
 
 
 
 
 
 c 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 [ 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 \p 
 
 
 
 
 
 
 ^*-'"*'^ " 
 
 "-k"** 
 
 ■^s 
 
 
 
 
 t^ 
 
 
 Si 
 
 
 X 
 
 
 
 X 
 
 Ar 
 
 
 
 
 
 
 
 
 ^^ " I 
 
 ) 
 
 
 n 
 
 A X 
 
 
 - ^; 
 
 s^^ 
 
 
 
 y 
 
 
 
 
 _'^-^»: 
 
 -^ 
 
 
 
 
 
 
 pTi 
 
 5^ 
 
 
 
 
 
 
 - T- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 7- 
 
 
 
 
 
 
 
 
 
 
 
14^-144 J GRAPHIC liEPRE^EN TA TION OF EQ UA TIONS 229 
 
 EXERCISE XCVIll 
 
 Solve graphically the following systems of equations, and check 
 your results as the teacher directs : 
 
 ^ {x-^y = 3, ^^ ja.'- + 2/' = 25, 
 
 5. 
 
 6. 
 
 8. 
 
 10. 
 
 11. 
 
 12. 
 
 x — y = 3. 
 
 
 12/ = ^- 
 
 2x-y = 5, 
 
 14. 
 
 ■x' + f^ = 2o, 
 
 4.x = W-y. 
 
 
 x-\-y=^l. 
 
 4?/4-3a; = 5, 
 
 15. 
 
 y = 3x + 2, 
 
 0^=5. 
 
 
 x^ = 4.-y\ 
 
 x + y = A, 
 y = 2-x, 
 
 5x-10y = S6, 
 2x-\-3y=-S. 
 
 16. 
 17. 
 
 { . 4.x 
 
 x = l. 
 
 xy=- 10, 
 x+y = 2. 
 
 4a.- + |2/ = 6, 
 ix-^y=S. 
 
 18. 
 
 ix'+9==y, 
 
 \y = x^-5x-\-G. 
 
 2x^y\ 
 
 19. 
 
 'x'-^y^ = 25, 
 
 2y=.x. 
 
 
 [xy=-^. 
 
 2a;-/-l = 0, 
 
 20. 
 
 '4.x'-9y' = S6, 
 
 2a; + 6v/ + T = 0. 
 
 
 y4-r = 25. 
 
 x-2y=-12, 
 
 21. 
 
 y = Bx—15, 
 
 y-x^=-2x-2. 
 
 
 x^-9x- + 23x-15 = 'y. 
 
 By referring to the graphs in the above exercises, find the 
 number of solutions of a system consisting of: 
 
 22. Two simple equations. 
 
 23. A simple and a quadratic equation. 
 
 24. Two quadratic equations. 
 
 144. Graphic solution of equations containing but one 
 unknown number. By slightly extending the method of 
 § 143, we may find graphic solutions for quadratic equations 
 in one unknown number. 
 
230 
 
 HIGH SCHOOL ALGEBRA 
 
 [Ch. XUI 
 
 IT 
 
 -\ \- 
 
 w jfe 
 
 M -j^ 
 
 r- ^-4 
 
 X t 
 
 \ I 
 
 ^ f 
 
 
 "^r lo j_ :S 
 
 -^^ m 7^ 
 
 _^ L 
 
 ^-4- 
 
 v/ ^ 
 
 F -^ 
 
 |2/ = ^= 
 12/ = 0. 
 
 Thus, the roots of x- — 2x — 2 = are manifestly the values of 
 X found b}^ solving the pair of simultaneous equations : 
 
 2x-2, (1) 
 
 (2) 
 Now the graph of (2), viz., the a>axis, 
 cuts the graph of (1), viz., the curve MQS, 
 in the points P and E, whose coordinates 
 are (approximately) ic = 2.75, y = 0, and 
 x= — .75, y=0. And since (§ 143) each 
 of these pairs of values constitutes an 
 approximate solution of (1) and (2), there- 
 fore 2.75 and — .75 are approximate roots 
 of a;2_2,^_2::=0. 
 
 EXERCISE XCIX 
 
 Find graphic solutions of : 
 
 1. x:'-6x + S = 0. 3. 6a;H-5x-4 = 0. 
 
 2. x''-Sx-\-5 = 0. 4. a^_3a;2-6a; + 8 = 0. 
 
 5. Show graphically that Ax^— 4:X-—llx-\-6 = has one 
 root between and 1, and a second root between — 1 and — 2. 
 What is the third root of this equation ? 
 
 6. Show that one root of cc^ — 7 x- + 9 .^• = 1 lies between 1 and 
 2. Between what integers do each of the other two roots lie? 
 
 7. Corresponding to any given value of x, how does the value 
 of y \\\ y = x' — 6x-\-6 compare with its value in y = x^ — 6 x-{-7 ? 
 Could, then, the graph of the second equation be obtained by 
 merely moving that of the first upward through one division ? 
 
 8. Compare the graphs of 2/=2a7^—10a;— Sand ?/=2ic^—10aj+l; 
 also those of y = 3-\-4:X — x''^ and y = 10-\-4:X — x"\ 
 
 9. By first constructing the graphs of y = oi^ — 6x + 6, 
 
 y = x^ — 6 X -\- 7, etc., compare the roots of 
 
 X- 
 
 6 a; + 6 = 0, 
 
 X' - 6 X -\-7 = 0, x' - 6 X + S = 0, x^ - 6 X -{-9 = 0, X- - 6x -{-10 = 0, 
 anda;2-6a;4-ll =0. 
 
144-14.-.] linAPlllr UEritESENTATION OF EQUATION t> 281 
 
 10. As in Ex. 9 compare the two smaller roots of 
 ic3 - 7 a;2 + 9 a; - 1 = with those of a;"^ - 7 a;^ _^ 9 ^ _ 3 ^ q ^^^^ 
 jB3_7a;2_j_9^_5^() 
 
 Note. Exercises 9 and 10 illustrate how, by changing the absolute 
 term in an equation, a pair of unequal roots can be made gradually to become 
 equal and then imaginary. 
 
 11. Show that the roots of x'^-2x-2 = (§ 144) can be 
 found from the graphic solution of the system 
 
 y = ^, (1) 
 
 y-2x-2 = 0. (2) 
 
 12. Show that the graph of Eq. (1) in Ex. 11 may be used in 
 the solution of other quadratic equations {e.g., x^ -f 5 a? = 7) also. 
 
 13. Is the method given at the top of p. 230, or that suggested 
 in Exs. 11 and 12, to be preferred when we have several quadratic 
 equations to solve graphically ? Explain. 
 
 By the method of Ex. 11 solve : 
 
 14. a;^-2a^-2 = 0. 17. x' = x-\-^. 
 
 15. a;"^ — a; = 0. 18. 12 a; — 4 a;"' = 5. 
 
 16. x' + x-4. = Q. 19. 2a;^-a; = -3. 
 
 145. Use of graphs in physics, engineering, statistics, etc. 
 
 Descartes's plan for graphically representing equations has 
 now been adopted by practically all scientific men to repre- 
 sent simultaneous changes in related quantities. Physicists, 
 chemists, engineers, physicians, statisticians, etc., all find 
 that this graphic representation of related changes often 
 gives at a glance information which could be secured other- 
 wise only by considerable effort, and that it often brings out 
 facts of importance which might otherwise escape notice. 
 
 As a simple example of the use of graphs in this way let us 
 consider the following temperature readings, taken from the 
 U. S. Weather Bureau report, for 28 hours beginning at noon on 
 Feb. 5, 1906, at Ithaca, N.Y. 
 
^32 
 
 UIGII SCHOOL ALGEnnA 
 
 [Ch. xm 
 
 IIR. 
 
 Tem. 
 
 - 
 
 -11 
 
 ^°H 
 
 - 
 
 
 
 
 
 
 
 
 T 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■V, 
 
 s 
 
 
 
 
 
 
 
 
 
 
 12 
 
 1 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 1 
 
 2 
 
 3 
 
 4 
 
 10° 
 
 9 
 
 8.5 
 
 7 
 
 6 
 
 5 
 
 3.5 
 
 3 
 
 2 
 
 1 
 
 0.5 
 
 
 -5 
 -6 
 -7 
 -6 
 -6.5 
 -6 
 -8 
 -9 
 -11 
 
 -3 
 
 2 
 3 
 
 3.5 
 5 
 3 
 
 
 Si 
 
 
 s 
 
 
 
 
 
 
 
 
 
 
 — 
 
 3 
 
 =1 
 s 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 ^i 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 Z^ 
 
 
 
 
 
 V 
 
 
 
 
 
 
 " z:^ 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 z 
 
 -Ji 
 
 ^, 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 I i£ 
 
 
 
 
 
 
 
 
 
 ^O 1 1 
 
 HOURS l_ 
 
 - ^ 
 
 
 
 O i 
 
 
 4 
 
 6 
 
 8 
 
 JO \ 1;2 1 
 
 2 4 6 8 10/ 
 
 12 2 4 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 - T- 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 r 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 - ^ - 
 
 
 
 
 _^o 
 
 
 
 
 
 
 
 
 \ 
 
 7 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 -^^^ / - 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 sy^'^s, 1 _ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5 J 
 
 
 
 
 
 
 
 
 
 
 
 
 
 : si: 
 
 
 
 
 -10° 
 
 
 
 
 
 
 
 
 
 \t 
 
 
 
 
 
 
 
 
 
 
 
 
 
 yz . 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 the 
 mi 
 hi^ 
 
 tal 
 sti 
 
 Her 
 
 3sa 
 ich, 
 ;hes 
 ^Ids 
 nila 
 idie 
 
 e 
 
 Ql 
 
 a 
 ,t 
 
 t 
 Lt( 
 d 
 
 t 
 
 e 
 .n 
 
 a 
 h 
 id 
 
 a 
 
 d 
 n 
 is 
 \ 
 nc 
 
 } t 
 
 les 
 he 
 
 a ] 
 
 ir 
 igi 
 Ic 
 
 ab 
 
 tic 
 
 )W 
 
 o^^ 
 ifo 
 ire 
 om 
 
 ul 
 )n 
 r 
 
 re 
 rn 
 s 
 P 
 
 at 
 
 s 
 
 H 
 
 st 
 aa 
 
 y 
 
 ai 
 
 ed fig 
 as to i 
 )idly ] 
 point 
 Ltion £ 
 iekl i 
 ed. 
 
 'ures and the graph answer 
 :he temperature — when, how 
 t rose or fell, what were its 
 s, etc. The graph, however, 
 it a mere glance, while the 
 b only after they have been 
 
 Note. For other interesting applications of graphs see Tanner and 
 Allen's Analytic Geometry, pp. 73-78. Also, and especially, " Graphic Meth- 
 ods in Elementary Algebra," by Prof. William Eetz, in /School Science and 
 Mathematics, vol. 6, pp. 683-687. This article gives many good suggestions 
 as well as valuable material and references. 
 
 EXERCISE C 
 
 By reference to the above temperature graph (usually called 
 thermograph)^ answer the following questions : 
 
 1. Between what hours was the temperature below 0° ? When 
 was it lowest ? 
 
 2. When was the temperature falling most rapidly ? Explain. 
 
145 J GRAPHIC liEPBElsENTATlON OF EQUATION H 233 
 
 The following tables give the population (in millions) of the 
 countries named, for certain years between 1800 and 1900. 
 
 British Isles 
 
 Lands now in the 
 German Empire 
 
 France 
 
 Unitei 
 
 States 
 
 Year 
 
 Population 
 
 Year 
 
 Population 
 
 Year 
 
 Population 
 
 Year 
 
 Population 
 
 
 (millions) 
 
 
 (millions) 
 
 
 (millions) 
 
 
 (millions) 
 
 1801 
 
 15.9 
 
 1816 
 
 24.8 
 
 1801 
 
 27.3 
 
 1810 
 
 7.2 
 
 1811 
 
 17.9 
 
 1837 
 
 31.5 
 
 1821 
 
 30.4 
 
 1820 
 
 9.6 
 
 1821 
 
 20.9 
 
 1847 
 
 34.7 
 
 1841 
 
 34.2 
 
 1830 
 
 12.8 
 
 1831 
 
 24 
 
 1856 
 
 36.1 
 
 1861 
 
 37.3 
 
 1840 
 
 17 
 
 1841 
 
 26.7 
 
 1865 
 
 39.4 
 
 1866 
 
 38 
 
 1850 
 
 23.2 
 
 1851 
 
 27.8 
 
 1872 
 
 41 
 
 1872 
 
 36.1 
 
 1860 
 
 31.4 
 
 1861 
 
 28.9 
 
 1876 
 
 42.7 
 
 1876 
 
 36.9 
 
 1870 
 
 38.5 
 
 1871 
 
 31.4 
 
 1885 
 
 46.8 
 
 1881 
 
 37.6 
 
 1880 
 
 50.1 
 
 1881 
 
 34.9 
 
 1895 
 
 52.2 
 
 1891 
 
 38.3 
 
 1890 
 
 62.6 
 
 1891 
 
 37.7 
 
 
 
 1896 
 
 38.5 
 
 
 
 3. Taking the number of years after 1800 as the cc-coordinate 
 and the population (in millions) as the ^/-coordinate, locate the 
 several points represented by the above table for the British Isles, 
 and join these points by straight lines. Similarly, draw graphs 
 for the remaining tables. 
 
 4. By reference to your graphs, compare the population of the 
 countries named in 1830; in 1856; in 1880. By reference to the 
 tables compare the populations in 1871. 
 
 5. In making comparisons like those of Ex. 4, is it easier to 
 use the tables or to use the graphs ? Why (cf. § 145)? 
 
 6. By reference to the graphs answer the following questions : 
 
 (1) When did the population of the United States first exceed 
 that of the British Isles ? that of France ? that of Germany ? 
 
 (2) During what years has the population of the United States 
 increased most rapidly ? 
 
CHAPTER XIV 
 IRRATIONAL NUMBERS - RADICALS 
 
 146. Preliminary remarks and definitions. A number 
 which may be expressed as the quotient of two integers, 
 positive or negative, is called a rational number. 
 
 J^.^., 3( = f); -7l( = ^); 2. 75( = |f|), etc., are rational 
 numbers. 
 
 Nearly all the numbers thus far used have been rational, 
 although we have met a few such forms as V2 and V — 5. 
 
 In this and the next chapter we shall examine more 
 closely such numbers as V2 and V— 5. These numbers are 
 particular cases of a/^, which is defined (§ 113) by the 
 equation (VaY^a; hence ( V2)2 = 2 and (V^=~5)2=-5. 
 
 The numbers V2 and V — 5 resemble each other in that 
 neither of them is rational (since no rational number squared 
 is 2 or — 5), but, as we shall soon see, they differ widely 
 in another regard. 
 
 By squaring 1 and 2, we find that V2 is greater than 1 and 
 less than 2 ; then by squaring 1.1, 1.2, 1.3, •••, we find that 
 V2 is greater than 1.4 and less than 1.5; similarly, V2 is 
 greater than 1.41 and less than 1.42 ; etc. 
 
 Since V2 lies between 1 and 2, 1.4 and 1.5, 1.41 and 1.42, 
 etc., therefore, if 1.4 (or 1.5) is taken for V2, the error is 
 less than 0.1 ; if 1.41 (or 1.42) is taken, the error is less than 
 0.01; and, by continuing this process, we can find rational 
 numbers which approximate V2 to any required degree of 
 accuracy. 
 
 234 
 
140-147] IRRA TIONAL NVMBEIiS — liADTCA LS 235 
 
 On the other hand, since the square of any rational num- 
 ber is positive, therefore we cannot express V — 5, even 
 approximately, by means of rational numbers. 
 
 Numbers like V2, which are not rational, but which may 
 be expressed approximately to any required degree of accuracy 
 by means of rational numbers, are called irrational numbers. 
 
 ^.g., V2, 5 — V7, and 10 4- v2 are irrational numbers. 
 
 Numbers like V — 5, wliich cannot be expressed, even 
 approximately, by means of rational numbers, are called 
 imaginary numbers (cf. § 114, Note). Rational and irra- 
 tional numbers taken together are called real numbers. 
 
 Note to the Teacher. Emphasis should be laid upon the fact that 
 although such numbers as V2 can be expressed only approximately by means 
 of rational numbers, they are, nevertheless, just as exact and definite as are 
 integers and fractions. n 
 
 Thus, let ABCD be a square whose side AB is 1 foot long, 
 and let x represent the number of feet in its diagonal AC, 
 then it is easily proved by geometry that 
 cc2 = 2, i.e., that x=y/2. 
 
 The numbers 1, 1.4, 1.41, 1.414, 1.4142, etc., are successive 
 approximations to the length of this diagonal, but its exact 
 length is V2 ; hence the necessity of including such numbers as y/2, in our 
 number system. 
 
 It will be worth while also to connect this latest extension with the exten- 
 sions previously made (see p. 16, footnote). Thus fractions arose from gen- 
 eralizing division ; negative numbers arose from generalizing subtraction ; and 
 in the present article it appears that generalizing evolution introduces two 
 further new kinds of numbers, viz., the irrational and the imaginary. 
 
 In other words : while the direct operations (viz., addition, multiplication, 
 and involution) with positive integers always produce results that are positive 
 integers, the inverse operations (viz., subtraction, division, and evolution) 
 lead respectively to negative, fractional, and irrational and imaginary num- 
 bers, and demand for their accommodation that the primitive idea of number 
 be so enlarged as to include these new kinds of numbers. 
 
 147. Further definitions. An indicated root is usually 
 called a radical; the number whose root is indicated is 
 called the radicand. If the root is irrational, but the radi- 
 cand rational, the expression is often called a surd. Thus, 
 
 HIGH SCH. ALG. — 16 
 
236 HIGH SCHOOL ALGEBRA [Ch. XIV 
 
 V2, a/8, 6-v/45, V — 2, and V5 + VlO are radicals, whose 
 respective radicands are 2, 8, etc. ; of these radicals V2 and 
 6V45 alone are called surds. 
 
 The coefficient of a radical is the factor which multiplies 
 it, and the order of the radical is determined by the root 
 index. Two radicals which have the same root index are 
 said to be of the same order. Thus, the surds 12V5 ax^ and 
 m2V674 are of the same order, viz., the 7th, and their co- 
 efficients are 12 and m^ respectively. 
 
 Surds of the second and third orders are usually called 
 quadratic and cubic surds, respectively. 
 
 Radicals which, when simplified, are of the same order 
 and have their radicands exactly alike are called similar 
 (also like) radicals ; otherwise they are dissimilar (unlike). 
 Expressions which involve radicals, in any way whatever, 
 are called radical expressions ; they are monomial, binomial, 
 etc. (cf. § 20), depending upon the number of their terms. 
 Thus, V5 and 3V5 are similar, monomial, quadratic surds, 
 while b a -\- 3 V7 and 2 V9 + 3 Va: are binomial surds. 
 
 148. Principal roots. We have already seen that a number 
 has two square roots (e.^., V9 is + 3 or — 3), and we shall 
 see later that every number has three cube roots, four fourth 
 Yoot^^ five fifth roots, etc. 
 
 E.g., a/8 = 2, -1+V":r3, or - 1 -V^^, since the 
 cube of each of these numbers is 8 ; and VI6 = 2, — 2, 2V— 1, 
 or - 2V^^. 
 
 Although the number of roots always equals the order of 
 the radical, not more than two of these roots can be real; and 
 when there are two real roots, they are opposite numbers. 
 By the principal root of a number is meant its real root., if 
 there is but one real root, and its real positive root if there 
 are two real roots. 
 
 E.g.., if attention is confined to principal roots, V9 = 3 
 (and not - 3), -V^^ = - 2, ^125 = 5, a/T6 = 2, etc. 
 
147-148] IRRATIONAL NUMBERS — RADICALS 237 
 
 EXERCISE CI 
 
 1. What is a rational number ? Use your answer to show- 
 that 7, f , — 8|-, and V36 are all rational. 
 
 2. What is an irrational number ? Is ^/8 an irrational num- 
 ber ? Why ? 
 
 3. By the method used in § 146 for V2, find two approximate 
 values for V3 (one larger and the other smaller than the true 
 value) which differ from V3 by less than 0.001. 
 
 4. Find two successive approximations to the value of V5. 
 Compare these approximations with the result of extracting the 
 square root of 5 by the method of § 118. 
 
 5. What is an imaginary number ? Give several illustrations. 
 For what values of n is-v^'— 5 imaginary ? 
 
 6. Is the number 2l4-Vl7 rational or irrational? Why? 
 What kind of number is 84 V5 - ^^-^ ? Why ? 
 
 7. Are both ^'21 and v 2 + V7 radicals ? Are they surds ? 
 Are all radicals surds ? Are all surds radicals ? 
 
 8. In Exs. 44-54, p. 242, point out the coefficient of each surd. 
 May the coefficient of a surd be fractional ? negative ? 
 
 9. Write a surd of each of the following orders : 2d, 5th, 
 3d, 7th. 
 
 10. Define similar surds, and illustrate your definition. May 
 the coefficients differ and the surds still be similar ? 
 
 11. What factor do two similar surds necessarily have in com- 
 mon? What kind of a number, then, is the quotient of two 
 similar surds ? Illustrate your answer. 
 
 12. Write a monomial cubic surd ; a binomial quadratic surd ; 
 a trinomial surd of the 5th order. 
 
 13. How many values has Vl6? What are they? What is 
 the principal square root of 16 ? What is the principal fifth root 
 of — 32 ? Define the principal root of a number. 
 
238 HIGH SCHOOL ALGEBRA [Ch. XIV 
 
 149. Principles involved in operations with radicals. If 
 
 we exclude imaginary numbers, the principles employed 
 in operations with radicals may be symbolically stated 
 thus: 
 
 (i) Vlcy = ^Hc • Vy^ 
 
 (ii) 4-=—^ 
 
 (iii) 'Vx = ^'Vx = ^x, 
 
 wherein n and t are positive integers, while x and y may 
 have any values whatever, except that they cannot be nega- 
 tive when the root index is even. 
 
 150. Proof of the principles in § 149. For the sake of sim- 
 plicity, we shall (1) limit the proofs to principal roots, and 
 (2) assume that a change in the order of the factors of a 
 product, even when these factors are irrational numbers, 
 leaves the product unchanged (cf. § 42). 
 
 With these restrictions the correctness of (i), (ii), and 
 (iii), § 149, follows from the meaning of the symbol Va 
 (§ 113). Thus, to prove (i) we proceed as follows : 
 
 Qy/xVyY = VxVy • Vx Vy- •••to n factors [§ 9 
 
 = xy ; [since ( Va)" = a 
 
 whence Vx^y^^'xy, [§113 
 
 which was to be proved. 
 
 This principle may be translated into words thus : 
 
 The nth root of the product of two numbers equals the product 
 of the nth roots of these numbers. 
 
 The proofs of (ii) and (iii) are left as an exercise for the 
 pupil. 
 
149-101] IHRATIONAL NUMBERS — liADICALS 239 
 
 EXERCISE Cll 
 
 Verify the following equations : 
 
 1. V9- V25 = V9.25. 3. Vi6T9 = Vl6. V9. 
 
 2. ^/38.^27=^-8.27. 4. ■v'lOOO a« = ^125 a' • -y/S. 
 
 5. Show that Exs. 1-4 are special cases of § 149 (i). 
 
 6. Find V5 • v3 correct to two decimal places (§ 118) ; then 
 find Vl5 (i.e., VS • 3) correct to two decimal places, and compare 
 results. Does this exercise illustrate any jwactical advantage in 
 knowing that -\/x • -s/y = -Vxy ? Explain. 
 
 7. By means of § 149 (ii), show that V35-^V7=V5, and 
 that VlOa^-^- V— 2a = V — 8a. State § 149 (ii) in words. 
 
 8. Find (correct to two decimal places, § 118) V7 -^ V5, also 
 Vl.4 (i.e., VT^S), and compare results. Does this exercise 
 
 n/ ni 
 
 illustrate any practical advantage in knowing that ^^ = -y- ? 
 
 -\/y ^y 
 
 9. Show that f^X^^ and thus prove that V*/^-^. 
 
 Verify that "Vx = yj</x = ^ VS [cf. § 149 (iii)] when 
 
 10. n = 2, p = 2, a; = 81. 12. n = 4, p = 3, a; = c-*d^l 
 
 11. n = 5, p = 2, x = m^. 13. n = 2, p=^S, a; = 64. 
 
 14. Use § 149 (iii) to find -^49 (i.e., \/49), correct to two 
 decimal places ; also find Vl44. How may you find the 9th 
 root of any given number? the 8th root? 
 
 15. Prove the correctness of § 149 (iii). What do n and t 
 represent in this principle ? May nt, then, equal 11 ? Explain. 
 
 151. Special cases of § 149. Besides the three principles 
 given in § 149, these four others are often useful : 
 
 (i) V^ = jrV/, 
 
 (ii) -^jr^... =Vx-V/- Vz--, 
 
 (iii) ^^=(^xy, 
 
 (iv) V]r=A/i^. 
 
240 HIGH SCHOOL ALGEBRA [Ch. XIV 
 
 The correctness of these principles may be established by 
 the method used for the proof of (i), § 149 ; it is easier, how- 
 ever, to regard them as special cases of § 149, thus: 
 
 ■v/i^ = V^V^, [§149(i) 
 
 which establishes (i) above. 
 
 So, too, ^xyz ••• = Vx ' ^yz ••• [§ 149 (i) 
 
 = ^x ' -\/y ' -Vz ••• = etc., 
 which proves (ii) above ; and (iii) follows from (ii) by let- 
 ting x = y = z = '•', and supposing the number of these fac- 
 tors to be t. 
 
 Again, %/^=a/^ [§ 149 (iii) 
 
 , . , ,. . =V:c% [since </(xy = x'- 
 
 which proves (iv). *- ^ ^ 
 
 EXERCISE cm 
 
 1. Is S VE equal to VS^Ts? Why? May 2-^ be written 
 as V2^ • 6 ? Why ? How may the coefficient of a radical be 
 inserted under the radical sign [cf. § 151 (i)]? 
 
 2. By means of § 151 (i) show that V20 = 2V5, and also that 
 V— 54=— 3V2. How may we simplify a square root which 
 contains a square factor? a 5th root which contains a factor 
 raised to the 5th power ? 
 
 3. By the method of § 150, show that x^y = V^. 
 
 4. Verify that V2 • V3 • V5 = V30 ; find at least two deci- 
 mal places in each member of the equation (cf. Ex. 6, p. 239). 
 How find the product of several radicals of the same order ? 
 
 5. By the method of § 150, show that ■\/x • -Vy • -\/z = -Vxyz. 
 
 6. Find, correct to two decimal places, (V6)^, also V6^, and 
 compare results. How would you raise Vl5 to the second 
 power? to the 5th power? Explain. 
 
 7. Show, using the method of § 150, that (^x) = -y/af. 
 
151-152] IRRATIONAL NUMBERS — RADICALS 241 
 
 8. Verify that ■\/aP = -^a^, and that -\/a^^ = Va\ In each 
 of these equations compare the exponents of a, also the root- 
 indices. 
 
 9. Is a radical changed in value if we multiply its root- 
 index and also the exponent of its radicand by the same factor 
 [cf. § 151 (iv)]? if we divide both by any factor common to 
 them ? Illustrate. 
 
 10. Using § 151 (iv), show that -^9 (i.e./^3^) = V3 = ^8i. 
 Which is more easily computed, V32, or its equal, V2? 
 
 152. Reduction of radicals to their simplest forms. A 
 
 radical is said to be in its simplest form when the radicand 
 is integral, when the index of the root is as small as possible, 
 and when no factor of the radicand is a perfect power corre- 
 sponding in degree with the indicated root. 
 
 The following examples illustrate the application of the 
 foregoing principles in the reduction of radicals to their 
 simplest form. 
 
 Ex. 1. Reduce VSoV to its simplest form. 
 
 Solution. V8 aV = V4 a V . V2 ax [§ 149 (i) 
 
 = 2 ax^ V2 ax. 
 Ex. 2. Reduce V4 a^x^y^ to its simplest form. 
 
 Solution. -s/A.o'xY = </{2 aa^i/f = ^2^^. [§ 151 (iv) 
 Ex. 3. Reduce Vf to its simplest form. 
 
 Solution. -^ - = -J — 1 
 
 5_2 
 5.52 
 
 =^^\>m=Ujm, [§i5i(i) 
 
 EXERCISE CIV 
 Reduce each of the following to its simplest form : 
 4. V18 (^.e., V9^). 6. V45. 8. S^IG. 
 
 5 V24. 7. V75. 9. S^^^^. 
 
242 
 
 HIGH SCHOOL ALGEBRA 
 
 [Ch. XIV 
 
 29. 
 
 
 45 
 
 /27 a? 
 
 10. 2\/54. 
 
 11. ^32. 
 
 12. VI 0-.e.,ViT7). 30. Va^ (cf. Ex. 2). 46. i^f. 
 
 13. VS 
 
 15. ^^ ^ 
 
 16. VJ (cf. Ex. 3). 34. V216. 
 
 17. vi- 
 
 31. V^ 
 
 32. Va;y. 
 
 33. a/25. 
 
 47. 3^125 a^a^. 
 
 48. y\^±y. 
 ^ x-y 
 
 18. lOVf 
 
 19. </l 
 
 20. 2V|. 
 
 21. V^^ 
 
 22. v=i: 
 
 35. V32m^7i'«. 
 
 36. 3^36 h-x'\ 
 
 49. ■>/a2"a;"+\ 
 50. 
 
 64 m« 
 
 37. V 32 a VI 
 39. 
 
 125 
 
 51. 6^320. 
 
 52. V- 486^2^ 
 
 53. |V2|. 
 
 23. V27a;l 
 
 24. VSw?. 
 
 \9^ 
 
 54. Vl8a-9. 
 
 25. Va«6^'. 
 
 26. VV^/*. 
 
 40. 
 
 41. 
 
 yV 
 
 55. Var+Wy^\ 
 
 11, 
 2x 
 
 56. Va2''ft.""+^ 
 
 4 25a« 
 
 57. </ - 40 a^^-^y\ 
 
 27. 
 
 r 
 
 58. 4 Va^" — a*"aJ". 
 59. 
 
 28. J^ 
 
 V 
 
 L 3a^^ 
 
 -2 6a; 
 
 2a 
 
 60. V3a;2— 6«2/+32/^. 
 
 42. V162. 
 
 43. a/36. 
 
 44. 2V|V. 
 
 In the following, insert the coefficients under the radical 
 signs : 
 
 66. -4^1. 
 
 61. 3V7. 
 
 62. 5^4. 
 
 63. 2a/6. 
 
 64. -2 Vs. 
 
 65. y\V40, 
 
 70. (c4-l)V5c. 
 
 67. iVlW-. 
 
 71. 
 
 72. 
 73. 
 
 1 , 
 
 O/. ^ V X24. 
 
 -VaMo^-i). 
 
 68. -V72a^l 
 
 4 w3vV4 wv^ 
 
 69- ~^12a'x. 
 
 -2y'-2^<Jyh\ 
 
152-153] IRRATIONAL NUMBERS — RADICALS 243 
 
 153. Addition and subtraction of radicals. Similar radicals 
 (§ 147) may evidently be added and subtracted by regarding 
 the common radical factor as the unit of addition. The sum 
 or difference of dissimilar radicals can, of course, only be 
 indicated, and this is done by connecting the radicals with 
 the proper signs (cf. § 23). 
 
 The radicals to be added or subtracted should first be 
 reduced to their simplest forms ; the following examples 
 will illustrate the procedure. 
 
 Ex. 1. Find the sum of V75 and 3 Vl2. 
 
 Solution. Since V75 = V25T3 = ^25 • V3 = 5 V3, [§ 149 (i) 
 and 3 Vl2 = 3 V473 = 3 V4 . V3 = 6 V3, [§ 149 (i) 
 
 therefore V75 + 3 Vr2 = 11 V3. 
 
 Ex. 2. Find the sum of 5 Vl8 and — VOX 
 
 Solution. Since 5 Vl8 = 5 V9^ =5-3 V2 = 15 V2, 
 
 and -Vo:5^-VT=:--v/-^ = -|V2, 
 
 therefore 5 Vl8 + ( - Vo:5) = (15 - 1) V2 = 14i V2. 
 
 Ex. 3. Find the sum of V9 x - 18, 6 V4 a; + 8, V36 x - 72, and 
 
 -V25a; + 50. 
 
 Solution. V9 a; - 18 + 6 V4a;4-8 + V36a;-72 - V25a;-f 50 
 
 = 3 V^^^ + 12 V^T2 + 6 V^^^ - 5 a/^T2 
 =9 Vx'^ + 7 V^T2^ 
 
 EXERCISE CV 
 Find the sum of : 
 
 4. V5, 7 V5, and - 3 V5. 8. 2 V20, -^ V45, and Vsa 
 
 5. Vl8, V50, and V98. 9. ^250, -v^, and -^U. 
 
 6. VT2, V75, and V27. 10. \/500, -y/WS, and ^^^^32: 
 
 7. V28, - 2 V63, and V700. 11. <^cdf*, 3 -\/cdf, V - 8 c'df. 
 
244 HIGH SCHOOL ALGEBRA [Ch. XIV 
 
 12. Find (correct to two decimal places) the value of each of 
 the surds in Ex. 7 (cf. § 118) and add your results. How does 
 this sum compare with that previously found for Ex. 7 ? Is there, 
 then, any practical advantage in simplifying surds before adding 
 them? 
 
 13. Find the sum of a, 2 b, and c ; oi S x, 4: y, 2 x, and —5y. 
 
 14. What isjhe sum of 3V2 and 5^/7? of 3 V2, 5^/7] 
 - 2 V7, and -V2? 
 
 15. Write a rule for the addition and subtraction of radicals, 
 providing both for those cases in which the given radicals are 
 similar and for those in which they are dissimilar. 
 
 Simplify the following expressions as far as possible (cf . § 152), 
 and explain your work in each case : 
 
 16. ^/i35 + -^625-■^/320. 22. -^128^+ a/375^- -v^Sl^ 
 
 17. -^40 + V28 4- V175 +^25: /«;,/«_ ja 
 
 19. V5 + V75-V12 + 2V3. 4»_ jW^ /£^ 
 
 20. Vl47-Vi + iv/3+J^9: • ^bY \ bf ^^bf' 
 
 21. 6^|o+4^/I|-8^/|||: 25. V(a4-6)^c-^(a+6)V. 
 
 26. ^l92^-2-^3^-^/5^+v40^. 
 
 27. -y/abx -\- -^^^m^ - \^S~aW^. 
 
 28. V3a;3 + 30ar^ + 75a5-V3a^-6a^ + 3a;. 
 
 29. V5a^ + 30a* + 45a3-V5a^-40a^ + 80a^ 
 
 30. V50 + -V/9-4 VJ + -v/24 + ^27-a/64. 
 
 31. V| + 6V|-iVl8 + ^36-^+^125-VS. 
 
 32. Va^ — a^x — Vaa?? — a^ — V(a + a;)(a^ — a?^. 
 
 154. Reduction of radicals to the same order. By (iv) of 
 §151, 
 
 ^5 = ^-^5^ =^5^ =^625, 
 and ^7 = ^^P = -^^'P=-^^3"43, 
 
153-154] IRRATIONAL NUMBERS — RADICALS 245 
 
 i.e., the radicals V5 and -^/l are equivalent, respectively, to 
 ^625 and v'348 ; and these last two radicals are of the 
 same order, viz., the twelfth. 
 
 Moreover, it is evident that, by a similar application of 
 §151 (iv), any two or more radicals whatever may be 
 reduced to equivalent radicals of the same order. This new 
 order must, of course, be some common multiple (preferably 
 the L. C. M.) of the orders of the given radicals. 
 
 EXERCISE cvi 
 
 1. Is -^^ equal to Vx? Is ^3aV equal to ^9aV? Using 
 the method of § 150, prove the correctness of your answer to each 
 of these questions. Compare also § 151 (iv). 
 
 2. Eeduce ■y/25 mV and a/8 a^6V to equivalent radicals of 
 the 4th order, and explain (cf. Exs. 8-10, p. 241). 
 
 Reduce to equivalent radicals of the order indicated, and ex- 
 plain your work : 
 
 3. -s/ay^, 9th order. 7. 3 ax, 4th order. 
 
 4. -^SM, 6th order. 8. ^2 m\ 12th order. 
 
 5. -^2sf, 10th order. 9. ^AV, 12th order. 
 
 6. V2 sf, 10th order. 10. VciVa^, 8th order. 
 
 11. Express as equivalent radicals of the 6th order: -\/' 
 
 m, 
 
 ■\/9 m^n\ and mn. 
 
 Eeduce the following to equivalent radicals of the same order: 
 
 12. V5 and ^/li. 19. Vl4, ^25, and ^95. 
 
 13. -v/T and V3. 20. 3V3, 5V2, and 2-\/20. 
 
 14. V27and-\/3^l 21. 2 V3, 3^2, and 2-v/39. 
 
 15. V6and3. 22. 3Va&, a^2^, and -^^^^ 
 
 16. -Vp, V5j^ and -y/f^. 23. x'</2^, V^,and2^^^ 
 
 17. i/W, V2, and ■^. 24. </x, 5 xVy, and V^. 
 
 18. ~^2¥y, \^3a? and xy. 25. ^d^ + 6"^, and aVa ~ b. 
 
246 inan school algebra [Ch. xiv 
 
 26. Can the radicals in Ex. 22 be reduced, to equivalent radi- 
 cals of the 6th order ? of the 12th order ? of the 9th order ? 
 Give the reasons for your answer in each case. 
 
 27. What is the lowest common order to which you can reduce 
 the radicals in Ex. 23 ? those in Ex. 24 ? in Ex. 25 ? 
 
 28. By first reducing Vl5 and V6 to the same order, show 
 that -\/W is greater than V6» 
 
 29. Which is greater, V5 or Vl2 ? Explain your answer 
 (cf. Ex. 28). _ 
 
 30. Which is greater, 3 VlO or 2 a/100 ? 
 
 Hint. First insert coefficients under radical signs (cf. Ex. 1, p. 240). 
 
 Arrange the following radicals in order of magnitude : 
 
 31. 3 V3, 5 V2, and 2 v'20. 32. 2 ^39, 3 -y/2, and 2 V3. 
 
 155. Product of monomial radicals. If two or more radi- 
 cals are of the same order, their product may be written 
 down immediately by § 151 (ii) ; while if they are of differ- 
 ent orders, they should first be reduced to equivalent radi- 
 cals of the same order (§ 154). 
 
 Ex. 1. Multiply a/5 by V2^ 
 
 Solution. The L. C. M. of the orders of the given radicals is 
 6, and by §154 ^=V5^, 
 
 and V2=-v/2^; 
 
 therefore </5' V^ = </¥' ''^¥ = y/W^' [§ 149 (i) 
 
 = a/200. 
 
 Ex. 2. Find the product of 4 h-\/ax and V5 a^, and simplify 
 the result. 
 
 Solution. As in Ex. 1, 4 h^ax = 4 6 VaV, 
 
 and -\/5 a^ = V5^ a^ ; 
 
 therefore 4 b^ax • V5 a^ = 4 ft Va V . V 5^a* 
 
 = 4 6^52^V [§ 149 (i) 
 
 = 4a6A^25^^. [§151(i) 
 
154-156] IRRATIONAL NUMBERS — RADICALS '247 
 
 EXERCISE evil 
 Find the following products, and simplify the results: 
 
 4. 2Vi5 . V5. '^2y^y' 
 
 6. -^ . ^^. 11. (8 - 2 Vl5) . 2 V6. 
 
 13. How may we find the product of two or more radicals 
 which are of the same order? 
 
 14. How may we find the product of two or more radicals of 
 different orders ? Illustrate, using the surds V3 and V2. 
 
 Find the following products, and simplify the results : 
 
 15. -Vab • V&c. 
 
 16. V3 . 3^. 
 
 17. V2 . </I. 
 
 18. 2\/5 . 7-^10. 26. V^' • Vl2^ . V75^^ 
 
 23. 
 
 ^f. 
 
 </2f. 
 
 24. 
 
 2V2 
 
 ■ ^256. 
 
 25. 
 
 V2- 
 
 ^1-^. 
 
 19. a/i . ^(if. 27. V2a& . ^abc • ^4:a^b\ 
 
 20. \/^ . aP^ 
 
 3/]^ J/W 28. V9(c-A;)=^.-lV3(c-A;). 
 
 29. (V72+^12-3 V98) • V2. 
 
 21. ^4a2 . WSa^ 
 
 22. ^^ . V^. 
 
 156. Multiplication of polynomials containing radicals. The 
 
 product of two polynomials containing radicals is obtained 
 by multiplying each term of the multiplicand by each term 
 of the multiplier and adding the partial products, just as in 
 the case of rational polynomials. 
 
248 HIGH SCHOOL ALGEBRA [Ch. XIV 
 
 Ex. 1. Multiply 5V2-2V3 by 3V2 + 4V3. 
 
 Solution. 5 V2 — 2 V3 
 
 3 V2 + 4 V3 
 30 - Q^Q_ 
 
 + 20V6-24 
 30 + 14 V6 - 24 = 6 + 14 V6. 
 
 Ex. 2. Expand (2V3— ■v2)^ by the binomial theorem. 
 
 Solution. (2V3-a/2)2==(2V3)2-2(2V'3)^2+(a/2)2 
 
 = 12-4^108 + ^4. 
 
 EXERCISE CVIII 
 
 Multiply, and express your results in the simplest form : 
 
 3. V5 — 5 by V5 + 1. fa S 
 
 /a 4b\ , fa 4/6\ 
 
 4. 2V7-3by 2V7 + 4. 
 
 5. 5Vn-h4by 5Vil-4. 13. V2 + V3 + V6 by V2+4V3. 
 
 6. Vc + V2d by Vc - V2d. 14. x - Vxyz + 2/^ by Vx + V^. 
 
 7. (-Vac-VS^by, ^^ V2^-^3^by ^^-^^. 
 a (3Vi + 5yi)l _ ^^ ^^ 
 9. (a - a6 V2 + 62)2. 16- 3 V3- V2 by 5 V4 4- V2. 
 
 10. ^3-3^9 by ^/3 + 3-^9. 17- 2 V3-^2 by 2 V3- ^4. 
 
 11. (</9 - 4 ^6 y. 18. ^ls-2^1 by V8 + 2V7. 
 
 19. VlO-3V5-4V3by 4V3-3V5. 
 
 20. Vic?/ + 2 ViC2! — V2/2! by ^xy — 3 ViC2!. 
 
 21. -\/m^v?- + Vm7i by -\JmT? -f- -^mhi. 
 
 22. 2-^/^-3 vi^s i^y a/16^2^^^. 
 
 23. Vaf" — VlO aj"+^ + V^ by V5^ + V2^«+^. 
 
 24. V(.'C+?/)*-V(a;+2/)'»+2by V(a:+2/)""+'+V(a;4-2/)'- 
 
 25. .+|-^Z^by.. + | + ^/Z±S". . 
 
156-157] IRRATIONAL NUMBERS— RADICALS 249 
 
 157. Rationalizing factors.* Conjugate surds. The factor 
 by which ii given surd (radical) must be multiplied in order 
 to obtain a rational product is called its rationalizing factor. 
 Thus, of the surds V5 a^ and V 25 a, each is the rationalizing 
 factor of the other, since their product, 5 a, is rational ; the 
 same is true of the surds 2Va— V3 and 2 Va + VS. (Why ?) 
 
 Of two such binomial quadratic surds as 2 Va — V3 and 
 2Va + V3, which differ only in the sign of one term, each 
 is called the conjugate of the other. Moreover, since the 
 product of any two conjugate quadratic surds is rational 
 (§ 53), therefore each of them is the rationalizing factor 
 of the other. 
 
 EXERCISE CIX 
 
 1. Is V2a the rationalizing factor of V2a? Why? Show- 
 that VS kH and 2 V3 — V5 are the rationalizing factors of V'^ kH^ 
 and 2 V3 + V5, respectively. 
 
 2. Is V5-2V3 the rationahzing factor of 2V3-hV5? 
 Explain. Are these surds conjugate to each other ? 
 
 Find the rationalizing factor of : 
 
 3. V3. 10. Vf. 16. 3a-V5^. 
 
 4 ^^7 2 6/"^ 17. ^5x — V2ay. 
 
 5. 2ViO. "' "'^^' ^«- Va» + 2^6. 
 
 6. 
 
 ^■27/ X.. ^. - xi-vs- 
 
 '• ^4*^- 13. V2-V7. 20. ^Va-cJl. 
 
 3 _ b ^ d 
 
 8. V-i. 14. 4 + 5V3. ,14-3 
 
 9. 5^^^ 15. 2V3 + V8. ^^- \"r'^4^" • 
 22. How may we find the rationalizing factor of any binomial 
 
 quadratic surd? Why? Does the same method answer for a 
 binomial cubic surd ? Explain. 
 
 * See also § 177. 
 
250 HIGH SCHOOL ALGEBRA [Ch. XIV 
 
 158. Division of monomial radicals. If the dividend and 
 divisor are of the same order, tlieir quotient may be written 
 down immediately by § 149 (ii), while if they are of differ- 
 ent orders, they should first be reduced to equivalent radicals 
 of the same order (§ 154). 
 
 E.g., to divide -^4 a:(?y^ by ■\/2 a^x, we proceed thus : 
 
 ■V4 aoi?y- _ V16 a-^m^y^ 
 
 -V2a'x V8a' 
 
 [§154 
 
 4 
 
 ^ CH49(ii) 
 
 W2^^1^^^;^_ [§152 
 
 ^ 
 
 159. Division of polynomials containing radicals. If the 
 
 divisor is a monomial, then, manifestly, the quotient may be 
 obtained by dividing each term of the dividend by the 
 divisor. 
 
 Ex. 1. Divide 3V2 + 4V3 by V2. 
 
 Solution. ^V2 + 4V3 ^ 3 _^ ^^3 ^ 3 _^ 2V6. [§ 152 
 
 We may also solve Ex. 1 as follows : 
 
 3V2 + 4V3 ^ (3V2+4V3)V2 ^ 6 + 4V6 ^o ^ /^ 
 V2 (V2)^ 2 -^ ^ ' 
 
 i.e., we may, before dividing, multiply both dividend and divisor 
 by the rationalizing factor (§ 157) of the latter. This method 
 is known as " division by means of rationalizing the divisor " ; in 
 many examples it is easier than the one used in the first solution 
 above (cf. Exs. 28, 29, and 37 below). 
 
 EXERCISE CX 
 
 Eind the following indicated quotients and simplify each : 
 
 2. V20--V5. 4. 6V5-T-V4(). 
 
 3. V2l6-f-Vl2. 5. V^-f-V3^'. 
 
1 -is- lot) J inilATlONAL NUMBERS — RADICALS 251 
 
 6. V'M-^V^. 13. 6-r-3^4. 
 
 7. 2-v^54--v/216. 14. VI^TS^f. 
 
 8. Sv^lS-lS^J. 15. ^57i2^^-j- V2^. 
 
 9. </(a-6)---V(a + 6)l 4a; 3/^ 
 
 10. 2V6-=-^/3. 2/ ^2/' 
 
 11. 2\/l2^V8. 1^- a^4ar*2/^-^2feA/2a;2/. 
 
 12. 10 -- V5. 18. 3 aV2~W'^^ -^ 2 6^3^^. 
 
 19. Find the value of 1 h- V2, correct to two decimal places : 
 (1) by finding V2 and dividing, and (2) by first rationalizing the 
 divisor. Which is the easier process ? 
 
 20. Find (correct to two decimal places) the value of : 
 
 -^ ; -^ ; ^ ; and -^. (Cf. Ex. 19.) 
 
 V3 ' V2 ' V2 ' V(5 ^ ^ 
 
 Find the following indicated quotients and simplify each : 
 
 21. ( Vl5 — V3) -^ V3. 24. (x-Vy'z — ofyz^) -7- ^xyz. 
 
 22. (V6 + 2V3)--V2. 25. (-\/aFb--\/'b^ + -y/d^t)^^abi^. 
 
 23. (4-7V5)--V6. 26. (5-v/i2-2V6 + 4)-v--^4. 
 
 27. In w^hich of the above exercises is division by means of 
 rationalizing the divisor easier than direct division ? 
 
 Divide by means of rationalizing the divisor, and simplify: 
 
 28. -^ . 31. t 34. 3V2-4V5 . 
 
 V3-V2 Vc-Vd 2V3 + V7 
 
 29. ^ . 32. 8:t2V15. 35 g+V^M^- . 
 3^^-'^ ' V3+\/5 ' a_V^iM^ 
 
 30. 2 + V3. 33 27-4V7 3^ V^+^^V^^ 
 2-V3 ■ 2V7-1 ' V^+V^^i 
 
 37. If the result in Ex. 26 were wanted correct to 3 decimal 
 places, show in detail that it is far simpler first to rationalize the 
 divisor than to extract roots and divide by the ordinary arithmeti- 
 cal method. Is this true in Ex. 28 ? in Ex. 33 ? 
 
 HIGH. SCH. ALG. — 17 
 
252 BIGH SCHOOL ALGEBRA [Ch. XIV 
 
 38. What is the product of (2 + V3) - V5 by (2 + V3) + V5 ? 
 of this result by 2 — 4 V3 ? What, then, is the rationalizing 
 factor of 2 + V3 - V5? Divide 2 by 2 + V3 + V5; also by 
 1 4. V3 - V2. 
 
 160. Powers of surds. Roots of monomial surds. Powers 
 of surds, being merely products of two or more equal surd 
 factors, may be found by the method of § 155 ; and roots of 
 monomial surds may be found by § 149 (iii). 
 
 Ex. 1. Find the fifth power of v'2 a^. 
 
 Solution. (^27')' = V(2^' [§ 151 (iii) 
 
 = ^2^^= 2 a^^4^ [§ 152 
 
 Ex. 2. Extract the square root of -v/4 a^x. 
 
 Solution. yj'^Ta^x= i/4r a^x. [§ 149 (iii) 
 
 This result may also be written in either of the following forms : 
 
 ^j■V^o^ or '\J2 aVx. [§ 149 (iii) 
 
 EXERCISE CXI 
 9. 
 
 Simplify: 
 
 3. (V'r?)2. g (^FA\ 15. (V2^+V3^)^ 
 
 4. (2-^aby, ' ^ ^J 16. (5V3-3V2)^. 
 5 (</6^m:ii?^' ^^ (-V2V^)3. 
 
 12 
 3,^,, 12. (^^F^^v^)\ ^^' (1-^V3). 
 
 7. 
 
 13. (9+V5)l 19. (Vm^-Vm7i)2. 
 
 8. (iViey. 14. (V7-V2)2. 20. (V2-3^3)l 
 
 Express each of the following by means of a single radical 
 sign, and simplify : 
 
 21. VV^. 24. ^-27V^. 27. VtV-V^^- 
 
 / o , 3; 7= 28. -x/VS^ 
 
 22. V\/3cdl 25. -^-^V^. ^ 
 
 ___^^ . 29 x\ 'fZHH 
 
 23. V^25mV. 26. yV(r + s)i«^ • A\8(e-/) 
 
159-102] IRRATIONAL NUMBERS — RADICALS 253 
 
 161. An important property of quadratic surds. Neither 
 the sum nor the difference of two dissimihir quadratic surds 
 (§ 147) can be a rational number ; for, if possible, let 
 
 Vx±Vy = r^ (1) 
 
 -Vx and V^ being dissimilar surds, and r rational, and not 
 zero. 
 
 From Eq. (1) ±V^=r-Vi, (2) 
 
 whence, squaring, y = r^-~2r ^x -h a;, (3) 
 
 and, solving for V a:, ^x = — ^ ^ ; 
 
 2 r 
 
 i.e.^ if Eq. (1) were true, then the surd ^x would equal the 
 
 rational number — ^ ^, which is impossible; hence Eq. (1) 
 
 cannot be true. 
 
 From what has just been shown it at once follows that 
 if x-{-^/y = a-\-^^ where x and a are rational^ and V?/ and 
 V6 are quadratic surds^ then x = a and y—h. 
 
 For, if re + V?/ = a + VJ, 
 
 then V^ — V6 = a — x\ 
 
 which, by the above proof, can be true only if each member 
 is zero, i.e.^ ii a = x and ■Vy=\^b. In other words, the 
 equation x + Vy = a-\- V3 is equivalent to the two equations 
 x= a and y = h. 
 
 162. Square roots of binomial surds. Some binomial quad- 
 ratic surds are exact squares; the following examples show 
 how to extract the square root of such surds. 
 
 Ex. 1. Extract the square root of 8 + V60. 
 
 Solution. If 8+V60 is the square of a binomial surd, let 
 Va;+ V2/ represent that surd, i.e., let 
 
 then, squaring, 8 + V60 = x-{-2 Vxy -{-y=x-\-y-\- 2^xy ; 
 
 therefore S = x-\-y and V60 = 2 ^/xy, [§ 161 
 
254 HIGH SCHOOL ALGEBRA [Cii. XIV 
 
 and combining these last two equations (after squaring the second) 
 easily leads (§ 131) to the solution 
 
 x = 8 and 2/ = 5 ; 
 
 therefore ^js + V60 = V3 + V5, 
 
 as is easily verified by squaring the expression V3 + VB. 
 
 Note . This example might also have been solved by inspection ; for, 
 writing 8 + V60 in the form 8 + 2^V'l5, and then comparing it with 
 (\/x+ \/?/)2, i.e., with x -\- y -\- 2 y/xy, we see that we have only to find 
 two numbers whose sum is 8 and w^hose product is 15, and take the sum of 
 their square roots as the required root. 
 
 Ex. 2. Ky inspection, find the square root of 18 — Q-y/^. 
 
 Solution. Writing 18 — ^y/b in the form 18 — 2 •\/45, we see 
 that we need to find two numbers whose sum is 18 and whose 
 product is 45, and take the difference of their square roots. These 
 numbers are evidently 3 and 15, hence 
 
 Vl8-2V45 = V3-Vl5. 
 
 EXERCISE CXIl 
 
 Find (by inspection where practicable) the square root of each 
 of the following expressions, and check your results : 
 
 3. 4 + 2V3. a 30-20V2. 13. e4-4/-4Ve/: 
 
 4. 16 + 2j^l5. 9. 39-12V3. 14. 2 m + 9 n - 6 V2m?i. 
 
 5. 12 + 8V5. 10. 47-12VI1. 15. 146-56V6. 
 
 6. 17-12V2. 11. 63 + 24V5. 16. m + 2Vm, 
 
 7. 27-4V35. 12. Sxy-4.xyV^. 17. a — Ve. 
 
 18. In the first solution of Ex. 1 above, why does a; + 2/ = 8, 
 and2V^= VeO? 
 
 19. If the numerical value of '\21-f 8V5 is required, is it 
 easier to find first the binomial whose square is 21 + 8V5, or to 
 begin by extracting the square root of 5 ? Explain. Also answer 
 the question if 12 + 3V5 is substituted for 21 -f- 8 V5, 
 
102-10:1] IRRATIONAL NUMBERS — RADICALS 255 
 
 163. Irrational equations. Equations which contain indi- 
 cated roots of the imkyioivn numbers are called irrational 
 equations (also radical equations) . Thus 6 ■\/x — 25 a; + 88 = 0, 
 
 ViTl+a; = 8,^^7r-4-l = 0, and 3 + iVi = ^a;2-l are ir- 
 -vx 
 
 rational equations, but such an equation -d^ x— V3 = 5 a; is 
 
 rational. 
 
 The solution of irrational equations is illustrated by the 
 following examples : 
 
 Ex. 1. Solve the equation V.^• + 1 + .t = 11. 
 
 Solution. On transposing, the given equation becomes 
 
 ■\/x -h 1 = 11 — a?, 
 
 whence, squaring both members (Ax. 3), 
 
 fl; + l = 121-22a; + «2, 
 i.e., a;2_23a; + 120 = 0, 
 
 whence (§ 126), a; = 15 or 8 ; 
 
 and, on substitution, it is found that 15 satisfies the given equa- 
 tion if Vcc + 1 means the negative value of this root, while 8 satis- 
 fies it if the positive value of this root is intended. 
 
 Ex. 2. Solve the equation V5 x-\-l — wx + 2 = 3. 
 Solution. On transposing, the given equation becomes 
 V5 a; 4- 1 = 3 + Va? + 2, 
 whence, squaring both members (Ax. 3), 
 
 5 i» + 1 = 9 + 6 V^T2 + a; + 2, 
 i.e., ' 4a;-10 = 6V^T2; 
 
 whence, dividing through by 2, then squaring and simplifying, 
 
 4a^-29aj + 7 = 0, 
 from which (§ 126), a; = 7 or ^. 
 
 On substitution it is found that the given equation is satisfied 
 by x = 7 if each radical is regarded as positive, and by a; = ^ if 
 V5 a; + 1 is taken as positive and Va; + 2 as negative. 
 
256 HIGH SCHOOL ALGEBRA [Ch. XIV 
 
 EXERCISE CXIII 
 
 Solve the following equations, and show what restrictions, if 
 any, must be made on the signs of the radicals in order that your 
 results shall be roots : 
 
 16. -\/3-\-x-\-Vx=—=' . 
 
 3. 
 
 V2a;-f 6 = 4. 
 
 4. 
 
 V4aj + 5 = 7. 
 
 5. 
 
 V3 a; - 8 = Va;. 
 
 6. 
 
 -y/x'-4x = 2^4:. 
 
 7. 
 
 i/x-\-oc' = -2c. 
 
 8. 
 
 9. 
 10. 
 
 ^5 — x = x—5. 
 
 X 4- V^ =4:X — 4 Va;, 
 
 2/4- V^-20 = 0. 
 
 17. Vm + 5+ Vm— 8 = V3. 
 
 18. Vl+sVs2 + 12 = l+s. 
 
 19. 
 
 20. 
 
 ^v — 8 Vv —4 
 
 V -v -}- 1 -Vv—2 
 
 2^ V3a; + l + V3^ ^o^ 
 
 11. V 4y + 17 + Vy + 1=4. ^■^- V3¥+l-V3^- 
 
 12. V25-65=8-V25+66. ^^ V^-2 ^ V^4-1 . 
 
 13. cV^— d\/a?=c- + d2— 2cd. ' ^x-\-3 Vx-\-2 
 
 14. Va;+1+— 4^ = 2. 23. r+V^"^^ 
 
 Va;+1 vr — c^ 
 
 15. V^^^ ^ = 0. 24. a- Va^^-a^ ^_ V3-l 
 
 Vs + 7 a + Va^-a^ V3 + 1 
 
 ^=-:^ 
 
 25. V4 a; +1 — Vaj + 3 = Va; — 2. 
 
 26. Va; + a + Va: + ft = V2 a; + a + 6. 
 
 27. Va; + 3 + V4a.' + 1 = Vl0a; + 4. 
 
 28. ^^-^^^-^ = ^+V^^38. 
 
 a; - Va;^ - 8 
 
 29- y'-y- V2/'-2/ + 4 = 8 (cf. Ex. 18, p. 206). 
 
 30. a + 10 = 2 Va~+10 + 5. 
 31. How many roots has the equation in Ex. 27, if the radicals 
 are unrestricted in sign? How many, if each radical must be 
 taken positively ? 
 
CHAPTER XV 
 IMAGINARY NUMBERS* 
 
 164. Definitions. In the solution of quadratic equations, 
 and elsewhere, such numbers as V— 5 and 6 — V— 10 fre- 
 quently present thenjselves; these numbers cannot be ex- 
 pressed, even approximately, as the quotient of two integers 
 (cf. § 146). _ 
 
 Numbers of the form V— 5, where h represents a positive 
 number, are called pure imaginary numbers, while numbers 
 of the form a ± V — 6 are called complex imaginary num- 
 bers, or complex numbers. Two complex numbers are said to 
 be conjugate if they differ only in the signs of their imaginary 
 terms. Thus, V — 5, 2 V — 6, and V— |^ are pure imaginary 
 numbers, while 2— V— 3, 7 — 2V— 5, and 7-f-2V— 5 are 
 complex numbers, the last two being conjugates of each other. 
 
 From the definition of the symbol Va (§ 113) it follows 
 that (V^=T)2=-5; (1) 
 
 and by the method of proof used in § 150, it is easily shown 
 that V^=V6.V"^^. (2) 
 
 Remark. The second member of Eq. (2) may be regarded as 
 a standard form; and it will be found that operations with imagi- 
 nary numbers are usually much simplified by first reducing such 
 numbers to this standard form. The symbol V— 1 is often 
 called the imaginary unit, and is represented by /. 
 
 * Teachers who prefer less work in imaginaries than is here given may 
 omit §§ 166-168 ; the entire chapter should be read if time permits. 
 
 257 
 
258 HIGH SCHOOL ALGEBRA [Ch. XV 
 
 165. Powers of V— 1, i.e., of i. As a particular case of 
 Eq. (1), § 164, we have 
 
 similarly (V- 1)3= (V- 1)2. V-l=-V-l, " i^= -i, 
 
 (V^:n)*=(V^^i)3. V^i:=l, " ^4=l, 
 
 (V^a)5 = (V^^)*- V^^ = V^, " ^5 = ^, 
 
 and so on for the higher powers of V — 1 ; any one of these 
 powers, when simplified, will be found to have one or another 
 of the four values : V — 1, —1, — V — 1, and 1. 
 
 EXERCISE CXIV 
 
 1. Define an imaginary number (of. §§114 and 146). 
 
 2. Which of the following are* imaginary numbers : V— 3, 
 V^, -V^^ V5, ^^^ 3-\/^ 4 aV^, and I + i V^=^ ? 
 
 3. Is V— a; imaginary when x represents a positive number ? 
 when X represents a negative number ? Answer the same ques- 
 tions for -^ — x. 
 
 4. Eeduce to the standard form (§ 164, Kemark) : V— 9 ; 
 V^^; V"=^l0"; V^^l3; V^^12; and V-i:5c^. 
 
 5. Show by the method of proof suggested in § 150 that 
 V^^T^ V7 . V^l. 
 
 6. Show that if ^ = V— 1, then 
 
 ^2=-l, ^«=-l, ^l»=? 
 
 ^3= -I, i'^=-i, ^"= ? 
 
 i*=l, i^ = l, i^'=? 
 
 7. Since any even number may be written in the form 2 n, 
 where n is an integer, and since a^"* = (ci^)"? show that every even 
 power of i is real. 
 
 8. When n is even, does i'-^" equal 1 or — 1 ? Why? Answer 
 the same questions if 7i is odd. 
 
 9. Give the values of the following even powers of i: 
 
 ^8[ = (^2y]; P; |-32. ^-18. ^'64. ^100. ,^-6. ^^^ 
 
166-167] 
 
 IMAGINARY NUMBERS 
 
 269 
 
 10. Show that every odd power of i is either ^ or — i (cf. Ex. 7). 
 
 11. Find by inspection the value of the following odd powers 
 
 of i : i"^ ; f 
 
 i'\ 
 
 12. Distinguish between pure and complex imaginary numbers, 
 and give three examples of each. 
 
 -B 
 
 A' 
 
 I I I. 
 
 B' 
 
 166. Graphical representation of imaginary numbers. Since 
 opposite numbers, such as + 3 and — 3, 
 are represented graphically by opposite 
 distances, such as OA and OA' ; and since 
 multiplying +3 by —1 gives —3; there- 
 fore we may regard the multiplier — 1 as 
 an operator which rotates OA through 180° 
 about 0, into the position OA' , 
 
 Again, since i • ^= —1, therefore i may be regarded as an 
 operator which when applied twice in succession rotates OA 
 through 180°; hence using i as a multiplier once (instead of 
 twice) should rotate OA through 90° into the position OB. 
 
 In other words : 3v^ — 1 {i.e.^ OA • i) may be represented 
 graphically by OB. Similarly, any pure imaginary number 
 whatever may be laid off on the line B' OB, above the origin 
 if the number is positive., below the origin if it is negative. 
 
 E.g., if each division on the lines JT and RN in 
 the figure represents a unit, then OS=l, 0J= — 2, 
 OL = V^ ON = SV"^^, 0Q=- 2 V^ etc. 
 
 The lines JT and BN are often called 
 the axis of real numbers and the axis of 
 imaginaries, respectively. 
 
 J H 
 
 o 
 p 
 
 Q 
 
 R 
 
 S T 
 
 -i — H- 
 
 167. Graphical representation of complex 
 numbers. A complex number, such as 5 + 3V— 1, may be 
 graphically represented as follows : lay off OA, 5 units on 
 the axis of real numbers, and OB, 3 units on the axis of 
 imaginaries, tlien complete the parallelogram AOBR, and 
 draw its diagonal OR ; this diagonal is a graphical repre- 
 
260 
 
 HIGH SCHOOL ALGEBRA 
 
 [Ch. XV 
 
 sentation of the complex number 5 + 3V— 1, i.e., of the sum 
 
 of 5 and 3V^=a. 
 
 Moreover, it is evident that any com- 
 plex number vrhatever may be graphic- 
 ally represented by the above method. 
 
 B 
 
 R 
 
 -/^i 
 
 a 
 
 O 1 1 1 J^ 
 
 Note. The appropriateness of calling OR the 
 sum of OA and OB will be evident to pupils who 
 have an elementary knowledge of physics. Thus, if 
 two forces, represented in amount and direction by OA and OB, respectively, 
 act simultaneously on a body at 0, the result is the same as though a single 
 force represented in amount and direction by OB were acting on this body ; 
 i.e., the sum of the forces OA and OB is the force OB. 
 
 168. Graphical representation of the sum of complex num- 
 bers ; also of their difference. The sum of two complex 
 R numbers, such as 7 + 2V— 1 and 
 1 + 4v^— 1, may be graphically repre- 
 sented as follows : let OP and OQ 
 be the graphical representations of 
 7+2 V^^nr and 1 + 4V - 1, respec- 
 tively (§ 167) ; complete the par- 
 allelogram POQR, and draw its 
 diagonal OR ; then OB is the graph- 
 ical representation of the sum of 7 + 2V— 1 and 1 + 4V— 1 
 (cf. §167, Note). 
 
 Obviously the sum of any two complex numbers may be 
 represented by this method. 
 
 Again, to find the difference of two complex numbers 
 graphically, we have only to reverse the sign of the subtra- 
 hend, and proceed as in addition. 
 
 EXERCISE CXV 
 Represent graphically the following imaginary numbers : 
 
 1. 2V^^. 3. -7 1. 5. 2AL 1. V^5. 
 
 2. -5V^. 4. \i. 6. V^9. a V^. 
 
167-160] 
 
 IMAGINARY NUMBERS 
 
 261 
 
 Perform the following additions and subtractions graphically 
 9. 4 + 2 [cf. § 4]. 13. 3-7. 
 
 10. 4.i + 2i. 14. 3^-10^. 
 
 11. Ai-2i. 15. 3-8-6. 
 
 12. 7 
 
 5. 16. V-25 4-2V-36-V-49. 
 
 Represent by drawing each of the following complex numbers : 
 
 17. 2 + 4aA3i. 21. 4 + 3l 25. 
 
 18. 4 + 2V^^. 22. -6-6i. 26. 
 2 
 
 V-i + 6. 
 
 V-2-7. 
 
 19 
 
 23. 3+V-36. 
 
 2 + V^^. 
 
 27. lO+V^^OS. 
 ~8. 
 
 5V-1. 
 
 20. -7 + 1. 24. -:^ + V-9. 28. -i^- 
 
 Perform the following indicated operations graphically : 
 
 29. (l+4i) + (4 + 2i). 33. (5-V^~9) + (-3 + V^^). 
 
 30. (3 + 2i) + (5 + 3i). 34. -2^ + (-3 + V^^). 
 
 31. (2-50 + (8 + 0. 35. (4 + 3t)-(2 + i). 
 
 32. (3 + V^4) + (-2 + V^). 36. (8-20-(5-3i). 
 
 37. (10+V-9)-(-2+V-16). 
 
 38. (|_V38)_(7-V^. 
 
 169. Fundamental operations with pure and complex imagi- 
 nary numbers. If complex numbers are first reduced to the 
 standard form a + 5V— 1, and if we are careful to remember 
 that V — 1 • V — 1 = — 1 and not + 1, then the operations of 
 addition, subtraction, etc., with these numbers may be per- 
 formed exactly as are the corresponding operations with 
 real numbers. The following examples will illustrate these 
 operations. 
 
 Ex. 1. Find the sum of 2 + V- 9, 8 -V^i, and - V - 25. 
 
 Solution. Since 2 + V^9 = 2 + 3 V^^, 
 
 8-V=l = 8-2V"^, 
 
 and -V^r25= -5V^^ , 
 
 10_4V^ i.e., 10-V^=36. 
 
 therefore the sum is 
 
262 Jiiaii SCHOOL algebra [Ch. xv 
 
 Ex. 2. Multiply r>V^^ by W^^. 
 
 Solution. 5V^ = 5 V2 • V^, 
 
 and 4 V^^ = 4 V7 . V^=l, 
 
 hence the product is 20 V2 • V7 (V — 1)^ i.e., — 20Vl4. 
 
 Note. Observe that this product is not +20Vl4, as it would be if tlie 
 factors were i-eal numbers. Beginners should be especially careful to guard 
 against errors in the sign of a product of imaginary numbers. 
 
 Ex. 3. Multiply 3 + V^^ by 2 - V^^. 
 
 Solution. Writing these imaginary numbers in terms of the 
 imaginary unit, the work may be arranged thus : 
 
 3H-V5. V^ 
 
 2-V3. yiTi 
 
 6 + 2V5.V^^ 
 
 - 3 V3 ■ V^ - Vl5 (V3i)2 
 6+ (2 V5-3 V3) . V=3 + Vi5. 
 
 Ex.4. Divide 12 + V- 25 by 3- V- 4. 
 
 Solution. Such divisions are easily performed by first mul- 
 tiplying both dividend and divisor by the conjugate of the 
 divisor, thus : 
 
 12 + V- 25 ^ 12 + 5 V ^=n: ^ (12 + 5 V^I)(3 + 2 V^:i) 
 S-V"^^ 3-2V^=^ (3-2V^(3 + 2V^ri) 
 ^ 36 + 39 V^i: + 10 ( V"^)' 
 9 _ 4 ( V^^)' 
 26 + 39V^^ 
 
 9 + 4 
 
 2 + 3V-1. 
 
 170. Important property of complex numbers. By a method 
 altogether like that used in § 161 it may be shown that if 
 a + 5V— 1 = ^ + (^V— 1, then a = c and b = d. Moreover, 
 this fact may be used, as in § 162, to extract the square roet 
 of any complex number (cf. Ul. Alg, §§ 151, 182). 
 
16l>-170j IMAGINARY NUMBERS 263 
 
 EXERCISE CXVI 
 
 5. Add 3 + 5 i and 7-|- v — 4 as in § 169; then add these 
 numbers graphically, and check your work by comparing results. 
 
 Simplify Exs. 6-13 below, and check as teacher directs : 
 
 6. 7-6^+2 + 3^. 8. (3 + 2 i) - (3 - 2 *). 
 
 7. (3 + 2*) + (3-2i). 9. (-4-A/^^=^)+(-4-V^. 
 
 10. V-4 + 4V-9 + V-25. 
 
 11. 3 + V-16 + V-4-5-V-9. 
 
 12. 3 + V-36-(l+2 V-25) + 3 V-16. 
 
 13. V-49 + 5 V-4-(6 + 2 V-9). 
 Simplify each of the following expressions (cf. § 162): 
 14. V^--(2V^ + o~3 V^24) + 3V^^18. 
 
 15. V-16 a^x" +VI-5 + 2V0-3O-V-9 a2«2 + V- aV. 
 
 16. a; V— 4+V — ar' — 2if — 1 — V— 32. 
 
 17. Solve the equation x^ — 1 = (cf. § 72) and find the sum of 
 the roots; check your addition graphically. Similarly find the 
 sum of the 3 cube roots of 8; of the 3 cube roots of —27. 
 
 Find the product of : 
 
 18. 3 V^=^ by 5 V-12. 20. 2 V^^ by V-4aV. 
 
 19. 5 V^^ by 2 V^^. 21. —1-6 r^ + i*^ by i\ 
 
 22. V^^ + V^=^ by V"^ - V^=^ 
 
 23. 3+2V^^by 5-4V^=n:. 
 
 24. V^r50_2V^^n^by V^^-5V^^. 
 
 25. 3P-4.i^hy2i^-3i^\ 
 
 26. Show that the sum, and also the product, of a-\-bi and 
 a — pi is real (a and h being any real numbers). Show that the 
 same is true also for V— 4 — 3 and — V — 4 — 3. 
 
 27. Show that both the sum and also the product of any two 
 conjugate complex numbers is real. 
 
 28. Multiply V^a + V^^ + V"^ by V^^ — V^^ + V^^ 
 
264 HIGH SCHOOL ALGEBRA [Ch. XV 
 
 29.. (l-}-V^)'=? 30. (2-3if=? 31. (2a-3a;V^)'=? 
 
 32. Find the product of a V— ^ + 6 V— ot, a V— a + 6 V— 6, 
 and h V— h — a^ — a. 
 
 33. Show that — ^ + i V— 3 and — i — i V — 3 are conjugates 
 of each other, and also that each is the square of the other. 
 
 34. Reduce -^ ~ -\ — ^ to its simplest form. 
 
 3-V^4 3 + 2i 
 
 35. Simplify each of the following indicated quotients : 
 
 V^T5 . V^24. V^. Vc V^. V84 v:r6-i-2V^:r8~ 
 V^^^' V"^^^' V^=^' V^^' V5 ' 2V^^' v^^ 
 
 36. Show that ^ + ^^zl = «c + ?>^ + (^c-ad)V-l . 
 
 c+dV-1 ^^^ 
 
 Perform the following indicated divisions (cf. Ex. 4), and check 
 your work by multiplying the quotient by the divisor : 
 
 37. -i-,. 40. ^-±^^. 
 
 l^i 5-2z 
 
 38 — 41 V2a;-3ai 
 
 ^■' + ^' ' V2^ + 26^ 
 
 39. 2W^. ^2 V^-lV6. 
 
 3 + V-2 ?:V6H-Va 
 
 43. If a and h are positive and unequal numbers, show that 
 V— a ± V— 6 cannot equal a real number (cf. § 161). 
 
 44. Show that if cc + V— 2/ = aH-V— 6, wherein y and h are 
 positive numbers, then x = a and ?/ = ft. 
 
 45. Find the square root of 5 —12 V— 1. 
 
 Hint. Let y/x-y/y. V^^l = Vs - 12 V^HT (cf. § 162). 
 
 Find the square root of : 
 
 46. 10-6V^^. 48. 3 + 2V^=l0. 
 
 47. 6V^^-17. 49. 5|-3}V^^. 
 
CHAPTER XVI 
 
 THEORY OF EXPONENTS 
 
 ZERO, NEGATIVE, AND FRACTIONAL EXPONENTS 
 
 171. Introductory, (i) As originally defined (§9) an 
 exponent is necessarily a positive integer, and it is in this 
 sense only that we have thus far used it. Under this restric- 
 tion we have established the following exponent laws (§ 110), 
 wherein a is any real number except : 
 
 I dT" 'dP' = a"^+», 
 
 II {^ory = a*»% 
 
 III {ahy = d^ . 5% 
 
 IV a"' : a» = a^-**. 
 
 (ii) We now propose to extend the meaning of an expo- 
 nent so as to include such symbols as a^ a~^, and a^, along 
 with our former exponent expressions. 
 
 In extending the meaning of any symbol already in use, 
 however, the extended meaning should be such as not to 
 disturb any rules of operation already established for the 
 symbol in question. Hence, we shall admit such symbols as 
 a^ a"^ etc., into our algebraic notation if, and only if, we can 
 assign to each of these symbols a meaning consistent with the 
 above exponent laws. 
 
 172. Meaning of such symbols as a^, a~^ and ai (1) If, 
 
 following the plan given § 171 (ii), wa let ?^ = in law I, 
 § 171, we obtain 
 
 a"* ^0 = a"*, [since a^^^ = a^ 
 
 hence a^ = 1 \ 
 
 i.e., if law I is to admit the symbol a^ then this symbol must 
 have the value 1. 
 
 266 
 
266 HIGH SCHOOL ALGEBRA [Ch. XVI 
 
 (2) Again, if in law I, § 171, we let m = 5 and n=—3, 
 we obtain a^ - a~^ = a^ = 1, [since a^~^ = a^ 
 
 whence a~^ = — ; 
 
 aP 
 
 i.e., if law I is to admit the symbol a~^, then this symbol 
 must have the same meaning as — . 
 
 (3) And finally, if law I, § 171, is to remain valid for such 
 symbols as a^, then 
 
 a^ -a^ -a' = a\ [since a^^'^'^^ = or 
 
 and therefore a^ = Va\ [§11^ 
 
 i.e., if law I is to admit the symbol a% then this symbol must 
 have the same meaning as ~va^. 
 
 p 
 173 Definitions of a^, a"^ and a^. Using as a basis the 
 
 special cases considered in § 172, we shall now define the 
 
 p 
 symbols a^ a~\ and a"" ?ls follows: 
 
 (1) aO=l, 
 
 (2) «-' = -,. 
 
 a'' 
 
 and (3) a^-=</'^; 
 
 wherein a represents any number whatever, and k, p, and r 
 
 are positive integers. 
 
 Moreover, these definitions — since they are based upon 
 one exponent law only — must be regarded as tentative until 
 they are shown (§§ 174, 175) to satisf}^ all of the exponent 
 laws. 
 
 EXERCISE CXVII 
 
 Assuming the validity of § 173 (1) and (2), show that : 
 
 1. 3c-^=3.i=^^. 3. a%'c=c. 
 
 c c 
 
 
172-174] TIIEOliY OF EXPOXIJXTS 267 
 
 Keduce the following to equivalent expressions free from zero 
 and negative exponents : 
 
 5. ««. 9. 5A-2. 13. Sx'^-r-y-*. 
 
 6. cc-3. 10. ifc-^. 14. f-^y-\ 
 
 7. Qa\ 11. 7--r-3. 15. 2/'-2/"^ 
 
 8. -4-!-a°. 12. Sa^V'- 16. 3 a^-^^ -- ic^/-^. 
 
 Assuming the validity of § 173 (3), translate the following into 
 equivalent radical expressions : 
 
 17. ai 20. t^. 23. (ia-^^s)!. 
 
 18. ai 21. {axy^. ^*' (^^"'>^- 
 19 a;l 22. (4c^d)i 25. (^^'y. 
 Write in the fractional exponent notation : 
 
 26. a/^. 30. -v/oV. 34. -V-^m^. 
 
 27. -v/m. 31. %V. 35^ &-^'l6^. 
 
 28. V 6. 36. 2\x y-^. 
 
 ^\ 23 _i 
 
 29. V^24. 33. \-^,' 37. acv^a^c-^^. 
 
 38. Find the numerical value of: 5« ; 3"^; 4'^; 9* ; (i)^ ; 4-2-2^. 
 
 39. Show that if law II, § 171, is to admit the symbol a^, then 
 a^ must equal 1 (cf. § 172). 
 
 40. Show that if law IV, § 171, is to admit the symbol a"'*, 
 
 then a~^ must equal —• 
 a 
 
 41. Show that if law II, § 171, is to hold for a^, then a^ must 
 equal Va^. 
 
 174.* The symbols a^ and a~* obey all the exponent laws. 
 
 That a^ and a^^ as defined in § 173, satisfy all the exponent 
 laws may be shown by assigning zero and negative integral 
 
 *The proofs given in §§ 174 and 175 may, if the teacher prefers, be omitted 
 until the subject is reviewed. 
 
 HIGH 8CH. ALG. — 18 
 
268 HIGH SCHOOL ALGEBRA [Ch. XVI 
 
 values to m and n^ both separately and together, in the equa- 
 tions which express those laws. 
 
 Thus, if we let w = in law II (n remaining a positive 
 integer), we obtain (a^y = a^ [since a" • « = a" 
 
 i.e., r = l, 
 
 which is correct ; hence a^ =1 is consistent with law II. 
 
 Again, let m = in law I V (ri remaining a positive integer), 
 and we obtain a^ : a"" = a~\ [since aO-"= a~" 
 
 i.e., l:a^ = a-% 
 
 which is a correct equation [§ 173 (2)] ; hence a^ = l and 
 
 a"" = — are consistent with law IV. 
 
 Once more, let m = — r and n= —s (where r and s are 
 positive integers) in law II, and we obtain 
 
 (a-0"* = «""""' = «'■*; 
 but this is consistent with § 173 (2), for 
 
 <-=(r=^ 
 
 [§ 173 (2) 
 [§§ 109, 92 
 
 or 
 
 Moreover, if we similarly test the remaining combinations 
 of positive and negative integral and zero values of m and 
 71, in the four exponent laws, we find that definitions (1) and 
 (2) of § 173, and the exponent laws of § 171, are entirely 
 consistent. Hence we need no longer regard the definitions 
 of a^ and a~* as tentative (cf. § 173, last part). 
 
 The testing of some or all of these remaining cases may 
 be assigned as an exercise to the pupil. 
 
 p. 
 175. The symbol a' satisfies all the exponent laws. That 
 
 p 
 
 a^=:^aP is consistent with the exponent laws may be shown 
 
 as follows : 
 
174-175] THEORY OF EXPONENTS 269 
 
 Let p, r, s, and t represent any positive integers, then 
 
 PI . _ 
 
 = Va^^ • Va''^= VaP' • a'' [§§ 154, 149 (i) 
 
 pt+rs P,i 
 
 ^.e., law I holds good for such symbols as a^, 
 
 p s^ s 
 
 Similarly, (</')' = ( V^)' = ^( Va^)^ 
 
 = V^ [§ 149 (iii) 
 
 ps Pi. 
 
 P 
 i.e., law II holds good for such symbols as a^, 
 
 p . 
 
 Again, (a6)^= </(^ahy=<JaP • 6^ 
 
 = 7^.V^ [§149(i) 
 
 p p 
 
 2 
 
 i.e., law III holds good for such symbols as a^. 
 
 The proof for law IV, being closely similar to that for law 
 I, is left as an exercise for the pupil. 
 
 Observe that the above proofs remain valid: 
 
 (1) if r (or t) takes the value 1, in which case ^ [or - j be- 
 comes an integer. 
 
 (2) if p (or 8) is negative or zero (cf. § 174), in which 
 
 ^( or - ) becomes a negative fraction or integer, or zero. 
 r\ tj 
 
 Therefore these proofs, taken in connection with those 
 
 previously given, include all possible combinations of positive 
 
 and negative, integral, fractional, and zero values of m and n, 
 
 in the exponent laws of § 171. We need, therefore, no longer 
 
 p 
 regard the definition of a'^ [§ 173 (3)] as tentative. 
 
 p 
 Note. Observe also that a!" represents the principal rth root of a^, since 
 
 that is the meaning of the symbol y/(f (§ 150). 
 
 case 
 
270 HIGH SCHOOL ALGEBRA [Cii. XVI 
 
 EXERCISE CXVIII 
 
 1. Does a^ equal x^ even when a and x are unequal ? Explain. 
 
 2. By means of § 173 (1) show that a"* -r- a" = a"*'" when n = 
 (cf. § 171, IV). 
 
 3. By means of § 173 (2) show that (a^)-^ = a-^ and that 
 a-'^a-' = a-' (cf. § 171, II and IV). 
 
 4. Show that 6a!^^ = 6aY ^^^^ ^^^^^ 3:Var^ ^ 2_:^^3 ^ 
 
 y~* XT 2*xy~^ S^n~^x 
 
 5. By proceeding as in Ex. 4 show that, by changing the sign 
 of the exponent, di factor may be transferred from the numerator 
 to the denominator of a fraction, and vice versa. 
 
 6. Is ^^^-^ equal to ^L±A? Explain. Observe carefully 
 
 4 a? 4:Wx 
 
 that a factor but not a part may be transferred as in Ex. 5. 
 
 Free the following expressions from negative exponents, and 
 explain your work in each case : 
 
 7. ^. 
 8. 
 
 5-2 
 
 11. 
 
 52 . 12-1 
 10-1 . 3* 
 
 15. 
 
 (82_1)-1 
 
 2d' 
 
 12. 
 
 a-' + 2. 
 
 16. 
 
 9-V 
 6x-' 
 
 a-'x-' , 
 
 13. 
 
 x-'-\-y-^ 
 5 
 
 17. 
 
 9-V 
 
 h-'x-' 
 
 (-6)-i(a^2/)-^ 
 
 3.2-2 
 
 8-2 
 
 14. 
 
 m~3 — n"^ 
 m-3 . w-« 
 
 18. 
 
 3r-2-4s 
 
 10. 
 
 19. Is . ^ equal to 3 aa;-^ ? Why ? 
 
 a;2 
 
 20. As in Ex. 19, write the following fractions in integral form : 
 o^. 4:a-^c-\ 4-ia-V . m-i-3n-* 
 
 6^2/' ac ' 5a-id~i' m~^n 
 
 21. What is the diiference in meaning between mJ and m^? 
 between m-^ and m"'^ z.e., between m^ and m ^ ? 
 
175-176] THEORY OF EXPONENTS 271 
 
 Write Exs. 22-26 below as radical expressions, and write Exs. 
 27-30 with positive fractional exponents : 
 
 22. a~T^6TV. ^^ 6~«/?. 28. -Vr-Ps^ 
 
 23. 2^dh~'^. 1 n 
 26. a'^ — b^. 
 
 29. V-32r-V«. 
 
 24. — - . __ 3/ — . ._«,,_o, _. * 
 
 3^' 27. -v/-ic-«d-'e. 30. V(c+d)«-f--v^(c+d)*. 
 
 Find the numerical value of : 
 31. A 35. 6° -2-*. /27xY 
 
 32.4- 36. (.09)1. 'V.OOSr 
 
 40. (32-^^(32*)-^. 
 
 33. 9t. 37. (256).. 41. (-,W-(169)i 
 
 34. 8-^.25^ 38. (Hl)"^ 42. (64)-^ . (16-^)^ 
 
 -(i)'-(r-(f )■'-"■»-■ 
 
 176. Operations with negative, zero, and fractional expo- 
 nents. From the definitions of negative, zero, and fractional 
 exponents (§§ 173-175) it follows that in all operations 
 with such symbols as a^ a~^ and «s their exponents obey 
 the exponent laws of § 171 ; that is, these exponents behave 
 just as though they were positive integers. 
 
 When working with fractional-exponent expressions it is 
 frequently necessary to change the exponents to higher or 
 lower terms ; this does not change the value of such an 
 expression ; 
 for since V^= %^, [§ 151 (iv) 
 
 therefore a'' = a^'^ . 
 
 Operations with radicals may often be greatly simplified 
 by first converting the various radical expressions into their 
 fractional-exponent equivalents (cf. Exs. 29, 41, and 57 in 
 the following exercise). 
 
272 - HIGH SCHOOL ALGEBRA [Ch. XVl 
 
 EXERCISE CXIX 
 
 .5 
 
 1. Simplify : V2 • ^4 • 2^ • 2-^ 
 
 Solution. \/2 . ^ • 2' • 2"^ = 2^ • 2^ . 2^ . 2"^ [§ 173 (3) 
 
 = 2MH-|=2'3 = ^4. [§171, I 
 
 2. Simplify: {ah)^ - (p^)^. 
 
 Solution. (a&)^ • {h'^c)^ = (ab^c)^ = ahc^. [§ 171, III, II 
 
 6^r 
 
 3. Simplify: (- 
 
 s„.„„o.. (_) =(_) =(-) =_ f5„i,„ 
 
 ~ &2 - 52 • 
 
 Perform the following indicated operations, and express your 
 results in their simplest form : 
 
 4. 8t . 8^ . si 7. (^)f . A y . M\ 10. s-^i^-^s=^ 
 
 8. 2x^ -h4: x^' 
 6. 2L8-i.27i 9.^?.^?. 12. (36a-tci)-i. 
 
 13. By means of fractional exponents, reduce Va^ and V^ to 
 equivalent radicals of the same order (cf. § 154). 
 
 2. 5 
 
 Solution. The given radicals are, respectively, equivalent to g^ and cc^' 
 
 and these expressions are, respectively, equivalent to g^ and o;^, i.e., to 
 \/g4 and v^, each of which is of order 6. 
 
 14. Solve Exs. 12-16, p. 245, by means of fractional exponents. 
 
 15. By means of fractional exponents, solve Exs. 19-22, p. 247, 
 Exs. 10-14, p. 251, and Exs. 9-12, 25-29, p. 252. 
 
 By means of fractional exponents, simplify : 
 
 16. V^-^Sv"^ . -y/x. 18. ("v/r^ . Vi . -v/O"" 
 
 17. ("v/m^ . VOi 19. (8^ . 8-^)-2. 
 
176] 
 
 THEOR 
 
 20. 
 
 
 21. 
 
 -^4a-2-^-8a-^. 
 
 22. 
 
 (^?)-^.(3s-V^-)l 
 
 23. ViC"^ • Vit*"^. 
 
 24. (Vo'^VO"'""- 
 i 
 
 THEORY OF EXPONENTS 273 
 
 26 (a"" + o^^)V^ . 
 
 27 m^ — m~^ + a/7 m^ 
 
 28. !;! 
 
 o 
 
 25. f "" '\ ] • V ^ + 
 
 0,2 
 
 29. Eind the product of 3Va — BVy by 2Va + V^/. 
 
 Solution. Since SVa - 6\/y = S a^^ -^vK and 2 Va + S^y = 2a^+ ?/3, 
 therefore this product becomes 
 
 2a^ + y'^ 
 qJ-^^ - 10 Jy^ 
 
 + 3 Jy^ - 5 y^'^^ 
 
 Qa-7 a^y^ -6y\ 
 If it is desired, this product may, of course, be written in either of the 
 following forms :6a- lVa\/y — 5\/«/2, or 6 a - iVcfiy^^ — 5v^. 
 
 Perform the following multiplications : 
 
 30. a^ + 6^ by a^ — &i 32. m^ — m^r^ -f- n^ by m^ + ?i^. 
 
 31. oj^ — Q^y^ + 2/^ ^y i»^ + 2/^- 33. m^ — m^n~^ + n" ^ by m"^ + n ^. 
 
 34. ia^^-TV^^/HTVa^^^Z-aV^/^byiaj^ + i^/^. 
 
 35. 81 yy^ - 27-s/'^</y + 9^/^Vf - 3 2/a/^4- 2/Hy 3 "V^ + y\ 
 
 36. V« — 4-v/a^a; + 6^/ax — 4 A^aa;^ H- Vx by \/a — 2 Vax + V«. 
 
 37. '>n ^ + m"^ — 2 m^ + 4 m~^ by 1 + 2 m^ — -^— • 
 
 Vm 
 
 3 *_ 4 — i 
 
 38. ^r^ + q-^-^ — p-'^^q ^ by p~-^^ + Q" ^. 
 
 39. If Jx^/x + 2 n Vn + f a.'^« + 6 n^/^ by V7i - 3 a;^ + ^,«^- 
 
 40. 5 a-'x* + 3 a%"x-' - b'-V by x-^ - 3 ^"~%-' + ah 
 
274 HIGH SCHOOL ALGEBRA [Ch. XVI 
 
 41. Divide x^ — y^hj Va? + Vy. 
 
 Solution. Since \/x + v^ = a;^ + y^, this solution may be put into the 
 following form : 
 
 x2 — ?/3 
 
 x3 + y^ 
 
 x^ — x^y^ + xy - x'^y^ + x^y^ - y^ 
 
 — x^y^ — y^ 
 
 — x^y^ — x^y 
 
 x^y - y^ 
 x% + xy^ 
 
 - xy^ - y^ 
 
 3 2. 
 
 — xy^ — x^y"^ 
 
 1 
 
 X^y^ — y3 
 
 a; V + x^y^ 
 - x^y^ ■ 
 -x^y^ 
 
 r 
 
 
 The above quotient may also be written thus : 
 
 VxP — Vx^ Vy -\- xy — y/o^Vy^ + Vx • y^ — y/y^. 
 
 Note. To appreciate one of the advantages of fractional exponents the 
 student has only to perform the division in Ex. 41, using the radical notation, 
 and compare his work with the above solution. 
 
 Perform the following divisions : 
 
 42. a + x^hy a • + x^. 43. m^ — n^ by m^ — n®. 
 
 44. x-^ + 3y~^ — 10 xy-'^ by x''^ Vy — 2. 
 
 45. J +2 ^ah-'^ + ^ by ^a + 6"^. 
 
 b 
 
 46. x^ + x^ -\/y — X -y/x y'^ — xy-{- -Vx y^ + y'^ by Va; 4- -y/y. 
 Simplify the following expressions : 
 
 47. / Va? + -v^.v Y. y/x-Vy ^^ x-y y^-yt 
 
 a;— +r'*'aj-"-r"** ' 2/ + V2/+I ' yt-1 
 
176J 
 
 51. 
 
 THEORY OF EXPONENTS 
 1 . 1 
 
 275 
 
 a^ 
 
 + 
 
 Write down, by inspection, the square root of each of the 
 following expressions: 
 
 52. 1 — 2i6^ + wi 54. p^ — 4 + 4p"i 
 
 53. x^ + 4 x^ + 4. 55. ax^ + 2 a^x^ -\- a^x, 
 
 56. 771 -\- n -{- p — 2 m^n^ + 2 n^p^ — 2 m^pK 
 
 57. Extract the square root of ^x* — 2 Va^ '+ 5 Va^ —4 Va? + 4. 
 Solution. This expression written in the equivalent fractional-exponent 
 
 form is cc^ — 2 a;'^ 4- 5 x^ — 4 x^ + 4, and in this form its square root may be 
 extracted just as though it were a rational expression (cf. § 117) ; thus : 
 
 2 x^ + 5 x5 -ix^ + A\x^ -xM- 2 
 
 x^ — x^ 
 
 -2x^ 
 
 + 5x^ 
 
 2x^ + x 
 
 2x^ 
 
 2x^ + 2 
 
 4x^+4 
 
 4x^ 
 
 4 x^ — 4 x^ 4- 4 
 
 
 
 hence the required root is xs — x? + 2, i.e., Vx^ _ vx + 2. 
 
 Extract the square root of each of the following expressions : 
 
 58. ar^ + 2 a;^ + 3 a; 4- 4 a;^ + 3 + 2 a."2- 4- x'K 
 
 59. a3 — 4 ot^ + 4 a 4- 2 a^ — 4 a^ + aK 
 
 60. n'^ — 2 wr^ii^' + 2 m^n^ + imT^n 5—2 mW 4- m^ 
 
 Extract the cube root of the following expressions ; write the 
 results with all exponents positive, and then replace all fractional- 
 exponent forms by radical signs (cf. footnote, p. 185): 
 
 61. 8 4-12a;t4-6a;^4-aj2. 
 
 62. 8a;-i-12a;"%4-6aj-V-2/^. 
 
 63. r^_6ri4-15r^-204-15«^-6^4-^i 
 
 64. 8 a^h--^ 4- 9 a&^ 4- 13 a^ 4- 3 a^6 4- 18 a^h'^^ 4- ^' 4- 12 ah-\ 
 
276 HIGH SCHOOL ALGEBRA [Ch. XVI 
 
 Solve the following equations: 
 
 65. m^ = 4. 67. x~^ = 5. 69. a;"^ = — 27. 
 
 66. ^^ = 8. 68. \yi = 25. 70. yjm^ = 3\js. 
 
 71. 2r^ + 5r^ — 3 = 0. [Hint. Put?/ = ri]. 
 
 72. £c"^ 4- 5 a;"^ 4- 4 = 0. 
 
 73. (2A;-3)-2 + 7(2A:-3)-i-8 = 0. 
 
 177. Rationalizing factors of binomial surds. Another 
 advantage of the fractional-exponent notation is that it fur- 
 nishes an easy method for finding the rationalizing factor of 
 any binomial surd whatever, — only quadratic binomial surds 
 were considered in § 157. 
 
 Ex. 1. Eind the rationalizing factor of x^ + 2/^- 
 
 Solution. Since (x'^y — (y^y is exactly divisible by x'^ -j- y^ 
 when n is any positive even integer [§ 66 (ii)], and since 6 is the 
 smallest value of n for which both (x^y and (y'^y are rational, 
 therefore the rationalizing factor (cf. § 157) is 
 
 (X^y—(yh'^ X^—f 5 4 1 2 3 1, 6 
 
 -^^ — 7 1 — = ^ 7 = a?3 — x^y^ -\- xy — x^y^ -^ x^y^ — y^ ' 
 
 .^3 -f ?/^ x^-\-y^ 
 
 Ex. 2. Find the rationalizing factor of x^ + y'^. 
 
 Solution. As in Ex. 1 [cf. %66 (iv)], we find this factor to be 
 
 (X^y^ + (yh^^ X^ -f V^^ M 13 2 , J 2 4 11 
 
 x'5 _|_ 2/^ a;^ -f y^ 
 
 EXERCISE CXX 
 
 
 Find the rationalizing factors of the following expressions : 
 
 3. s^+A 7. Sv^ — 2vK 
 
 11. ah'^-^3v\ 
 
 4. s^ — t^. 8. s~^-\-t\ 
 
 12. x~^-\-2y\ 
 
 5. a^ — xK 9. 2m^ — ?ii 
 
 13. .'i;"* — Zi 
 
 6. m^ + ni 10. 2x^-3yk 
 
 14. 2r~h-'^-t' 
 
CHAPTER XVII 
 
 QUADRATIC EQUATIONS 
 [Supplementary to Chapter XII] 
 
 178. Solution of quadratic equations by means of a formula. 
 
 Since every quadratic equation in one unknown number 
 may be reduced to the form ax^ -\-hx-{- c= ^ (§ 122), and 
 
 since the roots of tliis equation are ^whatever 
 
 the number 8 represented hy a, 6, and c (Ex. 3, p. 195), therefore 
 the roots of any particular quadratic equation may be found 
 by merely substituting for «, 5, and <?, in the roots of the 
 above general equation, those values which these coefficients 
 have in the particular equation under consideration. 
 
 E.g., since the roots of ax^ + hx+ c = {) are — ^ "^ —, therefore 
 
 the roots of 3 ic2 + 10 x - 8 = (in which a = 3, & = 10, and c = - 8) are 
 
 -10±VlO-^-4.3.(-8) ,,,.^-10^14 ,.,.,2,^^_4, 
 2.3 ' ' 6 ' 3 
 
 So, too, the roots of 6 y^ 4. 19 ^ _ 7 = are 
 
 19±Vl9^-4.6(-7) 1 ^^^ _ 7 
 
 2.6 ' ' 3 2 
 
 and the roots of a;2 - 3 x + 5 = are , -(-3)=tr V(- 3)2 — 4.1-5 ^ 
 
 i& " 1 
 
 i.e., ^ 
 
 Note. While the student should, of course, be able to solve quadratic 
 equations without the use of the formula, he is advised to commit this for- 
 mula to memory, and henceforth to employ it freely ; he will find this well 
 worth his while, because roots of quadratic equations are so often required 
 in mathematical investigations. 
 
 277 
 
278 HIGH SCHOOL ALGEBRA [Ch. XVII 
 
 179. Character of the roots of ax" -\-bx + c = 0. Discriminant. 
 
 As we have already seen (§ 126), the roots of the equation 
 
 ^ , ^ -b-h -VP-^ac , _5_ V62-4a(? 
 
 aar + oa; + <? = are ^ and t: 
 
 2a 2a 
 
 Hence it follows that, if a, 5, and c represent real and 
 rational numbers, the roots can be imaginary or irrational 
 only if V52 — 4ac is imaginary or irrational; ^.e., 
 
 If h^ — 4tac is positive^ the roots are real and unequal; 
 ifb^~4:ac = 0, the roots are real and equal ; 
 if l^ — ^ ac is negative^ both roots are imaginary/ ; 
 and the roots are rational only when b^—iac is an exact square, 
 (Let the pupil fully explain each case.) 
 
 The expression b'^ — 4:ac^ which determines the character of 
 the roots, is usually called the discriminant of the equation. 
 
 Thus, without actually solving the equation 3 cc^ _ 5 aj — l = 0, we know 
 that its roots are real, irrational, and unequal because, for this equation, 
 \/&2 _ 4 ac = V37, and VST is real and irrational. 
 
 Similarly, we see that the roots of 2 r.^ + 5 a; — 8 = 4a; — 11 are imaginary, 
 because for this equation h"^ — Aac = — 23. 
 
 EXERCISE CXXI 
 
 1. By means of the formula of § 178, solve Exs. 6-17, p. 196. 
 
 Without first solving the following equations, tell whether 
 their roots are real, imaginary, rational, equal, etc., and explain : 
 
 2. a?-^x + Q> = 0. 5. 3^^4-ll^ + 17 = 0. 
 
 3. a;2-6a; + 9 = 0. 6. 1(3.^2+ 2) -i = i (a; -5). 
 
 4. ^^-1^-11=0. 7. 25?^2_20^^ + 7 = 0. 
 
 8. In each of Exs. 4-11, p. 197, determine the character of the 
 roots without solving the equation. 
 
 9. For what value of h will the roots of 3 .t^ — 10 ic + 2 A; = 
 be equal ? 
 
 Suggestion. The roots will be equal if (- 10)2 _ 4 • 3 . 2 A; = 0. Why ? 
 
179-180] QUADBATIC EQUATIONS 279 
 
 For what values of m will Exs. 10-15 have equal roots ? 
 
 10. mic^— 6x4-3=0. 13. my^ — 5my-\-ll = m. 
 
 11. ^ + 3ma; + 7 = 0. 14. y\l - m) -\-7 y = 9 -3my. 
 
 12. 3a^-4mic-f 2 = 0. 15. -4i/2-32/-3 = m(2/-f 2/ + 4). 
 
 16. Translate into verbal language the conditions for the char- 
 acter of the roots of a^i? -\-hx-\-G — 0. 
 
 17. Show that if one root of a quadratic equation is imaginary, 
 then both are imaginary, and each is the conjugate of the other. 
 
 18. For what values of A; are the roots of 36 a?^ — 24 A;a; -f- 15 A; 
 = — 4 imaginary ? 
 
 Solution. Here the discriminant Ifi — ^ac= ( — 24 ky- — 4 • 36(15 Jc -\- i) 
 = 144(4 A;2 - 15 A; - 4) = 144(4 A; + 1) (k - 4) [cf. § 64]. Hence the roots are 
 imaginary for those values of k which make {4:k + l)(k — 4) negative, and 
 for those values only. Now (4A; + l)(^• — 4) is negative only when one of 
 its factors is positive and the other negativ^; hence the roots of the given 
 equation are imaginary when k lies between — ^ and 4. (Why ?) 
 
 19. For what values of k are the roots of 36 f — 24:kt-\-15k 
 = — 4 real ? How do the roots compare if k= — ^? k = 4:? 
 
 20. Without actually solving the equation, find the values of 
 m for which the roots of 4 mV + 12 m^x -\-10 = m are equal ; also 
 those values of m for which the roots are real ; also, those for 
 which the roots are imaginary. 
 
 180. Relation between roots and coefficients. If we let r 
 
 and / represent the roots of ax^ -^ bx -\- c — 0^ i.e,^ if 
 
 
 _54.V52-4a(? ■. , -b-Vh^-iac 
 
 
 2 a 2 a 
 
 then r + r' = 
 
 _5 + V^2_4^^ _5_V52_4ac_ 6, 
 2a 2a a 
 
 and r-r'^ 
 
 _ J 4- V52 _ 4 ac -b- V62 -4:ac_c 
 
 2a 2a a 
 
 (2) 
 
 Let the pupil work out (1) and (2) in detail, and translate 
 each of these equations into verbal language. 
 
280 HIGH SCHOOL ALGEBRA [Ch. XVII 
 
 EXERCISE CXXII 
 
 In the following equations, name the sum of the roots, also 
 their product ; then check your answers by solving the equations. 
 
 1. x' + 5x-2 = 0. 4. a;2_30a; + 25 = 0. 
 
 2. a^-10x = — 16. 5. a^-\-px = -q. 
 
 3. 4s2-6s=3. 6. ax^-{-2bx^c = 0. 
 
 7. In each of Exs. 4-11, p. 197, write down, without solving the 
 equation, the sum and the product of its roots; explain your 
 work in each case. 
 
 8. If one root of ic^ + 5 a; — 24 = is known to be 3, how may 
 the other root be found from the absolute term ? from the co- 
 efficient of the first power of a; ? Do the results agree ? 
 
 9. If one root of any given quadratic equation whatever is 
 known, how may the other root be most easily found ? 
 
 10. What is the sum of the roots of 3 mV -j-(Sm — l)x 4-5 = 0? 
 For what value of m is this sum 3? 
 
 11. If one root of2a^ — 3(2A: + l)a;-}-9A; = is the reciprocal 
 of the other, find the value of k. 
 
 Hint. Equate one root to the reciprocal of the other and solve for k. 
 
 12. For what value of k will one root of the equation in 
 Ex. 11 be zero ? With this value of Zc, what will be the value of 
 the other root? 
 
 13. Answer the questions of Exs. 11 and 12 for the equation 
 
 2(k+iyx'-S(2k + l)(k + l)x-{-9k=0. 
 
 14. Show that if one root of ax^ -\-bx-\-c = (whatever the 
 values of a, b, and c) is double the other, then 2 6^ = 9 ac. 
 
 181. Values of simple expressions containing the roots. 
 
 If r and / are the roots of a given quadratic equation, 
 § 180 enables us to find the value of such expressions as 
 
 - + — , r^ + r'^, etc., without first solving the equation. 
 
180-182] QUADRATIC EQUATIONS 281 
 
 E.g., the value of - + — , for the equation fl:;^ — 5 a; -f 3 = 0, may 
 be found thus : 
 
 1 l_r|-f-r. 
 
 r r rr 
 but, for this equation, r' + r = 5 and ri^' = 3, 
 
 therefore - + - = -. 
 
 r r' 3 
 
 Similarly, since r^ + r'^ = (r-{- r'y — 2 rr'j 
 
 therefore, for the above equation, 
 
 7^4-/2 = 25-6 = 19. 
 
 182. Formation of equations whose roots are given numbers. 
 
 (i) Sum and product method. If r and r' are the roots of 
 
 the equation ax^ -\- bx + e = 0, i.e., of a^-\--x-\--=0, then 
 
 a a 
 
 (§ 180) this equation may be written in the form : 
 2^ — (r + r'}x -\- rr' = 0. 
 And from this we learn how to write down a quadratic 
 equation whose roots are any two given numbers. 
 E.g., if the roots are to be 2 and — 5, we have 
 — (r + r') = 3, and rr' = — 10, 
 whence the equation is 
 
 a^4_3aj-10 = 0. 
 
 (ii) The factor method. An equation whose roots are any 
 given numbers may be written down as follows (cf . § 72) : 
 
 The roots of (^x — r}(x — r') = 
 
 are evidently r and / ; hence the equation whose roots are 
 2 and - 5 is (x - 2)(x + 5) = 0, 
 
 i.e. , as before, a^ -^Sx — lO = 0. 
 
 EXERCISE CXXIII 
 
 1. If r and r' denote the roots of x^ — lx + 12 = 0, find, with- 
 out solving the equation, the value of 
 
 J- J- -I- , -L 9 ■ 19 
 
 ——,, —,, - + -, r^-\-r". 
 
 ly. _l_ y.1 n.iy.1 n. y>^ 
 
282 HIGH SCHOOL ALGEBliA [Cii. XVII 
 
 Find for each of tlie following equations the value of - + - , 
 11 r r" 
 
 T^ + ?''^, r — r', and '- 
 
 r r 
 
 2. f~-12t = -m. 5. a;2 + 2a; = 4. 
 
 3. 6m2 — m = 2. 6. si? -{-px-\-q = 0. 
 
 4. 32/^-162/ + 5 = 0. 7. ax^ + hx-\-c==0. 
 
 8. Solve each of the above equations and thus verify the 
 results in Exs. 2-7. 
 
 By each ol the methods given in § 182, form the equations 
 whose roots are : 
 
 21. 1±V5. 
 
 16. r^, r'^. 2 
 
 17. V6±4. 22. 5, -^. 
 
 18. 5±2V3. 23. Vc±V^. 
 
 14. r, — r'. 19. 2 ± 5 ?:. 24. r±-,- 
 
 r 
 
 25. The roots of x' — ^x-\-2 — being ?♦ and ?•', form a new 
 
 equation whose roots are - and -. (Cf. Exs. 1 and 15 above.) 
 
 r r' 
 
 26. If r and r' are the roots of 3z^ — llz-20 = 0, form an 
 
 equation whose roots are - and _ ; also one whose roots are r^ 
 
 r r' 
 
 and ?''^; also one whose roots are r -\ — andr' + -. 
 
 r' r 
 
 Write the equations whose roots are : 
 
 27. - 1, 2, - 5. 29. 1 ± V5, 5. 
 
 28. -a, -b, —c. 30. ± Vc, c + c?. 
 
 183. Factors of quadratic expressions. As in § 182, if r 
 and r' are the roots of the equation ax^ -{-bx-{-c = 0, then 
 
 a^ + - X + - = x^ — (r -{- r') X -\- rr' = (x — r^Cx — rO, 
 a a V yv ^ 
 
 9. 
 
 5, -3. 
 
 10. 
 
 -4,4. 
 
 11. 
 
 -a, -6. 
 
 12. 
 
 i|. 
 
 13. 
 
 i-f 
 
182-184] qUADUATlC EQUATIONS 288 
 
 ^.e., multiplying by a^ 
 
 ao^ -\- hx -\- c = a (x — r)(x — r^) ', 
 hence, if the roots of the equation aa^ + bx-i- c = are r and 
 r\ then the factors of the expression ax^ -{- bx -\- c are «, x — r^ 
 and x — r'. 
 
 184. A quadratic equation has two roots, and only two. 
 
 By actually solving the equation ax^ -\- bx + c = (§ 126) 
 we find that it has two roots, say r and r^ That it can 
 have no other root, as r", is evident if we write the equation 
 in the form 
 
 a(x-r')(x-r^)=0 [§ 183 
 
 and observe that a (r" — r}(r" — r'} cannot be zero if r'' 
 differs from both r and r', 
 
 EXERCISE CXXIV 
 
 1. Since 2 and 7 are the roots of x^ — 9 x-\- 14: = 0, what are 
 the factors of a^ — 9 ic + 14 ? 
 
 2. By first finding the roots of the equation 15 a^ — 4a?— 3 = 0, 
 find all the factors of the expression 15 a^ — 4 a; — 3. Check your 
 answer by finding the product of these factors. 
 
 3. Write a carefully worded rule for factoring quadratic 
 expressions by the method of § 183. 
 
 Find (in accordance with the rule just made) all the factors of 
 the following expressions, and check your results : 
 
 4. 
 
 5a;2-12a;-9. 
 
 
 11. 
 
 (2y-iy-5(y + l)+8. 
 
 5. 
 
 Sz'-4:5z-lS. 
 
 
 12. 
 
 a^+px-^q. 
 
 6. 
 
 a)2 + 9. 
 
 
 13. 
 
 ax^ + bx-\-c. 
 
 7. 
 
 s^+s+l. 
 
 
 14. 
 
 3m2 ^ 
 -— m — 5. 
 
 8. 
 9. 
 
 4m2-24m-13. 
 8a^_2a;-3. 
 
 
 15. 
 
 10. 
 
 (^ + l)(2-a^)+9 
 
 HIGH 8CH. ALG.— 
 
 — X. 
 •19 
 
 16. 
 
 m^ + 6 m + 13. 
 
284 HIGH SCHOOL ALGEBRA [Ch. XVII 
 
 17. Are the expressions in Exs. 4-16 equal to 0? What justi- 
 fication have we, then, for writing them so ? 
 
 18. How many roots has a quadratic equation ? Verify your 
 answer for the equations 28 x^ -f- 29 a; +6 = and m^— 10 m= —25. 
 
 19. Show that the cubic equation 27 2/^ — 1 = has three roots 
 and only three (cf. Ex. 17, p. 263). 
 
 REVIEW EXERCISE- CHAPTERS XJ-XVIi 
 
 1. Find the square root of x^ -\- 20 a^ -\- 16 — 4: oc^ -{- 16 x ; also 
 of a;'* — 2 x^y~^ — 4 cc^ + y~^ + 4 x^y + 4 xy^. 
 
 2. Find (correct to three decimal places) the value of Vll-7 ; 
 VA; V23561. 
 
 3. Expand by the binomial theorem : (a^ — 2/^" ; ( 9" ~^ ^ 
 (l + c-y- (l-Sm^y-, ^-l 
 
 + a;i 
 
 4. Use the binomial theorem to find (correct to five decimal 
 places) the value of (10.001)^ i.e., of (10 + .001)^ 
 
 Simplify : 
 
 5. ^1 + -^^+^. 7. </ab^(ah-'c^)(a-h^c-'). 
 
 prnj 
 
 \ z-l ^ z + 1 Xc** V'c^ V 
 
 9. (a3-62)^(-^^-V6). 
 
 ^c-d^c-d ^c-{-d^c + d 
 
 11. (2V-3-3V^=^)(6V-2 + 4V-3). 
 
 13. By rationalizing the divisor perform the following indi- 
 cated divisions : 
 
 1 + r + Vl^^ . 1 V^-V^^3y . 3 
 
 l+r-Vl-r^' V6-V3-I' V^^-V^' 3*-f5"^ 
 
184] 
 
 q UADRA ric Eq ua tiojsi s 
 
 285 
 
 14. Express -^^89 — 28 VlO as the difference of two surds. 
 
 15. Find the sum of V- 25, - 1- V^ 5-lOV^, and 
 — 2 — 7 i, graphically ; also, by the same method, subtract the 
 fourth of these numbers from the third. 
 
 Solve the following equations and check as the teacher directs 
 
 20. m + 5 -h Vm -\-5 = 6. 
 
 16. f-| + 7| = 
 
 17. ^x-^l + x = 
 
 
 18. i-?l 
 
 = 1. 
 
 19 3 a^ 4- V4 a? — a?^ ^ 2 
 
 21. ^a-{-z-{--\/a — z = -y/b. 
 
 22. aj^ + 4 ic^ — 117 = 0. 
 
 23. 19a;*4-216a;^ = a;. 
 
 24. 144r2-l + 6V9r2-r=16r. 
 
 3 aj — V4 a; — x^ 
 
 25. (7-4 V3)a:2+(2-V3)a;-2 = 0. 
 Solve the following systems 
 0^2 + 2/2 = 85, 
 , xy = 42. 
 
 26. 
 
 29. 
 
 (2 
 
 xy = 4:- y\ 
 0^-2/2 = 17. 
 
 27. 
 
 r 0^-2^^ = 215, 
 I a^4-a^?/ + 2/' = 43. 
 
 28. ^ 
 
 f 3r2-2rs = 15, 
 
 30. 
 
 31. 
 
 a? 2/ 
 
 ,a^ r 
 
 mo 4- Vv -h ty = 11, 
 
 t2r + 3s = 12. 
 
 2vw— Vv + w = 13. 
 
 32. Show that the difference of the roots of the equation 
 ic^ +pa; + g = is the same as that of a52 -|- ^px + 2 p^ + g = 0. 
 
 33. For what values of m will a^ - 2 (1 +3 m) «+ 7 (3 + 2 7^) = 
 have real roots ? equal roots ? imaginary roots ? 
 
 34. Form the equation whose roots are a, — a, and h ; also the 
 equation whose roots are the negative reciprocals of the roots of 
 ax^ — 6a; 4- c = 0. 
 
 35. If the roots of a^ — 2>a5 + g = are two consecutive integers, 
 show that 2>' — 4 g — 1 = 0. 
 
286 HIGH SCHOOL ALGEBRA [Ch. XVII 
 
 36. A rectangular plot of ground contains 42 acres; find its 
 sides if its diagonal measures 1243 yards. 
 
 37. In a regiment drawn up in the form of a solid square, 
 the number of men in the outside five rows is -fj of the entire 
 regiment. Find the size of the regiment. 
 
 38. In a quarter of a mile drive, the fore wheel of a carriage 
 makes 22 revolutions more than the hind wheel ; if the circum- 
 ference of each wheel were 2 ft. less than it now is, the fore wheel 
 would make 33 revolutions more than the hind wheel. Find the 
 circumference of each. 
 
 39. A crew can row a certain course upstream in 8^ minutes, 
 and were there no current, they could row it in 7 minutes less 
 than the time it now takes them to drift downstream. How long 
 would it take them to row the course downstream ? 
 
 40. Two men, A and B, have a money box containing $ 210, 
 from which each draws a certain fixed sum daily, the two sums 
 being different. Find the sum drawn daily by each, knowing 
 that A alone would empty the box 5 weeks earlier than B alone, 
 while the two together empty it in 6 weeks. 
 
CHAPTER XVIII 
 INEQUALITIES 
 
 185. Definitions. The symbols > and < stand for "is 
 
 greater than," and " is less than," respectively ; thus, the 
 expression a<b is read ''a is less than 5." One real number, 
 a, is said to be greater than another, 6, U a — b is positive ; 
 if a — 5 is negative then a is less than b. 
 
 E. g., 5 < 8, since 5 - 8 = - 3 ; and - 4 >- 9, since -4-(— 9) = +5. 
 
 A statement that one of two numbers is greater or less 
 than the other is called an inequality ; thus, 5a;— 3>2yis 
 an inequality, of which 5 a; — 3 is the first member, and 2 y 
 the second. 
 
 Two inequalities are said to be of the same species (or to 
 subsist in the same sense') if the first member is the greater in 
 each, or if the first member is the less in each ; otherwise 
 they are of opposite species. 
 
 Thus, the inequalities a > 6 and c + <? > e are of the same species, while 
 2-2 ^ y'2 -> 2;2 and m^<,n^ + mn are of opposite species. 
 
 186. General principles in inequalities. Before memoriz- 
 ing the following principles (1-7), the pupil should illus- 
 trate each by one or more numerical examples; he should 
 also try to invent a proof of his own for each principle before 
 reading the printed proof. 
 
 Principle 1. If the same number is added to, or sub- 
 tracted from, each member of an inequality, the result is an 
 inequality of the same species. 
 
 287 
 
288 HIGH SCHOOL ALGEBRA [Ch. XVIII 
 
 For, if a<h, i.e.^ if a — h is negative, then a-{- c —(b -\- c^^ 
 which equals a — b^ is negative, and therefore 
 
 a -f c<b-^ c, 
 and similarly a— c<b — c. 
 
 So, too, if « > 5, then a-\- c>b -\- c^ and a— c>b — c. 
 
 Principle 2. If each member of an inequality is multi- 
 plied or divided by the same positive number^ the result is an 
 inequality of the same species. 
 
 For, if a>b^ and n is any positive number, then an — bn, 
 i.e.^ (a — b^n^ is positive (why?), and therefore an>bn. 
 
 Similarly if we divide by n ; and so, too, \i a<b. 
 
 Principle 3. If each member of an inequality is multi 
 plied or divided by the same negative number^ the result is an 
 inequality of opposite species. 
 
 Hint. If « > 6, and n is negative, then an — hn is negative (why ?) and 
 therefore an < hn. 
 
 Principle 4. If several inequalities of the same species 
 are added^ member to member^ the result is an inequality of the 
 same species. 
 
 Hint. Let a<b, c<d, e</, ..., then (a - b)-\-(c - d) + (e -/)+ •••, 
 i.e., (a -f c + e + •••) — (6 + d +/+ •••)) ^^ negative (why ?), and therefore 
 
 Principle 5. If an inequality is subtracted from an equa- 
 tion^ or from an inequality of opposite species.^ member from 
 member., the result is an inequality whose species is opposite to 
 that of the subtrahend. 
 
 Hint. Let a<Cb, and c = d or c^d, then in either case, b — a -\- c — d, 
 which equals c — a —(d — b),is positive (why ?), and therefore c — a^d — b. 
 
 Principle 6. If the first of three numbers is greater than 
 the second., and the second is greater than the third., then the 
 first is greater than the third ; and conversely. 
 
 Hint. Let a > 6 and 6 > c, then (a — &) + (& — c), i.e. , a — c, is positive 
 (why ?), and therefore a>c. 
 
186-187] INEQUALITIES 289 
 
 Pkinciple 7. If two or more inequalities of the same 
 species^ whose members are positive^ are multiplied together^ 
 member by member^ the result is an inequality of the same species. 
 
 Hint. Let a > 6 and c> d, then, by Prin. 2, ac > he and he > hd^ whence, 
 by Prin. 6, aC^^hd -, and similarly for three or more such inequalities. 
 
 187. Conditional and unconditional inequalities. An iden- 
 tical or unconditional inequality is one which is true for all 
 values of its letters. Thus, a + 4 > a and (x — yY + 1 > 
 are unconditional inequalities. 
 
 A conditional inequality is one which is true only on con- 
 dition that certain restricted values are assigned to its letters. 
 Thus, a;-f4<3a;— 2 only on condition that x>'^. 
 
 A conditional inequality is solved by means of the princi- 
 ples of § 186, and in much the same way that an equation is 
 solved by means of the ordinary axioms. 
 
 Ex. 1. If 3 .'» — -2^ > -^ — X, find the possible values of x. 
 
 Solution. On multiplying each member of this inequality by 
 3, it becomes 9»=-25>ll -3x, [§186,2 
 
 whence 9 a; + 3 a; > 11 + 25, [§ 186, 1 
 
 I.e., 12 a; > 36, 
 
 whence ic> 3 ; [§ 186, 2 
 
 i.e., if the given inequality is true, x must be greater than 3. 
 
 By means of the principles established in § 186 the student may show that 
 each step in the above reasoning is reversible, and hence that the converse 
 is also true ; viz. , that if a; > 3, then 3 x — ^^ > ^^^ — ic. 
 
 rx- ^i_ . 1 .• (2a; + 32/>5, (1) 
 
 Ex. 2. Given the two relations \ . / /o\ 
 
 I a; + 42/ = 6; (2) 
 
 to find those values of x and y that will satisfy them both. 
 
 Solution. On multiplying each member of (1) by 4, and each 
 member of (2) by 3, we obtain 
 
 8a;-}-122/>20, 
 and 3a;-|-122/ = 18; 
 
 whence, subtracting, 5 a? > 2, [§ 186, 1 
 
 and therefore a^ > f • [§ 186, 2 
 
290 HIGH SCHOOL ALGEBRA [Ch. XVIII 
 
 Now substitute for x in (2) above, any number greater than |, 
 and find the corresponding value of y (this value of y will 
 always be less than J ; why ?) ; these values of x and y, taken 
 together, will satisfy both (1) and (2). 
 
 EXERCISE CXXV 
 
 3. Using the definitions of "greater" and "less" in § 185, 
 show that 5 > 2 ; that - 23 < - 12 ; and that 2 > - 9. 
 
 4. It x-^y > z — iv, show that x-\-w ^ z —y. 
 Hint. Apply Principle 1 twice. 
 
 5. May terms be transposed from one member of an inequality 
 to the other ? If so, how and why (cf . Ex. 4) ? 
 
 ^ j^ m-n^ m-\-2n ^ ^^^^ ^^^^ 3(m -ii) < 2(m + 2 n). 
 
 How may an inequality be cleared of fractions ? Why ? 
 
 7. Show, from Principles 2 and 3, how to remove a common 
 factor from both members of an inequality. 
 
 8. What happens if the signs in each member of an inequality 
 are reversed ? AVhy ? 
 
 Hint. In the proof of Principle 3, put — 1 for n. 
 
 9. If a > 6 and c — d, show that G — a<d — h. 
 
 10. Illustrate by numerical examples that 
 
 (1) if a > & and c < d, then the sum of these inequalities may 
 be either a + c = 64-d, ora + c>&4-(?, or a + c<54-d; 
 
 (2) if a > 6 and c > d, then the difference of these inequalities 
 may be either a— c = h^d, a — c^h — d, ova — c<h—d. 
 
 11. Translate (1) and (2) of Ex. 10 into verbal language. 
 
 12. If a < 6 and c < d, and if d alone is positive, show that 
 ac > hd. Is this inconsistent with Principle 7 ? 
 
 13. If a, b, c, and d, are positive numbers, and if a > 6 while 
 c<d, which is the greater, ac or bd ? Why ? Illustrate your 
 answer numerically. 
 
 14. What operations Avith or upon inequalities lead to results, 
 one of which is certainly greater or less than the other ? 
 
187] INEQUALITIES 291 
 
 15. Name and illustrate some operations with inequalities 
 which lead to results whose relations are uncertain. 
 
 16. Show that a^ + b->2 ah except when a — h. 
 Hint, (a — 6)2 is positive whether a> 6 or a < 6. 
 
 17. Distinguish between a conditional and an unconditional 
 inequality. To which of these classes does a^ + h^ -\-l>2 ah 
 belong ? Why ? 
 
 18. To which class of inequalities does 6 £c — 5 > 3 a? belong ? 
 AVhy ? Solve this inequality. 
 
 19. If 3 a; < 5 X — 9, show that x is greater than 4^ (cf. Ex. 1). 
 
 20. If x^ -\-2^ <C 11 Xj show that the range of values of x is be- 
 tween 3 and 8. i.e., that x must be greater than 3 and less than 8. 
 
 Hint. In order that (x - 3) (8 — x), i.e., \\x- x'^ - 24, may be positive, 
 both factors must be positive or both negative. 
 
 Find the range of values of x in each of Exs. 21-26 : 
 
 21. x" < 9. 
 
 22. a.-2-h24>lla;. 25. 
 
 Q^ 10 — a;>5. 
 
 23. 30>x-f4^>25. ) 
 
 2 r3-4a^<7, 
 
 24. 28>3a; + x2. 26. |^_^2<4. 
 
 27. Show that no positive number plus its reciprocal is less 
 
 than 2 5 i.e., n being any positive number, that n 4- - < 2.* 
 
 n 
 
 28. Show that 4 ic- + 9 < 12 ic. 
 
 29. Show that 2 6 (6 a - 5 6) > (2 a + &)(2 a-h). 
 If a, &, and c are positive and unequal, show that 
 
 30. a' 4- ?>' > a%\ 32. a^ + 6^ > a^ft + ah\ 
 
 g- a4-26 a4-3 ?> 33. a" -{■ h" -\- c^ > ah -\- he + ca. 
 
 ' a-\-Sb a + 46 34. a^-{-]/-^cy^>3ahc(Gt'Ex.30,]).51). 
 35. If a=^ -f &2 = 1^ and (r' + cJ- == 1, prove that ah-\-Gd>l. 
 
 * Cf. Ex. 16. The symbol < stands for " is not less than," and > stands 
 for " is not greater than," 
 
 4x-ll>| 
 
292 HIGH SCHOOL ALGEBRA [Ch. XVIII 
 
 36. If both m and *i are positive, which is the greater, 
 m-i-n ^^ 2mn ^ 
 
 2 m 4- ?i 
 
 Solve the following systems : 
 
 37 (2x-Sy<2, (y-x>9, 
 
 \2x-\-5y=6. 39. |7^_^j_^^ 
 
 r3 a; + 2 2/ = 42, I 20 15 
 
 I 7 1 4 a; < 3 2/. 
 
 41. Find the smallest integer fulfilling the condition that ^ of 
 it decreased by 7 is greater than J of it increased by 6. 
 
 42. The sum of three times A's money and 4 times B's is $ 1 
 more than 6 times A's ; and if A gives $ 5 to B, then B will have 
 more than 6 times as much as A will have left. Find the range 
 of values of A's money and B's. 
 
CHAPTER XIX 
 
 RATIO, PROPORTION, AND VARIATION 
 
 I. RATIO 
 
 188. Definitions. The ratio (direct ratio) of two numbers 
 is the 'quotient obtained by dividing the first of these num- 
 bers by the second. The numbers themselves are usually 
 called the terms of the ratio, the first being the antecedent, 
 and the second the consequent. 
 
 E.g.^ the ratio of 15 to 5 is 15 -^ 5, i.e., 3 ; the antecedent is 15, and the 
 consequent is 5. 
 
 The ratio of a to 5 may be vsrritten as « : 6, a -^ 5, or - ; 
 it is read "the ratio of a to 6," or "a divided by 6." 
 
 The inverse ratio of 6k to 5 is 6 -h a, i.e.^ it is the reciprocal 
 of the direct ratio of these numbers. 
 
 Two numbers are said to be commensurable or incommen- 
 surable with each other according as their ratio is rational or 
 irrational (cf. § 146), i.e.^ according as they have or have not 
 a common measure. 
 
 E.g.^ 1.5 and f are commensurable with each other; so also are 3\/2 and 
 5 v'2 ; but 3V^ and 5 are incommensurable. 
 
 189. Ratio of like quantities. The concrete quantities 
 with which algebra is concerned are expressed by means of 
 numbers, and the ratio of two like * quantities is therefore the 
 ratio of the numbers which represent these quantities. 
 
 E.g., the ratio $6 : $9 is the same as 6 : 9, i.e.., as 2 : 3. 
 
 * Unlike quantities can, of course, have no ratio to each other. 
 293 
 
.294 TIIGU SCHOOL ALGEBRA [Ch. XIX 
 
 190. Properties of ratios. Since ratios are quotients, i.e.^ 
 fractions, therefore they have all the properties of fractions. 
 Thus, a ratio is not changed if both antecedent and 
 consequent are multiplied or divided by any given number. 
 Again, if a, ^, and k are positive, and a<h, then since 
 
 a a + k . (a—b)k re oo 
 
 is negative (a — h being negative), therefore (§ 185), the ratio 
 a-\-k '. h -{-k is greater than the ratio a : h. [Translate this 
 important fact into words : (1) calling a ^ b ix proper fraction ; 
 and (2) calling a-i- b a ratio less than unity.] 
 
 EXERCISE CXXVI 
 
 What is the ratio of : 
 
 1. 6 to 8? 3. -10 to 24? 5. -I to 11? 
 
 2. 50 to 15 ? 4. 6.3 to .7 ? 6. 21 a;^ to 9 a;? 
 
 7. Which term of a ratio corresponds to the divisor ? What 
 is the other term called ? Illustrate from Exs. 1-6. 
 
 8. Form the inverse of each of the ratios in Exs. 1-6. 
 
 9. The ratio of x to 5 equals 2 ; find x, and check your work. 
 
 10. If the ratio of two numbers is f and the consequent is 
 6, find the antecedent. 
 
 In each of Exs. 11-14, find (and check) the value of x : 
 
 11. x^ :2 = ^. 13. 64 : a; = a;. 
 
 12. a; + 5: 2a; = -7. 14. 25:a;2 = 9. 
 
 15. What is the ratio oi x to y when 7(x~y)=S(x-\-y)? 
 when af -[-6y^ = 5xy? 
 
 16. A yard measure is divided into two parts whose lengths 
 are in the ratio 7 : 11 ; how many inches in each part ? 
 
 17. A and B divide $ 100 between them so that A receives $13 
 out of every $ 20. What is the ratio of A's share to B's share ? 
 How many dollars does each receive ? 
 
190-191] llATIO, PROPORTION, AND VARIATION 295 
 
 18. What number must be added to each term of ^^ in order 
 that the resulting ratio shall be 2:3? Does this addition in- 
 crease or diminish the given ratio ? 
 
 19. Which is the greater ratio, 5 + 2 : 17 + 2 or 5 : 17 ? 
 21 + 8 : 11 -f 8 or 21 : 11 ? Does the addition of the same positive 
 number to both terms of a ratio always increase the latter's value 
 (cf. § 190)? Explain. 
 
 20. If a, b, and k represent positive numbers, translate (1) and 
 (2) below into words ; then give three numerical illustrations of 
 each: 
 
 (1) ^^^ < 7 when a<b', 
 
 — K 
 
 (2) ^'^>r when a>b. 
 b —k b 
 
 21. By a method similar to that used in § 190, show the cor- 
 rectness of (1) and (2), Ex. 20. 
 
 22. If X, y, and z are positive numbers, which is the greater 
 
 ,. , -, i r>x 2ic4-5?/ ic-|-2?/o x — y-\-z x-\-y-\-Zo 
 ratio (and why ?), — — ^ or —^ — ^ ? ^L-L_ or ^ ^ ^ ? 
 
 2x-[-ly x-\-3y x-\-y — z x — y — z 
 
 23. Show that the following ratios are all equal: $12 : $9; 
 8 bu. oats : 6 bu. oats ; 4 T. of coal : 3 T. of coal ; 10 in. : 7^ in. ; 
 4:3; i:^^. 
 
 24. Eind the value. of each of the following ratios: 4V2 : V2; 
 4V2:2; 7V3 in. : 14V2 in.; $5.80 : 29 cents. 
 
 25. Eind two integers whose ratio equals 15| : 9f . Can the 
 ratio of any two numbers whatever be expressed as the ratio of 
 two integers (cf . Ex. 24, also § 188) ? 
 
 26. Which of the pairs of numbers (or quantities) in Ex. 24 
 are commensurable ? Which are incommensurable ? Why ? 
 
 II. PROPORTION 
 
 191. Definitions. If a, 5, c, and d are any four numbers 
 such that a : b — c : d, then these numbers are said to be 
 proportional, or to form a proportion ; i.e., a proportion is 
 a statement that two ratios are equal. 
 
296 HIGH SCHOOL ALGEBRA [Ch. XIX 
 
 The proportion a :b = e : d (sometimes written a : b :: c : d^ 
 is read : " the ratio oi a to b equals the ratio of c to c?," and 
 also "a is to 5 as <? is to c?." In this proportion a and d are 
 called the extremes, while b and e are called tlie means. 
 
 Ji a : b = e : d, then d is said to be the fourth proportional 
 to a, 5, and c ; while ii a : b = b : c, then c is called the third 
 proportional to a and 5, and 6 is called the mean proportional 
 between a and e. 
 
 A succession of equal ratios, in which the consequent of 
 each is also the antecedent of the next, is called a continued 
 proportion ; thus a : b = b : c = e : d = d : e = • • -^ is a continued 
 proportion. 
 
 EXERCISE CXXVII 
 
 1. Is it true that 8 : 12 = 10 : 15 ? Why ? How is this pro- 
 portion read ? What does it mean ? 
 
 2. In Ex. 1 name the means and the extremes of the propor- 
 tion, also the fourth proportional to 8, 12, and 10. 
 
 3. Is it true that 8 : 10 = 12 : 15 ? How does this proportion 
 compare with that in Ex. 1 ? Does a proportion remain true 
 after its means have been interchanged ? Try several numerical 
 examples and compare Principle 4, p. 298. 
 
 4. By arranging the numbers 3, 4, 6, and 8 in different ways, 
 make three different proportions. 
 
 5. Is 6 a mean proportional between 4 and 9? between 18 
 and 2 ? Is the same thing true of — 6 ? Name the third 
 proportional in each case. 
 
 6. Show that 2 : 6 = 6 : 18 = 18 : 54 = 54 : 162 is a continued 
 proportion in which each ratio equals ^. Write a continued 
 proportion of five ratios each of which equals f. 
 
 192. Important principles of proportion. Since a propor- 
 tion is merely an equation whose members are fractions, 
 therefore the principles of proportion may be derived from 
 those governing equations and fractions. 
 
101-192] RATIO, PROPORTION, AND VARIATION 297 
 
 Note. Before memorizing the following principles (1-8) the pupil should 
 illustrate each by one or more numerical examples ; he should also try to 
 invent a proof of his own for each principle, before reading the printed proof. 
 
 Principle 1. If four numbers are in proportion, then the 
 product of the extremes equals the product of the means. 
 
 For, let a, h, c, and d be any four numbers which are in 
 proportion, then a :h = c : d\ 
 
 a c 
 
 whence ad = he, [clearing of fractions 
 
 which was to be proved. 
 
 Principle 2. If the product of two numbers equals the 
 product of two others, then either pair may be made the 
 extremes of a proportion in which the other pair are the means. 
 
 For, if ad = be, 
 
 a o 
 then 7 ~ ;j' [dividing by hd 
 
 ct 
 
 i.e., a '.b = c : d. 
 
 In the same way it may be shown that, if ad = he, then 
 
 h : a = d : c, c : a = d : h, etc. 
 
 Eemark. From the proof just given it follows that the correct- 
 ness of a proportion is established when it is shown that the product 
 of the 7neans equals the product of the extremes ; this test is very 
 useful. 
 
 Principle 3. i/" four numbers are in proportion, then 
 they are in proportion by inversion ; i.e., the second is to the 
 first as the fourth is to the third. 
 
 For, ii a : h = c : d, then ad = he (why ?) ; hence h : a = d : c 
 (cf. Principle 2, Remark). 
 
 Suggestion. Let the pupil state Principles 4-7 below entirely in verbal 
 language, and prove each in detail (cf. statement and proof of Principle 2). 
 
298 IIJGH SCHOOL ALGEBRA [Ch. XIX 
 
 Principle 4. If four numhers are in proportion^ then they 
 are in proportion hy alternation; ^.e., if a:h — c:d^ then 
 a : c = b : d. 
 
 Principle 5. If four rmmbers are in proportion^ then they 
 are in proportion by composition ; i.e.^ ii a : b = c : d, then 
 (a + ^) : a= (^c -{- d') : e ; [also, (a + 6) : 5 = (c + 6?) : c?]. 
 
 Principle 6. If four numbers are in proportion^ then they 
 are in proportion by division or separation ; i.e.,if a : b = c : d, 
 then (a — b^ : a= (^c — d} : c; [also, (^a — b^ : b= (^c — d^ : dj^. 
 
 Principle 7. If four numbers are in proportion, then 
 they are in proportion by composition and division ; i.e.^ if 
 a : b = e : d, then (^a + 5) : (a — 6) = ((? + t?) : (c — c?). 
 
 Principle 8. In a series of equal ratios the sum of the 
 antecedents is to the sum of the consequents as any antecedent 
 is to its own consequent. 
 
 Thus, if a'.b — c:d=e:f=g:h = --' = x'.y., 
 
 then <ia + c + e+g-\--" + x): (^b -j- d-hf+ h+ ■-■ -^ y} = e :f. 
 
 To prove this theorem, let each of the given equal ratios 
 be represented by a single letter, say r ; 
 
 , •, a c e a -, x 
 
 then -=r, -=r, - = r, ^ =r, •••, and - =r, 
 
 b d f h y 
 
 hence a = h\ c = dr^ e =fr^ g=hr., •••, and x = yr^ 
 
 and, adding these equations, member to member, 
 
 a^c^e-\-g-\-"-^x = Q)^d^f-\-h-\--'^y)r, 
 
 and therefore ^ + ^ + ^ +.^+ •>• +^ ^^^e 
 b+d+f-\-h^-- + y f 
 
 which proves the principle. 
 
 Note. As in the proof just given, so it will often be found advantageous 
 to represent a ratio by a single letter. 
 
192] RATIO, PROPORTION, AND VARIATION 299 
 
 EXERCISE CXXVIII 
 
 Using Prin(3iple 1, find .*; in each of Kxs. 1-4 : 
 
 1. 14 : 3 = 56 :x. 3. - 16 : .^• = 18 : 7. 
 
 2. aj : - 5 = 20 : - 2. 4. J : ic = x : g^^ . 
 
 Find a mean proportional between each of the following pairs 
 of numbers (cf. Ex. 4) : 
 
 5. 3, 27. 7. 25, — 4. - 9. ain'^, a^m. 
 
 6. -2,-5. 8. .25, .09. 10. a + h,a-b. 
 
 11. How many answers has each of Exs. 5-10 ? Why ? 
 Show that the mean proportional between any two numbers 
 equals the square root of their product. 
 
 12. Find the third proportional to 1 and 4 ; to — 25 and — 40 ; 
 also the fourth proportional to m — n, m^ — rr, and m + n. 
 
 13. Using Principles 2-7, make seven different proportions 
 from the equation cd = mn. 
 
 14. Add 1 to each member of the equation a:b = c: d; write 
 the result as a proportion and thus prove Principle 5. 
 
 15. Prove that like powers (also like roots) of proportional 
 numbers are proportional, i.e., prove that if a:b = G: d, then 
 a" : ft** = c" : d\ 
 
 16. If a: b = c: d and e:f = g:h, show that ae :bf= eg: dh ; 
 also translate this principle into verbal language. 
 
 17. li a: b = c:d and e : /= a : h, show that - : - = - : -• 
 
 e f g h 
 
 Hint. Use a single letter to represent a ratio (cf. proof of Principle 8). 
 
 li p : q = r : s, Siud m and n are any numbers whatever, show 
 that the proportions in Exs. 18-25 are true. 
 
 18. mp : 7iq=mr : ??s(cf. § 190). 21. s^: q^ = r'^ : p^ 
 
 19. op:r = 5q:s. 22. p -\-q :2 p = r + s : 2 r. 
 
 20 r:s=-: ' 23. pr : qs= r^ : s\ 
 
 q p 
 
 24. j9'^-4r-:^"-4s'»=-|": -^• 
 
 HIGH ^ClI. ALG. — 20 
 
300 HIGH SCHOOL ALGEBRA [Ch. XIX 
 
 25. 2) + q: r+ s = Vp~ -f g^ ■ V?^T^. 
 
 26. li a:b = c: d = e :f=g : h= •", and I, m, 7i, p, -" are any 
 numbers whatever, show that 
 
 (ma -\- Ic — ne -i- pg -^ •••) -.{mb -\- Id — nf -\- ph -{ ) = a:h. 
 
 Hint. Compare § 190, also Principle 8. 
 
 27. If {p-\-q^-r+s){p—q—r-{-s) = {p-q^r-s){p + q—r—S), 
 show that p : q = r : s. 
 
 28. If ax + cy ^ay + cz^ az + cx ^ ^^^^ ^^^^. ^^^,^ ^j jj^^^^ 
 
 ?>.?/ -|- dz bz +dx bx+ dy 
 
 ratios equals ^LjL^ (cf. Principle 8). 
 b -\-d 
 
 By the principles of proportion solve the equations : 
 
 29. a;:15 = aj-l:12. 33. a: : 27 = 2/ : 9 = 2 : a; -?/. 
 
 30. a;-:32 = a; + 2:12. a;+Va;-l 13 
 
 34. ^=iiiz = — 
 
 31. {\cx+V)rdx'=\c\ x--Jx-l 7 
 
 'x-y=2, 
 
 Hint. Apply Principle 7. 
 
 32. \ Q^-^f ^5 32 Va; + 7 + V^ ^ 4 + Va? ^ 
 
 (a? + 2/f 9 ■ Va; + 7 - V^ 4 — Vx 
 
 36. AVhat number must be added to each of the numbers 7, 9, 
 11, and 21 in order that the four sums may be proportional ? 
 
 37. In the triangle ABC^ ^/^ divides BC into two parts, 5^ and 
 KC^ respectively proportional to AB and AC. If ^J5 = 10 in., 
 AC = 16 in., and 5(7= 20 in., find BK and KC (Draw a figure 
 to illustrate.) 
 
 38. Find two different numbers, m and w, such that 
 
 m-\-n\m — n\ m^ + n^ = 5:3: 51.* 
 
 39. The perimeter of a triangle whose sides are in the ratio 
 5 : 6 : 8, is 57 meters ; find the lengths of the sides. 
 
 40. How may $10 be divided among three boys so that for 
 every dollar the first receives, the second shall receive 15 cents 
 and the third 10 cents ? 
 
 * The expression a:h:c = x:y'.z, means that a : h ■= x : y, a : c = x : z^ 
 and b :c = y : z; and also the equivalent statement a : x = b : y = c : z. 
 
192-194] RATIO, PROPORTION, AND VARIATION 301 
 
 41. Two rectangles are equal in area. If their widths are as 
 2 : 3, find the ratio of their lengths. 
 
 42. The sides of a certain rectangle are in the ratio 7 : 3. 
 Compare the area of the rectangle with that of a square which 
 has the same perimeter. 
 
 43. If a:b, c : d, e\f, g : h, ••• are unequal ratios, in which 
 a, b, c, '" are positive numbers, and if a: b is the greatest and 
 e : / the least among these ratios, show that 
 
 is less than a : b, but greater than e : / (cf. proof of Principle 8). 
 
 III. VARIATION 
 
 193. Variables, constants, and limits. Many questions in 
 mathematics are concerned with numbers whose values are 
 changing ; such numbers are usually called variables, while 
 numbers whose values do not change are called constants. 
 
 If the difference between a variable (in the course of its 
 changes) and a constant may become and remain less than 
 any assigned number however small, then this constant is 
 said to be the limit of the variable. 
 
 Thus, your own age, the height of the mercury column in a thermometer 
 tube, the length of the shadow cast by a given flagstaff, etc., are variables ; 
 while the difference between the ages of two given men, the weight of the 
 mercury in a given thermometer, the length of a certain flagstaff, etc., are 
 constants. 
 
 Again, the decimal .3333 ••• (i.e., .3 + .03 + .003 H — ) is a variable whose 
 limit is I ; this decimal grows larger and larger as more and more places are 
 included, and may thus be made to differ from i by less than any assigned 
 number however small. 
 
 So, too, 1 + J + I + I + Jg ••• is a variable whose limit is 2 (cf. § 202). 
 
 n n 
 
 194. Interpretation of the forms -, -, and -• Two of these 
 
 oo 
 
 forms were first hiet with in § 41, and were there inter- 
 preted by assuming that the definition of division, given in 
 § 8, remains valid for infinitely large numbers and for zero. 
 
302 HIGH SCHOOL ALGEBRA [Ch. XIX 
 
 It is better, however, to interpret these forms from the 
 standpoint of variables and limits. 
 
 (i) If the values 1, -jIq-, y^^, -f^foo^, •••, are successively 
 
 assigned to x^ what are the corresponding values of -? 
 
 of -, where a is any finite constant whatever ? Answer the 
 
 same questions when x takes the successive values 1, 10, 
 102, 103, .... 
 
 These examples illustrate two important facts, viz. : 
 
 (1) ^8 the divisor grows smaller and smaller^ approaching 
 zero as a limit (the dividend beirig a finite constarit^^ the quotient 
 increases without limit. 
 
 (2) As the divisor increases without limit (the dividend being 
 a finite constant^ ^ the quotient approaches zero as a limit. 
 
 For the sake of brevity, (1) and (2) are often expressed 
 by the equations 
 
 - = 00 and — = 0, 
 
 CX) 
 
 respectively ; but the interpretation of these equations is as 
 stated in (1) and (2) above. 
 
 (ii) In the fraction -— , as x takes the successive 
 
 x—1 
 
 values 1.1, 1.01, 1.001, 1.0001, what limit is approached by 
 
 the numerator? by the denominator? by the value of the 
 
 fraction? Answer the same questions for — -, as x 
 
 approaches 2 as a limit. 
 
 These examples illustrate the fact that as a dividend and 
 its divisor each approach zero as a limit, the quotient may 
 approach any value whatever ; this is often expressed by 
 saying that 
 
 - is indeterminate. 
 
194] RATIO, PROPORTION, AND VARIATION 303 
 
 EXERCISE CXXIX 
 
 1. Which of the following quantities are constants and which 
 are variables : (1) the circumference of a growing orange ? (2) the 
 length of the shadow cast by a certain church steeple between 
 sunrise and sunset? (3) the length of the steeple itself? (4) the 
 time since the discovery of America ? (5) the interest earned by 
 an outstanding note ? (G) the principal of the note ? 
 
 2. A point P moves through half the distance AB {i.e., to P'), 
 then through half the remaining distance p pn pu 
 
 (i.e., to P"), then through half the remain- i i i 
 
 ing distance (i.e., to P"), and so on. Show ^ 
 
 that the distance from A to P is a variable whose limit is AB. 
 
 3. In Ex. 2, is the distance from P to J5 a constant or a 
 variable ? What is its limit ? Explain both answers. 
 
 4. If X takes in succession the values .6, .06, .006, .0006, •••, 
 what is its limit ? Why ? 
 
 5. What is the limit of the variable sum .6 + .06 -f- .006 -\ 
 
 (i.e., of the decimal .6666 ••-) ? Explain. 
 
 6. If r is any finite constant, trace the changes in the quotient 
 r/s as s passes through the values, 3, 1, ^, ^, -^j, •••; also as s 
 passes through the values 3, 9, 27, 81, •••. Is there a limit to 
 the quotient in the first case ? in the second ? Explain. 
 
 7. Translate into verbal language [cf . § 194 (i)] : 
 
 (1) ^ = 00; (2) ^=0. 
 
 8. As X approaches the limit 1, what limit does approach 
 
 X —1 
 
 in form f in value f Answer the same questions for the fractions 
 a; — 1 x^ — x 1 3 a^ — 2a; — 1 
 
 a^-1' a;-l' 
 
 ar 
 
 9. By means of your answers to the questions in Ex. 8, illus- 
 trate the fact that - is indeterminate in value. 
 
304 IlIGIl SCHOOL ALGEBRA [Cii. XIX 
 
 195. Direct and inverse variation; etc. Of two variables 
 which are so rehited that, during all their changes, their ratio 
 remains constant, each is said to vary (also to vary directly) 
 as the other. The symbol employed to denote variation is ^ ; 
 it stands for the words " varies as," and the expression acch 
 is read '•^ a varies as 5." 
 
 If a Qc ^, ^.g., if a : 6 = A;, a constant, then a = kb (why ?) ; 
 hence a variation statement may be converted into an 
 equation. 
 
 E.g., if a tank contains v cii. ft. of water, each cubic foot weighing 02.5 lb., 
 and if the total weight of the water is to lb., then : 
 
 (1) When V changes (as it must, for example, while the tank is filling), 
 w changes also. 
 
 (2) Since, no matter how the quantity of water changes, to = 62.5 -w, or 
 w : V = 62.d, therefore wcav; i.e., the weight of watisr varies as its volume. 
 
 One of two numbers is said to vary inversely as the other 
 if the ratio of the first to the reciprocal of the second is 
 constant. If a varies inversely as 5, then a ' h = k: let pupils 
 fully explain why. 
 
 Again, if x, y, and z are variables such that x = ki/z^ where 
 k is Si constant, then x is said to vary jointly as ^ and z ; and 
 
 if a; = -^, then x is said to vary directly as y and inversely as z. 
 
 z 
 
 E.g., the time required for a railway train to travel a given distance varies 
 inversely as the speed ; for if t, r, and d represent, respectively, the time, rate, 
 
 and distance, then t 'r = d, i.e., t: ~ =d, where d is constant. 
 
 r 
 
 Again, the cost of a railway journey varies jointly as its length and its 
 cost per mile ; while the number of posts required to build a certain fence 
 varies directly as the length of the fence, and inversely as the distance be- 
 tween the posts. 
 
 Note. It should be remarked in passing that such an expression as wccv 
 above (i.e., the weight of water varies as its volume) is merely an abbreviated 
 form of the proportion 
 
 w:w' =v:v', 
 
 wherein w and w' stand for the respective weights, and v and v' for the 
 volumes, of any two quantities of water. 
 
195] RATIO, PROPORTION, AND VARIATION 305 
 
 The theory of variation is, therefore, substantially included in that of 
 ratio and proportion, and the only reason for even defining the expressions 
 "varies as," "varies inversely as," etc., here, is that this convenient 
 phraseology is so well established in physics, chemistry, etc. 
 
 EXERCISE CXXX 
 
 1. Explain and illustrate the following statements : 
 
 (1) The interest earned by a certain principal varies as the time. 
 
 (2) The circumference of a circle varies as its radius. 
 
 2. State (1) and (2) of Ex. 1 as equations (cf. § 195), also as 
 proportions (cf. § 195, Note). 
 
 3. It xccy and if a? = 12 when y = 3, find the equation connect- 
 ing X and y ; also find x when y==7. 
 
 Solution. Since xccy, therefore x = ky where k is constant (why?) ; 
 moreover, when x = 12 and ?/ = 3, the equation x = ky gives k = 4. There- 
 fore, under the given conditions, x = 4y; hence, when y = 7, x = 28. 
 
 4. li aocb and if a = 89 when b = —3, find a when 6 = 2; also 
 when & = I ; also find b when a = — 65. 
 
 5. If ^ X -B and BccC, show that AccC. 
 Hint. Show that A = kC\ where k is some constant. 
 
 6. If m cc n and pccriy prove that m±pccn. 
 
 7. If 3 m^— 18 oc 2n + 1, and if m = 4 when n = 2, find m when 
 n = 23.5. 
 
 8. The area of a circle varies as the square of its radius. If 
 a circle whose radius is 10 ft. contains 314.16 sq. ft., find the 
 area of a circle whose radius is 5 ft. ; of one whose radius is 12 ft. 
 
 9. Find the radius of a circle whose area is twice that of a 
 circle 10 ft. in radius (cf. Ex. 8). 
 
 10. If X varies inversely as y, how is the value of x aifected if 
 y is doubled ? if ?/ is multiplied by 10 ? ii y is divided by — 6 ? 
 Explain. 
 
 11. Give three numerical examples of inverse variation. 
 
306 HIGH SCHOOL ALGEBRA [Cii. XIX 
 
 12. If X varies inversely as y, show that : 
 
 (1) xy = k (where k is constant). 
 
 (2) x' :x" = y" : y' (where x' and y', x" and y" are correspond- 
 ing vahies of the variables). 
 
 13. If x varies inversely as y, and if x = 4 when y = 2, find y 
 when x = — 8; when x = l^ ; when x = 2.5. 
 
 14. If x varies directly as y and inversely as z, and if a; = — 12 
 when 2/ = 2 and z = 7, find y when x = 2 and 2 = 3. 
 
 15. Solve Ex. 13 by drawing the graph of the equation con- 
 necting X and y (cf. § 141), and then measuring the ^/-coordinates 
 of the points whose respective .^-coordinates are — 8, 1^, and 2.5. 
 Also show, from the graph, that any change in x makes an oppo- 
 site change in y. 
 
 16. If the volume of a pyramid varies jointly as its base and 
 altitude, and if the volume is 20 cu. in. when the base is 12 sq. in. 
 and the altitude is 5 in., what is the altitude of the pyramid 
 whose base is 48 sq. in. and whose volume is 76 cu. in. ? 
 
 17. The distance (in feet) through which a body falls from a 
 position of rest, varies as the square of the time (in seconds) 
 during which it falls. If a body falls 257^ ft. in 4 sec, how far 
 will it fall in 5 sec. ? how far during the 5th second ? how far 
 during the 7th second ? 
 
 18. If the intensity of light varies inversely as the square of 
 the distance from the source of light, how much farther from a 
 lamp must a book, which is now 2 ft. away, be removed so as to 
 receive just one third as much light ? 
 
 19. The weight of a body comparatively near the earth's sur- 
 face varies inversely as the square of its distance from the earth's 
 center. Assuming that the radius of the earth is 4000 mi., find 
 the weight of a 4-lb. brick 2000 mi. from the earth's surface. 
 (Two solutions.) 
 
 20. The number of oscillations made by a pendulum in a given 
 time varies inversely as the square root of its length. If a pen- 
 dulum 39.1 inches long oscillates once a second, what is the 
 length of a pendulum that oscillates twice a second? 
 
CHAPTER XX 
 SERIES — THE PROGRESSIONS 
 
 196. Definitions. A series is a succession of numbers 
 which proceed according to some definite law. The num- 
 bers which constitute the series are called its terms. 
 
 E.g.^ in the series 1, 2, 3, 5, 8, 13, eacli term after the second is the sum 
 of the two preceding terms ; in the series 2, 6, 18, 54, 162, each term after 
 the first is 3 times the preceding term ; and in the series 1, 4, 9, 16, ••., 81, 
 each term is the square of the number of its place in the series. 
 
 A series which consists of an unlimited number of terms 
 is called an infinite series ; otherwise it is a finite series. 
 
 The present chapter considers only the simplest kinds of 
 series — the so-called "progressions." 
 
 I. ARITHMETICAL PROGRESSION 
 
 197. Definitions and notation. An arithmetical series, or 
 arithmetical progression (designated by A. P.), is a series 
 in which the difference found by subtracting a term from 
 the next following term is the same throughout the series. 
 This constant difference, whether positive or negative, inte- 
 gral or fractional, is known as the common difference of the 
 series. 
 
 E.g.^ the series 2, 5, 8, 11, 14, ••. is an A. P. whose common difference is 
 3. So, too, the series 18, 11, 4, — 3, — 10, is an A. P. whose common differ- 
 ence is — 7. 
 
 The elements of any given A. P. are the first term (desig- 
 nated by a), the last term (Z), the common difference (c?), 
 the number of terms (n), and the sum of all the terms (s). 
 
 Thus, in the series 2, 5, 8, •••, 32, the elements are « = 2, Z = 32, (?= 3, 
 11 = 11, s= 187. 
 
 307 
 
308 HIGH SCHOOL ALGEBRA [Ch. XX 
 
 EXERCISE CXXXI 
 
 1. Does a row of numbers written down at random constitute 
 a series ? Explain. 
 
 2. Show that 1, 7, 13, 19, 25 is an A. P. What are its ele- 
 ments ? 
 
 3. What is d in the A. P. 7, 11, 15, 19 ? Extend this series 
 four terms to the right ; also three terms to the left. 
 
 4. If the 1st, 3d, and 5th terms of an A. P. are 18, 24, and 30, 
 respectively, find d and write 8 consecutive terms of the series. 
 
 5. Write 10 consecutive terms of the series in which 19, 9, 
 and 4 are the 1st, 5th, and 7th terms, respectively. 
 
 6. What are the elements of the A. P. 5, 5 + 3, 5 + 6, 5 + 9, 
 5 + 12 ? How is any term of this series formed from the pre- 
 ceding term ? 
 
 7. Show that x,x+y, x-\-2y, x-{-3y, ••• is an A. P. What 
 is d in this series ? How many times must d be added to the 
 first term to make the 2d term ? to make the 3d term ? the 7th 
 term ? the 10th term ? the nth term ? 
 
 8. Show from the definition of an A. P. that such a series 
 may be written in the form 
 
 a, a+d, a-j-2d, a + 3d, '•-, I — 2d, l—d, I, 
 wherein a, d, and I represent, respectively, the first term, com- 
 mon difference, and last term. 
 
 198. Formulas. The elements of an A. P. are connected 
 by the two fundamental equations (formulas) numbered (1) 
 and (2) below. 
 
 Since each term of an A. P. may be derived by adding d 
 to the preceding term (cf. Exs. 6-8, above), therefore, if 
 I stands for the nth term 
 
 l = a-}-(n-l)d. (1) 
 
107-198] SERIES — THE PPiOGnESSIONS 309 
 
 Again, since the sum of the terms of an A. P. may be 
 written in each of the two following forms, 
 
 s = a-^ (a-\-d)-^(a-\-2d)-h--' + (I- 2 d) -h (I- d) -^ I 
 and s = I i- {I - d) -{- {I - 2 d) -\- •" + (a -^ 2 d) -{- {a -\- d) -^ a, 
 therefore, by adding these equations, term by term, 
 
 i.e.^ 2s = n(^a-\- Z), [n terms 
 
 whence * = -^-^ — ^ ; (2) 
 
 or, substituting the value of I from (1), 
 
 n [Za + (n-l)<n , 
 "-^ 2 
 
 Note. If any three of the five elements of an A. P. are given, the other 
 two can always be found from (1) and (2) above, because, in that case, the 
 two unknown elements will be connected by two independent equations. 
 
 Ex. 1. Find the sum of 8 terms of the A. P. — 3, — 1, 1, 3 •••. 
 
 Solution. Here a=— 3, d = 2, w = 8; 
 
 whence, from (1), Z = -3 + (8 - 1) 2 = 11, 
 
 and from (2), s = ^(-^ + ^^) = 32 ; 
 
 Li 
 
 i.e., the sum of 8 terms of the A. P. is 32. 
 
 EXERCISE CXXXII 
 
 Verify formulas (1) and (2) above for the following series : 
 
 2. 10, 13, 16, 19, 22, 25, 28. 
 
 3. 26, 19, 12, 5, -2,-9, -16, -23, -30. 
 
 4. _8,-5|, -3i, -1,H,3|,6. 
 
 5. By means of § 198 (1), find the 17th term of 7, 11, 15, ... ; 
 then by § 198 (2), and without writing all the terms, find the sum 
 of the first 17 terms of this series. 
 
 6. As in Ex. 5, find the 12th term of 1, 3.5, 6, 8.5, •..; also 
 the sum of the first 8 terms. 
 
310 • HIGH SCHOOL ALGEBRA [Ch. XX 
 
 Find the sum of : 
 
 7. Ten terms of 4, 11, 18, .... 
 
 8. Thirty terms of - 2, - 0.5, 1, 2.5, .... 
 
 9. Nineteen terms of 2, 5, 8, •.-. 
 
 10. k terms of 2, 5, 8, •••. 
 
 11. n terms of 5, 5 -f A', 5 + 2 A:, 5 -f 3 A;, • ... 
 
 12. t terms of li, 2 h, 3h, •••. 
 
 13. Find the sum of the even numbers from 2 to 100 inclusive. 
 Compare your result with that in Ex. 12 when h = 2 and t = 50. 
 
 14. How many strokes are made in a day (24 hours) by a clock 
 which strikes the hours only ? 
 
 15. Suppose that 50 eggs are placed in a row, each 2 yd. from 
 the next^ and a basket 2 yd. beyond the last egg ; how far would 
 a boy, starting at the basket, walk in picking up these eggs and 
 carrying them, one at a time, to the basket ? 
 
 16. If a body falls 16.1 feet during the first second, 3 times as 
 far during the next second, 5 times as far during the third second, 
 etc., how far will it fall daring the 8th second? how far during 
 the first 8 seconds ? 
 
 17. By means of § 198 (1) find n when a = 2, cZ = 4, 1 = 66', 
 also find s for this series. 
 
 18. If a = -10,d = 3, s = 35, findZandn. 
 
 Hint. Substitute in § 198 (1) and (2) and solve the resulting equations 
 for I and n (cf. § 198, Note). 
 
 19. If a = l, d = -|, 71 = 9, findZand.9. 
 
 20. If 1 = -^, n = lS, s = -45i, find a and d. 
 
 21. How many consecutive odd integers (beginning with 1) 
 must be added to give the sum 225 ? 441 ? (Cf. Ex. 18.) 
 
 22. If s = 112 and n = 7, determine the unknown elements in 
 the series ..«, 10, 13, 16, .-., and write the series. 
 
 23. If s, n, and d are given, find a and I; i.e., find a and I in 
 terms of s, n, and d (cf. Ex. 22). 
 
198-191)] SERIES—- THE PEOGliESSIONS 311 
 
 24. Find a and ?i in terms of d, I, and s. Make up and solve 
 eight other examples of this kind. 
 
 25. Show that an A. P. is fully determined when any three of 
 its elements are given. 
 
 26. If the 6tli and 11th terms of an A. P. are 17 and 32, 
 
 respectively, find the common difference, and also the sum of 
 
 the first 11 terms. 
 
 Hint. Since the 6tli term is 17, therefore 17 = a + 5 (Z. Similarly, 
 32 = a + 10 d 
 
 27. Show that if each term of an A. P. is multiplied (or 
 divided) by any given number, the resulting products (or 
 quotients) are themselves in arithmetical progression. 
 
 28. If each term of an A. P. is increased or diminished by any 
 given number, will the results be in arithmetical progression ? 
 Explain. 
 
 199. Arithmetical means. The two end terms of an A. P. 
 are called its extremes, while all the other terms are called 
 arithmetical means between these two. 
 
 E.g., in the A. P. 5, 9, 13, 17, 21, between the extremes 5 and 21 there 
 are 3 arithmetical means (viz., 9, 13, and 17). 
 
 In an A. P. of 3 terms the (one) arithmetical mean 
 between the extremes equals half their sum ; for if A is 
 the arithmetical mean between a and 5, then 
 
 A — a = h — A^ [definition of an A. P. 
 
 whence 4 =?-LA. % 
 
 Ex. 1. Find the arithmetical mean between 3 and 27. 
 Solution. The arithmetical mean between 3 and 27 
 
 = 3 + 27^ 15_ 
 
 2 
 
 Ex. 2. Insert 5 arithmetical means between 3 and 27. 
 
 Solution. In this series a = 3, 1 = 27, and (since there are to 
 be 5 means) n = 5-}-2 = 7; whence, from § 198 (1), d = 4, and 
 the required series is 3, 7, 11, 15, 19, 23, 27. 
 
312 HIGH SCHOOL ALGEBRA [Ch. XX 
 
 EXERCISE CXXXIII 
 
 3. Find the arithmetical mean between 14 and 9; between 
 -5 and 17; -3 and -4; fandf; - 2.75 and 11.4. 
 
 4. Insert 4 arithmetical means between 12 and 27. 
 
 5. Insert 15 arithmetical means between 19 and 131. 
 
 6. Insert 20 arithmetical means between 16 and — 40. 
 
 7. Insert 12 arithmetical means between — f and f . 
 
 8. Insert 7 arithmetical means between — .08 and — .0032. 
 
 9. If m arithmetical means are inserted between two given 
 numbers, such as a and b, show that the common difference of 
 the series thus formed is (6— a) -j- (m -f- 1). 
 
 10. What does the formula of Ex. 9 become when m = 1 ? Is 
 this consistent with the formula for A obtained in § 199 ? 
 
 11. Without actually finding the means asked for in Ex. 2, 
 find the sum of the series formed by inserting them. 
 
 12. Find three numbers in A. P. whose sum is 15, and the sum 
 of whose squares is 107. 
 
 Hint. Let x — y, x, and x -{- y represent the required numbers. 
 
 13. Find three numbers in A*. P. whose sum is 18 and whose 
 product is 202i. 
 
 14. The sum of the first seven terms of an A. P. is 105, and the 
 sum of the third and fifth terms is 10 times the first term. Find 
 the series. 
 
 15. The product of the extremes of an A. P. of three terms is 4 
 less than the square of the mean, and the sum of the series is 24. 
 Find the series. 
 
 16. The sum of four numbers in A. P. is 14, and the product 
 of the means is 12. What are the numbers ? 
 
 Hint. Let x — 3y,x—y,x + y,x-\-Sy represent the series. 
 
 17. The sum of an A. P. of five terms is 15, and the product of 
 the extremes is 3 less than the product of the second and fourth 
 terms. Find the series. 
 
 18. How many arithmetical means must be inserted between 4 
 and 25 so that the sum of the series may be 116 ? 
 
199-200] SERIES— THE PROGRESSIONS 813 
 
 19. A number, expressed by three digits in A. P., equals 30.4 
 times the siim of its digits ; but if 9 is added to the number, the 
 units' and tens' digits will be interchanged. Find the number. 
 
 20. In the series 1, 3, 5, •• •, what is the n\h. term ? Prove that 
 the sum of the first n odd numbers, beginning with 1, is nl 
 
 II. GEOMETRIC PROGRESSION 
 
 200. Definitions and notation. A geometric series, or geo- 
 metrical progression (designated by G. P.), is a series in 
 which the quotient of each term (after the first) divided by 
 the next preceding term is the same throughout the series. 
 This constant quotient is called the common ratio, or simply 
 the ratio, of the series. 
 
 E.g.^ the numbers 2, 6, 18, 54, ••• form a geometric series whose ratio is 3 ; 
 while |, — 1, f , - I, V", ••• is a G. P. whose ratio is - |. 
 
 The elements of any given G. P. are the first term (a), 
 the last term (Z), the number of terms (n)^ the ratio (r), 
 and the sum of all terms (s). 
 
 E.g., in the G. P. 2, -6, 18, -54, 162, -486, 1458, a = 2, Z = 1458, 
 n = 7, r = - 3, and s = 1094. 
 
 EXERCISE CXXXIV 
 
 Which of the following are geometric series ? Explain. 
 
 1. 7, 21, 63, 189, 567. 
 
 2. 1, 4, 16, 64, 192, 576. 
 
 3. -6, 12, -24, 48, -96, 192, -384. 
 
 4. What are the elements of the G. P. in Ex. 3 ? Extend this 
 series two terms to the right, also five terms to the left. 
 
 5. If a is the first term of a G. P., and r the ratio, what is the 
 second term? the third? the fourth? the fifth? the fourteenth? 
 the twenty-third ? the nth ? Explain. 
 
 6. Are x, xy, xy\ xf, -■, and p'q-', 2A PV, Pg\ Q% P~V geo- 
 metric series ? If so, name a and r in each. 
 
314 RIQH SCHOOL ALGP.nnA tCn. XX 
 
 7. What is r in the G. P. 2, |, |, ••• ? in the G. P. 21, 7, J, ••• ? 
 Are these two series merely parts of the same series ? Explain. 
 
 8. If the 1st, 3d, and 6th terms of a G. P. are 12, 3, and |, 
 respectively, find r and then write down the first 8 terms of the 
 series (cf. Ex. 26, p. 311). 
 
 201. Formulas. The elements of a G. P. are connected by 
 the two fundamental equations shown below (cf. § 198). 
 
 Since each term of a G. P. may be obtained by multiply- 
 ing the preceding term by r (cf. Exs. 5 and 6, p. 313), 
 therefore, if I represents the n{\\ term of such a series, then 
 
 l = ar-\ (1) 
 
 Again, if s represents the sum of a G. P. of n terms, then 
 sz=a-{- ar -\- ar^ -\- ai^ + • • • + «r"~^ + ar^~^ ; 
 whence, multiplying by r, we obtain 
 
 sr = ar -^ ar^ + a7^ -\- h ar^~^ + ar", 
 
 and, therefore, by subtracting the second of these equations 
 from the first, member from member, 
 
 s — sr = a— ar" ; 
 
 hence ^~~/ * (^) 
 
 EXERCISE CXXXV 
 
 1. By means of § 201 (1) write down the 6th term of the geo- 
 metric series 4, 12, 36, • • • . 
 
 2. As in Ex. 1, write the 7th term of the G. P. 3, 6, 12, .•• ; 
 then by § 201 (2) find the sum of the first 7 terms of the series. 
 Check results by writing down and adding these first 7 terms. 
 
 3. Find the 8th term of 24, 12, 6, .••; also the 11th term of 
 TTJinr? ~ Tu^y ttj"? *"• 
 
 Find the sum of the following series: 
 
 4. 1, 2, 4, . • • , to 10 terms. 7. 1, — 2 a;, 4 a^, • • • , to 7 terms. 
 
 5. 1, 1.5, 2.25, •••, to 6 terms. 8. -5, -2, -.8, •••, to A: terms. 
 
 6. 2, — I, I, ••• , to 7 terms. 9. x, x~^, x~^j ••• ,to n terms. 
 
200-201] SERIES — THE PIWGEESSIONS 315 
 
 10. Find the G. P. whose 3d term is 18, and whose 8th term 
 is 4374. 
 
 Hint. By § 201 (1), 18 = ar^, and 4374 = ar'' ; hence, dividing the second 
 of these equations by the first, 243 = r^, i.e., r = 3. 
 
 11. Find the G. P. whose oth term is |- and whose 9th term is 
 i||. Also find the sum of the first 9 terms of this series. 
 
 12. Show that § 201 (2) may be written in each of the follow- 
 ing forms : 
 
 g (1 — r") a — rl rl — a ar" — a -, __a ar'' 
 
 1 — r ' 1 — r' r— 1 ' r — 1 ' 1—r 1 — r 
 
 13. By actually dividing a(l — r") by 1 — r, verify the correct- 
 ness of § 201 (2). 
 
 14. If a = 4, I — 972, and n = C, find r, and write the series. 
 
 15. If 71 = 12, r = - 2, and 6- = - 1365, find a and I 
 
 16. Find the sum of a G. P. of 6 terms whose ratio is f and 
 whose last term is 32. 
 
 17. If r, n, and I are given, find a and s ; that is, find a and s 
 in terms of r, n, and I (cf. Ex. 16). 
 
 18. Find a and I in terms of n, r, and s; also r and s in terms 
 of a, n, and /. 
 
 19. Is a G. P. fully determined when any three of its elements 
 are given (cf. § 198, Note)? 
 
 20. Three numbers whose product is 216 form a G. P., and the 
 sum of their squares is 189. What are the numbers ? 
 
 Hint. Let -, a, and ar represent the required numbers, 
 r 
 
 21. Divide 38 into three parts which are in G. P., and such 
 that when 1, 2, and 1 are added to these parts, respectively, the 
 results shall be in A. P. 
 
 22. Find an A. P whose first term is 3, and such that its 2d, 
 4th, and 8th terms shall be in G. P. 
 
 23. If the population of the United States was 76,000,000 in 
 1900, and if it doubles itself every 25 years, what will it be in 
 the year 2000? 
 
 HIGH SCH. ALG. — 21 
 
316 HIGH SCHOOL ALGEBRA [Cn. XX 
 
 24. Thinking $1 per bushel too high a price to pay for wheat, 
 a man bought 10 bu., paying 3 cents for the first bushel, 6 cents 
 for the second, 12 cents for the third, and so on. What did the 
 tenth bushel cost him, and what was the average price per 
 bushel ? 
 
 25. Show that the amount of $ ^ for n years at a given rate 
 {R) of compound interest is the nth term of a G. P. whose first 
 term is A and whose ratio is (1 + i?). 
 
 26. A gentleman loaned a friend $250 at the beginning of 
 each year for 4 years. If money is worth 5 % compound interest, 
 how much should be paid back to him at the end of the fourth 
 year to discharge the obligation ? 
 
 27. The president of a charity organization starts a " letter 
 chain " by writing three letters, each numbered 1, requesting each 
 recipient to remit 10 cents to the society, and. also to send out 
 3 other letters, each numbered 2, with a similar request, the chain 
 to close with the letters numbered 20. Should every recipient 
 comply, how much money would be realized ? 
 
 202. Infinite decreasing geometric series. A decreasing 
 G. P. is one in which r < 1, numerically, and in which, 
 therefore, the terms grow smaller and smaller as we pass 
 from left to right in the series. Thus, 6, 3, |, f, f, ••• is a 
 decreasing G. P. 
 
 If we let s„ represent the sum of the first n terms of this 
 series, then 
 
 6 _ 6(i)^ |-g^ j2, p. 315 
 
 l-\ 1-1 
 
 = 12-6(1)-^ 
 
 and, since 6(J)"~^ approaches zero as a limit when n increases 
 without limit [§ 194 (i)], therefore s„ approaches 12 as a 
 limit when n becomes infinite ; this is often expressed thus : 
 
 8 =12. 
 
201-202] SEBIES — THE PROGBESSIONS 317 
 
 Similarly, whenever r < 1, numerically, and 7^ = oo, the 
 
 formula 
 
 _ a _ gy" 
 
 becomes s = ; 
 
 * 1-r 
 
 since r" approaches the limit zero as n becomes infinite. 
 
 EXERCISE CXXXVI 
 
 1. If in Ex. 2, p. 303, AB = 2 ft., show that the successive 
 distances traversed, when expressed in inches, form the G. P. 
 
 12fi.S3 3 3 3 3 ... 
 
 ±^, D, O, 2"? 4? 8^? T6J 3T' ' 
 
 2. By § 201 (2), find s^ for the series in Ex. 1 ; also find Sg, Sg, 
 ^10, and s„. 
 
 3. From Ex. 2, p. 303, show that in the series of Ex. 1 above 
 s„ < 24, no matter how large 7i may be. How near to 24 will s„ 
 approach as n is made larger and larger ? Explain. Also find 
 s^ by § 202. 
 
 4. Find s^ for the series 0.6, 0.06, 0.006, •••, and thus show 
 that 0.6 {i.e., 0.666 •••) equals f ; similarly, show that O.iS (i.e., 
 0.151515...) equals 3%. 
 
 Find s^ for each of the following series : 
 
 5. 1, -hh'-- 9- ^•^- 13. l,k,k',^" 
 
 6. l,i,i,.... 10. 0.12. (wherein A: <1). 
 
 7. 1,-1, A,-. 11- 1.362. 14. ^,-,-,, 
 
 1 1^ 
 
 8. V2, 1, V(X5, .... 12. 4.7523. (wherein a; > 1). 
 
 15. If, in a G. P., r is positive and less than 0.5, show that any 
 given term of the series is greater than the sum of all the terms 
 that follow it. 
 
 16. A point traversing a straight line moves in any given 
 second 75 % as far as in the preceding second ; if it moves 24 ft. 
 in the first second, how far will it move before coming to rest ? 
 
318 HIGH SCHOOL ALGEBRA [Ch. XX 
 
 17. If a sled runs 80 ft. during the first second after reaching 
 the bottom of a hill, and if its distance decreases 20 % during 
 each second thereafter, how far will it run on the level before 
 coming to rest? 
 
 18. If a ball, on being dropped from a tower window 100 ft. 
 above the pavement, rebounds 40 ft., then falls and rebounds 
 16 ft., and so on, how far will it move before coming to rest ? 
 
 19. Although s^ for the series ^, i, ^, ••• is 1, show that for the 
 series -J, J, J, ^, •••, 6\ grows larger beyond all bounds, by suffi- 
 ciently increasing 7i. 
 
 Suggestion. Write the series thus : s« = I +(i + i) + (^ + i + 7 + i) -i , 
 
 putting 8 terms in the next group, 16 in the next, and so on, and show that 
 each group is greater than |. 
 
 203. Geometric means. The two end terms of a finite G. P. 
 
 are called its extremes, while all the other terms are called 
 
 geometric means between these two. 
 
 E.g., in the series f, i, ^, |, and ^, the extremes are | and ^4^, and |, ^, 
 and I are geometric means between them. 
 
 In a G. P. of three terms, the (one) geometric mean be- 
 tween the extremes equals the square root of their product ; 
 for if Gr is the geometric mean between a and 5, then 
 
 — = -77, [definition of a G. P. 
 
 a Gr 
 
 whence G- = ± Vab, 
 
 Ex. 1. Find the geometric mean between 6 and 24; also between 
 10 and 8. 
 
 Solution. The geometric mean between 6 and 24 is ± V6 • 24, 
 i.e. ±12; and the geometric mean between 10 and 8 is ± VlO • 8, 
 i.e. ±4V5. 
 
 Ex. 2. Insert four geometric means between f and — -^. 
 
 Solution. In this series, a = i Z = — y, and (since four means 
 are to be inserted) n = 4 + 2 = 6 ; hence by § 201 (1), - -V- = | • ?-^, 
 whence ?*^ = — 2^^- and r = — |. Therefore the required series is 
 4 _2 1 _4. 9 __2 
 
 3? 
 
 1. -I. I. -¥• 
 
202-204] SERIES— THE PROGRESSIONS 319 
 
 EXERCISE CXXXVII 
 
 Find the geometric mean between the following number-pairs : 
 
 3. 18,8. 5. }, -ff 7. (a + b),(a-by, 
 
 4. 5,20. 6. 0.5,3.5. 8. 2x-3, {x-{-:iy. 
 9. Insert 4 geometric means between 3 and 96. 
 
 10. Insert 3 geometric means between 2 and ^ (two answers). 
 
 11. Insert 6 geometric means between — 3125 x^^ and ^^y • 
 
 12. If m geometric means are inserted between a and b, show 
 that r for the series thus formed is "''^b -^ a. 
 
 13. What does the formula of Ex. 12 become when m = l? 
 Is this consistent with the formula for G obtained in § 203 ? 
 
 14. Two numbers differ by 24, and their arithmetical mean 
 exceeds their geometric mean by 6. Find the numbers. 
 
 204. Harmonic series. An harmonic series, or harmonic 
 progression (H. P.), is a series of numbers whose reciprocals 
 form an A. P. A supposed H. P. may therefore be tested, 
 and problems in H. P. be solved, by an appeal to our 
 knowledge of A. P. 
 
 E.g., the numbers f, ^, -^j, ^'j, ••• form an H. P. because their reciprocals, 
 viz., I, 4, J^, -2/, ..., form an A. P. 
 
 Again, if we were asked to extend the H. P. f, \, y\, f^, ••• one or more 
 terms toward the right, we should need merely to form the corresponding 
 A. P., viz., I, 4, -^5% ■^, ..-, extend it as required (cf. Ex. 3, p. 308), and 
 then write the reciprocals of its terms. 
 
 EXERCISE CXXXVIII 
 
 1. If the 6th term of an H. P. is ^, and the 17th term is ^, 
 
 find the 37th term. 
 
 Hint. First find the 37th term in an A. P. whose Oth and 17th terms 
 are 3 and ^, respectively. 
 
 2. Insert 5 harmonic means between 2 and — 3. 
 
320 HIGH SCHOOL ALGEBRA [Ch. XX 
 
 3. Assuming x to be the harmonic mean between a and 6, show 
 
 that = , and hence that x = — ^— • 
 
 X a b X a-^b 
 
 4. The arithmetical mean between two numbers is 5, and their 
 harmonic mean is 3.2. What are the numbers? 
 
 5. The difference between two numbers is 2, and their arith- 
 metical mean exceeds their harmonic mean by ^. Find the 
 numbers. 
 
 6. Given (b — a): (c — b) = a: x, prove that x equals a, b, or c, 
 according as a, b, and c form an A. P., a G. P., or an H. P. 
 
 7. If a and b are two unequal positive numbers, and ^ is their 
 arithmetical mean, G their geometric mean, and .H their harmonic 
 mean, show that : (1) A> G>Hj and (2) A:G=G: H. 
 
CHAPTER XXI 
 MATHEMATICAL INDUCTION — BINOMIAL THEOREM 
 
 205. Proof by induction. An elegant and powerful form 
 of proof, and one that is very useful in many branches of 
 mathematics, is what is known as "proof by induction." 
 
 To illustrate : suppose it to have been found by trial that 
 a; — ^ is a factor of x^ — ^^, oi^ — «/^ and a;* — ^*, and that we 
 wish to know whether it is a factor of x" — y^^ x^ — y^^ ••• 
 also. Actual trial with any one of these, say ^ — y^, would 
 show that it is exactly divisible hj x — y, but besides being 
 somewhat tedious, this division gives no information as to 
 whether a; — ^ is or is not a factor of x^ — y'^, ••• also ; each 
 successful trial increases the probability of the success of the 
 next, but it proves nothing beyond the single case tried. 
 
 That X — y is a factor of x^ — ^™, for every positive inte- 
 gral value of 7i, may be shown as follows : 
 
 Since a^^ - ^» = x(x''-^ - y''-^) + y""'^ (x - y), 
 
 therefore, if a; — ^ is a factor of x^~^ — y^~\ then it is a fac- 
 tor of the second member of this equation, and therefore of 
 x^—y^ also (why ?) ; i.e.^ if x — y is a factor of the differ- 
 ence of any two like integral powers of x and y, then it is a 
 factor of the difference of the next higher powers also. 
 
 But since, by actual trial, a; — ^ is already known to be a 
 factor of a;* — y^^ therefore, by what has just been proved, it 
 is a factor of a:^ — ^^ also ; again, since it is now known to be 
 a factor of a^^ — ^, therefore it is a factor oioc^ — y^ -, and so 
 on without end : ^.e., x — y is a factor of ot^—y"^ for every 
 positive integral value of n, 
 
 321 
 
322 BIGH SCHOOL ALGEBRA [Ch. XXI 
 
 The proof just given is an example of what is known as a 
 proof by mathematical induction ; such a proof consists essen- 
 tially of two steps, viz. : 
 
 (1) Showing hy trial or otherwise the correctness of a given 
 law when applied to one or more particular cases^ and 
 
 (2) Proving that if this law is true for any given case, then 
 it is true for the next higher case also. 
 
 From (1) and (2) it then follows that the proposition 
 under consideration is true for all like cases.* 
 
 EXERCISE CXXXIX 
 
 1. Prove that the sum of the first n odd integers is n^. 
 
 Solution. (1) By trial it is found that 1 + 3 = 2^ and 1 + 3 + 5 = 32. 
 
 (2) Moreover, if i + 3 + 5 + . . . + (2 A: - 1) = A;^, 
 
 then, by adding the next odd integer to each member, we obtain 
 
 1+ 3 + 5 ... + (2 A: - 1) + (2 A; + 1) =: ^•2 + (2 A: + 1) = (A; + 1)2 ; 
 i.e., if the law in question is true for the first k odd integers, then it is true 
 for the first k + I odd integers also. 
 
 But, by actual trial, this law is known to be true for the first 3 odd inte- 
 gers, hence it is true for the first 4 ; and, since it is now known to be true 
 for the first 4, therefore it is true for the first 6 ; and so on without end : 
 hence the sum of any number of consecutive odd integers, beginning with 1, 
 equals the square of their number. 
 
 By mathematical induction prove that : 
 
 2. 14-2 + 3+.. -+71 = 1.71(71 + 1). 
 
 3. 2 + 4 + 6+.-. + 27i = 7i(7i+l). 
 
 4. 12 + 2^ + 32+.. . + 7l2 = ^n(71 + l)(271 + l). 
 
 * The student should carefully distinguish between mathematical induc- 
 tion, as here defined, and what is known as inductive reasoning in the 
 natural sciences. A proof by mathematical induction is, from its very nature, 
 absolutely conclusive. On the other hand, the inductive method in physics, 
 chemistry, etc., consists in formulating a statement of a law which will fit 
 the particular cases that are known, and regarding it as a laio only so long 
 as it is not contradicted by other facts not previously taken into account. 
 From the nature of the case step (2) above cannot be applied in physics, etc. 
 
205-206] MATHEMATICAL INDUCTION 323 
 
 5. l« + 2« + 3«+"-+^' = i-n2(n + l)2 = (l+2 + 3+--+ri)'. 
 
 6. A + A4-A+..-+ ' 
 
 1.2 2.3 3-4 n(n-\-l) n + 1 
 
 7. 1.2 4-2.3 + 3.4+...+w(n + l) = in(n + l)(w.-f-2). 
 
 8. a + ar + ar^-] f- ar^'-i = ^^i^— 1^ . 
 
 1 — r 
 
 9. a;" — ?/'' is divisible by ic + 2/ when 71 is even. 
 
 10. Having established (1) and (2) in the inductive proof of 
 any law, show the generality of the law by showing that there 
 can be no Jirst exception, and therefore no exception whatever. 
 
 206. The binomial theorem. The method of induction 
 furnishes a convenient proof of what is known as the bino- 
 mial theorem; this theorem, which was presented without 
 formal proof in § 112, may be symbolically stated thus : 
 
 wherein x-\-y represents any binomial whatever, and n is 
 any positive integer. 
 
 To prove this theorem by mathematical induction, observe 
 first that it is correct when n = % for it then becomes 
 
 2 ^ • 1 
 
 (x + ?/)2 = a;2 _|_ ^^ _^ rL_^ a;V ; i'e.,{x -^ yy^ = x^ -\-1xy -^ ?/2, 
 
 which agrees with the result of actual multiplication. 
 
 Again, if (1) is true for any particular value of n^ say for 
 n = k^ i.e., if 
 
 {X -f yy = x^-\-\ x'-^y + ^^f=^ x^'Y 
 
 + ^^^-y-^> .-¥+-, (2) 
 
324 HIGH SCHOOL ALGEBRA [Ch. XXI 
 
 then, on multiplying each member of (2) by x-\-y^ it be- 
 comes 
 
 {X + yy^^ = x'+^ + J x'y + ^^^"^^ x'-y 
 
 1.2-3 "^ 
 
 which is of precisely the same form as (2), merely having 
 k-\-l wherever (2) has k. Moreover, (3) is obtained from 
 (2) by actual multiplication, and is therefore true if (2) is 
 true ; hence, if the theorem is true when the exponent has any 
 particular value (sa}^ A;), then it is also true when the exponent 
 has the next higher value. 
 
 But, by actual multiplication, the theorem is known to be 
 true when 7i= 2, hence, by what has just been proved, it is 
 true when ?i = 3 ; again, since it is now known to be true 
 when n = 3, therefore it is true when n = 4 ; and so on with- 
 out end : hence the theorem is true for every positive inte- 
 gral exponent, which was to be proved. 
 
 EXERCISE CXL 
 
 1. In the expansion of (x -f- ?/)", what is the exponent of y in 
 the 2d term? in the 3d term? in the 4th term? in the i2th 
 term ? in the rth term ? What is the sum of the exponents of 
 X and y in each term? 
 
20r,-207] BINOMIAL THEOREM 325 
 
 2. In the expansion of (x + ?/)** what is the largest factor in 
 the denominator of the 3d term ? of the 4th term ? of the 10th 
 term ? of the rth term ? In any given term, how does this factor 
 compare with the exponent of y? 
 
 3. In the expansion of (aj-f?/)'*, what is subtracted from n in 
 the last factor of the numerator in the 3d term ? in the 4th term ? 
 in the 5th term ? in the 9th term ? in the rth term ? 
 
 4. Based upon your answers to Exs. 1-3, write down the 6th 
 term of (x + yy. Also write the 10th term ; the 17th term ; and 
 the ?-th term. 
 
 207. Binomial theorem continued. Strictly speaking, all 
 that was really proved in § 200 is that, for every positive 
 integral value of the exponent, the first four terms of the 
 expansion follow the law expressed by (1) ; that all the 
 terms follow this law will now be shown. 
 
 In multiplying (2) of § 206 by a; + ?^, the 2d term of the 
 product (3) is x times the 2d term plus i/ times the 1st term 
 of (2) ; so, too, the 10th term of (3) would be found by 
 adding x times the 10th term to ?/ times the 9th term of (2), 
 and the rth term of (3) by adding x times the rth term to 
 1/ times the (r — l)th term of (2). 
 
 But the (r — l)th and the rth terms of (2) are, respec- 
 tively, 
 
 1.2.3....(r-2) ^ 
 
 and K^-l)(^-2)...(^-r+3)(y^-r + 2) ,_,^, , . 
 1.2.3. •(r-2Xr-l) ^ ' 
 
 therefore the rth term of (3) is 
 
 ' k(k - l)(Jc - 2) ... (k - r -\- ^) 
 1.2.3. ...(r-2) 
 
 ^(^-l)(^-2)...(^-r + 3)(^-r + 2) 1 ,_,^, ^ 
 1.2.3. ...(r-2)(r-l) J ^ ' 
 
 (k+l}k(k-l) ... (^-r+ 3) ,_,^, 1 
 1.2.3. ...(r-1) "^ ^ ' 
 
326 IIIGII SCHOOL ALGEBRA [Cn. XXI 
 
 wliicli conforms to the law for the rth term expressed by (1) 
 of § 206. Hence the rth term, i.e.^ every term, in (3) con- 
 forms to the law expressed by (1), which was to be proved. 
 
 EXERCISE CXLI 
 
 1. Write down the expansion of (a+by-, also of {p — qf. 
 Explain why the alternate terms in the expansion of (^ — qf are 
 negative. 
 
 2. Write down the 1st, 2d, 3d, and 8th terms of {x + yf^. 
 
 3. Write down the 4th and 7th terms of {a — xy\ 
 
 4. How many terms are there in the expansion of (x+yy^? 
 Write down the first three, and also the last three terms of this 
 expansion, and compare their coefficients. 
 
 5. Write down the coefficient of the term containing ay, in 
 the expansion of (a — yy^. 
 
 6. Expand (3 a^ - 2 xff ; compare Ex. 2, § 57. 
 
 7. Write down the 4th and 9th terms of (f ic — | ?/)". 
 
 8. How many terms are there in [a; ) ? Write down the 
 
 \ xj _ 
 
 10th term. Also write down the 5th term of (J^-^J'^J. 
 
 9. Write down the term of (3 a^-^- 2 x'y, i.e., of (xy(S a^-2)^ 
 which contains a^^. 
 
 10. Write down the term of f a^ ) which contains a". 
 
 3 a, 
 
 11. Expand (a^ + 3 a^x'^y, and write the result with positive 
 exponents. 
 
 12. Expand (l — x-{- of)* by means of the binomial theorem 
 (cf. Exs. 40-41, p. 176). 
 
 13. By applying the law expressed in (1) of § 206 show that 
 the coefficient of the (n + l)th term of (x -f yy is 1 ; also show 
 that the coefficient of every term thereafter contains a zero factor, 
 and hence that {x+yy contains only n-\-l terms. 
 
207-208] MATHEMATICAL INDUCTION^ ETC, 327 
 
 14. Show that the sum of the binomial coefficients, ^.e., of 1, 
 71 n(n-l) n(n-l)(n-2) ■ o,, 
 
 r 2~~' 1.2.3 '••'^'^- 
 
 Hint. After expanding (a; -f y)**, let cc = y = 1. 
 
 15. Show that the sum of the even coefficients (i.e., the 2d, 
 4th, . . •) in Ex. 14 equals the sum of the odd coefficients, and that 
 each sum is 2"~\ 
 
 Hint. In (x + 2/)** let x = 1 and y =— 1. 
 
 16. Show that the coefficient of the rth term in (x 4-?/)" may be 
 
 obtained by multiplying that of the (r— l)th term by ^~^"'~ , 
 
 ?' — 1 
 and thus show that the binomial coefficients increase numerically 
 in going from term to term toward the center. 
 
 17. Show that the coefficient of the rth term is numerically 
 greater than that of the (r — l)th term so long as r < ^ (n 4- 3) ; 
 and thus write down the term whose coefficient is greatest in the 
 expansion of (x + 3/)" ; and also in (x + ijy^. 
 
 208. Binomial theorem extended. It may be remarked in 
 passing that the binomial theorem (§ 206), which has thus 
 far been restricted to the case where the exponent is a posi- 
 tive integer, is greatly extended in Higher Algebra, where 
 it is shown that under certain restrictions it admits negative 
 and fractional exponents also. Although the proof of this 
 fact is beyond the limits of this book, its correctness may be 
 assumed in the following exercises. 
 
 EXERCISE CXLII 
 Using the binomial theorem, write the first 5 terms of : 
 
 1. (a;4-2/)i 3. (a-c)i 5. {l-as^^. 
 
 2. (l + w)i 4. (a-^b)-\ 6. (2m-Jc)-K 
 
 7. Write the 6th term of (3 r — sy^ ; also the 5th term of 
 {\-3x)\ 
 
 8. Show that in such cases as the above the binomial theorem 
 leads to infinite series (cf. Ex. 13, p. 326). , 
 
328 HIGH SCHOOL ALGEBRA [Ch. XXI 
 
 9. Expand (1 — x)~'^ to 8 terms by the binomial theorem and 
 compare the result with the first 8 terms of the quotient 
 1^(1-0;). 
 
 10. Show that, when expanded by the binomial theorem and 
 
 simplified, (25 + 1)^ = 5 + ^^ - wo^ + sirio^ > compare this 
 
 result with VW as found by the usual method. 
 
 11. By expanding (9 — 2)^, find an approximate value of V7; 
 similarly, find an approximate value of VM (i.e., V27 + 4), and 
 of ^40 (i.e., ^^2 + 8). 
 
 209. The square of a polynomial. In § bQ it was pointed 
 out that, by actual multiplication, the square of a polyno- 
 mial consisting of 3, 4, or 5 terms equals the sum of the 
 squares of all the terms of the polynomial, plus twice the 
 product of each term by all those that follow it. It will 
 now be shown that if this theorem is true for polynomials of 
 n terms, then it is also true for those of w + 1 terms ; and 
 from this it will follow, as in § 205, that it is true for poly- 
 nomials of any finite number of terms whatever, since it is 
 already known to be true for polynomials of five terms. 
 
 Let a+h-[- c-\- -•■ -\- p -[- qhQ 2i polynomial of n terms, and 
 
 let (a+h-\-c-\- ••• +Jt?4-g)2= «2 + h^-\ \- q^ -\- 2 ah -{■ 2 ac+'-' 
 
 -\-1aq-\-1hc-\ h 2^g + [-2jt?g. 
 
 In this identity replace a everywhere by ic + ^ ; then the 
 number of terms in the polynomial in the first member will 
 become n + 1, and the second member will still consist of 
 the sum of the squares of all the terms of the polynomial, 
 plus twice the product of each term by all those that follow 
 it (the student should work this out in detail) ; therefore, 
 if the theorem is true for polynomials of n terms, then it is 
 also true for those of n + 1 terms, which was to be proved. 
 
CHAPTER XXII 
 LOGARITHMS 
 
 210. Introduction. Early in tlie seventeenth century, two 
 British mathematicians, Lord Napier and Henry Briggs, con- 
 ceived the idea of expressing all real positive numbers as 
 powers of 10,* arranging the exponents of these powers in a 
 table for convenient reference, and then employing this table 
 to simplify certain arithmetical computations, especially 
 multiplication. 
 
 E.g., to find the product of 3.578, 7.986, and 48.67, we find 
 from the table that 
 
 3.578 = 10°^^, 7.986 = 10«-^23^ and 48.67 = lO^^^^^ 
 whence 3.578 x 7.986 x 48.67 = 10«^^ x lO"^^^ x lO^-^^ 
 
 __ -j^QO.5536+0.9023+1.6873 Tfi 3Q 
 
 we now find from the table that 
 
 10«-i^ = 1390.6, 
 whence 3.578 x 7.986 x 48.67 = 1390.6. 
 
 Thus, by performing an addition (of the exponents), we have 
 found the product of the given numbers. 
 
 Other advantages of such a table of exponents (loga- 
 rithms) will be shown later (§ 218) ; some necessary defini- 
 tions and principles must now be given. 
 
 211. Definitions. The logarithm of a number (iV) to any 
 given base (5) is the exponent (x) of the power to which 
 this base must be raised to equal the given number. 
 
 * That it is possible to do this, either exactly or to any required degree of 
 approximation, will be assumed in this chapter. 
 
 329 
 
330 HIGH SCHOOL ALGEBRA [Ch. XXII 
 
 The logarithm of iV to the base b is usually written 
 logjiV; and the two statements 
 
 iV=5^ and log(,N=x 
 are, therefore, only different ways of saying the same thing. 
 
 E.g., •.•2^ = 8, .•.log28 = 3; •.•3^ = 243, .•.log8243 = 5 
 and •.• 10i-^'3 = 48.67, .-. logjo 48.67 = 1.6873. 
 
 EXERCISE CXLIII 
 
 1. rrom the equation 3^* = 81, find logg 81 . 
 Translate into logarithmic equations (cf. Ex. 1) : 
 
 2. 4^=64. 5. 2^ = 32. 8. 10« = 1. 
 
 3. 9^ = 81. 6. (|)' = ^V 9- 10-' = .001. 
 
 4. 10-* = 1000. 7. 2-^ = ^2- 10- (i)"' = 125. 
 Express the following statements in the exponent notation, 
 
 and then verify the correctness of each : 
 
 11. log7 49 = 2. 14. Iogiol0 = l. 17. log3i = -2. 
 
 12. log2l6 = 4. 15. logiol = 0. 18. logio.0001 = -4. 
 
 13. log.5 .125 = 3. 16. Iogiol0000 = 4. 19. loga256 = -8. 
 
 20. Find the value of the following logarithms : logg 27 ; 
 log2 64; log_8 64; log_6(-216); log4l; logio.l; log.ilO; 
 
 21. Between what two consecutive integers does each of the 
 following logarithms lie: logio83; logio2224; logio4; logio.007; 
 logio .1256 ? Explain your answers. 
 
 22. May the base of a set of logarithms be fractional ? nega- 
 tive ? May a logarithm itself be fractional ? negative ? May 
 negative numbers have logarithms ? Illustrate your answers. 
 
 212. Principles of logarithms. Since logarithms are expo- 
 nents (§ 211), therefore the principles of logarithms are 
 easily obtained from those governing exponents (§§ 171-175). 
 
211-212] LOGARITHMS 331 
 
 Principle 1. TJie logarithm of 1 to any base is 0, and the 
 logarithm of the base itself is 1 ; i.e.^ 
 
 logj 1 = and log^ 5 = 1. 
 The correctness of tliis principle follows at once from the 
 definition of a logarithm (§ 211), and from the fact that 
 
 Z>o=l and 51 = 6. [§§ 173, 9 
 
 Principle 2. The logarithm of a product equals the sum 
 of the logarithms of the factors ; i.e.^ 
 
 log, (M/if) =log, M -{-log, /if. 
 
 For, if M= b"^ and iV^= ^>^ 
 
 then M]sr= b^'^\ [•/ b'' - by = b'^+y 
 
 whence log<. {MN) =x-\-y = log^ M-\- log^ N. 
 
 Similarly, log (MZVP • • •) = log^ M+ log^, iY+ log, P + • • •• 
 
 Let the pupil translate Principles 3-5 below into verbal 
 language, and prove each in detail (cf. Principle 2 above). 
 
 Principle 3. log^^=log;, Af-log^/l^. 
 
 Hint. 11 M=h'' and N = &", then M^ N= b''-^. 
 
 Principle 4. log^ N^ = p' log^ N. 
 
 Hint. liN= &^, then N^ = (b^y = b^». 
 
 Principle 5. log^ ^N = ~ • logj N. 
 
 r 
 
 1 _ 1 
 
 Hint. If iV^= ^)^ then Vi\^=(^,-)^ 1</N=N' 
 
 EXERCISE CXLIV 
 
 Using Principles 1-5, express the following logarithms in 
 terms of log a, log c, and log e, the base h being understood 
 throughout : 
 
 1. log(ac). 3. log (ace^). ^ ^^ 
 
 2. log(a^). 4. log (cV). * c* 
 
 HIGH 8CH. ALG. — 22 
 
332 HIGH SCHOOL ALGEBRA [Ch. XXII 
 
 6. log-. alogVc. 11. log(aV^). 
 
 a 
 
 6/ lo 1 3 6 
 
 9. log ^ace. 12. log 
 
 lo i. 12 ^"^ 
 
 a 10. log (a^c^). 13. logVce~l 
 
 Express each of the following by means of a single logarithm, 
 and explain [e.g., log c + log e = log (ce)] : 
 
 14. logc + loge. 16. 2 log a + 3 log e. is. i(loge — log 5 a). 
 
 15. log c — log e. 17. 4(logc — loga). 19. f log a -f 4 log 2 c. 
 
 If logio 2 = 0.3010, logio 3 = 0.4771, and logi„ 7 = 0.8451, find the 
 logarithms, to the base 10, of : 
 
 20. 
 
 6(i.e., 
 
 3. 
 
 2). 
 
 25. 
 
 63. 
 
 
 30. 
 
 400. 
 
 
 21. 
 
 14. 
 
 
 
 26. 
 
 f. 
 
 
 31. 
 
 tV 
 
 
 22. 
 
 42. 
 
 
 
 27. 
 
 2i. 
 
 
 32. 
 
 V3. 
 
 
 23. 
 
 49. 
 
 
 
 28. 
 
 5(/.e., i^y 
 
 
 33. 
 
 <m. 
 
 
 24. 
 
 12(i.e. 
 
 ,2^ 
 
 '.3). 
 
 29. 
 
 30(?.e., 3 . 
 
 10). 
 
 34. 
 
 5-2^. 
 
 • (i)^. 
 
 213. Common logarithms ; characteristic and mantissa. 
 Logarithms to the base 10 (called common or Briggs loga- 
 rithms) possess many advantages over those having any other 
 base, and are used in all practical computations. In the 
 following pages the base 10 will be understood when no base 
 is written ; thus log 25 will mean log^^ 25. 
 
 Now since 100 = 1, 10^ = 10, 102 = 100, 10^=1000, etc., 
 therefore log 1 = 0, log 10 = 1, log 100 = 2, log 1000 = 3, 
 etc.; and therefore the logarithm of any number between 1 
 and 10 lies between and 1, i.e., it is plus a decimal ; the 
 logarithm of any number between 10 and 100 is 1 plus a 
 decimal ; the logarithm of any number between 100 and 
 1000 is 2 plus a decimal ; etc. 
 
 Again, since 10-i=.l, 10-2=. 01, 10-3 =.001, etc., 
 therefore log.l = -l, log .01 = -2, log .001 = -3, etc., 
 and therefore the logarithm of any number between 1 and .1 
 is — 1 plus a decimal ; the logarithm of any number between 
 .1 and .01 is — 2 plus a decimal ; etc. 
 
212-214] LOGARirUMS 333 
 
 The integral part (whether positive or negative) of the 
 logarithm of a number is called the characteristic of the 
 logarithm, and the decimal part (always positive) is called 
 the mantissa. 
 
 E.g.^ log 685 is 2.8357 ; the characteristic is 2, and the mantissa is .8357. 
 
 214. Advantages of the base 10. (i) It follows from § 213 
 that the characteristic of the logarithm of any number 
 between 10 and 100 is 1 (why ?); between 100 and 1000, 2 ; 
 between 10,000 and 100,000, 4; between .01 and .001, -3 
 (why?); between .001 and .0001, —4; etc.; i.e. (let pupil 
 fully explain why), 
 
 ( 1) The characteristic of the logarithm of any number greater 
 than unity is less by one than the number of digits in its integral 
 part ; and (2) the characteristic of the logarithm of any number 
 less than unity is negative, and (^numerically} greater by one 
 than the number of ciphers preceding the first significant figure 
 of the given number. 
 
 (ii) Another great advantage of logarithms to the base 
 
 10 is that moving the decimal point to the right or left in 
 
 any given number makes no change in the mantissa of the 
 
 logarithm of that number. 
 
 ^.^., if log 57.32 =1.7583, 
 
 then log 573.2 = 2.7583 ;. r§ 212, 
 
 for log 573.2 = log (57.32 x 10)= log 57.32 + log 10 LPrin. 2 
 
 = 1.7583 + 1=2.7583. 
 
 EXERCISE CXLV 
 
 1. Which of the following logarithms have negative charac- 
 teristics : log 79; log .315; log 5228; log 4.07; log .00098 ; 
 log .0231 ; log 14865.01 ? Explain. 
 
 2. How many units in the characteristic of each of the above 
 logarithms ? Explain. 
 
 3. How many places to the left of the decimal point has a 
 number if the characteristic of its logarithm is 2? 5? 0? 7? 
 Explain your answers. 
 
334 HIGH SCHOOL algebra [Ch. XXII 
 
 4. How many ciphers has a decimal to the left of its first 
 significant figure if the characteristic of its logarithm is — 5 ? 
 — 1 ? — 3 ? Explain your answers. 
 
 5. If log 469 = 2.6712, show that log 469000 = 5.6712 and that 
 log 4.69 = 0.6712 [cf. § 214 (ii)]. Also point out the characteristic 
 and the mantissa in each of these logarithms. 
 
 6. If log 8.93 = 0.9509, find log .0893; log 893000; log 89.3; 
 and log .000893. 
 
 7. Show that logarithms to the base 10 have at least two prac- 
 tical advantages over logarithms to other bases. 
 
 215. Table of common logarithms. The mantissas (with 
 decimal points omitted) of the logarithms of all integers be- 
 tween 1 and 1000 are given in tabular form on pp. 341, 342. 
 
 This table omits the characteristics of the logarithms be- 
 cause these can be supplied by inspection (§ 214) ; but it 
 includes the mantissas of the logarithms of decimal fractions 
 as well as of integers — the mantissa of log 26.5, or of log 
 .00265, for example, is the same as the mantissa of log 265. 
 
 Note. Except where a number is an integral power of 10, the mantissa 
 of its logarithm is an endless decimal. Hence the mantissas in a table are 
 only approximate values, correct to three or more decimal places. The table 
 on pp. 341, 342 is called a "four-place table " because the mantissas are com- 
 puted to four decimal places. Such a table gives results less accurate than 
 those obtained from a six-place table, for example ; the degree of accuracy 
 required in any given computation determines the choice of table. 
 
 216. Use of tables. Given a number, to find its logarithm. 
 Write down the characteristic by § 214, before consulting the 
 table. Then : 
 
 (a) The number consisting of not more than three significant 
 figures. Find the first two figures of the number in the 
 column headed N in the table ; opposite these, and in the 
 column headed by the third figure of the given number, find 
 the required mantissa. 
 
214-210] LOGARITHMS 335 
 
 Ex. 1. Find log 374. 
 
 Solution. By § 214 the characteristic is 2. On p. 341 oppo- 
 site 37 (in the N column), and in the column under 4, we find the 
 mantissa .5729 ; hence log 374 is 2.5729. 
 
 Ex. 2. Find log .835. 
 
 Solution. The characteristic is — 1 (§ 214). Now the man- 
 tissa of log .835 is the same as the mantissa of log 835 (§ 214), 
 which, found as in Ex. 1, is .9217 ; hence log .835 is — 1 + .9217. 
 
 Note. The mantissa being always positive, the sign of a negative charac- 
 teristic is (to prevent confusion) written over the characteristic. Thus we 
 write log .835 not as - 1 +.9217, but as 1.9217. 
 
 Another common notation is to add 10 to the negative characteristic and 
 then to indicate the subtraction of 10 from the entire logarithm. Thus 
 log .836 may be written 9.9217 - 10. 
 
 Ex. 3. Find log 6. 
 
 Solution. The characteristic is (§ 214)- The mantissa of 
 log 6 is the same as that of log 600, which is .7782 ; hence log 6 
 is 0.7782. 
 
 (5) The number consisting of more than three significant 
 figures. In this case we assume that the logarithm of a 
 number varies directly as the number itself. While this as- 
 sumption is not entirely correct (doubling 50, for example, 
 multiplies its logarithm by 1.17 H- instead of by 2), still for 
 small changes in a number, it leads to results sufficiently 
 accurate for many purposes. 
 
 Ex.4. Find log 2547. 
 
 Solution. The characteristic is 3, and the mantissa is the 
 same as that of log 254.7 (§ 214). 
 
 Now, mantissa of log 254 is .4048, 
 
 and mantissa of log 255 is .4065, 
 
 i.e., adding 1 to 254 adds .0017 to the mantissa of its logarithm, 
 hence adding .7 to 254 should add, approximately, .7 of .0017, i.e., 
 .0012, to its logarithm, and hence 
 
 log 2547 = 3.4048 + .0012 = 3.4060. 
 
836 IIIGTI SCHOOL ALGEBIIA 
 
 Ex. 5. Find loff 74.326. 
 
 [Ch. XXII 
 
 SoLUTio:^. The characteristic is 1, and the mantissa equals 
 the mantissa of log 743.26, which equals (let pupil explain why) 
 mantissa of log 743 -f .26 X (log 744 -log 743). 
 
 = .8710 + . 26 X. 0006, 
 
 = .8712; 
 hence log 74.326 = 1.8712. 
 
 
 EXERCISE 
 
 CXLVI 
 
 
 By reference to the table 
 
 verify 
 
 that : 
 
 
 
 6. log 416 
 
 = 2.6191. 
 
 
 9. 
 
 log 
 
 .00972 = 3.9877. 
 
 7. log 5 
 
 = 0.6990. 
 
 
 10. 
 
 log 
 
 5268 =3.7216. 
 
 8. log 83000 
 
 = 4.9191. 
 
 
 11. 
 
 log 
 
 .7436 =1.8714. 
 
 Find the logarithm of : 
 
 
 
 
 
 12. 513. 
 
 19. 
 
 7. 
 
 
 
 26. .1008. 
 
 13. 692. 
 
 20. 
 
 .009. 
 
 
 
 27. 3.141. 
 
 14. 3.47. 
 
 21. 
 
 4000. 
 
 
 
 28. 22220. 
 
 15. .81. 
 
 22. 
 
 36.02. 
 
 
 
 29. .000694. 
 
 16. 27.8. 
 
 23. 
 
 6215. 
 
 
 
 30. .011111. 
 
 17. .055. 
 
 24. 
 
 .3972. 
 
 
 
 31. 437910. 
 
 18. 200. 
 
 25. 
 
 851.3. 
 
 
 
 32. .0018952. 
 
 33. Write the logarithms in Exs. 24, 2&, 29, 30, and 32 in 
 two different forms (cf. Ex. 2, Note). 
 
 217. Given a logarithm, to find the corresponding number. 
 The number to which a given logarithm corresponds is 
 called its antilogarithm. Thus, 
 
 •. • log 53 = 1.7243, . •. antilog 1.7248 = 53. 
 
 Antilogarithms are found by reversing the processes of 
 § 216 ; a few examples will make the procedure plain. 
 
216-217] LOGARITHMS 337 
 
 Ex. 1. Find antilog 2.5587. 
 
 Solution. On consulting the table we find that .5587 is the 
 mantissa of log 362, and the characteristic 2 tells us that there 
 must be one cipher between the decimal point and the first sig- 
 nificant figure [§ 214 (ii)] ; hence 
 
 antilog 2.5587 = .0362. 
 
 Ex. 2. Find antilog 1.7493. 
 
 Solution. On consulting the table we find that 
 .7490 = mantissa of log 561, 
 and .7497 = mantissa of log 562, 
 
 these being the mantissas next smaller and next larger, respec- 
 tively, than the given mantissa. Hence antilog 1.7493 lies be- 
 tween 56.1 and 56.2 (the characteristic being 1). 
 
 Again, since the given mantissa, viz., .7493, is f of the way 
 from .7490 to .7497, therefore the required antilogarithm is 
 approximately f of the way from 56.1 to 56.2, 
 i.e,, antilog 1.7493 = 56.1 + f of 0.1 
 
 = 56.1 -f .043 
 = 56.143. 
 
 Ex. 3. Find antilog 3.1188. 
 Solution. antilog 3.1206 = 1320, 
 
 antilog 3.1173 = 1310 
 whence, subtracting, we obtain .0033 and 10 
 also 3.1188 - 3.1173 = .0015 ; 
 
 therefore antilog 3.1188 = 1310 + ^f of 10 
 
 = 1310 4- 4.5 = 1314.5. 
 
 EXERCISE CXLVII 
 
 Verify from the table that : 
 
 4. antilog 0.1875 = 1.54. 6. antilog 1.8454 = 70.05. 
 
 5. antilog 1.6021 = .4. 7. antilog 2.5221 = .03328. 
 Find the antilogarithm of : 
 
 8. 2.9605. 10. 1.8451. 12. 6.4983. 
 
 9. 0.5963. 11. 1.8401. 13. 8.0755-10, 
 
338 HIGH SCHOOL ALGEBRA [Ch. XXII 
 
 14. 3.3997. 18. 3.7361. 22. 1.3019. 
 
 15. 4.2226. 19. 0.9002. 23. 5.9754-10. 
 
 16. 2.6512. 20. 2.9068. 24. 9.5327-10. 
 
 17. 1.8846. 21. 5.8049. 25. 4.6831 - 10. 
 
 218. Computation by means of logarithms. 
 Ex. 1. Find p, if j> = 47.45 x 3.514 x .0064. 
 
 SOLUTION 
 
 log p = log 47.45 + log 3.514 + log .0064 ; [§ 212, Prin. 2 
 but log 47.45= 1.6763, [§216 
 
 log 3.514= 0.5458, 
 and log .0064 = 7.8062 - 10 ,^ [§ 216, Note 
 
 therefore log p = 10.0283 - 10 
 
 = 0.0283; 
 and therefore p = 1.067. [§ 217 
 
 This product found in the ordinary way is .10671+. 
 
 Ex. 2. Find 3.041^ 
 
 Solution, log (3.041^) = 4 x log 3.041 [§ 212, Prin. 4 
 
 = 4 X 0.4830 = 1.9320 ; [§ 216 
 
 therefore 3.041^ = antilog 1.9320 = 85.5. [§ 217 
 
 Obtained by ordinary multiplication 3.041* = 85.5196+. 
 Ex. 3. Find ^:0572. 
 
 Solution, log V.0572 = ^ x log .0572 [§ 212, Prin. 5 
 
 = I X 2.7574 
 
 = ix (1.7574 -3) t 
 
 = 0.5858-1 = 1.5858; 
 
 therefore V.0572 = antilog 1.5858 = .3853. 
 
 Obtained by the method of § 120, -^;0572 = .38529+. 
 
 * The form 7.8602 - 10 (instead of 3.8062) is used for log .0064 because, 
 in computation, negative characteristics increase the danger of errors. 
 
 t In order to divide 2.7574 by 3 without mixing positive and negative 
 numbers it is well first to write 2.7574 in one of the following forms: 
 1.7574 - 3, 4.7574 - G, 7.7574 - 9, etc., i.e., to add (and then subtract) some 
 multiple of 3 which will make the characteristic positive. 
 
217-218] 
 
 LOGARITHMS 
 
 339 
 
 „ , J. 37.22 X (-19.86) n , 
 '^•*- ^'^= (12.33y -^"^"' 
 
 Solution. In such examples we first find the numerical value 
 of the result by regarding all the factors as positive, and then 
 prefix the proper sign as determined by §§ 18 and 19. Thus, 
 ignoring the minus sign, we have 
 
 log X = log 137.22 + log 9.86 - 2 x log 12.33 [§ 212, Prin. 2 and 3 
 = 1.5707 + 1.2980 - 2 x 1.0910 
 = 0.6847 ; 
 
 therefore x=- antilog 0.6847 = - 4.84. 
 
 Ex. 5. Given 47.5^ = 293.64 ; find x. 
 
 Solution. On taking the logarithm of each member of this 
 equation we obtain 
 
 X . log 47.5 = log 293.64 
 ^ log 293.64 . 
 log 47.5 ' 
 2.4678 
 
 whence 
 
 I.e.. 
 
 x = 
 
 1.6767 
 
 1.472. 
 
 Note. Equations in which the unknown number appears as an exponent 
 are called exponential equations. Such equations cannot be solved by the 
 methods given in the preceding pages, but are easily solved by the method 
 illustrated in the above solution of Ex. 5. 
 
 EXERCISE CXLVIII 
 By logarithms find the value of : 
 
 6. 376x58. 12. 380.7 -^ 9.8. 
 
 7. 2.29x8.7. 13. 10 -^ 3.141. 
 
 8. 69.5 x. 00543. 14. 3 -r- 5.963. 
 
 9. -42.37 X. 236. 15. 30.07 -?- .002121. 
 
 10. .2912x3.141. 16. .005918 -f- .0009293. 
 
 11. .0695 x .002682. 17. 13 x 753 ^ .06238. 
 
340 HIGH SCHOOL ALGEBRA [Ch. XXII 
 
 By logarithms simplify : 
 
 18. 23\ 23. (2)8. 28. V675. 
 
 19. .08^1 24. (If)^. 29. ^:0500l. 
 
 20. .395^-1^ 25. (62)i 30. ^(.3192)«. 
 
 21. (-3.813)'. 26. (991.7)^. 3^^ -;/:i277^l7. 
 
 32. '^18^V2574. 
 
 22. (1.228)10. 27. (.1183)? 
 
 33. 
 34. 
 
 19x(-700) 4635^« X 200.4* 
 
 970 X 1.4 X .0616 
 
 36. 
 
 10123 
 
 3-1^1 X .0711 ^^ 13^n^2^5 
 
 .8331x51 • 57o^7)j2l 
 
 33^ 1.78 X. 0052x16. 2^x(#X^f 
 ^ .339x4.315 38. ^2J^ ^. 
 
 39. If a, b, and c are the sides of a triangle, and s is one half 
 their sum, the area of the triangle is -\/s(s — a)(s — b)(s — c). 
 Find, by logarithms, the area of the triangle whose sides are 
 13.6 ft., 15.1 ft., and 20.1 ft. ; also the area of the triangle whose 
 sides are 260 ft., 319 ft., and 464 ft. 
 
 Solve for x (cf. Ex. 5) : 
 
 40. 16-"= 354. 43. 6"^ = 5'=+\ 
 
 41. 7^ = 9.59. 44. 2'^ = 113-+!. 
 
 42. 28.8^ = 12750. 45. 152»<^«^ = 3275. 
 
 46. From (1), § 201, show that in a G. P. log r = ]2K1zl}^.^ 
 
 n — 1 
 also find r when a = 10, w = 10, and I = 196830. 
 
 47. If A is the amount of P dollars at r % compound interest 
 for n years, show that A = P(l + r)" ; also solve this equation for 
 each letter it contains. (Cf. Ex. 25, p. 316, also Ex. 46 above.) 
 
 48. Find the amount of $700 for 5 years at 4% compound 
 interest; also the amount of $450 for 10 years at 3% compound 
 interest. 
 
 49. In what time will $ 800 amount to $ 1834.50 if put at com- 
 pound interest at 5 % ? 
 
218] 
 
 LOGARITHMS 
 
 341 
 
 Table of Common Logarithms 
 
 N 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 lO 
 
 CXXX) 
 
 0043 
 
 0086 
 
 0128 
 
 0170 
 
 0212 
 
 0253 
 
 0294 
 
 0334 
 
 0374 
 
 II 
 
 0414 
 
 0453 
 
 0492 
 
 0531 
 
 0569 
 
 0607 
 
 0645 
 
 0682 
 
 0719 
 
 0755 
 
 12 
 
 0792 
 
 0828 
 
 0864 
 
 0899 
 
 0934 
 
 0969 
 
 1004 
 
 1038 
 
 1072 
 
 1 106 
 
 13 
 
 1 139 
 
 1173 
 
 1206 
 
 1239 
 
 1271 
 
 1303 
 
 1335 
 
 1367 
 
 1399 
 
 1430 
 
 14 
 
 I46I 
 
 1492 
 
 1523 
 
 1553 
 
 1584 
 
 1614 
 
 1644 
 
 1673 
 
 1703 
 
 1732 
 
 15 
 
 I76I 
 
 1790 
 
 1818 
 
 1847 
 
 1875 
 
 1903 
 
 1931 
 
 1959 
 
 1987 
 
 2014 
 
 i6 
 
 2041 
 
 2068 
 
 2095 
 
 2122 
 
 2148 
 
 2175 
 
 2201 
 
 2227 
 
 2253 
 
 2279 
 
 17 
 
 2304 
 
 2330 
 
 2355 
 
 2380 
 
 2405 
 
 2430 
 
 2455 
 
 2480 
 
 2504 
 
 2529 
 
 18 
 
 2553 
 
 2577 
 
 2601 
 
 2625 
 
 2648 
 
 2672 
 
 2695 
 
 2718 
 
 2742 
 
 2765 
 
 19 
 
 2788 
 
 2810 
 
 2833 
 
 2856 
 
 2878 
 
 2900 
 
 2923 
 
 2945 
 
 2967 
 
 2989 
 
 20 
 
 3010 
 
 3032 
 
 3054 
 
 3075 
 
 3096 
 
 3118 
 
 3139 
 
 3160 
 
 3181 
 
 3201 
 
 21 
 
 3222 
 
 3243 
 
 3263 
 
 3284 
 
 3304 
 
 3324 
 
 3345 
 
 3365 
 
 3385 
 
 3404 
 
 22 
 
 3424 
 
 3444 
 
 3464 
 
 3483 
 
 3502 
 
 3522 
 
 3541 
 
 3560 
 
 3579 
 
 3598 
 
 23 
 
 3617 
 
 3(>3^ 
 
 3655 
 
 3674 
 
 3692 
 
 37" 
 
 3729 
 
 3747 
 
 3766 
 
 3784 
 
 24 
 
 3802 
 
 3820 
 
 3838 
 
 3856 
 
 3874 
 
 3892 
 
 3909 
 
 3927 
 
 3945 
 
 3962 
 
 25 
 
 3979 
 
 3997 
 
 4014 
 
 4031 
 
 4048 
 
 4065 
 
 4082 
 
 4099 
 
 4116 
 
 4133 
 
 26 
 
 4150 
 
 4166 
 
 4183 
 
 4200 
 
 4216 
 
 4232 
 
 4249 
 
 4265 
 
 4281 
 
 4298 
 
 27 
 
 43H 
 
 4330 
 
 4346 
 
 4362 
 
 4378 
 
 4393 
 
 4409 
 
 4425 
 
 4440 
 
 4456 
 
 28 
 
 4472 
 
 4487 
 
 4502 
 
 4518 
 
 4533 
 
 4548 
 
 4564 
 
 4579 
 
 4594 
 
 4609 
 
 29 
 
 4624 
 
 4639 
 
 4654 
 
 4669 
 
 4683 
 
 4698 
 
 4713 
 
 4728 
 
 4742 
 
 4757 
 
 30 
 
 477^ 
 
 4786 
 
 4800 
 
 4814 
 
 4829 
 
 4843 
 
 4857 
 
 4871 
 
 4886 
 
 4900 
 
 31 
 
 4914 
 
 4928 
 
 4942 
 
 4955 
 
 4969 
 
 4983 
 
 4997 
 
 501 1 
 
 5024 
 
 5038 
 
 32 
 
 5051 
 
 5065 
 
 5079 
 
 5092 
 
 5105 
 
 5"9 
 
 5132 
 
 5145 
 
 5159 
 5289 
 
 5172 
 
 33 
 
 5185 
 
 5198 
 
 5211 
 
 5224 
 
 5237 
 
 5250 
 
 5263 
 
 5276 
 
 5302 
 
 34 
 
 5315 
 
 5328 
 
 5340 
 
 5353 
 
 5366 
 
 5378 
 
 5391 
 
 5403 
 
 5416 
 
 5428 
 
 35 
 
 5441 
 
 5453 
 
 5465 
 
 5478 
 
 5490 
 
 5502 
 
 5514 
 
 5527 
 
 5|3? 
 
 555' 
 
 36 
 
 
 5575 
 
 5587 
 
 5599 
 
 5611 
 
 5623 
 
 5635 
 
 5647 
 
 5658 
 
 5670 
 
 37 
 
 5682 
 
 5694 
 
 5705 
 
 5717 
 
 5729 
 
 5740 
 
 5752 
 
 5763 
 
 5775 
 
 5786 
 
 38 
 
 5798 
 
 5809 
 
 5821 
 
 5832 
 
 5843 
 
 5855 
 
 5866 
 
 ^Hl 
 
 5888 
 
 5899 
 
 39 
 
 59" 
 
 5922 
 
 5933 
 
 5944 
 
 5955 
 
 5966 
 
 5977 
 
 5988 
 
 5999 
 
 6010 
 
 40 
 
 6021 
 
 6031 
 
 6042 
 
 6053 
 
 6064 
 
 6075 
 
 6085 
 
 6096 
 
 6107 
 
 6117 
 
 41 
 
 6128 
 
 6138 
 
 6149 
 
 6160 
 
 6170 
 
 6180 
 
 6191 
 
 6201 
 
 6212 
 
 6222 
 
 42 
 
 6232 
 
 6243 
 
 6253 
 
 6263 
 
 6274 
 
 6284 
 
 6294 
 
 6304 
 
 6314 
 
 6325 
 
 43 
 
 6335 
 
 6345 
 
 6355 
 
 6365 
 
 6375 
 
 6385 
 
 6395 
 
 6405 
 
 6415 
 
 6425 
 
 44 
 
 6435 
 
 6444 
 
 6454 
 
 6464 
 
 6474 
 
 6484 
 
 6493 
 
 6503 
 
 6513 
 
 6522 
 
 45 
 
 6532 
 
 6542 
 
 6551 
 
 6561 
 
 6571 
 
 6580 
 
 6590 
 
 6599 
 
 6609 
 
 6618 
 
 46 
 
 6628 
 
 6637 
 
 6646 
 
 6656 
 
 6665 
 
 6675 
 
 6684 
 
 6693 
 
 6702 
 
 ^?'^ 
 
 47 
 
 6721 
 
 6730 
 
 6739 
 
 6749 
 
 6758 
 
 6767 
 
 6776 
 
 6785 
 
 ?Zt4 
 
 6803 
 
 48 
 
 6812 
 
 6821 
 
 6830 
 
 6839 
 
 6848 
 
 6857 
 
 6866 
 
 6875 
 
 6884 
 
 6893 
 
 49 
 
 6902 
 
 691 1 
 
 6920 
 
 6928 
 
 6937 
 
 6946 
 
 6955 
 
 6964 
 
 6972 
 
 6981 
 
 50 
 
 6990 
 
 6998 
 
 7007 
 
 7016 
 
 7024 
 
 7033 
 
 7042 
 
 7050 
 
 7059 
 
 7067 
 
 51 
 
 7076 
 
 7084 
 
 7093 
 
 7101 
 
 7110 
 
 7118 
 
 7126 
 
 7135 
 
 7H3 
 
 7152 
 
 52 
 
 7160 
 
 7168 
 
 7177 
 
 7185 
 
 7193 
 
 7202 
 
 7210 
 
 7218 
 
 7226 
 
 7235 
 
 53 
 
 7243 
 
 7251 
 
 7259 
 
 7267 
 
 7275 
 
 7284 
 
 7292 
 
 7300 
 
 73Sf 
 
 7316 
 
 54 
 
 7324 
 
 7332 
 
 7340 
 
 7348 
 
 7356 
 
 7364 
 
 7372 
 
 7380 
 
 7388 
 
 7396 
 
 N 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
342 
 
 HIGH SCHOOL ALGEBRA 
 
 Table of Common Logarithms 
 
 N 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 55 
 
 7404 
 
 7412 
 
 7419 
 
 7427 
 
 7435 
 
 7443 
 
 7451 
 
 7459 
 
 7466 
 
 7474 
 
 56 
 
 7482 
 
 7490 
 
 7497 
 
 7505 
 
 7513 
 
 7520 
 
 7528 
 
 7536 
 
 7543 
 
 7551 
 
 57 
 
 7559 
 
 7566 
 
 7574 
 
 7582 
 
 7589 
 
 7597 
 
 7604 
 
 7612 
 
 7619 
 
 7627 
 
 58 
 
 7634 
 
 7642 
 
 7649 
 
 7657 
 
 7664 
 
 7672 
 
 7679 
 
 7686 
 
 7694 
 
 7701 
 
 59 
 
 7709 
 
 7716 
 
 7723 
 
 7731 
 
 7738 
 
 7745 
 
 7752 
 
 7760 
 
 7767 
 
 7774 
 
 60 
 
 7782 
 
 7789 
 
 7796 
 
 7803 
 
 7810 
 
 7818 
 
 7825 
 
 7832 
 
 7839 
 
 7846 
 
 61 
 
 7853 
 
 7860 
 
 7868 
 
 7875 
 
 7882 
 
 7889 
 
 7896 
 
 7903 
 
 7910 
 
 7917 
 
 62 
 
 7924 
 
 7931 
 
 7938 
 
 7945 
 
 7952 
 
 7959 
 
 7966 
 
 7973 
 
 7980 
 
 7987 
 
 63 
 
 7993 
 
 8000 
 
 8007 
 
 8014 
 
 8021 
 
 8028 
 
 8035 
 
 8041 
 
 8048 
 
 8055 
 
 64 
 
 8062 
 
 8069 
 
 8075 
 
 8082 
 
 8089 
 
 8096 
 
 8102 
 
 8109 
 
 8116 
 
 8122 
 
 65 
 
 8129 
 
 8136 
 
 8142 
 
 8149 
 
 8156 
 
 8162 
 
 8169 
 
 8176 
 
 8182 
 
 8189 
 
 66 
 
 8195 
 
 8202 
 
 8209 
 
 8215 
 
 8222 
 
 8228 
 
 8235 
 
 8241 
 
 8248 
 
 8254 
 
 67 
 
 826i 
 
 8267 
 
 8274 
 
 8280 
 
 8287 
 
 8293 
 
 8299 
 
 8306 
 
 8312 
 
 8319 
 
 68 
 
 8325 
 
 8331 
 
 8338 
 
 8344 
 
 8351 
 
 8357 
 
 8363 
 
 8370 
 
 8376 
 
 8382 
 
 69 
 
 8388 
 
 8395 
 
 8401 
 
 8407 
 
 8414 
 
 8420 
 
 8426 
 
 8432 
 
 8439 
 
 8445 
 
 70 
 
 8451 
 
 8457 
 
 8463 
 
 8470 
 
 8476 
 
 8482 
 
 8488 
 
 8494 
 
 8500 
 
 8506 
 
 71 
 
 8513 
 
 8519 
 
 8525 
 
 8531 
 
 8537 
 
 8543 
 
 8549 
 
 8555 
 
 8561 
 
 8567 
 
 72 
 
 8573 
 
 8579 
 
 8585 
 
 8591 
 
 8597 
 
 8603 
 
 8609 
 
 8615 
 
 8621 
 
 8627 
 
 73 
 
 8633 
 
 8639 
 
 8645 
 
 8651 
 
 8657 
 
 8663 
 
 8669 
 
 8675 
 
 8681 
 
 8686 
 
 74 
 
 8692 
 
 8698 
 
 8704 
 
 8710 
 
 8716 
 
 8722 
 
 8727 
 
 8733 
 
 8739 
 
 8745 
 
 75 
 
 8751 
 
 8756 
 
 8762 
 
 8768 
 
 8774 
 
 8779 
 
 8785 
 
 8791 
 
 8797 
 
 8802 
 
 76 
 
 8808 
 
 8814 
 
 8820 
 
 8825 
 
 8831 
 
 8837 
 
 8842 
 
 8848 
 
 8854 
 
 8859 
 
 77 
 
 8865 
 
 8871 
 
 8876 
 
 8882 
 
 8887 
 
 8893 
 
 8899 
 
 8904 
 
 8910 
 
 8915 
 
 78 
 
 8921 
 
 8927 
 
 8932 
 
 8938 
 
 8943 
 
 8949 
 
 8954 
 
 8960 
 
 8965 
 
 8971 
 
 79 
 
 8976 
 
 8982 
 
 8987 
 
 8993 
 
 8998 
 
 9004 
 
 9009 
 
 9015 
 
 9020 
 
 9025 
 
 80 
 
 9031 
 
 9036 
 
 9042 
 
 9047 
 
 9053 
 
 9058 
 
 9063 
 
 9069 
 
 9074 
 
 9079 
 
 81 
 
 9085 
 
 9090 
 
 9096 
 
 9101 
 
 9106 
 
 9112 
 
 9117 
 
 9122 
 
 9128 
 
 9133 
 
 82 
 
 9138 
 
 9143 
 
 9149 
 
 9154 
 
 9159 
 
 9165 
 
 9170 
 
 9175' 
 
 9180 
 
 9186 
 
 83 
 
 9191 
 
 9196 
 
 9201 
 
 9206 
 
 9212 
 
 9217 
 
 9222 
 
 9227 
 
 9232 
 
 9238 
 
 84 
 
 9243 
 
 9248 
 
 9253 
 
 9258 
 
 9263 
 
 9269 
 
 9274 
 
 9279 
 
 9284 
 
 9289 
 
 85 
 
 9294 
 
 9299 
 
 9304 
 
 9309 
 
 9315 
 
 9320 
 
 9325 
 
 9330 
 
 9335 
 
 9340 
 
 86 
 
 9345 
 
 9350 
 
 9355 
 
 9360 
 
 9365 
 
 9370 
 
 9375 
 
 9380 
 
 9385 
 
 9390 
 
 87 
 
 9395 
 
 9400 
 
 9405 
 
 9410 
 
 9415 
 
 9420 
 
 9425 
 
 9430 
 
 9435 
 
 9440 
 
 88 
 
 9445 
 
 9450 
 
 9455 
 
 9460 
 
 9465 
 
 9469 
 
 9474 
 
 9479 
 
 9484 
 
 9489 
 
 89 
 
 9494 
 
 9499 
 
 9504 
 
 9509 
 
 9513 
 
 9518 
 
 9523 
 
 9528 
 
 9533 
 
 9538 
 
 90 
 
 9542 
 
 9547 
 
 9552 
 
 9557 
 
 9562 
 
 9566 
 
 9571 
 
 9576 
 
 9581 
 
 9586 
 
 91 
 
 9590 
 
 9595 
 
 9600 
 
 9605 
 
 9609 
 
 9614 
 
 9619 
 
 9624 
 
 9628 
 
 9633 
 
 92 
 
 9638 
 
 9643 
 
 9647 
 
 9652 
 
 9657 
 
 9661 
 
 9666 
 
 9671 
 
 9675 
 
 9680 
 
 93 
 
 9685 
 
 9689 
 
 9694 
 
 9699 
 
 9703 
 
 9708 
 
 9713 
 
 9717 
 
 9722 
 
 9727 
 
 94 
 
 9731 
 
 9736 
 
 9741 
 
 9745 
 
 9750 
 
 9754 
 
 9759 
 
 9763 
 
 9768 
 
 9773 
 
 95 
 
 9777 
 
 9782 
 
 9786 
 
 9791 
 
 9795 
 
 98CX) 
 
 9805 
 
 9809 
 
 9814 
 
 9818 
 
 96 
 
 
 9827 
 
 9832 
 
 
 9841 
 
 9845 
 
 9850 
 
 9854 
 
 9859 
 
 9863 
 
 97 
 
 9868 
 
 9872 
 
 9877 
 
 9881 
 
 9886 
 
 9890 
 
 9894 
 
 9899 
 
 9903 
 
 9908 
 
 98 
 
 9912 
 
 9917 
 
 9921 
 
 9926 
 
 9930 
 
 9934 
 
 9939 
 
 9943 
 
 9948 
 
 9952 
 
 99 
 
 9956 
 
 9961 
 
 9965 
 
 9969 
 
 9974 
 
 9978 
 
 9983 
 
 9987 
 
 9991 
 
 9996 
 
 N 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
INDEX 
 
 [Numbers refer to pages.] 
 
 Absolute, term, 190. 
 
 value, 19. 
 Addition, 8, 21, 29, 30, 115, 243. 
 Algebraic, expressions, 27. 
 
 fraction, 109. 
 
 numbers, 19. 
 
 sentence, 60. 
 
 sum, 21. 
 Alternation, 298. 
 Antecedent, 293. 
 Antilogarithm, ,336, 
 Approximate root, 181. 
 Arithmetical, mean, 311. 
 
 numbers, 19. 
 
 progression, 307. 
 
 series, 307. 
 Arranged polynomial, 43. 
 Associative law, 54. 
 Axioms, 56. 
 Axis, of coordinates, 219. 
 
 of imaginaries, 259. 
 
 of real numbers, 259. 
 
 Base, of logarithms, 329. 
 
 of power, 12. 
 Binomial, 27, 236. 
 , cube of, 76. 
 
 square of, 71. 
 
 theorem, 174, 323. 
 Brace, bracket, etc., 15. 
 Briggs logarithms, 332, 
 
 Character of roots, 278. 
 Characteristic of logarithm, 333. 
 Checking results, 31, 56. 
 Clearing of fractions, 126. 
 Coefficients, 27, 190, 236. 
 
 Commensurable numbers, 293. 
 Common, difference, 307. 
 
 factor, 98, 
 
 logarithms, 332, 
 
 multiple, 105. 
 
 ratio, 313. 
 Commutative law, 54. 
 Completing the square, 194. 
 Complex, fractions, 123. 
 
 numbers, 257. 
 Composite numbers, 77. 
 Conditional, equation, 55. 
 
 inequality, 289. 
 Conjugate, complex numbers, 257. 
 
 surds, 249. 
 Consequent, 293. 
 Consistent equations, 146. 
 Constant term, 190. 
 Constants, 301. 
 Continued, product, 25. 
 
 proportion, 296. 
 Coordinate axes, 219. 
 Cube, 13. 
 
 of a binomial, 76. 
 
 root, 176, 185, 187. 
 Cubic equation, 126. 
 
 Decreasing series, 316. 
 Degree, of term, 43. 
 
 of equation, 125. 
 Denominator, 109. 
 Determinate system, 165. 
 Difference, 8. 
 Discriminant, 278. 
 Dissimilar, radicals, 236. 
 
 terms, 28. 
 Distributive law, 54. 
 343 
 
U4: 
 
 INDEX 
 
 Division, 11, 26, 45, 121, 250, 298. 
 Divisor, dividend, 12. 
 
 Elements of A. P. and G. P., 307 
 
 313. 
 Elimination, 147, 149. 
 Equations, conditional, 55. 
 
 consistent, 146. 
 
 determinate, 165. 
 
 equivalent, 127. 
 
 exponential, 339. 
 
 fractional, 130, 151, 198. 
 
 graph of, 221. 
 
 homogeneous, 208. 
 
 identical, 55. 
 
 inconsistent, 146. 
 
 independent, 146. 
 
 indeterminate, 144, 165. 
 
 in quadratic form, 205. 
 
 integral, 125. 
 
 irrational, 255. 
 
 linear, 125. 
 
 literal, 125, 128, 154. 
 
 locus of, 221. 
 
 numerical, 125. 
 
 of tlie problem, 62. 
 
 quadratic, 126, 190, 277. 
 
 radical, 255. 
 
 simple, 125. 
 
 simultaneous, 146, 206. 
 
 solution of, 57. 
 
 symmetric, 212. 
 Equivalent equations, 127. 
 Even, power, 170. 
 
 root, 176. 
 Evolution, 176. 
 Exponent, 12. 
 
 fractional, negative, zero, 265 
 laws, 38, 45, 171, 265. 
 Exponential equation, 339. 
 Extraneous roots, 131. 
 Extremes, 296, 311, 318. 
 
 Factoring, 78, 90, 282. 
 
 solving equations by, 94. 
 Factors, 10, 77, 78. 
 
 of quadratic expressions, 282. 
 Factor theorem, 92. 
 
 Finite, numbers, 52. 
 
 series, 307. 
 Formulas, for A. P., G. P., 308, 314. 
 
 for solving equations, 140, 277. 
 Fourth proportional, 296. 
 Fractional, equations, 130, 151. 
 
 exponent, 265. 
 Fractions, 12, 109. 
 
 clearing of, 126. 
 
 lowest terms of. 111. 
 
 General problem, 139. 
 Geometric, infinite G. P., 316. 
 
 means, 318. 
 
 series, 313. 
 Graph of an equation, 221. 
 Graphic solutions, 227. 
 Graphical representation of complex 
 
 numbers, 259. 
 Greater than, 287. 
 
 Harmonic series, 319. 
 Highest common factor, 98. 
 Homogeneous equations, 208. 
 
 Identical equations, 55. 
 Imaginary, numbers, 177, 235, 257. 
 
 unit, 257. 
 Improper fraction, 109. 
 Incommensurable numbers, 293. 
 Inconsistent equations, 146. 
 Independent equations, 146. 
 Indeterminate, equations, 144, 165. 
 
 systems, 165. 
 
 Index of a root, 176. 
 
 Induction, mathematical, 322. 
 
 ^ Jnequalities, 287. 
 
 -^Infinite series, 307, 316. 
 
 Infinitely, large, 52. 
 
 small, 52. 
 Insertion of parentheses, 35. 
 Integral, equation, 125. 
 
 expressions, 42. 
 Interpretation, of results, 189. 
 
 of the forms 
 
 «-, .5, 301, 
 
 00 
 
 Inverse, operations, 9. 
 ratio, 293. 
 
INDEX 
 
 345 
 
 Inversion of proportion, 297. 
 Involution, 170. 
 Irrational, equation, 255. 
 numbers, 235. 
 
 Known and unknown numbers, 125. 
 
 Laws, of exponents, 38, 45, 171, 265. 
 
 of operations, 54. 
 
 of signs, 25, 26, 177. 
 Less than, 287. 
 Letter of arrangement, 43. 
 Like, and unlike, radicals, 236. 
 
 terms, 28. 
 Limit, 301. 
 
 Linear equations, 125. 
 Literal, coefficients, 27. 
 
 equations, 125, 128, 154. 
 
 numbers, 1, 3, 
 Locus of an equation, 221. 
 Logarithms, 329. 
 
 table of, 341. 
 Lowest common multiple, 105. 
 
 Mantissa of logarithm, 333. 
 Mathematical induction, 322. 
 Mean proportional, 296. 
 Members of an equation, 55. 
 Minuend, 9. 
 Mixed expression, 109. 
 Monomials, 27. 
 Multiples, 105. 
 Multiplicand, multiplier, 10. 
 Multiplication, 10, 38. 
 
 Negative, exponent, 266. 
 
 numbers, 18. 
 
 term, 28. 
 Numbers, absolute value of, 19. 
 
 commensurable, etc., 293. 
 
 complex, 257. 
 
 constants and variables, 301. 
 
 finite and infinite, 52. 
 
 imaginary, 177, 235, 257. 
 
 known and unknown, 125. 
 
 literal, 1, 3. 
 
 negative and positive, 18. 
 
 opposite, 19. 
 
 Numbers, prime and composite, 77. 
 
 rational and irrational, 234, 235. 
 
 real, 177, 235. 
 Numerical, coefficient, 27. 
 
 equation, 125. 
 
 Odd, power, 170. 
 
 root, 176. 
 Operations, with literal numbers, 1. 
 
 with imaginary and complex num- 
 bers, 261. 
 Opposite, numbers, 19. 
 
 species, 287. 
 Order, of operations, 14. 
 
 of radicals, 236. 
 
 Parentheses, 15, 35. 
 Polynomials, 27. 
 
 square of, 75, 328. 
 Positive numbers, 18. 
 
 terras, 28. 
 Power, 12. 
 
 Powers of imaginary unit, 258. 
 Prime, numbers, 77. 
 
 to each other, 98. 
 Principal roots, 236. 
 Principles, of clearing of fractions, 
 126. 
 
 of elimination, 147, 149. 
 
 of inequalities, 287. 
 
 of logarithms, 303. 
 
 of proportion, 296. 
 Problems, 62. 
 
 directions for solving, 62. 
 
 general, 139. 
 Products, 10, 24, 38, 40, etc. 
 
 of fractions, 119. 
 
 of sum and difference, 72. 
 Progression, arithmetical, 307. 
 
 geometric, 313. 
 
 harmonic, 319. 
 Proof by induction, 321. 
 Proper fraction, 109. 
 Property, of complex numbers, 262. 
 
 of quadratic surds, 253. 
 Proportion, 295. 
 Pure, quadratic, 190. 
 
 imaginary numbers, 257. 
 
346 
 
 INDEX 
 
 Quadratic equation has two roots, and 
 
 only two, 283. 
 Quadratic equations, 126, 190, 277. 
 
 form of, 205. 
 
 graphs of, 229. 
 
 roots of, 278, 279, 
 
 simultaneous, 206. 
 
 solution by formula, 277. 
 
 special devices for, 211. 
 
 surds, 253. 
 Quotient, 12. 
 
 Radicals, radical equations, 235, 236, 
 
 241, 255. 
 Radicand, 235. 
 Ratio, 293, 294, 313. 
 Rational numbers, 234. 
 Rationalizing factor, 249, 276. 
 Real numbers, 177, 235. 
 Reciprocal of a number, 110. 
 Relation between roots and coeffi- 
 cients, 279. 
 Remainder, 8. 
 
 theorem, 92. 
 Removal of parentheses, 35. 
 Review exercises, 67, 166, 284. 
 Root, principal, 236. 
 
 to n terms, 181. 
 Roots, of an equation, 56. 
 
 character of, 278. 
 
 extraneous, 131. 
 
 relation between coefficients and, 
 279. 
 Rule of signs, 25, 26. 
 
 Series, 307. 
 
 Signs, of aggregation, 14. 
 
 of deduction, 3. 
 
 of inequality, 287. 
 
 of operation, 1 , 10, 19. 
 
 of quality, 19. 
 Similar, radicals, 236. 
 
 terms, 28. 
 Simple equations, 125. 
 Simultaneous equations, 146, 206. 
 Solution of equations, 57, 146, 206, etc. 
 
 by factoring, 94. 
 
 Species of inequalities, 287. 
 Specitic gravity, 139. 
 Square, 13. 
 
 of a binomial, 71. 
 
 of a polynomial, 75, 328. 
 Square root, 176, 180, 188. 
 
 of binomial surd, 253. 
 
 of complex number, 262. 
 Standard form, of complex number, 
 257. 
 
 of quadratic, 190. 
 Subtraction, 8, 22, 32. 
 Subtrahend, 8. 
 Sum, 8, 21. 
 Summands, 8. 
 Surds, 235. 
 
 conjugate, 249. 
 Symbols, •.• and .-.,3, 
 
 > and <, 287. 
 Symmetric equations, 212. 
 System of equations, 146. 
 
 indeterminate, 165. 
 
 Table of logarithms, 341. 
 
 Terms, 27, 109, 203, 307. 
 
 Theorem, binomial, 174, 323. 
 
 Thermograph, 232. 
 
 Third proportional, 296. 
 
 Translation of common language into 
 
 algebraic language, and vice 
 
 versa ^ 60. 
 Transposing, 58. 
 Trinomial, 27. 
 Type forms, 71. 
 
 Unconditional inequality, 289. 
 Unknown numbers, 125. 
 Unlike terms, 28. 
 
 Variable, variation, 301. 
 Vary, 304. 
 
 directly, 304. 
 
 inversely, 304. 
 
 jointly, 304. 
 Vinculum, 15. 
 
 Zero, 52. 
 
 exponents, 266. 
 
^N INITIAL FINE OF 25 CEKTS 
 
 OVERDUE. ■= 
 
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 THE UNIVERSITY OF CALIFORNIA LIBRARY