mm^ ,<,h- ESSENTIALS OF ALGEBRA FOR SECONDARY SCHOOLS BY WEBSTER WELLS, S.B. PROFESSOR OF MATHEMATICS IN THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY VERSITY OF LEACH, SHEWELL AND COMPANY NEW YORK BOSTON CHICAGO Copyright, 1897, By WEB8TEE WELLS. NorhJooU 53resg J. S. Cushing & Co. - Berwick & Smith Norwood Mass. U.S.A. PREFACE. The cordial reception which the author's other Algebras have received at the liands of the educational public, their extensive use in schools of the highest rank in all parts of the country, the appreciative reconnnendations which have come to him from instructors of reputation, lead him to believe that this latest attempt to adequately meet the demands of the best secondary schools will be cordially welcomed. Our teachers are progressive, and the author who fails to keep abreast of the times, and in sympathy with the best educational thought and methods, will appeal in vain for the patronage and sympathies of his fellow-teachers. Fully conscious of the above truth, the author earnestly recommends " The Essentials of Algebra " to the attention of the educational public. It affords a thorough and complete treatment of elemen- tary Algebra, and attention is especially invited to the fol- lowing features : — The introduction of easy problems at the very outset ; § 5. The Addition and Multiplication of Positive and Nega- tive Numbers ; §§ 14 to 19. The Addition of Similar Terms ; § 31. The discussion of Simple Equations, not involving Frac- tions, directly after Division ; Chap. VII. The suggestions in regard to the solution of problems ; §§ 76, 77. The discussion of the theoretical principles involved in the handling of fractions ; §§ 129, 136, 143, 145. 184017 iv PllEFACE. The examples on page 176. The discussion of square roots and cube roots of arith- metical numbers; §§ 197, 198, 203, 204. The examples at the end of § 229. The solution of equations by factoring ; §§ 266, 267. The factoring of a quadratic expression when the co- efficient of fl^ is a perfect square ; § 286. Great care has been taken to state the various definitions and rules with accuracy, and every principle has been dem- onstrated with strict regard to the logical principles in- volved. As a rule, no definition has been introduced until its use became necessary. The examples and problems have been selected with great care, are ample in number, and thoroughly graded. They are especially numerous in the important chapters on Fac- toring, Fractions, and Radicals. The latest English practice has been followed in writing Arithmetic, Geometric, and Harmonic, for Arithmetical, Geometrical, and Harmonical, in the progressions. The author wishes to acknowledge, with hearty thanks, the many suggestions and the assistance that he has received from principals and teachers of secondary schools in all parts of the country, in improving and perfecting the work. WEBSTER WELLS. Massachusetts Institute of Technology, March, 1897. CONTENTS. PAGE I. Definitions and Notation 1 Solution of Problems by Algebraic Methods .... 2 Algebraic Expressions 6 II. Positive and Negative Numbers 9 Addition of Positive and Negative Numbers .... 11 Multiplication of Positive and Negative Numbers . . 12 III. Addition and Subtraction of Algebraic Expressions . 15 Addition of Monomials 16 Addition of Polynomials 20 Subtraction 21 Subtraction of Monomials 22 Subtraction of Polynomials 23 IV. Parentheses 26 Removal of Parentheses 26 Introduction of Parentheses 28 V. Multiplication 29 Multiplication of Monomials 30 Multiplication of Polynomials by Monomials .... 32 Multiplication of Polynomials by Polynomials ... 32 VI. Division 37 Division of Monomials 38 Division of Polynomials by Monomials 39 Division of Polynomials by Polynomials 40 VII. Simple Equations 48 Properties of Equations 49 Solution of Simple Equations 50 Problems 52 VIII. Important Rules in Multiplication and Division . . 59 IX. Factoring 67 V vi CONTENTS. PAGE X. Highest Common Factor 81 XI. Lowest Common Multiple 91 XII. Fractions 96 Reduction of Fractions 97 Addition and Subtraction of Fractions 105 Multiplication of Fractions Ill Division of Fractions 113 Complex Fractions 115 . XIII. Simple Equations (Continued) 120 Solution of Equations containing Fractions . . . 120 Solution'of Literal Equations 124 Solution of Equations involving Decimals .... 12(j Problems 127 Problems involving Literal Equations 136 XIV. Simultaneous Equations Containing Two Unknown Quantities ..... 138 XV. Simultaneous Equations Containing more than Two Unknown Quantities . . 160 XVI. Problems Involving Simultaneous Equations 154 XVII. Inequalities 165 XVIII. Involution . * 170 Involution of Monomials 170 Square of a Polynomial 171 Cube of a Binomial 172 — XIX. Evolution 174 Evolution of Monomials 174 Square Root of a Polynomial 176 Square Root of an Arithmetical Number . . , . 179 Cube Root of a Polynomial 183 Cube Root of an Arithmetical Number 186 XX. Theory of Exponents 191 XXI. Radicals 201 Reduction of a Radical to its Simplest Form . . . 201 Addition and Subtraction of Radicals 205 To Reduce Radicals of Different Degrees to Equiva- lent Radicals of the Same Degree 206 CONTENTS. vil XXI. Radicals (Continued). page Multiplication of Radicals 207 Division of Radicals 210 Involution of Radicals 212 Evolution of Radicals 212 To Reduce a Fraction having an Irrational Denom- inator to an Equivalent Fraction whose Denom- inator is Rational 213 Properties of Quadratic Surds 215 Imaginary Numbers 218 Solution of Equations containing Radicals . . 222 XXII. Quadratic Equations 224 Pure Quadratic Equations 224 Affected Quadratic Equations 226 Problems 238 XXIII. Equations Solved like Quadratics 243 Equations in the Quadratic Form 243 XXIV. Simultaneous Equations Involving Quadratics 248 Problems 258 **Jlp XXV. Theory or Quadratic Equations 261 ' Factoring . 263 Discussion of the General Equation 268 XXVI. Zero and Infinity 270 Variables and Limits 270 The Problem of the Couriers 272 XXVII. Indeterminate Equations 275 XXVIII. Ratio and Proportion 278 Properties of Proportions . . ' 279 XXIX. Variation 287 -^XXX. Progressions . 291 Arithmetic Progression 291 Geometric Progression 299 Harmonic Progression 307 -AZ^XXI. The Binomial Theorem 310 ^^ ^' Positive Integral Exponent 310 *' viii CONTENTS. PAGE XXXII. Undetermined Coefficients 317 Convergency and Divergency of Series . . . . 317 The Theorem of Undetermined Coefficients . . . 320 Expansion of Fractions into Series 321 Expansion of Radicals into Series 323 Partial Fractions 324 Reversion of Series 330 XXXIII. The Binomial Theorem 332 Fractional and Negative Exponents .... 332 XXXIV. Logarithms .339 Properties of Logarithms 341 Use of the Table 346 Applications 351 Arithmetical Complement 353 Exponential Equations 357 Answers to the Examples. ALGEBRA, I. DEFINITIONS AND NOTATION. 1. Ill Algebra, the operations of Arithmetic are abridged and generalized by means of Symbols. 2. Symbols which represent Numbers. The symbols generally employed to represent numbers are the figures of Arithmetic and the letters of the Alphabet. Known Numbers are usually represented by the first letters of the alphabet, as a, 6, c. Unknown Numbers, or those whose values are to be determined, are usually represented by the last letters of the alphabet, as a?, y, z. 3. Symbols which represent Operations. The following symbols have the same meaning in Alge- bra as in Arithmetic : +, read ^^plusy — , read ^'^ninus.'^- X , read " times^" " mto," or " multiplied by^ -!-, read ^'divided by.^^ The sign of multiplication is usually omitted in Algebra, except between arithmetical figures. Thiis, 2 X a; is written 2 x. Division is usually indicated by a horizontal line. Thus, a -J- 6 is written -• b 2 ALGEBRA. 4. The Sign of Equality, =, is read ^'equalsj^' or 'Hs equal toJ^ An Equation is a statement that two numbers are equal. SOLUTION OF PROBLEMS BY ALGEBRAIC METHODS. 5. The following examples will illustrate the use of Algebraic symbols in the solution of problems. The utility of the process consists in the fact that the unknown numbers are represented by symbols, and that the various operations are stated in Algebraic language. 1. The sum of two numbers is 30, and the greater exceeds the less by 4 ; what are the numbers ? We will represent the less number by x. Then the greater will be represented by x + 4- By the conditions of the problem, the sum of the greater number and the less is 30 ; this is stated in Algebraic language as follows : a; + 4 + X = 30, (1) But the sum of x and x is twice x, or 2x; whence, equation (1) may be written 2 x + 4 = 30. Now it2x plus 4 equals 30, 2 x must equal 30 - 4, or 26. Whence, 2x = 26. • But if twice x is 26, x must be one-half of 26, or 13. Hence, the less number is 13, and the greater is 13 + 4, or 17. The written work will stand as follows : Let X = the less number. Then, x + 4 = the greater number. By the conditions, x -\- 4 + x = SO. Or, 2x + 4 = 30. Whence, 2 x = 26. Dividing by 2, x = 13, the less number. Whence, x + 4 = 17, the greater number. DEFINITIONS AND NOTATION. 3 2. The sum of the ages of A and B is 109 years, and A is 13 years younger than B ; find their ages> Let X represent the number of years in B's age. Then, x — 13 will represent the number of years in A's age. By the conditions of the problem, the sum of the ages of A and B is 109 years. Whence, a; + x - 13 = 109. Or, 2 X - 13 = 109. Now if 2 X minus 13 equals 109, 2 x must equal 109 + 13, or 122. Whence, 2 x = 122. Dividing by 2, x = 61, the number of years in B's age. And, X — 13 = 48, the number of years in A's age. The written work will stand as follows : Let X = the number of years in B's age. Then, x — 13 = the number of years in A's age. By the conditions, x + x — 13 = 109. Or, 2x-13 = 109. Whence, 2 x = 122. Dividing by 2, x = 61, the number of years in B's age. Therefore, x — 13 = 48, the number of years in A's age. 3. A, B, and C together have %^^. A has one-half as much as B, and C has 3 times as much as A. How much has each ? Let X = the number of dollars A has. Then, 2 x = the number of dollars B has, and 3 X = the number of dollars C has. By the conditions, x + 2x + 3x = 66. But the sum of x, twice x, and 3 times x is 6 times x, or 6 x. Whence, 6x = 66. Dividing by 6, x = 11, the number of dollars A has. Whence, 2 x = 22, the number of dollars B has, and 3 X = 33, the number of dollars C has. ALGEBRA. PROBLEMS. 4. The greater of two numbers is 4 times the less, and their sum is 70. What are the numbers ? 5. Tlie sum of the ages of A and B is 116 years, and A is 18 years younger than B. What are their ages ? 6. Divide 123 into two parts, such that the greater exceeds the less by 67. 7. The sum of the ages of A and B is 102 years, and A is 26 years older than B. What are their ages ? 8. Divide $ 93 between A and B, so that A may receive $ 23 less than B. 9. Divide $ 56 between A and B, so that A may receive 6 times as much as B. 10. Divide 85 into two parts, one of which shall be 19 less than the other. 11. Divide $ 72 between A and B, so that A may receive one-third as much as B. 12. A certain hall contains 425 persons ; there are 3 times as many men as women, and 4 times as many women as children. How many are there of each? 13. A man had ^4.95. After spending a certain sum, he found that he had left 4 times as much as he had spent. How much did he spend? 14. A, B, and C together have $96. B has twice as much money as C, and A has as much as B and C together. How much has each ? 15. The sum of three numbers is 168. The second is 23 less than the first, and the third is 3 times the second. What are the numbers ? 16. A, B, and C together have $ 230. A has $ 21 more than B, and $ 17 less than C. How much has each ? DEFINITIONS AND NOTATION. 5 17. A watch and chain are together worth ^5G, and tlie chain is worth one-sixth as much as the watch. What is the value of each ? 18. Divide 169 into three parts, the first of which is one- half of the second, and the second one-tifth of the third. 19. Divide $ 144 into three parts such that the second is one-third of the first, and one-fourth of the third. 20. A man bought a cow, a sheep, and a hog for ^.84. The price of the hog was one-fifth the price of the cow, and $ 7 less than the price of the sheep. What was the price of each ? 21. The sum of three numbers is 127. The first is one- half of the third,' and 17, greater than the second. What are the numbers ? 22. At a certain election, two candidates, A and B, to- gether received 508 votes; and A had a majority of 136. How many did each receive ? 23. The sum of the ages of A, B, and C is 101 years. A is 17 years younger than B, and 15 years older than C. What are their ages ? 24. Divide $ 174 between A, B, and C, so that A may receive 4 times as much as B, and $ 42 more than C. 26. My horse, carriage, and harness are together worth $ 456. The carriage is worth 8 times as much as the har- ness, and $ 48 less than the horse. Find the value of each. 26. Divide $ 155 into three parts such that the first shall be 5 times the second, and one-fifth of the third. 27. At a certain election, three candidates. A, B, and C, together received 512 votes. A received 28 less than B, and 64 less than C. How many did each receive ? 28. Divide $ 69 between A, B, C, and D, so that A may receive ^5 more than B, C as much as A and B together, and D as much as A and C together. 6 ALGEBRA. 29. The sum of four numbers is 160. The first is 3 times the second, the second 3 times the third, and the third 3 times the fourth. What are the numbers ? DEFINITIONS. 6. If a number be multiplied by itself any number of times, the result is called a power of that number. An Exponent is a number written at the right of, and above another number, to indicate what power of the latter is to be taken. Thus, o?, read '' a square/^ or " a second power/' denotes a x a ; a^, read " a cube,'' or " a third power," denotes a x a x a; a*, read " a fourth," or " a fourth power," denotes axaxaxa, and so on. If no exponent is expressed, the first power is understood. Thus, a is the same as a^. 7. Symbols of Aggregation. The parentheses ( ), the brackets [ ], the braces {. \ , and the vinculum , indicate that the numbers enclosed by them are to be taken collectively ; thus, (a -\-b) X c, \_a-\-b~\x c, \a-\-b\x c, and a -f- 6 x c all indicate that the result obtained by adding 6 to a is to be multiplied by c. ALGEBRAIC EXPRESSIONS. 8. An Algebraic Expression, or simply an Expression, is a number expressed in algebraic symbols ; as, 2, a, or 2x^-^ab-{-5. The Numerical Value of an expression is the result obtained by substituting particular numerical values for the letters involved in it, and performing the operations indicated. DEFINITIONS AND NOTATION It' Find the numerical value of the expression when a = 4, b = S, c = 5, and d = 2. We have, 4a + ^-rf3 = 4x4+ ^-^ - 23 b • 3 = 16 + 10 - 8 = 18, Ans. EXAMPLES. Find the numerical value of each of the following when = 3, b = o, c = 2, d = 4, m = 4, and 7i = 3 : 8. a'^d''. 9. ^ + 5-1 be a 10. Sa'-9(f. 11. i+ui+i. a 6> c tt 12 ^ — .^ ' 2c2 4^2' 13. ^V^- + ^. 36" • 62^c2^f/2 If the expression involves parentheses, the operations indicated ivithin the parentheses must be performed first. 14. Find the numerical value, when a = 9, 6 = 7, and c = 4, of {a-h)(b + c)-pdi. — c We have, a - 6 = 2, 6 + c = 11, a + 6 = 16, and Z> - c = 3. Then the numerical value of the expression is 2xll-l« = 22_l« = 50, ^„,. 3 3 3 2. acZ^ - b(^. 3. Sabcd. 4. 4 a^fZ - 5 6c - -6cd. 5. ab _^cd To" 6. a 6 c 7. 5a- g . ALGEBRA. Find the numerical value of each of the following expres- sions, when a= Of b = 3, c = 4, and d = 2 : 1-5. f^-ff- 17. Sa\c-d)-2b\c-{-d). 16. (a'-{-b'-cy. 18. 5(a + by -9(c- ay. 19. (a - 5) (6 + c) - (c - d) {d + a). 20. (a - 6 + c - d) (a + 6 - c - d). 21 3a — 56 + 7 c go ^ + 6 6 + c c-fc/ '4a — 36H-6c b -{- c c.-\- d d -\- a Find the numerical value of each of the following expres- sions, when a = -J, 6 = ^, c = |, and a; = 3 : . 23 g + 6 _ a — b 24 16 a — 18 6 + 15 c j ' a-6 a + 6* " 44 a -32 6 -27c" 25. a^-(4a-5c)a^+(2a + 6)a;-12a6c. \6 ay \a 6 cj 9. Axioms. An Axiom is a truth which is assumed as self-evident. Algebraic operations are based upon the following axioms : 1. If the same operation be performed upon equal number Sy the resulting numbers will be equal. 2. If the same number be both added to, and subtracted from another, the value of the latter will not be changed, S. If a number be both multiplied and divided by another, the value of the former will not be changed. 4. Numbers which are equal to the same number, are equal to each other. POSITIVE AND NEGATIVE NUMBERS. II. POSITIVE AND NEGATIVE NUMBERS. 10. Many concrete magnitudes are capable of existing in two opposite states. Thus, in financial transactions, we may have gains, or losses; in the thermometer, we may have temperatures above zero, or beloiv zero ; a place on the surface of the earth may be in north latitude, or south latitude ; etc. The signs -f- and — , besides indicating the operations of addition and subtraction, are also used in Algebra to distin- guish between the opposite states of magnitudes like the above. Thus, in financial transactions, we may indicate gains or assets by the sign -f , and losses or debts by the sign — ; for example, the statement that a man's property is — f 100, means that he has debts or liabilities to the amount of $ 100. Again, in the thermometer, we may indicate temperatures above zero by the sign -f , and temperatures beloiv zero by the sign — ; for example, + 25° means 25° above zero, and — 10° means 10° below zero. Also, we may indicate north latitude and west longitude by the sign +, and south latitude and east longitude by the sign — ; thus, a place in latitude — 30°, longitude + 95°, would be in latitude 30° south of the equator, and in longi- tude 95° west of Greenwich. EXERCISES. 11. 1. At 7 A.M. the temperature is — 13°; at noon it is 8° warmer, and at 6 p.m. it is 5° colder than at noon. Re- quired the temperatures at noon and at 6 p.m. 2. At 7 A.M. the temperature is +6°; at noon it is 14° colder, and at 6 p.m. it is 8° warmer than at noon. Required the temperatures at noon and at 6 p.m. 10 ALGEBRA. 3. What is tlie difference in latitude between two places whose latitudes are + 67° and — 48° ? 4. A man has bills receivable to the amount of $ 480, and bills payable to the amount of $ 925 ; how much is he worth ? 6. A vessel sails from the equator due north 28°, and then due south 57° ; what is her latitude at the end of the voyage ? 6. At 7 A.M. the temperature is — 7°, and at noon -f- 9°. How many degrees warmer is it at nooii than at 7 a.m. ? 7. What is the difference in longitude between two places whose longitudes are -f 29° and — 86° ? 8. The temperature at 6 a.m. is +14°; and during the morning it grows colder at the rate of 4° an hour. Required the temperatures at 9 a.m., at 10 a.m., and at noon. 12. Positive and Negative Numbers. If the pot-xtive and negative states of any concrete mag- nitude be expressed without reference to the unit, the results are called positive and negative numbers, respectively^ Thus, in + $ 5 and — $ 3, + 5 is a positive number, and — 3 is a negative number. For this reason the sign + is called the positive sign, and the sign — the negative sign. If no sign is expressed, the number is understood to be positive ; thus, 5 is the same as + 5. The negative sign can never be omitted before a negative number. 13. The Absolute Value of a number is the number taken independently of the sign affecting it. Thus, the absolute value of — 3 is 3. POSITIVE AND NEGATIVE NUMBERS. H ADDITION OF POSITIVE AND NEGATIVE NUMBERS. 14. The result of Addition is called the Sum. We shall retain for Addition in Algebra its arithmetical meaning, .so long as the numbers to be added are positive. We may then attach any meaning we please to addition involving other forms of number, provided the new meaning is not inconsistent with principles which have been pre- viously established. 15. If a man gains $ 5, and then loses $ 3, he will be worth $2. If he owes $5, and then gains $S, he will be in debt to the amount of ^2. If he owes $5, and then incurs a debt of $3, he will be in debt to the amount of $S. Now with the notation of § 10, losing $3, or incurring a debt of ^ 3, may be regarded as adding — $ 3 to his property. Whence, the sum of + $ 5 and - $ 3 is -f ;^ 2 ; the sum of -$5and -h$3 is -$2; and the sum of — $ 5 and — f 3 is - $ 8. Or, omitting reference to the unit, (+5) + (-3) = +2; (_5) + (+3)=-2; (_5) + (_3)=-8. We then have the following rules : To add a positive and a negative number, subtract the less absolute value (§ 13) from the greater, and prefix to the result the sign of the number having the greater absolute value. To add two negative numbers, add their absolute values, and prefix a negative sign to the result. 12 ALGEBRA. 16. 1. Find the sum of + 10 and — 3. Subtracting 3 from 10, the result is 7. Whence, ( + 10) + ( - 3) = + 7 , Ans. 2. Find the sum of — 12 and + 6. Subtracting 6 from 12, the result is 6. Whence, ( - 12) + ( -f 6) = - 6, Ans. 3. Add - 9 and - 5. The sum of 9 and 5 is 14. Whence, ( - 9) + ( - 5) = - 14, Aiis. EXAMPLES. Find the values of the following : 4. (_7) + (-5). 10- (-61) + (+28). 5. (+9) + (-4). 11. (-!) + (+! 6.(-8) + (+2, l.(-|) + (-f). 7. (+6) + (-15). - 13. (\4+\Joj). A 8. (-11) + (-16). u. (-18t75.) + (+12|).'"J* 9. (+52) + (-37). 15. (+20Jy) + (-13A).> MULTIPLICATION OF POSITIVE AND NEGATIVE NUMBERS. 17. The terms Multiplicand, Multiplier, and Product have the same meaning in Algebra as in Arithmetic. We shall retain for Multiplication, in Algebra, its arith- metical meaning, so long as the multiplier is a positive number. That is, to multiply a number by a positive integer is to add the first number as many times as there are units in the second. POSITIVE AND NEGATIVE NUMBERS. 13 For example, to multiply — 4 by 3, we add — 4 three times. That is, (-4) X (4-3) = (-4) + (-4) +(-4) =- 12. We may then attach any meaning we please to multipli- cation by a negative number. 18. In Arithmetic, the product of two numbers is the same in whatever order they are taken. Thus, 3x5 and 5x3 are each equal to 15. If we assume this law to hold universally, we have (+3)x(-4) = (-4)x(4-3). But by § 17, (- 4) X (H- 3) = - 12 = - (3 X 4). Whence, (-f 3) x (- 4) = - (3 x 4). (§9, 4) We then have the following definition : To imdtiply a number by a negative number is to multiply it by the absolute value (§ 13) of the multiplier^ and change the sign of the result. Thus, to multiply + 4 by — 3, we multiply + 4 by -f 3, giving 4- 12, and change the sign of the result. That is, (4-4) x (-3) = -12. Again, to multiply — 4 by — 3, we multiply — 4 by 4- 3, giving — 12 (§ 17), and change the sign of the result. That is, (- 4) X (- 3) = -f 12. ; 19. From §§ 17 and 18 we derive the following rule : To multiply one number by another, multiply together their absolute vcdues. Make the product plus ichen the multiplicand and multiplier are of like sign, and minus when they are of unlike sign. 14 ALGEBBA. 1. Multiply + 8 by - 5. Bytherule, (+ 8) x (- 5) = -(8 x 5) = - 40, Ans. 2. Multiply - 7 by - 9. Bytherule, (- 7) x (- 9) = + (7 x 9) = + 63, Ans. EXAMPLES. Find the values of the following : 3. (+6)x(-3). 10. (-24)x(-18). 4. (-10)x(+5). U. f+4X /_5 5. (-7)x(-6). I Tj I 9 12. f-^^>,f-± 6. (-12) X (+4). V ISy ^ 14 7. (-8)x(-8). 13.-(-|)x(+|). 8. (_15)x(+9). . 14. (+,9|)^(_2|). 9. (+ll)x(-16). 15. (-I«)x(-1A). ADDITION. 15 III. ADDITION AND SUBTRACTION OF ALGEBRAIC EXPRESSIONS. DEFINITIONS. 20. A Monomial, or Term, is an expression (§ 8) whose parts are not separated by the signs -f or — ; as 2 a^, —Sab, or 5, 2a^, — 3 ah, and + 5 are called the tei'7ns of the expression 2 X- — 3 ab -\- 5. A Positive Term is one preceded by a -f sign ; as + 5 a. If no sign is expressed, the term is understood to be posi- tive. A Negative Term is one preceded by a — sign ; as —Sab. The — sign can never be omitted before a negative term. 21. If two or more numbers are multiplied together, each of them, or the product of any number of them, is called a Factor of the product. Thus, a, b, c, ab, ac, and be are factors of the product abc. 22. Any factor of a product is called the Coefficient of the product of the remaining factors. Thus, in 2 ab, 2 is the coefficient of ab, 2 a of b, a ot 2b, etc. 23. If one factor of a product is expressed in numerals, and the other in letters, the former is called the numerical coefficient of the latter. Thus, in 2 db, 2 is the numerical coefficient of ah. If no numerical coefficient is expressed, the coefficient 1 is understood ; thus, a is the same as 1 a. 16 ALGEBRA. 24. By § 19, (- 3) X a = - (3 X a) = - 3a. That is, —3a is tlie product of — 3 and a. Then, — 3 is the numerical coefficient of a in —3 a. Thus, in a negative term, the numerical coefficient includes the sign. 25. Similar or Like Terms are those which do not differ at all, or else differ only in their numerical coefficients ; as 2 sc^y and — 7 x^y. Dissimilar or Unlike Terms are those which are not simi- lar ; as 3 x^y and 3 xy^. ADDITION OF MONOMIALS. 26. The sum of a and & is a -f 6 (§ 3) ; and the sum of a and —b is expressed a + (— 6). 27. Required the sum of a and — b. By § 10, if a man incurs a debt of $ 4, we may regard the transaction either as adding — $ 4 to his property, or as subtracting $ 4 from it. That is, adding a negative number is equivalent to subtract- ing a positive number of the same absolute value (§ 13). Thus, the sum of a and — 6 is obtained by subtracting 6 from a. Or, a + (-&)= a -6. 28. It follows from §§ 26 and 27 that the addition of monomials is effected by uniting them with their respective signs. Thus, the sum of a, —b,c,— d, and — e is a — b-i-c — d — e. It is immaterial in what order the terms are united, pro- vided each has its proper sign. ADDITION. 17 Hence, the above result may also be expressed c + a — e — d — b, — d — b-\-c — e + a, etc. 29. If the same number be both added to, and sub- tracted from another, the value of the latter will not be changed (§ 9). That is, a + b — b = a. Hence, terms of equal absolute value, but opposite sign, in an expression, neutralize each other, or cancel 30. To multiply 4 by 5 + 3, we multiply 4 by 5, and then 4 by 3, and add the second result to the first. In like manner, to multiply a by 6 + c, we multiply a by b, and then a by c, and add the second result to the first. That is, a(b -h c) = ab -\- ac. 31. Addition of Similar Terms (§ 25). 1. Find the sum of 5 a and 3 a. We have, 6 a + 3 a = (5 + 3)a (§30) = 8 a, Ans. 2. Find the sum of — 5 a and — 3 a. We have, (- 5a) + (- 3a) = (- 6) x a + (- 3) x a (§19) = [(-5) + (-3)]xa (§30) = (-8)xa (§15) = -8a, Ans. (§19) 3. Find the sum of 5 a and — 3 a. We have, 6a +(- 3)a =[5 +(- 3)] x a (§30) = 2 a, Ans. (§16) 18 ALGEBRA. 4. Find the sum of — 5 a and 3 a. We have, (_ 5)a + 3a =[(- 5)+ 3] x a (§30) = (-2)x« (§15) = — 2a, Aus. Therefore, to add ttvo similm' terms, find, the sum of their numerical coefficients (§§ 15, 24), and affix to the result the commoyi letters. EXAMPLES. Add the following : 5. 5a and —12a. 9. —be and 6bc. 6. — 7 m and — 8 m. 10. xyz and — 9 xyz. 7. 15 X and -llx. 11. -18??iV and -27mV. 8. -lOa^andlal 12. 86 a^ftc^ and - 19 a^ftc^. 13. Eequired the sum of 2 a, — a, 3 a, — 12 a, and 6 a. Since the order of the terms is immaterial (§ 28), we may add the positive terms first, and then the negative terms, and finally combine these two results. The sum of 2 a, 3 a, and 6 a is 11 a. The sum of — a and — 12 a is — 13 a. Then the required sum is 11 a + (— 13 a), or — 2a, Ans. Add the following : 14. 9 a, — 7 a, and 8 a. 15. 13 x, — x, — 10 Xy and 5 x. 16. 12 abc, abc, — 6 abc, and — 17 abc. 17. 15 m^, — 11 m^, — 4 m^, m^, and 14 ml 18. 21 a^2/^ - 16 x^, - x^y\ 3 x^y\ and - 19 3^y\ If the terms are not all similar, we may combine the similar terms, and unite the others with their respective signs (§ 28). ADDITION. 19 19. Kequired the sum of 12 a, —5 a;, —Sy^, —5 a, 8 a;, and — 3 x. The sum of 12 a and — 5 a is 7 a. The sum of - 5 x, 8 a;, and - 3 a; is (§ 29). Then the required sum is 7 a — 3 y'^, Ans. Add the following : 20. Sab, —7 cd, — 5 a6, and 3 cd. 21. Qx, —lOz, 2 y, 4 2, — 9 y, and — x. 22. 12"m% - 2 wi, - 8 n, 5, - 3 n, - 7 m^, and 11 n. 23. 10a, -Gd, -5c, 126, - a, c, -3c, and -9a. 24. 7 X, —4:y, — Sz, 9y, — 2x, — Sx, — 5 2;, 6 ?/, and — z. DEFINITIONS. 32. A Polynomial is an algebraic expression consisting of more than one term ; as a -f 6, or 2 a;^ — 3 a;y — 5 y'i A Binomial is a polynomial of two terms ; as a -h b. A Trinomial is a polynomial of three terms. 33. A polynomial is said to be arranged according to the descending powers of any letter, when the term containing the highest power of that letter is placed first, that having the next lower immediately after, and so on. Thus, x^-\-Safy-2a^y"-{-3xy -4:y* is arranged according to the descending powers of x. Note. The term — 4 y^, which does not involve x at all, is regarded as containing the lowest power of x in the above expression. A polynomial is said to be arranged according to the ascending powers of any letter, when the term containing the lowest power of that letter is placed first, that having the next higher immediately after, and so on. Thus, x''-\-3^y-2xY + 3^-^y^ is arranged according to the ascending powers of y. 20 ALGEBRA. ADDITION OF POLYNOMIALS. 34. A polynomial may be regarded as the sum of its separate monomial terms (§ 28). Thus, 2a — 3& + 4cis the sum of 2 a, —3b, and 4 c. Hence, the addition of polynomials may he effected by uniting their terms ivith their respective signs. 1. Required the sum of Ga — 7oc^, 3a^ — 2a-\-3y^, and 2x^ — a — mn. It is convenient in practice to set the expressions down one under- neath the other, similar terms being in the same vertical column. We then add the terms in each column, and unite the results with their respective signs. Thus, 6 a - 7 x2 — a + 2 a;2 — mn 3 a — 2 a;2 + 3 2/3 — mn, Ans. EXAMPLES. Add the following : 2. 3. 4. 7a_55 _ 8m2+ 6n^ -19a6- led - 9 a + 2 & 12 m^ - 16 n« Sab -lied 3a- b - Gm^ 4-14^3 6ab-\-13cd 5. 4a-66H-3cand5a + 26-9c. 6. m^ + 2 mw -f n^, m^ — 2mn-\- n^, and 2m^ — 2n\ 7. 3x — Sy,ly — 6z, and 5z — 2x. 8. 2a^-^ab-b% la''+3ab-^b\ and - 4a2_6a6 -f- 8 61 9. 4«-3a;2-ll + 5ar^, 12ar^-7-8ar«-15a;, and 14 + 6a^ + 10a;-9cc^. ADDITION. 21 Note. It is convenient to arrange the first expression in descending powers of x (§ 33), as follows : 5a;3_3^2_,.4a,_n. and then write the other expressions underneath the first, similar terms being in the same vertical column. 10. 2a-3b-5c, Sb -\-6c-{-7 d, -4a-3c4-2d, and 7 a — b — 9d. 11. x'-3xy'-2a^y, 3x^y - 5if - Axf, 5xy'' - 6 f -7 j^, and Sf-j-7x^-dxry. 12. 6a-Sb-2c, 12c-\-9d-7a,llb-10c-5d, and —3b — 4:d-\-a. 13. 15a'-2-9a'-3a, 13a - 5a^ -6 -7 a^, 8-f 4a-8a3-7a2, and IGa^ + Sa^- 10a - 2. 14. 9a2-1362-18c2, lo c' -^ 12 b' - S d', 19d^-Ua'-\-3c% and - 2^^- 16^^ + lla^. 15. 12a^-c^-{-4:ax'-5a\ ISa^ -2a'x-3a^ -13ax', 15a'x - lla^ - 17 a^ -\- 3as^, and Gax^-Sd-x-7x^-i-9 a?. 16. 13.T2+3-4.r4-8a^, - 9a; + 5 4- 16a^ H- a^, -15-6i«2-7a;3 + lla', and - lOar^ - 12a; + 14a;2 _ 17. SUBTRACTION. 35. Subtraction, in Algebra, is the process of finding one of two numbers, when their sum and the other number are given. The Minuend is the sum of the numbers. The Subtrahend is the given number. The Remainder is the required number. 22 ALGEBRA. 36. The remainder when b is subtracted from a is ex- pressed a — 6 (§ 3) ; and the remainder when — b is sub- tracted from a is expressed a — (— 6). 37. Let it be required to subtract — b from a. By § 35, the sum of the remainder and the subtrahend is equal to the minuend. Therefore, the required remainder must be such an ex- pression tliat, when it is added to — b, the result shall equal a. Now if a + 6 be added to — b, the result is a. Hence, the required remainder is a + b. That is, a — (— b) = a -\- b. 38. From §§36 and 37, we have the following rule: To subtract one number from another, change the sign of the subtrahend, and add the result to the minuend. SUBTRACTION OF MONOMIALS. 39. 1. Subtract 5 a from 2 a. It is convenient to place the subtrahend under the minuend. We then change the sign of the subtrahend, giving — 5 a, and add the result to the minuend. Thus, 2a — ba — 3 «, Ans. 2. Subtract —5a from —2a. The student should perform mentally the operation of changing the sign of the subtrahend ; thus, in Ex. 2, we mentally change - 6 a to 5 a, and then add 5 a to —2 a. -2a — 6a 3 a, Ans. 9 from - -25. 9. -5 from 16. 5 from 5. 10. 12 from -17. 26 from -18. 11. -14 from 13. 14. 15. 16. -lab 14 m^n 21xyz 11 ah -Sm^w Ma^z SUBTRACTION. 23 EXAMPLES. Subtract the following : 3. 7 from 4. 6. 4. 4 from - 11. 7. 5. -15 from -9. 8. 12. 13. 15a -12r^ 6a -Six" 17. — xy from xy. 21. — 45aaJ*from — 19aa^. 18. -16a« from -Ua'. 22. SI a'b^ from Sa^ft'*. 19. 21 m'7i^ from 39 mV. 23. From 8 a take - 12 b. 20. 19 abc from - 6 a6c. 24. From - 3 m^ take 4 n^ 25. From —23 a take the sum of 19 a and —5 a. Note. A convenient way of performing examples like the above is to write the given expressions in a vertical column, change the sign of each expression which is to be subtracted, and then add the results. 26. From the sum of — 18 xy and 11 ooy, take the sum of — 29 an/ and 17 ocy. 27. From the sum of 26 a^ and — 7 a^, take the sum of -15a2 and 48a2. 28. From the sum of 33 n^x and — 16 n% take the sum of 49 n\ — 27 n\ and — 39 n^x. SUBTRACTION OF POLYNOMIALS. 40. A polynomial may be regarded as the sum of its sep- arate monomial terms (§ 28) ; hence, To subtract one polynomial from another, change the sign of each term of the subtrahend, and add the result to the minuend. 24 ALGEBRA. 1. Subtract 7ah^-9 arh + 8 6^ from 5 a^ - 2 a^^b + 4 ab\ It is convenient to place the subtraliend under the minuend so that similar terms shall be in the same vertical column. We then mentally change the sign of each term of the subtrahend, and add the result to the minuend. Thus, 5 a^ - 2 a^h + 4 ah"^ - 9 a2& + 7 alfi + 8 ft^ 5 a3 + 7 a% - 3 aft^ _ 8 6^, Ans. EXAMPLES. Subtract the following : 2. 12a2-9a-7 3. 2a64- 56c-3c« 8 ^2 _ (5 ^ _l_ 13 — a6 + 11 6c — 4 ca 4. From x- — 2xy -{-y^ subtract o? -\-2xy -\- y^. 5. From 5a — 36 + 4c subtract 5 a + 3 6 — 4 c. 6. From 4aj3-9a;2+llaj-18 take 3a^-8a^+17a;-25. 7. Fi'om 8a7 — 3?/ — 42; take — 2; + 11 a^ — 6 ^. 8. Take 76-9c-2cZ from 6 a - 5 6 + 12 c. 9. Take 12 a^ + 4 a - 9 from 3 a^ + 8 a^ - 6. 10. Subtract a^- 7 - 2a; -Ga.-^ from 5a;2-12 H-9a^-2a;. (See Note to Ex. 9, page 21.) 11. Subtract 1 + a^ — a — a^ from 3 a — 3 a^ + 1 — a^. 12. Take 81 5^ + 4 a^ - 36 ah from - 30 a6 + 9 a^ + 25 h^. 13. From lOa^ - 21a^ - 11a; take - 15 a;^ _ 20 a; + 12. 14. From 17 a» - 12 ah'' + 5 6^ take 8 a=^ - 3 a-6 + 13 h\ 15. Take -o^^Z^y-'^xy^^f from x'-^x^y-lxf^-f. 16. Take 6c-5(^-96-4a from -106-2c+3a-9cZ. 17. Subtract 4-3a;-a;2 + 8a;3_^lQ^4 from 9-7a; + 6a;2_i2a;3^5^4 SUBTRACTION. 25 18. Subtract 20:^ - xy -\-Si/ -9x- Uy from 3x^-5xy-\-2y'^-2x + 7y. 19. From 7 a - 11 a"^ - 8 -f 6 a* subtract 16 a^ _ 9 + 2 a^ + lo a - 10 a\ 20. From a^ + Sx^y -xY' -\-oa^if-4tX7/*' subtract Sx^y-7 xY - Q 3?f -\- 11 xy^ - f. 21. From d^-^2ab-^b'^ subtract the sum oi —a^^2ab-b^ and -2d' + 2b\ Note. Write the expressions one underneath the other, similar terms being in the same vertical column, change the signs of the terms of each expression which is to be subtracted, and add the results. 22. From the sum oi So?-\-2ab- b^ and a^- 8 a6 -f 6 6^ take 6a--oab-\-5b\ 23. Subtract the sum of 9x^ — Sx-\-x^ and 5—Jt^-{-x from 6x^ — 7 X — 4i. 24. Subtract the sum of x-\- y — Sz and — 4:X-\-9y from the sum of dx — 2y — z and — 5x -^ 6y — 7 z. 25. Take the sum of 6 — 4 x-^ — a; and 5 ic — 1 — 2 ic^ from the sum of 2x^ -\-7 — 4:X — 5x^ and 3a^-63^-2-^Sx. 26. From the sum of 2a -36 + 4(i and 2 6 + 4c-3d, take the sum of — 4a — 46 + 3c — 2d and 3 a — 2 c. 27. From the sum of 9 a^ — a^ — 5 and 3 a^ — a + 1, take the sum of -8a3+13a + 3 and 5a''+2a2-6a. 26 ALGEBRA. IV. PARENTHESES. REMOVAL OF PARENTHESES. 41. The expressions a — b-\-(c — d) and a — b — (c — d) indicate that the expression c — d is to be respectively- added to, and subtracted from, a — b. If the operations be performed, we have by § § 33 and 40, a — b + {c — d) = a — b-{-c — d, and a — b — (c — d) = a — b — c-]-d. In the first case, the signs of the terms within the paren- thesis are not changed when the parenthesis is removed; while in the second case, the sign of each term within is changed, from + to — , or from — to +. We then have the following rules : A parenthesis preceded by a -\- sign may be removed without changing the sigyis of the terms enclosed. A parenthesis preceded by a — sign may be removed if the sign of each term enclosed be changed, from -\- to —, 'or from — to +. 42. The above rules apply equally to the removal of the brackets, braces, or vinculum (§ 7). It should be noticed in the case of the latter that the sign apparently prefixed to the first term underneath is in reality prefixed to the vinculum ; thus, -{-a — b means the same as + (a — b), and —a—b the same as — (a — b). 43. 1. Remove the parentheses from 2a -36 -(5a -46) + (4a -6). By the rules of § 41, the expression becomes 2a — 36 — 5a + 46+4a — 6 = a, Ans. PARENTHESES. 27 Parentheses are often found enclosing others; in this case they may be removed in succession by the rules of § 41 ; and it is better to remove first the innermost pair. 2. Simplify 4a;-J3a;4-(-2x-a;-a)|. Removing the vinculum first, and the others in succession, we have Ax - {^x -\- {- 2x - X - a)} = 4x -{Sx+(-2x-x-{- a^} = ix—{Sx — 2x — x-{-a} = ix — 3x +2x-\- X — a = Ax — a, Ans. EXAMPLES. Simplify the following expressions by removing the parentheses, etc., and uniting similar terms: 3. 8a + (56-a)-(-76 + 2a). 4. 4m-[2mH-9n] -S-5>?i-6Vij. 5. x-i-y — z-\-y — z — X — z — x-\-y. 6. ab - 4.b^ - {2a' - b^ -\- 5a^ -\-2 ab -Sb^ 7. m^ — 3 mn -\-5m' — mn — 6 w^ — [8 m^ — 4 mn — 7 n^]. 8. 4a; -(5a;- [3a; -1]). 9. a-(6 — c + cZ + e). 10. 5 ab - [(3 ab - 10) -(-4:ab- 7)]. 11. 7ar^ |-(-3ar^H-2a;-5)-(4ic2-6a;-2). 12. m-(6m-7w) -j-3m + 4:W-(2m-3w)j. 13. 17 - [45 - (9 - 23 - 32)]. 14. 3(i-(5a-J-7a + [9a-4]|). 15. a; - [2 a; - (- a; -f 1) + 3] - J6a; - [- (a; - 3) - a;] J. 16. X - (y -^ z - [x ~ {- X - y) -{- zj) -\-lz - 2x-y\. 17. 27i-[37i-J4n-n-4|-(-5n-9)]. 18. 28-j-16-(-4+[55-31-f47])S. 28 ALGEBRA. 19. a-(2a-lSa-\4.a-5a-l\J). 20. c-[2c-(6a-6)-Jc-5a+2 6-(-5a+6a~3 6)S]. 21. x—[y — \x — z — x — y-{-z\-\-{2x — \ — x-\- y\)]. 22. 5x-l2x-(-x-{2x-^r^\-Sx)-Sx']. 23. a-\-a-[-a-(-a-\-a-a-l\)]\. INTRODUCTION OF PARENTHESES. 44. To enclose any number of terms in a parenthesis, we take tlie converse of the rules of § 41 : Any number of terms may be enclosed in a parenthesis preceded by a -\- sign, without chayiging their signs. Any number of terms may be enclosed in a parenthesis pre- ceded by a — sign, if the sign of each term be changed, from + to —, or from — to +. 1. Enclose the last three terms of a — b-\-c — d-\-e in a parenthesis preceded by a — sign. Result, «— 6— (— c + d — e). EXAMPLES. In each of the following expressions, enclose the last three terms in a parenthesis preceded by a — sign: 2. a-[-b-c-d. 6. o? -{-f + z^ -3xyz. 3. a;3-5a^-8a; + 7. 7. a-b-c + d-^e. 4. m^ + m^n + mn^ -f 7i3. 8. a* + Ba^ -f- a^- 9a -f- 2. 5. a^ — b^ -\-2bc— cl 9. ^ — m^ — 2 mn — n^. 10. In each of the above results, enclose the last two terms in parenthesis in brackets preceded by a — sign. MULTIPLICATION. 29 V. MULTIPLICATION. 45. The Law of Signs. If a and b are any two numbers, we have by § 19, (4- a) xi+b) = -{- ah, (-f- a) x(-b) = - ab, (— a) X (-\- b) = — ab, (— a) x (— b) = -\- ab. From these results, we may state the Rule of Signs in Multiplication as follows : -f- multiplied by +, and — midtiplied by — , prodxice + ; + multiplied by — , and — multiplied by +, produce — . Or, as it is usually expressed with regard to the product of two terras. Like signs produce -{-, and unlike signs produce — . 46. The Index Law. Let it be required to multiply a^ by a\ By § 6, a^ = a X a X a, and a^ = a X a. Whence, a^xar = axaxaxaxa=a^. Therefore, the exponent of a letter in the jyroduct is equal to its exponent in the multiplicand plus its exponent in the i multiplier. Or in general, if ni and n are any two positive integers, a*" X a" = «'"+". A similar result holds for the product of three or more powers of a. Thus, a^xa'x a' = a^^'+' = a''. 30 ALGEBRA. MULTIPLICATION OF MONOMIALS. 47. Let it be required to multiply 7 a by —2 b. We have, -2h = {-2)xh. (§45) Whence, Tax (-2&) = 7ax (-2) x h. Then since the order of the factors is immaterial (§ 18), Tax (-2Z>) = 7x(-2) xax5 = -14a6. (§ 19) 48. From §§ 45, 46, and 4T, we derive the following rule for the multiplication of two monomials : To the product of the. absolute values of the yiumerical coeffi- cients, annex the letters; giving to each an exponent equal to its exponent in the multiplicand plus its exponent in the mtdtiplier. Make the product -f- when the midtiplicand and multiplier are of like sign, arid — when they are of unlike sign. 1. Multiply 2 a^ by 9 al By the rule, 2a5x9a4 = 2x9x a^+^ = 18 a?, Ans. 2. Multiply a^Wc by -^w'bd. We have, a^&'^c x (- 5 a%d) = - 5 a^b^cd, Ans. 3. Multiply - Taj"* by 4^3 We have, ( - 7 x'») x 4 x^ = -28 x'"+% Ans. 4. Multiply -3ic"by -8a;^ We have, ( - 3 a:") x ( - 8 x«) = 24 a;"+» = 24 x^», Ans. EXAMPLES. Multiply t!ie following : 5. 7a^ by 3a^ 7. 5xyz by —llxyz. ?, -}4a6b^2cd 8. - 15 a% hj - 4: ah' , MULTIPLICATION. 31 9. -9mVby7mV. 13. - a'»6"c^ by - a6V. 10. -6a^b^ by - 6V. 14. -Sary*" by Ux'^y^ 11. Sa^z' by -8/2^. 15. 10a*6V by 9a'^-4:X^y-3xy^ + 2f hy 2x' + xy-2y\ MULTIPLICATION. 35 Find the product of the following : 36. x — 2, x — S, and ic — 4. 37. a -\- 5, 2 a — 3, and 4 a — 1. 38. x — y, x^ -\-xt/ -{- y^, and x" -\-if 39. 3?i — 8, 4 7t + 7, and hn — ^. 40. a — x^ Oj-\-x, o? -\- x^f and a'^ -\- a^. 41. m — 4 ?j, 2 m + 3 n, and 2 7?i^ -f 5 m7i — 12 7i^ 42. a -f 1, a — 1, a^ + a -h 1, and a^ — a + 1. 43. a^ -H a; + 1, a^ — a; + 1, and a^ — a:^ + 1. 44. a-\-h, a — b, 2a — 3 b, and 2a + 36. 46. a; + 3, 2a;-|-l, 2a;-l, and 4a^- 12ar^-h x- 3. 53. 1. Simplify (a - 2a;)2- 2(a;-f 3a)(a - a:). To simplify the expression, we should first multiply a-2x by itself (§ 6) ; we should then find the product of 2, x + 3 a, and a - x, and subtract the second result from the first. a -2x a -2x 3a +x a -X a2-2ax - 2 ax + 4 x2 3a2+ ax -Sax- x2 a2-4ax f 4x2 3a2_2ax- x2 2 6a2-4ax-2x2 Subtracting the second result from the first, we have a2 - 4 ax + 4x2 - 6a2 + 4ax + 2x2 = - 5a2 4- 6x2, AiiS. EXAMPLES. Simplify the following : 2. (3a;-8)(a; + 6) + (2a;-7)(4a;4-9). 3. (2a + 6)(3a-7)-(2a-5)(3a + 7). 36 ALGEBRA. 4. (a — m) (b + n) + (a + m) (6 — n). 5. (x-y-\-zy-(x + y - zf. 6. (a-b-c + ay. 7. (2x + 3y(2x-3y. 8. (a + &) (a' - 6^) -(«-&) (a^ + b'). 9. (3i«-52/)'-5(a;-?y)(aj-52/). 10. (a + x) (a^ 4- a^'') [a' - .t (a - a;)]. 11. (a - b) (a' + a'b + a5^ + 6^) [(a^ + &2)2 _ 2^252]. 12. (x -\-l)(x + 2) (a; + 3) - (a; - 1) (x -2)(x- 3). 13. (x -y){y-z)- (x-z) (y-z)-(x- y) (x - z). 14. (aJ^b + cy- {a + by - c(2 a + 2 6 + c). 15. (a + l)(2a + 5)(4a-3) + (a-l)(2a-5)(4a + 3). 16. (aj + 2/-^)' + (2/ + ^-a')'H- (24-05-2/)'. 17. 2(a + 2x)(a-2x)l(a + 2xy-\-{a-2xy]. 18. (a + & + c)2 - (a + & - c)2 _ (a - 6 + c)^ + (ct - 5 - c)^ 19. [(m + ny-\-{7n-7iy]l{2m-\-7iy-(m-2 7iyi 20. (a + ft - c) (6 + c - a) (c + a - 6). 21. (a-}- by -(a -by. 22. (^• + 2/ + 2)'-3(aj24-2/' + 2;2)(a; + 2/ + 4 DIVISION. 37 VI. DIVISION. 54. Division, in Algebra, is the process of finding one of two numbers, when their product and the other number are given. The Dividend is the product of the numbers. The Divisor is the given number. The Quotient is the required number. 55. The Law of Signs. Since the dividend is the product of the divisor and quo- tient, the equations of § 45 may be written as follows : (4- ah) - (4- a) = + 6, (- a6) ^ (-h a) = - h, {— ab) ^ {- a) = -\- h, {-\- ah) ^ {— a) = - h. * From these results, we may state the Rule of Signs in Division as follows : -f divided by +, and — divided hy — , produce + ; + divided hy —, and — divided by -j-, produce — . Hence, in Division as in Multiplication, Like signs produce -j-, and unlike signs produce ~. 56. The Index Law. Let it be required to divide a* by a^ The quotient must be a number which, when multiplied by the divisor, a-, will produce the dividend, a^. Now if d^ be multiplied by a^, the product is a^. Whence, ^ = a'. a^ Hence, the exponent of a letter in the quotient is equal to its exponent in the dividend minus its exponent in the divisor. 38 ALGEBRA. DIVISION OF MONOMIALS. 57. Let it be required to divide — 14 a?h by 7 a^. We find a number which, when multiplied by 7 a^, will produce — 14 a?h. That number is evidently — 2 6. Whence, rLl4^ = _2 6. i Qi 58. From §§ 55, ^Q, and 57, we derive the following rule for the division of two monomials : To the quotient of the absolute values of the nuinerical coefficients, annex the letters; giving to each an expo7ient equal to its exponent in the dividend minus its exponent iii the divisor, and omitting any letter having the same exponent in the dividend and divisor. Make the quotient + when the dividend and divisor are oj like sign, and — when they are of unlike sign. 1. Divide 54 a^ by -9a*. By the rule, -^^^ = - 6 a'-* = - 6 a*, Ans. — 9 a* 2. Divide - 2 a^b^c^' by abd\ We have, =il^^P^ = -2a-^bc, Ans, abd^ 3. Divide - 91 x^'^y^'z' by - 13 x'^y''^. We have, ~ ^^ ^^"^^"^ = 7 oc^^r^-m^-^ = 7 x^z^-^ Ans. — 13 x^y"z^ EXAMPLES. Divide the following : 4. 35 by -5. 6. -64 by -4. 5. -44 by 11. 7. -84 by 7. DIVISION. 39 8. -144 by -8. 17. 40 mV by 5mhi. 9. 168 by -12. 18. -33aVy* by -3a'y. 10. 16 a' by A a*. 19. -36a=^+^ by 12 a^— 3. 11. -18 0^2/ by 2a^. 20. 81a*6V by 9 5V. 12. 2mV by - m V. 21. 65af'2r^2' by -13a^. 13. - a«6V by - a^b^c. 22. - a^b' by - a«6*. 14. -6xy^ by 6a^/. 23. 540^1^ by 9iB«y". 15. -24a*62by _8a^6l 24. 98aVc« by - 14 a^jc^. 16. 28ar'2/23 by ^ja^z\ 25. -USmVp^ by llmVi>3. DIVISION OF POLYNOMIALS BY MONOMLAXS. 59. We have, a(b-{- c) = ab-{- ac. Since the dividend is the product of the divisor and quo- tient (§ 54), we may regard ab + ac as the dividend, a as the divisor, and 6 + c as the quotient. Whence, ?^±^=b + c. a We then have the following rule : Divide each term of the dividend by the divisor, and unite the results with their proper signs. 1. Divide 9 a362_ 6 a*c + 12 a^ftc' by -Sa\ ^am-Qa^c+l2a%c^ = _ Safe^ + 2 a^c - 4 6c3, Ans. — oa^ EXAMPLES. Divide the following : 2. 16a^ + 282^-24.^3 by 4a:«. 3. 104m7i3-39m3n by - 13m7i. 4. 6a26V-15a«6V + 3a^6«c by -Sa^ft^ 40 ALGEBRA. 5. -63a^/;22_ 34^32^4^7 ^^^ 7 a^yz\ 6. 20 m^rv' — 45 m'^n' — 35 wi%^ by — 5 mW. 7. - 24 a" + 108 a-* -f 84 a' by 12 a«. 8. 40 a^hc - 24 ah\ - 32 aftc^ by - 8 ahc. 9. 72aji« - 9a^ + 54a;« - ^9x^ by - 9a;^ 10. - 2x^ + 6a^/ - Gic'^/^ + 2x1/^ by - 2a?2/. 11. 60 a}"^ - 30 a^ + 15 a}^ - 45 o? by 15 a\ 12. a^'"6'' - 3 a^'^ft^'* + 2a'»6^ by a'"6". 13. 48 a'h^d" + 36 a^h^(^ - 30 a«6V by 6 a^6^c^ 14. - 88 xYz" +'6b xfz^ + 66 i^f^ by - 11 a;?/V. 15. i»'*+ V+^2:^ — icy^;'' — a?'"!/*^;'" by —x^'if^^. DIVISION OF POLYNOMIALS BY POLYNOMIALS. 60. Let it be required to divide 12 + 10 ar — 11 a? — 21 a? by 2x-2-4-3a;. Arranging each expression according to the descending powers of x (§ 33), we are to find an expression which, when multiplied by the divisor, 2 a^ — 3 a? — 4, will produce the dividend, 10 x^ - 21 x"" - 11 a; + 12. It is evident that the term containing the highest power of X in the product is the product of the terms containing the highest powers of x in the multiplicand and multiplier. Therefore, 10 a^ is the product of . 2 a;^ and the term con- taining the highest power of x in the quotient. Whence, the term containing the highest power of x in the quotient is 10 m? divided by 2 oi?, or 5 x. Multiplying the divisor by 5 a;, we have the product 10 a;^ — 15 a^ — 20 a; ; which, j^^hen subtracted from the divi- dend, leaves the remainder — 6 a?^ + 9 a? + 12. This remainder must be the product of the divisor by the rest of the quotient ; therefore, to obtain the next term of the quotient, we regard — 6 x^ + 9 x + 12 as a new dividend. DIVISION. 41 Dividing the term containing the highest power of x, — 6 ic^, by the term containing the highest power of x in the divisor, 2 a^, we obtain — 3 as the second term of the quotient. Multiplying the divisor by — 3. we have the product — 6 ic^ -f 9 .X -}- 12 ; which, when subtracted from the second dividend, leaves no remainder. Hence, 5 a? — 3 is the required quotient. It is customary to arrange the work as follows : 10ar^-21ar^-llaj-M2 10ar^-15a^-20ic Is? — '6x — 4, Divisor. 5 a; — 3, Quotient. 60^2+ 9a; + 12 6a;2+ 9a; +12 Note. The example might have been solved by arranging the dividend and divisor according to the ascending powers of x. From the above example, we derive the following rule : Arrange the dividend and divisor in the same order of powers of some common letter. Divide the first term of the dividend by the first term of the divisor, and write the remit as the first term of the quotient. M^dtiply the ivhole divisor by the first term of the quotient, and subtract the product from the dividend. If there be a remainder, regard it as a neio dividend, and proceed as before; arranging the remainder in the same order of poivers as the dividend and divisor. 61. 1. Divide dab'' -\- a^ -9 b^ - 5 a^bhj Sb^ -\- a^ -2 ab. Arranging according to the descending powers of a, «8 - 6 a2& + 9 a62 - 9 68 I g"^ - 2 gfe + 3 &2 gs _ 2 ggft 4- 3 ab^ \ a-Sb, Ans. - 3 g26 + 6 gftg - 9 b^ Note 1. In the above example, the last term of the second divi- dend is omitted, as it is merely a repetition of the term directly above. 42 ALGEBRA. Note 2. The work may be verified by multiplying the quotient by the divisor, which should of course give the dividend. 2. Divide 8 +18 0^^-56.^2 by - ex' + 4.-\-8x. Arranging according to the ascending powers of x, 4 + 8^-6x2)8 - 56a;2 + 18x*(2 - 4x - 3x-2, Ans. 8 + 16 X - 12 x-^ -16x -44x2 + 18x* -16x -32x2 + 24x3 - 12x2-24x3+ 18x* - 12x2-24x3+ 18x4 EXAMPLES. Divide the following : s/ 3. Wx'-llx-U hj 3x-\-2. T 4. 25m2 + 40mn-h 16 7^2 by 5m+4n. ^ 5. 12^2- 28a + 15 by 6a -5. 3. 15. a^ - 52 + 2 6c - c^ by a + 6 - c. 16. 4^- 16ic2?/ + 60;/ + 6a^ by 3a;2-/-2aj^. ^ 17. 39mn2 + 30m3-20n3-43m2n by 6m -5n. DIVISION. 43 18. 4a^-9a2-^30a-2o by 2a- -^ 3a -5. ^19. 4:X + x*-\~3 by Si-x^-2x. 20. n' - 16 by 2^^ + 8 -\-^7i-{-n^ ^ 21. 6m^-19m^-{-22m-\-5 by 3??i-5. 22. a.-* + y^ + a^/ by y^ -\- x^ — xy. 23. l-16a« by lH-2al 24. 16a;^-8l2/' by 2x-3y. 25. - 9m2 - 16 + ??i^ - 24 m by 3 m + ?/i2 + 4. 26. 9a;*+4-13a^ by 3ar'-2 + a;. 27. 2a* -a^.^ 8a -5 by 2a2-3a + 5. 28. 13a^ + 71a;-70a^-20+6a:* by 4 + 3ir'-7a;. 29. 4 mV + w» + 16m* by 2mn2 + 4:m^ + w*. 30. a^ + 32 by a + 2. . 31. 120a* + 26a3 - llla^ - 14a + 24 by (3a + 2)(4a - 3). 32. (2 m^ - m -1)(3 m^ -h m - 2) by (2m + 1)(3 m - 2). 33. a^ + 243 by 9a2 + 81-3a3-27a4-a*. 34. 4ar^'«+y-16af+y+^ + 12ar'i/^"-^ by af^+^y-3xy''-\ 35. 6a'-6ab' by -36 + 3a. 36. a'* - a*6 - a6* + 6* by a^ - 2 a6 -f b\ 37. 8m*-14m2-18m + 21 by 4m3 + 6m-7. 38. 16a*-96a3 + 216a2-216a-h81 by(2a-3)2. 39. 7a^-6a^-28 + 81a^ + 3a'-25a;* by 4-3ar^-5a;. 40. 2a;«-6a;^-a;*-9a:2_^3a._9 ]^y 2a;^-a; + 3. 41. 70a-50-a^-37a2 by 6a-5-a3-2a2. 42. ar'-81a;^ + 243/-3a;*2/ by 9xy^-\-x^-{-27f-{-3x^y. 43. 14a;*-23a; + 6a:«4-6a^-llar' + 5-12«3 by 5a;-3ar^ + 2a^-l. 44 ALGEBRA. 44.- 4a«-49a^ + 76a2-lG by 2a-^ + 5a- - 6a - 4. 45. m^ — 6 mV + 9 m^n^ — 4 w^ by (m + n)(m — 7i)(m -\-2n). 46. 8a2 - 10a6 + 18 ac - 36^ + 86c - Sc^ by 2a - 36 + 5c. 47. r^'' — 2/-' + 2 ^/'^^ — z^' by a^^ — ?/« -f- s;'-. The operation of division may be abridged in certain cases by the use of parentheses. 48. Divide a? +{a — o -\- c)x^ -\-(—ab — bc + ca)x — abc by X + a. x^+(a — b-^c)x^+{ — ab — bc + ca)x—abc \ x -{- a x^ +ax2 \ x^-\-(-b-\-c)x-bc, Ans. (_6 + c)a:2 (-6 + c)x2+(-a& 4-ca)a; — bcx — bcx—abc Divide the following : 49. ar^+(— « + 6 — c)a?^-h(— a6 — 6c + ca)a;4- a6c by X- + (— a -\- b)x — ab. 50. a^ + (a + 6 + c)a^ -f (a6 + 6c + ca)a.- + a6c by x + c. 51. a^+(3a-26 - c)^^ +(- 6a6 + 2 6c - 3ac)a; + 6a6c by a;^ + (3 a — c)a7 — 3 ac. 52. a(a + 6)a;2+(a6 + 62-f-6c)a;-c(6-fc) by aa;H-(6 + c) 53. m (m — n) x^ -\- (— mn + n^ — np) x -\- p {n —p) by maj — (n — ^). 54. a^ + (a — 6 — c)a^ + (— a6 + 6c — ca) a? + a6c by a:^ — (6 + c) aj 4- 6c. 55. a;^ — (a + 6 + c) x^ -f (a6 + 6c + ca) a; — a6c by x — a. 56. a'(b-c)d-{-a(-b'-{-c' + d')-(b + c)d by ad — (6 + c). 57. d^ -\- (m -{- n) a — 2 m^ -\- 11 mn — 12n^ by a — m + 4 n. DIVISION. 45 EXAMPLES FOR REVIEW. 62. 1. Find the numerical value when « = 4, b =— 7, c = — 3, and d = 5, of (a -h by 2 c — d c-^d We have, (a + 6)2 = (4 - 7)(4 - 7) = (- 3)(- 3) = 9, and c-j^-3-5^-8^_^ c + fZ -3 + 5 2 Then, (a + 6)2 -^^li?= 9 -(- 4)= 9 + 4 = 13, Ans. c + d Find the numerical value of each of the following when a = 5, 6 = — 4, c = — 2, and d = S: 2. {a-b)(b-\-c)(c-d). 3. b^ - c- + 2cd - d-. 4. (a + 3 6) (4 c - d) + (a - c) (2 6 -h d). 5. rt-^-3a26 + 3a62_63. 8. 3a26 _ 5 6«c + 4cU g. Sad 6ab n / i.\'i / jn^ 7 ct + 2fe a-56 -^ 26a + 236 + 64 c ' 4:C + d 6c-d ' lla + 246-7c* -J 2a — 6 36 — c 4c — d 6— c c — d d — a 12. Add 9(a - 6) - 8(6 - c), - 3(6 - c) - 7(c - d), and 4 (c — d) — 5 (a — 6). 9(a-6)- 8(6 -c) _ 3(6-c)-7(c-(?) -5(a-6) +4(c-d) 4(a- 6)- 11(6 -c)-3(c-(0, ^«s- 13. Add 4a2(a + a;)-6(6-^), - Sa\a + x) -2(b - y), ' and -7a^(a + x)-\-S(b-y). 14. Add 18 (;r - yf - 11 (a^ + yf, - 9{x-yy + 7 (x -{- yf, and - 4 (a; — y)^ — 5 (a; + yy. 46 ALGEBRA. 15. Subtract 5 (a - 5) - 8 (c + d) from 2 (a - 6) - o(c + d). 16. Multiply 3 (a; + 2/) - 5 by 3 (x 4- 2/) + 5. 17. Multiply 7 (a - 6) + 4 by 9 (a - 6) - 8. 18. Divide 6 (m + ^)2 - (m -h w) - 15 by 3 (?w, + ^0 - 5. 19. Divide (x-yf + l by (a;-?/) + l. 20. Add |a + |ft-ic and ia-|6-hfc. 21. Add 4a-f6 + fc and ^aH-|6-fc. 22. Add |a;-|?/-y2^2; and -ix-j-^y-^z. 23. From i a - f /> + | c take ia-^6-fc. 24. Subtract -y3^x 4- 12/ + i2; from -ix-{-^y-^z. 25. Multiply f a;^ + ^ a? -f ^9^ by | a^ - f. 26. Multiply ^ a^ - i a6 + tV ^^ by ^ a - ^ 6. 27. Divide -\'-^ + jh by fa^ + |. 28. Divide |a^ --Ja^ft + M«^' - i^' by ^a-^b. 29. Multiply a^^+^ft^ - a%^+^ by a^^"^ - b^-\ 30. Divide a^^m-l _ ^.3y4n+2 ^y ic"» 2_^y2n+l_ 31. Divide a'^+'' - ab^~^ by aP+^ - ft^^-i. 32. Add 3(a; + 1)^ - 2(.t -f 1), 5(x + l)-7, and -(x + l)2-3(a;4-l)+4. 33. From 7(x-{-yy-9x(x-\-y)-h4: take 12 (a; + 2/)^ + ir(a} + ?/) - 11. 34. Simplify 5x-[3x-\x-(7 x-Sx-4:)\ -(9x-5x-2)'\. 35. AMj\x'-ix-j\, -^x' + ^^^x-i, and |x2-|x+tV 36. Multiply x^ + (b — c)x — bc by oj + a. DIVISION. 47 37. Divide a^m+s ^3 _ ^252^-5 ^^y ^m+.i _ 5n-4 38. Subtract j\ a^ - i « + Jg. from f^ a? + y\ a - 3^. 39. Multiply {m-ny -\-2{m -n) + 1 by (7?i — 7i)- — 2 (m — ?i) + 1. 40. Multiply a*-^"-a"6" + 52'* by a^+^ft^ + aft^+l 41. Simplify (a + 5)^ - 2 (a + 5) (a - 6) + (a - 6)2. 42. Simplify a-[2a-(6-6c)- |a-(-2&-5c)-36-cS]. 43. Multiply fa^ - Ja - | by f a^ - a - f. 44. Divide ^a"" - ^a^ -{■ ^a" - ^ hy ^a?-a- J. 45. Divide a' -W-^ah (a^ - b^) 4- 10 a^b"" (a - b) by (a + bf-Aab. 46. Divide 12 x^+hj- ^ _ ^3 ^+4^«-4 _ 35 ^+7y5n-6 by 4 y?^+^if ^ -I- 5 ic'^+'V"-^ 47. Multiply (a + &) x - 2 a6 by a; + (a + 6). 48. Divide (a - 6)' - 3 (a - bfc + 3 (a - 6) c^ - c^ by (a-6)-c. 49. Divide X*^ -f a:-"*//" _^ y^n ]^y ^^m _^ ^myn _^ yZn 50. Multiply J a^ - f ax - 1- x^ by | a^ + f aa; + J a;^. 51. Multiply ar^ + (— a + 6)a; — ctd by x — c. 52. Multiply a^^ — «« + x"" by a;P — a;' + a;''. 53. Divide |a;^- Jx^ + fa)-^ by Ja^H-|a;-|. 54. Divide oi? -\- {a — b — c)x^ -\- {— ab + be — ca)x -{- abc by X— c. 55. Simplify (a; + 2/ + ^) [(x + y+ zf - 3 (xy -hyz-^- zx)]. 56. Simplify (a + 6 + c)(-a + 6 + c) (a-6 4-c)(a + 6 -c). 48 ALGEBRA. VII. SIMPLE EQUATIONS. 63. The First Member of an equation is the expression to the left of the sign of equality, and the Second Member is the expression to the right of that sign. Thus, in the equation 2ic — 3 = 3iC + 5, the first member is 2x — 8, and the second member is 3 aj -j- 5. . Any term of either meijiber of an equation is called a term of the equation. The sides of an equation are its two members. 64. An Identical Equation, or Ide^itity, is one whose members are equal, whatever values are given to the letters involved ; as (a + b) (a — b)=o? — b^. 65. An equation is said to be satisfied by a set of values of certain letters involved in it when, on substituting the value of each letter wherever it occurs, the equation becomes identical. Thus, the equation x — y = b is satisfied by the set of values a; = 8, 2/ = 3 ; for on substituting 8 for x, and 3 for y, the equation, becomes 8-3 = 5, or 5 = 5; which is identical. 66. An Equation of Condition is an equation involving one (^r more letters, called unknown qumitities, which is not satisfied by every set of values of these letters. Thus, the equation a; + 2 = 5 is not satisfied by every value of Xy but only by the value aj = 3: An equation of condition is usually called an equation. SIMPLE EQUATIONS. 49 67. If an equation contains but one unknown quantity, any value of the unknown quantity which satisfies the equation is called a Root of the equation. Thus, 3 is a root of the equation x + 2 = 5. To solve an equation is to find its roots. 68. A Numerical Equation is one in which all the known numbers are represented by Arabic numerals; as, 2x-17=zx-5. 69. A monomial is said to be rational and integral when it is either a number expressed in Arabic numerals, or a single letter with unity for its exponent, or the product of two or more such numbers or letters. Thus, 3, a, and 2 a%c^ are rational and integral. 70. If each term of an equation, involving but one un- known quantity x, is rational and integral, and no term con- tains a higher power of x than the first, the equation is said to be of the first degree. us, X — — I equations of the first degree, and d'x + h^=c) ^ A Simple Equation is an equation of the first degree. PROPERTIES OF EQUATIONS. 71. It follows from § 9, 2 and 3, that : 1. The same number may be added to, or subtracted from, both members of an equation, without destroying the equality. 2. Both members of an eqhation may be multiplied, or divided, by the same number, without destroying the equvvdty. 72. Transposition of Terms. A term may be transposed from one member of an equation to the other by changing its sign. 50 ALGEBRA. Let the equation be x -{- a = b. Subtracting a from both members (§ 71, 1), we have x= b — a. In this case, the term + a has been transposed from the first member to the second by changing its sign. Again, consider the equation X — a = b. Adding a to both members, we have x = b -\-a. In this case, the term — a has been transposed from the first member to the second by changing its sign. 73. It follows from § 72 that If the same term occurs in both members of an equation affected with the same sign, it may be cancelled. 74. The sign of each term of an equation may be changed without destroying the equality. Let the equation be a — a? = 6 — c. (1) Transposing each term (§ 72), we have — b-\-c= — a-\-x. That is, x — a = c — b] which is the same as (1) with the sign of each term changed. SOLUTION OF SIMPLE EQUATIONS. 75. 1. Solve the equation ' Bx — 1 = ^x-\-l. Transposing 3 a; to the first member, and — 7 to the second, we have 5x-3a; = 7 + l. Uniting similar terms, 2 re = 8. SIMPLE p:quations. 51 Dividing both members by 2 (§ 71, 2), we have X = 4, Ans. From the above example, we derive the following rule : Transpose the unknown terms to the first member, and the known terms to the second. Unite the similar terms, and divide both members by the coefficient of the unknown quantity. 2. Solve the equation 14-5a;=19 + 3a;. Transposing, — 5 a; -- 3 x = 19 - 14. Uniting terms, — 8 x = 5. Dividing by - 8, x = - -, Ans. 8 Note 1. The result may be verified by putting x = — - in the given equation ; thus, U-5(-|) = ,9 + 3(-|). That is, 14 + 2^=19-1^. 8 8 Or, 137 ^ 137 ^^^^^ ^ identical. 8 8 EXAMPLES. Solve the following, in each case verifying the answer : 3. 9a; = 7a; + 28. 10. 7a; - 29 = 16a; - 17. 4. 8a;-5=-61. 11. 13 - 6a;= 13a; - 6. 5. 6a; + ll = a; + 31. 12. 19 - 16a; = 27 -28a;. 6. 9a;-7 = 3a;-37. 13. 9a;- 23 = 20a; - 18. 7. 4a;-3 = 8a;-|-33. 14. 30 + 17a;= 27a; + 22. 8. 12-13a; = 6-10a;. 15. 24a; - 11 = 28 + 11a;. 9. 5a;H-9 = 14-2a;. 16. 33 a; + 25 = 41 + 51 a;. 17. 14a;-|- 21-35= -29a; + 44a;-22. 52 ALGEBRA. 18. 32x-S9 = 25x-10x-Ul. 19. 12a.'-23a5-h55 = 15aj-75. 20. Solve the equation (2x- ly = 2(a; + 3) (2a; - 3) - 3(6 a; - 1). Expanding (Note 2), 4 ^2 - 4 x + 1 = 4 a;2 + 6 x - 18 - 18 ic + 3. Transposing, 4x2 -4x- 4x2 -6x4- 18x= - 18 + 3-1. Uniting terms, 8 x = - 16. Dividing by 8, x= —2, Ans. Note 2. To expand an algebraic expression is to perform the operations indicated. Solve the following equations : 21. 2(5x + l)-4.=:3(x-7)-16. 22. 10aj-(3cc + 2)=9a;-(5x-4). 23. 8a;-5(4a;4-3)=-3-4(2a;-7). 24. 5.T-6(3-4a.-)=x-7(4-f .t). 25. 6x(3x-5)-\-Ul = 2x(9x-{-l)-{-lS. 26. 19-5a?(4a;-M)=40-10ic(2a;-r). 27. 2(4a;-h7)-8(3x-4) = 6(2a; + 3)-7(2a;-3). 28. (5x + 7)(3x-8) = (5x-\-A)(Sx-5). 29. (4.x-7y=(2x-5)(Sx-\-3). 30. (5-3a;)(34-4a;) -(7 + 3rc)(l-4ic)= -1. 31. (l-3x)2-(a^ + 5)2=:4(i» + l)(2a;-3). 32. 6(4:-xy~5(2x + 7)(x-2) = 5-(2x + 3y. PROBLEMS. 76. For the solution of problems by algebraic methods, no general rule can be given, as much must depend upon the skill and ingenuity of the student. SIMPLE EQUATIONS. 53 The followiug suggestions will, however, be found of service : 1. Represent the unknown quantity, or one of the un- known quantities if there are several, by x. 2. Every problem contains, explicitly or implicitly, pis- cisely as many distinct statements as there are unknown quan- tities involved. All but one of these should be used to express the other unknown quantities in terms of x. 3. The remaining statement should then be used to form an equation. The beginner will find it useful to write out the various statements of the problem, as shown in Exs. 1 and 2, § 77 ; after a little practice he will be able to dispense with these aids to the solution. 77. 1. Divide 45 into two parts such that the less part shall be one-fourth of the greater. Here there are two unknown quantities, the greater part and the less. In accordance with the first suggestion of § 76, we will represent the less part by x. The two statements of the problem are, implicitly : 1. The sum of the greater part and the less part is 46. 2. The gi-eater part is 4 times the less part. In accordance with the second suggestion of § 76, we will use the second statement to express the greater part in terms of x. Thus, the greater part will be represented by 4 x. We now in accordance with the third suggestion of § 76 use the first statement to form an equation. Thus, ix + x = 45. Uniting terms, 5x = 45. Dividing by 5, X = 9, the less part. Whence, 4x = 36, the greater part. 1 r >- or THE A UNIVERSITY J\ 54 ALGEBRA. 2. A had twice as much money as B ; but after giving B f 35, he had only one-third as much as B. How much had each at first ? Here there are two unknown quantities : the number of dollars A had at first, and the number B had at first. Let X represent the number of dollars B had at first. The first statement of the problem is : A had twice as much money as B at first. Then 2 x will represent the number of dollars A had at first. The second statement of the problem is, implicitly : After A gives B ^ 35, B has 3 times as much money as A. Now after giving B $35, A has 2 a; — 35 dollars, and B x -}- 35 dollars ; we then have the equation x + 35 = 3(2x-35). Expanding, aj + 35 = 6 x — 105. Transposing, — 5 x = — 140. Dividing by — 5, x = 28, the number of dollars B had at first ; and 2 X = 56, the number of dollars A had at first. Note 1. It must be carefully borne in mind that x can only rep- resent an abstract number; thus, in Ex. 2, we do not say, "let x represent what B had at first," nor "let x represent the sum that B had at first," but "let x represent the number of dollars that B had at first." 3. A is 3 times as old as B, and 8 years ago he was 7 times as old as B. Required their ages at present. Let X = the number of years in B's age. Then, 3 x = the number of years in A's age. Also, X -r- 8 = the number of years in B's age 8 years ago, and 3 X — 8 = the number of years in A's age 8 years ago. But A's age 8 years ago was 7 times .B's age 8 years ago. Whence, 3x-8 = 7(x-8). Expanding, 3x — 8 = 7x — 56. Transposing, — 4 x = — 48. Dividing by — 4, x = 12, the number of years in B's age. Whence, 3 x = 36, the number of years in A's age. SIMPLE EQUATIONS. 55 Note 2. In Ex, 3, we do not say, "let x represent B's ag'e," but " let x represent the nxmiber of years in B's age." 4. A sum of money amounting to $ 4.32 consists of 108 coins, all dimes and cents; how many are there of each kind ? Let X = the number of dimes. Then, 108 — x= the number of cents. Also, the X dimes are worth 10 x cents. But the entire sum amounts to 432 cents. ^ '^. "V Whence, 10 a; + 108 - x = 432. Transposing, 9x = 324. Whence, x = 36, the number of dimes ; and 108 — a; = 72, the number of cents. PROBLEMS. 5. Divide 19 into two parts such that 7 times the less • shall exceed 6 times the greater by 3. '^ 6. What two numbers are those whose sum is 246, and whose difference is 72 ? 7. Divide 38 into two parts such that twice the greater . shall be less by 22 than 5 times the less. 8. Divide $22 between A, B, and C, so that A may receive $2.25 more tlian B, and $1.75 less than C. 9. A is 5 times as old as B, and in 13 years he will be . only 3 times as old as B. What are their ages ? 10. B is twice as old as A, and 35 years ago he was 7 times as old as A. What are their ages? 11. A had one-third as much money as B;, but after B had given him $ 24, he had three times as much money as B. How much had each at first ? 12. A sum of money, amounting to $2.20, consists en- tirely of five-cent pieces and twenty -five-cent pieces, there being in all 16 coins. How many are there of each kind? 56 ALGEBRA. 13. A is 68 years of age, and B is 11. In how many years will A be 4 times as old as B? 14. A is 25 years of age, and B is 65. How many years is it since B was 6 times as old as A ? 15. A man has two kinds of money ; dimes and fifty-cent pieces. If he is offered $4.10 for 17 coins, how many of each kind must he give? 16. Divide 76 into two parts such that if the greater be taken from 61, and the less from 43, the remainders shall be equal. 17. What two numbers are those whose sum is 13, and the difference of whose squares is 65 ? 18. Find two numbers whose difference is 6, and the difference of whose squares is 120. 19. A is 14 years younger than B; and he is as much below 60 as B is above 40. Kequired their ages. 20. A drover sold a certain number of oxen at $ 60 each, and 3 times as many cows at $35, realizing $ 1485 from the sale. How many of each did he sell ? 21. A man has $4.35 in dollars, dimes, and cents. He has one-fourth as many dollars as dimes, and 5 times as many cents as dollars. How many has he of each kind ? 22. A garrison of 4375 men contains 4 times as many cavalry as artillery, and 7-^ times as many infantry as cavalry. How many are there of each kind ? 23. At an election where 5760 votes were cast for three candidates. A, B, and C, B received 5 times as many votes as A, and C received twice as many votes as A and B together. How many votes did each receive ? 24. Divide $ 115 between A, B, C, and D, so that A and B together may have $ 43, A and C $ 65, and A and D $57. SIMPLE EQUATIONS. 57 25. A man divided $ 1656 between his wife, three daugh- ters, and two sons. The wife received 4 times as much as either of the daughters, and each son one-third as much as each daughter. How much did each receive ? 26. Divide $ 125 between A, B, C, and D, so that A and B together may have $ 6o, B and C $ 52, and B and D f 54. 27. A man has 4 shillings in three-penny pieces and farthings; and he has 23 more farthings than three-penny pieces. How many has he of each kind ? 28. Divide 71 into two parts such that one shall be 4 times as much below 55 as the other exceeds 37. 29. A square court has the same area as a rectangular court, whose length is 9 yards greater, and width 6 yards less, than the side of the square. Find the area of the court. 30. Two men, 84 miles apart, travel towards each other at the rates of 3 and 4 miles an hour, respectively. After how many hours will they meet ? 31. Find three consecutive numbers whose sum is 108. 32. In 7 years, A will be 3 times as old as B, and 8 years ago he was 6 times as old. What are their ages ? (Let X represent the number of years in B's age 8 years ago.) 33. A sum of money, amounting to $ 24.90, consists en- tirely of $ 2 biUs, fifty -cent pieces, and dimes ; there are 5 more fifty -cent pieces than $ 2 bills, and 3 times as many dimes as $ 2 bills. How many are there of each kind? 34. Find two consecutive numbers such that the difference of their squares, plus 5 times the greater number, exceeds 4 times the less number by 27. 35. Find four consecutive numbers such that the product of the first and third shall be less than the product of the second and fourth by 9. 58 ALGEBRA. 36. A laborer agreed to serve for 32 days on condition that for every day he worked he shoukl receive f 1.75, and for every day he was absent he should forfeit ^1. At the end of the time he received $ 28.50. How many days did he work, and how many days was he absent ? 37. A merchant has grain worth 5 shillings a bushel, and other grain worth 9 shillings a bushel. In what pro- portion must he mix 24 bushels, so that the mixti^re may be worth 8 shillings a bushel ? *) i ]4 38. A general, arranging his men in a square, finds that he has 43 men left over. But on attempting to add 1 man to each side of the square, he finds that he requires 108 men to fill up the square. Kequired the number of men on a side at first, and the whole number of men. 39. In a school of 535 pupils, there are 40 more pupils in the second class than in the first, and one-half as many in the first as in the third. The number in the fourth class is less by 30 than 3 times the number in the first class. How many are there in each class ? 40. A man gave to a crowd of beggars 15 cents each, and found that he had 80 cents left. If he had attempted to give them 20 cents each, he would have had too little money by 10 cents. How many beggars were there ? 41. A tank containing 120 gallons can be filled by two pipes, A and B, in 12 and 15 -iainutes, respectively. The pipe A was opened for a certain number of minutes ; it was then closed, and the pipe B opened ; and in this way the tank was filled in 13 minutes. How many minute's was each pipe open ? 42. A grocer has tea worth 70 cents a pound, and other tea worth 40 cents a pound. In what proportion must he mix 50 pounds, so that the mixture may be worth 49 cents a pound ? IMPORTANT RULES. 59 VIII. IMPORTANT RULES IN MULTIPLICA- TION AND DIVISION. 78. Let it be required to square a -\- b. a + b a + b a^ -\-ab ab + b'' Whence, (a -\- bf = a^ -\- 2 ab -\- b\ That is, the square of the sum of two quantities is equal to, the square of the first, plus tivice the product of the two, plus the square of the second. Example. Square 3 a + ^ be. We have, (3 a + 2 &c)2 = (3 a)2 + 2 x 3 a x 2 6c + (2 hey = 9 a2 + 12 a6c 4- 4 h'^c'^, Ans. 79. Let it be required to square a — b. a- -b a -b a' -ab - ab + b' Whence, (a -b)- = a^-2ab-\- b\ That is, the square of the difference of two quantities is equal to the square of the first, minus twice the product of the two, plus the square of the second. Example. Square 4 a; — 5. We have, (4 x - 5)2 = (4x)2 - 2 x 4ic x 5 + 52 = 16 x2 - 40 X + 25, Ans. 60 ALGEBRA. 80. Let it be required to multiply a -{- b by a — 0. a -\-b a — b o? + ab — ab Whence, (a + b){a-b) = o? - b\ That is, the product of the sum and difference of two quanti- ties is equal to the difference of their squares. Example. Multiply 6 a^ + 6 by 6 a^ — 6. We have, (6 a^ + h) (6 a^-l)) = (6 a'^y - h'^ = 36 a* _ &2^ j^^s. 81. In connection with the examples of the present chapter, a rule for raising a monomial to any power whose exponent is a positive integer will be found convenient. Let it be required to raise 5 a^b^c to the third power. We have, (5 a'b'cf = 5 a'b'c x 5 a'b'c x 5 a'b'c = 125 a^6V. We then have the following rule : Maise the numerical coefficient to the required power, and multiply the exponent of each letter by the exponent of the required power. EXAMPLES. 82. Find by inspection the values of the following : 1. (aj + 4)2. 9. (8 + 3m37i2)2. 2. (a -3)1 10. (ab' + 2a'by. 3. (6a- 56)2. II {(dxy-lxzf. 4. {2xy+^)\ 12. (4> + 116c)l ; 5. (3m + 4ti)(3m-4?i). 13. {9fxf + 2z'){^xf-2z^). 6. (1-2 ay. 14. {lab-^cd)\ 7. (5x2 + 8)(5ar^-8). 15. (6a^ + ll/)(6aj5- 11/). 8. {a'-Qaf, 16. (Qa^ + Sa^. IMPORTANT RULES. 61 17. (7m^+12 7i)(77n^-12n). 20. (3 a*" + 4 6'*)2. 18. (Sx' + Ty^y. 21. {5xP-Sj:^y. 19. (13 a'x- 6 byy. 22. (a^ -^ a') {a^ - a"-). 23. Multiply a + b-\-chya + b — c. ia+ b + c)(a-^ 6 -c) = [(a + 6)+c][(a + 6)-c]- = (a + 6)2-c2 (§80) = a2 + 2 aft + 62 _ c2, ^us. (§ 78) 24. Multiply a + 6 — c by a — 6 -h c. (a + & - c)(a - ft 4- c) = [a +(6 -c)][a -(ft - c)] = rt-2_(5_c)2 (§80) = rt2_(62_26c + c2) (§79) = a- - b^ + 2bc- c2, ^ns. Expand the following : 25. (a + ft + c)(a-ft4-c). 28. (a'' -\-a-l){a^ - a-{-l). 26. (aj-2/-f 2)(a?-?/-2). 29. (x' -\- x - 2) (x" - x - 2). 27. (a 4- 6 + c)(a - 6 - c). 30. (1 + a + 6) (1 - a - 6). 31. (x'-{-2x-hl)(x''-2x-\-l). 32. (a + 2ft-3c)(a-2ft-h3c). 33. (a^-f aft4-?^')(a2-«ft + ft2). 34. (3x-{-Ay-\-2z){3x-4:y-2z). \. We find by multiplication : x + 5 x-5 x + 3 x-3 a^-\-5x x'-Bx + 3a; + 15 -3x4-15 x2 + 8a; + 15 • x'-Sx-^-W x-f-5 x-5 x-3 x-\-3 x^-\-5x a^-5x -3x-W + 3a;-15 a;2 + 2a;-15 x'-2x-15 62 ALGEBRA. In these results it will be observed that : I. The coefficient of x is the algebraic sum of the second terms of the multiplicand and multiplier. II. The last term is the product of the second terms of the multiplicand and multiplier. By aid of the above laws, the product of any two binomial r of the form x -\- a, x -\-h may be written by inspection. 1. Required the value of (x + 8) (x — 5). The coefficient of x is + 8 — 5, or 3. The last term is 8 x (- 5), or - 40. Whence, (a; + 8) (a; - 5) = ic^ + 3 x - 40, Ans. 2. Required the value of (a — 6 — 3) (a — 6 — 4). The coefficient of a — & is — 3 — 4, or — 7. The last term is (- 3) x (- 4), or 12. Whence, {a-h- 3)(a - 6 _ 4) = (« - 6)2 - 7 (a - 5) + 12, Ans. EXAMPLES. Find by inspection the values of the following : 3. (a; + 6) (ic + 4). 14. (a + 6 - 7>(a + 6 + 8). 4. {x-2)(x + Z). 15. (x-^a){x-na). 5. (a; - 10) (a; - 1). 16. {x -ir y) (x - 2 y). 6. (x-}-5)(a^-6). 17. (a + lU)(a-66). 7. (a + l)(a + 9). 18. {a + 1 x){a + ^x). 8. (a-7)(a + 4). 19. {x -y - 4.){x- y + 10). 9. (m + 5)(m-l). 20. {x -llz){x + 9z). 10. {x'-7){x'-2). 21. {x^ + 3y){^ + ^y). 11. (ti^ + 3) (71^ _ 10). 22. (aT^-9m2)(a^-6m2). 12. (a6 4- 2) (a6 + 11). 23. {ah + Scd) {ah -12 cd). 13. {xy-12){xy-S). 24. (x + y + 12)(aj + ^ - 9). IMPORTANT RULES. 63 84. We have by § 80, =a — h, and = a-\-h. a -\-b a — b That is, if the difference of the squares of two quantities he divided by the sum of the quantities, the quotient is the dif- ference of the quantities. If the difference of the squares of tivo quantities be divided by the difference of the quantities, the quotient is the sum of the quantities. 1. Dividel6a'6^-9by 4a62-h3. We have, 16 a^b* = (4 ab'^y. (§ 81 ) Whence, \6a'^b*-9 ^ 4 ^52 _ 3, Ans. 4 ab-^ + 3 EXAMPLES. Write by inspection the values of the following : 2 ^-^ 5 25 <^'- ^6 8 1-64 nv'n^ ' x-\-l' ' 5a^-{-6 ' ' 1+8 mw" 3 ^-^' 6 ^^'-^^y' 9 4ci^6^-c^ ' 2-a ' 3x-\-iy ' ' 2ab'-c'' •4 16 m- -49 y 2Da--b\ ^^ 49 m-- 100 u" 4m-7 5a-b' 7m-10n^ .. 81/-196a;^ ,3 144x-y - 169^*^ ■ 9?/-hl4a^ * ■ 12xy'-lSz' ' 12 121&V-64a'^ ^^ 225a^Q-64 6^¥ ■ llbc + Sa ' ' 15a^-h8 6V 85. We find by actual division : t±^ = a^-ab + b^ a + b and 9LJ=L^=a'^ab-\-b\ a — b 64 ALGEBRA. That is, if the sum of the cubes of two quantities he divided by the sum of the quantities, the quotient is the square of the first quantity, minus the product of the two, plus the square of the second. If the difference of the cubes of two quantities be divided by the dijfference of the quantities, the quotient is the square of the first quantity, plus the product of the two, plus the square of the second. 1. Divide 1 -f- 8 a^ by l-f 2 a. We have, l±^^ l^{2aY = 1 -2a + (2a)2 = 1 - 2 a + 4 a2, Ans. 2. Divide 27 3^-64 /by 3 iK- 4?/. We have, 27^^-64^ ^ {^xY-j^vY 3x — 4y Sx — 4:1/ = (3a;)2 + (3x)(4?/) + (4i/)2 = 9x^ + l2xy -\- 16^/2, Ans. EXAMPLES. Find the values of the following : 3 «''-! 8 ?L^^Z^. 13 8a;^-125/ ' a^b — (? 2x — hy'^ g 1 + 64 m\ ^^ a%'' + 512 & l-|-4m ' a& + 8c' 5 !!^!_±§. 10 ?lizi^. 15 64 mhv' + 343 6 — X 4 mn + 7 ^^ g^ + 125 ^g 729a^-125a-' '^ a 4- 5 9 a^ — 5 £c ^ a;« + /' .g l-343a^6^ ,« 512ajy + 27/ x^^f ' l-7ab' ' ' Safy-\-3z' ' a'-l a-1 1 + a^ l-\-x m^ + 8 m-f 2 27 - a'' 3-a a^« + /' IMPORTANT RULES. 65 86. We find by actual division : a-\-b a'b 4_ ab- - h\ ^!i^I±=:a^ + a'b + ab' + b^ a — b a^-\-b^ a-\-b a'-b' = a'- cv'b + d'b- - aW + b\ = a* -h o?b -h d-b- -\- aW + b^ ; etc. In these results we observe the following laws : I. The number of terms is the same as the exponent of a in the dividend. II. The exponent of a in the first term is less by 1 than its exponent in the dividend, and decreases by 1 in each succeeding term. III. The exponent of b in the second term is 1, and increases by 1 in each succeeding term. IV. If the divisor is a — b, all the terms of the quotient are positive ; if the divisor is a-\-b, the terms of the quo- tient are alternately positive and negative. 87. The following principles are of great importance. If n is any positive integer, it will be found that : I. a" — 6" is always divisible by a — b. Thus, a^ — b^, a^ — b^, a^ — b\ etc., are divisible hj a — b. II. a" — 6" is divisible by a -\- b if 7i is even. Thus, a^ — b^, a^ — b*, a^ — b% etc., are divisible by a + 6. III. a** -f 6"* is divisible by a-\-b ifn is odd. Thus, a^ + b% a^ + b^, a' + b'^, etc., are divisible by a -f 6. IV. a" + b" is divisible by neither a -\-bi nor a — b if n is even. Thus, a^ -h 6^, a* + b^, cf + b^, etc., are divisible by neither a -f 6 nor a — b. 5 ALGEBRA. 88. 1. Divide cd - h' by a — b. Applying the laws of § 86, we have, ^6 _|. a^l) ^ a^l)2 _j. Qj3^3 + (^2^4 ^ ^55 ^ 56^ ^,^5. a — b 2 Divide 16 ic^ - 81 by 2x + 3. We have, 16^-81^(2^)4-34 2 X + 3 • 2 a; + 3 (2 a:)3 - (2 :r)2 x 3 + (2 a;) x 32 - 3^ :8x3 - 12^2+ 18iC-27, ^?is. EXAMPLES. Find the values of the following : a' -1 a + 1 X' -1 X -1 a« -b' a -hb 1- -x' 1- -X a« -b' 0? -b' x^': f + z^' g 16 -x' 2-x 10. l-16a^ l + 2a 11. a' + b' ■ a-\-b 12. 1-m' 1 — m 13. 32 + a^ 2 + a 14 m^ — 71^ 15. 16. 18. a^y -\-^ m + n 20. 64a«-6« 2a-& 81 a;^- 2/' 3a; + 2/ ■ a^-243a^ a — 3x 81 a^- 256 6^ 3a-46 243 0^^ + 32?/^ 3a; + 22/ 128m^-n" 2m — Ti^ FACTORING. 67 IX. FACTORING. 89. To Factor an algebraic expression is to find two or more expressions which, when multiplied together, will produce the given expression. 90. A Common Factor of two or more expressions is an expression which will exactly divide each of them. 91. A monomial can always be factored ; thus, 12 a^bc- = 2x2xSxaxaxaxbxcxc. It is not always possible to factor a polynomial ; but there are certain types which can always be factored, the more important of which will be taken up in the present chapter. 92. Case I. When the terms of the expression have a common monomial factor. 1. Factor Uxy^-353^y\ Each term contains the monomial factor 7 xi/^. Dividing the expression by 7 xy^^ the quotient is 2y^ - 6 y?-. Whence, 14 x^ - 35 y?t = 7 xy^(2 y^ - 5 x^) , Ans. EXAMPLES. Factor the following : 2. a3 + 4a. 7. 12 a^ - 20 a'^ + 4 a^. 3. 6a5^-14aj3. 8. a^6 V + a^^V + a'6c«. ^ 4. 30wi2-5m3. 9. 12 a^^^ + 24 a;/ - 42 a;y. 5. l^a^h^ + ^ah'^ ■ 10. 14 a«6* + 21 a^d^ _ 49 ^352 6. 56a^2/'-32a;y. 11. 81 m^n + 54 mV + 9 m V. 12. 48 a^y- 144x3/ + 108 a^^. 13. 70 aV - 126 aV ^ 112 aV. 68 ALGEBRA. 93. Case II. When the expresdon is the sum of tivo bino- mials which have a common binomial factor. 1. Factor ac — bc^ ad — bd. • By § 92, {ac - be) + {ad - bd) = c{a - 6) + d{a - b). The two binomials have the common factor a — b. Dividing the expression by a — 6, the quotient is c + d. Whence, ac — 6c + ad — bd = (« — &) (c + d), Aiis. If the third term of the given expression is negative, as in the following example, it is convenient to enclose the last two terms in a parenthesis preceded by a — sign. 2. Factor6ic3-15x2-8ic + 20. 6^3- 15x2 -8x + 20 =(6x3- 15x2)- (8x - 20) zz3x2(2x-5)-4(2x-5) = (2x-5)(3x2-4), Ans. EXAMPLES. Factor the following : 3. ab + an + bm + mn. 9. 3 x"' + 6 x^ + ic + 2. 4. ax — ay -{-bx — by. 10. 10 mx — 15 nx — 2 m -f 3 n. 5. ac — ad — bc-\- bd. 11. a^x + abcx — a^by — b^cy. 6. a3 + a2 + (* + l. 12. a'bc - ac'd -{- abH - bed'. ^ 7. 4a^-5a;2-4a;-f5. 13. 30a2- 12^^ - 55a + 22. 8. 2 + 3a-8a2-12a^ 14. 56 - 32 a^ + 21 a^ - 12 ic^. 15. a^b^ -h a'^bcd' -\- ab'c'd -\- cH^ 16. Sax — ay — 9bx-{-S by. 17. 4 a^ + x^y' —16xy — 4:y^. 18. 20 ac + Wbc + Aad-^Sbd. 19. 16 mx — 56 my + 10 7ix — 35 ny. 20. 45 a^ - 20 a^ft^ _ 53 ^ J ^ 28 6^. FACTORING. 69 94. If an expression can be resolved into two equal factors, it is said to be a perfect square, and one of the equal factors is called its square root. Thus, since 9 a^6^ is equal to 3 a% x 3 a^b, it is a perfect square, and 3 a^b is its square root. Note. 9 a4&-^ is also equal to ( - 3 a^b) x ( - 3 a^b) ; so that - 3 a'^b is also its square root. In the examples of the present chapter, we shall consider the positive square root only. 95. The following rule for extracting the square root of a perfect monomial square is evident from § 94 : Extract the square root of the numerical coefficient, and divide the exponent of each letter by 2. Thus, the square root of 25 a*6V is 5 a^b^c. 96. It follows frouL §§78 and 79 that a trinomial is a perfect square when its first and last terms are perfect squares and positive, and the second term twice the product , of their square roots. Thus, in the expression 4:X^ — 12xy -\-9y^, the square root of the first term is 2 x, and of the last term 3 y ; and- the second term is equal to 2(2x)(Sy). Whence, 4:X^ — 12xy-{-9y^ is a perfect square. 97. To find the square root of a perfect trinomial square, we simply reverse the rules of §§ 78 and 79 : Extract the square roots of the first and last terms, and connect the results by the sign of the second term. Thus, the square root of4ar' — 12a^-f9y^is2a; — 3?/. 98. Case III. When the expression is a perfect trinomial square (§ 96). 1. Factor a^-\-2ab^+b\ By § 97, the square root of the expression is a + b^. Whence, a^ -\- 2 ab'^ + 6* =(a + b'^y- = (a + b"^) (a + b"-), Ans. 70 ALGEBRA. 2. Factor 25 x^ - 40 xy"" + 16 y\ The square root of the expression is 5x — 4?/3. Whence, 25 x2 - 40 xy^ -\- \Qy^ ^{px - ii/y Note. The expression may be written 16 y^ — 40 xy^ + 25 ic^ ; in which case, according to the rule, its square root is 4«/3 — 5ic. Thus, another form of the result is 162/6 _ 40iC2/^ + 25x2 =(4 «/3 - 5a:)(4y3 _ 5^)^ EXAMPLES. Factor the following : 3. wr + 2mn + n\ 15. Ua'b^ + lQahcd + (?d\ 4. o?-2ah + h\ 16. 100a^-60a^4-9x^ 5. 9 + 6flJ + a;2. 17. 49 m* + 112 m^ii^ + 64 ti^. 6. a2-8a + 16. 18. 121 a^fts ^ 132 ^^c^ + 36 c^ 7. 49a.-2 + 14a;2/ + 2/2. 19. 144i»y-120a;y+25icy. 8. m''-l()mn + 2bn\ 20. 64 a^fts.^ 175 ^52^ .1.12162^2 9. 4 a* - 4 a'hh + 6V. 21. 49a^2/' - 168iC2/«^ + 144^^ ^' 10. mV + 18ma;4-81. 22. 36aV-156aV+169a2a!4. 11. 4a2 - 20aa; + 25a^. 23. (a + 6)2 _ 4 (a + 6) + 4. 12. 9a2 + 42a6+4962. 24. (x - 2/)' + 10 (a; - ^) + 25. 13. 81 a^- 72 0^2/ + 16 2/'. 25. 16(a + a;)2 + 8(a + a^) + l. 14. a^ + 12 x^yz + 36 yh\ 26. 4(a - hf - 12(a - 6) + 9. 99. Case IV. When the expression is the difference of tivo perfect squares. By § 80, aP -h'' = {a + h)(a ~ h). Hence, to obtain the factors, we reverse the rule of § 80 : Extract the square root of the first square, and of the second square ; add the results for one factor, and subtract the second result from the first for the other. FACTORING. 71 1. Factor 36 a2 - 49 6\ The square root of 36 a^ is 6 a, and of 49 M is 7 6^. Whence, 36 d^ - 49 6^ = (6 a + 7 62) (6 a - 7 62) , ^ ns. EXAMPLES. Factor the following : 2. a'-W. 8. 49m*-16ii2. 14. 144mV-49. 3. x^-l. 9. 25 a^- 64 6V. 16. 36a«-169a:». 4. 9 -ml 10. 100a^/-92*. 16. Ux'^-imfz\ 5. KSx'-f. 11. 64m*-81n«. 17. 64 a^^^s _ 225 c>«. 6. 4. a' -25. 12. 121 a^fo^ - 4 c^d^. 18. 169-144a^y^ 7. l-36a26l 13. 81x«-100/. 19. 196aV-1216y. 20. Factor (2x-Syy-(x- y)\ We have, (2x — ^yy-{x-yY = l(2x-Zy)-V{x-y)^[{2x-^y)-{x-y)] = (2x-Sy + X -y){2x- Sy -x + y) = {Sx-iy)(x-2y), Ans. Factor the following : 21. (a -f by - c\ 28. (a + hf - (c - d)\ ^ 22. {m-nf-x'. 29. {a - xf - (h - yf. 23. d'-ih-cf. ^ 30. (x + yy-{m + nf. 24. aj2_(2, + 2)2. 31. (8 a -5)2 -(3 a + 7)2. 25. m2-(t»-p)2. 32. (4 a; + 1)2 -(a; + 6)2. 26. {lx-2yf-y\ 33. (7a - 56)2 - (5a- 26)2. 27. {a-bf-{x + y)\ 34. (9 a; + 82/)'- (2 a; - 3?/)2. A polynomial may sometimes be expressed in the form of the difference of two perfect squares, when it may be factored by the rule of Case TV. 72 ALGEBRA. 35. Factor 2 nm + m^ — 1 + n^ Since 2 mn is the middle term of a perfect trinomial square whose first and third terms are w^ and n' (§ 96), we arrange the given ex- pression so that the first, second, and last terms shall be grouped together, in the order m^ + 2 mn + n^ ; thus, 2 wm -h w2 - 1 + w2 = (w2 + 2 mn + n^) - 1 = (m + w)2 - 1, by Case III. = (m + w -I- l)(m + w — 1), JlWS. 36. Factor 12 1/ + aj^ - 9 2/^ - 4. We have, I2y + x^ - 9y^ - ^ = x^ -9y^ -^ 12y - 4: = x2-(9?/2_ 12 2/ + 4) = x'^ - (3 ?/ - 2)2, by Case III. = lx + iSy-2)][x-iSy-2)^ = (x-}-Sy-2){x-3y-\-2),Ans. 37. Factor a2-c2 + 62_(^2_2cd-2a6. We have, a^ - c^ -}- b^ - d^ - 2cd - 2 ab = a^-2ab-^b'^-c^-2cd-d^ = (a2 - 2 a6 + 62) _ (c2 + 2 c(Z + c?^) = (a - 6)2 - (c + d)2, by Case III. = [(a - 6) + (c + d)J[(a - 6) - (c + <^)] = {a — b + c + d)(a — b — c — d), Ans. Factor the following : 38. a2-2a6 + 62_c2. 43. 2 mn - jv" -{- 1 - m\ 39. m'-{-2mn-{-n--p\ 44. Oa^ - 24 a5 + 16 6^ _ 4c'-, 40. o?-x'-2xy-f. 45. 16 a^- 4/ + 20?/^ -25.:' 41. a^_2/2_22^22/;2. 46. 4.n^ + m^ -x" -4.m7i.^\ 42. 62-44-2a6 + a2. 47. 4a2- 66-9-62. 48. 10052/ -92^ + 2/2 + 25a^. 49. a2-2a6 + 62-c2-f 20^-^2.'"^ 50. a^ - 62 -f a^ - / + 2 aa; + 2 by. FACTORING. 73 51. a? 4- ?yi^ — \f — ii^ — 2 mx — 2 ny. 53. 4a2 + 4a6 + 62-9c2 4-12c-4. 54. 16 / - 36 - 8 xy- z^ + a^ - 12 2. 55. m2-9/i2 + 25a2-62-10a??H-66«. 100. Case V. When the expression is a trinomial of the form a^ -\- ax -\- b. We have by § 83, (a; -f. 5)(a; + 3) = ar^ -f 8a; + 15, (x-5){x _ 3) = ;r2 _ 8x -f 15, (a; H- 5) (a; - 3) = a^ + 2 x - 15, and (a; - 5)(a; + 3) = ar^ - 2x - 15. In certain cases it is possible to reverse the process, and resolve a trinomial of the form x- -\- ax -{-b into two binomial factors. The first term of each factor will obviously be x ; and to obtain the second terms, we simply reverse the rule of § 83. Find two numbers whose algebraic sum is the coefficient of x, and whose product is the last term. 1. Factor ar^ + 14 a; H- 45. We find two numbers whose sum is 14, and product 45. By inspection, we determine that the numbers are 9 and 6. Whence, a;^ + 14 x + 45 = (x + 9) (x + 5), Ans. 2, Factor ar^ - 5 a; + 4. We find two numbers whose sum is — 5, and product 4. Since the sum is negative, and the product positive, the numbers must both be negative. By inspection, we determine that the numbers are - 4 and — 1. Whence, x^ - 5 x + 4 = (x - 4) (x - 1), Ans. 74 ALGEBRA. 3. Factor aj2 + 6 a; - 16. We find two numbers whose sum is 6, and product — 16. Since the sum is positive, and product negative, the numbers must be of opposite sign, and the positive number must have the greater absolute value. By inspection, we determine that the numbers are -f 8 and — 2. Whence, x^ -\- 6 x - 16 = (x -\- S) {x - 2), Ans. 4. Factor cc^ _ a; _ 42. We find two numbers whose sum is — 1, and product — 42. The numbers must be of opposite sign, and the negative number must have the greater absolute value. By inspection, we determine that the numbers are — 7 and + 6. Whence, x^ - x - 42 = (x - 7) (cc + 6), Ans. Note. In case the numbers are large, we may proceed as follows : Required the numbers whose sum is — 26, and product — 192. One number must be + , and the other — . Taking in order, beginning with the factors + 1 x — 192, all possible pairs of factors of — 192, one of which is + and the other — , we have : + 1 X - 192, + 2 X - 96, + 3 X - 64, + 4 X - 48, + 6 X - 32. Since the sum of -f 6 and — 32 is — 26, they are the numbers required. EXAMPLES. Factor the following : 5. aj2 + 6x + 8. 6. cc2 - 13 aj -h 22. 7. x'-\-6x-7, 8. x'-4:x-21. 9. a^-lla; + 24/' 10. a^-\-Sx-20. 11. x^-x-6. 12. x' + lOx-^-d. 13. a'-7a-U. 14. a''-^a-2. 15. m2 + 11m 4-30. 16. n'-Tn + e, FACTORING. 75 17. a^ + 3a?-40. 31. z'—21z + 110. 18. 2/'+18?/ + 77. 32. x^ + 17 x^ - 84. 19. a2 - 15 a + 54. 33. a* -^2oa'-\- 150. • 20. m2-2m-48. 34. m«-5m3-36. 21. c2h-15c + 36. 35. n« + 10n*-96. 22. a^-12ic + 32. 36. a^y^ _ ^g ^ _^ 34 23. a^-6a;-55. 37. a'b' -^ 2S ab -h 160. 24. n'-\-2n-63. 38. icy - 27 a^i/ + 50. 25. m2-18??i + 72. 39. a^o;* + 5 a V - 126. 26. a2-3a-70. 40. mV - 11 m?i3 - 152. 27. x' + 4.x-96. 41. (a + &)'+ 23(a + 6)+ 60. 28. if2 + 24» + 95. 42. (a;-2/)=^+3(ic-2/)-180. 29. 62-106-24. 43. (a -6)^-22 (a- 6) +112. 30. c2 + 20c + 84. 44. {x -\- yf - 2 (x -\- y) - US. 45. Factor .t^ + 6 abx - 27 a^t^. We find two quantities whose sum is 6 aft, and product — 27 a^b^. By inspection, we determine that the quantities are — Sab and 9 ab. Whence, x^ ^Qabx- 27 a^fts =(x - 3 a6) (x + 9 a6), ^ws. 46. Factor 1 _ 3 a - 88 al We find two quantities whose sum is -3 a, and product - 88 a^. By inspection, we determine that the quantities are 8 a and - 11 a. Whence, 1 - 3rt - 88a2 =(1 + 8a)(l - 11a), Ans. Factor the following : 47. a2 + 12a6-h35 6l 51. a'i.5am-66m\ 48. x^-llax-\-2Sa'. 52. m^ + 16 w?i + 48 ^i^. 49. ar^ H- 4 xy — 5y^. 53. x^ — mx — 12 m\ 50. l-2a-3a2. 54. l-14a + 33al 76 ALGEBRA. 55. a^-4:ab-60 h\. 61. 1 + 18 a5 + 80 a^h\ 56. l-\-x-12x\ 62. x'-\-lxy -my\ 67. x" - 15 xy + 50 y\ 63. a"})' + 16 a?>c + 28 c\ 58. i»2 4. 20 aaj + 99 al 64. x" -21 xyz + lQSyh\ 59. m2-16mn4-15n^ 65. l+llxy-2&x-y\ 60. a^-aft- 20 51 66. a« - 6 a-^ftc^ - 160 6 V. 101. If an expression can be resolved into three equal factors, it is said to be a perfect cube, and one of the equal factors is called its cube root. Thus, since 27 a%^ is equal to 3 a^6 x 3 orb x 3 d^b, it is a perfect cube, and 3 a^b is its cube root. 102. The following rule for extracting the cube root of a perfect monomial cube is evident from § 101 : Extract the cube root of the numerical coefficient, and divide the exponent of each letter by 3. Thus, the cube root of 125 a^6V is 5 a^Wc. 103. Case VI. When the expression is the sum or differ- ence of two perfect cubes. By § 85, the sum or difference of two perfect cubes is divisible by the sum or difference, respectively, of their cube roots. In either case, the quotient may be obtained by aid of the rules of § 85. 1. Factor a^ + l. The cube root of a^ is a, and of 1 is 1 ; hence, one factor is a + 1. Dividing a^ + 1 by a + 1, the quotient is a^ — a + 1 (§85). Whence, a^ + 1 = (a + l)(a2 - a + 1), Ans. 2. Factor 27 a^- 64/. The cube root of 27 x^ is 3 x, and of 64 y^ is 4 tj (§ 102). Hence, one factor is 3 x — 4 y. Dividing 27 x» - 64 y^ by 3 x - 4 y, the quotient is 9 x^ + 12 xy + 16 y^ (§ 85). Whence, 27 x^ - 64 2/3 = (3 x - 4 y) (9 x^ + 12 xy + 16 y^) , Ans. 3. m^ -\- n^. 9. 64a^ + l. 4. a' - h\ 10. l-125al 5. a?-l. 11. 27a^-8/. 6. a^ - f7^. 12. 8 a^ly" -h 125. 7. 8. a^ + Qi^. 1 + m\ 13. 14. 216 - m\ 125-64ary.'' FACTORING. 77 EXAMPLES. Faxjtor the following : 15. m«4-343n3. 16. 125 6^- 216 c^. 17. M^m^-]-^x^. 18. 27a«-f 343 6«. 19. 512a^4-27yz«. 20. 64 a«6« - 729 c». 104. Case VII. When the expression is the sum or differ- ence of two equal odd powers of ttvo quantities. By § 87, the sum or diiference of two equal odd powers of two quantities is divisible by the sum or difference, respectively, of the quantities. In either case, the quotient may be obtained by aid of the rules of § 86. , C y ST i^a a -f 7 ^ (^ 1. Factor a^ + 2>^. I y ^ ^ X "^ By § 87, one factor is a + 6. Dividing a^ + 6^ by a + 6, the quotient is a* - a^h + aW - ab^ + 6*. (§ 86) Hence, n^ -\- b^ = (a -\- b) (a* - a^b + a^b^ - ab^ + 6*), Ans. EXAMPLES. Factor the following : 2. x^-f. 6. aJ-hbl 10. l4-32ic5. 3. a-' + l. 7. 1-xl 11. 243m^-l. 4. l-m\ 8. m^ + 1. 12. a:^ - 128. 5. .ry4-2'. 9. 32 -a\ 13. 32 a* -h 243 6^ 105. By application of the rules already given, an ex- pression may often be resolved into more than two factors. 78 ALGEBRA. If the terms of the expression have a common monomial factor, the method of Case I should always be applied first. 1. Factor 2 a^if — 8 axy^. We have, 2 ax^y'^ — 8 axy^ = 2 axy'^(x^ — 4 ?/-), by Case I. Whence by Case IV, 2 ax^y"^ - 8 axy^ = 2 axy'^ix -h 2y)(x - 2y), Ans. If the given expression is in the form of the difference of two perfect squares, it is always better to first apply the method of Case IV. 2. Factor a«-6«. We have, a^ - b^ = (a^ + 63) (^^3 _ ^3)^ by Case IV. Whence by Case VI, a^-b^= (rt + b) (a2 -'a6 + b^) (a - b) (a^ + ab + b^) , Ans. 3. Factor ic^ — i/. We have, x^ — y^ = (x^ + y^) (x* — y^), by Case IV = (x4 + ?/4)(x2 + ?/2)(a:2-?/2) = (ic* + y*) (a;2 + y^) (x + y) (x-y), Ans. MISCELLANEOUS AND REVIEW EXAMPLES. 106. Factor the following : 1. 35a*b^-^9Sa'b^-49a'b. 10. 4. a'b' + ^ a'b'. 2. 25 aV- 816V. 11. a^ + 15 a6 + 56 61 3. a^ + 11^ + 18. 12. xhf - 23 xy + 132. 4. a'bc-\-acH-abH-bcd^ 13. lOS x' - 36 x" + 3 x". 5. 6x^-6x\ 14. 64: a^b- 121 a'b\ 6. A97n^ + 567nn + 16n\ 15. x^-1. 7. o? -10a + 24.. 16. x^ + o^y + xy"^ + f. 8. ar'4-17a;2-38a;. 17. a=^6^ - 3 aft^ _ 180. 9. a?-(p^c)\ 18. 2aj2 + 20a^-782/^ FACTORING. 79 Id. SO x' -DO 3^-^-65x^-20 x\ 25. 27d'-6'kxf. 20. l-a«. 26. 32a^ + 2/i«. 21. 16 x' - 1. 27. 8 a^b- 12 a%'' 4- 162 ah^. 22. 64 a2^2 _ 80 ahc + 25 &. 28. 1-11 m/i - 60 mV. 23. 15ac+18arf-356c-42 M 29. {x - yf-{m - n)\ 24. 100 iB« - 49 1^. 30. (1 + ny- 4 n\ 31. 64 0.^2"- 56 a;yz* + 72 ar^2/22^ 32. 3a«62_3air 5O. ^x? ■\-2of -\6z^ -^SOxy. 33. ?M* - 81. 51. 343 m^ + 216 n\ 34. 8 ary + 125. 62. (9 a^ + 4)" - 144 a\ 36. {yii^-n)^^-l{yii^n)-\\\. 53. (.x-^ -fa;- 9)^ - 9. 36. a^^ - 15 ahxy - 54 ^y. 64. {a^-2af+2 {o?-2a) + l. 37. 25 ar* + HO xy + 121 /. 55. a^¥ -f aV - 6V - x^- 38. 4a«-8a^-2a^-f 4«l 66. a;«-256. 39. (5a;-8y)2-(4.x-9^)2. 57. 36a2-462_ 49c2 + 28 6c. 40.5ar'4-5ar^. 58. m'' - 62b. 41. (a2 + 9)=^ - 36 a\ 69. (a.-2-f3 a;)^ +4 (ar^+3 a;) +4. 42. a;^ - (a; -h 2f. 60. a« - 7a« - 8. 43. aV-46V-9a2d2-f 36^2^2 61. 27 a« - 1000 ftW 44. (a;2-5a;)2-2(ar^-5a;)-24. 62. 128 - m^ 45. 16 a;* - 72 ar^^z' + 81 y". 63. 2 a26c-2 hh-A Wc^-2 h&. 46. a« - 2 a^ -f 1. 64. (a2-f7a)2+4(a2+7a)-96. 47. 64-a;«. 66. a;^« + 2af + 1. 48. 45 ar^+18 a;*+60 a.-3+24 3?. 66. {:^ - 4)^ -(x-{- 2)\ 49. 9 (wi-n)2-12 (m-w)4-4. 67. (a^ - 6^ 4. c^)^ - 4a2c2. 68. Resolve x^ — 7/ into two factors, one of which is x-\- y. 69. Resolve a^ — b^ into two factors, one of which is a — b. 80 ALGEBRA. 70. Resolve x^ -\- y^ into two factors by the method of § 104. 71. Resolve a^ + 2/^ into three factors by the method of §103. 72. Resolve 1 — w? into two factors by the method of § 104. 73. Resolve a^ — 1 into three factors by the method of §103. 74. Faxjtor 3 {m -f nf - 2 (m^ - n^). 3(w + n)2 - 2(m2 - ii^) = S{m + n)'^ - 2(m + ?i)(wi - n) = (m + n) [S(m + «) - 2(m - n)] = (m 4- w)(w + 5w), Ans. 75. Factor (a + 6)^ - (a - bf. By the method of § 103, we have {a + by-ia-by = l{a + b)-(a- &)][(« + 6)2 + (a + b)(a - b)+{a - 6)2] = (^aj-b -a + 6)(a2 + 2a6 + 62 + a2 - 62 + a'^- 2ab + 62) = 2 6(3a2 4- 62), Ans. Factor the following : 76. (m - xf 4- Sx-^. 84. a:'-b'+x'-y'-^2ax-\-2by. 77. a^ - (a - bf. 85. (a? - m)'^ -x(x^- m^. 78. 5 (aj2 - 2/') 4- 4 (.t - 2/)'. 86. (x + 2/)-^ - (a? - y)^. 79. (a' + 53) - 2 a6 (a + b). 87. a^° - 1. 80. a^-\-b^-d'-d^+2ab-2cd. SS. x' -{- x' - a^ - 1, 81. (x + 1)^' +{x- ly. 89. (a^ - 1) - (a - 1)^ 82. (a^ + f)-\-x(x + yf. 90. (3m- 2f + (2 m + 1)^. 83. a' -a'- a' + 1. 91. (c^ -f-%y-^ yh\ 92. a2 + 2562_16c2-9d2_10a6-24cd 93. (l + a^)+2(l-a)(l + a)2. HIGHEST COMMON FACTOR. 81 X. HIGHEST COMMON FACTOR. , 107. The Degree of a rational and integral monomial (§ 69) is the number of letters which are multiplied to- gether to form its literal portion. Thus, 2 a is of the first degree ; 5 a6 of the second degree ; Sa^b^, being the same as Saabbb, is of the fijlh degree; etc. The degree of a rational and integi-al monomial is equal to the sum of the exponents of the letters involved in it. Thus, a*b(^ is of the eighth degree. 108. A- polynomial is said to be' rational and integral when each term is rational and integral ; as 2a^b —3c-\- d^ The degree of a rational and integral polynomial is the degree of its term of highest degree. Thus, 2 a^b — 3 c + di^ is of the third degree. 109. A Prime Factor of an expression is a factor which cannot be divided without a remainder by any expression except itself and unity. Thus, the prime factors of 6a^(x^ -^ 1) are 2, 3, a,a,x-\- 1, and x — 1. 110. The Highest Common Factor (H. C.F.) of two or more expressions is the product of all their common prime factors. It is evident from this definition that the highest common factor of two or more expressions is the expression of high- est degree (§ 108) which will divide each of them without a remainder. 111. Two expressions are said to be priine to each other when unity is their highest common factor. 82 ALGEBRA. 112. Eequired the H. C. F. of a'b'c", a^W&, and a^hc\ Resolving each expression into its prime factors, we have a'^h'^i? = aaaabbccc, o?W& _ aabbbccccc, and a^bd^ = aaabcccc. Here the common prime factors are a, a, b, c, c, and c. Whence, the H. C. F. = aabccc = a^b&. It will be observed, in the above result, that the exponent of each letter is the lowest exponent with which it occurs in any of the given expressions. 113. In determining the highest common factor of alge- braic expressions, we may distinguish two cases. 114. Case I. When the expressions are monomials, or 'Dolynomials which can be readily factored by inspection. 1. Find the H. C. F. of 28 a'W, 42 ab% and 98 a^Wd^. We have, 28 a%^ = 2^x1 x a'^b^ 42 ab^c = 2 X 3 X 7 X ab^c, and 98 a%^d^ = 2xT^x a%^cP. By the rule of § 112, the H. C. F. = 2 x 7 x ab^ = 14 ab^, Ans. EXAMPLES. Find the highest common factor of : 2. 2a%5a^b\ ^ 4. 45cr6^120aV. 3. 20a^y,15xy'. 5. lS2a^yz', S4:a^fz. 6. 16mV, 56 m%2, 88 mV. 7. 36 a'bc% 72 a^b% ISO ab^c^ 8. 126 aV, 21 a'xV, 147 a'x^z. 9. 140 mVa;2, 175m^n% 105 mhis^. 10. 117a^62c«, 104a*6V, 156a^6V. HIGHEST COMMON FACTOR. §3 11. Find the H. C. F. of 5 x^y - 45 y?y and 10 y^y" + 40 ^y'' - 210 xy".. We have, 5 a^y — 45 x^y = 5 yhf {y?- - 9) = 6x2y(x + 3)(x-3), (§99J and 10 xhP' + 40 xV _ 2IO xy"^ = 10 x^a (x2 + 4 x -21) = 2 X 5 X X2/2 (x + 7) (X - 3). (§ 100) By the rule of § 112, the H. C. F. is 5xy (x - 3), Ans. 12. Find the H. C. F. of 4 a^ — 4 a + 1, 8 a^ — 1, and 2am — m — 2a]i-\- n. We have, 4rt2 _ 4^ _|. 1 = (2a _ i)2, (§98) 8a3- 1 = (2a-l)(4a2 + 2a+l), (§103) and 2 am — »n — 2 an + n = (2 a— 1) {m — 71). (§ 93) By the rule of § 1 12, the H. C. F. is 2 a - 1, Ans. Find the highest common factor of : 13. 6 a^b^ - 15 a^b\ 12 a'h + 21 a^h\ 14. 68 (m + nf (m - n)\ 85 (m + nf (m - n). 15. ar^-9/, a.-2-6a^ + 9/. ^ 16. 3a3-21a2_a + 7, a2 + 6a-91. ^ 17. 2a8a; + 4aV + 2aar8, 3a^x + 3aa;^ 18. m« - 27, m^ - 11 m + 24. 19. ac-^ad — bc — bd, a^ — 6ab-\-5 b\ 20. a; + 4a^4-4a^, 4 + 44a;4-72iB2. 21. 80n'-5n«, 20ri^4-5 7i2. ^ 22. a2 + 62_^_j..2a6^ a2-62_c2 4-26c. ^ 23. ar^H-2a;-24,y-14a;-^40, ar^-8a;-f 16. 24. 9,a2 _ 12 a -f- 4, 9 a^ - 4, 18 a^ - 12 a\ 25. iB2-6a;-27, a.'24.6a;4-9, 3^^21. 84 ALGEBRA. 26. a^ -f 13 cr + 40 a, a^ - a^ - 30 a', a' -\- 2 a' - 15 a^ 27. m^ - 4 m, m^ + 9 m^ - 22 m, 2 m^ - 4 1??;^ - 3 m^ + 6 m. 28. x'^-^f, x'-4.y', x'-9xy-^Uy\ 29. Sa'-a'b-^-Sab- b% 27 a^ - 6^ 9 a^ _ 6 a?> + 61 30. 27a.'3 + 125, 9a?2-25, 9 a;^ ^ 39 3. _^ 25. 31. ^y-:f?f-2^xf, 2 x'y'' ^-22 o^f ^m xy\ Zx'^y-^^:f?f. 32. 16 m^ — wS 16 m'' — 8 mV + ri^ 2 ma; + 2 m?/ — nic — wi/. 33. (X' — ^, a^ — a^x — aa^ + .t^, 3 a^ — 3 a^a; 4- 5 ax^ — 5 x^. 115. Case II. IF/ieii the expressions are polynomials which cannot be readily factored by inspection. The rule in Arithmetic for the H. C. F. of two numbers is: Divide the greater number by the less. If there be a remainder, divide the divisor by it; and con- tinue thus to make the remainder the divisor, and the preceding divisor the dividend, until there is no remainder. The last divisor is the H. C. F. required. Thus, let it be required to find the H. C. F. of 169 and 546. 169)546(3 507 ~39)169(4 156 ~13)39(3 39 Then, 13 is the H. C. F. required. 116. We will now prove that a rule similar to that of § 115 holds for the H. C. F. of two algebraic expressions. Let A and B be two polynomials, the degree of A (§ 108) being not lower than that of B. HIGHEST COMMON FACTOR. 85 Suppose that B is contained in A p times, with a remain- der (7; that C is contained in B q times, with a remainder D ; and that D is contained in C r times, with no remainder. To prove that D is the H. C. F. of A and B. The operation of division is shown as follows : B)A{p pB ~C)B(q qC D)C{r rD We will first prove that Z) is a common factor of A and B. Since the minuend is equal to the subtrahend plus the remainder (§ 35), we have A=pB+C, (1) B=:qC-\-D, (2) and C = rD. Substituting the value of G in (2), we obtain B = qrD + D = D(qr -\- 1). (3) Substituting the values of B and C in (1), we have A=pD (qr -{-l)-^rD = D (pqr -\-p + r). (4) From (3) and (4), Z> is a common factor of A and B. We will next prove that every common factor of A and B is a factor of D. Let F be any common factor of A and B ; and let ^ = mi^ and ^ = nF. From the operation of division, we have C=A-pB, (5) and D=B- qC. (6) 86 ALGEBRA. Substituting the values of A and B in (5), we have C= mF — pnF. Substituting the values of B and C in (6), we have . D = nF — q (mF — piiF) = F(n — qm-{- jyqn). Whence, i^ is a factor of D. Then, since every common factor of A and 5 is a factor of D, and since D is itself a common factor of A and B, it follows that D is the highest common factor of A and B. 117. Hence, to find the H. C. F. of two polynomials, A and B, of which the degree of A is not lower than that ofB, Divide Ahy B. If there he a remainder, divide the divisor by it; and con- tinue thus to make the remainder the divisor, and the 2^receding divisor the dividend, until there is no remainder. The last divisor is the H. C. F. required- Note 1. Each division should be continued until the remaiudei- is of a lower degree than the divisor. Note 2. It is of the greatest importance to arrange the given polynomials in the same order of powers of some common letter (§ 33), and also to arrange each remainder in the same order. 1. Find the H. C. F. of 6 a;2 _ 13 a; _ 5 and 18 a^ - 51 a^ + 13 a; + 5. 6 ic2 - 13a; - 5)18 a;3 - 51 a;2 + 13 a; + 5(3 a; _ 2 18x3 -39x2 -15 a; - 12 x2 + 28 X - 12 x2 + 26 X + 10 2x- 5)6x2 -13x-5(3x + l 6x2 -15x 2x 2x-5 Whence, 2x — 5 is the H. C.F. required. HIGHEST COMMON FACTOR. 87 Note 3. If the terms of one of the given expressions have a common factor which is not a common factor of the terms of the other, it may be removed ; for it can evidently form no part of the highest common factor. In like manner, we may divide any remainder by a factor which is not a factor of the preceding divisor. 2. Find the H. C. F. of 6a^ -25x^-^Ux and 6ax^-{-llax- 10 a. In accordance with Note 3, we remove the factor x from the first expression, and the factor a from the second. 6x2 - 25x -f- 14)6x2 + llx - 10(1 6x2-25x+14 36x- 24 divide this remainder by 3x -2)6x2- 6x2- 12 (Note 3). -25x + 14(2x - 4x -21x - i!l X + 14 . ;-7 Whence, 3 x - 2 is the H. C. F. required. Note 4. If the given expressions have a common factor \yhich can be seen by inspection, remove it, and find the H. C. F. of the ' resulting expressions. The result, multiplied by the common factor, will be the H. C. F. of the given expressions. 3. Find the H. C. F. of 2a^-Sa^b-2 ab^ and 2 a» -f- 7 a^ft + 3 ab\ In accordance with Note 4, we remove the common factor a, and find the H. C.F. of 2a2 -Sab - 2 62 and 2 a2 + 7 ab + 362. 2a2-3a6-262)2a2+ lab-{-Sh^(l 2a2- 3ab-262 5 6)10a6 + 5 62 2a + b 2 a + 6)2 a2 - 3 a6 - 2 62(a - 2 6 2 a2 4- ab -4ab -4 ah -2 ir^ Multiplying 2 a + 6 by a, the required II. C. F. is «(2 a + 6). 88 ALGEBRA. Note 5. If the first term of the dividend, or of any remainder, is not divisible by tlie first term of the divisor, it may be made so by multiplying the dividend or remainder by any term which is not a. factor of the divisor. Note 6. If the first term of any remainder is negative, the sign of each term of the remainder may be changed. 4. Find the H. C. F. of 2a^-3a;2^2aj-8and3a^-7a^ + 4a;-4. 3x3- 7x=^ + 4a;-4 2 2x^-Sx^-\-2x~S)6x^-Ux^-\-Sx- 8(3 6x3- 9x2 4-6x-24 - 5x-^ + 2x + 16 2x3-3x2+ 2x- 8 5 6x2 -2x- 16)10x3- 15x2+ lOx- 40(2x 10x3- 4x2- 32 X -11x2+ 42 X- 40 ' ' 5 - 55x2 + 210 x-200(- 11 -55x2+ 22X+176 188) 188 x - 376 X- 2 X- 2)5x2- 2x-16(5x + 8 5x2 -lOx 8x 8x-16 Whence, x — 2 is the H. C. F. required. In the above example, we multiply 3 x3 — 7 x2 + 4 x — 4 by 2 in order to make its first term divisible by 2 x3. We change the sign of each term of the first remainder (Note 6), and multiply 2 x3 — 3 x2 + 2 x — 8 by 5 to make its first term divisible by 5x2. We multiply the remainder — 1 1 x2 + 42 x — 40 by 5 to make its first term divisible by 5 x2. HIGHEST COMMON FACTOR. 89 EXAMPLES. Find the H. C. F. of : 6. 2 a^ + 7 a + 6, 6 «- + H a + 3. 7. 4x2 + 13a; + 10, Gx-^H-Sa^-U. 8. ar^ + 5a;-24, a^ + 4a!2-26a;-f 15. 9. 3m2 + m-2, 4m3 + 2m2_mH-l. 10. lSa^-^9ab-ob-, 24:a'-29ab-{-7b'. 11. 12 a^ — 5 a^x — 11 aor + 6 af^, 15 a*^ + 11 arx — 8 aa^ — 4 ic^. 12. 4a;«-12ar'4-5a;, 2a;^ + ur^-7x'2-20a;. 13. S3^-{-13x'y-\-12xy', 9a^y-22xf-Sy\ 14. 4a*-lla2 + 5a + 12, 6a'^-lla^ + 13a'^-4a2. 15. 2 m^-f 5 m^M— 2 ??iV-f 3 7nn% 6 m^7i— 7 ??t^;A^+5 mn^— 2 ^^^ 16. 3ar'-4aj-4, 3a;* - 7a^4-6ar^-9a; + 2. 17. 3a* + 5a3 + 12a2-f 8, 6a* + lOa--^ + 19a- ^ 10a -4. 18. 2m^-3m^x-Smx^-3a^, 3 m* — 7 w?x — 5 m V — mar^ — 6 a;"*. 19. 2a*-a»-4a2 + 3a, 4a*-6a=' + a- + 4a-3. 20. m^ + 8 m2, w^•^ - 2 »i* - 15 ^u^ - 14 m\ 21. 4a*-22a=^64-6a2/r'+20a6^ 9 a''b-^2 tv^b^-l^aif+l^b^ 22. 4ar' + 9x--9, 2.r* + 11 ar'^ + 14.i-- 5a;- 6. 23. 3a*-6a^+4a=^+4a-4, 3 a^+ 3 a*- 11 a^- 2 a^ + 6 a. 24. a« + 2a2-2a + 24, a* + 2a3 - 11 a^ - 6a + 24. 25. 2x^-3x''y^-3x'y''-3xf + y', 2 a;* + ar^y — 3 a;^2/^H" 5 a;^ — 2 ?/*. 26. 2a;* + a;3-9a;2_^a;4-l, 2a;* - 9a.'^4- 12a;2 _ 3^_2. 27. 2y?-7x^ + 7 x-2, x^-3x''-^nx'-4.x + 4t. 28. a« a; - aV - a?:t? - aV - 2 ax^, a^x + 3 a*ar^ — a^a^ — 4 a V — oar*. 90 ALGEBRA. 118. The H. C. F. of three expressions may be found as follows : Let A, B, and C be the expressions. Let G be the H. C. F. of ^ and B\ then, every common factor of G and (7 is a common factor of A, B, and C. But since every common factor of two expressions exactly divides their highest common factor (§ 116), every common factor of A, B, and C is also a common factor of G and C. Whence, the highest common factor of G and C is the highest common factor of A, B, and C. Hence, to find the H. C. F. of three expressions, find the H. C. F. of two of them, and theii of this result and the third expression. We proceed in a similar manner to find the H. C. F. of any number otf expressions. 1. Find the H. C. F. of iK3_7a;4-6, ;x?-{-Zx^-Ux-^12, and o?-Bx^ + lx-^. TheH.C.F. of ic^ - 7 a: + 6 and x^ + 3:^2- 16x + 12 isx2-3x + 2. The H. C. F. of x^ - 3 x + 2 and a;^ - 5 x'^ + 7 x - 3 is x - 1, Ans. EXAMPLES. Find the H. C. F. of: 2. ^x'-llx^-S^, 4.x^-12x-27, 6a^-31a;+18. 3. Sa^ + 22a-{-5, 12a^-lSa-4:, 20 a' + 29 a -\- 6. 4. 15m' -4.771 -32, 18m2 + 3m-28, 21m' + 25 m -A. 5. 5a'-\-23ab-10b', 5 a^ -\- S3 a'b -\- ^6 ab' - 24: b% 5a^-{- SSa'b + 59 ab' - 30 61 6. a^-{-x^-14:X-24, x^ - 3x^ -6x-{-8, a^ -\-4:X^ -i- x - 6. 7. a'-a'-5a-3, a' + 2a'-a-2, a^ - 2a' -2a-\-l. 8. 2m^ + 97n'-6m-5, 3m^ + 10m'-23m-\-10, 6m'-7m'-m-\-2. 9. 2x^-x'y-27xf + 36f, 2 a^ - 5 x'y - 37 xif -\- 60 f, 2x^- 19a^^ + 51xf - 4.5 f. LOWEST COMMOX MULTIPLE. 91 XI. LOWEST COMMON MULTIPLE, 119. A Common Multiple of two or more expressions is an expression which can be divided by^each of them with- out a remainder. 120. The Lowest Common Multiple (L. C. M.) of two or more expressions is the product of all their different prime factors (§ 109), each taken the greatest number of times that it occurs as a factor in any one of the expressions. 121. Required the L. C. M. of a'b^c, ah'd\ and 6Vd*. Here, the different prime factors are a, h, c, and d; a occurs twice as a factor in a?bh ; h five times as a factor in a6'd^; c three times as a factor in 6Vd*; and d four times as a factor in h^&d*. Whence, the required L. C. M. is a^ftVd^ (§ 120). It will be observed, in the above result, that the exponent of each letter is the highest exponent icith which it occurs in any one of the given expressions. 122. It is evident from the definition of § 120 that the lowest common multiple of two or more expressions is the expression of loivest degree (§ 108) which can be divided by each of them without a remainder. 123. If two expressions are prime to each other (§ 111), their product is their lowest common multiple. 124. In determining the lowest common multiple of algebraic expressions, we may distinguish two cases. 125. Case I. When the expressions are monomials, or polynomials which can he readily factored by inspection. 92 ALGEBRA. 1. Find the L. C. M. of 2Sa''h% Uh(?, and 63 cU We have, 28 a^h^ = 2^ x 7 x a^ft^, 54 6c3 = 2 X 33 X 6c3, and 63 cH = 32 x 7 x cH. By the rule of § 121, the L. C. M. = 2^ x 3^ x 7 x a^h'^cH = 756 a'^b^c% Ans. EXAMPLES. Find the lowest common multiple of : 2. 5ab% 7a^b\ 6. 55 xy, 70 yz, 77 zx. 3. 12 xy% 54. yz\ ■ 7. 50 a'b% 60a'b% 75a'b\ 4. 24 m^ 4.5 n\ 8. Wx'f, 21 fz, 33 a^;?^. 5. 72 a%, 96 6V. 9. 20 a6^ 27 6V, OOc^cZ^. 10. 36m^nic, 40mwy, 48 7iV?/. 11. 56a26c«, 84a36«d^ 126 aVt^^^ 12. Find the L. C. M. of x'-^-x-Q, a^-4a; + 4, and x^ — 9x. We have x"^ -h x - 6 =(x + S)(x - 2), (§ 100) aj2_4a; + 4=(x-2)2, (§98) and x^-9x = x(x + S)(x-S). (§99) By the rule of § 121, the L. C. M. = x(x - 2)2(a; + 3) (x - 3), Ans. Find the lowest common multiple of : 13. a" - 62, a^ _|_ 2 a6 + 61 14. m^ + '^n, mn — ti^. 15. a^-9, ar^ + lOaj + 21. 16. a;^-18a^ + 81aj^ x'3-13a;2H-36a;. 17. a^ - 3a6 + 2 6^ ac + ad -be- bd. 18. a2-h2aa; + a^, a^ + a^. 19. l-8ar^, l + 9a;-22a;l LOWEST COMMON MULTIPLE. 93 20. m^ 4- 13 m^n + 40 mn^, mhi — mii^ — 30 n^. 21. 4a;2_25^ 2aj3-5a^-4a; + 10. 22. x" -^^ax'-lS a% ax" + 15 a-x + 54 a\ 23. 4 a^ - 2 «6, 4 a6 + 2 6^, 4 a^ - h\ 24. 6a:2_^-^Q3^^ 9a,'2/-15/, ^Qs^y-imxif. 25. 4m2-8i^ + 4, 6/?i2 + 12?yi + 6, m^ - 1. 26. a2 - 12 a + 35, a^ + 2 a - 63, a^ - 3 a - 108. 27. a;* — 4 aa^ + 4 aV, a^ + 4 aa; H- 4 a^, oa;^ — 4 a?x. 28. 3a:2_g,p_72, 4ar^4-8x-192, 2a^-24a; + 72. 29. x^y-xi^, a^-ff x^-2xy + y^. ■^ if 30. ar^ + 2/2 _ ^2 _ 2a^, a^ _ y-' _ ^2 ^2;. ^ 31. 16 m^ - 9 ?i^ 8 ahhn - 6 aft^^i, 16 m^ - 24 m^i + 9 n\ 32. a-^-a, a^-9a^-10a, a* - a^ -\- a^ - a. 33. aj2 + 4.ri/ + 4/, a.'^-h a;?/ - 2/, ar^ + 82/^. 34. 2a^-2a'--ia, 3a*-6a^-9a% 4 a^ -f 20 a^ -f 16 a^. 35. 27a;«-8, 9a.'2-4, 9ar'-12a; + 4. 36. 4ar^-4m2, ex-{-6m, Sx^-^-Sm^ 9a;-9m. 37. x*-y\ x^-\-2x'fVy*, x'-2a^y^ + y\ 38. a^ 4- b% a^ - b% (a" + ft?)^ - ci'b'. 39. a2- 11 aa; + 18^2, a^- 5aa; - Ua:^^ a*- 8aV+ 16 a;*. 40. m^ — 71^, m^ — m^n — mn^ + n^, m^ + m^^i — mn^ — n^. 41. a2+62_c2+2a6, a^- 62_ c2_2 6c, a^- 6^+ c^- 2ac. 126. Case II. When the expressions are polynomials which cannot be readily factored by inspection. Let A and B be any two expressions. Let F be their H. C. F., and M their L. C. M, ; and sup- pose that A = aF, and B = bF. 94 ALGEBRA. Then, Ax B== ahF\ (1) Since F is the H. C. F. of A and B, a and b have no com- mon factors ; whence, the L. C. M. of aF and bF is abF. That is, M=abF. Multiplying each of these equals by F, we have FxM= abF\ (2) From (1) and (2), AxB = FxM. (§ 9, 4) That is, the product of two expressions is equal to the prod- uct of their H. C. F. and L. C. M. Therefore, to find the L. C. M. of two expressions, Divide their product by their highest common factor ; or, Divide one of the expressions by their highest common fac- tor, and multiply the quotient by the other expression. ' 1. Find the L. C. M. of 6x^-17 aj + 12 and 12a^-4a;-21. 6x2- 17 x + 12)12x2- 4x-21(2 12x2- 84x + 24 15)30 X- 45 2x- 3)6x2 - 17 X -f 12(3 X- 6x2- 9x -4 - 8x - 8x4-12 Then the H. C. F. of the expressions is 2 x - 3. Dividing 6 x2 - 17 x + 12 by 2 x - 3, the quotient is 3x - 4. Whence, the L. C. M. =(3x - 4) (12 x2 - 4x - 21), Ans. EXAMPLES. Find the L. C. M. of: 2. 2aj2-3ic-35, 2a;2-19a; + 45. 3. 3a2-13a + 4,3a2-f 14a-5. 4. 6a2 + 25a6 + 2462, 12a2 + 16a6-362. 5. &:»?^-llx'y-2xy\^x'y + 21xy''-\-10f. LOWEST COMMON MULTIPLE. 95 6. 12m2-21m-45,4m3-llm2-6m4-a 7. 2a^-5a^-lSa-9,3a^-Ua'-a-\-6. 8. 2a^x-\-a^x^-[-2aa^-hSx\ 2 a^x -\- 5 a'sc^ -\- 2 as^ - x\ 9. 2 a^ - 5 a6 + 3 b% a* -\- a^b - 5 a'b'' + 2ab^-\- b\ 10. 6a^-7ic2 + 5a;-2,4a^-5ar^ + 4a;-3. 11. 2a-^-5a2 + a + 2,4a3-9a-4. 12. 3 m^ — 7 m^n + 4 mn^, 6 mhi — 4 m^7i^ — 14 mn^ — 4 n*. 13. a' + 2a*-5a^-hl2a%Sa^-\-lla'-6a*-7a'-{-4a\ 14. 3a^-2a.-3-12a^-a; + 6, 3a;* + 7353 + 60^2 _ 2a; _ 4. 127. The L. C. M. of three expressions may be found as follows : Let A, B, and C be the expressions. Let M be the L. C. M. of A and B ; then, every common multiple of M and C is a common multiple of A, B, and C. But since every common multiple of two expressions is exactly divisible by their lowest common multiple, every common multiple of A, B, and C is also a common multiple of M and C. Whence, the lowest common multiple of M and C is the lowest common multiple of A, B, and C. Hence, to find the L. C. M. of three expressions, find the L. C. M. of two of them, and then of this result and the third expression. We proceed in a similar manner to find the L. C. M. of any number of expressions. EXAMPLES. Find the L. C. M. of : •^ 1. 2a^ + a;-15, 2a!- + 7a: 4-3, 2a;2 + 9a; + 9. 2. 3a2 + a-2,6a- + ll(H-5,9a2 + 5a-4. 3. 2m2-5m4-2, 3 wi" - 10 ?n, + 8, 4 m^ + 10 m - 6. 4. 2a^- 5.^ ^ 3a;, 4a;* - 11 a^-Sop',ex^- x^ -2x\ . 5. a3-2a2-5a + 6,a3-3a2-a + 3,a3-|-4a2 + a-6. 96 ALGEBRA. XII. FRACTIONS. 128. The quotient of a divided by b is Avritten - (§ 3). The expression - is called a Fraction ; the dividend a is b called the numerator, and the divisor b the denominator. The numerator and denominator are called the te7'ms of the fraction. 129. Let ^ = x. (1) Then since the dividend is the product of the divisor and quotient (§ 54), we have a = bx. Multiplying each of these equals by c (§ 9, 1), ac = bcx. Regarding ac as the dividend, be as the divisor, and x as the quotient, this may be written Tc = - (^) From (1) and (2), ^ = ^. (§9,4) be b That is, if the terms of a fraction be both multiplied, or both divided, by the same expression, the value of the fraction is ^lot altered. 130. By the Law of Signs in Division (§ 55), ■j-a _ — a _ -]- a _ —a That is, if the signs of both terms of a fraction be changed, the sign before the fraction is not changed ; but if the sign of either one be changed, the sign before the fraction is changed. FRACTIONS. 97 If either term is a polynomial, care must be* taken, on '.^hanging its sign, to change the sign of eoc/i of its terms. Thus, the fraction ~ , by changing the signs of both c — d , _ numerator and denominator, can be written (§ 41). d — c 131. It follows from §§49 and 130 that If either term of a fraction is the indicated product of two or more expressions, the signs of any even number of them may he changed without changing the sign before the fraction ; but if the signs of any odd number of them be changed, the sign before tJie fraction is changed. Thus, the fraction ^^^ may be written a -^b b — a b — a , • , etc. {d - c) (/- e)' (d - c) {e -/)' {d - c) (/- e)' REDUCTION OF FRACTIONS. 132. To Reduce a Fraction to its Lowest Terms. A fraction is said to be in its lowest terms when its numer- ator and denominator are prime to each other (§ 111). 133. Case I. When the mimerator and denominator can be readily factored by inspection. By § 129, dividing both terms of a fraction by the same expression, or cancelling common factors in the numerator and denominator, does not alter the value of the fraction. We then have the following rule : Resolve both numerator and denominator into their factors, and cancel all that are common to both. 24 o^b^c 1. Reduce — — to its lowest terms. ^Od'bH We have 24 gSftgc ^ 28 x 3 x gSft^c ' 40 a'^hhl 28 X 5 X a%'^d ^Uh 98 ALGEBRA. Cancelling the common factor 2^ x a^h'^, we obtain 24 a%'^c 3 ac 40 o^hH 5 d Ans. a^ — 27 2. Reduce — to its lowest terms. nr — 2x — 'd We have, _^slIL. = (x - S)(x^ ^ Sx + 9) ^^^ x+1 Note. If all the factors of the numerator be cancelled, unity re- mains to form a numerator ; thus, -^-^ = If all the factors of the denominator be cancelled, the division is exact. EXAMPLES. Reduce each, of the following to its lowest terms : 3 ^^. 6 ^^^^' 9 ^^^^^ a6V 12a%' lOSa^ftV 7m%«p - 56a%V -^ 60 54a;y 120a.Vg^ -. 126a^6V 45^* ISit-V ■ * 98a6V* j2 3a^6-6a^6^ ^y m^-m^-SGrn 4a262 - 8a63 m^ + m^ - 42m2 ^g 6a?^i/ + 8ar^y^ ^g^ 0^ + 2/^ 15i»y + 20a^/ ' 2a^y-2x'y^ + 2xf j^ a^ + 7a + 10 jg 64a^ + 112 a^a; + 49 ga;^ a^ -f- 4 a — 5 64 a^cc — 49 ic^ jg a^-8a^ + 12a; gQ a^-Umx + ABm^ a^-12a; + 36* ' of - 2 7nx -15m^' jg 25a^ + 20a6 + 4&^ ^^ a^-S 25a2-462 a«_2a2 + a-2 FRACTIONS. 99 6?/i-^ + 8m2-9m-12 ((i'-\-6a-\-d)(a--a-6) a^-/ + g^ + 2a^ og (a + 6)^-(c + cZ)^ '*''• x'-y^-z^-\-2yz ' (a -rf)--(6 - c)^ 04 27a-^ + 646-^ 2^ 12ar^ + 8ar^ -3a; -2 9a2-f-24a6 + 1662* * 18af^- 9a;2 _ g^.^ 4* 28. Reduce j^ ^ to its lowest terms. }r — (V- ___ , ax-hx-ay -\- by (a - b) (x -y) ... - ^ ._. Changing the signs of the factors of the numerator (§ 131), we have ax - hx — ay -\- by _ {b - d){y - x) _ y - X 62 -a2 ~ (6 + a)(& - a) ~ 6 + a' Reduce eax;h of the following to its lowest terms : 29. „'^-'"' . ■ 32. Ans. 30. 7^2 -7m 4- 12 14ar^-4ar^ 4ar'-28a; + 49* 2ac-26c-od-|-6d d'-4.<^ 1- -lla + 18a2 8a3-l a?- -(b-^cy 134. Case II. When the numerator and denominator can- not he readily factored by inspection. Since the H. C. F. of two expressions is the product of all their common prime factors (§ 110), we have the following rule: Divide both numerator and denominator by their highest common factor. 2 a^ — 5 a 4- 3 1. Reduce ^^-— to its lowest terms. ■6a2_a_l2 By the rule of § 117, we find the H. C. F. of 2a2-5a + 3 and 6 a2 - a - 12 to be 2 a - 3. 100 ALGEBRA. Dividing 2 a^ - 5 a + 3 by 2 a - 3, tlie quotient is a - 1. Dividing 6 a^ — a — 12 by 2 a — 3, the quotient is 3 a + 4. Whence, 2a^-6a + S ^ a-1 ^^^ 6a2_a_i2 3a + 4' EXAMPLES. Reduce each of the following to its lowest terms : • 5a;2-23a;-42* 3 2a^ + ft-10 g • 4a2 + 8a-5* . 2oiy^ — xy — 15 y^ q 2x'-15xy + 2'7y^' f. 6 m^ — 13m + 6 ^q • 9m2 + 6m-8* 6 a.-^ + 3a.--10 ^^ aj3 + 2a.-2-14a; + 5* 135. To Reduce a Fraction to an Integral or Mixed Ex- pression. An Integral Expression is an expression which has no fractional part ; as 2 xy, or a + 6. An integral expression may be considered as a fraction whose denominator is 1 } thus, a + 6 is the same as -^ — A Mixed Expression is an expression which has both integral and fractional parts; as a-\--, or ^ + ^-3 — 136. We have by § 30, . ax(- + -)=ax- + ax-=b + c. (§9,3) \a aj a a 4a2 + 15a6- -4?>2 3aj3-17x2 + 4a; + 4 3aj3-14a)2- -llaj-2 2a' + 9a'- -2a-3 6a3 + 23a2- -22a + 3 m^ + m^ + m + 6 m^ + 6m^4-6m — 4 a^-\-2a'x- ■2aaf-x^ FRACTIONS. 101 Kegarding 6 + c as the dividend, a as the divisor, and - 4- - as the quotient (§ 54), this may be written a a ^ h + c_ 6 c a ~a a 137. A fraction may be reduced to an integral or mixed expression by the operation of division, if the degree (§ 108) of the numerator is equal to, or greater than, that of the denominator. 1. Reduce - — "*" ^^~ to a mixed expression. •^ ^ ' 3x 3x 3x 3a; 3x 12ar' — 8a^-l-4a; — 5 2. Reduce — to a mixed expression. 4ar^H-3 ^ 4x2 + 3)12x« - 8x2 + 4x - 5(3a; - 2 12x8 +9x -8x2-5x -8x2 -6 -5x+ 1 A remainder of lower degree than the divisor may be written Over the divisor in the form of a fraction, and the result added to the quotient. Thus, 12x3-8x2 4-4x-6^3^ _ ^ ^ -5x+l. 4x2 + 3 4x2 + 3 If the first term of the numerator is negative, it is usual to change the sign of each term of the numerator, at the same time changing the sign before the fraction (§ 130). Thus, 12x«-8x2 + 4x-5 ^ 3^ _ ., _ 6x^ ^^^ * 4x2 + 3 4x2 + 3 EXAMPLES. Reduce each of the following to a mixed expression: 3 12 a^- 16 a; +-7 ^ 15 a» +■ 6a^ - 3a - 8 4 a; ' 3 tt 102 ALGEBRA. 5. ^^ + 1 6 a^-\-f y a3-263 205 + 3 ^• X- -y a + 6 g 15a^ + lla'-15a -6 12 12 m^ + 19 ^7^2 _ 7 ^ 3a4-4 4m2 + l 9. ^^^' 2m — 5/1 13. ic^ + 2/'' 05 + 2/ x^ — x — 1 14. 18a3-3a2 + 38 3a2-4a + 5 ^^ 12a^-5a-5 15. a^+6^ 4a jg 8 a;^ 4- 16 a^ - 10 a^ - 28 a; + 11 2aj2 + a;-3 138. To Reduce a Mixed Expression to a Fraction. The process being the reverse of that of § 137, we have the following rule : Multiply the integral part by the denominator. Add the numerator to the product when the sign before the fraction is +, and subtract it when the sign is — ; and write the result over the denominator. 1. Reduce — — h x — 2 to a fractional form. 2aj-3 Wehave, ^ + 5 ^ ^ x + 5 +(x - 2)(2a. - 3) 2x-3 2x-3 2x-3 2x-S If the numerator is a polynomial, it is convenient to en- close it in a parenthesis when the sign before the fraction is — . FRACTIONS. 103 2. Reduce a — h to a fractional form. a + b We have, ^ _ 5 _ «^ - «?> - ^^ ^ (« + &)(« - ^) - («^ - «& - f ) a + 6 a -\- h a-\-b ab a-{-b Ans. EXAMPLES. Reduce each of the following to a fractional form : x + 2y 12. 4m'-9+ ^'"(^'"-^) . 2 m 4- 3 13. 2a? + 3a-i^^2^^. 2a-l 15. . + , ^ + ^ 3 a 1 1 "^"^"^ " '"3a 4. a; — 2/ 5. 5a 1 1 ^ . 2 a - 3 6. 3^-2-11 ^^ + 7. 5a; 7. ^ 3a-6 3a + & 8. 2 . 2 2n« m^ — mn 4-n'' m + n 9. 2a + 5a; ^ 2a-5a; ' 10. 3. + 4 + ^f + 15 2 6* 16. a3_^a25_|_^^2^53^ 17. _(^zd}!__(ar^_a;H-l). 18. m4-3»i — 3 a; -4 m^-Smn-i-dn^ 139. To Reduce Fractions to their Lowest Common De- nominator. To reduce fractions to their Lowest Common Denominator (L. C. D.) is to express them as equivalent fractions, having for their common denominator the lowest common multiple of the given denominators. 104 ALGEBRA. Let it be required to reduce ^^, ^^, and ^^ to 3 a^b 2 ab^ 4 a^b their lowest common denominator. . The L. C. M. of 3a% 2ab% and Aa^b is 12a^b' (§ 125). By § 129, if both terms of a fraction be multiplied by the same expression, the value of the fraction is not altered. 4cd Sa'b Multiplying both terms of -^-j- by Aab, both terms of -^ by 6 a^, and both terms of — ^ by 3 b, we have 2 ab^ 4 a^b 16 abed 18 a^mx ^ Wbny 12 a'b' ' "12^^' ^^ 12^2* It will be seen that the terms of each fraction are multi- plied by an expression which is obtained by dividing the L. C. D. by its own denominator ; whence the following rule : Find the lowest common multiple of the given denominators. Divide this by each denominator separately, multiply the correspondiiig numerators by the quotients, and write the results over the common denominator. Before applying the rule, each fraction should be reduced to its lowest terms. 140. 1. Reduce -^^ and - — — to their lowest a^ — 4 a^ — 5a + 6 common denominator. ^ Wehave, a2_4 = («_|.2)(a-2), and a'^-^a+Q ={a-2){a-^). Then the L. C. D. is (a + 2)(a - 2) (a - 3). (§ 125) Dividing the L. C. D. by (a + 2)(a - 2), the quotient is a — 3 ; and dividing it by {a — 2) (a - 3), the quotient is a + 2. Multiplying 4 a by a - 3, the product is 4 a (a - 3) ; and multiply, ing 3 a by a 4- 2, the product is 3 a{a + 2). Then the required fractions are ^ "(«-«) and 3«(«±2) _, Am. (a + 2)(o-2)(o-3) (a + 2)(a-2)(a-3) FRACTIONS. 105 EXAMPLES. Keduce the following to their lowest common denominator : 5x ^ 5xy 3xz 4:yz g 3 a; ~6"' TT' 2r' ' 6ar^-f-2ic 9a^-l g 1 2 6 y aa; 6?/^ ca;^!/ 2m^w 3??i7i^ ^m^n^ ' x-\-y (x-\-y)^ (aj+y)* 4. 2a-5c^ 4a + 36 q 2a 4:b^ da'b 12 ac^ a'-b' a^-\-b^ K Ta^^ 9 6y _8c^ g 3 6 10 82^2- iOa^2 i52/;22 a + 1 a-1 a^ + 1 X2 11. 12. 3a;«-12ar^ x'-Qx-\-d> 3?-% x-{-y a — b ax — bx — ay -\- by a^ — 2xy -^ y^ a + 5 a + 3 a — 2 a^-a-6 a^ + Ta-hlO a^-\-2a-15 ADDITION AND SUBTRACTION OF FRACTIONS. 141. We have by § 136, In like manner, a a 6 , c _ b -\-c a a a b c b — c Whence the following rule : To add or subtract fractions, reduce them, if necessary, to equivalent fractions having the lowest common denominator. Add or subtract the numerator of each resulting fraction, according as the sign before the fraction is -\- or —, and write the result over the loivest common denominator. The final result should be reduced to its lowest terms. 106 ALGEBRA. ^XL 142.1. Simplify i|^+i^. /;i;<^^ The L. C. D. is 12 a^b^ Multiplying the terms of the first fraction by 3 b^, and the terms of the second by 2 a, we have 4 a + 3 1 - 6 ?>-^ ^ 12 ab^ + 9 b^ _^2 a- 12 ab^ ^a% 6ab^ 12 a'b^ 12 a'^b^ ^ 12 ab-^ + 9 b^ + 2 g - 12 gft^ ^ 9 ft'^ + 2 g 12a2fe3 12 a^b-^ ' J.WS. If a fraction whose numerator is a polynomial is preceded by a — sign, it is convenient to enclose the numerator in a parenthesis preceded by a — sign, as shown in Ex. 2. If this is not done, care must be taken to change the sign of each term of the numerator before combining it with the other numerators. 2. Simplify g-^ j^. The L. C. D. is 42. 5x-4?/ lx-2ij 35X-28?/ 21x-6.v Whence, 6 14 42 42 _ 35 a; - 28 y - (21 x - 6 y) - 42 35«-28?/-21x + 6?/ 42 Ux-22y _ 7x- 1 1 y 42 ~ 21 , Ans. EXAMPLES. Simplify the following : 3 5ct-6 3a + 7 g 3x-\-4: 2x-\-5 8 12 * ' 12 16 -46 c a — 4:X 7x — 6a Sxy^ 5a^y 6ax^ Oa^a; FRACTIONS. 107 „ x — 3m . 4:X-\-m g 2a— 9 8a— 5 4a+T ' 24 m 32 a; * " 7 14 28~' « 2a — b 2b — c 2c — a .^ x-^1 3a;— 4 5:^+7 ab be ca 2x bx^ 8a;^ jj 5a + l 26 + 3 7c-4 Qa 86 12c 12 3g;-y 4a;-5y 6a;^ + 2y^ 5a; "•" 10?/ 15a;3/ ' 13 6a; + l 5a;-2 8a;-3 7a; + 4 3 6 9 12 * 14 3a + 4 4a-3 5a + 2 6a-l • K 2ft — 36 3a + 6 4ft — 56 5 a -f76 9 18 27 > 36 ' 1 16. Simplify 01? + X 'J? We have, cc- + x = x(x + 1), and x^ — x = x{x — 1). Then the L. C. D. is x(x + l)(a; - 1), or x(x2 - 1). Multiplying the terms of the first fraction by x — 1, and the terms of the second by x -f 1 , we have 1 1 ^ x-l x + 1 X2 + X X2 - X X(X2 - 1) X(X2 - 1) ^ x-l-(x + l) ^ x-l-x-l ^ -2 ^^^^ x(x2-l) x(x2-l) a;(x2-l)' By changing the sign of the numerator, at the same time chang- ing the sign before the fraction (§ 130), we may write the answer 2 x(x2-l)' Or, by changing the sign of the numerator, and of the factor x2 — 1 2 of the denominator (§ 131), we may write it x(l - x2) 108 ALGEBRA. 17. Simplify -^4:-^ - ... f „ , o + TF ^ Wehave, a2_3a+2 = (a-l)(a-2), a2-4a + 3 = (a-l)(a-3), and a2-5a + 6=:(a-2)(a-3). Then the L. C. D. is (a- l)(a - 2)(a - 3). Whence, — ^ — + a^-Sa-\-2 a2_4rt^3 ^^-Sa + G 0-3 2(a-2) a-1 (a-l)(a-2)(a-3) (a-l)(a-2)(a-3) (a-l)(a-2)(a-3) a-8-2(a-2)+a-l _ a-3-2a + 4 + a-l _Q ^^^-^ (a-l)(a-2)(a-3) (a - l)(a - 2)(a - 3) Simplify the following : 18. ^ I ^ - 23. ^ + ^ ^m^^. 3 a + 5 4 a — 7 in — n m -\- n jg __m 1^^ 24 1 — ^ _ ^ + ^ , m — 1 m + l l+ic 1— ic 20 -^^ i-. 26. 4a^ + l_2a-l. ■ 2aj + l 5x-6 4a2-l 2a + l 21, ^ I ^ 26. ^^-2/ y(y-^^) . ' a-{-b a — b x x^ — xy 22 ^^ _ 2a^ — 6a — 3 27 ^ + ^ ^ ~" ^ aH-4 a2_3a-28 4:0" -9b' (2 a 4- 3 6)2 28. 1 a^ _l_ 4 a; - 12 1 c^-Sx-54: 29. x'-6ax-\-9a^ X ! a;2 + 4 aa? - 21 a2 30. a'-\-b' a b 32 a 6 2 62 ^ a^-^ab a + b a a — 6 a + 6 a' — b' 31. ^ + 3x 6a^ 33. * -^ 1. 1 + a; 1 — x 1— a^ a; — 2/ a; + 3/ 38. 34. FRACTIONS. 109 1 . 2x a{a + x) a(a — x) or — x^ 35 1 2x 3a^ + 4 ' x + 2 {x^2f {x^2Y 36. ^ 1 i. 39. -i (^^-^n a-3 a + 6 a 2a4-& Sa^^-^' qy a; 4-2 _ a? — 2 _ 16 ac\^-^^ a — x 4: ax x — 2 x-\-2 a^ — 4 a — X a-\- x a- — x^ x-\-y x' + f ^^ ^ 2(x-\-y) ^ (x -j- yf X — y x^ — y^ x — y (^ — 2/)^ 43. m — ?i (m — ?i)^ (m — Jii)-' x-\-\ a; — 3 a; — 5 x + 2 a;-4 a^-2a;-8" 44 1 I 1 I 1 (a - 6) (6 _ c) (6 - c) (c - a) ^ (c - a) (a - 6) 45. + a;-3 a;-2 a^-5a; + 6 46.-1^+ 1 2a 47. a + 6 ' (1-6 0,2 + 62 1 3a; oa; a — a; a^ — m? (f — x 4g ^^ g a^-A * a 4-1 a^-a + l a^ + l' 49. a; + z ?/ + g .T + ?/ (^-y)(y-^) (p^-y)(x-^) (^-^)(y-z) 50 a; + 2 2(a;-l) a;-3 a^ + 4a; + 3 a^^x-6 af-x-2 In certain examples, the principles of §§ 130 and 131 enable us to change the form of a fraction so that the given denominators shall be arranged in the same order of powers. 110 ALGEBRA. 51. Simplify _A_ + 25ia^ ^ ^ a-b h^-oj" Changing the signs of the terms in the second denominator, at the same time changing the sign before the fraction (§ 130), we have 3 2h + a a-b cfi- 62* The L. C. D. is now d^ - b^. Whence, -^ 2b±a^ S{a + b)-(2h + a) a-b a'^-b'^ a^ - h'' _ 3a + 3& -26 - g _ 2 a + 6 . 52. Simplify 111 {x-y){x-z) (y-x){y-z) (z-x)(z-y) By § 131, we change the sign of the factor y — x in the second de- nominator, at the same time changing the sign before the fraction ; and we change the signs of both factors of the third denominator. The expression then becomes 1,1 1 (x-y)(x-z) (x-y){y-z) (x-z)iy-z) The L. C. D. is now (x — y){x — z) (y — z) ; whence the result - (y-^)-\-(^-^)-(^-y) ^ y-z+x-z-x+y {x-y){x-z){y~z) (x-y)(x-z)(y-z) 2y-2^ ^ 2(y-z) ^ 2 ^^^ {z-y){x-z){y-z) (x-y){x-z)(y-z) (x-y){x-z)' Simplify the following: 53. -y — ^-^. 57. Qir — xy y^ — xy 54. ^- + 5 _2x-l_ 5g 3a;-6 8-4a! a^ — 9 3 — a 56. _J-- + _1_. 60. a 1 1 1 ab-W b — a b a a 2 a + l ' 1-a a?-l X X ^ , 2 + x 2-x a^-4 X y 2/ 4 w — m^ m^ — 16 ^ + 2/ a; — 1/ y'^ — o^ FRACTIONS. Ill 61 1 _ 1 1 ^g'^ — 9 62 ^ ^ ^ ^^ 63. a 2a -S 9a- Aa^ m + 2 m-2 i-m^ 1 1 (a - 6) (a + c) (& - a) (6 + c) 64. -^+ 2a. 1 65. , 1 , t ,+ (^-y)(y-^) {y-x)(x-z) {z-x)(x-y) 66. ^^; + (a-b)(a-c) (b-c){b-a) {c-a)(c-b) MULTIPLICATION OF FRACTIONS. a c 143. Required the product of - and -• a Let ?x^ = a;. (1) b a Multiplying each of these equals by 6 x c? (§ 9, 1), we have - X- X b X d = X X b X d. b d Or, since the factors of a product may be written in any order, f^ xb\xf^x d\=x X b X d. Whence, (a) x (c) = x x b x d. (§ 9, 3) Dividing each of these equals by 6 x ci (§ 9, 1), we have axe b X d (2) From (1) and (2), f x | = ^. (§ 9, 4) ^ b d b X d We then have the following rule for the multiplication of fractions : Multiply the numerators together for the numerator of the product, and the denominators for its denominator. 112 ALGEBRA. Common factors in the numerators and denominators should be cancelled before performing the multiplication. Integral or mixed expressions should be expressed in a fractional form (§ § 135, 138), before applying the rule. 144. 1. Multiply^ V 1^. -^. We have «^^^y^ 2 x 5 /H a^hH^ _ U H . ^bx" ^d^^ 32 X 22 X a8&x2y8 In this case,-Hhe factors cancelled are 2, 3, a^, b, x^, and y. 2. Find the product of — -, 2 ^^, and x^ — 9. a^ + fl7 — 6 x — 3 We have, ^ x f2 -^^:i^^ x (x2 - 9) x^ + x-6 x-S X _ X ^ X (X + 3) (X - 3) = ^^^±11, Ans. (x + 3)(x-2) x-3 ^ ^^ x-2 ' In this case, the factors cancelled are x + 3 and x — 3. . EXAMPLES. Simplify the following : 3 l^^^Txv' 8 27 m^ 15n^ 7 x' UxY ^' 20n'x 2SxY ISm'y , 6a^m 20 6V o a^-\-a-SO ^^ 5 a ^' — : X — — :: — r* *'• 25 5V 3aV 3 a a^-4.a-5 K 5x 3y_ Sz ^Q 9m^ — 1 m^-\-5m ^y lOz^x ' m3-25m' 3m-l' « 4a^ 156^ 21c^ ,, a^-h3a;-18 2a.-8-4a^ 96^^ 7c^ lOa^* ' aj2_8x4-12 a^^ - 36 * y. 3a^&% 6 5V lOc^g -o x y + y^ x" ^- xy -2y^ 4c* 5a« 96^ ^'*- 0? - xy^ x" + 2xy + y^ 13. FRACTIONS. 113 a^-ab~6b'' ^ a'+^ab ' ar^H-8 a^-^a^-\-x' -. 5.^•-f-15 3a;-9 ^ 8ar-2 AO. -— — X T7. z X 8a; -4 10a; + 5 3ar^-27 Ifi a^ -{-2 a a- — 16 a^ -\- a a^ — 3 a — 4 a? — a a^ + 6 a + 8 17. f6 2-^ + 3 ■' A 2^ + 33// • (x-yf-z' a?-{y + zf 19. ±^Ax2-:^'x^ 2«6 a' + 6* « + <' V a'' + a6 + ^> 20 g^a; + ox'^ g^ -h 2 oa; + a;^ g'' — 2 ga; + ag^ g* — 2 g V + a;* g^ -h ar* ga; '*^* 16a;*-9ar'''2ar^ + 2'^V ^a;-iy DIVISION OF FRACTIONS. 145. Required the quotient of - divided by -• b d Let 1^1 = .. (1) Then since the dividend is the product of the divisor and quotient (§ 54), we have b d Multiplying each of these equals by - (§ 9, 1), we have 2x^ = ^xxx^ = «. (2) bed c From (1)" and (2), ^ ^ ^ = ^ x -• (§ 9, 4) d c 114 ALGEBRA. Therefore, to divide one fraction by another, multiply the dividend by the divisor inverted. Integral or mixed expressions should be expressed in a fractional form (§§ 135, 138) before applying the rule. 146. 1. Divide — — by q-r^n- 5 oi?y^ 10 x-y' 5xV i0a;2i/7 5a;V 9«^''' Sft'^x' 2. Divide 9+^ by 3 + ^. We have, «-^)-(^-.-^) _ 9x2-4y2 a; -2/ _ (3ic + 2y)(3x -2y) aj2-y2 ^3x + 2^~ (a; + y)(x-«/) 3x-2w , = -, Arts. x + y EXAMPLES. Simplify the following : 3. ?i^^8a362. 7. ^^_?V^|^ + - 21 an^ . 14 aV g a^ + lQa + 21 . a^-9 5 3 . 2 g a?^+4fl?y+4y^ . xy-\-2y\ x^—ex-\-S ' Q(?—x—V2, ' x — y ' Q^ — xy' g 4m^-25n^ . 2mn-hn^ ^^ y? - x . a^-2a; + l j^ a3_8 ^ a2-^2a + 4 12. a2 + 7 a + 10 a^ + 2 a a2_5^5_1452 ^ a2_3a5-286^ a2 4.5a6-2462 ' a' - S ab -{- 15 b'' 13. 14. FRACTIONS. (.. a- a- -5x -2x^ a2- -b'- -(^-2 be . a -b — c 115 a2-62_c2-4-2dc a + 6.^c COMPLEX FRACTIONS. 147. A Complex Fraction is a fraction having one or more fractions in either or both of its terms. It is simply a case in division of fractions, its numerator being the dividend, and its denominator the divisor. 14a 1. Simplify — ^. , ~r JC We have, -!L- = -J!— = a x :rj— (§146)=-^, AiisT^^ , c bd-c bd — c^" ' bd — € It is often advantageous to simplify a complex fraction by multiplying its numerator and denominator by the lowest common multiple of their denominators (§ 129). 2. Simplify —J- —' a — b a-{-b The L. C. M. of rt + 6 and a-b is (a + 6) (a - 6). Multiplying both terms by (a + 6)(a — 6), we have a(a + b)- a(a — b) _ a^ -{- ab — a^ -\- ab _ 2 ah ^ b(a + b)-ha(a-b) ab + b^ + a^ - ab~ a^ + b^' EXAMPLES. Simplify the following : a c 1,1 1 r — 3 ^~^o — ^ — ::^ ^ b a . 2m _ ar o. • 4. T — • 0. T— b d 4m X 116 ALGEBRA. ™_:!L ?+l_^ ^-13+^ mn y X x ^_2-i-^ a_a^-6 a; a? — ^ ,y % 2^^ ^^ 5 a + 6 jg a;4- y a^ 23 b , a — b x x-\- y X a a + b ^ — y ^ a-x ^ 82/ 26^ a-- ~^ + ^- ^ Z 8 i±^. 11. y \ ■ 14. — ^. a^ — ax \^- __2 + ^i a + l-faa? 2/ ^ a + 25 a^ + a + 1 H — r -I- 15. T-^— ^- 16. a + 36 a — 36 1 2a4-6 2a-6 ^■^a-1 a-36 a + 36 17. Simplify r 1+ — -*\ Wehave, __^ = —^ = -^.^^ = ^^^^ , ^«.. 1 a,+ l X In examples like the above, begin by simplifying the lowest complex 1 X fraction; first multiply both terms of by x, giving ., , and 1 + 1 ''+' 1 "^ X 4- 1 then multiply both terms of bv x + 1, giving ^- . 1 I ^ ' X + 1 + x X + 1 Simplify the following : 2 1 18. 3 i-j— 19. 1 -rr—' 5+ ^ - ' 7+- 3- X FRACTIONS. 117 ^ S{a'-\-b^ x^y x^ + f 20. ^^'-^' 23 ""-y "^-^ ^ 2 (g + 2 6) ■ x" xjx' + y^) 3a + 6 x + y {x + yf x-\- a X — a 2n(m — n) x — a x-^a g. m +n or* + a^ _ -l ' ?^^ H- ?i- l-x" l-i-x" g + 6 _ g^ + &« 1 + ar^ 1-a^ g - 6 g« - 6« 1-a; 1 + a; ' a + 6 g« + 6« ' l + a; 1-ic a-6"'"g3-63 MISCELLANEOUS AND REVIEW EXAMPLES. 149. Reduce each of the following to a fractional form : 4g + 3 d^-ab + b^ ^ ^ Simplify the following : Q 1 2g , Qdx 2a-Sx (2a-Sxy (2a-3xy 4 (l-x)(l-y) 5 (a^-2Y-a^ (l-^xyy-(x-\-yy ' a*-3g'^-4 ft oi-\-b . c-{-d 2 (gc — bd) a — b c — d (b —a){c — d) g 6 xy-(x + 2yy .q (a^ - 6 a; - 4)^ - 144 a? + Sf ' ' (a^ + a;-ll)2-81 ■ 118 11. ALGEBRA. 3a 4a 3a + 2a2-7a4-10 a^-Ga-f-S 13. 14. 12. (2ai-S-,^^(2a-3 + -^^. \ 2a-\-3j\ 2a -3j oc^ -{-y^ Qi? — if yf- (^_^\y x^ — y^ a?^ 4- 2/^ ^ + ^ 15. :^_- + ^_ a^ H- ci'^h — a^h^ — ah^ jg ahd + abd^ — ah(? — 6^cc? a'h - a^b' + a6^ - b'' ' ahd - abd^ - abc^ + b'cd 17. f^ + -^ \fl; — 2 a; — 18. 19. 20. Sj\Sx-S x-\-2 1 \ / 1 x^-^3x-\-2 x+1 3a; + 2 x-\-2j m m 2{m — n) 2(m H- n) iii?{w? — inF) (g 4- ^ + c)^ - (a + 6 - c)^ (a - 6 + c)2 - (a - 6 - c)2* 2a(2a-3&a;)+ 36(3 & + 2aa;) (2a-36x)2+(36 + 2aa;)2 a? — y , a; + g _ (y + g)^ aj+g ic — 2/ (ic — 2/)(a; + z) a'b - ab^ b- 23. (a-\-by a-{-b , a^ + d^ 24. a— b 1 25. aV a? — ct a; + — V -2ay 4a:^-16a^ + 17a;-3 6a3-17a;2 + 8a; + 6* 3 a?^ -h aa; — 2 a^ 6a6 + 962 _ /g2_95: a6 — 6 6 a2_4^5_j_452 \^a2-462 a'-^ab-Qb gy a;^-2a^y^ + / Y ^' ~ '^ ♦ (a; + y)^-2a^y a;* + 2a^/ + 2/* ' V ajy * a;y 5' FRACTIONS. 119 nn m^ -f 2 7nn + 4 n^ 4 ?>i^ — 9 n^ _^ 2 m -\-Sn . 2 m — 3 n m^ — Sn^ m'- — 4 n^ x-[-y a^ — f 1 I 1 31. "'"^ "^"^^ a(a — 6) (a — c) 6(6 — c) (6 — a) ' a.-^ + 2/^ _ ic — 2/ a;3 _ 2^ a; + 2/ (^-y)(^-^) (y-2;)(2-a;) (2-«)(a;-y) 33 A ^^ V i ^^ ^ >c ^'^ "^ ^^ • ^^ a2-a6 + 6VV a' + 2a6 + 6V «' - 6^ 34. 14-^- + .-^+ ^ 14-251+3^1 + a;' (First add the first two fractions ; to the result add the third frac- tion, and to this result add the fourth fraction.) 35. -H -^r+ „ . + a-2a + 2a2H-4a* + 16 36. 1 1.1 1 -1 x-^1 x — 2 a;4-2 (First combine the first two fractions, then the last two, and then add these results. ) 37 1 1 2a 2a a - 6 a -f 6 a^ - 6^ a^ + 6^ 38 1 1 3ar^ 3a^ x-1 x + 1 aj3 + l a^_l 39 4a; — 3 2 a; — 5 6ar^ + 13 a; - 5 12 ar^ H- 5a; - 3 (Find the H. C. F. of the denominators by the method of § 117.) ^Q 3a 4- 2 5a -1 6a2- a- 12 10a2-19a + 6 120 ALGEBRA. XIII. SIMPLE EQUATIONS (Continued). SOLUTION OF EQUATIONS CONTAINING FRACTIONS. 150. Clearing of Fractions. ^.j,i .. 2x 5 5x 9 Consider the equation = -• ^ 3 4 6 8 The lowest common multiple of 3, 4, 6, and 8 is 24. Multiplying each term of the equation by 24 (§ 71, 2), we have ^^ ^^ ^ 16aj-30 = 20a;-27, where the denominators have been removed. We derive from the above the following rule for clearing an equation of fractions : Multiply each term by the lowest common multiple of the given denominators. 151. 1. Solvetheequation ^-1 = ^-^. 6 3, 5 4 The L. C. M. of 6, 3, 5, and 4 is 60. Multiplying each term of the equation by 60, we have 70a;- 100 = 36a; -15. Transposing, 70 a; - 36 a; = 100 - 15. Uniting terms, 34 a; = 85. Dividing by 34, ^ = It = % ^^*- EXAMPLES. Solve the following equations : 2. »+|_^ = 9. 3. ^-^ + ^^0. 9. 3a; 5 x 7x 4x 2 14 3 ~ 6 7 ' 10. 2x X 3 _7x Sx 5 2 10 8 4 11. 2 3 4 5 _ 1 3a; 4a; 5u; 6a; 20 ^^JPALirofy#^ SIMPLE EQUATIONS. 121 4 ^_? = ^ + l 8 -5 L = 1_A '234 8* ' 18a; 6x 4 9a;* 1^ 5 i£_? = ^_^ '9 3 6 2 * 6 '^'^ 4a;2a;_ 11 ■ 2 3 5 ~ 6* ^ 7. A_i = l__i.. 5a; 10 4a; If a fraction whose numerator is a polynomial is preceded by a — sign, it is convenient, on clearing of fractions, to enclose the numerator in a parenthesis, as shown in Ex. 12. If this is not done, care must be taken to change the sign of each term of the numerator when the denominator is removed. 12. Solve the equation ?^fll - i^ = 4 + ^4±i. 4 5 10 The L. C. M. of 4, 5, and 10 is 20 ; multiply ing • each term by 20, we have 16a;-5-(16x-20) = 80+ 14x + 10. Whence, 16x - 6 - 16x + 20 = 80 + 14x+ 10. Transposing, 15 x - 16 x - 14 x = 80 4 10 + 5 - 20. Uniting terms, — 15x = 75. Dividing by — 15, x =— 5, Ans. Solve the following equations : 13. 4x + »''-^^ = ^. 16. a,-3^±I = 8^Ili-l. 14. S^ 2.-2^^ 2. 3 9 17. 2x-5 3a;-8 2 7 6 3 15. 2.-^-^^^-^- + l. 18. a;-|-2_ 9 3a; + 14 10 35 14 . 19 nx + i 14 a;-f-3 10a;4-7 ^ ^ 2 4 8 122 ALGEBRA. 2 8^^ 12 5^ ^ 20 5 2 22 1^(^ + ^) 5a; -4 5a; + 12 ^^T 9 12 6 "" '' 23. li^-l(8^-5) + l(10a;-7) = ^(i^L±ll. ^ O D 4 24. ll^L^-l(3a)-l) = lIi-+I-?(7a;-2). 3 2^ ^6 9^ ^ 25. 2a; + 4 7a;- 1 ^ 13.-c + 5 lla;-3 5 2 3 10 ' oc 7a; — 8 7a; + 6 x-5 4a; + 9 «o. ■ ■ - = *-• 14 4a; 2 7a; o- 3(a;-3) 2(a;^-5) 5a.'^- 12 ^ 9 2 3a; Gx" 2 2 5 28. Solve the equation x-2 x-^2 a;2-4 The L. C. M. of X - 2, x + 2, and x-^ - 4 is x^ - 4. Multiplying each term by x^ — 4, we have 2 (X + 2) - 5 (X - 2) - 2 = 0. Whence, 2x + 4 - 5x + 10 - 2 = 0. Transposing, 2x — 5x = — 4 — 10 + 2. Uniting terms, — 3x = — 12. Dividing by — 3, x = 4, Ans. If the denominators are partly monomial and partly poly- nomial, it is often advantageous to clear of fractions at first partially; multiplying each term of the equation by the L. C. M. of the monomial denominators. 29. Solve the equation SIMPLE EQUATIONS. 123 6x + l 2.T-4 2a;-l 15 7x- 16 The L. C. M. of the monomial denominators is 15. Multiplying each term by 16, we have , 30x-60^,^_3 7 X - 16 Transposing and uniting terms, 4 = — ^-^ 7x — 16 Multiplying by 7 x - 16, 28 x - 64 = 30x - 60. Transposing, 28 x - 30 x = 64 - 60. Uniting terms, — 2 x = 4, Dividing by —2, x = — 2, ^ns. Solve the following equations : 30. -^ ^ = 0. 35. I^±li^-24__A_=7. 6x-\-2 3x-\-4: (x-\-iy ic+l 3«-4 6a;-l 6 2(2a;+l) 3a^ + 6a;4-4 '^' * 3 3a;-4 9 33 6a; + 5 ^ 3a;-2 38 " o i 2x(x — 1) ar^ — 1 3 if — 5 x — 2 x — S 34 3a; 2x ^ 2x-—5 gg 2a;-fT 5a; — 4 ^ a; + 6 2a;-h3 2a;-3 4ar^-9* 14 3a?+l 7 * 40. 3 4 2a;-l 3a; + 2 41 2(.T-7) a;-2 x-\-3^q x' + Sx-2S x-4: a; + 7 42 ^^ + ^ _ 4a; + 7 _ 3a; — 2 _ ^ 43. 6 6a; + ll 3 1.3 6 2a; + 3 3a;-2 4a; + l 124 44. 45. 2x + 1 ALGEBRA. 2a^-l 9 a.- + 17 2x X — -16 2ic + 12 x^- •2 x-1 2x-\-4._ -2a; -48 :0. a; + 2 x-\-l x^ — 1 ^g 2 + 3a; 2-3a; ^ 36-4a; 3-x 3 + x ~ x'-9 ' -- 2a^ + 3a;-l 2a^-3a; + l _^ (^+l)(a? + 3) _a;-6 7(3a^-8) 3(a;-2) _ (a.4-5)(aj + 7) x-{-2' 3(a!-3) "^ 3aj- 1 49 3a^+5a;-4 ^ 3a;+5 ^^ 2x±7__3^--5^r[x±2 4.x'-3x + 2~4.x-3 ' 6x-4: 9x+6 9a^-4* 2 a?- 3 oj- 5 X — 6 48. 52. 3 X-4: (First combine the fractions in tlie first member ; tlien the fractions in the second member. ) eg Sx-{-5 3x-2 ^ 6x-5 Tx-^S 7 14 28 4(4 a; -3)* SOLUTION OF LITERAL EQUATIONS. 152. A Literal Equation is one in which some or all of the known quantities are represented by letters ; as, 2x-\-a = bx^-10. 153. 1. Solve the equation (6 — cxy— (a — cxy= b(b — d). Performing the operations indicated, we have 62-2 bcx + c2x2 - (a2 _ 2 acx + c^^) = 62 _ ab. Whence, b'^-2 bcx + 02^2 -a^+2acx- d^^ = b^-ab. Transposing and uniting terms, 2 acx — 2 bcx = a^ — ab. Factoring both members, 2 cx(a — b)= a(a ^ b). Dividing by 2 c(a -b), x = ^f^^ = §-^, ^»»- SIMPLE EQUATIONS. 125 EXAMPLES. Solve the following equations : 2. a(3bx-2a)=b(2a-3bx). 3. (x + ay + (b + cy = (x - ay+Q) - cy. - X — a , 2x r, a ^ — ^ b —x 2x b 4. 1 = o. o. = • X X— a ax a a X ^ Sx — 4 5 7n — 2n „ -, x — 2 1 0. = • 7. X rr i- = — 5* 3x-\-4: 5m-\-2n m mr g 5x-2a 9x-5a^ S( x-^2a^) 5x^^ 2a 3a« a^ 6a q ax — b bx-{-a _2 , a — b bx ax abx 10. 2(x-b){2a-3b-3x)-(2a-3x){b + 2x)=0. 11 . (x-\- m) (x -\- n) — (x — m) (x — n)=2{m-\- ny. 12 x~^ x-{-b _ 4:a^ — b^ x — 2a X -\-2a ar*— 4a^ 13 2x-\-3a ^ 3x-\-A:b -^ 2nx — 3 ^^ 9nx + 2 2x—3b 3a; — 4a na; — 1 37ix — l 1- a — b.b — c.c — a ^ X — c x — a X 16. (x-i-ay-\-(x-ay = 2x(x'-a^)-24:a\ ,- 3 x(a — b) a — 2b a — b _r. s? — h^ x + b b — X 18 a? g — 2 5ca; __ 5 a; 8 ac — 8 ^o; — a ' 2 ~ 4 6c ~ 6 c 12 6c 19. (a; - 2 a -f 3 6)2 - (a; - 2 a) (x + 3 6) - 6 a6 = 0. 126 ALGEBRA. 2^ __a b__ _ b^ — a^ nn (2 x — 3 vrif _ x — 3m -""'" ' * ■ {2x-Sny~ x-Sn' a % b b'-a' X — l x-b W-bx ax - + bx = a4-b. 21. _ii!l_ + _i^ = a + 6. 23. 2(a+&) ^x4:5_^--a^ x-{-b x-{- a X x—b x-\-a 24 1 , 1 2x — a—b X — a x — b x{x — a — b) ^• + 4a4-6 4:X-^a-\-2 X -\- a + b x-\-a — b gg g; + 4 g + ^ _^ 4:X-^ai-2b ^ ^ SOLUTION OF EQUATIONS INVOLVING DECIMALS. 154. 1. Solve the equation .17 a; - .23 = .113 x + .112. Transposing, .17 x - .llSx = .23 + .112. Uniting terms, .057 x = .342. Dividing by .057, x = '^^ = 6, Ans. .057 EXAMPLES. Solve the following equations : 2. 2.9 a; - 1.98 = 1.4 a; - 1.845. 3. .05a; + .117 = .186cc-.2a;-.139. 4. .6 a; -.265 + .03 = .4 + .66 a; -.187 a;. 5. .4(1.7aj-.6) = .95x + 5.16. 6. .08(35 aj - 2.3) = . 9(7 a5 + . 18) -.997. 7. 2.8.-'^^^+ -^^^^ = .5. -.064. 8 3 39 .4 a; + .708 ^18 .3 2a; 5 x a .7 a; + .371 .3a; -.256^ ^^ 10. .9 .6 2-3a; 3a;-14 a;-2 10a;-9 1.5 9 1.8 2.25 3 7^^ ^^ Q SIMPLE EQUATIONS. 127 PROBLEMS. 155. 1. Divide 43 into two parts such that three-eighths of one part may equal two-ninths of the other. Let X = one part. Then, 43 — x = the other. By the conditions, ^ = - (43 - x) . J 8 9 Clearing of fractions, 27 x = 16 x 43 - 16 x. Transposing, 43 x = 16 x 43. Dividing by 43, x = 16, one part. Whence, 43 — x = 27, the other part. 2. The fifth part of a number exceeds its eighth part by 3 ; what is the number ? 3. What number is that from which if four-sevenths of itself be subtracted, the result will equal three-fourths of the number diminished by 18 ? 4. What number exceeds the sum of its third, sixth, and fourteenth parts by 18 ? 5. Divide 45 into two parts such that the sum of four- ninths the greater and two-thirds the less shall equal 24. 6. Divide 56 into two parts such that five-eighths the greater shall exceed seven-twelfths the less by 6. 7. Divide $ 124 between A, B, and C so that A's share may be five-sixths of B's, and C's nine-tenths of A's. 8. A man travelled 768 miles. He went four-fifths as many miles by water as by rail, and five-twelfths as many by carriage as by water. How many miles did he travel in each manner ? 9. A's age is three-eighths of B's, and eight years ago it was two-sevenths of B's age ; find their ages at present. 128 ALGEBRA. 10. A has $ 52, and B $ 38. After giving B a certain sum, A has only three-sevenths as much money as B. What sum was given to B ? 11. I paid a certain sum for a picture, and the same price for a frame. If the picture had cost $ 4 more, and the frame 30 cents less, the price of the frame would have been one-third that of the picture. Find the cost of the picture. 12. A can do a piece of work in 8 days which B can per- form in 10 days. In how many days can it be done by both working together ? Let X = the number of days required. Then, - = the part both can do in one day. Also, - = the part A can do in one day, 8 and — = the part B can do in one day. 10 By the conditions, - H — = — 8 10 « 5 ic + 4 ic = 40. 9x = 40. Whence, x = 4|, the number of days required. 13. The second digit of a number exceeds the first by 2 ; and if the number, increased by 6, be divided by the sum of its digits, the quotient is 5. Find the number. Let X = the first digit. Then, x -\- 2 = the second digit, and 2 X + 2 = the sum of the digits. The number itself is equal to 10 times the first digit, plus the second, Then, lOx + (ic + 2), or 11 x + 2 = the number. By the conditions, li^-+l±^ = 5. 2x + 2 llx + 8 = 10x + 10. Whence, x = 2. Then, 11 x 4- 2 = 24, the number required. SIMPLE EQUATIONS. 129 14. A can do a piece of work in 18 days, and B can do the same in 24 days. In how many days can it be done by both working together ? 15. A can do a piece of work in 3^ hours which B can do in 3| hours, and C in 3| hours. In how many hours can it be done by all working together ? 16. A tank can be filled by one pipe in 9 hours, and emptied by another in 21 hours. In what time will the tafik be filled if both pipes be opened ? 17. A vessel can be filled by three taps; by the first alone in 7| minutes, by the second alone in 4^ minutes, and by the third alone in 4| minutes. In what time will it be filled if all the taps be opened ? 18. The first digit of a number is 4 less than the second ; and if the number be divided by the sum of its digits, the quotient is 4. Find the number. 19. The second digit of a number is one-fourth of Ihe first; and if the number, diminished by 10, be divided by the difference of its digits, the quotient is 12. Find the number. 20. If a certain number be diminished by 23, one-fourth of the result is as much below 37 as the number itself is above 56. Find the number. 21. What number is that, seven-eighths of which is as much below 21 as three-tenths of it exceeds 2\ ? 22. B is 24 years older than A ; and when A is twice his present age, B will be f as old as he now is. How old is each? 23. The denominator of a fraction exceeds the numerator by 5. If the denominator be decreased by 20, the resulting fraction, increased by 1, is equal to twice the original frac- tion. Find the fraction. 130 ALGEBRA. 24. Divide 44 into two parts such that one divided by the other shall give 2 as a quotient and 5 as a remainder. Let X = the divisor. Then, 44 — x = the dividend. Now since the dividend is equal to the product of the divisor and quotient, plus the remainder, we have 44 - X = 2 cc + 5. -3cc = -39. Whence, x = 13, the divisor, and 44 — X = 31, the dividend. 25. Two persons, A and B, 63 miles apart, start at the same time and travel towards each other. A travels at the rate of 4 miles an hour, and B at the rate of 3 miles an hour. How far will each have travelled when they meet ? Let 4x = the number of miles that A travels. Then, Sx = the number of miles that B travels. By the conditions, 4ic + 3x = 63. 7x = 6S. x = 9. Whence, 4 a; = 36, the number of miles that A travels, and 3x = 27, the number of miles that B travels. Note. It is often advantageous, as in Ex. 25, to represent the unknown quantity by some multiple of x instead of by x itself. 26. Divide 49 into two parts such that one divided by the other may give 2 as a quotient and 7 as a remainder. 27. Two men, A and B, 66 miles apart, set out at the same time and travel towards each other. A travels at the rate of 15 miles in 4 hours, and B at the rate of 9 miles in 2 hours. How far will each have travelled when they meet? SIMPLE EQUATIONS. 131 28. Divide 134 into two parts such that one divided by the other may give 3 as a quotient and 26 as a remainder. 29. The denominator of a fraction is 7 less than the numerator ; and if 5 be added to the numerator, the value of the fraction is f. Find the fraction. 30. The second digit of a number exceeds the first by 4 ; and if the number, increased by 39, be divided by the sum of its digits, the quotient is 7. Find the number. 31. I paid a certain sum for a horse, and seven-tenths as much for a carriage. If the horse had cost $ 70 less, and the carriage $ 50 more, the price of the horse would have been four-fifths that of the carriage. What was the cost of each? 32. A can do a piece of work in 15 hours, which B can do in 25 hours. After A has worked for a certain time, B completes the job, working 9 hours longer than A. How many hours did A work ? 33. A man owns a horse, a carriage worth $100 more than the horse, and a harness. The horse and harness are together worth three-fourths the value of the carriage, and the carriage and harness are together worth $ 50 less than twice the value of the horse. Find the value of each. 34. The rate of an express train is | that of a slow train, and it covers 180 miles in two hours' less time than the slow train. Find the rate of each train. 35. Two men, A and B, 57 miles apart, set out, B 20 minutes after A, and travel towards each other. A travels at the rate of 6 miles an hour, and B at the rate of 5 miles an hour. How far will each have travelled when they meet? 36. A grocer buys eggs at the rate of 4 for 7 cents. He sells one-fourth of them at the rate of 5 for 12 cents, and the remainder at the rate of 6 for 11 cents, and makes 27 cents by the transaction. How many eggs did he buy ? 132 ALGEBHA. 37. At what time between 3 and 4 o'clock are the hands of a watch opposite to each other ? Let X = the number of minute-spaces passed over by the minute- hand from 3 o'clock to the required time. Then, since the hour-hand is 15 minute-spaces in advance of the minute-hand at 3 o'clock, x — 15 — 30, or x — 45, will represent the number of minute-spaces passed over by the hour-hand. But the minute-hand moves 12 times as fast as the hour-hand. Whence, x = 12(x — 45). x = 12X-540. -llx = -540. X = 49j-V. Then the required time is 49yx minutes after 3 o'clock. 38. At what time between 1 and 2 are the hands of a watch opposite to each other ? 39. At what time between 6 and 7 is the minute-hand of a watch 5 minutes in advance of the hour-hand ? 40. At what time between 4 and 5 are the hands of a watch together? 41. At what time between 5 and 5.30 are the hands of a watch at right angles to each other ? 42. The sum of the digits of a number is 15 ; and if the number be divided by its second digit, the quotient is 12, and the remainder 3. Find the number. 43. A man has 11 hours at his disposal. How far can he ride in a coach which travels 4|- miles an hour, so as to return in time, walking back at the rate of 3| miles an hour ? 44. A, B, and C together can do a piece of work in If days ; B's work is one-half of A's, and C's three-fourths of B's. How many days will it take each working alone ? 45. At what time between 9 and 10 are the hands of a watch together ? SIMPLE EQUATIONS. 133 46. A, B, C, and D found a sum of money. They agreed that A should receive $ 4 less than one-third, B $ 2 more than one-fourth, C $3 more than one-iifth, and D the remainder, $ 25. How much did A, B, and C receive ? 47. At what time between 8 and 9 are the hands of a watch opposite to each other ? 48. A vessel can be emptied by three taps ; by the first alone in 90 minutes, by the second alone in 144 minutes, and by the third alone in 4 hours. In what time will it be emptied if all the taps be opened ? 49. A and B start in business, B putting in f as much capital as A. The first year, A loses $ 500, and B gains ^ of his money; the second year, A gains ^ of his money, and B loses $ 205 ; and they have now equal amounts. How much had each at first ? 50. A man buys two pieces of cloth, one of which con- tains 6 yards more than the other. For the larger he pays at the rate of $ 7 for 10 yards, and for the smaller at the rate of $ 5 for 3 yards. He sells the whole at the rate of 9 yards for $ 11, and makes $ 5 on the transaction. How many yards were there in each piece ? 51. A man loaned a certain sum for 3 years at 5 per cent compound interest; that is, at the end of each year there was added ^ to the sum due. At the end of the third year, there was due him $2130.03. Find the amount loaned. 62. At what times between 7 and 8 are the hands of a watch at right angles to each other ? 53. At what time between 2 and 3 is the hour-hand of a watch one minute in advance of the minute-hand ? 54. Grold is 19} times as heavy as water, and silver 10|- times. A mixed mass weighs 1960 oz., and displaces 120 oz. of water. How many ounces of each metal does it contain ? 134 ALGEBRA. 55. A merchant increases Ms capital annually by one- third of it, and at the end of each year takes out $ 1800 for expenses. At the end of three years, after taking out his expenses, he finds that his capital is $ 3800. What was his capital at first ? 56. A and B together can do a piece of work in 2| days, B and C in 2j§j- days, and C and A in 2|- days. How many days will it take each working alone ? 57. A alone can do a piece of work in 15 hours ; A and B together can do it in 9 hours, and A and C together in 10 hours. A commences work at 6 a.m. ; at what hour can he be relieved by B and 0, so that the work may be completed at 8 P.M. ? 58. A man invests j^ of a certain sum in 4J per cent bonds, and the balance in 3^ per cent bonds, and finds his annual income to be f 117.50. How much does he invest in each kind of bond ? The annual income from p dollars, invested at r per cent, is rep- resented by ^^. \ 59. An express train whose rate is 36 miles an hour starts 54 minutes after a slow train, and overtakes it in 1 hour 48 minutes. What is the rate of the slow train ? 60. At what time between 10 and 11 is the minute-hand of a watch 25 minutes in advance of the hour-hand ? 61. A woman sells half an egg more than half her eggs. She then sells half an egg more than half her remaining eggs. A third time she does the same, and now she has sold all her eggs. How many had she at first ? 62. A man invests two-fifths of his money in 6 J per cent bonds, two-ninths in 5^ per cent bonds, and the balance in 3| per cent bonds. His income from the investments is f 915. Find the amount of his property. SIMPLE EQUATIONS. 135 63. A man starts in business with $ 8000, and adds to his capital annually one-fourth of it. At the end of each year he sets aside a fixed sum for expenses. At the end of three years, after deducting the fixed sum for expenses, his capital is reduced to $ 6475. What are his annual expenses ? 64. If 19 oz. of gold weigh 18 oz. in water, and 10 oz. of silver weigh 9 oz. in water, how many ounces of each metal are there in a mixed mass weighing 127 oz. in air, and 117 oz. in water ? 65. A fox is pursued by a hound, and has a start of 63 of her own leaps. The fox makes 4 leaps while the hound makes 3 ; but the hound in 5 leaps goes as far as the fox in 9. How many leaps does each make before the hound catches the fox ? (Let 4 a; = the number of leaps made by the fox, and 3 a; = the number made by the hound.) 66. A merchant increases his capital annually by one- third of it, and at the end of each year sets aside $ 2700 for expenses. At the end of three years, after deducting the sum for expenses, he has || of his original capital. Find his original capital. PROBLEMS INVOLVING LITERAL EQUATIONS. 156. 1. Divide a into two parts such that m times the first shall exceed n times the second by b. Let X = one part. Then, a — x = the other part. By the conditions, inx = n(a -x) + b. mx = an — nx -{■ b. mx -{- nx = an + b. x(jtn + w) = an + b. 136 ALGEBRA. Whence, x — ^^ "^ , one part, m + n and ^ _ ^ ^ ^ _ an + b ^ am + an - an - b m + n m + n am — b m + n the other part. Note. The results can be used as formulae for solving any prob- lem of the above form. Thus, let it be required to divide 25 into two parts such that 4 times the first shall exceed 3 times the second by 37. Here, a = 25, m = 4, w = 3, and 6 = 37. Substituting these values in the results of Ex. 1 , thefirstpart ^25x3 + 37^75 + 37^112^ 7 7 7- and the second part = 26x^^l37 ^ 10^-37 ^ 63 ^ , 7 7 7 2. Divide a into two parts such that m times the first shall equal n times the second. 3. A is m times as old as B, and a years ago he was n times as old. Find their ages at present. 4. A can do a piece of work in m hours, which B can do in n hours. In how many hours can* it be done by both working together ? 5. A vessel can be filled by three taps ; by the first alone in a minutes, by the second alone in b minutes, and by the third alone in c minutes. In how many minutes will it be filled if all the taps be opened ? 6. A has m dollars, and B has n dollars. After giving A a certain sum, B has r times as much money as A. What sum was given to A ? 7. A gentleman distributing some money among beggars, found that in order to give them a cents each, he would need b cents more. He therefore gave them c cents each, and had d cents left. How many beggars were there ? SIMPLE EQUATIONS. 137 8. A man has a hours at his disposal. How far can he ride in a coach which travels h miles an hour, so as to return home in time, walking back at the rate of c miles an hour ? 9. A courier who travels a miles in a day is followed after n days by another who travels h miles in a day. In how many days will the second overtake the first ? 10. What principal at r per cent interest will amount to a dollars in t years ? 11. In how many years will p dollars ainount to a dollars at r per cent interest ? 12. At what rate per cent will p dollars amount to a dol- lars in t years ? 13. Divide a into two parts, such that one divided by the other may give 6 as a quotient and c as a remainder. 14. Two men, A and B, a miles apart, start at the same time, and travel towards each other. A travels at the rate of m miles an hour, and B at the rate of n miles an hour. How far will each have travelled when they meet ? 15. A grocer mixes a pounds of coffee worth m cents a pound, h pounds worth n cents a pound, and c pounds worth p cents a pound. Find the cost per pound of the mixture. 16. A banker has two kinds of money. It takes a pieces of the first kind to make a dollar, and h pieces of the second kind. If he is offered a dollar for c pieces, how many of each kind must he give ? 17. Divide a into three parts, such that the first may be m times the second, and the second n times the third. 18. A and B together can do a piece of work in m hours, B and C in n hours, and C and A inp hours. In how many hours can each alone do the work ? X38 ALGEBRA. XIV. SIMULTANEOUS EQUATIONS. CONTAINING TWO UNKNOWN QUANTITIES. 157. If a rational and integral monomial (§ 69) involves two or more letters, its degree ivith respect to them is denoted by the sum of their exponents. Thus, 2 a^b^xy^ *is of the fourth degree with respect to X and y. 158. If each term of an equation containing one or more unknown quantities is rational and integral, the degree of the equation is the degree of its term of highest degree. Thus, if X and y represent unknown quantities, ax — by = c is an equation of the first degree. ic^ + 4 ic = — 2 is an equation of the second degree. 2 x^ — Sxy^ = 5 is an equation of the third degree ; for the term 3 xy"^ is the term of highest degree, and 3 xy^ is of the third degree. 159. An equation containing two or more unknown quan- tities is satisfied by an indefinitely great number of sets of value of these quantities. Consider, for example, the equation x -\-y = 5. If a; = 1, we have 1 -\-y = 5, or ?/ = 4. If a; = 2, we have 2 -\-y = 5, ov y = 3; and so on. Thus the equation is satisfied by any one of the sets of values x = l, y = 4:; x=:2, y = 3; etc. For this reason, an equation containing two or more un- known quantities is called an indetermmate equation. SIMULTANEOUS EQUATIONS. 139 160. Two equations, each containing two unknown quan- titieSj are said to be Independent when one of them is satis- fied by sets of values of the unknown quantities which do not satisfy the other. Consider, for example, the equations x-\-y = 5, x — y = 3. The first equation is satisfied by the set of values x = 3. y = 2, which does not satisfy the second. Therefore, the equations are independent. But the equations x -\-y = 5, 2 x -\- 2 y = 10, are not inde- pendent; for the second equation can be reduced to the form of the first by dividing each term by 2; and hence every set of values of x and y which satisfies one equation will also satisfy the other. 161. Let there be two independent equations (§ 160), each of the first degree, containing the unknown quantities X and y, SiS x + y = 5, x — y = 3. By § 159, each equation considered by itself is satisfied by an indefinitely great number of sets of values of x and y. But there is only one set of values of x and y, i.e., a; = 4, y = 1, which satisfies both equations at the same time. A series of equations is called Simultaneous when each contains two or more unknown quantities, and every equa- tion of the series is satisfied by the same set of values of the unknown quantities. 162. To solve a series of simultaneous equations is to find the set of values of the unknown quantities involved which satisfies all the equations at the same time. 163. Two independent, simultaneous equations may be solved by combining them in such a way as to form a single equation containing but one unknown quantity. This operation is called Elimination. There are three principal methods of elimination. 140 4( ALGEBRA. 1. Solve the equations 164. I. Elimination by Addition or Subtraction. 5x-3y = 19. 7x-\-4:y= 2. Multiplying (1) by 4, 20 x - 12 y = 76. Multiplying (2) by 3, 21 x + 12 ?/ = 6. Adding (3) and (4), 41 x = 82. Whence, z= 2. Substituting the value of x in (1), 10 — Sy = 19. -.Sy= 9. Whence, y = — S. The above is an example of elimination hj» addition. 1. (1 (2) (3) (4) u 4 1> 2. Solve the equations Multiplying (1) by 2, Multiplying (2) by 3, Subtracting (4) from (3), Whence, 15x-\-Sy y llOa^-7 -24. 30x+ 16?/ = 30x-21z/ = 2. 72. (1) (2) (3) (4) 37 ?/ = 74. ?/= 2. Substituting the value of y in (2), 10 x — 14 = — 24. 10x = - 10. Whence, • x=:— 1. The abote is an example of elimination by subtraction. Rule. If necessary, multiply the given equations by such numbers as will make the coefficients of one of the unknown quantities in the resulting equations of equal absolute value. Add or subtract the resulting equations according as the coefficients of equal absolute value are of unlike or like sign. Note. If the coefficients which are to be made of equal absolute value are prime to each other, each may be used as the multiplier for the other equation ; but if they are not prime to each other, such multipliers should be used as will produce their lowest common mul- tiple. Thus, in Ex. 1, to make the coefficients of y of equal absolute value, we multiplied (1) by 4 and (2) by 3 ; but in Ex. 2, to make the coefficients of a 3f equal absolute value, since the L. C. M. of 10 and 15 is 30, we multiplied (1) by 2 and (2) by 3. SIMULTANEOUS EQUATIONS. 141 SX+'+ii.^'^'^-^ EXAMPLES. Solve by the method of addition or subtraction ■■( 10. 5x-\- 4^=22. 3a; + y = 9. X- 6y = -10. 2x- 7^ = -15. 7x- 2y = Sl. 4a; -h 3y = -S. 6x-^lly = -2S. 5y-lSx = S. 6x-\- 2y = -S. 5x- 3y = -6. 4:X-\-15y = 7. 14a; -f 6^ = 9. 12a;- 5y = 10. 30a;+lly = -69. 3a;H- 7y = 2. 7x-{- Hy = -2. 11. 12. 13. 14. 15. 16. 17. 18. rl7a; 113 a; 17a;-|-102/ = -30. 35 2/ = -40. rlla;— 5y= 4. 1 9a; 4- 6^ = 10. Sx-\- 9y = -4:. Sy- 9a; = 77. 5x— 9y = 1. Sx-10y = -5. i21x- Sy = 92. [ 9a; + 17y = 19. 10a;-ll2/ = -27. 10?/ -11 a; = 36. ^22x-\-Wy = 9. .18a; + 252/ = 71. r 5a;-242/ = -123. 119a; -36?/ = -81. 165. II. Elimination by Substitution. (7x-9y 1. Solve the equations Transposing — 6x in (2), Whence, Sy- y 15. 5a; = -17. 5X-17. 5X-17 ' "8 Substituting this in (1), 7x- q/ ^^-^^ N ^ ^^ Clearing of fractions, Expanding, Whence, Substituting the value of x in (3) 66x-9(5x-17)=120. 56x-45x + 163 = 120. llx = -33. x = -3. 15- y = 17 8 -4. \y I (1) (2) (3) n .'?<^r invf'V 142 ■^ ALGEBRA. Rule. From one of the given equations find the value of one of the unknown quantities in terms of the other, and substitute this value in place of that quantity in the other equation. Ml 3 EXAMPLES. Solve by the method of substitution : ^x+ 2y = 17. _ r 8a; x-\- y = 16. X— 6y = 2. 3y- 8aj = 29. 2x- 3?/ = -14. 3x-\- 7y = 4S. Sx-\- 5y=:5. 3x- 2y = 29. ( 2x+ 5y = 13. Ix- 4j/ = -19. 3x+ 7y = -23. 5x-\- Ay = -23. 5x+ 9y = 8. 6y- 9a; = -7. 5x+ Sy = -6. 10x-12v = -5. e-\ 10. 11. 12. 13. 14. 15. 16. 17. r ««— 32/ = — 6. I ix-{-^d.y = \.i I 7x^- 'sV^-io. '- llla;+ 6?/ = -19. 6 X — 10 y = 5. 15 2/ — 14 ic = — 15. ( 9a;+ Sy = -6. [l2x-{-10y = -7. 16x — lly = o6. 12 a;- 7y = 37. 7x- 82/ = -43. 5y- 6a; = 35. 6x- 9y = 19. 15 a; + 7 2/ = — 41. 5 a;- 8 2/ = 60. ex-\- 7y = -ll. 166. III. Elimination by Comparison. y - 1. Solve the equations f 2a; -52/ = -16. 3a; + 72/ = 5. (1) (2) Transposing - 5?/ in (1), 2x = 5y-16. Whence, 2 (3) Transposing 7 ?/ in (2), 3x = 6-72/. SIMULTANEOUS EQUATIONS. 143 Whence, Equating the values of x, Clearing of fractions, Whence, Substituting the value of y in (3) Rule 7y 3 5_y_ -16 5-' Ll. 2 3 15?/ -48 = = 10- Uy. 29y^ = 58. y-- = 2. X - 10- J6_ 3. From each of the given equations find the value of the same unknown quantity in terms of the other, and pkice these values equal, to each other. EXAMPLES. Solve by the method of comparison : I 2a; -f y = 9. OX+ 3 2/ = 25. x-^ 2y = -2. 4.x- 7y = 37. ( 6x— oy = — 10. 1 ox- 2y = -17. rll.^•+ 42/ = 3. 1 8a;+ 9y = -10. 7X+ 32/ = -9. 6y- 9.x- = 28. 12x-25y = l. 10. llOa; 12. 13 fl2a;- 6 7/ =19. 4«/- 3a;^-ll. Qx— 7 y = — 12. 92/ = -12. {Wx-\- Sy = -U. 6x-\-12y = l. 5x-\- 3y = 27. 3a; = -26. r 5a;H- * 1 82/- ^^ r 2x-\- 5y = -27 ' [nx-\- 62/ = -41 (rZx — -Zby = l. -- f oa;— y?/ = (5. 1 4:X-j-10y = -7. ' I 7x-\- 42/ = 29. 9. 6 a; — 5y = 1. 9a; + 10?/ = 12. 3a;- Sy = -17. 7a;+ 62/ = -15. 16. 17. 10a;-f 182/ = -ll . 14 2/ — 15 a; = — 4. 9a;_ 72/ = -85. 4 a; — 11 2/ = — 93. 144 ALGEBRA. MISCELLANEOUS EXAMPLES. 167. iB^fore applying either method of elimination, each of the given equations should be reduced to its simplest form.- r_7 3 1. Solve the equations 0. (1) ajH-3 y + 4. x(y-2)-y(x-5)=-lS. (2) From (1), 7i/ + 28-3x-9 = 0, or 1y -Sx=-\9. From (2), xy - 2x - xy -{- ^y = - IS^ or by — 2x = —lS. Multiplying (3) by 2, Multiplying (4) by 3, Subtracting (5) from (6), Substituting the value of y in (4), Whence, • Solve the following : '2x Sy^_7 3 4 2* x_2y^n 4 5 2' (3) (4) (6) (6) 2. -! 4. (Sx-\-7y = 12. x + 2y , 2x-\-y ^^ 4 3 f x-\-5y 2y -\-x _ -. 13 11 ~ 6. 7. 2a;-3 2y-\-13 6±x--y_ 7 1-x-y 2x-\-Sy = Sx-y 8 2. = 2. 3-2a; 4 + 5y 5 11 4. (x + l)(y-h9)-(x + 5)(y-7) = 112. 2x-\-3y-\-9 = 0, SIMULTANEOUS EQUATIONS. 10. 2 3 3 4 X — y _25 X -\-y 2 ~'~6 3 x-\-y-9_y--x-6^Q 2 3 11. 12. 10a;-^^^=ll. 7 82,-^±^ = _17. x-\-y-2 ^ 1 a;-^ 3 3.r + y-3 ^ 1 2y-;r 11* i 145 3 13. 14. 15. 16. 17. 18. ^ + 8 = 20.-^. 3^_2j^-3^^^ 3 ^ x-^y 7 X — 5y 11 + 3 = 0. ?-?l=l 5 7 ' 8-y 2a; + 3 ^ y + 3 5 4 4 * "^i 3~-'^- 3a5-|(^4aj + 2/ + |) = 52/. Sx-^(2x-\-y-{-6) = 5y. l(Sx + 2y)-h2y-x) = -S. aj + 22/ + 4 ~2a5-3(2/+i)1=0. 1K^04<^^^) 11 lo' 146 * ALGEBRA, x-Sy y-3x q^^ 2 2 19. 21. 22. 23. 24. 26. 1 ,3 on |.8a^ + .052/ = .6365. 5 4^ 7 \mx-Ay=.l. 2 3^ 7 ^ 4 7a^ + 3y + 12 ^ 3 x~ 5y — 4: 2 2x + Sy ^ 5x-2y - 3 ^ 17 3a; + 4.^ ^—0 3x-4.y 13 5x-\-6 _lly — 5 _^^ 10 ~"^i ~ ■ 7 a; _ 55 y — 12 _ orr a;-2 10 -a; y-lO ^^ 5 3 4' 2/ + 2 2a; + ?/ a; + 13 _ 0. 6 32 16 r 2a;y-3 4y + 5^^ 25. a; + 2 a; - 3 ^ [(2aj-32/ + l)(3a; + ll2/) + 252/^= (3aj + 8^)(2a;-2/> .322/-2.4^-:505l±2:6 = _.8a.-:56^±^. ^ .25 .5 . 07 a; + .1 I -Q^y + .l ^Q .6 "^ .3 SIMULTANEOUS EQUATIONS. 147 168. 'Literal Simultaneous Equations. In solving literal simultaneous equations, the method of elimination by addition or subtraction is usually the best. 2. 4. 1. Solve the equations dx -^ by = c, a'x + b'y = c'. (1) (2) Multiplying (1) by 6', Multiplying (2) by 6, ab'x + bb'y = b'c. a'bx + bb'y = be'. Subtracting, Whence, ab'x - a'bx = b'c - be'. ^_b'c- be' ab' - a'b Multiplying (1) by a', Multiplying (2) by a, aa'x 4- a'by = a'c. aa'x + ab'y = ae'. (3) (4) Subtracting (3) from (4), ab'y - a'by = ae' — a'e. Whence, ae'-a'c^ ab' - a'b EXAMPLES. Solve the following : (Sx-\-4:y = 7a. 7. 8. .ab. 9- •|±M = „. 1. ax — by — 1. bx-{-ay = l. mx 4- ny = p. bx-\-ay_ ^ m n m'x-\-n'y =p'. ax -\-by = m, ex — dy = n. X .y _ 2 3m 6n 3 a b c r (a — b)x — by = a^ — \x-\-y = 2a. ^ + 2/^1. a'^b' c' 10. (^«-*)- [ay + bx c - (a -f b) = 0. y = a'-^b'. 148 11. ALGEBRA. ax — by = 2 ah. 2bx-\-2ay = Sb^ 19 (x-ay = b(l-{-ab). a\ ^^' [b bx-\-y = a{l-\- ab). 13. 14. 15. 16. 17. 18. 19. y ra^tb=^-''' x-y = 2{a^- 62). (b — a)x — {a — c) y = be — a^ (b — c)x — ay = — ac. (b -h c)x -\- (b — c) y = 2 ab. (a -\- c) X — (a — c)y = 2 ac. mx -^ny = mn (m^ -|- n^. x-\-y = mn (m + n). ax — by = 2 b. bx-\-ay ^aPj-a^b_±ab^ ab (a + b)x— (a — b)y = 3 ab. r (a -f- 0) ic — (a \(a — b)x— (a (a — b)x— (a -\-b)y = ab. 2x — b _ Sx — y a ~ a-\-2b 2x — b _ a — 2y a b 169. Certain equations in which the unknown quantities occur in the denominators of fractions may be readily solved without previously clearing of fractions. = 8. 1. Solve the equations fl0_9 x y X y (1) (2) SIMULTANEOUS EQUATIONS. 149 Multiplying (1) by 5, Multiplying (2) by 3, Adding, Then, Substituting the value of a: in (1), Then, = 40. 50 45 X ~Y X y 74 = 37. 74 = 37 a;, and x = 2. y - ? = 3, and «/ = - 3. y EXAMPLES. Solve the following 2. 4. X y x^ y 10 9 = 4. a; y 8_15^9 X y 2 Sx y 9* X 4:y 8 A_±=l. 2a; Sy 2 2 L^^. 3a5 2?/ 72* 2a; + - = -ll. 2/ A 3 21 2 2/ a 6_ 6 . a - 4- - = c. a; y m n _ 'x~y~'^' a; 2/ hx ay 10. ax by a±_b_ X 1^ ay b*-a'' a'b' ' b a ab-b' l^ a'-\-3ab X y a-\-b 11. a + b . a = 5b-a. X « + * = 2a-36. X y ALGEBRA. 10 ^il ^1? % J-lVV 7725 XV. SIMULTANEOtJS^ EQUATIONS. CONTAINING MORE THAN TWO UNKNOWN ^j, ^ 3^ ^, QUANTITIES. '^ ^ (^ 170. '^tf we have three independent simple equations, con- ^ ' taining three unknown quantities, we may combine any two ^^ of them by one of the methods of elimination explained in J^ §§ 164 to 166, so as to obtain a single equation containing /*^ only two unknown quantities. i^ We may then combine the remaining equation with either r^i of the other two, and obtain another equation containing (4 the same two unknown quantities. v'^ By solving the two equations containing two unknown quantities, we may obtain their values; and substituting them in either of the given equations, the value of the remaining unknown quantity may be found. We proceed in a similar manner when the number of equations and of unknown quantities is greater than three. The method of elimination by addition or subtraction is usually the most convenient. ^ Qx-4:y- 7z = 17. (1) dx-7y-iez = 29. (2) I0x-5y^- 3z = 23. (3) 1. Solve the equations Multiplying (1) by 3, Multiplying (2) by 2, Subtracting, 2y + llz=-7. (4) Multiplying (1) by 5, 30 x - 20 «/ - 35 s = 85. (5) Multiplying (3) by 3, 30a; -15y- 9z= 69. (6) Subtracting (5) from (6), 5y + 26z = -16. (7) Multiplying (4) by 5, Multiplying (7) by 2, Subtracting, ' Sz = — 3. 18x- -I2y -21z = 51. 18x- -14?/ -S2z = 58. ^y + 11;^ = -7. 30x- -20y -350 = 85. 30x- -16y - 9z = 69. , 52/ + 26z = -16. 10 2/ + 55;^ = -35. 10 y + 52^ = -32. SIMULTANEOUS EQUATIONS. 151 Whence, Substitutmg in (4), WKence, Substituting in (1), ,^hence, In certain cases the solution may be abridged by means of the artifice which is employed in the following example. 2. Solve the equations u -{- X -\- y = 6. x-\-y-\-z = 7. y-{-z -{-u=zS. z -{- u -\- X = 9. Adding, Su + Sx + Sy -^ 3z = 30. Dividing by 3, u + x-\-y-^z = 10. Subtracting (2) from (6), u = 3. Subtracting (3) from (5), x = 2. Subtracting (4) from (6), y = 1. Subtracting (1) from (5), « = 4. (1) (2) (3) (4) (5) Solve the following : 3x-\-2y = lS, 3. |32/-2z = 8. 2x-Sz = 9. 4. EXAMPLES. 6. 3a; 4-41/4-52 = -21. x-\-y — z = — ll. 7. y-Sz = -20. 12x-^y-{-z = 3. X — y — 2z = ~l. 8. 5x-2y = 0. l2x — y-\-z = — 9. x — 2y-^z = 0. x-y-\-2z^-ll. X — y -\-z = 9. x-2y-\-3z = 32. x — 4:y-{-5z = 62. 3x — y — z = 7. x-3y-z = 21. x-y-3z = 27. 152 ALGEBRA. 10. 11. 12. 13. 14. 16. 16. 17. 2a;-32/ = 4. 4:y-\-2z = -3. x-\- 2y — Sz = 5. 3x-22y-\-6z = 4:. 7x- 6y-Sz = 15. 5x -\- y -\- 4:Z = — 5. 3x-5y-h6z = -20. x-3y-4:Z = -21. 2x-3y-4.z = -10. 3x-\-4:y + 2z = — 5. 4.x-\-2y + 3z = -21. 6x-\-^y-\-3z = l. 9a;- 2/ + 62 = -39. ^x-ly-12z = -2. 2x — Qty — 6z = —ll. 10x + ^y-3z = m. 18. 19. 20. 21. x-{-3y-lz = 31. 3x-{- ?/ + 5 ;2 = — 49. 22. 20x-\-2y-6z = -3b. 9a; + 42/ = 102-f 11. 121/ -5^ = 6a; -9. 15;3 4-3a; = -8y-16. 5a; + 16?/ + 6:3 = 4. 10x-\.4.y-12z = -l. [15a; -12^- 32 = -10. 23. ?+ 1 = 5. y X §-+ 5=1. X z 1 3^^9 X 2y 5 1 + A=5 y 3z 3 i + A=I. z 4a; 4 X — ay = a(a^ + 1). az — nVn'i a\a' - 1). y z — ax = —. a^(a + 1). a(x — c) + b(y — c) = 0. 6(2/ — a) + c(z — a) = 0. 0(2; — 6) + a(a; — b)=0. w + a; + 2/ = 7. a; + 2/ + 2;=-8. 2/ + 2J + w = 5. 2; + tfc + a; = — 10. 1_1 1 . X y 1_1 y ^ i_i 2; X X y 2; 1_ 2/ 1 2; -1=6 a; 1 _i_ -l = c. SIMULTANEOUS EQUATIONS. 153 24. 25. 26. 27. 28. 29. u-2x = -13. x — 3y = 13. 1 1_1 X y~c' y — 4:Z =5. z -5u = 23. 7x-{-4:y-3ii=0. 30. y z a z^x b 5x-\-4:y-{-4:Z — 5u= 0. 2x-{-z-u = 0. 2x-\-Ay—3z—u=- -8. a c 2 3 3 ^23 31. ? + 2 = 2a. 6 c | + ^ = 2c. 16 a 2 3 3 9a; -26?/ -16:2 = - 12a;-8i/ + 15z = - -44. -15. 32. Z X 8a;-92/ + 132 = - -44. 12/ 2; -M=^^- ..|.6.^-±^. 33. x+y+az=a-\-2. ay-\-az-{-a^x=a^-\-a-\-l [3 ^ 2 2 ^az-\-ax-\-a^y=2a^-^l. 2^3 4 (3x-y 5y-\-4.z 19 5 ' 2 ~ 2 3 4 2 34. 2x-^3z x-4:y 7 6 4 4 4 2 3 4aj-2 3y-5z 49 L 3 2 3 154 ALGEBRA. XVI. PROBLEMS. INVOLVING SIMULTANEOUS EQUATIONS. 171. In solving problems where two or more letters are used to represent unknown quantities, we must obtain from the conditions of the problem as many independent equations (§ 160) as there are unknown quantities to he determined. 172. 1. Divide 81 into two parts such that three-fifths of the greater shall exceed live-ninths of the less by 7. Let X = the greater part, and y = the less. Since the sum of the greater and less parts is 81, we have x + y = 8l. (1) And since three-fifths of the greater exceeds five-ninths of the less by 7, f = f+7. (2) Solving (1) and (2), x = 45, y = S6. 2. If 3 be added to both numerator and denominator of a fraction, its value is | ; and if 2 be subtracted from both numerator and denominator, its value is ^. Kequired the fraction. Let X = the numerator, and y = the denominator. By the conditions. and X + 3 ^ 2 y + s 3' x-2 1 y-2 2 Solving these equations, x = 7, y == 12. Therefore, the fraction is ■^^. PROBLEMS. 155 K^'^'^T^.^"''^ V PROBLEMS. '^. Divide 59 into two parts such that two-thirds of the ^ '/I fess shall be less by 4 than four-sevenths of the greater. [J\ 4. Find two numbers such that two-fifths of the greater v£>^ exceeds one-half of the less by 2, and four-thirds of the less us^^' exceeds three-fourths of the greater by 1. 5. If 5 be added to the numerator of a certain fraction, tit its value is | ; and if 5 be subtracted from its denominator, its value is f. Find the fraction. 6. If 9 be added to both terms of a fraction, its value is f ; and if 7 be subtracted from both terms, its value is f . Find the fraction. 7. A grocer can sell for $ 57 either 9 barrels of apples and 16 barrels of flour, or 15 barrels of apples and 14 bar- rels of flour. Find the price per barrel of the apples and of the flour. 8. A's age is f of B's ; but in 16 years his age will be ^ of B's. Find their ages at present. 9. If twice the greater of two numbers be divided by the less, the quotient is 3 and the remainder 7 ; and if five times the less be divided by the greater, the quotient is 2 and the remainder 23. Find the numbers. 10. If the numerator of a fraction be trebled, and the denominator increased by 8, the value of the fraction is f ; and if the denominator be halved, and the numerator de- creased by 7, the value of the fraction is \. Find the fraction. 11. Three years ago A's age was f of B's ; but in nine years his age will be ^ of B's. Find their ages at present. 12. A and B can do a piece of work in 9 hours. After working together 7 hours, B finishes the work in 5 hours. In how many hours could each alone do the work ? 156 ALGEBRA. 13. A man invests a certain sum of money in 4^ per cent stock, and a sum $ 180 greater than the first in 3^ per cent stock. The incomes from the two investments are equal. Find the sums invested. 14. My income and assessed taxes together amount to $ 64. If the income tax were increased one-fourth, and the assessed tax decreased one-iifth, they would together amount to $ 63.80. Find the amount of each tax. 16. If B gives A $ 12, A will have f as much money as B ; but if A gives B $ 12, B will have J as much money as A. How much money has each ? 16. A man pays with a f 5 note two bills, one of which is six-sevenths of the other, and receives back in change seven times the difference of the bills. Find their amounts. 17. Find three numbers such that the first with one-third the others, the second with one-fourth the others, and the third with one-fifth the others may each be equal to 25. 18. A sum of money was divided equally between a cer- tain number of persons. Had there been 3 more, each would have received $ 1 less ; had there been 6 less, each would have received $5 more. How many persons were there, and how much did each receive ? Let X = the number of persons, and y = the number of dollars received by each. Then, xy = the number of dollars divided. The sum of money could be divided between cc + 3 persons, each of whom would receive y — 1 dollars ; and between x — 6 persons, each of whom would receive ?/ + 5 dollars. Whence, (x -I- 3)(y - 1) and (x -Q)(y + 5) will also represent the number of dollars divided. Then (x + S)(y - 1) = xy, and (x — 6) (?/ -h 5) = xy. Solving these equations, X = 12, y = 5. PROBLEMS. 157 19. A man bought a certain number of eggs. If he had bought 56 more for the same money, they would have cost a cent apiece less ; if 24 less, a cent apiece more. How- many eggs did he buy, and at what price each ? 20. A boy spent his money for oranges. If he had got 15 more for his money, they would have cost li cents each less; if 5 less, they would have cost 1|- cents each more. How much did he spend, and how many oranges did he get ? 21. A sum of money is divided equally between a certain number of persons. Had there been m more, each would have received a dollars less; if n less, each would have received b dollars more. How many persons were there, and how much did each receive ? 22. A purse contained $6.55 in quarter-dollars and dimes; after 6 quarters and 8 dimes had been taken out, there remained 3 times as many quarters as dimes. How many were there of each at first? 23. A dealer has two kinds of wine, worth 50 and 90 cents a gallon, respectively. How many gallons of each must be taken to make a mixture of 70 gallons, worth 75 cents a gallons ? 24. A grocer bought a certain number of eggs at the rate of 22 cents a dozen, and seven-fifths as many at the rate of 14 cents a dozen. He sold them at the rate of 20 cents a dozen, and gained 24 cents by the transaction. How many of each kind did he buy ? 25. A and B can do a piece of work in 10 days, A and C in 12 days, and B and C in 20 days. In how many days can each of them alone do it ? 26. A resolution was adopted by a majority of 10 votes ; but if one-fourth of those who voted for it had voted against it, it would have been defeated by a majority of 6 votes. How many voted for, and how many against it ? 158 ALGEBRA. 27. The sum of the three digits of a number is 13. If the number, decreased by 8, be divided by the sum of its second and third digits, the quotient is 25; and if 99 be added to the number, the digits will be inverted. Find the number. Let X = the first digit, y = the second, and z = the third. Then, 100 x -\- 10 y -\- z = the number, and 100 z -^r lOy -\-x = the number with its digits inverted. By the conditions of the problem, x + y + z = lS, 100x + lOy + ^ -8 _o^ y -\- z and 100x + 10y-\-z-h99 = 100z-^10y-^x. Solving these equations, x = 2, y = S, z = 3. Therefore, the number is 283. 28. The sum of the two digits of a number is 16 ; and if 18 be subtracted from the number, the digits will be inverted. Find the number. 29. The sum of the three digits of a number is 23 ; and the digit in the ten's place exceeds that in the unit's place by 3. If 198 be subtracted from the number, the digits will be inverted. Find the number. 30. If the digits of a number of two figures be inverted, the sum of the resulting number and twice the given num- ber is 204 ; and if the number be divided by the sum of its digits, the quotient is 7 and the remainder 6. Find the number. 31. If a certain number be divided by the sum of its two digits, the quotient is 4 and the remainder 3. If the digits be inverted, the quotient of the resulting number increased by 23, divided by the given number, is 2. Find the number. PROBLEMS. 159 32. Two vessels contain mixtures of wine and water. In one there is three times as much wine as water, and in the other five times as much water as wine. How many gallons must be taken from each to fill a third vessel whose capacity is 7 gallons, so that its contents may be half wine and half water ? 33. If a lot of land were 6 feet longer and 5 feet wider, it would contain 839 square feet more ; and if it were 4 feet longer and 7 feet wider, it would contain 879 square feet more. Find its length and width. 34. A and B are building a piece of fence 189 feet long. After 9 hours A leaves off, and B finishes the work in 12 1 hours. If 12 hours had occurred before A left off, B would have finished the work in 4^ hours. How many feet does each build in one hour ? 35. The sum of the three digits of a number is 17. The sum of 3 times the first digit, 5 times the second, and 4 times the third is 70 ; and if 297 be added to the number, the digits will be inverted. Find the number. 36. The rate of an express train is five-thirds that of a slow train, and it travels 36 miles in 32 minutes less time than the slow train. Find the rate of each in miles an hour. 37. Divide $396 between A, B, C, and D so that A may receive one-half the sum of the shares of B and C, B one- third the sum of the shares of C and D, and C one-fourth the sum of the shares of A and D. 38. A merchant has two casks of wine, containing together 56 gallons. He pours from the first into the second as much as the second contained at first; he then pours from the second into the first as much as was left in the first ; and again from the first into the second as much as was left in the second. There is now three-fourths as much in the first as in the second. How many gallons did each contain at first? 160 ALGEBRA. 39. A crew can row 10 miles in 50 minutes down stream, and 12 miles in an hour and a half against the stream. Find the rate in miles an hour of the current, and of the crew in still water. Let X — the number of miles an hour rowed by the crew in still water, and y — the number of miles an hour of the current. Then, x ■\ij — the number of miles an hour of the crew rowing down stream, and X — y = the number of miles an hour of the crew rowing up stream. Since the number of miles an hour rowed by the crew is equal to the distance divided by the time in hours, we have D and x-w=12-^- = 8. ^ 2 Solving these equations, a: = 10, y = 2. 40. A crew can row a miles in m hours down stream, and b miles in ii hours against the stream. Find the rate in miles an hour of the current, and of the crew in still water. 41. A vessel can go 63 miles down stream and back again in 20 hours ; and it can go 3 miles against the current in the same time that it goes 7 miles with it. Find its rate in miles an hour in going, and in returning. 42. If a number of two figures, diminished by 3, be divided by the sum of its digits, the quotient is 5. If the digits be inverted, the quotient of the resulting number increased by 18, divided by the sum of the digits, is 7. Find the number. 43. The digits of a number of three figures have equal differences in their order. If the number be divided by one-half the sum of its digits, the quotient is 41 ; and if 594 be added to the number, the digits will be inverted. Find the number. PROBLEMS. 161 44. If I were to make my field 5 feet longer and 7 feet wider, its area would be increased by 830 square feet ; but if I were to make its length 8 feet less, and its width 4 feet less, its area would be diminished by 700 square feet. Find its length and width. 45. A certain sum of money at simple interest amounts in 3 years to $ 420, and in 7 years to $ 480. Required the sum and the rate of interest. 46. A certain sum of money at simple interest amounts in m years to a dollars, and in n years to b dollars. Re- quired the sum and the rate of interest. 47. A and B together can do a piece of work in 8 J days ; but if A had worked f as fast, and B | as fast, they would have done it in 7| days. In how many days could each alone do the work ? 48. A sum of money at simple interest amounts to $ 2080 in 8 months, and to $ 2150 in 15 months. Find the sum and the rate of interest. 49. A train running from A to B meets with an accident which causes its speed to be reduced to one-third of what it was before, and it is in consequence 5 hours late. If the accident had happened 60 miles nearer B, the train would have been only 1 hour late. Find the rate of the train before the accident, and the distance to B from the point of detention. Let 3 a; = the number of miles an hour of the train before the accident. Then, x = the number of miles an hour after the accident. Let y = the number of miles to B from the point of detention. The train would have done the last y miles of its journey in ^ V ^^ hours ; but owing to the accident, it does the distance in - hours. T"-' l = ii + ^- CD 162 ALGEBRA. If the accident had occurred 60 miles nearer B, the distance to B from the point of detention would have been y — QO miles. Had there been no accident, the train would have done this in ^ — hours, and the accident would have increased the time to 3x X hours. Then, — = ^ + 1. (2) X Sx ^ ^ Subtracting (2) from (1), — = ^—+4, or — = 4. X o X X Whence, x = 10. Then the rate of the train before the accident was 30 miles an hour. Substituting in (1), ^ = ^ + ^' ^^ T^ = ^• Whence, y z=76. 50. A train running from A to B meets with an accident which delays it 45 minutes ; it then proceeds at five-sixths its former rate, and arrives at B 75 minutes late. Had the accident occurred 45 miles nearer A, the train would have been 90 minutes late. Find the rate of the train before the accident, and the distance to B from the point of detention. 51. The unit's digit of a number of three digits is 7. If the digits in the hundreds' and tens' places be interchanged, the number is decreased by 180. If the digit in the hun- dreds' place be halved, and the other two digits interchanged, the number is decreased by 273. Find the number. 52. A, B, C, and D play at cards, having together $ 46. After A has won one-third of B's money, B one-fourth of C's, and C one-fifth of D's, A, B, and C have each $ 10. How much had each at first? 53. A, B, and C have together f 24. A gives to B and C as much as each of them has; B gives to A and C as much as each of them then has ; and C gives to A and B as much as each of them then has. They have now equal amounts. How much did each have at first ? PROBLEMS. 163 54. The fore-wheel of a carriage makes 8 revolutions more than the hind-wheel in going 180 feet ; but if the cir- cumference of the fore- wheel were | as great, and of the hind- wheel f as great, the fore- wheel would make only 5 revolutions more than the hind-wheel in going the same distance. Find the circumference of each wheel. 55. A and B together can do a piece of work in m days, B and C in n days, and C and A in p days. In what time can each alone perform the work ? 56. A piece of work can be completed by A working 3 days, B 7 days, and C 1 day ; by A working 5 days, B 1 day, and C 7 days; or by A working 1 day, B 5 days, and C 11 days. In how many days can each alone perform the work ? 57. A man has a sum of money invested at a certain rate of interest. Another man has a sum greater by $ 3000, invested at a rate 1 per cent less, and his income is $ 45 less than that of the first. A third man has a sum less by $ 2000 than that of the first, invested at a rate 1 per cent greater, and his income is f 40 greater than that of the first. Find the capital of each man, and the rate at which it is invested. 58. A and B can do a piece of work in a hours. After working together b hours, B finishes the work in c hours. In how many hours could each alone do the work ? 59. A crew row up stream 26 miles and down stream 35 miles in 9 hours. They then row up stream 32 miles and down stream 28 miles in 10 hours. Find the rate in miles an hour of the current, and of the crew in still water. (Let X and y represent the number of miles an hour of the crew rowing up and down stream, respectively.) 60. A sum of money, at 6 per cent interest, amounts to $ 5900 for a certain time, and to ^ 7100 for a time longer by 4 years. Find the principal and the time. 164 ALGEBRA. 61. A gives to B and C twice as much money as each of them has; B gives to A and C twice as much as each of them then has ; and C gives to A and B twice as much as each of them then has. Each has now f 27. How much did each have at first ? 62. A party at a tavern found, on paying their bill, that had there been 4 more, each would have paid 75 cents less ; but if there had been 4 less, each would have paid $ 1.50 more. How many were there, and how much did each pay ? 63. An express train travels 30 miles in 27 minutes less time than a slow train. If the rate of the express train were f as great, and of the slow trairi f as great, the express train would travel 30 miles in 54 minutes less time than the slow train. Find the rate of each in miles an hour. 64. A and B run a race of 450 feet. The first heat, A gives B a start of 135 feet, and is beaten by 4 seconds ; the second heat, A gives B a start of 30 feet, and beats him by 3 seconds. How many feet can each run in a second ? 65. A sum of money consists of quarter-dollars, dimes, and half-dimes. Its value is as many dimes as there are pieces of money ; its value is also as many quarters as there are dimes ; and the number of half-dimes is one more than the number of dimes. Find the number of each coin. 66. A man invests $ 5100, partly in 3|- per cent stock at $ 90 a share, and partly in 4 j3er cent stock at $ 120 a share, the par value of each share being ^100. If his annual income is ^185, how many shares of each stock does he buy? 67. A and B run a race of 336 yards. The first heat, A gives B a start of 28 yards, and beats him by 2 seconds ; the second heat, A gives B a start of 12 seconds, and is beaten by 48 yards. How many yards can each run in a second ? INEQUALITIES. 165 XVII. INEQUALITIES. 173. Definitions. The Signs of Inequality, > and <, are read "is greater than " and " is less than," respectively. Thus, a > 6 is read " a is greater than b" ; a < 6 is read " a is less than 5." The Sign of Continuation, ••♦, signifies "and so on/^ or " continued by the same law." * 174. One number is said to be greater than another when the remainder obtained by subtracting the second from the first is a positive number ; and one number is said to be less than another when the remainder obtained by subtracting the second from the first is a negative number. Thus, if a — 6 is a positive number, a > 6 ; and if a — 6 is a negative number, a 6 and c>d subsist in the same sense. 177. An inequality will continue in the same sense after the same quantity has been added to, or subtracted from, both members. 166 ALGEBRA. For consider the inequality a^b. Then by § 174, a — b is a positive number. Hence, each of the numbers (a + c) — (6 + c), and (a — c) — (6 — c) is positive, since each is equal to a — b. Therefore, a -f c > 6 -f- c, and a — c> b — c. (§ 174) 178. It follows from § 177 that a term may be transposed from one member of an inequality to the other by changing its 179. If the signs of all the terms of an inequality be changed, the sigyi of inequality must be reversed. For consider the inequality a — b > c — d. Transposing every term, we have d-c>b-a. (§ 178) That is, b — ab. By § 174, a — b is a positive number. Hence, if m is a positive number, each of the numbers m(a — b) and , or, ma — mb and , is positive. ' mm Therefore, ma > mb, and — > — ' 'mm' 181. It follows from §§ 179 and 180 that if both members of ayi inequality be multiplied or divided by the same negative number, the sign of inequality must be reversed. INEQUALITIES. 167 182. If any number of inequalities, subsisting in the same sense, be added meynber to member, the resulting inequality will also subsist in the same sense. For consider the inequalities a>b, a' > b', a" > b", •••. Then each of the numbers a — b, a' — b', a" — b", •••, is positive. Therefore, their sum a-b + a'-b'-\-a"-b"+'", or, a -f- a' + a" H (6 + 6' + b" + .••), is a positive number. Whence, a-^a' -\- a" -+-...> 6 -f- 6' + b" + •••. 183. It is to be observed that, if two inequalities, subsist- ing in the same sense, be subtracted member from member, the resulting inequality does not necessarily subsist in the same sense. Thus, if a>6 and a'>b', the numbers a—b and a'—b' are positive. But (a — b) — (a' — b'), or its equal (a— a')— (b — b'), may be positive, negative, or zero ; and hence a— a' may be greater than, less than, or equal to b — b'. EXAMPLES. 184. 1. Find the limit of x in the inequality '-!<¥-»■ Multiplying both members by 3 (§ 180), we have 21x-23<2x + 16. Transposing (§ 178), and uniting terms, 19a; < 38. Dividing both members by 19 (§ 180), X < 2, Ans. 168 ALGEBRA. 2. Find the limits of x and y in the following: Sx-j-2y>37. (1) 6x> 45. x>2 1. Qx + ^y> 74. 6x + 9y = 99. 2x-{-3y = 33. (2) Multiplying (1) by 3, . 9x + 6ij> 111. Multiplying (2) by 2, 4 x + 6 «/ = 66. Subtracting (§ 177), Whence, Multiplying (1) by 2, Multiplying (2) by 3, Subtracting, — 5 ?/ > — 25. Dividing both members by — 5 (§ 181), y < 5. -^ Find the limits of x in the following : 3. (6x-iy-2S<(4.x-3)i9x-\-2). 4. (3a; + 2)(4a;-5)>(2a;-3)(6a;-hl) + 5. 5. (5x + iy + W>(3x-2y-{-(4.x-{-3f. 6. (x -2){x- 3) (a; -h 4) < (a; + 1) (a; + 2) (a; - 4). 7. 6 ma; — 5 an > 15 am — 2 Tiic, if 3 m + n is negative. 8. — i ^^ < 2, if a and b are positive, and a> b. a b Find the limits of x and ?/ in the following : 9_ i4.x + 9y<4:0. ^^ (7x-^2y>25. \6x-y = 2. ' \3x-\-5y = 19. 11. Find the limits of x when 5a; + 7<9a;-13, and 11a; - 20 < 6a; + 25. 12. A certain positive integer, plus 21, is greater than 8 times the number, minus 35 ; and twice the number, plus 11, is less than 7 times the number, minus 19. Find the number. INEQUALITIES. 169 13. A teacher has a number of his pupils such that 8 times their number, minus 31, is less than 3 times their number, plus 69 ; and 13 times their number, minus 45, is greater than 7 times their number, phis 57. How many pupils has he ? 14. A shepherd has a number of sheep such that 4 times the number, minus 7, is greater than 6 times the number, minus 89 ; ^nd 5 times the number, plus 3, is greater than twice the number, plus 114. How many sheep has he ? 15. Prove that if a and b are unequal positive numbers, K + ->2- a Since the square of any number is positive, (a-6)2>0. That is, a2-2a6 + &2>0. Transposing - 2 ab, a^ ■^b'^>2 ah. Dividing each term of the inequality by a6 (§ 180) , we have ^+^>2. h a 16. Prove that for any value of x, except 1, x^ + 1 > 2 x. 17. Prove that for any value of x, except |, 9 a;^ -f- 4 > 12 a;. " In each of the fallowing examples, the letters are under- stood as representing positive numbers. 18. Prove that -^ + ?^ > 2, if 6 is not equal to \ a. 26 a 19. Prove that (ci + 26)(a - 26) > 6(6a - 13 6), if h is not equal to \a. 20. Prov.3 that a(9a - 46) >46(2a - 6), if h is not equal to fa. / 21. Prove that (a^ - W) {c" - dF) < (ac - hdf, if he does not equal ad. 170 /- ALGEBRA. XVIII. INVOLUTION. 185. Involution is the process of raising a given expres- sion to any required power whose exponent is a positive integer. This may be effected, as is evident from § 6, by taking the expression as a factor as many times as there are units in the exponent of the required power. INVOLUTION OF MONOMIALS. 186. 1. ¥mdtheY3i\iieoi(5a%y. By § 6, (5a352)3 ^6(^352 x 6a^b^ x ^a%^ = 126 a^¥, Ans. 2. Find the value of (- ay. (-ay= (- a) X (- a) X (- a) X {- a)= a^ (§49), Ans. 3. Find the value of (- 3 my. (-3m4)3= (-3TO4)x(-3TO4)x(-3m4) = -27mi2(§49), Aiis. From the above examples, we derive the following rule : liaise the absolute value of the numerical coefficient to the required power, and multiply the exponent of each letter by the expoyient of the required power. Give to every power of a positive term, and to every Even poiver of a negative term, the positive sign, and to every Odd power of a negative term the negative sign. EXAMPLES. Find the values of the following : 4. (a'b^c'f. 8. (2m^n'y. 12. (pq'^r'y\ 5. {:f^fzy\ 9. (-a26V)l 13. {-QxY^y. 6. {-mhipy. . 10. {x^yz'y. 14. {^a'^x'^. 7. (-12a2«63n-^2^ 11. (^-Sx'y'^y. 15. (-5m/iy)^ INVOLUTION. 171 A fraction may be raised to any required power by rais- ing both niwierator and denominator to the required 2)ower, and dividing the first result by the second. 16. Find the value oi ( - — Y- V 32/V V 3y2; (3j/2)4 812/8' Find the values of the following : "(f^" -C-^)' "-(^S)' SQUARE OF A POLYNOMIAL. 187. We find by multiplication : a -\-b -\- c a -\-b -\- c a^ + ab + ac -\- ab -\-b^-\- be -\- ac -\- bc-\- -13. 2a;-7a;3. 6. m-4?i. 10. 4m-3n^ 14. 5a« + 66^ The cube of a trinomial may be found by the above method, if two of its terms be enclosed in a parenthesis and regarded as a single term. 15. Find the cube of ar^ — 2 a; — 1. (x2 - 2 X - 1)3 = [(a;2 - 2 x) - l.]3 = (x2_2x)8-3(x2-2x)2 + 3(x2-2x)- 1 = x6-6x5+12x4-8x3-3(x*-4x3+4x2) + 3(x2-2x)-l = x6-6x5+12x4-8a;3-3x*+12x3_i2x2+3x2-6a;-l = x6-6x5 + 9x4 + 4x3-9x2-6x- 1, Afis. . Cube each of the following : ^16. a-{-b-c. 18. x-^J-\-2z. 20. 2x^-\-x-3. 17. ay' + x + l. ^9. a2-3a-l. 21. 3-4a; + ar^. 174 ALGEBRA. XIX. EVOLUTION. 189. If an expression when raised to the 7ith power, w being a positive integer, is equal to another expression, the first expression is said to be the nth. Root of the second. Thus, if a"" = b, a is the nth root of b. 190. Evolution is the process of finding any required root of an expression. 191. The Radical Sign, y, when written before an ex- pression, indicates some root of the expression. Thus, Va indicates the second^ or square root of a ; Va indicates the third, or cube root of a ; Va indicates the fourth root of a ; and so on. The index of a root is the number written over the radical sign to indicate what root of the expression is taken. If no index is expressed, the index 2 is understood. EVOLUTION OF MONOMIALS. 192. 1. Required the cube root of a^b^c^. We have, (abH^)^ = a%^(^. Whence, \/a%^c^ = ahH^. (§ 189) 2. Required the fifth root of — 32 a^. We have, ( - 2 a)^ = - 32 a^. Whence, \/-32a5 = -2a. 3. Required the fourth root of a^. We have either (+ a)* or (— a)* equal to a^. Whence, Va!^ = ±a. The sign ±, called the double sign, is prefixed to an ex- pression when we wish to indicate that it is either -f or — . EVOLUTION. 175 193. From § 192 we derive the following rule : Extract the required root of the absolute value of the numeri- cal coefficient, and divide the exponent of each letter by the index of the required root. Give to every even root of a positive term the sign ± , ayid to every odd root of any term the sign of the term itself. EXAMPLES. 1. Find the square root of 9(X*6V^. By the rule, V9 a^ft^cio = ^ 3 a%^&, Ans. 2. Find the cube root of — 64 Qifiy^". V— 64 a^y^'* = — 4 x^y, Ans. Find the values of the following : ■3. V49a«6^. 4. ^125a!«2/'. 5. V — m}^nJp^^. 6. -yiGA^. 9. 10. ^243a^6». --11. «y^«;2". To find any root of a fraction, extract the required root of both numerator and denominator, and divide the first result by the second. 15. Find the value of ^'/-?I^. We have, ;C^1^ = -<^^^ = -^-^, Ans. Find the values of the following : i« liea^ i« 4/ 81mV^ „n 6/64m^ ,7^ ^^343^^ 13_ JZ^ 21. VC| \ 64 \ 32 2^^ \128 176 ALGEBRA. The root of a large number may sometimes be found by resolving it into its prime factors. 22. Find the square root of 254016. We have, \/254016 = V26 x 3* x 72 = 2^ x 82 x 7 = 504, Ans. 23. Find the value of ^^72 x 75 x 135. We have, ^72 x 75 x 136 = y/(2» x 32) x (3 x 52) x (S^ x 5) = V28 x 36 X 58 = 2 X 32 X 5 = 90, Ans. Find the values of the following : "^4. V3l36. 26. V63504. 28. V42 x 56 x 147. 25. V18225. 27. V48 x 54x72.-29. ^13824 30. Vl5a6x216cx35ca. 31. V 213444. 32. ^91125. 33. a/20736. 34. a/7776. -^ 35. a/63 X 162 X 196. — 36. a/56 x 98 x 112. 37. V(a2 -^5a-\- 6){d' + 2a - 3)(a2 -\-a-2). . SQUARE ROOT OF A POLYNOMIAL. 194. Since (a + by = a^ -^2ab -\- b% we know that the square root of a^ ■}- 2 ab -\- b'^ is a + b. It is required to find a process by which, when the e;c- pression a^ -{-2ab + b^ is given, its square root may be determined. a^ -\-2ab -^b^ a 4- 6 The first term of the root, a, is found ^2 by taking the square root of the first "^ — , , 2 term of the given expression. Subtracting the square of a from Z ao -\- ^jjg given expression, the remainder is 2a& + 6'^ or (2a + h)b. If we divide the first term of this remainder by 2 a, that is, by twice the first term of the root, we obtain the second term of the root, 6. 2a-\-b EVOLUTION. 177 Adding this to 2 a, we obtain the complete divisor, 2 a + 6. Multiplying this by 6, and subtracting the product, 2 a6 -f 6'^, from f he remainder, there are no terms remaining. From the above process, we derive the following rule : Arrange the expression according to the powers of some letter. Extract the square root of the first temij write the result as the first term of the root, and subtract its square from the given expression, arranging the remainder in the same ordei' of pow- ers as the given expression. Divide the first term of the remainder by twice the first term of the root, and add the quotient to the part of the root already found, and also to the trial-divisor. Multijyly the complete divisor by the term of the root last obtained, and subtract the product from the remainder. If other terms remain, proceed as before, doubling the part of the root already found for the next trial-divisor. EXAMPLES. '^^^ ^^Ji^ 195. 1. Find the square root of^9 a^- 30 a^3i? + 25 a\ 9a;4-30a*x2 + 25a' 6*2 _ 6 a* -30a8x2 + 25a6 - 30 a3x2 + 26 a6 The first term of the root is the square root of 9 x* or 3 x^. Subtracting the square of 3 x^, or 9 x*, from the given expression, the first term of the remainder is — 30 a^x^. Dividing this by twice the first term of the root, or 6 x^, we obtain the second term of the root, — 5 a*. Adding this to 6x2, we have the complete divisor, 6x2 — ba^. Multiplying this complete divisor by — 5 a^, and subtracting the product from the remainder, there is no remainder. Hence, 3 x2 — 5 a^ is the required square root. 2. Find the square root of 12ar^ - 22a;3 -f 1 - 20.r^ + 9a;« + 8a; + 12x2. 178 ALGEBRA. Arranging according to the descending powers of x, we have 9x6 + 12x5-20aj4_22xH12a:2+8ic + l 3x3+2x2-4x-l, Ans. 9x6 6x8+2x2 12x5 12x5+ 4x4 6x3+4x2-4x -24x4 -24x4-16x3+16x2 6x3+4x2-8x-l 6x3- 6x3- 4x2 4x2+8x+l It will be observed that each trial divisor is equal to the preceding complete divisor, ivith its last terin doubled. To avoid needless repetition, the last five terms of the first remainder, the last four terms of the second, remainder, and the last two terms of the third remainder are omitted. Note. Since every square root has the double sign (§ 192), the result may be written in a different form by changing the sign of each term. Thus, in Ex. 2, the answer may be written 1 + 4 x — 2 x2 — 3 x'**. Find the square roots of the following : , , 4. l-6a-{-na^-6a^-{-a\ 5. 9a;*-24aj34-4a^ + 16a;+-4. 6. 20o^-70x+^x* + 4:9-Sx\ 7. a^-^b^-\-c^-2ab-2ac-h2bc. 8. 9a^ + l-4a3+-4a«-6a2 + 12a^ ^* 9. a;«-4ajV + 10a^a3+-4fl^a^-20rf + 25a«. 10. 9a^ -\-25y^ + 16z^ -\- SOxy -24:xz - ^Oyz. 11. 49 m* — 14 m^n — 55 mV + 8 mn^ + 16 7i\ 12. 49a2-30a3 + 16 + 9a*-40a. — 13. 25x'-20i^y-26aff + 12xf + 9y', 14. 16m^4-8mV-23mV-6ma;« + 9a;«. EVOLUTION. 179 15. 20 ab^-\-9 a' -26d'b^-\-2ob'- 12 a^b. 16 4 16. m2 + 8m + 12 + — 2- m Tur 17. I_2a5 + 3x2-4a^ + 3a;^-2a75-f a;«. -V 18. 12a;^4-12a;-8af + 94-28a,-2 + a^ + 10aj3. ly. x^ xy ^ ^2x^4.a^ 20 a;^ ■«" 31 a.-^ 2.r 4 ■ 9 3"^ 60 5 "^25* 21. 4 a«+ 12 a'b + 25 a*62+ 4 a%'- 14 a^ft*- 40 06*+ 25 6«. - 22. - + _ + ^^ + — + -. 23. 28a.y 4. 9356 _ 15 ^^ - 12 ar^?/ - 8 xy' - 2a^/ + 16/. ' 9 3a Sa' a' '^ a' ' Find to four terms the approximate square roots of: ^25. H-4x. 27. 1-x. 29. a^-^6. 26. l4-2a. 28. l-3a. ^30. 4:a'-2b. SQUARE ROOT OF AN ARITHMETICAL NUMBER. ■ 196. The square root of 100 is 10 ; of 10000 is 100 ; etc. Hence, the square root of a number between 1 and 100 is between 1 and 10 ; the square root of a number between 100 and 10000 is between 10 and 100 ; etc. That is, the integral part of the square root of a number of one or two figures, contains one figure ; of a number of three or four figures, contains tivo figures; and so on. Hence, if a j)oint be placed over every second figure of any integral number, beginning with the units' jjlace, the number of points shows the number of figures in the integral part of its square root. 180 ALGEBRA. 197. Let it be required to find the square root of 4624 + 8 a-\-b d'-j-2ab-\-b^ = 4624 a' = 3600 120 + 8 = 2a-\-b go 4- 8 Pointing the number ac- cording to the rule of § 196, we find that there are two 1024 = 2ab-\-b figures in the integral part of lQ24 its square root. Let a denote the greatest multiple of 10 whose square is less than 4624 ; this we find by inspec- tion to be 60. Let h denote the digit in the units' place of the root ; then, the given number is denoted by (a + 6)^, or a^ + 2ab -{■ b^. Subtracting a^, or 3600, from 4624, the remainder is 1024. That is, 2a6 + ?)2 = 1024. (1) Since b^ is small in comparison with 2 ab, we may obtain an ap- proximate value of b by neglecting the b'^ term in (1). Then, 2ab = 1024, and b = 1^ = 1^ = 8 +. 2 a 120 This suggests that the digit in the units' place is 8. If this be correct, 2ab -\- b^, or (2 a + b)b, must equal 1024. Adding 8 to 120, multiplying the sum by 8, and subtracting the product from 1024, there is no remainder. Hence, 60 + 8, or 68, is the required square root. Omitting the ciphers, for the sake of brevity, and con- densing the operation, it will stand as follows : 4624 36 68 128 1024 1024 From the above example, we derive the following rule : Separate the number into periods by pointing every second figure, beginning with the units^ place. Find the greatest square in the left-hand period, and write its square root as the first figure of the root; subtract the square of the first root figure from the left-hand period, and to the result annex the next period. ji^VOLUTIOy. 181 Divide this remainder, omitting tJte last figure, by twice the j)art of the root already found, and annex the quotient to the root, and also to the trial-divisor. Multiply the complete divisor by tJie root-figure last obtained, and subtract the jyroduct from the remainder. If other periods remain, jyroceed as before, doubling the part of the root already found for the next trial-divisor. Note 1. It sometimes happens that, on multiplying a complete divisor by the figure of the root last obtained, the product is greater than the remainder. In such a case, the figure of the root last obtained is too great, and one less must be substituted for it. Note 2. If any root-figure is 0, annex to the trial-divisor, and annex to the remainder the next period. 198. Required the square root of 4944.9024. 1 We have, V4944.9024 = ^^■ ^^11:49 49449024 ^ V49449024 10000 Vioooo 49449024 I 7032 1403 4490 4209 14062 28124 28124 Since 14 is not contained in 4, we write as the second root-figure, annex to the trial-divisor 14, and annex to the remainder the next period, 90. (See Note 2, § 197.) 7032 Then, V4944.9024 = t2^^ = 70.32. 100 The work may be arranged as follows : 70.32 4944.9024 49 1403 44 90 42 09 14062 2 8124 2 8124 182 ALGEBRA. It follows from the above that, if a point he placed over every second figure of any number, beginning with the units^ place, and extending in either direction, the ride of § 197 may be applied to the result and the decimal point inserted in its proper position in the root. EXAMPLES. 199. Find the square roots of the following : 1. 4225. 2. 21904. 3. 508369. 4. 65.1249. 5. .156816. 6. .064516. 7. 3956.41. 8. 96.4324. 9. .00321489. 10. 12823561. 11. 75570.01. 12. .16216729. 13. 2666.6896. 14. .0062504836. 15. 86.825124. If there is a final remainder, the number has no exact square root ; but we may continue the operation by annex- ing periods of ciphers, and thus obtain an approximate root, correct to any desired number of decimal places. 16. Find the square root of 12 to four decimal places. 3.4641 +, Ans. 12.00000000 9 64 3 00 2 56 686 4400 4116 6924 28400 27696 69281 1 70400 Find the first five figures of the square root of: 17. 7. 20. 13. 23. .2. 26. .009. 18/ 8. 21. 48. 24. .056. 27. .00074. 19. 10. 22. 64.7. 25. .39. 28. SM4:5. EVOLUTION. 183 The square root of a fraction may be obtained by taking the square root of the numerator, and then of the denomi- nator, and dividing the first result by the second. If the denominator is not a perfect square, it is better to reduce the fraction to an equivalent fraction whose denomi- nator is a perfect square. 29. Find the value of \/- to five decimal places. Wehave, ^ =Jl = ^=^-^^^^= .612S7 -.^ Ans. Find the first four figures of the square root of: 30. ?. 32. ^. 34. 5. 36. ^. 38. ^. 4 25 5 32 12 33 25* ■^i- '-3^ 1 2* 3= J. -H 31. I 33.1. 35_7. 3,, 15. gg, 10. CUBE ROOT OF A POLYNOMIAL. 200. Since (a + bf = a^ + S a'b -f 3 aft^ -f b% we know that the cube root of a* -f 3 a^b + 3 a6^ -f 6* is a + b. It is required to find a process by which, when the expres- sion a^ -{- S a^b + 3 ab^ -{- b^ is given, its cube root may be determined. a^^Sa^b + Sab^' + b^ a-\-b 3a^-\-3ab-^b' 3a26 + 3a62 + 63 3a^b-\-3ab^-{-b^ The first term of the root, a, is found by taking the cube root of the first term of the given expression. Subtracting the cube of a from the given expression, the remainder is Sa^b + S ab'2 + ¥, or (3 a^ + 3 a6 + b'^)b. If we divide the first term of this remainder by 3 a^, that is, by three times the square of the first term of the root, we obtain the second term of the root, b. 184 ALGEBRA. . Adding to the trial-divisor 3 a&, that is, three times the product of the first term of the root by the second, and h^^ that is, the square of the second term of the root, we obtain the complete divisor, 3 a2 + 3 a6 + &2. Multiplying this by 6, and subtracting the product, Za^h+Zah'^-\-h^, from the remainder, there are no terms remaining. From the above process, we derive the following rule : Arrange the expression according to the powers of some letter. Extract the cube root of the first term, write the result as the first term of the root, and subtract its cube from the given expression; arranging the remainder in the same order of powers as the given expression. Divide the first term of the remainder by three times the square of the first term of the root, and write the result as the next term of the root. Add to the trial-divisor three times the product of the term of the root last obtained by the part of the root previously found, and the square of the term of the root last obtained. Multiply the complete divisor by the term of the root last obtained, and subtract the product from the remainder. If other terms remain, proceed as before, taking three times the square of the part of the root already found for the next trial-divisor. EXAMPLES. 201. 1. Find the cube root of 8x^-36x*y + 54:x'y'-27f. 8 ic6 _ 36 x^y + 54 x^y^ - 27 y^ 2x^-3y, Ans. 12x*-lSx^y-{- 9?/2 -36x4?/ + 54x2?/2-27 2/8 -36a:^y + 54a;2y2-27y8 The first term of the root is the cube root of 8 x^, or 2 x^. Subtracting the cube of 2ic2, or Sx^, from the given expression, the first term of the remainder is — 36 ic*y. EVOLUTION. 185 Dividing this by three times the square of the first term of the root, or 12 X*, we obtain the second term of the root, — 3 y. Adding to the trial-divisor three timeg the product of the term of the root last obtained by the part of the root previously found, or — 18 x'^y, and the square of the term of the root last obtained, or 9 y% we have the complete divisor, 12 x-* — ISx^y -\- 9y^. Multiplying this complete divisor by — 3 y, and subtracting the product from the remainder, there is no remainder. Hence, 2x^ — Zy is the required cube root. 2. Find the cube root of 28aj3 - 54a;4- a^ + 3a;* - 9a^ - 27 - 6a:*. Arranging according to the descending powers of x, we have x6_6a^-f 3a;4+28x3-9x2_54x-27 x2-2x-3. 3x*-63c8+4x2 6x^ 6x6+12x4- 8x8 3 X*- 12x8+ 12x2 - 9x2+18x+9 3x*-12x8+ 3x2+18x+9 - 9x*+36x8 - 9x4+36x8-9x2-54x-27 The second complete divisor Is formed as follows : The trial-divisor is three times the square of the part of the root already found ; that is, 3(x2 - 2 x)^, or 3x* - 12 x^ + 12x2. Three times the product of the term of the root last obtained by the part of the root previously found is 3( — 3) (x2 — 2 x), or — 9 x2 + 18 x. The square of the term of the root last obtained is ( — 3)2, or 9. Adding these, the complete divisor is 3x* — 12x8 + 3x2 + 18x + 9. The last five terms of the first remainder and the last three terms of the second remainder are omitted. Find the cube roots of the following : 4. l-12a3 + 48a«-64a». 6. 27m« + 135m*?i.+ 225?MV4-125w8. 6. 294a62_84a26-343 63 + 8al 7. 3:f^-6oc^-{-9x^-{-4:a^-9a^~6x-l. 186 ALGEBRA. 8. Sa^ -\-36a' + 66a' + 63a^ -\- SSd" -\- 9a -{-1. 9. 30 y' + 27/ + 12 y - 4.5y' - 8 - 35 f -\- 21 f, * 8 4 "^ 6 27' 11. 9a«-36a + a^+21a^-9a^-8-42a2. 12. 174 a;^ + 8 + 174 a;2 - 60 x' - 245 x^ + 8 a;^ - 60 x. 13. 27 a«~ 54 a«6 + 63 a'})"- 44 a^^^.^ 21 a^ft^- 6 a6^4- &'• 14. 6 a.-^^ + 96aj/+ my?f^- o^ + 24icy + 6^f + 96x2/. 15. £!-^V^-9 + ??-^ + ^. 27 33 X :^ x' CUBE ROOT OF AN ARITHMETICAL NUMBER. 202. The cube root of 1000 is 10 ; of 1000000 is 100 ; etc. Hence, the cube root of a number between 1 and 1000 is between 1 and 10 ; the cube root of a number between 1000 and 1000000 is between 10 and 100; etc. That is, the integral part of the cube root of a number of one, two, or three figures, contains one figure ; of a number of four, five, or six figures, contains two figures ; and so on. Hence, if a point he placed over every third figure of any integral number, beginning with the units' ijlace, the number of points shows the number of figures m the iyitegral part of its cube root. 203. Let it be required to find the cube root of 157464. a^^3a% + 3 ab^ + W= 157464 a^ = 125000 50 + 4 = a + 6 3a2=7500 3ab=: 600 b'= 16 3a2-h3a6 + 62 = 8116 32^64: = 3a'b-\-3ab'-\-b^ 32464 Pointing the number according to the rule of § 202, we find that there are two figures in the integral part of its cube root. EVOLUTION. 187 Let a denote the greatest multiple of 10 whose cube is less than 157464 ; this we find, by inspection, to be 50. Let b denote the digit in the units' place of the root ; then, the given number is denoted by (a + 6)^, or a^ + 3 a^b + 3 ab^ + b^. Subtracting a^ or 125000, from 157464, the remainder is 32464. That is, 3 a^b + 3 aft^ + 53 = 32464. (1) Since 3 ab^ and b^ are small in comparison with 3 a^b, we may obtain an approximate value of b by neglecting the Zab'^ and b^ terms in (1). Then, 3 a% = 32464, and 6 = §^464 ^ 32464 ^ 3a2 7500 This suggests that the digit in the units' place is 4. If this be correct, 3a^b + Sab^-{- b\ or (Sa^ + 3a6 -f b^)b, must equal 32464. Adding to 7500 3 a6, or 600, and 62, or 16, the sum is 8116 ; multi- plying this by 4, and subtracting the product from 32464, there is no remainder. Hence, 50 + 4, or 54, is the required cube root. Omitting the ciphers for the sake of brevity, and con- densing the operation, it will stand as follows : 54. 157464 125 7500 600 16 8116 32464 32464 From the above example, we derive the following rule : Sejmrate the number into periods by pointing every third Jigu7^e, beginning with the units' place. Find the greatest cube in the left-hand period, and write its cube root as the first figure of the root; subtract the cube of the first root-figure from the left-hand period, and to the result annex the next period. Divide this remainder by three times the square of the part of the root already found, ivith two ciphers annexed, and write the quotient as the next figure of the root. 188 ALGEBRA. Add to the trial-divisor three times the product of the last root-figure by the ^yart of the root previously found, with one cipher annexed, and the square of the last root-figure. Multiply the complete divisor by the figure of the root last obtained, and subtract the product from the remainder. If other periods remain, proceed as before, taking three times the square of the part of the root already found, with two ciphers annexed, as the next trial-divisor. Note 1. Note 1, p. 181, applies with equal force to the above rule. Note 2. If any root-figure is 0, annex two ciphers to the trial- divisor, and annex to the remainder the next period. 204. If, in the example of § 203, there had been more periods in the given number, the next trial-divisor would have been three times the square of a +5, or 3a^+6a6-h36^. We observe that this may be obtained from the preceding complete divisor, 3 a^ -f 3 a6 + b^, by adding to it its second term, 3 ab, and twice its third term, 2 W. Hence, if the first number and twice the second number required to complete any trial-divisor, be added to the comr plete divisor, the result, with two ciphers annexed, will be the next trial-divisor. 205. Required the cube root of 8144.865728. We have, ^ 8144.865728 = ^ ^^^^^^^^^^ = ^^1^^^^^^^^ . ^ 1000000 ^1000000 8144865728 1 8 1 120000 144865 600 1 120601 120601 600 24264728 2 12120300 12060 4 1213236 4 24264728 2012 EVOLUTION. 189 Since 1200 is not contained in 144, the second root-figure is ; we then annex two ciphers to the trial-divisor 1200, and annex to the remainder the next period, 865. The second trial-divisor is formed by the rule of § 204. Adding to the complete divisor 120601 the first number, 600, and twice the second number, 2, required to complete the trial-divisor 120000, we have 121203 ; annexing two ciphers to this, the result is 12120300. Then, \/8144. 865728 =?^ = 20.12. 100 The work may be arranged as follows : 20.12 8144.865728 8 120000 600 1 120601 144 865 120 601 600 2 24 264728 12120300 12060 4 12132364 24 264728 It follows from the above that, if a point he placed over eve^-y third figure of any number , beginning with the units' place, and extending in either direction, the rule of § 203 may be applied to the result, and the decimal point inserted in its proper position in the root. EXAMPLES. 206. Find the cnbe roots of the following : 1. 19683. 2. 148877. 3. 59.319. 4. .614125. 5. 2515456. 6. 857.375. 7. .224755712. 8. 46.268279. 9. 523606616. 10. 187149.248. 11. .000111284641. 12. 788889.024. 13. 444.194947. 14. 338608873. 15. .001151022592. 190 ALGEBRA. Find the first four figures of the cube root of : 16. 3. 18. 9.1. 20. ^. 22. -. 8 9 17. 7. 19. .02. 21. ii. 23. -• 27 5 207. If the index of the required root is the product of two or more numbers, we may obtain the result by successive extractions of the simpler roots. For by § 189, fVa)'"" = a. Taking the nth root of both members, C-Var=^a. (1) Taking the mth root of both members of (1), mn/— ^/ n/— ■\a= V S/a. Hence, the mnth root of an expression is equal to the mth root of the nth. root of the expression. Thus, the fourth root is the square root of the square root ; the sixth root is the cube root of the square root, etc. EXAMPLES. Find the fourth roots of the following : 1. 81 a^ + 216 a^y + 216 a^ft^ + 96 aW + 16 h\ 2. l-12aj + 50a^-72a^-21a;*4-72a^+50aj6+12aj^+a^. 3. 16 a«-32 a^-40 a«+88 a'-\- 49 a^-88 a^-4:0 a2+32 a+16. Find the sixth roots of the following : 4. ici« 4- 6 x^Y + 15 ^'y + 20 ^f + 15 a;y + 6 ^tf + y'^ 5. a«-12a^ + 60a^-160a3 + 240a2-192a4-64. 6. Find the fourth root of 209727.3616. 7. Find the sixth root of .009474296896. THEORY OF EXPONENTS. 191 XX. THEORY OF EXPONENTS. 208. In tlie preceding chapters, an exponent has been considered only as a positive integer. Thus, if m is a positive integer, aJ^ = axaxay. • • • to m factors. (§ 6) 209. Let m and n be positive integers. Then, «"' = axaxax •••to m factors, and a"" = a K a xax • • • to n factors. Whence, cr x a'* = a x a x a x • • • to m -\- n factor^. That is, or x a" = a'"+". (1) This proves the law stated in § 46 for all positive integral values of the exponents. Again, (a*")" = a*" x a"* X a"* x • • • to n factors ^m+m+m+—to n terms That is, {ory = a""*. (2) This proves the first paragraph of the law stated in § 186 for all positive integral values of the exponents. 210. It is found convenient to employ exponents which are not positive integers; and we proceed to define them, and to prove the rules for their use. It will be convenient to have all forms of exponents sub- ject to the same laws in regard to multiplication, division, etc. ; and we shall therefore find what meanings must be attached to fractional, negative, and zero exponents in order that equation (1), § 209, may hold for all values of m and n. 192 ALGEBRA. 211. Meaning of a Fractional Exponent. 1. Required the meaning of aj^ If (1), § 209, is to hold for all values of m and n, we have 5 5 5 5 I 5 ,5 Hence, a^ is such an expression that its third power is a^. Then/ a^ must be the cube root of a^ ; or, a^ = -v/o^. p 2. Required the meaning of a', where p and q are any positive integers. If (1), § 209, is to hold for all values of m and n, we have p p p p p p p a^ xa^ X a^ X ••• to g factors = a^'^^'^^'^'"^''^^""''= a^""" = a^. p Hence, a^ is such an expression that its qth power is a^ p p Then, a* must be the gth root of a^ ; or, a' = va^. Hence, in a fractional exponent, the numerator denotes a power, a7id the denominator a root. For example, a^ = -\/'a^; 6^ = V6*; x^ = -y/x; etc. EXAMPLES. 212. Express the following with radical signs ; 1. a^. 3. 4.x\ ^-5. aM. ^7. 6x^yK 9. abh'd'^. m 1 q _ 2. b\ 4. 9 abK 6. m%i 8. 8 aW. ^0. 3 x^^^:^^. Express the following with fractional exponents : 11. v^3. 13. Vm^ 15. 2-^«. 17. ^/^^/6"^ 12. ~l 11. 6m-2n"V- 4. 2aV9. 8. a"^6-V. ^12. cj-^^-Vto. Transfer the literal factors from the denominators to the numerators in the following : 1/. — i». „ — 18. -iCl_. 20. ^^lM- 2/3^ 7 m-^a?^ 4 h'^c 8 6~^?/^' Transfer the literal factors from the numerators to the denominators in the following : /^ 21. — . 23. ^- 25. '^^. 27. ^^l!^. 2^ 5 x^2/~ 13. 1 15. 2 14. a;-t 16. 1 2 3 6* 24. 6 22. ^. 24. ^^i^. 26. ^l!^. 28. ^^"'^l 2/"* 96-^71 3 217. Since the definitions of fractional, negative, and zero exponents were obtained on the supposition that equation (1), § 209, was to hold universally, we have for all values of m and n or xa'' = «"*+". For example, a^ x a~^ = o?~^ = a~^ ; a^ X a~^ = a^~^ = a^^ ; a X Va* = a X a^ = a^^^ = a^ • etc. THEORY OF EXPONENTS. 195 EXAMPLES. Find the values of the following : 1. a' X a-\ 4. m^ X m i 7. 5 ic-2 X 4 a; i 2. o? X a-^. 5. 2 J ■ X n"i 8. m^ X ^/m. 3. x-^ X x-^. 10. n-^x-^ n ^ 6. a X 3a-V 9. c^x^y^^. 13. £c-«/ X ^x'y-\ 14. m^w"^ X \ m-*n~K 11. iVa'x^ 12. 3a\/Fx 15. ^ X a-'x-K a-'x^ 16. Multiply a + 2 a^ - 3 a^ by 2 - 4 a"^ - 6 a"i 2 _4a"^- 6a~^ 2a + 4a^ - 6a^ 2 a -20a^ + 18a"^ Ans. Note. It must be carefully borne in mind, in examples like the above, that the zero power of any quantity is equal to 1 (§ 213). Multiply the following : 17. J + ah^ 4- b^ by a^ -bK 18. 4a;"^ -6a;"^ + 9 by 2a;~^ + 3. 19. 2a-i-7-3a by 4a-i4-5. 20. x~^ + 2 a;"^ + 4 rc"^ 4- 8 by x~^ - 2. 21. x^ -{■ xhj^ -\- y^ by x^ — x^y^ + y^- 22. m — 2 m^w"^ + m^n~^ by m^n~^ — 2 m^w"^ + w"». 196 ALGEBRA. 23. a~25-3 _^ ^-3^-4 _ ^-4^-5 by ^-15-2 _ ^-2^-3 _ ^-35-4_ 24. m-^4- 2 m"3n-i+ 3 ??r^n-2 by 2 /?r^— 4 n"^ + 6 m^-^. 25. 2 ah-' + a^ - 4 ^-^6^ by 2 a^ - 5^ _ 4 ^-^54 01 21 12 11 ^ 26. 3m%3_477ia;3-f-m^ic by 6m^ic~3 + 8m~%~3-|-2m~*. 218. To prove — = a*^"" for all values of m and n. By § 215, - = a*" X a-" = a*"-", by (1), § 209. For example, -I. a a" = 2Ll = a-^i a"'^; etc. EXAMPLES. Divide the following : 1. a^ by a\ 2. ic by x^. 3. m^ by ?)i~». 2 1 4. a 3 by a^. 5. 6-2 by V6^. 6. 2Vx by ic" 1 7. n^ by ^n^ 8. lOa-^6-^ by 5 a^h'\ 9. 6 by 2^ 3. {x^f. ■ 9. (a-)l t^ 4- (--¥. 10. (^)e ''' ^^ ">" ^ , 4 11 f - 1 !?-i -!!L 6. (a-^"^. ' W * 16. (a;- )"--. 221. The value of a numerical quantity affected with a fractional exponent may be found by first, if possible, ex- tracting the root indicated by the denominator, and then raising the result to the power indicated by the numerator. 1. Find the value of (- 8)1 We have, (- 8)^ = [(- 8)^]2 = (v/38)2 = (_ 2)2 = 4, Ans. EXAMPLES. Find the values of the following : 2. 25l 6. 49"^ 10. 16-1 14. 32-1 3. 9l 7. (-27)-^ 11. (-32)1 15. (-64)1 4. 8l 8. 4i 12. 64l 16. (-243)^ 5. Sll 9. 343l 13. (-125)i 17. (-128)-^ U' THEORY OF EXPONEXTS. 199 222. To j)rove {ahy = al'h'^ for any value ofn. I. Let n be a positive integer. Then, (aby = ab x ab x ab x -" to n factors = (a X a X • • • to 71 f actors)(6 x & X • • • to n factors) = a"6". V) II. Let n = -, where p and q are positive integers. Then by § 219, [(aft)']' = (aby = a^b^, by § 222, I. And by § 222, I., [a^b^y = (Jy(b^y = a^b^. Therefore, [(a^)']' = [a'6']'. Taking the ^th root of both members, we have p p p {aby = a^¥. III. Let w = — s, where s is any positive number. Then, (a6)-=-i-=: J- (§ 222, L or IL)=a-6-. ^ {aby a'b' ^ ' ^ MISCELLANEOUS EXAMPLES. 223. Square the following by the rule of § 78 or § 79 : 1. 2a? + 36i 2. 5x-Y-2a^y-'. 3. 3aV^-4a"V. Extract t'he square roots of the following : 4. a-*bi 5. 25mn-U 6. ^^. 7. ^!^. 8. 4a* + 4a*-19-10a-^ + 25a-i 9. 9a;"t_12a;-2 + 10a;-^-4a;"^4-l. 10. a'b-^ - 8 ah'' + 10 a'b-' + 24 ah'' + 9 ab-\ 200 ALGEBRA. Extract the cube roots of the following : 11. a^h-\ 12. xy^z-^. 13. -27mV^. 14. ^^. 15. 8a4-12aV^ + 6aV3-6-^. 16. x-^ + 6a;-* + 3a;* - 28aj^ - 9a;^ + 54a;^ - 27a;l Simplify the following : ^2»»— 3n sy «— Sm-n 1 1 17. ^-^^ 20. lirf-^f^K 1 1 1 _*_ ae-y x+y 19. (a;^i X a;^)". 22. (a=^+^^a ^ ) " . 23. (2"+-^ - 2 X 2«) X (2-2 X 2-"-2). 24. (a;^r"'^^^(|y- 25. (c.*-a*y+(l + a^*a*y ^V 1 + a*a;*(a~*a;* - a^x~^ - a^x^) ^ 26. A±l_4._£zil_. 28. a;-^ - 2/"^ ^~^ + ^"1 a;i_2/i a;^H-2/^ ' a;"* + 2/"* a;"* - 2/~* 3n _3» 3n 27. ^'-^ ' 29. ^^-1 a^"-l . n M n n 30. ^' + y-\^-'-y-'_x, 31 <^^ + 2 6^ a^-2a^6^H-4 6^ a^ -2h^ a* + 2 a^ 6* + 4 6^ .^^ RADICALS. 201 XXI. RADICALS. 224. A Radical is a root of an expression, indicated by a radical sign ; as Va, or V» + 1. If the indicated root can be exactly obtained, the radical is called a rational quantity ; if it cannot be exactly obtained, it is called an irrational quantity, or surd. 225. The degree of a radical is denoted by the index of the radical sign ; thus, -\/x + 1 is of the third degree. 226. Most problems in radicals depend for their solution on the following principle : 111 For any value of n, (abf = a" x 6«. (§ 222) That is, -\/ab = -s/d x ^/b. REDUCTION OF A RADICAL TO ITS SIMPLEST FORM. 227. A radical is said to be in its simplest form when the expression under the radical sign is integral, is not a perfect power of the degree denoted by any factor of the index of the radical, and has no factor which is a perfect power of the same degree as the radical. 228. Case I. When the expression under the radical sign is a j)erfect power of the degree denoted by a factor of the index. 1. Reduce VS to its simplest form. We have, v/8 = \^ = 2^ (§ 211) = 2^ = \^, Ans. EXAMPLES. Reduce the following to their simplest forms : 2. -im. 3. -^. 4. ^25. 5. ^^64. m ^4. 9. 10. 11. ^216. ^100. ^243. ALGEBRA. 15. 16. 17. 6. 12. = /l5, Ans. Reduce the following to their simplest forms : 27. VSei. 30. V125 X 135. 33. -J/4ll6. 28. y/2625. 31. V98 x 336. 34. ^196 x 392. 29. V3528. 32. ^/1125. 35. -v/40 x 45 x 48. 36. V75a=^xl05a6 x 189 61 230. Case III. When the expression under the radical sign is a fraction. In this case, the radical may be reduced to its simplest form by multiplying both terms of the fraction by such an expression as will make the denominator a jierfect jyotver of the same degree as the radiccd, and then proceeding as in § 229. 1. Reduce \-^—^ to its simplest form. 'Oft Multiplying both terms of the fraction by 2 a, we have /X = J9xI^^J_0_x2« = JIIIxV2^=^V2^,^ns. EXAMPLES. Reduce the following to their simplest forms : '■4 ^-4 -4- '-4 204 • ALGEBRA, W| "^ "4 ^'-^'^■ 'a/I "€ "€ "x'ff- 8. # 12. 4 16. Jl. » Jf. 9. /i- 13. 4 17. V^. 21. 3 21 3/ 50^ 10 or^' ' ^16f' 22 - f"- -r ^ o«5 - . /2 a^ - 8 « + 8 I. J^H. 23. ^^ J 231. To Introduce the Coefficient of a Radical under the Radical Sign. The coefficient of a radical may be introduced under the radical sign by raising it to the power denoted by the index. 1. Introduce the coefficient of 2aV3x^ under the radical sign. , 2 aVSx^ = \/8a3^3x2 = y/Sa^ xS x'^ (§ 226) = v/24 a^x'^, Ans. Note. A rational quantity may be expressed in the form of a radical by raising it to the power denoted by the index, and writing the result under the corresponding radical sign. . EXAMPLES. Introduce under the radical signs the coefficients of : 2. 5V2. 5. 5^/4. 8. 4aV8a. 11. ^f^^y\ 3. 8V3. 6. 2^5.- 9. 7x'^/6a^. 12. 3m2'23x33 A'23x3~>24* 2. Divide VlO by \/40. We have, vlO = 10^ = 10^ = VW = y/(2 x 6)8. -471S. Whence, :^ = ^?i2LM = ^ = 5I = 5* = ^, 4,«. Divide the following : 3. V84 by V7. 6. 4. Vl2 by Vl5. 6. 9. ^8l by ^9. 10. -y/W^ by ^39a3. 11. '=(V^)'x V-1 = 1 xV^=V^; etc. Thus, the first four positive integral X30wers of V — 1 are V— 1, —1, — V— 1, and 1; and for higher powers these terms recur in the same order. 251. Addition and Subtraction of Imaginary Numbers. Imaginary numbers may be added or subtracted in the same manner as other radicals. (See § 233.) 1. Add V^4 and V-36. V34 4- ^/33B = 2V^n. + 6 V^^ (§ 249) = 8 V'^l, Ans. EXAMPLES. Simplify the following : 2. V^Ii6+V^r25. 7. V^+V^^^-V=^. 3. V^+V^^27. 8. V^36-V=300 + V-81. 4. V-18-V-8. 9. V-a2-V-4a2-V^^9^ 5. V^ra2-V-(a-6)2. 10. V-20-h V^-45- V-^. 6. V-aj2-f-V-y'4-V^2'. 11. V-l-2a;-iB2-V-4^. 252. Multiplication of Imaginary Numbers. The product of two or more imaginary square roots may be found by aid of the principles of § § 249 and 250. 1. Multiply V^^ by V^^. Vir2 X V^^ = V2 V^n[ X VS V^ (§ ii49) = v^ V3(V^)2 = V6 (-!)(§ 260) = - V6, Ans 220 ALGEBRA. 2. Multiply together y/~9 X V^n^ X V^25 = SV^n; X 4\/^l X 5y/^^ = 60(\/^=T)3 = 60(- V^T) (§ 250) 3. Multiply 2 V^^ + V^^ by V^^ - 3 V"^. y/^2 - 3 V^=^ 2(-2) + V2 V5(V^nr)2 - 6\/2 V5 ( V^^)2 -3(- 5) 4-5\/l0(-l) + 15 = 11 + 5\/l0, ^ns. Note. It should be remembered that to multiply an imaginary square root by itself simply removes the radical sign ; thus, V-2 x\/-2 2. EXAMPLES. Multiply the following : 4. V~^^4^ by V^^. 8. - V^^by V-12. 5. V- 9^2 by -^-16ai 9. - V-72 by - V-50. 6. V^=^ by V^=l0. 10. 2-5V^ by 3+4V^. 7. -V'^T^ by -V^=~&. 11. 8+-V^by 7-5V^^. 12. 4V^=^ - 7 V^^ by 2V^=^ - V^^. 13. 2 V^^ + 3 V^^ by 4 V^^ - 6 V^=^. 14. V— aj^, V— 2/^ and V—z^. 15. _8, _V-18, and V-32. "3+V^^. 16. V- 10 + 5V- 5 by 3 17. 2+V-3 by V^r8-V-12. RADICALS. 221 18. V-1, V^36, V-04, and V- 100. 19. V^"2, V^ -V"^ and V^^. Expand the following by the rules of §§ 78, 79, or 80: 20. (H-V'=^)2. 23. (2^/^^-SV^^y. 21. (5V^^ + 2V^'. 24. (a;+V^)(a;-V^). 22. (4-V^l 25. (5 4-6V^n:)(5-6V^). 26. (3V^^ + 2V^(3V^^t-2V^. 27.. (7V^=^ + 4V^^)(7V^^-4V^. ^ 28. (V:r2 + ^v^-+(V^=^-2V'^)^ Reduce each of the following to an equivalent fraction having a rational denominator : 5 V^^ + 4 V^^ 29. 30. 1+V^^ 1-V^ 31. 32. 5V-2-4V^^ 3V^=^ -h 2 V^Ts Expand the following by the rule of § 188 : 33. (1-hV'^^)^ 34. (2-V^^ 253. Division of Imaginary Numbers. 1. Divide V^^^^ by V^^. We have, ^^31 = VT2x^ ^ Vl2 ^ ^ ^ 2, Ans. v^=r3 vs V- 1 V3 2. Divide VlO by V- 2.' Vio _ - VTo (-!)_- VIo ( V^ )-^ V-2 V2\/^ \/2V-l 250) = ->/5>/-l; 222 ALGEBRA. EXAMPLES. Divide the following : 3. V^^^25 by V^=^. 8. -Va by V^=^l 4. V^=r32 by V^^. 9. x' + ^ = -XXx, 228 . ALGEBRA. 260. If the coefficient of o? is a perfect square, it is con- venient to complete the square directly by the principle stated in § 258; that is, by adding to both members the square of the quotient obtained by dividiyig the coefficient of x by twice the square root of the coefficient of x^. 1. Solve the equation 9 a;^ — 5 a: = 4. Dividing 5 by twice the square root of 9, the quotient is f . Adding to both members the square of f , we have \Ql 36 36 5 IS Extracting the square root, 3 a: — = ± — Transposing, 3x = - ± — = 3 or - -• 6 6 3 4 Whence, a; = 1 or — , Ans. 9 If the coefficient of a^ is not a perfect square, it may be made so by multiplication. 2. Solve the equation 8 a^ — 15 a; = 2. Multiplying each term by 2, 16 x^ — 30 a: = 4. Dividing 30 by twice the square root of 16, the quotient is -3/, or J^^.. Adding to both members the square of Y-, we have I 4 / 16 16 A/ 4'~^T y i/ Extracting the square root, 4 x = ± — '^ Transposing, 4x = — ± — = 8or 4 4 2 Whence, x = 2 or — , Ans. Note. If the coefficient of x^ is negative, the sign of each term must be changed. QUADRATIC EQUATIONS. 229 EXAMPLES. Solve the following equations : 3. 4:x' + 7x = 2. 10. 49af'-7a- = 12. .4. Wx'-\-S2x = -15. ^11. 25a.-2-f 25.^•^-6 = 0. 5. 9i«^-lla; = -2. 12. 120^ + 8a; = - 1. 6. Sx'-j-2x = 3. 13. 32ic2^i^_12a... 7. 5x'-{-16x = -3. 14. 28 + 5a;-3iB2^0 8. 36ar^-36cc = -5. 15. x' + l = 20ar'. 9. 64a;2^48^.^7 jg 4 + 3a;- 27a.'2 = 0. 261. Second Method of Completing the Square. Every affected quadratic can be reduced to the form aa^ -\-bx = c. Multiplying both members by 4 a, we have 4 a V 4- 4 abx = 4 ac. Completing the square by adding to both members the squai-e of - — ^^ (§ 260), or b, we obtain 4aV + 4 abx + b^=^b^ + 4ac. Extracting the square root, 2 oa; + 6 = ± VfeM^Toc. Transposing, 2ax = — b± V6^ + 4 ac. Whence, ^ = :^^±V6^ + 4 2 a , From the above example, we derive the following rule : Y^ Reduce the equation to the form ax^ + bx = c. Multiply both members by four times the coefficient of a^^ and add to each the square of the coefficient of x in the given equation. 230 ALGEBRA. Extract the square root of both members, and solve the simple equation thus formed. The advantage of this method over the preceding is in avoiding fractions in completing the square. 262. 1. Solve the equation 2ic^ — 7 a; = — 3. Multiplying both members by 4 x 2, or 8, 16x2-56x=-24. Adding to both members the square of 7, we have 16a;2 _ 56x + 72 = - 24 + 49 = 25. Extracting the square root, 4 a; — 7 = ± 5. 4x = 7±5 = 12 or 2. Whence, x = 3 or ^, Ans. If the coefficient of x in the given equation is even, frac- tions may be avoided, and the rule modified, as follows : ■ Multiply both members by the coefficient of x^, and add to each the square of half the coefficient ofx in the given equation. 2. Solve the equation 15 aj^ + 28 x = 32. Multiplying both members by 15, 15%2+ 15(28 x)= 480. Adding to both members the square of 14, we have 152x2 + 15(28 x) + 142 = 480 + 196 = 676. Extracting the square root, 15 x + 14 = ± 26. 15x=-14±26=:12 or -40. 4 8 Whence, x = - or - -, Ans. 6 EXAMPLES. Solve the following equations : 3. 0^-7 a; = 30. 6. 8a;2 + 14a; = -3. 4. 2a^-f-5a; = 18. 7. 10a^+7a; = -l. 5. 3aj2-2a; = 33. 8. ^x'-2x = 12. QUADRATIC EQUATIONS. 231 9. 4a^-7x = -3. 14. 6a^4-17ic = - 10. 10. 6a^-llaj=10. 15. 5 x' -\- 15 = 2S x. 11. 4a^-f 24aj + 35 = 0. 16. 9 a.-^ = 32 a; - 15. 12. 4:x + 4. = 15x'. 17. 3-5a;-12x2 = 0. 13. 4-15a;-4ar^ = 0. 18. 9ar^ -f 15a; + 4 = 0. MISCELLANEOUS EXAMPLES. 263. The following equations may be solved by either of the preceding methods, preference being given to the one best adapted to the example under consideration. 3 1 13 ^ 1 Sx" 24a; 2* 2. ^-A = — ^. 4. ? + ^=-??. 4^a; 10 5. (3 a.' -f- 2) (2 a; 4- 3) = (a; -3) (2 a; -4). 6. 9(aj-l)2-4(a;-2)2 = 44. 7. 4(a5-l)(2aj-l) + 4(2a:-l>(3a;-l) + 4(3x-l)(4a;-l) = 53ajl 8 ?2__M_ = i —10 a; + 2 4-a; ^7 ' X x-\-l ' . ' x-l 2x 3* 9 _3 ?_=i 11 a; + l 2a; + 1 ^17 a;-6 a;-5 ' ' 2a;-|-l 3a; + l 12* 12. (2aj + l)2-(3.'c-2)2-(a;-f 1)^ = 0. 13. V6 + 10a;-3ar^=2a;-3. 17 a; x" - 6 ^6 00^ 11 3 3(a; + 4) x 14 2 — 3a; 4 — a; _ll ^ ^ 4 a;_2~4 18. Va; 4-2-1- V3a; + 4 = 8. 15. (.-3)3- (.-,2)3= -65. ^^ vi2^ 3 x' 3 X 2" 35 6' 1 2' 5 6.2 = - 7 12a; 16. Va;-lH-V3a;-|-3 = 4. 5 2-hVl2-a; 232 ALGEBRA. *^ 21 1 I ^ '^ I ~_^ =::0 ■ 3a.'4-l (3a; + l)(7a;-f 1) 22 1 4 ^2 • a;_6 3(a?-l) 3a; 4i»-3 2» ~ * 2^ a; + 2 , a; - 2 ^ 6 a; +16 a;_2'^a; + 2 3a; 12 26. V5 + a;+V5- V5 — a; 26. Va; + 3-Va; + 8 = -5Va;. 27. ^.+ 1 1 ' 28. 3 l_a;2'l + a; 1-a; 8 13 5 a;-h2 2(2 a; -3) (a; + 2) (2 a; - 3) 29 ^^ ~ ^ _ ^ ~4a ; _ o 7 — X 2a; + 1~ 3a;-6 7 ll-2a; 30. 5 -a; 2 2(5 -2a;) gj 3-2a; 2 + 3a; ^l 16a; + ay^ 2 + a; 2 -a; 3 . a;^ - 4 ' 32 ^ + ^ _L ^ + ^ ^ 2 a; + 13 a; — 1 a; — 2 a; + l 33. V2^+9^T9+V2^T7^T5 = V2. - 35 34. V3x + 1 - V4a; + 5 + Va; - 4 = 0. 1 15 a; (3a;+l)(l-5a;) 2(1 - 5 a;) (7 a; +1) (3a;+l)(7a;+l) 36 Va; _ Va; + 2 _ 5 Va; + 2 Vi ^ QUADRATIC EQUATIONS. 233 37. Vo - 2X + V15 - Sx =^/26 - 5x. x^8 x-1^ x-2 264. Solution of Literal Quadratic Equations. For the solution of literal affected quadratic equations, the methods of § 262 will be found in general the most convenient. 1. Solve the equation x^ -\- ax — bx — ab = 0. The equation may be written rc2 -f- (a - b)x = ah. Multiplying both members by 4 times the coeflBcient of x^, 4ic2 + 4(a - b)x = Aah. Adding to both members the square of a — ft, 4 a;2 + 4 (rt - 6)x + (a - 6)2 = 4 a6 + a2 - 2a6 + 62 = a2 + 2 a6 + 62. Extracting the square root, 2x4-(a-6) = ±(a + 6). 2«=-(a-6)±(a + 6). Therefore, 2x = -a + 6 + a + 6 = 2&, or 2x = -af6-a-6 = -2a. Whence, x = 6 or -a, Ans. Note. If several terms contain the same power of x, the coeffi- cient of that power should be enclosed in a parenthesis, as shown in Ex. 1. 2. Solve the equation {m — V)y? — 2 m^x = —4:m^. Multiplying both members by m — 1, (m - 1)2x2 - 2 m^{m - l)x = -4 m2(w - 1). Adding to both members the square of m^, (m - 1)2x2 r- 2 m2(m - l)x + m* = m* - 4 m^ + 4 m^. 234 ALGEBRA. Extracting the square root, (m — \)x — mP- = ± {m- — 2 m). {m — l)x = m2 + m^ -- 2 m or m^ — m^ + 2 »7i = 2 77i(m — 1) or 2 m. Whence, a: = 2 m or — ^, ^?is. m — 1 EXAMPLES. Solve the following equations : 3. aj2-4aa; = 962_4a2. 6. x^ -\- ax -{- bx + ah = i). 4. a;^ + 2 mx = 2 m + 1. 7. a^ — m^ic — m%* = — m'\ 5. x^ — (« — l)aj = a. 8. aco;^ + hex — ada; = 6d. 9. x'-2ax-12x = Sa^-\^a-^^. 10. (a-5)a;2-(a + 6>' = ~26. 11. (a - x)(a' + 6^ _^ ax) = a^ + 6a^. 12. l+g ^ l-g ^-^ 13^ _^_4._fL_=2. 1 — cia; 1 -\- ax x — a x — b 14. (aj4-2a)«-(a.'-3a)^ = 65a^ 15. (l-a^(x-{-a)-2a(l-xF)=0. 16. V(a-26)x + 8a6 = a;H-46. 17. 6a^-(5ct + 5)aj = -a2_a6 + 262. 18. Vx-{-7a-{-Vx — 2a=-V5x-\-2a. -Q ic — aaj + a 5aa; — 3a — 2_ IS. ; 1 5 -g — U. .T-f-a ic — a a^ — ar 20. Vx-12a6 = ^^'~^' . 21 ^!_±1 ^ 2(aMi&!)^ 22.^±^ + ^-^ = -. a? o? — b^ ' ' x-\-b x-\- a 2 23. V2a;-4a + V5ajH-3a 3a; V2 a; — 4 a QUADRATIC EQUATIONS. 235 24. x^-(a-b)x = (a-c){b-c). 25. V3 a; + 2 a — V4 ic — 6 rt = v2a. 26. f^^Y-7f^^^Vl2 = 0. rtiy a; 4-V12a — a; _ Va + 1 X — Vl27t — a; Va — 1 28. (a-^/)aj2 + (6-c)a; + (c-a)=0. 29. (a*-l)x'-2(a*-\-l)x = -a*-{-l. 30. eiJLl = .^Jzi + _^. 31. -l- + _i- = l + l. x— a a; — /> a b 32. (c + a - 2 6) ar2 -f (a + ^ - 2 c) a; + /^ + c - 2 « = 0. 265. Solution of Quadratic Equations by a Formula. It was shown in § 201 that, if a^ + bx = c, then ^ ' ^^ -6±V6^4-4ac c^ y J (^) This result may be used as a formula for the solution of any quadratic equation in the form oar^ + bx = c. 1. Solve the equation 2ar^ + 5a; = 18. In this case, a = 2, 6 = 5, and c = 18 ; substituting in (1), 5±V25Tig ^-5±Vl69^-5i:l3^^ ^^ _9 ^^^^^ 4 4 4 2 2. Solve the equation 110 a?^ - 21 a^ = - 1. In this case, a = 110, 6 = - 21, and c = - 1 ; substituting in (1), ^^21±V441-440^2_^±_1^^ ^^ 1 ^^^ 220 220 10 11 Note. Particular attention must be paid to the signs of the coeffi- cients ill making the substitution. 236 ALGEBRA. EXAMPLES. Solve the following equations :. 3. 2x^-{-x=G. 4. x'-5x = 36. 5. a^-\-Ux-{-4:S = i). 6. 5x'-13x = -0. 7. 6x''-x = 5. 9. 4:X^-21x = -27. 10. 280^2 -hl6.^' = -l. 11. Sa^-\-4:lx-\-5 = 0. 12. 16a-24- 16.1'- 5 = 0. 13. 30a?-8 = 25a;2. 14. 12x^-^7 = -25x. 15. 2-3a;-54.T2=:0. 16. 3 + 14a;-24a^ = 0. 266. Solution of Equations by Factoring. Let it be required to solve the equation (x-3)(2x-^5)=0. It is evident that the equation will be satislied when x has such a value that one of the factors of the first member is equal to zero ; for if any factor of a product is equal to zero, the product is equal to zero. Hence, the equation will be satisfied when x has such a value that either a; - 3 = 0, (1) or 2x-\-5 = 0. (2) 5 Solving (1) and (2), we have x = S or It will be observed that the roots are obtained by placing the factors of the first member separately equal to zero, and solving the resulting equations. 267. 1. Solve the equation y?—Bx — 2i = 0. Factoring the first member, (a: - 8) (cc + 3) = 0. (§ 100) Placing the factors separately equal to zero (§ 266), we have X - 8 = 0, and x + 3 = 0. Whence, a; = 8 or - 3, Ans. • QUADRATIC EQUATIONS. 237 2. Solve the equation 2 a^ — a; = 0. Factoring the first member, x(2x - 1) = 0. Placing the factors separately equal to zero, X = 0, and 2 X - 1 = 0. Whence, x = or -, Ans. 3. Solve the equation y? -\- \q^ — x — \ = ^. Factoring the first member, (x + 4) (x^ - 1 ) = 0. (§ 93) Therefore, x + 4 = 0, and x'^ — 1 = 0. Whence, x = — 4 or ±1, Ans. 4. Solve the equation or^ — 1 = 0. Factoring the first member, (x - 1) (x^ + x + 1) = 0. (§ 103) Therefore, x - 1 = 0, and x^ + x + 1 = 0. Solving the equation x — 1 = 0, we have x = 1. Solving the equation x^ -f x + 1 = 0, we have EXAMPLES. Solve the following equations : 5. a.'2 + 3a;-28 = 0. 10. 3 a.-^ + 24 .r^ = 0. 6. aj2_i4a;-|.45 = 0. 11. 16a^-9a;=0. 7. .T2_|_lia;^24 = 0. 12. (2a; + 5)(9ar^-49)=0. 8. a;2-6a;-72 = 0. 13. \2^ -1 :^ -\^x = f). 9. 5a;2-7a; = 0. 14. (.^•2- 8)(.'c24-4)= 0. 16. {x - 3)(2 ^ -f- 13 a; + 20) = 0. 16. {x - 3)(a; + 4)(a; _ 5) - 60 = 0. 17. (ar»-9a2)(2a;2 + aa;-a2)=0. 238 ALGEBRA. 18. x« + l = 0. 22. 8a^ + 125 = 0. 19. a^-27 = 0. 23. x« - 64 = 0. 20. 16 aj^- 81 = 0. 24. ^xf - a^ -\-x -1 = 0. 21. 27a^-64a3 = 0. 25. ^x" - ^2x -{-1 = x -1. 26. 5a^-a;2- 125a; + 25 = 0. 27. 8x3_^20a;2-18a;-45 = 0. 28. 4:a^+5x'-\-72x-{-90 = 0. 29. Va -\-x -\- V" oii nd substituting tliift,. a; = 4 or — 2. ' "' Subtracting (3) from (6), 2 ?/ = ± 6 - 2 = 4 or - 8. Whence, ?/ = 2 or — 4. X = 4, y = 2; or, x = — 2, ?/ = — 4. ^ws. Note 3. The above equations are not symmetrical according to the definition of § 272 ; but the method of Case III. may often be used in cases where the given equations are symmetrical except with respect to the signs of the terms. x'-^-f-^ 50. (1) xy = -7. (2) Multiplying (2) by 2, 2xy = - 14. (3) Adding (1) and (3), x'^-^2xy + y^ = 36. Whence, x-{- y = ±Q- (4) Subtracting (3) from (1), x^ -2xy + y^ =^64. Whence, x — y = ±S. (6) Adding (4) and (5), 2x = G± 8, or -6± 8. Whence, x = 7, — 1, 1, or — 7. Subtracting (5) from (4), 2 y = 6 T 8, or - 6 T 8. Whence, y = — 1, 7, — 7, or 1. « = ± 7, y = T 1 ; or, X = ± 1, y = T 7, Ans, ( X^ -\-y- = , 3. Solve the equations \ [xy = -7. SIMULTANEOUS EQUATIONS. 253 EXAMPLES. e the following equations : .'2/ = 48. ic -|- 2/ = 14. ar^-f 2/2=101. a; + 2/ = - 9. f.T3_2/-^ = 37. " [:^-^xy + if = Sl. |.r2/ = 45. yx —y = — 4. f a^ = 12. lar^ + 2/^ = 40. a;3_2/3^133 x-y = l. a5» + 2/« = ^17. «" + ajy + ?/" = 39. X -h y = - 2. 12. 13. 14. 15. 16. 17. 18. 19. r .r2 + 2/' = 260. (^ .r — 2/ = — 14. f iB2/ = - 80. |..'-y=:24. {^Jrf = 504. j a^ _ a;2/ + 2/- - 84. I ar^ - an/ + 2/' = 63. |a;-2/ = -3. .^2 + 2/' = 305. a; -2/ = 21. a^H- 2/2^218. a^ =•- 91. a!»4-3^ = -^35. a^ — a^ -I- 2/2 = 67. «2/ = - 150. a; - y = - 31. 279. Case IV. When each equation is of the second degree, ■I ■ omogeneous (§ 273). "-.e 1. Certain equations which are of the second degree and eiieous may be solved by the method of Case I. or Case III. -X. 1, § 276, and Kx. .3, § 278.) method of Case IV. should be used only when the example !- be solved by the methods of Cases I. or III. I Solve the equations aj2 — 2 a;?/ = 5. ar^ 4- 2/' = 29. lilting y = vx in the given equations, we have «2_2vx2 = 5; or, x^ x2 + v^x^ = 29 ; or, x^ 1 -2v' 29 (1) 1 +t?2 Divide the first equation by the second. 254 Equating the values of x^, Or, Or, Solving this equation, ALGEBRA. 5 29 1 - 2 1; 1 + ^2 5 + 5v2 = 29-58v. 5 u2 + 58 1? =z 24. 12. V =- or 5 Substituting these values in (1), x^ 5 5 l+24 ^^"^25- Whence, X = ± 5 or i: V5 Substituting the values of v and x in the equation y = vx, If v = - andiK = ±5, ?/=?(±5) = ±2. 6 o If v = -12 and ^ = ±^' 2/ = -12^±^^ = T 12 V5 iN'ote 2. In finding ?/ from the equation y = vx, care must be taken to multiply each pair of values of x by the corresponding value otv. EXAMPLES Solve the following equations : (2x^-xy = 2S. 1 x'-{-2y' = lS. ( x^ -{- xy = — 6. 6. 7. [xy-y^ = - 35. fc^-^xy-\-y' = 6S. 1 a^ - 2/2 = - 27. 1^2 + 32/' = 28. I a^ 4- a;?/ + 2/= 16. - a;2 - 2 a;2/ = 84. . 2 a;^/ — 2/' = — 64. 8. 9. 10. 11. 3a52-}-a;2/-32/' = 33. 2a^-2/2 = 23. x^-{- 6 xy — 2/2 7. x^-^3xy-2y^ = -4:. (x^-xy-12y' = S. [af^-\-xy-10y' = 20. 5x^-4.xy=33. 27x'-32xy-4:y'=55. |3aj2 + a?2/ + 2/2 = 47. [4.x^-3xy-y^ = -39. SIMULTANEOUS EQUATIONS. 255 MISCELLANEOUS AND REVIEW EXAMPLES. 280. No general rules can be given for the solution of examples wliich do not come under the cases just considered. Various artifices are employed, familiarity with which can only be gained by experience. ' x^-f^ 19. (1) a^-xy^ = 6. (2) Multiplying (2) by 3, Sx'^y - Sxy^ = 18. (3) Subtracting (.3) from {I), x^ - Sx^y -\- ^xtf- - y^ = I. Extracting the cube root, x — y = 1. (4) Dividing (2) by (4), xy = 6. (5) Solving equations (4) and (5) by the method of Case III., we find X = S, y = 2 ; or, x = - 2, y = - 3, Ans. 1. Solve the equations | ( Qir^ -\- y^ = 9 xy. 2. Solve the equations \ [x-\-y = 6. Putting X = ?( + V and y = u — v, vre have (M + vy + (u - vy = 9(n+v)(ti-v), (1) and (M+t>) + (M-v)=6. (2) Keducing (1), 2 «» + 6 uv^ = 9(m2 - v^). (3) Reducing (2), 2 m = 6, or u = 3. Substituting the value of u in (3), 54 + 18«2 = 9(9 - v^). Whence, u^ = 1, or « = ± 1. Therefore, x = M + v = 3±l=4or2, and ?/ = «-v = 3=Fl = 2or4. X = 4, y =2; or, x = 2, r/ = 4, Ans. Note. The artifice of substituting u + v and u - v for x and y is applicable in any case where the given equations are symmetrical with respect to x and y (§ 272). See also Ex. 4, p. 256. (x'^f + 2x + 2y = 2S. (1) 3. Solve the equations ] ^ ,^. [ xy = 6. (^) Multiplying (2) by 2, 2xy = 12. (3) Adding (1) and (3), x"^ + 2xy + y^- + 2x-{-2y= 35. 256 ALGEBRA. Or, (a; + ,y)2 + 2(x + y)=35. Completing the square, (x+y)24-2(x +«/) + != 36. Whence, (x + y) + 1 = ± 6, or X + ?/ = 5 or — 7. (4) Squaring (4), x^ + 2 xj/ + 2/^ = 25 or 49. Multiplying (2) by 4, ixy = 24, Subtracting, x^ — 2xy + y^= 1 or 25. Whence, x — y = i 1 or ± 5. (5) Adding (4) and (5), 2x = 5 ± 1, or - 7 ± 5. Whence, x = 3, 2, - 1, or - 6. Subtracting (5) from (4), 2?/ = 5 T 1, or -7^5. Whence, ?/ = 2, 3, - 6, or - 1. x=3, y = 2; x=2, y=S; x=— 1, y=—Q; or, x=-6, ?/= - 1, ^ws. 4. Solve the equations ] " [ x-\-y = -l. Putting X = u -{■ V and y = u — -y, we have (u-^vy-{-(u-vy = 97, (1) and (u + v) + {u-v) = -l. (2) Reducing (1), 2 Jt* + 12 ?<'V-^ + 2 v* = 97. (3) Reducing (2), 2 w = - 1, or m =- -. Substituting in (3) , ^ + 3 ?j2 + 2 v* = 97. 8 Solving this equation, v^ = — or — -• 4 4 Whence, v = ± - or ± ~ • Then, and 1^5 1 ^ V^niT o „ „, -1 tV-31 j, = «_« = _-T- or --T^- = -3, 2, or ^ _ x=2, 2,= -3; a;=-3, y=2 ; or, ^^ -1±V-3T y^ -lTV^^ , SIMUI^ANEOUS EQUATIONS. 257 EXAMPLES. Solve the following equations : 12 h'^ = -25- r car + jf ^x-y = 26. \xy:=12. • (2x'-3xy = -4:. \ 4 xy — 5y^ = 3. 1 4 ic2 _ 5 ^.y ^ 19 I ^jy + / = 6. 1-1 = 1. a; 2/ 2 18* a^ + 2 / = 47 + 2 a;, a.-^ - 2 2/2 = _ 7. jj p + a^ + 2/2 = 97. \x-y = 19. (x^-\-f = 756. a.-2 - xy + 2/' = 63. xY + 28 .i-2/ ~ 480 = 0. 2x + y=:ll. 10 12. 13. 14. 15. i + i = A. x' f 16 1_1 a; 2/ 16. 17. 18. 19. 20. 21. 22. 23. [x-\-y=l. 25. 26. (x^-y^ = 3. (a^-\-4:y'-\-3x = 22. ■ \2xy-^3y-\-9 = 0. 3x^-5xy-\-2f = -3 4: X — oy = 10. i. xy = a^ — 1. [ a; -h 2/ = 2 a. '^+?=^^- a; y x-\-y x — y ^lO x — y X -{-y 3 I ar^ + 2/^ = 45. .^-h2/' = 2a3 4-6a62. a.'y(a; + 2/)=2a='-2a6l 2x"2-3a^=15a-10a2. 3;r + 22/=12a-13. x^ -f a^2/' + 2/' = 91. x^ + xy + y^ = 13. x' + f = 13(a' + l). x-\-y = ^a — l. 2ar^+3x2/-42/'=-20. 5aj2-72/'=-8. * Divide the first equation by the second. 268 ALGEBRA. 28. 27. x'^xy + y^ = 3a^-Sab-^3b\ y = ^• 29. \^^^-^y-y = ^^- [ —5xy-\-y^-\-3x = 81. 30. 31. 32. 33. 34. 35. 36. -\/x 2/ = 19. 2 7 y X 2 x-\-y = l. aj - 2/ = 1. f x^y — ic = — 14. I .'cy + aj2 = 148. x^ — xy = 27 2/. ajy — 2/^ = 3 ic. x-\-y 2x—y ^15 X — y x-{- 2y 4 x-3y = -2. y(x — a)=2ab. x(y — b)=2 ab. 37. 38. 39. 40. 41. 42. 43. 44. = -T-' 45. 46. (xP-y' = [x-y=l X^ = X -{- I f = Sy- 31. I Va;2 + 7 = 6-2/. I V^^+227 = 22 - ic2. 5SxP-12Sxyi-64.y^=5. 26x^-62xy + 32y'=:5, xy-(x-y) = l. xY + (« - yf = 13. i»2/+ C^' — 2/) =—5. xy{x-y)=-SL y? — xy -\-y^ = 12. 0,-3 _|_ 2/3 _j_ 3 a;^/ = 48. :^^rxy^y^ = 7. a: + 2/ = 5 H- a;?/. 2a^ + 22/2 = 5 a;?/. ar^-2a;2/ + 3a;2 = -16. 2x-3y^l. 3 a; + 5)3 = -14. PROBLEMS. Note. In the following problems, as in those of § 268, only those answers are to be retained which satisfy the given conditions. 281. 1. The sum of the squares of two numbers is 52, and their difference is one-fifth of their sum. Find the numbers. 2. The difference of the squares of two numbers is 16, and their product is 15. Find the numbers. SIMULTANEOUS EQUATIONS. 259 3. If the length, of a rectangular field were increased by 2 rods, and its width diminished by 5 rods, its area would be 80 square rods ; and if its length were diminished by 4 rods, and its width increased by 3 rods, its area would be 168 square rods. Find its length and width. 4. The difference of the cubes of two numbers is 218, and the sum of their squares is equal to 109 minus their prod- uct. Find the numbers. 5. If the product of two numbers be multiplied by their sum, the result is 70 ; and the sum of the cubes of the num- bers is 133. Find the numbers. 6. A farmer bought 4 cows and 8 sheef) for $ 600. He bought 5 more cows for $490 than sheep for $80. Find the price of each. 7. Find a number of two figures such that, if its digits be inverted, the difference of the number thus formed and the original number is 9, and their product 736. 8. The sum of two numbers exceeds the product of their square roots by 7; and if the product of the numbers be added to the sum of their squares, the result is 133. Find the numbers. 9. The sum of the terms of a fraction is 13. If the numerator be decreased by 2, and the denominator increased by 2, the product of the resulting fraction and the original fraction is y\. Find the fraction. 10. A rectangular mirror is surrounded by a frame 3^ inches wide. The area of the mirror is 384 square inches, and of the frame 329 square inches. Find the length and width of the mirror. 11. A crew row up stream 18 miles in 4 hours more time than it takes them to return. If they row at two-thirds their usual rate, their rate up stream would be one mile an hour. Find their rate in still water, and the rate of the stream. 260 ALGEBRA. 12. A rectangular field contains 2\ acres. If its length were decreased by 10 rods, and its width by 2 rods, its area would be less by an acre. Find its length and width. 13. A distributes $ 180 equally between a certain number of persons. B distributes the same sum between a number of people less by 40, and gives to each $ 6 more than A does. How many persons are there, and how much does A give to each ? 14. A, B, and C together can do a piece of work in one hour. B does twice as much work as A in a given time; and B alone requires one hour more than C alone to per- form the work. .In what time could each alone do the work ? 15. If the length of a rectangular field were increased by one-eighth of itself, and its width decreased by one-sixth of itself, its area would be decreased by 60 square rods, and its perimeter by 2 rods. Find its length and width. 16. If the product of two numbers be added to their difference, the result is 26; and the sum of their squares exceeds their difference by 50. Find the numbers. (Represent the numbers hy x -h y and x — y.) 17. A sets out to walk to a town 21 miles off, and one hour afterwards B starts to follow him. When B has over- taken A, he turns back, and reaches the starting-point at the same instant that A reaches his destination. B walked at the rate of 4 miles an hour. Find A's rate, and the dis- tance from the starting-point to where B met A. 18. A tank can be filled by three x^ipes. A, B, and C, when opened together, in 2^^ hours. If A filled at the same rate as B, it would take 3 hours for A, B, and C to fill the tank ; and the sum of the times required by A and C alone to fill the tank is double the time required by B alone. In what time can each pipe alone fill the tank? 19. The sum of two numbers is 4, and the sum of their fifth powers is 244. Find the numbers. THEORY OF QUADRATIC EQUATIONS. 261 XXV. THEORY OF QUADRATIC EQUA- TIONS. 282. Sum and Product of the Roots. Let ?*i and rg denote the roots of the equation x^ -{-px = q. By § 265, n = -P+-iP' + ^? —riX — r^-{- TiV^ = 0. That is, (x - r,) (x - r^) = 0. (§ 93) Hence, any quadratic equation can be written in the form (x-r,)(x-r,) = 0, ^ (2) where Vi and rg are its roots. Therefore, to form a quadratic equation which shall have any required roots. Subtract each of the roots from x, and place the product of the resulting expressions equal to zero. and 1. Form the quadratic equation whose roots shall be 4 7 4* By the rule, (x - 4)(x + -\ = 0. Multiplying by 4, (aj — 4) (4 cc + 7) = 0. Whence, 4 ic2 - 9 a; - 28 = 0, Ans. EXAMPLES. Form the quadratic equations whose roots shall be : 2.6,9. 4.1,-1 6.1, I 8. -f,0. 3. 2,-3. 6. -4,-|. 7. -|,|. 9. -| -|. THEORY OF QUADRATIC EQUATIONS. 263 10. 2a + b,a-3b. 12. ;; -f 7V2, 3 - 7V2. 11. a + Sm,a-3m. 13. ^(- Va + V^),i(- Va- V6). FACTORING. 284. Factoring of Quadratic Expressions. A quadratic expression is an expression of the form ax^ -\- bx + c. The principles of § 283 serve to resolve sncli an expres- sion into two factors, each of the first degree in x. We have, aa^ -h bx -\- c = afa^ -h — ^-X (1) \ a aj Now let rj and r^ denote the roots of the equation a a By § 283, (2), the equation can be written in the form (x-r{)(x-r2) = 0. Hence, the expression a^-h— +- can be written a a (x-ri)(x-r2). Substituting in (1), we have ax^ -\-bx-\-c = a(x — r,) (x — rg). hx c But 7\ and ?*o are the roots of the equation a^-] 1- - = 0, a a or oor^ -f 6aj -(- c = ; which, we observe, is obtained by placing the given expression equal to zero. We then have the following rule : To factor a quadratic expression, place it equal to zero, and solve the equation thus formed. Then the required factors are the coefficient of o^ in the given expression, x minus the first root, and x minus the second root 264 ALGEBRA. EXAMPLES. 285. 1. Factor Gx'-^-lx- 3. Solving the equation 6 a:^ + 7 a; - 3 = 0, we have by § 265, 12 12 3 2 Then by the rule, 6x^-h 1 x - 5 = qIx -^\ Ix + -\ = (Sx- l)(2a; + 3), Ans. 2. Factor 4 + 13a;-12«2. Solving the equation 4 + 13 x — 12 x^ = 0, we have by § 265, ^ ^ - 13 -t- Vl69 + 192 ^ - 13 ± 19 ^ 1 ^^4^ -24 -24 4 3* Whence, 4 + 13 a;- 12x2 = - 12 (x + iVx- 1"! = 4(. + l)x(-3)(.-|) = (l + 4ic)(4-3a;), ^ws. Factor the following : 3. ic2-13a;4-42. 14. 6a^ -23mx + 21m\ 4. x' + Wx-^-U. 15. 14a^ + 25a;4-6. 5. a^-9x-36. 16. 18 ar^ - 15 a; + 2. 6. 3a^H-7a;-6. 17. 5-19a:-4a^. 7. 5a^+18a5 + 16. 18. 18a^ + 31a; + 6. 8. 6a^-llaj + 3. 19. 45 + 7x-12a^. 9. 15a^-14a:-8. 20. 42 + 23 a; - 10 o^. 10. 20-7a;-3a^. 21. 24a^-26a; + 5. 11. 35-lla;~6«2 22. 80^2^33^.^35 12. 12 + 28a;-5i«2 23. 21 a^ - 10 a^ - 24 2/1 .13. 3a^-i7aa;-28al 24. 7 ar^ + 37 a6a; - 30 a-6l THEORY OF QUADRATIC EQUATIONS. 265 25. Y^Qtov 2a^-Sxij-2y^-7x + 4:y + 6. Placing the expression equal to zero, we have 2x^-3xy -2y'^-7x-hiy + Q = 0, or " 2x^-(Pjy + 7)x = 2y2-iy - 0. Solving this hy the formula of § 265, ^_ 3y + 7,±V(3y + 7)-^+16y^-32y-48 ■ ^ 3y + 7:i:V25y2 4-10y+T _ 3y + 7zb(5y-f 1) 4 ~ 4 = -i^or pi- = 22, + 2 or --i^ Therefore, 2x2 - 3a;2/ - 22/2 _ 7a; + 4y + 6 = 2 [a; -C2y + 2)]rx -^=^^±-?l = (a; - 2y - 2)(2a; + y - 3), ^ns. Factor the following : 26. a^ + a;?/-12/ + 7a; + 72/ + 12. 27. a52_a^_22/2 + a7-52/-2. 28. a^-42/^-h3a; + 10i/-4. 29. 2»24.7a^_42/2-f a; + 132/-3. 30. 3a^-oab-2b^-7a-{-2. 31. 6-152/-5d; + 9/4-9a;?/-4a;2 32. 6a^-dxy + xz-Wy^-lSyz-2z'. 286. If the coefficient of x^ is a perfect square, it is con- venient to factor the expression by the artifice of completing the square (§ 260) in connection with § 99. 1. Factor 9x^-9x-4:. By § 260, the expression 9 rc2 - 9 x will become a perfect square by q o iing to it the square of — -, or — Then, 2V'9 2 9 x2 - 9 X - 4 = 9 a:2 - 9 X + ( § y - 2 _ 4 Z3 ( 3 X -r - ] "^ - — . 266 ALGEBRA. Factoring as in § 99, we have 9.-9.-4 = (3.-| + |)(.S.x-|-|) = (3x+ l)(3x-4), Ans. If the x^ term is negative, the entire expression should be enclosed in a parenthesis preceded by a — sign. 2. Factor S-12x-4.x\ 3-12ic-4x2 = -(4x2 + 12x-3) = _(4x2+12ic + 32-9-3) = _ [(2x + 3)2-12] = (2x + 3 + Vl2)x(- l)(2x + 3-Vl2) = (2V3 + 3 + 2 X) (2\/3 - 3 - 2 x), ^/is. EXAMPLES. Factor the following : ' 3. ic2-5a; + 4. 9. 36 x" -{- 24: x - 5. 4. 4a^ + 16x + 15. 10. 4.x' + 5x-6. 5. 9a;2_l8x + 8. 11. 25x'-\-30x + 6. 6. 16ar2+16a;-21. 12. 4 + 12a)-9a^. 7. a^ + 2a;-ll. 13. 49 a^^ + 56 ic + 12. 8. 4x2 + 4a^-l. 14. 5-\-SSx-16a^. 287. Certain trinomials of the form ax* + fta?^ + c, where a and c are perfect squares, may be resolved into two fac- tors by the artifice of completing the square. 1. Factor 9x^-28x^-^4:. By § 96, the expression will become a perfect square if its middle term is — 12 x^. Thus, 9ic4- 28x2 + 4 =(9^4 -12^2 + 4) -16x2 ^(3x2-2)2-(4x)2 = (3x2-2 + 4x)(3x2-2-4x) (§99) = (3x2 + 4x-2)(3x2-4x-2), Ans. THEORY OF QUADRATIC EQUATIONS. 267 2. Factor a' + aV/ + b\ = (a2+&2)2_a-2^2 = (a2 + 62 + «6)(a2 4.?,-2_a6) = (a2 + a6 + 62) (a2 _ a& + 6'0» ^««- 3. Factor a;^ + 1. X4 + 1 =(iC* + 2ic2 + 1)_ 2x2 = (a;2 + l)2_(a;\^)2 = (aj2 + a; v/2 + 1) (x2 _ a^ V2 + 1), ^ns. EXAMPLES. Factor the following : 4. x'-\-2a^ + 9. 12. a;* + 16. 5. a;^-19a;2-h25. 13. x'-5a^ + l. 6. 4a< + 7a262-|-166^ 14. 9 a* - 55 a^x^ _j_ 25 aj*. 7. 9a;^-28ic2/ + 4y. 15. 16 a* + 47 aV ^. 36 m*. 8. 16m^-mV4-n'. 16. 25 a^^ - 21 «2 + 4. 9. 4a^-53a2 + 49. 17. 25 m^ + 36 mV + 16 a.-*. 10. 9a;* + 5ar^ + 9. 18. 16 a^ - 60 a^^y. _^ 49 ^ 11. 4m^-13m2 + 4. 19. 36 a^ - 68 a^ft^ + 25 6^ Certain equations of the fourth degree may be solved by factoring the first member by the method of § 287, and then proceeding as in § 267. 1. Solve the equation a;* + 1 = 0. By Ex. 3, § 287, the equation may be written (x2 + X V2 + l)(a;2 - X V2 + 1)= 0. Then, as in § 267, a:2 + x V2 + 1 = 0, and x2 - x \/2 + 1 = 0. Solving the equation a;2 + x V'2 4- 1 = 0, we have by § 265, _ V2 ± V2^^ - V2 ± X = = = =^ 2 2 268 ALGEBRA. Solving the equation aj^ — x V2 + 1 = 0, we have EXAMPLES. Solve the following : 3. a?4-18a;2H-9 = 0. 6. .t^ - Ooj^ + 9 = 0. 4. 4aj^-5aj24-l = 0. 7. o.-^ + 81 = 0. DISCUSSION OF THE GENERAL EQUATION. By § 265, the roots of the equation x^ -\- px = q are We will now discuss these values for all possible real values of p and q. I. Suppose q positive. Since p^ is essentially positive (§ 186), the expression under the radical sign is positive, and greater than p^. Therefore, the radical is numerically greater than p. Hence, r^ is positive, and rg is negative. If p is positive, rg is numerically greater than Vi ; that is, the negative root is numerically the greater. If j9 is zero, the roots are numerically equal. If p is negative, 7\ is numerically greater than rgj that is, the positive root is numerically the greater. II. Suppose q = 0. The expression under the radical sign is now equal to p^. Therefore, the radical is numerically equal to p. If p is positive, r^ is zero, and rg is negative. If p is negative, Vi is positive, and rg is zero. THEORY OF QUADRATIC EQUATIONS. 269 III. Suppose q negative, and 4g niimericaUy p^. The expression under the radical sign is uow negative. Hence, both roots are imaginary (§ 248). The roots are both rational or both irrational, according as p'^ -\- ^q is or is not a perfect square. EXAMPLES. 290. 1. Determine by inspection the nature of the roots of the equation 2 ic^ — 5 a: — 18 = 0. The equation may be written x^ - = ^; here p = — - and g = 9. Since q is positive and p negative, the roots are one positive and the other negative ; and the positive root is numerically the greater. In this case, p'^ + ^ q = ^ Jf ZQ = — ; a perfect square. 4 4 Hence, the roots are both rational. Determine by inspection the nature of the roots of the following : 2. 6a52 + 7a;-5 = 0. 7. l&a^-^ = 0. 3. 10a^ + 17a; + 3 = 0. 8. ^:x?-l = 12x. 4. 4ar^-iB = 0. 9. 2bx' -\-^0x + ^ = 0. 5. 4a;2_20.^.-f 25 = 0. 10. 7.^•2 + 3a; = 0. 6. ar^- 21 a; + 200 = 0. 11. 41 a; = 20 ar^ + 20. 270 ALGEBRA. XXVI. ZERO AND INFINITY. VARIABLES AND LIMITS. 291. A variable quantity, or simply a variable, is a quan- tity which may assume, under the conditions imposed upon it, an indefinitely great number of different values. A constant is a quantity which remains unchanged throughout the same discussion. 292. A limit of a variable is a constant quantity, the dif- ference between which and the variable may be made less than any assigned quantity, however small, but cannot be made equal to zero. In other words, a limit of a variable is a fixed quantity to which the variable approaches indefinitely near, but never actually reaches. Suppose, for example, that a point moves from A towards B under the condition that it shall move, during succes- sive equal intervals of time, first from A to C, half-way f f f f T between A and B; then to D, half-way between C and B ; then to E, half-way between D and B; and so on indefinitely. In this case, the distance between the moving point and B can be made less than any assigned quantity, however small, but cannot be made equal to zero. Hence, the distance from A to the moving point is a vari- able which approaches the constant value AB as a limit. Again, the distance from the moving point to 5 is a variable which approaches the limit 0. 293. A problem is said to be indeterminate when the number of solutions is indefinitely great. (Compare § 159.) ZERO AND INFINITY. 271 294. Interpretation of ^• Consider the series of fractions 3' .3' .03' .003'"*' where each denominator after the first is one-tenth of the preceding denominator. It is evident that, by sufficiently continuing the series, the denominator may be made less than any assigned quan- tity, however small, and the value of the fraction greai^^er than any assigned quantity, however great. In other words. If the numerator of a fraction remains constant^ icliile the denominator approaches the limit 0, the value of the fraction increases without limit. It is customary to express this principle as follows : a - = QO. <) Note. The symbol oo is called Infinity. 295. Interpretation of — • Consider the series of fractions a a a a 3' 30' 300' 3000''"' where each denominator after the first is ten times the pre- ceding denominator. It is evident that, by sufficiently continuing the series, the denominator may be made greater than any assigned quantity, however great, and the value of the fraction less than any assigned quantity, however small. In other words, If the numerator of a fraction remains constant, while the denominator increases without limit, the value of the fraction approaches the limit 0. 272 ALGEBRA. It is customary to express this principle as follows; 00 296. It must be clearly understood that no literal meaning can be attached to such results as -=500. or — = 0:r for there can be no such thing as division unless the divisor is a finite quantity. If such forms occur in mathematical investigations, they must be interpreted as indicated in §§ 294 and 295. (Com- pare note to § 395.) THE PROBLEM OF THE COURIERS. 297. The discussion of the following problem will serve to further illustrate the form -, besides furnishing an inter- pretation of the form -• The Problem of the Couriers. Two couriers, A and B, are travelling along the same road in the same direction, RR\ at the rates of m and n miles an hour, respectively. If at any time, say 12 o'clock, A is at P, and B is a miles beyond him at Q, after how many hours, and how many miles beyond P, are they together ? R p q BT I I \ ^ I Let A and B meet x hours after 12 o'clock, and y miles beyond P. They will then meet y — o. miles beyond Q. Since A travels mx miles, and B nx miles, in x hours, we have ( y = mx. Xy — a = nx. ZERO AND INFINITY. 273 Solving these equations, we obtain a 1 am X = , and y = m — n m — n We will now discuss these results under different hypoth- eses. 1. m > n. In this case, the values of x and y are positive. Hence, the couriers will meet at some time after 12 o'clock, and at some point to the rigid of P. This corresponds with the hypothesis made ; for if m is greater than n, A is travelling faster than B ; and it is evi- dent that he will eventually overtake him at some point beyond their positions at 12 o'clock. 2. tn < n. In this case, the values of x and y are negative. Hence, the couriers met at some time before 12 o'clock, and at some point to the left of P. (Compare § 10.) This corresponds with the hypothesis made ; for if m is less than 7i, A is travelling more slowly than B ; and it is evident that they must have been together before 12 o'clock, and before they could have advanced as far as P. 3. m = ?i, or m — nz= 0. In this case, the values of x and y take the forms - and — , respectively. • If m — n approaches the limit 0, x and y increase with- out limit (§ 294) ; hence, if m = ii, no finite values can be assigned to x and y, and the problem is impossible. Thus, a result in the form - indicates that the problem is This interpretation corresponds with the hypothesis made ; for if m = n, the couriers are a miles apart at 12 o'clock, and are travelling at the same rate ; and it is evident that they never could have be§B^_and never will be together. CALIFO^J 274 ALGEBRA. 4. a = 0, and m > n or m < n. In this case, x = and 2/ = 0. Hence, the couriers are together at 12 o'clock, at the point P. This corresponds with the hypothesis made ; for if a = 0, and m and n are unequal, the couriers are together at 12 o'clock, and are travelling at unequal rates ; and it is evi- dent that they never could have been together before 12 o'clock, and never will be together afterwards. 5. a = 0, and m = n. In this case, the values of x and y take the form — If a = 0, and m = n, the couriers are together at 12 o'clock, and are travelling at the same rate. Hence, they always have been, and always will be together. In this case the number of solutions is indefinitely great ; for any value of x whatever, together with the correspond- ing value of y, will satisfy the given conditions. Thus, a result in the form - indicates that the problem is indeterminate (§ 293). NDETERMINATE EQUATIONS. 275 XXVII. INDETERMINATE EQUATIONS. 298. It was shown in § 159 that a single equation con- taining two or more unknown quantities is satisfied by an indefinitely great number of sets of values of these quanti- ties. If, however, the unknown quantities are required to satisfy other conditions, the number of solutions may be finite. We shall consider in the present chapter the solution of indeterminate equations of the first degree, containing two or more unknown quantities, in which the unknown quanti- ties are restricted to positive integral values. 299. Solution of Indeterminate Equations in Positive Integers. 1. Solve the equation 7x-\-5y = 118 in positive integers. Dividing by 5, the smaller of the two coefficients, we have 5 5 Or, lEp3 = 2S-x-y. 5 Since, by the conditions of the problem, x and y must be positive 2 X — 3 integers, it follows that must be an integer. 5 Let this integer be represented by p. Then, ^^f-^ =p, or 2 X - 3 = 5i). (1) 5 Dividing (1) by 2, x - 1 - i = 2j9 + ^. -,. - 2 2 Or, x-l-2p=P^- Since x and p are integers, x-l-2p is also an integer ; and there- fore ^ must be an integer. 2 Let this integer be represented by q. 276 ALGEBRA. Then, ^-i-i = q, or p = 2q-l. Substituting in (1), 2 x - 3 = 10 g - 5. Whence, 2ic = lOg^ - 2, and x = 5g - 1. (2) Substituting this value in the given equation, 35g-7 + 5?/=118. Whence, 5 y = 125 - 35 g, and y = 25 - 7 g. (3) Equations (2) and (3) form what is called the general solution in integers of the given equation. Now if q is zero, or any negative integer, x will be negative ; and if q is any positive integer greater than 3, y will be negative. Hence, the only positive integral values of x and y which satisfy the given equation are those arising from the values 1, 2, 3 of g-. If g = 1, 5C = 4, and y = 18; if g = 2, x = 9, and y = ll; if g = 3, X = 14, and y = 4. 2. In how many ways can the sum of $ 15 be paid with dollars, half-dollars, and dimes, the number of dimes being equal to the number of dollars and half-dollars together ? JiGt X = the number of dollars, y = the number of half-dollars, and z = the number of dimes. Then, lOx + by + z = 160, (1) and z = x + y. (2) Subtracting (2) from (1), 10a; + 5?/ = 150-x-?/, or llx-f6?/ = :150. (3) Dividing by 6, x -\- — -{■ y = 25. 5x Then, — - must be an integer ; or, x must be a multiple of 6. 6 Let X = 6p, where p is an integer. Substituting in (3), dSp + 6y = 150, or y = 25- Up. Substituting in (2), z = 6p -\- 26 — Up =26 - dp. The only positive integral solutions are when p = 1 or 2; ifp = l, X = 6, y=U, and 5; = 20 ; if p = 2, x = 12, y = 3, and z = 15. Then the number of ways is two ; either 6 dollars, 14 half-dollars, and 20 dimes ; or 12 dollars, 3 half-dollars, and 15 dimes. INDETERMINATE EQUATIONS. 277 EXAMPLES. Solve the following in positive integers : 3. 2x-^3y = 21. 9. 43 a; -|- 10 y = 719. 4. 7x-\-4:y = S0. 10. Sx-\-19y = 700. 5. 7a; + 382/ = 211. 6. 31 a; + 9 2/ = 1222. 11. 2x-^Sy — oz = -S. 5x-y-\-4:Z = 21. 7. 24a; + 72/ = 422. ( Sx-2y -z = -^ 57. 8. 8a;-f 67 2/ = 158. ^^- | 6a;-f- lly + 22 = 348. Solve the following in least positive integers : 13. 4.x-3y = 5. 16. 21a;-8y = -25. 14. 5x-7y = ll. 17. 13a; - 30.1/ = 61. 15.^9 a; -4 2/ = 128. 18. 17x-5Sy = -79. 19. In how many different ways can the sum of ^ 2.10 be paid with twenty-five and twenty-cent pieces ? 20. In how many different ways can the sum of f 3.90 be paid with fifty and twenty-cent pieces ? 21. Find two fractions whose denominators are 9 and 5, respectively, and whose sum .shall be equal to -y^'^. ^22. In how many different Ways can the sum of $ 5.10 be paid with half-dollars, quarter-dollars, and dimes, so that the whole number of coins used shall be 20 ? 23. A farmer purchased a certain number of pigs, sheep, and calves for $ 160. The pigs cost $ 3 each, the sheep f 4 each, and the calves ^ 7 each ; and the number of calves was equal to the number of pigs and sheep together. How many of each did he buy ? 24. In how many different ways can the sum of $5.45 be paid with quarter-dollars, twenty-cent pieces, and dimes, so that twice the number of quarters plus 5 times the num- ber of twenty-cent pieces shall exceed the number of dimes by 36? ^ 278 ALGEBRA. XXVIIL RATIO AND PROPORTION. 300. The Ratio of one number to another is the quotient obtained by dividing the first number by the second. Thus, the ratio of a to 6 is - ; and it is also expressed a : b. h 301. A Proportion is a statement that two ratios are equal. The statement that the ratio of a to 6 is equal to the ratio of c to d, may be written in either of the forms , T a c a:b = c: d, 01 - = -' b d ■ ' 302. The first and fourth terms of a proportion are called the extremes, and the second and third terms the means. The first and third terms are called the antecedents, and the second and fourth terms the consequeyits. Thus, in the proportion a\b = c:d, a and d are the ex- tremes, b and c the means, a and c the antecedents, and b and d the consequents. 303. If the means of a proportion are equal, either mean is called a Mean Proportional between the first and last terms, and the last term is called a Third Proportional to the first and second terms. Thus, in the proportion a:b = b: c, b is a mean propor- tional between a and c, and c is a third proportional to a and 6. 304. A Fourth Proportional to three quantities is the fourth term of a proportion whose first three terms are the three quantities taken in their order. RATIO AND PROPORTION. 279 Thus, in the proportion a: b = c: d, d is a fourth propor- tional to a, h, and c. 305. A Continued Proportion is a series of equal ratios, in which each consequent is the same as the following ante- cedent; as, a\h = h :c = c:d = d'.e. PROPERTIES OF PROPORTIONS. >J 306. In any proportion^ the product of the extremes is equal to the ])rodu(it of the means. Let the proportion be a : 6 = c : rf. Then by §301, ^=L d Clearing of fractions, ad = he. , 307. A mean proportional between two quantities is equal to the square root of their product. Let the proportion be a:b = b:c. Then, b^ = ac. (§ 306) Whence, b = Vac. 308. From the equation ad — be, we obtain a = — , and 6= — d c That is, in any proportion, either extreme is equal to the product of the means divided by the other extreme; and either mean is equal to the product of the extremes divided by the other mean. A 309. (Converse of § 306.) If the product of two quantities is equal to the ptroduct of two others, one pair may be made the extremes, and the other pair the means, of a proportion. 280 ALGEBRA. Let ad = be. T-k- -J- I. X. 7 ad be a c Dividing by bd, f:i = lZi^ ^^' i. = y bd bd b d Whence by § 301, a:b = e:d. In like manner, we may prove that a: e=:b: dj e:d = a:b, etc. 310. In any proportion, the terms are in proportion by Alternation; that is, the first term is to the third as the second term is to the fourth. Let the proportion be a: b = c: d. Then, ad = be. (§ 306) Whence, a:c = b:d. (§ 309) 311. In any proportion, the terms are in proportioyi by Inversion ; that is, the second term is to the first as the fourth temn is to the third. Let the proportion be a : 6 = c : d. Then, ad = be. (§ 306) Whence, b:a = d'.c. (§ 309) V 312. In any proportion, the terms are in proportion by Composition ; that is, the sum of the first two terms is to the first term as the sum of the last two terms is to the third term. Let the proportion be a:b = c:d. Then, ad = be. Adding each member of the equation to ac, ac -\- ad =^ ac ■\- be. Or, a{c + cZ) = c{a -f b). Whence, a ^ 6 : a = c + c« : c. (§ 309) In like manner, we may prove that a + b : b = c -\- d : d. RATIO AND PROPORTION. 281 '313. In any proportion, the terms are in i)roportion by Division; that is, the difference of the first two terms is to the first term as the difference of the last two terms is to the third term,. Let the proportion be a : 6 = c : d. Then, ad = he. Subtracting each member of the equation from ac, ac — ad = ac — he. Or, a{c — d) — c{a — h). Whence, a — b:a = c — d:c. Similarly, a — h : b = c — d : d. V 314. In any iiroportion^ the terms are in proportion hy Composition and Division ; that is, the sum of the first two terms is to their difference as the sum of the last two terms is to their difference. (1) (2) ^315. In a series of equal ratios, any antecedent is to its con- sequent as the sum of all the antecedents is to the sum of all the consequents. Let a:b = c:d = e:f. Then by § 306, ad = he, and af= be. u^ Also, ah = 6g. — ( Adding, a(h -\- d -\-f) = b(a -{- c + e). Whence, a:6 = a + c + e:6 + d-f/. (§309) Let the proportion be a:b = c:d. Then by § 312, a + b c-\-d a c And by §313, a — b c — d a c Dividing (1) by (2), a-\-b c-^d a — h c — d Whence, a-\-h : a — b = c -\- d : c 282 ALGEBRA. ^ a _ _c b~ ~d ma mb _nc J In like manner, the theorem may be proved for any num- ber of equal ratios. 316. In any proportion, if the first two terms be multiplied by any quantity, as also the last two, the resulting quantities loill be in proportion. Let the proportion be a:b = c: d. Then, Therefore, Whence, ma : mb — nc : nd. In like manner, we may prove that a b _c d m ' m n ' n Note. Either m or 71 may be unity ; that is, eitlier couplet may be multiplied or divided without multiplying or dividing the other. 317. In any proportion, if the first and third terms be mul- tiplied by any quantity, as also the second and fourth terms, the resulting quantities ivill be in proportion. Let the proportion be a: b = c: d. Then, Therefore, Whence, ma : 71b = mc : nd. In like manner, we may prove that a b _ c d m' n m' n Note. Either m or w may be unity. a _ V _ c ~d ma nb mc nd RATIO AND PROPORTION. 283 318. In any number of j^roportions, the products of the cor- responding terms are in proportion. Let the proportions hQ a:h = c:d, and e:f=g:h. Then, ^ = £,and« = f. b d f h Multiplying these equals, we have a e c q ae cq _ y .. — _ V — or = — ^« b^f~d^h' bf dh Whence, ae:bf—cg: dh. In like manner, the theorem may be proved for any num- ber of proportions. 319. In any propoi'tion, like powers or like roots of the terms are in proposition. Let the proportion he a:b = c: d. 1=1- ■ Therefore, ^ = ^. Whence, a"" -. b"" = c"" : d\ In like manner, we may prove that y/a : -Vb = -Vc : -^d. 320. If three quantities are in continued proportion, the first is to the third as the square of the first is to the square of the seco7id. Let a:b = b:c. Then, « = *. b c Therefore, ? x ^ = ? X |, or ?5 = ^'. b c b b c b^ Whence, a:c = a^:bK 284 ALGEBRA. 321. If four quantities are in continued proxiortion, the first is to the fourth as the mihe of the first is to the cube of the second. Let a: b = b : c = c: d. a_6_c b c^ d Then, rpi o a b c a a a a a^ Therefore, - x - X = - x - x -, or - = -• b c d b b b d b^ Whence, a : d = a^ : b^. PROBLEMS. 322. 1. Solve the equation 2x+3:2x-3 = 2b-\-a:2b-a. By §314, ix:6 = 4h :2a. Dividmg the first and third terms by 4, and the second and fourth terms by 2 (§ 317), we have X : 'S = b : a. Whence by § 308, x = —, A7is. a 2. li x:y =(x-^z)- : (y -{- %f^ prove that ^ is a mean pro- portional between x and y. From the given proportion, y{x + zY = x(y + zY- (§ 306) Or, . x^i/ + 2 xyz + yz"^ = xy^ -\- 2 xyz + xz^. Or, x^y — xy"^ = xz^ — yz^. ^\:^ Dividing hy x — y, xy = z^. , ^ "> Therefore, 2 is a mean proportional between x and y (§ 307). 3. If - = -, prove that b d a' - b' : a' - Sab = c^ - d' : c" - 3cd. Let -=- = x: whence, a = bx. "J I --1 a^-b^ &2x2 -b^ _ x"^-! _ d^ c^ - d^ - -^*^®^' a2-3a6~6%2_362a;-x2-3x~c^_3c~c2-3c(^* ^ d^ d Whence, a^ - b^ : a"^ - Z ab = c^ - d^ : c2 - 3 cd. RATIO AND PROPORTION. 285 4. Find a fourth proportional to 35, 20, and 14. 5. Find a mean proportional between 18 and 50. 6. Find a third proportional to ^ and ^. 7. Find the second term of a proportion whose first, third, and fourth terms are 5^, 4J, and 1|. 8. Find a third proportional to a^ — 9 and a — 3. ^' 9. Find a mean proportional between o^ and 18^. \j 10. Find a mean proportional between ; — and — — — X + 4: x + 2 Solve the following equations : h ^ 11. 5a;-3a:5a; + 3a = 7a-5:13a-5. I \i 12. 2a;-l:3a;-l = 7a; + l:5a;-3. "" 13. .x-^ - 16 : a^ - 25 = ar^ - 2a; - 24 : .-cf - 3a; - 10. "^ 14. 1-Vr^:l + VT=^=V6-V&^: V6+V6^. 15. ax — by:bx-^ay = a^ — b^:2ab. xy = a^b^. ) 16. Find two numbers in the ratio 16 to 9 such that, if each be diminished by 8, they shall be in the ratio 12 : 5. 17. Divide 36 into two parts such that the greater dimin- ished by 4 shall be to the less increased by 3 as 3 is to 2. 18. Find two numbers such that, if 4 be added to each, they will be in the ratio 5 to 3 ; and if 11 be subtracted from each, they will be in the ratio 10 to 3. 19. There are two numbers in the ratio 3 to 4, such that their sum is to the sum of their squares as 7 is to 50. What are the numbers ? M 20. If 7a;-42:8a;-32 = 4?/-72:3y-82, provethat 2 is a mean proportional between x and y. 286 ALGEBRA. 21. If ma -\- nb : pa -^ qb = mb -}- nc : pb -{- qc, prove that 6 is a mean proportional between a and c. 22. If 2a-6:4a + 36 = 2c-(^:4c-f 3d, prove that a:b = c\d. 23. If 8 cows and 5 oxen cost four-fifths as much as 9 cows and 7 oxen, what is the ratio of the price of a cow to that of an ox ? 24. Given (a^ + ab)x + (b^ - ab)y= (a" + b^)x - (a? - b^)y ; find the ratio of x to y. 25. Find a number such that if it be added to each term of the ratio 5 : 3, the result is f of what it would have been if the same number had been subtracted from each term. If - = -, prove that b d! ^ 26. 2a + 36:2a-36 = 2c + 3d:2c-3d 27. a^ + 2 a6 : 3 a& - 4 62 = c2 + 2 c(^ : 3 c(i - 4 dl 28. a« - a'b + ab^ :a^-W = & - cH -f cd^ : c^ - d\ 29. The population of a town increased 2.6 per cent from 1870 to 1880. The number of males decreased 3.8 per cent during the same period, and the number of females increased 10.6 per cent. Find the ratio of males to females in 1870. 30. Each of two vessels contains a mixture of wine and water ; in one the wine is to the water as 1 to 3, and in the other the wine is to the water as 3 to 5. A mixture from the two vessels is composed of wine and water in the ratio 9 to 19. Find the ratio of the amounts taken from each vessel. 31. The second of three numbers is a mean proportional between the other two. The third number exceeds the sum of the other two by 15, and the sum of the first and third exceeds twice the second by 12. Find the numbers. VARIATION. 287 XXIX. VARIATION. 323. One quantity is said to vary directly as another when the ratio of any two values of the hrst is equal to the ratio of the corresponding values of the second. Note. It is customary to omit the word "directly," and say simply that one quantity varies as another. 324. Let us suppose, for example, that a workman receives a fixed sum per day. The amount which he receives for m days will be to the amount which he receives for n days as m is to n. That is, the ratio of any two amounts received is equal to the ratio of the corresponding numbers of days worked. Hence, the amount which the workman receives varies as the number of days during which he works. 325. One quantity is said to vary inversely as another when the first varies directly as the reciprocal of the second. Thus, the time in which a railway train will traverse a fixed route varies inversely as the speed; that is, if the speed be doubled^ the train will traverse its route in one- half t\iQ time. 326. One quantity is said to vary as two others jointly when it varies directly as their product. Thus, the wages of a workman varies jointly as the amount which he receives per day, and the number of days during which he works. 327. One quantity is said to vary directly as a second and inversely as a third, when it varies jointly as the second and the reciprocal of the third. Thus, in physics, the attraction of a body varies directly as the quantity of matter, and inversely as the square of the distance. 288 ALGEBRA. 328. The symbol oc is read ^'varies as''-, thus, aozb is read "a varies as b.'" 329. If xccy, then x is equal to y multiplied by a constant quantity. Let x' and y' denote a fixed pair of corresponding values of X and y, and x and y any other pair. Then by the definition of § 323, X y xJ — = —., or x = —y. x' y'' y' x' Denoting the constant ratio — by m, we have y' X = my. 330. It follows from §§ 325, 326, 327, and 329 that : 1. Ifx varies inversely as y, x = — y 2. If X varies jointly as y and z, x = myz. 3. If X varies directly as y and inversely as z, x = 331. Problems in variation are readily solved by convert- ing the variation into an equation by aid of §§ 329 or 330. PROBLEMS. 332. 1. If a; varies inversely as y, and is equal to 9 when y = S, what is the value of x when y = 1S? If X varies inversely as y, we have x = — (§ 330). p " y (\M- Putting X = 9 and y = S, we obtain 9 = — , or m = 72. 8 Then, x = — ; and if y = IS, x= — = 4, Ans. % y 18 2. Given that the area of a triangle varies jointly as its base and altitude, what will be the base of a triangle whose altitude is 12, equivalent to the sum of two triangles whose bases are 10 and 6, and altitudes 3 and 9, respectively ? Let B, H, and A denote the base, altitude, and area, respectively, of any triangle, and B' the base of the required triangle. Vf Oy'A V > 'A\ m oX^^,^^ VARIATION. 289 Since A varies jointly as B and if, we have A = mBH (§ 330) . Then tlie area of the fii-st triangle is m x 10 x 3, or 30 m, and the area of the second is ?>i x 6 x 9, or 54 m. Whence, the area of the required triangle is 30 m -+- 54 wi, or 84 m. But the area of the required triangle is also m x JS' x 12. Therefore, 12 mB' = 84 m, and B' = 7, Ans. 3. li y^ X, and is equal to 40 when x = 5, what is its value when a; = 9 ? ^ 4. li ycc ^, and is equal to 48 when 2 = 4, what is the expression for y in terms of z^ ? 5. If a; varies inversely as y^ and is equal to ^ when y = ^, what is the value of y when a; = f ? 6. If z varies jointly as x and y, and is equal to f when y = * and a; = j, find the value of z when x = ^ and 2/ = f • 7. If X varies directly as y and inversely as z, and is equal to ^ when y = 27 and 2; = 64, what is the value of X when y = 9 and 2 = 32 ? 8. If 5 a; H- 8 oc 6 2/ — 1, and a; = 6 when ?/ = — 3, what is the value of x when y=7? 9. If a;* Qc y^, and a; = 4 when ^ = 4, what is the value of y when x = ^? 10. The distance fallen by a body from a iDOsition of rest varies as the square of the time during which it falls. If it falls 257 J feet in 4 seconds, how far will it fall in 6 seconds ? 11. Two quantities vary directly and inversely as x, respectively. If their sum equals — |i when a; = 1, and — ^ when a; = — 2, what are the quantities ? 12. The area of a circle varies as the square of its diame- ter. If the area of a circle whose diameter is 4 is ^^-j what will be the diameter of a circle whose area is y^ ? 13. If the volume of a pyramid varies jointly as its base and altitude, find the base of a pyramid whose altitude is 11, equivalent to the sum of two pyramids, whose bases are 13 and 14, and altitudes 6 and 7, respectively. 290 ALGEBRA. 14. Given that y is equal to the sum of two quantities which vary directly as x^ and inversely as x, respectively. If y = — ^ when x = l, and y = -^J- when x = — 2, what is the value of y when a; = — i ? 15. Three spheres of lead whose radii are 6, 8, and 10 inches, respectively, are melted and formed into a single sphere. Find its radius, having given that the volume of a sphere varies as the cube of its radius. 16. The volume of a cone of revolution varies jointly as its altitude and the square of the radius of its base. If the volume of a cone whose altitude is 3 and radius of base 5 is ^^, what will be the radius of the base of a cone whose volume is ^f ^ and altitude 5 ? 17. If 7 men in 4 weeks can earn $ 238, how many men will earn $ 127^ in 3 weeks ; it being given that the amount earned varies jointly as the number of men, and the number of weeks during which they work ? 18. If the volume of a cylinder of revolution varies jointly as its altitude and the square of its radius, what will be the radius of a cylinder whose altitude is 3, equiva- lent to the sum of two cylinders whose altitudes are 5 and 7, and radii 6 and 3, respectively ? 19. If the illumination from a source of light varies in- versely as the square of the distance, how much farther from a candle must a book, which is now 15 inches off, be removed, so as to receive just one- third as much light ? 20. Given that y is equal to the sum of three quantities, the first of which is constant, and the second and third vary as X and a^, respectively. If y = — 19 when x = 2, 2/ = 4 when a; = 1, and y =2 when a; = — 1, what is the expres- sion for y in terms of a;? (Represent the constant by I, and the other two quantities by mx and nac^.) PROGRESSIONS. 291 XXX. PROGRESSIONS. ARITHMETIC PROGRESSION. 333. An Arithmetic Progression is a series of terms each of which is derived from the preceding by adding a con- stant quantity called the common difference. Thus, 1, 3, 5, 1, 9, 11, ••• is an arithmetic progression in which the common difference is 2. Again, 12, 9, 6, 3, 0, —3, ••• is an arithmetic j)rogression in which the common difference is — 3. 334. Given the first term, a, the common difference, d, and the number of terms, n, to find the last term, I. The progression is a, a -\-d, a -f 2 d, a 4- 3 c?, • • • . It will be observed that the coelficient of d in any term is 1 less than the number of the term. Then in the ?ith or last term the coefficient of d is n — 1. That is, l = a-\-(n- 1) d. (I.) 335. Given the first term, a, the last term, I, and the num- ber of terms, n, to find the smn of the terms, S. JS = a +(a + d)-\-(a + 2 d)+ ••. +(l-d)-\-l I Writing the terms in reverse order, S = l-\-(l-d)-\-{l-2d)-\- ... -}-(a4-fZ)+a. ^ Adding these equations term by term, 2>Sf=(a-f-0 + (a + + (« + 0+ ••• +(a + 0-h(« + 0- Therefore, 2S = n(a + I), and S = '^ (a\ f)- (H-) 336. Substituting in (II.) the value of I from (I.), we have /S = ^[2 a -f (n - l)c^]. 292 ALGEBRA. EXAMPLES. 337. 1. Find the last term and the sum of the terms of the progression 8, 5, 2, ••• to 27 terms. In this case, a = 8, d = 5 — 8 = — 3, and n = 27. Substituting in (I.), Z = 8 + (27 - 1)(- 3)= 8 - 78 = - 70. Substituting in (II.), ;S' = —(8 _ 70) = 27 x ( - 31) = - 837. Z Note. The common difference may be found by subtracting the first term from the second, or any term from the next following term. Find the last term and the sum of the terms of : 2. 3, 9, 15, ••• to 12 terms. 3. -7, -12, -17, ••• to 15 terms. 4. -69, -62, -55, •.• to 16 terms. 5. I -| -| - to 17 terms. 6. I |, g, ...to 13 terms. 7. -^,h^y"' to 22 terms. 3' 2' 3' 3 5 11 8. --, --, -— , ... to 55 terms. 9. -% -% -%, ... to 19 terms. 5 2 5 10. 2a-56, 6a-26, lOa + h, ... to 9. terms. 11. -^, I -^2~' •" to 10 terms. 338. If any three of the five elements of an arithmetic progression are given, the other two may be found by sub- stituting the given values in the fundamental formulae (I.) and (II.), and solving the resulting equations. PROGRESSIONS. 293 5 5 1. Given a = — -, n = 20, S = — -; find d and /. o o Substituting the given values in (II.) > we have _§=10f_^ +Z V-or -^ = -^+Z; whence, Z = ^ - ^ = ?• 3 V 3 / 6 3 ' 3 6 2 Substituting the values of Z, a, and w in (I.), we have §. = -^^igd; whence, 19^=- + ^ = ^, and d = h 2 3 2 3 6 6 2. Given d = - 3, / = - 39, ^S = - 264; find a and n. Substituting in (L), - 39 = a + (w - 1)(- 3), or a = 37i - 42. (1) Substituting the values of S, a, and / in (II.), we have -264=-(3n-42-39), or -528=3 n2-81 n, or w2-27 n=-176. wi.^«o^ 27 ± \/729 - 704 27 ± 6 i^^, n Whence, n = — == = — == — = it) or il. 2 2 Substituting in (1), a = 48 - 42 or 33 - 42 = 6 or - 9. Therefore, a = 6 and « = 16 ; or, a = — 9 and 71 = 11, ^ns. Note 1. The interpretation of the two answers is as follows : If a = 6 and n = 16, the progression is 6, 3, 0, - 3, - 6, - 9, - 12, - 15, - 18, - 21, - 24, - 27, - 30, _ 33, _ 36, - 39. If a = — 9 and n = 11, the progression is - 9, - 12, - 15, - 18, - 21, - 24, - 27, - 30, - 33, - 36, - 39. In each of these the sum is — 264. 113 3. Given a = -, ^ — — zr^^ ^ — ~7y''> ^^^ ^ ^^^ ^' Substituting in (I.), ^ = | + (^^ " 1) ( " ^ (^^* ^^^ Substituting the values of S, «, and I in (II.), we have _3^n/l 5-_n\ ^^ _ 3 ^ J9-^A orn-^-9n = 36. 2 2V3 12 r - V 12 ] Whence, n ^ 9 ± VsTTTii ^ 9_^ ^ ,2 or - 3. 294 ALGEBRA. The value w= — 3 is inapplicable, for the number of terms in a progression must be a positive integer. Substituting the value 7i = 12 in (1), 1= ^^^li^ = _ -1 . 7 Therefore, I = and n = 12, Ans. 12 Note 2. A negative or fractional value of n is inapplicable, and must be rejected, together with all other values dependent upon it. EXAMPLES. 4. Given d = 5, 1 = 71, 7i = 15 ; find a and S. 5. Given d = - 4, n = 20, aS = - 620 ; find a and /. -^ 6. Given a = - 9, n = 23, 1 = 57; find d and S. 7. Given a = — 5, n = 19, S = — 950 ; find d and I. 8. Given a = -, 1 = —, 8 = ^^; find d and n. 4 4 2 3 -- 9. Given 1 = — , ?i = 19, 8 = 0; find a and d 5 10. Given d = —, S = —, a = -; find I and w. 1^ o o 15 1 11. Given a = -, 1 = ——, d = — — ; find n and >S'. ^ -LJ. ^^ — 12. Given c2 = i, n = 17, S=17; find a and /, z 13. Given Z = 6, d = -, aS = 24; find a and n. 6 14. Given Z = -5i n = 21, ^ = - 38i ; find a and d ■"-^ 16. Given a = , / = — ^, S = —91 ; find d and ii. 16. Given a = -, n = 15, 8 = ^7— ; find d and I. 4 8 17. Given a = —-, d = , >S' = -^ ; find w and I. PROGRESSIONS. 295 18. Given^ = -| d==-^, S = -^', find « and n. -^ 19. Given «= 5, c^ = -|, /S' = -80; find ?i and Z. o From (I.) and (II.), general formulce for the solution of examples like the above may be readily derived. 20. Given a, d, and S ; derive the formula for n. By § 336, 2 ^ = w[2 a + (n - 1) d], or dn^ +(2a-d)n = 2S. This is a quadratic in ?i ; and may be solved by the method of § 261. Multiplying by 4(Z, and adding (2 a - d)2 to both members, 4 dhi^ + 4 d(2 a - d)n + (2 a - rf)2 = 8 d^ + (2 a - d)\ Extracting the square root, 2dw + 2a - d = ±y/%dS + {2a - dy. Whence, n = d -2a ±VSdS -H^a - d)^ ^„^ 2d 21. Given a, I, and n ; derive the formula for d. 22. Given a, n, and S] derive the formulae for d and /. 23. Given c?, ?i, and iS' ; derive the formulae for a and I. 24. Given a, d, and Z ; derive the formulae for n and S. ^ 25. Given cZ, i, and »i ; derive the formulae for a and S. 26. Given /, n, and ^S' ; derive the formulae for a and d. 27. Given a, d, and S ; derive the formula for I. 28. Given a, I, and aS ; derive the formulae for d and n. 29. Given c?, ?, and S ; derive the formulae for a and ?i. 339. To insert any immher of arithmetic means between two given terms. 1. Insert 5 arithmetic means between 3 and — 5. We are to find an arithmetic progression of 7 terms, whose first term is 3, and last term — 5. 296 ALGEBRA. Tutting a = S, I = - 6, and n = 7, in (I.), § 334, we have 4 3" 4 — 5 = 3 + 6 (Z ; whence, 6cl =— S, and d = Hence, the required progression is 3, -, -, —1, — -, — — , —5, Ans. 3 3 3 3 EXAMPLES. 2. Insert 6 arithmetic means between 3 and 8. 10 5 3. Insert 4 arithmetic means between — and — -• 4. Insert 5 arithmetic means between and 1. o 3 9 5. Insert 7 arithmetic means between and — 2 2 6. Insert 8 arithmetic means between — - and — 5. 3 7. Insert 9 arithmetic means between - and — 11. 340. Let X denote the arithmetic mean between a and 6. Then, by the nature of the progression, X — a — h — X, or 2x = a-\-h. Whence, ^ = ^^' That is, tlie antlimetic mean between two quantities is equal to one-half their sum. EXAMPLES. . Find the arithmetic mean between : i5.,3 o2a — 1-, 2a + l 1. — and —■^- 3. and - — -^-• 12 20 2a + l 2a-l 2. (x + ly and (x - 7)1 4. ^^-±-^ and - 4^!' ^ a — 6 a^ — If PROGRESSIONS. 297 PROBLEMS. 341. 1. The sixth term of an arithmetic progression is 5 16 -, and the fifteenth term is — Find the first term. 6 3 ^ ^ i^ By § 334, the sixth term is a-\- &d, and the fifteenth term a + 14 d. \a+ 5d=| (1) Then by the conditions, I _ \a+Ud = ^± (2) I o Q 1 Subtracting (1) from (2), 9d = - ; whence, d = — 2 ^ Substituting in (1), a + - = - ; whence, a = — -, Ans. 2 6 3 2. Find four numbers in arithmetic progression such that the product of the first and fourth shall be 45, and the product of the second and third 77. Let the numbers be x — 3 y, x — y, x + y, and x + Sy. /• x^ — 9 v'-^ = 45 Then by the conditions, < ^ 2 — 77 K X y — II, Solving these equations, x=9, y=±2 ; or, a;= — 9, y =±2 (§ 276). Then the numbers are 3, 7, 11, 15 ; or, — 3, — 7, — 11, — 15. Note. In problems like the above, it is convenient to represent the unknown quantities by symmetrical expressions. Thus, if five numbers had been required to be found, we should have represented them by x — 2 y, x — y, x, x-\- y, and z -\-2y. 3. Find the sum of all the integers beginning with 1 and ending with 100. 4. Find the sum of all the even integers beginning with 2 and ending with 1000. 5. The 8th term of an arithmetic progression is 10, and the 14th term is — 14. Find the 23d term. 6. Find four numbers in arithmetic progression such that the sum of the first two shall be 12, and the sum of the last two - 20. 298 ALGEBRA. 7. Find the sum of the first 15 positive integers which are multiples of 7. 8. The 19th term of an arithmetic progression is 9 a;— 2 y, and the 31st term is ISx —Sy. Find the sum of the first thirteen terms. 9. Find four integers in arithmetic progression such that their sum shall be 24, and their product 945. 10. How many positive integers of three digits are there which are multiples of 9 ? 11. Find the sum of all positive integers of three digits which are multiples of 11. 12. The 7th term of an arithmetic progression is — I, the 16th term is ^, and the last term is ^-. Find the number of terms. 13. The sum of the 2d and 6th terms of an arithmetic progression is — f , and the sum of the 5th and 9th terms is — 10. Find the first term. 14. Find five numbers in arithmetic progression such that the sum of the second, third, and fifth shall be 10, and the product of the first and fourth — 36. 15. If m arithmetic means be inserted between a and b, what is the first mean ? 16. How many positive integers of one, two, or three digits are there which are multiples of 8 ? 17. How many arithmetic means are inserted between 4 and 36, when the second mean is to the first as 4 is to 3 ? 18. A man travels 3 miles the first day, 6 miles the second day, 9 miles the third day, and so on. After he has travelled a certain number of days, he finds his average daily distance to be 46^ miles. How many days has he been travelling ? PROGRESSIONS. 299 19. How many arithmetic means are inserted between | and — ^, when the sum of the first two is -^ ? 20. After A had travelled for 4i hours at the rate of 5 miles an hour, B set out to overtake him, and travelled 3 miles the first hour, 3^ miles the second hour, 4 miles the third hour, and so on; in how many hours will B over- take A ? 21. Find three numbers in arithmetic progression such that the sum of their squares is 347, and one-half the third number exceeds the sum of the first and second by 4i 22. The digits of a number of three figures are in arith- metic progression ; the sum of the first two digits exceeds the third by 3; and if 396 be added to the number, the digits will be inverted. Find the number. GEOMETRIC PROGRESSION. 342. A Geometric Progression is a series of terms each of which is derived from the preceding by multiplying by a constant quantity called the ratio. Thus, 2, 6, 18, 54, 162, ••• is a geometric progression in which the ratio is 3. Again, 9, 3, 1, ^, ^, ••• is a geometric progression in which the ratio is J. Negative values of the ratio are also admissible. Thus, — 3, 6, — 12, 24, — 48, ••• is a geometric progression in which the ratio is — 2. 343. Given the first term, a, the ratio, r, and the number of terms, n, to find the last term, I. The progression is a, ar, ai^, ai^, •••. It will be observed that the exponent of r in any term is 1 less than the number of the term. Then in the nih. or last term the exponent of r is w — 1. That is, I = ar^^-K (I.) 300 ALGEBRA. 344. Given the first term, a, the last term, I, and the ratio, r, to find the sum of the terms, S. >S' = a -f ar + ar'^ H h f«*"~^ + ct?*""^ + a>'""^ Multiplying each term by r, we have rS = ar + ar^ + ar^-\ \- af" + a>'"r^ + ar\ Subtracting the first equation from the second, rjSI—S = ar"" — a.-^y Whence, ^ ^ ar^ - a r — 1 But by (I.), § 343, rl = ar\ Therefore, ;S = ^^^-=-^. (II.) EXAMPLES. \ 345. 1. Find the last term and the sum of the terms of the progression 3, 1, -, ••• to 7 terms. o In this case, a = 3, r = -, and n = l. 3 Substituting in (I.), l = ^{^\' = l = A 243 ixJ--3 -L- 3 -2186 o V .-. *• • /TT N cr 3 243 729 729 1093 Substituting in (II. ), 8 = — j = — = 2~ "^ "243 ' 3"^ "3 -3 Note. The ratio may be found by dividing the second term by the first, or any term by the next preceding term. 2. Find the last term and the sum of the terms of the progression — 2, 6, — 18, ••• to 8 terms. ft In this case, a = - 2, r = = — 3, and w = 8. — Zi Then, ^ = - 2(- 3)7 = - 2 x (- 2187)= 4374. Ind, ^^ - 3 X 4374 -(- 2) ^ - 13122 + 2 ^ 3^3^^ _3_1 -.4 PROGRESSIONS. 301 Find the last term and the sum of the terms of ; 3. 1, 3, 9,... to 8 terms. 4. 6, 4, -, ••• to7 terms. 5. — 2, 10, — 50, ••• to o terms. 6. 2, 4, 8,..- to 11 terms. 7. -S, -,--,... to 9 terms. 8. --, -5, -10,... to 10 terms. r- 2 A 9. — 5, 2, — , ... to 6 terms. 5 10. --,-,--,'•• to 7 terms. 3' 2' 4' 11. ?,!,?... to 5 terms. 3' 2' 8' 12. _ ?, 3, - 12, ... to 6 terms. 4 346. If any three of the live elements of a geometric progression are given, the other two may be found by sub- stituting the given values in the fundamental formulae (I.) and (II.), and solving the resulting equations. But in certain cases the operation involves the solution of an equation of a degree higher than the second; and in others the unknown quantity appears as an exponent, the solution of which form of equation can usually only be affected by the aid of logarithms (§ 419). In all such cases in the present chapter, the equations may be solved by inspection. 1. Given a = — 2, n = 5, Z = — 32 ; find r and S. Substituting the given values in (I.), we have — 32 = - 2 »•* ; whence, r* = 16, and r = ± 2. 302 ALGEBRA. Substituting in (II. )» If r = 2, S = ^^~^P~^~^^ =-6i+2=~62, 2i — 1 _2- 1 -3 Therefore, r = 2 and .S = - 62 ; or, r = - 2 and aS^ = - 22, Ans. Note 1. The interpretation of the two answers is as follows : If r = 2, the progression is — 2, — 4, — 8, — 16, — 32, whose sum is - 62. If r = — 2, the progression is — 2, 4, — 8, 16, — 32, whose sum is -22. 2. Given a = 3, r = , & = —-— ; find n and I. ' 3 729 ' -lz-3 c ^, ^-^ *• • /TT N 1640 3 Z + 9 Substituting m (II. ) , -j^ = = -^. ~3~ Whence, Z 4- 9 = ; or, Z = 9 = • 729 729 729 Substituting the values of Z, a, and r in (I.), we have ' 729 V 3^ ' V 3; 2187 Whence, by inspection, n — 1 = 7, or w = 8. EXAMPLES. 3. Given r = 2, n = 9, ^ = 256 ; find a and >S. 4. Given ?' = -, ?i = 5, /S' = ; find a and I. 3 ' 27 ' 5. Given a = - 2, n = 6, Z = 2048 ; find r and /S'. ^ 6. Given a = 2, ?- = , Z = -: find n and aS'. ' 2/ 256' 7. Given r = i, 71 = 11, ^ = ^; find a and I 8. Given a = |, n = 9, ^ = ^; find r and /S. 3 128 PROGRESSIONS. 303 9. Given a = — S, 1 = — —, S = — '——; find r and ii. 10. Given «==-, ^* = ~o' '^~i7^' ^^^ ^ ^^^ ^^• 11. Given / = 192, r = -2, .S = 129; find « and n. 12. Given « = -^j ^"""199' "^"""19^' ^^^ '' ^"^ '^' From (I.) and (II.), general formulae may be derived for the solution of cases like the above. 13. Given a, ?•, and S ; derive the formula for I. 14. Given a, Z, and S ; derive the formula for r. 15. Given ?*, Z, and *S; derive the formula for a. 16. Given r, ?i, and Z ; derive the formulie for ci and S. 17. Given r, ri, and ^5; derive the formulae for a and /. 18. Given a, ?i, and I ; derive the formulae for r and aS. Note 2. If the given elements are u, Z, and /V, equations for a and r may be found, but there are no definite formuloe for their values. The same is the case when the given elements are a, w, and 8. The general formulae for n involve logarithms ; these cases are discussed in § 419. 347. The limit (§ 292) to which the sum of the terms of a decreasing geometric progression approaches, when the number of terms is indefinitely increased, is called the sum of the series to infinity. Formula (II.), § 344, may be written cr a — rl It is evident that, by sufficiently continuing a decreasing geometric progression, the last term may be made numeri- cally less than any assigned number, however small. Hence, when the number of terms is indefinitely increased, I, and therefore rl, approaches the limit 0. 304 ALGEBRA. Then the fraction ^ ~ ^ approaches the limit — — 1 — r 1 — r Therefore, the sum of a decreasing geometric progression to infinity is given by the formula S = -^. (III.) 1 — r EXAMPLES. 1. Find the sum of the series 4, — -, — , •••, to infinity. o J 2 In this case, a = 4, r = 4 12 Substituting in (III.), S = ^ = ^, Ans. 3 Find the sum of the following to infinity : 2. 3, 1, 1 .... 6. 7 21 63 4' 32' 256' •. 3. 16, -4,1, .... --7. 2 15 5' 3' 18' .... 4. _1 1 _JL ... ' 5' 25' • 8. 1 1 8' 18' 2 81' 5. 5 10 20 3' 9' 27'*"* ^ 9. 5 5 35 7' 8' 64' .... 348. To find the value of a repeating decimal. This is a case of finding the sum of a decreasing geometric series to infinity, and may be solved by formula (III.). 1. Findthe value of .85151.... We have, .85151 ... = .8 -I- .051 + .00051 + .... The terms after the first constitute a decreasing geometric pro- gression, in which a = .051 and ?' = .01. .051 .051 51 17 Substituting in (III.), S 1 - .01 .99 990 .330 8 17 281 Then the value of the given decimal is 1- — -, or — -, Ans. 10 ooO oov PROGRESSIONS. 305 EXAMPLES. Find the values of the following : 2. .8181.... 4. .69444.... 6. .11567567.... 3. .296296.... -5. .58686.... -7. .922828-... 349. To insert any number of geometric means between two given terms. 128 1. Insert 5 geometric means between 2 and — — • i ^u We are to fiud a geometric progression of 7 terms, whose first 128 term is 2, and last term -— • ' 729 Putting a = 2, 1 = — , and n = 7, in (I.), § 343, we have 1^ = 2r6 ; whence, r"^ = — , and r = ± ^. 729 ' 729 3 Hence, the required result is o j_4 8 ^16 32 , 64 128 .^ ^' ^3' 9' ^27' 81' ^24"3' 729' '^'''' EXAMPLES. 2. Insert 4 geometric means between 3 and 729. 1 64 3. Insert 6 geometric means between - and — — ■• 6 o 4. Insert 5 geometric means between 2 and 128. 2 125 5. Insert 3 geometric means between — - and -— 5 o 6. Insert 4 geometric means between — - and 3584. z 243 2 7. Insert 7 geometric means between — - and — • 350. Let X denote the geometric mean between a and h. Then, by the nature of the progression, - = -, or a^ = ab. a X 306 ALGEBRA. Whence, x — Vab. That is, the geometric mean between tivo quantities is equal to the square root of their product. EXAMPLES. Find the geometric mean between : 1. 2ijandlif. 2. 9 + 4V5 and 9 - 4 VS. 3. a- + 2 ab + b^ and a^-2 ah + b\ . 2x^ + 4:xy xy -\-2 y- xy — 2y^ 2x^ — 4:xy PROBLEMS. 351. 1. Find three numbers in geometric progression such that their sum shall be 14,. and the sum of their squares 84. Let the numbers be a, ar, and ar^. i a + ar + ar2 = 14. Then by the conditions, \ (1) (2) Dividing (2) by (1), a - ar -Y ar'^ = Q. (3) Subtracting (3) from (1), 2 ar = 8, or r = -• a (4) Substituting in (1), a + 4 + — = 14, or a2 _ 10 a = - 16. a Solving this equation, a = 8 or 2. Substituting in (4), r = | or | = 1 or 2. ' 8 2 2 Therefore, the numbers are 2, 4, and 8, Ans. 2. The 4th term of a geometric progression is — ^-, and the 7th term is |^f f . Find the second term. 3. The sum of the first and last of four numbers in geo- metric progression is 112, and the sum of the second and third is 48. Find the numbers. 4. The product of three numbers in geometric progres- sion is — 1000, and the sum of the squares of the second and third is 500. Find the numbers. PROGRESSIONS. 307 5. A man saves every year half as much again as he saved the preceding year. If he saved $ 128 the first year, to what sum will his savings amount at the end of seven years ? 6. A body moves 12 feet the first second, and in each succeeding second five-eighths as far as in the preceding second, until it comes to rest. How far will it have moved ? 7. The 5th term of a geometric progression is — f , and the 9th term is — ^^. Find the 11th term. 8. If m geometric means be inserted between a and b, what is the first mean ? 9. The sum of three numbers in arithmetic progression is 12. If the first number be increased by 5, the second by 2, and the third by 7, the resulting numbers form a geo- metric progression. What are the numbers ?*" 10. Divide $ 700 between A, B, C, and D, so that their shares may be in geometric progression, and the sum of A's and B's shares equal to $ 252. 11. There are four numbers, the first three of which form an arithmetic progression, and the last three a geometric progression. The sum of the first and third is 2, and of the second and fourth 37. What are the numbers ? 12. Find che ratio of the geometric progression in which the sum of the first ten terms is 244 times the sum of the first five terms. 13. There are three numbers in geometric progression whose sum is 19. If the first be multiplied by |, the second by |, and the third by -f, the resulting numbers form an arithmetic progression. What are the numbers ? HARMONIC PROGRESSION. 352. A Harmonic Progression is a series of terms whose reciprocals form an arithmetic progression. 308 ALGEBRA. Thus, 1, I, "l^, f, i, ••• is a harmonic progression, because the reciprocals of the terms, 1, 3, 5, 7, 9, •••, form an arith- metic progression. 353. Any problem in harmonic progression which is sus- ceptible of solution, may be solved by taking the reciprocals of the terms, and applying the formulae of the arithmetic progression. There is, however, no general method for finding the su7n of the terms of a harmonic progression. 354. Let X denote the harmonic mean between a and h. Then, - is the arithmetic mean between - and r (§ 352). X a b ^ ^ Whence by § 340, - = — rr— = ,. ^ , and x = — — j- •^ ' X 2 2ab' a + 5 EXAMPLES. 355. 1. Find the last term of the progression 2, |, ■§-, ••• to 36 terms. Taking the reciprocals of the terms, we have the arithmetic pro- 13 5 gression -i -» -> •••. Jj 2i 2i In this case, a = -, d=\^ and n = 36. Substituting in (I.), § 334, we have Z = ^- 4- (36 - 1) x 1 = — • Taking the reciprocal of this, the last term of the given harmonic progression is — , Ans. 2. Insert 5 harmonic means between 2 and — 3. We have to insert 5 arithmetic means between - and 2 3 Putting a = -, Z = — , and n — 7, in (I.), § 334, we have 2 3 _ 1 = 1 + 6 cZ ; whence, 6 d = - -, or (Z = - A. 3 2' ' 6' 36 PROGRESSIONS. 309 Then the arithmetic progression is 3. 1 13 2 1 1 7 1 2' 30' y' 12' 18' 36' 3' Therefore, the required harmonic progressior lis 2 36 9 ' 13' 2' 12, - 18, -f, - 3, Ans. Find the last terms of the following : 1 ?4 1 ' 5' 7' '' • •• to 13 terms. 4 4 12 5' 43' 12 71' *" to 25 terms. 6. -3, 2, -, ... to 38 terms. 4 6. -|-|-g, ...to43terms. « 5 2 5 ^ ,^ , 7. --,__,--, ... to 17 terms. 8. Insert 6 harmonic means between 2 and 9 2 2 9. Insert 7 harmonic means between — - and — 5 7 10. Insert 8 harmonic means between — - and — -. 5 5 Find the harmonic mean between : 11. 3 and 6. 12. l^H^ and - ^ ~ ^ 1-f-a; l+ar^ 13. The first term of a harmonic progression is x, and the second term is y ; continue the series to three more terms. 14. The arithmetic mean between two numbers is 1, and the harmonic mean — 15. Find the numbers. 15. The 5th term of a harmonic progression is — f , and the 11th term is — ^. What is the 15th term ? 16. Prove that, if a, b, and c are in harmonic progression, a: c = a — b :b — c. 310 ALGEBRA. XXXI. THE BINOMIAL THEOREM. POSITIVE INTEGRAL EXPONENT. 356. The Binomial Theorem is a formula by means of which any power of a binomial, positive or negative, inte- gral or fractional, may be expanded into a series. We sh3,ll consider in the "present chapter those cases only in which the exponent is a positive integer. 357. Proof of the Binomial Theorem for a Positive Inte- gral Exponent. By actual multiplication, we obtain: (a -\- xy = a^ -\- 2 ax -\- Qc^ ] (a -{-xy = a^-hS A -h 3 aa^ + a^ ; "* (a -^xy = a*-\-4: a^x + 6 cv^x^ -^ 4 ax^ + x* ; etc. In the above results, we observe the following laws : .1. The number of terms is greater by 1 than the expo- nent of the binomial. 2. The exponent of a in the first term is the same as the exponent of the binomial, and decreases by 1 in each suc- ceeding term. 3. The exponent of x in the second term is 1, and in creases by 1 in each succeeding term. 4. The coefficient of the first term is 1, and the coefficient of the second term is the exponent of the binomial. 5. If the coefficient of any term be multiplied by the exponent of a in that term, and the result divided by the exponent of x in the term increased by 1, the quotient will be the coefficient of the next following term. THE BINOMIAL THEOREM. 311 358. If the law« ui. s o57 l)e assuuit-d to hold for the expansion of (a + x)", where n is any positive integer, the exponent of a in the first term is n, in the second term 71 — 1, in the third term w — 2, in the fourth term n — 3, etc. The exponent of x in the second term is 1, in the third term 2, in the fourth term 3, etc. The coefficient of the first term i& 1 ,' of the second term n. Multiplying the coefficient of the second term, n, by w— 1, the exponent of a in that term, and dividing the result by the exponent of x in the term increased by 1, or 2, we have —\- — — ^ as the coefficient of the third term ; and so on. Then, (a + a!)» = a-. + wa'-'as+^i-fe-I^a—V -f- + !L(!Lzilfc21a-V+.... (1) Multiplying both members of (1) by a + x, we have I J. • ^ 4|a^4^ na-^ar^. + ^^ (^^ ~ •^) aP-^x^ + • • • . Collecting the terms which contain like powers of a and x, = a"+' + (?. + 1) a"a; + n r?i^ + llo"-'a!' * A point i? often used in place of the sign x ; thus, 1 • 2 is the same as 1 x 2. 312 ALGEBRA. Or, (a + xy+^ = a"+i + {n -\- 1) a"x -f 71 f^Lill a"-^a^ + 1.2 = a"+i + (w + 1) a^'x + ^^^+^)^ a" ^ V 1 • ^ I (^^ + l)^(n-l) 2 , 1.2.3 * It will be observed that this result is in accordance with the laws of § 357 ; which proves that, if the laws of § 357 hold for any power oi a + x whose exponent is a positive integer, they also hold for a power whose exponent is greater by 1. But the laws have been shown to hold for (a + xy, and hence they also hold for (a + xy ; and since they hold for (a + xy, they also hold for (a + xy ; and so on. Therefore, the laws hold when the exponent is any posi- tive integer, and equation (1) is proved for every positive integral value of 7i. Equation (1) is called the Binomial Theorem. Note 1. The above method of proof is known as Mathematical Induction. Note 2. In place of the denominators 1-2, 1-2 -3, etc., it is usual to write [2, [3, etc. The symbol \n, read '■'■ factorial w," signifies the product of the natural numbers from 1 to w inclusive. 359. Putting a = 1 in equation (1), § 358, we have (1 + a;)" = 1 4- nx + -^TS — ^x^ + — ^ 7^ ^^ + •••• EXAMPLES. 360. In expanding expressions by the Binomial Theorem, it is convenient to obtain the exponents and coefficients of the terms by aid of the laws of § 357, whi^-b have been proved to hold for any positive integral exconefit. THE BINOMIAL THEOREM. 313 1. Expand (a -|- xy. The exponent of a in the first term is 5, in the second term 4, in the third term 3, in the fourth term 2, in the fifth term 1. The exponent of x in the second term is 1, in the third term 2, in the fourth term 3, in the fifth term 4, in the sixth term 5. The coefficient of the first term is 1 ; of the second term, 5. Multiplying the coefficient of the second term, 5, by 4, the exponent of a in that term, and dividing the result by the exponent of x in the term increased by 1, or 2, we have 10 as the coefficient of the third term; and so on. Then, (a + x)^ = a& + 5 a^^ + 10 aV-^ + 10 «%» + 5 ax* + x^ Ans. Note 1. The coefl&cients of terms equally distant from the begin- ning and end of the expansion are equal. Thus the coefficients of the latter half of an expansion may be written out from the first half. If the second term of the binomial is negativey it should be enclosed, sign and all, in a parenthesis before applying the laws. In reducing jifterwards, care must be taken to apply the principles of § 186. 2. Expand (1 — xf. We have, (l-a;)6 = [l + (-x)]« = 16+6 . 15 . (_a;) + 15 . 1* . (- x)2+20 . 13 . (-x)3 + 15 . 12 . (- a-)* + 6 . 1 . (- xY + (- x)6 = 1 - 6 X + 15x2 - 20 x3 + 15 X* - 6 x^ + x6, Ans. Note 2. If the first term of the binomial is numerical, it is con- venient to write the exponents at first without reduction. The result should afterwards be reduced to its simplest form. If either term of the binomial has a coefficient or exponent other than unity, it should be enclosed in a parenthesis be- fore applying the laws. 5 — -^ ^^^ ^ _ rTvu z. aA>5 3. Expand (3 m' - Vn)\ !{_ o ^ (3 9»2 _ ^„.)4 ^ [ (3 ,^2) 4. ( _ ^*) ]4 = (3 ?n2)4 + 4(3?n2)3(- J) + 6(3 j?i2)2(_ n^)2 + 4(3 m2)(- 71^)3 + (-n^)4 = 81 7/18 - 108 mhi^ + 54 m%* - 12 m^n + n\ Ans. 314 ALGEBRA. Expand the following : 24. f2a^+-i^Y. 2aV 4. (x + iy. J. 15. (1+2 my. 5. (a + xf. 16. (l-x)«. 6. (a-o^y. ,^ 17. (^0^4 + ^ 1)5^ g^^ /^^2__L 7. (m-n)^ ^ 18. (a^- 2/. ^ ^^' 8- a+^-y. . 19. (3-f.^y. ^«- (2--^-v^)- 9. (a -by. ^^ ^1 ^_^6 ^^ fx-'^ ^5/-3^ (-!.»)• - (¥-«)• 10. (a'-\-b'cy. 11. (a.-2'« + /«)^. ^, 2^^ (4at-a;^y. ^8. (V^^ + 4^< 12. (2a-iy. ' , 4X5 / I 3 14. (a -3 by. 23. (m' + 5 a;-^. y 30. (2a -36)^ 31. (ah-i + a-hhl 32. (s^I^-2^jlJ. A trinomial may be raised to any power by the Binomial Theorem if two of its terms be enclosed in a parenthesis and regarded as a single term. 33. Expand (x^-2x- 2y. (a:2 _ 2 X - 2)4 = [ (a:2 _ 2 x) + ( - 2)]^ = (x^-2 xy + 4(a;2 - 2 x^i- 2) + 6(a;2 - 2 x)2( - 2)2 + 4(^2 -2a;)(- 2)3 + (-2)4 = a:8-8«7 4. 24 ic6_32x5+ 16x4 -8(x6-6xH 12^4-8x5) + 24 (x* - 4 x3 + 4 x2) - 32 (x2 - 2 x) + 16 s«x8-8x7+16x6 4- 1^x5-56x4-32x3 + 64 x2 + 64 X + 16, Ans. Expand the following : 34. (i_a; + a^)4. 37. (a^-2x-Sy. 35. (x' + x + 2y. y38. (l+x-x^y. 36. (l--i-3x-xy. 39. (x'-x + 2y. THE BINOMIAL THEOREM. 315 361. To find the rtk or general term in the expansion of (a -h xy. The following laws will be found to hold for any term in the expansion of (a -f xy\ in equation (1), § 358 : 1. The exponent of x is less by 1 than the number of the term. 2. The exponent of a is n minus the exponent of x. 3. The last factor of the numerator is greater by 1 than the exponent of a. 4. The last factor of the denominator is the same as the exponent of x. Hence, in the rth term, the exponent of x is r — 1. The exponent of a is ?i — (r — 1), or n — r -f 1. The last factor of the numerator is n — r 4- 2. The last factor of the denominator is r — 1. Therefore, the ?'th term of the expansion of (a + .^•)" is n(n - l)(n-2) ... (n - r + 2) ^._^,^_, EXAMPLES. 362. In finding any term of an expansion, it is convenient to obtain the coefficient and the exponents of the terms by aid of the laws of § 361. 1. FindtheSth termof (3a*-6-^)^\ We have, (3 a* - ft-i)" = [(3 a*) + (- 6-i)]".. In this case, n = 11 and r = 8. The exponent of (— 6-i) is 8 — 1, or 7. The exponent of (3 a^) is 11 — Y, or 4. The first factor of the numerator is 11, and the last factor 4 + 1, or 5. The last factor of the denominator is 7. Hence, the 8th term ^ 1^ • tO • ^ • 8 • 7 • 6 • 5 .3 oi)4(_ 5-1)7 1.2.3.4.5.6.7 ^ 2' ^ ^ = 330 . (81 a2)(- 6-") = -26730a26-7, Ans. 316 ALGEBRA. Note. If the second term of the binomial is negative, it should be enclosed, sign and all, in a parenthesis before applying the laws. If either term of the binomial has a coefficient or exponent other than unity, it should be enclosed in a parenthesis before applying the laws. Find the 2. 4th term of (a + xf. 3. 9th term of (m + 1)". 4. 5th term of (a - bf. 5. 10th term of (1 - x^. 6. 9th term of (m^ - ny\ 7. 7fhtevmoifa-' + ~\ 8. 4th term of \y ^J 9. lOth term of (a*" + a^'f^ 10. 8th term of (x-^- - 2 yfy^ 11. 6th term of (a~^ -f 3ar*)'°. / 3 1 \13 12. 7th term of f a^ ^] • \ WbJ 13. 8th term of (a"^ + 2^xy\ / 14. 5th term of (m^ + ^ 15. 6th term of fx'' - ^^ • 16. Find the middle term of e-0' UNDETERMINED COEFFICIENTS. 317 XXXII. UNDETERMINED COEFFICIENTS. CONVERGENCY AND DIVERGENCY OF SERIES. 363. A Series is a succession of terms so related that each may be derived from one or more of the preceding in accordance with some fixed law. A Finite Series is one having a finite number of terms. An Infinite Series is one the number of whose terms is unlimited. The progressions, in general, are examples of finite series ; but in § 347 we considered infinite geometrical series. 364. Infinite series may be developed by Division. Let it be required, for example, to divide 1 by 1 — x. l-x)l(l-\-x^x'+ ... 1-a; X X — x^ Therefore, — i-= 1 + a; + ar^+ ••• . 1 —X Infinite series may also be developed by Evolution (see Exs. 25 to 30, § 195), and by other methods, one of the most important of which will be considered in § 369. 365. A series is said to be convergent either when the sum of the first n terms approaches a certain fixed quantity as a limit (§ 292) ; when n is indefinitely increased ; or when the sum of all the terms is equal to a fixed finite quantity. A series is said to be divergent when the sum of the first n terms can be made to numerically exceed any assigned quantity, however great, by taking n sufficiently great. 318 ALGEBRA. 366. Consider, for example, the infinite series I. Suppose X = Xi, where Xi is numerically < 1. The sum of the first n terms is now l + x,-{- X,' + ••• + xr' = \^^^ (§ 86). If n is indefinitely increased, Xi" decreases indefinitely in absolute value, and approaches the limit 0. Then the fraction approaches the limit 1 — Xi 1 — a'l That is, the sum of the first n terms approaches a certain fixed quantity as a limit, when n is indefinitely increased. Hence, the series is convergent Avhen x is numerically < 1. II. Suppose x = l. In this case, each term of the series is equal to 1, and the sum of the first n terms is equal to n ; and this sum can be made to exceed any assigned quantity, however great, by taking n sufficiently great. Hence, the series is divergent when x = l. III. Suppose x = — l. In this case, the series takes the form 1 — 1 + 1— iH , and the sum of the first ii terms is either 1 or according as n is odd or even. Hence, the series is neither convergent nor divergent when a; = -l. IV. Suppose X = iCi, where x^ is numerically > 1. The sum of the first n terms is now 1 + a^i + o^i' + - + xr' = ^^^ (§ 86). iCj — 1 By taking n sufficiently great, — — can be made to numerically exceed any assigned quantity, however great. UNDETERMINED COEFFICIENTS. 319 Hence, the series is divergent when x is numerically > 1. 367. Consider the infinite series developed by the fraction (§ 364), i — X Let x= .1, in which case the series is convergent (§ 366). The series now takes the form 1 + -1 + -01 H-.OOl + •••, while the value of the fraction is — , or — • In this case, however great the number of terms taken, their sum will never exactly equal — ; but it approaches this value as a limit. (See § 347.) Thus, if an infinite series is convergent, the greater the number of terms taken, the more nearly does their sum approach to the value of the expression from which the series was developed. Again, let x = 10, in which case the series is divergent. The series now takes the form 1 + 10 -f- 100 + 1000 + .-., while the value of the fraction is — , or — - J. — J.U J In this case it is evident that, the greater the number of terms taken, the more does their sum diverge from the value — -• 9 Thus, if an infinite series is divergent, the greater the number of terms taken, the more does their sum diverge from the value of the expression from which the series was developed. It follows from the above that an infinite series cannot be used for the purposes of demonstration, unless it is convergent. 368. The infinite series a -\- hx -\- ca? -\- dx^ -[-••• is convergent when x = 0', for the sum of all the terms is equal to a when x = 0. 320 ALGEBRA. THE THEOREM OF UNDETERMINED COEFFICIENTS. 369. An important method for expanding expressions into series is based on the following theorem : The Theorem of Undetermined Coefficients. If the series A -\- Bx -\- Cx' -h Bx^ + ••• is always equal to the series A' -}- B'x + G'x^ + D'x^ + •••> ivhen x has any value which makes both series convergent, the coefficients of like powers of x in the series will he equal; that is, A = A', B = B', G=C',etc. For since the equation A -^ Bx + Cx" -\- Dx^ -i- '•' = A' -{- B'x -^ C'x^ -f D'x^ + ••. is satisfied when x has any value which makes both mem- bers convergent, and since both members are convergent when ic = (§ 368), it follows that the equation is satisfied when a; = 0. Putting x = 0, we have A = A'. Subtracting A from the first member of the equation, and its equal A' from the second member, we obtain Bx + Cx^ -\-Dx'-^'" = B'x-\- Cx" -{- D'x^ + .... Dividing each term by x, B-^Cx-\-Dx'-\-'" = B' + Cx + D'x" + •••• This equation also is satisfied when x has any value which makes both members convergent ; and putting x = 0, we have B = B'. In like manner, we iliay prove C = C, D = D', etc. 370. A finite series being always convergent, it follows from the preceding article that if two finite series A-]-Bx-j-Cx'+ ... +AV and A' + B'x-^C'x--{- .•• +K'x^ are equal for every value of x, the coefficients of like powers of X in the two series are equal. UNDETERMINED COEFFICIENTS. 321 EXPANSION OF FRACTIONS INTO SERIES. 2-;u ,2 _ a^ 371. 1. Expand -^ '— in ascending powers of x. 1 — 2x -\- Sx- Assume ^--Sg^-x^ ^ ^ ^ ^Jx + Cx^ + Dx^ + ^x* + ••• ; (1) 1 - 2 a; + 3 x"-^ where A, B, C, D, £*, etc. , are quantities independeut of x. Clearing of fractions, and collecting the terms in the second mem- ber involving like powers of x, we have 2-Zx^-x» = A+ B -2 A x-\- C -2B + 3^ x2+ D -2C + 35 x3+ E -2D 4-3C x*+.... (2) The second member of (1) must express the value of the fraction for every value of x which makes the series convergent (§ 307). Hence, equation (2) is satisfied when x has any value which makes both members convergent ; and by the Theorem of Undetermined Coefficients, the coefficients of like powers of x in the series are equal. Then, A = 2. B-2A = 0; whence, B = 2A =4. C-2B4-3.4=-3; whence, C = 2B-3^-3=-l. D-2C+SB = -l; whence, D = 2C-SB-\=-U. E-2D + SC = 0; whence, E = 2D-SC =-27; etc. Substituting these values in (1), we have 2-3a;2-a^ =2 + 4a; - a;^ - ISar^ - 27 a:^ + ..., Ans. l-2a; + 3a;2 The result may be verified by division. Note 1. A vertical line, called a bar, is often used in place of a parenthesis. Thus, + B \xis equivalent to (B — 2 A)x. -2a\ Note 2. The result expresses the value of the given fraction only for such values of x as make the series convergent (§ 367). If the numerator and denominator contain only even powers of x, the operation may be abridged by assuming a series containing only the even powers of x. 322 ALGEBRA. 2 4- 4:x'^ — X* Thus, if the fraction were — — , we should as- sume it equal to ^ + Bx^ + Cx'^ + Dx^ + j&V + • • •. In like manner, if the numerator contains only odd powers of x, and the denominator only even powers, we should assume a series containing only the odd powers of x. If every term of the numerator contains x, we may as- sume a series commencing with the lowest power of x in the numerator. If every term of the denominator contains x, we determine by actual division what power of x will occur in the first term of the expansion, and then assume the fraction equal to a series commencing with this power of x, the exponents of X in the succeeding terms increasing by unity as before. 2. Expand — -^ — — in ascending powers of x. O 3/ — X^ Dividing 1 by 3 x^, the quotient is — ; we then assume ^ = Ax-^ + Bx-^ + C-\- Dx + Ex'^ + .... (1) 3 a;2 - X Clearing of fractions, we have l = 3^ + 3J5|a; + 3C|a;2 + 3Z)|x3 + 3J5;|a;4+ .... -^I-^I - c\ - d\ Equating the coefficients of like powers of «, 3^ = 1. 35-^ = 0. 3 O - J5 = 0. 32)- = 0. 3 j^ - D = ; etc. Whence, A = \ B = -, C = —, D=—, E = —, etc. 3' 9' 27' 81' 243' Substituting in (1), we have V + ^ + -^ + £- + ^+ •••, Ans. 3 x2 - a;8 3 9 27 8i 243 UNDETERMINED COEFFICIENTS. 323 EXAMPLES. Expand each of the following to five terms in ascending powers of x : 3 1 -f 5a; o 2a; + 3r^ 1+a; ' . 3-2a; ■ 1-A.x - 2-f 7a;^ -a? 6. 7. 2-h3a;2 1-x- \-2x-x' 3ar^ 8. 9. 10. 11. 12. iJ^hx-lx"' 1 ^x'-bx^' l-2a; 2-3x-\-4:x'' l-4ar^ + 6a;^ 1 + 2a;-ic2 • 2 + a;-3a;^ 1 — 4:X-\- oay^ 13. 14. 15. 16. 17. 1 — 7 X^ — 4:X^ x'-5x*-2a^' x^-Sa^-^-x*' a^-4:X^ + 2a^ 2-Sx'-x^' 3-2x-\-a/ 3-4aT^ 2a;4-ar'-3iB*' EXPANSION OF RADICALS INTO SERIES. 372. 1. Expand Vl — a; in ascending powers of x. Assume Vl - x = A ^ Bx + Cx^ + Dx^ + ^x* + .... Squaring both members, we have by the rule of § 187, (1) 1 - X = ^2 I a; + ^ ^2Ab\ -\-2AC X2 -^2 AD + 2BC Equating the coefficients of like powers of x, ^•^ = 1 ; whence, ^ = 1. 2AB^-\ X8+ C2|X*+.. ^2Ae\ + 2Bd\ whence, B = = 2A 2 52 + 2^C = 2^2) + 25C=0 C^^2AE-¥2BD = whence, C 2A TiC whence, D = = A whence, E = — 1 8* J_ 16' C^-{-2BD Substituting these values in (1), we have x2 x3 5x* \/n^ = i 8 16 128 The result may be verified by evolution. 2A , Ans. 128 etc. 324 ALGEBRA. EXAMPLES. Expand each of the following to five terms in ascending powers of x : 2. Vl+4a;. 4. Vl-f2a;-a^. 6. = 0; etc. UNDETERMINED COEFFICIENTS. 331 Solving these equations, A = i^ = 1 8' C = 3 , D = 13 128' , etc. Substituting in . (1), X-. ^h 8^ ^ A ,3. 128^^ Ans. If the even powers of x are wanting in the given series, the operation may be abridged by assuming x equal to a series containing only the odd powers of y. EXAMPLES. Revert each of the following to four terms : 2. y = X — x^ + x^ — X* -\ . ^ 2 3 4 4. y = x-\-2x^-^Sa^-\-4:X*-{ . 5. y = x — Sx^-\-5x^— 7 X* -{-"-. a»2 /j«3 ^4 7. 2, = ? + ?' + ^ + ?'+.... * 2 4 6 8 8. y = x + 3? + 2x?-^Bx'-\ . 9. y = x+=f + t + =i+.... 357 332 ALGEBRA. XXXIII. THE BINOMIAL THEOREM. FRACTIONAL AND NEGATIVE EXPONENTS. 381. It was proved in § 359 that, if n is a positive integer, (l + ^)" = l + n^+^^^^:t^ + "("-|^("-^V + -.(l) I 382. Proof of the Theorem for a Fractional or Negative Exponent. I. When the exponent is a positive fraction. Let the exponent be -, where p and q are positive integers. p ^ \ By § 211, (1 + x). = -V(l + xy = -^l+pa^+..., by (1). It is evident that a process may be found, analogous to those of §§ 194 and 200, for expanding ■\/' 1 -i- px -\- • -- in ascending powers of x ; and the first term of the result will evidently be 1. Assume then, ^/l -\-px -]- -" = 1 -\- Mx-\- JVx^ -^ -". (2) Raising both members to the gth power, we have 1 +^^_ ... = [1 ^.(Mx + iVa^-h ...)? = 1 -^q(Mx + iV^a^ + •••)+ '", by (1). This equation is satisfied by every value of x which makes both members convergent; and by the Theorem of Unde- termined Coefficients (§ 369), the coefficients of x in the two series are equal. P That is, p = qM, or 3f = -• Substituting this value in (2), we have (l-faj>-=l+|aj+— (3) THE BINOMIAL THEOREM. 333 II. When the exponent is a negative integer or a negative fraction. Let the exponent be — s, where s is a positive integer or a positive fraction. By § 214, (1 + .)-. = ^. = ^^_, by (1) or (3). It is evident that can be expanded by division in a series proceeding in ascending powers of x-, thus, 1 + S«-|- '•')'^(X —SX+ '" lH-sa;-f--'- — SX— '" That is, (1 + it')" = 1 - SX -j- . . .. (4) From (3) and (4), we observe that, when n is fractional or negative, the form of the expansion is (1 -h a;)" = 1 4- nx -{- Ax^ -\- Bx^ + -". (5) Writing - in place of x, we obtain a M X .x^ ^a? Multiplying both members by a", (a + xy = a" + na'^-^x + ^a'*- V -f Ba^'-^a? H . (6) This result is in accordance with the second, third, and fourth laws of § 357 ; hence, these three laws hold for frac- tional or negative values of the exponent. We will now prove that the fifth law of § 357 holds for fractional or negative values of the exponent. Let P denote the coefficient of a;*", and Q the coefficient of ic*'+^, in the second member of (5). Then (5) and (6) may be written (1 + a;)" = 1 4- na; -f ••• + iV- + Qaf+i + •••, (7) and (a -f xy = a" + na'^^x -}-••• + Pa'^-'x'- + Qa'»-'- V+^ + • • .. (8) 334 ALGEBRA. In (8) put a = 1 + ?/ and x = z; then, (1 + 2/ + 2=)" = (1 + VT + ••• + P(l + 2/)"-'-2!'- + ..-. (9) Again, in (7) put x = z-\-y'^ then, (1 + z + yy = 1 + '" + P{z + yy + q(z ^ yy^^ 4- •••. Expanding the powers oi z -\-yhj aid of (8), we have (1 + 2; + y)'* = 1 + ... + P[z^ + rz'-^y + •••] 4- e[^''+^ + (r + lK2/ +...] + .... (10) Since the first members of (9) and (10) are identical, their second members must be equal for every value of z which makes both series convergent; and by the theorem of Unde- termined Coefficients, the coefficients of 2'" in the two series are equal. Or, P(l 4- yy-' = P+ Q{r + l)y-\- terms in y"^, f, etc. Expanding the first member by aid of (7), this becomes P[l 4- {n - r)y 4- ••.] = P + Qir + % +•... This equation is satisfied by every value of y which makes both members convergent, .and hence the coefficients of y in the two series are equal. That is, P{n - r) = Q(r + 1), or Q = P(n-r) ^ r-\-l But in the second member of (8), n — r is the exponent of a in the term whose coefficient is P, and r 4- 1 is the exponent of x in that term increased by 1. Hence, the fifth law of § 357 has been proved to hold for fractional or negative values of the exponent. By aid of the fifth law, the coefficients of the successive terms after the second, in the second member of (8), may be readily found as in § 358 ; thus, (a + xy = a" -h na^-^x 4- ^^^ ~ ^^ a" V n(n-l)(n_--2}^,.3^ ^..^ ^^^^ THE BINOMIAL THEOREM. 335 The second member of (11) is an infinite series ; for if n is fractional or negative, no one of the quantities n — 1, II — 2, etc., can become equal to zero. The result expresses the value of (a -\- x)" only for such values of a and x as make the series convergent (§ 367). EXAMPLES. 383. In expanding expressions by the Binomial Theorem when the exponent is fractional or negative, the exponents and coefficients of the terms may be obtained by aid of the laws of § 357, which have been proved to hold universally. If the second term of the binomial is negative, it should be enclosed, sign and all, in a parenthesis before applying the laws ; if either term has a coefficient or exponent other than unity, it should be enclosed in a parenthesis before applying the laws. 1. Expand (a + x)^ to four terms. The exponent of a in the first term is f ; in the second term, — ^ ; in the third term, — | ; in the fourth term, — | ; etc. The exponent of x in the second term is 1 ; in the third term, 2 ; in the fourth term, 3 ; etc. The coefficient of the first term is 1 ; of the second term, f ; multi- plying the coefficient of the second term, |, by — |, the exponent of a in that term, and dividing the result by the exponent of x in the term increased by 1, or 2, we have — ^ as the coefficient of the third term ; and so on. Then, (a + x)^ = a^ + f a'^x - I a'^x'^ + j\ a'^x^ , Ans. 2. Expand (1 — 2x~^-)-^ to five terms. (l-2a;~5)-2 =[!+(_ 2.x-^)]-1i = 1-2 -2.1-3. (._ 2a;"^)+ 3 • l"* • (-2x~h^ - 4 . 1-5 . (- 2a;"2)3 + 5 • l-« • (- 2x~h* - ... = 1 + 4 x~^ -f 12 x-^ + 32 x""2 -f 80 x-^ + • ••, Ans. 336 ALGEBRA. 3. Expand to five terms. 1 1 =[(«-!) + (3 x^)]-^ ^(a-i + Sx^) (a-i + 3x3)3 .4 1 _1 - if (a-i)"'^(3xb^ + 3j¥ir ia-^y'^\sx^y - - 1 4 1 7 2 10 lA 4 Expand each of the following to five terms : V 4. (a-^x)^. 5. {l + x)-\ 6. (i-x)-i 7. Va — X. fi 1 S/l+^ 9. 1 10. {xi-2y)i. ^15. .1 11. (m-2+vV'. , , ^,_3 y 12. (a'-2x^^yK 1 . 17. ^[(a.-t-32/t)-]. y . 18. ^- i_9 + 1, or - Y-. The last factor of tjie denominator is 6. THE BINOMIAL THEOREM. 337 Hence, the 7th term 1 _4 _7 10 13 3 " 3 ' 3 ' 3 * 3 1.2.3.4 16 3 _19 _3 Note. The note to § 362 applies with equal force to the examples in the present article. Find the 2. 5th term of (a + x)K 3. 7th term of (a + &)"^. 4. 12th term of (1 - x)-\ ^' 5. 6th term of {x-^^-^y^y. . 6. 9th term of {(i-^2xf. 7. 5th term of V(l - xf 8. 7th term of (a' - x^)^. 1 9. 10th term of {x -\- mf 10. 8th term of (m*- 2 ?i-^)"l 11. 9th term of V(a - x)\ 12. 6th term of (a^ - fe-^)"^. 13. 8th term of {x-^ -h S?/"^)"^. 14. 10th term of (x Vf - -r^V** 15. 11th term of (a^ + 3 6"*)i 385. Extraction of Roots by the Binomial Theorem. 1. Find V25 approximately to live places of decimals. We have, v^ = 25^ = (27 - 2)^ = (3* - 2)i Expanding by the Binomial Theorem, we have [(38) + (-2)p=(38)U|(38)-^(-2)-l(38)-f(-2)2 338 ALGEBRA. Or, . ^^ = 3--^- + 40 3.32 9.35 .81.38 Expressing the value of each fraction approximately to the nearest fifth decimal place, we have V'25 = 3 - .07407 - .00183 - .00008 = 2.92402 ... , Ans. Rule. Separate the given number into two parts, the first of ivhich is the nearest perfect poiver of the same degree as the required root. Expand the result by the Binomial Theorem. Note. If the second term of the binomial is small compared with the first, the terms of the expansion diminish rapidly ; but if the second term is large compared with the first, it requires a great many terms to ensure any degree of accuracy. EXAMPLES. ' Find the approximate value of each of the following to five places of decimals: 2. V26. 4. 7.962 3 "^ 7.962 3 ^ '^ log .03296 = 8.5180 - 10 log 7.962 = 0.9010 3 )27.6170 - 30 9.2057 - 10 = log .1606. Result, -/1606. Find the values of the following : .4 36. 4^x7l 3t 10/79 46* 38. 39. 40. V 1 4 307 J 7.543 50. (25.467)i« X (- .062)12. 51. a/5106.5 X .00003109. 52. (83.74 X .009433)7. 53. (4.8671)^ X (.17543)i g^ A./3:928 X A/eKi^ 55. V72L32 (.573)^ 49. V.004978 - (.25691) 8693.8 X m''--3Mj=l^ 22. x2+x?/+!/2. 23. l-2a2+4a*-8rt6. 24. 8xH12x2f/ + 18xy2+27y3. 25. w2_3 771-4. 26. 3x2-x-2. 27. a2+a-l. 28. 2x2+9x-5. 4 ALGEBRA. 29. 4m2-2w?i2+w*. 30. x*-2x^+ix^-Sx+16. 31. 10 a^+Sa-i. 32. m^-1. 33. a + 3. 34. 4 x'«+V - 4 a: V- 35. 2 a'^ + 2 a^b + 2 a%^ + 2 «&«. 36. a^ + a26 + ab^ + ft^. 37. 2m2-3. 38. 4a2-12a + 9. 39. 2x3 + 5a:2 - 8a; - 7. 40. x3 - 3x2 - 3. 41. a2 _ 2 a + 10. 42. x^ -6xy + 9y^. 43. 3x3 - x2 - 2 X - 5. 44. 2 a^ - 5 a2 _ 6 a + 4. 45. m^ — 2 m2« - mn^ + 2 n^. 46. 4a + b - c. 47. x? + 2/? - ^'•. 49. x-c. 50. x2+(a + &)x + a&. 51. x-2 6. 52. (a + 6)x-c. 53. (w — n)x— j9. 54. x + a. 55. x2 — (6 + c)x + 5c. 66. a(6-c)+d 57. a+(2m-3yi). § 62 ; pages 45 to 47. 2. 270. 3. -9. 4. 42. 5. 729. 6. -5. 7. -— • 15 8. -748. 9. 854. 10. — • 11. - -• 16. 9(x + t/)2-25. o 2 17. 63(a - &)2 - 20(a -b)- 32. 18. 2(m + n)+S. 19. (X - y)2 -(X - y)+ 1. 21. -V-a - ^jft + i§c. 22. -l^x + ify-T'iT^r. 23. -la-j\b-{-ii-c. 24- - tV* ~ I y - H^' 25. ^7X3 - -27. 26. ija^--j\a^b-hj\ab^-^\b^. 27. fx2-|x + .V 28. I ^2 _ I ^6 + ^ 62. 29. G54p54 _ 2 a2p+3533+2 + Qj656g. 30. x'«+i - x3?/2«+i. 31. a2i'+3 + ai'+2623-i + a6+i. 47. (a + 6)x2 + {a^ + 52^^^; -2ab{a-\- 6). 48. (a - 6)2 - 2 c(« - 6) + c^. 49. x2'« /x«y»* + 2/2n. 50. -4 a* - I a2x2 -^ax^- yV a:*- / 51. x^ + (- a + 6 — c)x2 +(— aj/— be + cd)x + a6c. 52. X2i' + X2? + X2'- - 2 XP+? + 2'XP+'' — 2 X9+»". 53. |x2 - 1 X + f. 54. x2 + (a - 6)x - ab. 55. x^ + ?/3 + ^3 - 3x^/0. 56. 2 a252 + 2 62^2 + 2 c2a2 - a^ - 6* - c*. ANSWERS. § 75 : pagres 51, 52. 3. 14. 8. 2. 13. 5 ll' 18. -6. 24. 2 7* 29. 32 11* 4. -7. 9. 5 7* 14. 4 5' 19. 5. 25. 4. 30. _1 4' 5. 4. 10. - 4 3* 15. 3. 21. -5. 26. 7 5' 31. -1, 6. -5. 11. 1. 16. 8 9' 22. 2. 27. 1, 2* 82. 10 3* 7. -9. 12. 2 3" 17. 8. 23. -10. 28. -6. § 77 ; pagres 55 to 58. 6. 10,9. 6. 169,87. 7. 24,14. 8. A, $7.50 ; B, $5.25 ; C, $9.25. 9. A, 65 ; B, 13. 10. A, 42 ; B, 84. 11. A, $ 12 ; B, $36. 12. 9 five-cent pieces, 7 twenty-five cent pieces. 13. 8. 14. 17. 15. 6 fifty-cent pieces, 11 dimes. 16. 47, 29. 17. 9, 4. 18. 13, 7. 19. A, 43 ; B, 57. . 20. 9 oxen, 27 cows. 21. 3 dollars, 12 dimes, 15 cents. 22. 3750 infantry, 500 cavalry, 125 artillery. 23. A, 320 ; B, 1600 ; C, 3840. 24. A, $25 ; B, $ 18 ; C, $40 ; D, $32. 25. Wife, $864 ; each son, $72 ; each daughter, $216. 26. A, $42; B, $23; C, $29; D, $31. 27. 13 three-penny pieces, 36 farthings. 28. 44, 27. 29. 324 sq. yd. 30. 12. 31. 35, 36, 37. 32. A, 68 ; B, 18. 33. 8 $ 2 bills, 13 fifty-cent pieces, 24 dimes. 34. 7, 8. 35. 3, 4, 6, 6. 36. Worked 22 days, was absent 10 days. 37. 6 bushels of first kind, 18 bushels of second kind. 38. 75 men on a side at first ; whole number of men, 5668. 39. First class, 75 ; second, 115 ; third, 150 ; fourth, 195. 40. 18. 41. A, 8 minutes ; B, 5 minutes. 42. 15 pounds of first kind, 35 pounds of second kind. § 82 ; pagres 60, 61. 25. a2 -I- 2 ac + c'^ - h\ 30. 1 - a^ - 2 a6 - h'^. 26. x^-2xy-\-y^-z^. 31. a;* -2x2 + 1. 27. rt2 - 62 _ 2 6c - c2. 32. a2 - 4 62 + 12 6c - 9 c2. 28. a*-a2 + 2a-l. 33. a4 + a262+64. 29. x*-5a;'2 + 4. 34. 9^2 - 16y2 _ 16?/^ - 4^2. 6 ALGEBRA. § 99 ; pag-es 72, 73. 38. (a-6 + c)(a-6-c). 44. (3 a-4 6 + 2c)(3a-4 &-2c), 39. (m-h n+p){m + n -p). 45. {4:X-\-2y- 5z)(4:X-2 y + 5 z). 40. (a + x-\-y){a-x- y). 46. {m-2n-{-x) (w - 2 « - x). • 41. (x + ?/-s)(x-?/ + ;2;). 47. (2a + 6 + 3)(2a-6-3). 42. (« + 6 + 2)(a + 6-2). 48. {bx -^ y -\-Zz){hx ^ y - ^z). 43. (1 + w -«)(!- wi + n). 49. (a-6 + c-fZ)(a-6-c+d). 50. (a + X + 6 - ?/)(« + X - 6+ ?/). 51. Cx — wi + y + n) (x — m — 2/ — n). 52. {X + y + a + h){x ^- y - a - h). 53. (2 a + 6 + 3 c - 2) (2 a + 6 - 3 c + 2). 64. (x - 4 y + + 6) (x - 4 y - - 6). 55. (5 a — m + & — 3 n) (5 a — «i — 6 + 3 n). § 106 ; pages 78 to 80. 30. (1 + 702(1 - ny. 45. (2x + ^y)\2x-^yY. 41. (a + 3)2(a - 3)2. 46. (a - l)2(a2 + « + 1)2. 42. (x+l)(x-2)(x2 + x + 2). 52. (3 « + 2)2(3 a - 2)2. 43. (a + 26X«-26Xc+3(?Xc-3fZ). 53. (x - 2)(x + 3)(x -3)(x + 4). 44. (x + l)(x-l)(x-4)(x-6). 54. (a - 1)*. 55. (a - x)(6 + y)(a2 + ax + x2)(62 _ &y + y'l), 57. (6a + 2fo-7c)(6a-25 + 7c). 59. (x + l)2(x + 2)2. 60. (a + l)(a-2)(a2_a + i)(a24.2a + 4). 63. 2 &c(a + 6 + c) (a - 6 - c). 64. (a-l)(rt + 3)(a + 4)(a + 8). 66. (x - l)(x + 2)2(x - 3). 67. {a-\-h-\-c){a-h + c)(a + 6 - c)(a - 6 - c). 76. (m + x) (m2 - 4 «ix + 7 x2). 77. &(3 a2 - 3 a6 + &2). 78. (x-y)(9x + y). 79. (a + 6)(a2 - 3a& + 6'0- 80. (a + ?> + c + d)(a + &-c-(Z). 81. 2x(x2 + 3). 82. (x + 2/) (2x2 + 2/2). 83. (a + l)2(a-l)2(a2 + i). 84. (a + X + 6 - ?/)(« + X - 6 + ?/). 85. m(x - m)(m - 3x). 86. 2^/(3x2 + 2/2). 88. (x + l)2(x-l)(x2+l)(x2-x+ 1). 89. 3a(a-l). 90. 7(5m - l)(w2 - »» + 1). 91. {x + y -z){x-y + z)(x^y -\- z){x-y - z). 92. (a-5 6+4c+3d)(a-5&-4c-3cZ). 93. (l + a)(3-a-a-). ANSWERS. 7 § 117 ; page 89. 5. a; - 1. 6. 2 a + 3. 7. x + 2. 8. x - 3. 9. m + 1. 10. 3 a - b, 11. 3a-^ + «x-2x2. 12. a;(2x-5). 13. 3x + 4y. 14. 2 a^ - 3 a2 - a 4- 4. 15. 2 )7i2 - mn + n"-. 16. x - 2. 17. a'^ + 2a + 4. 18. m'2 - 2 mx - 3x2. 19. a - 1. 20. 7n\m + 2). 21. a - 5 6. 22. x + 3. 23. 3 a^ - 2. 24. a + 4. 25. 2 x - y. 26. 2x2-3x-l. 27. x - 2. 28. ax{a + x). § 118 ; page 90. ^2. 2x-9. 3. 4a + 1. 4. 3m + 4. 5. 5a -26. 6. x + 2. 7. a + 1. 8. m- 1. 9. 2x-3y. § 125 ; page 93. 30. (x + y -h z)(x - y -\- z)(x - y - z). 40. (m + n)^ (m - n)2. 41. (^a + b + c)(a - b - c)(a -{- b - c). § 126 ; pages 94, 95. 2. (2x + 7)(2x2-19x + 45). 5. x^(ax - ?/)(8x2 + 21 xy + 10y2). 3. (a-4)(3a2 + 14a-5). 6. 3(4m + 5)(4 m^-ll «i2-6?)i + 9). 4. (3a+86)(12a2 + 16a6-362). 7. (2a + 3)(3a5 - 14a2 - a + 6). 8. x(2 a^ - ax + 3 x2) (2 a^ + 5 a-x + 2ax^- x^). 9. (2 a - 3 6)(a* + a^ft - 5 a262 + 2 aft' + 64). 10. (3x-2)(4x*-5x2+4x-3). 11. (a2-3a + 2)(4a3-9a-4). 12. 2 wn(3 m2 - mn - 2 n^) (3 m^ - 2 m^n - 7 mn2 - 2 n^). 13. a2(a2 - 2 a + 3) (3 a* + 11 a3 - 6 a2 - 7 a + 4) . 14. (x2-x-3)(3x4 + 7x3 + 6x2-2x-4). § 127 ; page 95. 1. 8x* + 20x^-46x2-117x-45. 2. 162 a* + 117 a3 - 147 a^ - 62 a + 40. 3. 12 m* - 10 to3 - 8G ??i2 + 140 7rt - 48. 4. 24 x7 - 70 x6 - 15 x5 + 25 x* + G x'. 5. a^ + 2 a* - 10 a3 - 20 a2 + 9 a + 18. 12. § 133 ; pages 98, 99. 3a 4 6* 13- ?1- 5 2/2 14 « + 2 j5 x(x-2) a-l x-6 jg 5 a + 2 6 5a-26 8 ALGEBRA. 17 in -S j^g X + y jg a( 8ff + 7 a;) g^ a; - 9?n m(w — 0) ' 2xy x(8a — 7x) ' x-\-Zm a-+2a + 4 gg 2 m-5 „, x + y-^z „^ U«--12«6 + 16 />2 21 25. 1. 26. ^ + ^ + ^ + ^ . 27. , a — b + c — d Sx 31. ^^^. Z/ + X 5x+7' ' 2a-r ' 2ic-9y' ' 3??z+4 "" x^-Sx+'l „ og-r ^M g 3x\2 g 2q! + 1 ' 3x + l' * 6a-l' ,Q ,jitr—_iii^rj^^ 11 g^ + 3 qx 4- x^ ■ m2 + 4 m - 2* " a2 - 2 ax - 4x2* § 137 ; page 102. 5. 4x — 6-^ 11. 3 a ■ — 2x + 3 4a-l 6. a;2 + xy + 2/2 + ^i^. 12. 37n2 + 4- '^ ^^ + ^ . X — y 4 m^ + 1 7. a2-a6 + &2_J^. 13, a;^ - x2y + x?/2 _ y3 + Jj^. a-\-b x + y 8. 5rt2_3rt-l ? 14. 6a + 7 2a-3 3. a- 2a- -2 -1* 3 a + 2 ?; 4a -6 7/l2 - m + 3 3 a + 4 * 3 a-2 - 4 a f 5 9. 2m + 5« + -^^-?^. 15. a^-\ra^b + a'^b'^-^abHbH^' 2 m — 5 n a — o 10. 2X-1+ ^'~^ — 16. 4x2 + 6x-2 ^^"^ x2 - X - 1 2 x2 + X - 3 § 138 ; page 103. g 3a2-lla + 2 ^ 2^ g 10a-^-13a-9 g 4x2-10x-7 3a x-y ' 2a -3 ' 5x 7 _?A_. 8 !^iilzJi!. 9 10^ 10 ^^^' 11 ^-^'^ 3a + & *w + n 2a-5x ■3x-4 x + 2y j2 8m34-24m-^-36m-27 ^3 4a'^+5a ^^ 5a2-23ay> + 86^ 2TO + 3 ' ' 2a-l ' ■ 4a-3 6 j^ 2x2y + 2ry 2 jg oM^t ^^ -3x2 ^g M n^ x^+xy + ^Z"-^ «-& x2+x+l ' ?n--3?nn + 9«- ANSWERS. 6. 10. 12. 8. 10. 13. 19. Qx^-Sx § 140 ; page 105. 4a62_4 53 2x(9x2-l)' 2x(9a;2-l)" ' {a-b)(ia^-hb^) \a-b){a^-{-b^) 3(a-l)(a2 + l) Q(a + l)(a^ + l) 9(a^ - 1) a* - 1 ' a*-l ' a* - 1 * 2 x3 - 16 3a;S-f 6a;^ + 12a;S 3x5_i2x* 3x2(x-4)(x3-8)' 3x2(a;-4)(x3-8)' 3x2(x - 4)(x3 - 8)" " y2 (g - 6)2 11. (a-6)(x-y)2' (a-6)(x-y)' (a+5)2 a2-9 q2_4 (a+2)(a-3)(a + 5)' (a + 2)(a-3)(a + 5)' (a+2)(a-3)(a + 6) 21a -4 . ^ 4x2 + 3m2 96 m:*: 20x8-4x2+57x+35 40x3 § 142 ; pages 106 to 111. 20x2- 18y . 6x+l 15 xV 48 g ab ■\-bc+ ca abc 4 6c-9ra+8a6 g 3a2-14x2 18a2x2 10a ^63 28 11. 24rt6c 12. 3x-10w 53 X 36* m2-l 14. 20 16. 5a 7X-22 (2x+l)(5x-6) 2m2 + 2n2 m2 — n2 2x x-y 24. 108 21. 4x x2-l 27 18. 30 X 11a -9 (3a + 5)(4a-7) «!±^. 22 «V15a + 3 a2-62' • a[-2_3fl5_28" 4a 25. 10 a?) 4a2-l 31. 35. - (x-2)(x + 6)(x-9) 32. 1. 12a4-18 a(a-3)(a+6) b a-\-b 4x2 (x+2)3' 4x 1 + x* 4^2 ^^^ . 40. 0. 41. 8aH&^^ {x-yY 42. (2a + 36)2(2a-3 6) Q 10 ax (x-3a)2(x-7a)' 2y2 x2-?/2' 8 x+2" 2n2 33. 37. (ni — n)^ 44. 48. 45. a2-a4-l x-3 49. 0. 46. 4a&2 47. 50. 11 (l+a;)(2-x)(3+x) • a(a2-x2)* ,Q 2x2?/ + 2xv2 38- ^S_y3 13 7-3x (24-x)(4-x) a3 - 2 x3 ^ (a + x)(a3-x3) - 53. -^^ + ^V xy(x-y) 10 ALGEBRA. 54 lOx-1 . 55 o:^. 56 4 5 ^ ^_ 12(x-2) a2-9 m(16-TO2) 1-a - ^' 60. ^^li^. 61. ^ "- 2 m 0:2 _ 4 ^ _j_ 1/ 9 _ 4 Qj2 ?)i + 2 64. ^ + ^f - 65. ?^ 66. 0. § 144 ; pages 112, 113. 4. -^. 5. I 6. 2abc. 7. ^. 8. ^^. 5m2 3 a3 32 y* 9. 5(a4-6) jQ 3m+l ^^ 2a:2(^-.3) ^^ ^C^+^y) 3(a + l) ■ m-5 " (a;-C)2 * ac(ic+?/) 13 (a-46)(a-2&) ^ ^^ -^(^^ - 1) . 15 l le -J^. rt(a-3&) ' (x + 2)(x2+a: + l) ' 2 ' a-l j^^ 2x-3y ^ jg^ (x + y-^)2 ^ ^g («Z1^. 20 1. 21 ^±^. a;-y (x-?/-2)2 (a + b)^ 2x § 146 ; pages 114, 115. 3 3g« 4 9m^^ 5 3Ca; + 3) g m(2w + 5w) 76xV' ' 4«*/ '2(x-2)' ■ n(im-3n)' ^ 3(2a-56) g ff(a + 7) ^ a;Cx + 2y) ^^ a;2 6(4a + 36)' ■ («-3)2* ' ^ " " x2,~ 1* jj a(a-2) _ j2 (a + 26)(a-56) jg 2« + a; ^^ ff + 6 + c a + 5 ' (a + 86)(rt + 46) ' a + 2x * a-6 + c § 148 ; pages 115 to 117. J, 2 5 3^±x±l^ g w-M Y 2x-3" 2 m — 1 X wi — H 9. ^+Ay. 10. «. 11. ^ + 2^ . 12. X + 2 y 6 ^/ 13. ^Zll. 14. ^'-^^' . 15. a + 1. X + y a2 - 62 jg 103x4-78 jg 2-3 a g^ « + 36 ' 39X+30* ■ 5-7a' ' 3a-6' 22 ^ ■ 23 2(^ + y> . 24 ItiUL^lJll. ' 1 + x2 ' (x-yy m a2 + &2 6 ^- 8. X. (X- 3)Cx + 2) X 16. 2 a2 _ 3 62 7a6 21 2(x-«) X 4- « S 5. «^ § 149 ; pages 117 to 119. «262 ^ 9X2 4a+3 "■ a'^-ab + b^' ' (2a-3x)2 " i + ar + y + xy J 79(1-31 2 a'^ft^ 3 9^^ A 1 ANSWERS. 11 1, 6. 2^-=^. 7. «^^1^. 8. 0. 9. a2 + 1 2 ic^^'+n'). 20. ^-±-^. 21. ^— • 22. 2 w'^C?)*''^ — 71^) a — b I + X- 23. («-^^^ 24. ^^'-^^-^^ 25. 2 26. 1, 2{a + b) Sx^-4x-2 a{x + 2a) 27. ^-=^- 28. a;2+i. 29. m + 2n. 30. " + ^~^ x'f+y^ ab{a-c){c-b) 31^ x^ + a;y + y\ ^^ 2{x + y) . 33^ Oj^ 34. -^. 36. 8^^ . 36. ^"""-^^ . 37. ^^<^^ + ^) a8-256 (x2-l)(x2-4) (a-6)(a2+62) 38. ^(^^-1> . 39. 20xl-_34 x4 + x2 + l (3x-l)(2x + 5)(4x + 3) 13 a 40. (2a-3)(3a + 4)(5a-2) § 151 ; pages 120 to 124. 2. 10. 13. 8 3* 23. 1 4* 35. -4. 45. 4 5* 3. -2. 14. — 5. 24. 2. 36. 4. 46. -2. 4. 3 15. 4 25. 5 37. 11 47. 1 2 3 2 6 2 6. 3 16. 2 26. 2 38. 7 48. _9 5 3 7 3 2 6. 5 7* 17. 6. 27. 3 6' 39. 3 5' 49. 1 11* 7. 1 2" 18. — 0. 30. 11 4* 40. 2 17' 50. 19 3 8. 4. 19. -1. 2 31. -1. 41. 2. 51. 2 3' 9. 5 8* 20. 7. 32. _4 5 42. 19, 9' 52. 6. 10. 4 3* 21. -1. 33. 1 3' 43. 43 7' 53. 2 6' 11. -1. 22. -4. •34. 1 3* 44. 3. 12 ALGEBRA. § 153; pages 125, 126. 2. II- 9. -2 a. 14. 1 3 6 n 3. -5^. 9. -?— . 15. ^. 21. - a a — b b 4. -a. 10. 2(a- &). 16. -3a. 5. 1^. 11. m + n. 17. ^. 3 w a 6. a-1. 12. 2^±^. 18. _^. 24. a + i!) 6 a + 6 2 3 ?7lJi ah a-b a-\-b m — \ 2 36 2 13. 12(a-5). 19. 2a -36. 25. -6. ■//* § 154; page 126. 2. .09. 4. 5. 6. — • 8. .6. 10. 0. 500 3. -4. 6. -20. 7. -.02. 9. -1.4. § 155 ; pages 127 to 135. 2. 40. 3. 56. 4. 42. 5. 27, 18. 6. 32, 24. 7. A, $40 ; B, $ 48 ; C, $ 36. 8. Water, 288 ; rail, 360 ; carriage, 120. 9. A, 24; B, 64. 10. $25. 11. $2.45. 14. 10 f. 16. l/j. 16. 15f hours. 17. If minutes. 18. 48. 19. 82. 20. 79. 21. 20. 22. A, 24; B, 48. 23. \' 26. 35, 14. 8 27. A, 30 miles ; B, 36 miles. 28. 107, 27. 29. — • 15 30. 59. 31. Horse, $250; carriage, $175. 32. 6. 33. Horse, $180; carriage, $280; harness, $30. 34. Express train, 45 miles an hour ; slow train, 30 miles an hour. 35. A, 32 miles ; B, 25 miles. 36. 120. 38. 38j2f minutes after 1. 39. 2,%^^ minutes after 6. 40. ^\^j minutes after 4. 41. lOi^f minutes after 5. 42. 87. 43. 22J miles. 44. A, 3 days ; B, 6 days ; C, 8 days. 45. 49Jj- minutes after 9. 46. A, $36 ; B, $32 ; C, $27. 47. 10}f minutes after 8. 48. 45 minutes. 49. A, $1200 ; B, $900. 60. Longer piece, 30 yards ; shorter, 24 yards. 61. $ 1840. 52. 2l^j and 54j-\ minutes after 7. 63. ANSWERS. 18 64. Gold, 1540 oz. ; silver, 420 oz. 65. $4725. 56. A, 4; B, 5; C, 6. 57. 2 p.m. 58. $ 1250 ill 41 per cent bonds, $ 1750 in 3^ per cent bonds. 69. 24 miles an hour. 60. 16j-\ minutes after 10. 61. 7. 62. $18000. 63. $2400. 64. Gold, 57 oz. ; silver, 70 oz. 65. Fox, 180 ; hound, 135. 66. $5400. § 156 ; pages 136, 137. an am ^ ^ am — amn „ a — an 2. , 3. A, years; B, years. m -\- n m -i- n m — n m —n 4. J?»iL_. 6. ^ 6. ^ - '''' dollars. 7. ^-^^• m-\- n ab + be + ca \ + r a — c 8. ^^ miles. 9. -^. 10. ^^'^^ dollars. 11. lM«:zP). 6 -t c b - a 100 + ri pr p« 6+16-1-1 14. A, -«^ miles ; B, -^^ miles. 15. ""''' + &^ + ^P cents. m-h n m -\- n a -\- b + c 16. First kind, ^li^^l^ ; second kind, ^i^~J^. 6 — a 6 — a amn «M uimt uii a 1 + n -|- mn' 1 + n + mn 1 + n -f- wi» 18 A — ^-^l^^^l— • B ^ w»^P o 2mnp mn -\- np — mp mp + np — mn mn -\- mp — np § 164; page 141. 3. x = 2. 3^^1 11. x = -2. 14. ^^_5. y = 3. 2 2 2 16. x = 4. 6. x = S. 5 io /.. — 2 , = _5. 9. . = -1 12.0. = -. ,=-1 6. « = -!. 16. x = - 6. -4. y = 2. y=_s. 6 ^ 17. x = -3. ,. ._3. 10. x = -|. 4 13. x = -5. ^3 ^4 ^*- ^--^- 18. x = 9. ^4* ^6 « = 4. y = 7. L4 ALGEBRA. §165; page 142. 2. 3. X=:S. y = 4. x = -4. 8. x = l. -1- 11. X = — 2. 14. 2/ =-4. 4. 5. y=-i. x = 2. y = 6. x = 5. y=-7. 9. 12. -1 15. 16. x = -5. y = \- 6. 7. x=-l. y = s. x=-S. y=-2. 10. 4 -1- 13. y = - 3, 2* 17. y = -3. x = 4. y=-b. § 166 ; page 143. 2. x = 2. 2/ = 5. 7. —I 10. ^-1- 13. x = 6. 2/=-l. 3. x = 4. 2/ =-3. — 1 y-l- 14. x = -l. 2/ =-5. 4. x = -6. 2^ =-4. 6. x = l. 2/ — 2. 6. x = -2. 8. -1- 11. 12. 2/ = 3. 15. 16. x = 3. 2/ = 2. -1 9. x = -3. • 2/ = l. -!• 17. x = -4. 2/ = 7. § 167 ; pages 144 to 146. 2. x = 6. 6. X = -8. 10. x = 4. 14. x = -5. 2/ =-10. y = 5. 2/ =-5. 2/ =-7. 3. x = 12. 7. X = -6. 11. x = l. 15. x=-7. 2/ = - 12. y = -3. 2/=-2. ^ = 8. 4. x = -l. 8. X = 3. 12. x=-l. ^=1- 2/ =-5. y = -5. 2/ = 5- 16. 6. x = 4. 2/ =-3. 9. X y = 18. = 6. 13. x = 5. 2/ =9. y-l- ANSWERS. 11 17. X = - 12. 20. x = .8. ?/=- .07. - ^=f- 25. x.lJ. 18. x = 5. 21. x = 2. y=-ll. ^=-1- --!■ -1- 24. x = 7. 2 19. x = -2. 22. x = 3. y = lO. 26. x = -10. y=-6. ?/=-l. y = 6. §168; pages 147, 148. ^ 35 g + 24 b 7. x=-2a. 14. x = a. ' ^ 23 * y=b. y = b. ^^14a^_186. 8. x=-3m. 16. x = a. ^^ y=-2n. y = a. 3. x=-^-i^. g ^^ aa'(bc'-hb'c) 16. x = wi^^. «^ + 62 • cc'(a'6 + a6') y = mn*. ^ a^+&2 y= ../L.^.;./N - 17. x = 5L±^ 4. x = w'n cc'{a'b-\-ab') y = -\ 6. »= = *• r= y = — 6. ^ m'p — mp' cia a2 + 6) + b'i c(a mn' -&) — wi'n n'p mn' -np' - w'n a ?i^p - up' 10. x=a. _ q- & mn' — m'n y= — b. ^~ 5 * mp' - m'p or n , y= ^^^ J^ ' 11. x=^. 18. x = « + 26- 2 *"• *- 2 - dm + bn y=--- .. a-26 0. x — ■ — • ^ o V = • ad-{-bc ^ ^2 * ^ _cm-an 12. x=a'^-\-b. ^--^dTbE' y=a-b^. 19. ^ = 4"^- 6. x = a + 6. 13. x=a(2a+6). y = ^~^ y = a-b. y=b(a-\-2b). 2 § 169 ; page 149. 2. x = -3. 5 ^^ a'^ + ft^ y = 5. 3. x = -. y = 4 7. x = -6. 2/ =-2. 8. x = 3. 2^ = 4. 9. ^=r --!• 16 ALGEBRA. 10. x = a-\-h. 11. X =^. ^ a+6 ^ 3 § 170 ; pages 151 to 153. 3. x = 3. 2/ = 2. i=-l. 10. x = l. 6 17. 22. w = 6. a; =-7. y = S. 4. a; = -5. ' = —s 1 z=-9. y=-4. 11. x = -'S. ^=3 b-hc 5. z = 2. X = 2. 2/ = 5. 2/ = 4. ^=1 18. a; = 2. 2/ = 4. 23. z = -l. 12. x = -5. z = Q. _ 2 6. x = -4. 2/ =4. 0=-3. 19. x = '-. a + 6 y=-S. z=-6. 13. a;=- 1. 2/ = 6. 4 -1 24. M = -5. x = 4. 7. x = -6. y^-7. z = S. 14. z=-4. x = 5. y = i. z = S. 20. x = a. 2/=-3. «=-2. 8. x = -2. t,=-5. z=-S. 15. x = -S. y=-b. 2 =-7. y =- a^. z =— a^. 25. w=10. x = 2. 2/ = 4. 9. ^4:- 16. -\ 21. a ;s = 6. y-l y-\ y=T 26. x = 6. 1 4 Z=^' 2/=-2. « = -. 4 ^ 5' c « =-4. 87. x = 2. 28. x = 6. 29. X = - 12. 2/ = 3. 2/ = 14 y = - 24. ;?=-!. z=- 12. z = 36. \n ^_ 2a&c 2a&c z= — at 2abc JU. ab-\-ac—bc ^~ah + bc- ac Q-i-bc-ab 31. 32. ANSWERS. 5c = ab. 33. x = a. y - be. y = l. z = ca. 1 X 2^^ a b ->r c- a V- ^'''' 34. x = 3. " c-\-a-b 2ab y=-l. a + b- c z = 6. 17 § 172 ; pagres 155 to 164. 3. 35, 24. 4. 20, 12. 6. — • 6. — • 7. Apples, ^1; flour, $3. C7 li7 8. A, 24 ; B, 40. 9. '26, 15. 10. -|. 11. A, 35; B,27. 12. A, 15; 3,22^. 13. §630 in 4^ per cent stock, $ 810 in 3^ per cent stock. 14. Income tax, $28 ; assessed tax, $36. 15. A, $60 ; B, $52. 16. $1.75, $1.50. 17. 13, 17, 19 19. 84, at 2^ cents each. 20. 45 cents ; 15 oranges. 21. ??»«i«+Al persons; each received «M»L±i^ doUars. bm — an bm — an 22. 21 quarter-dollars, 13 dimes. 23. 26 1 of first kind, 43 1 of second kind. 24. 45 of first kind, 63 of second kind. 25. A, 16 ; B, 30 ; C, 60. 26. 32 for, 22 against. 28. 97. 29. 896. 30. 83. 31. 59. 32. 4 from the first, 3 from the second. 33. 85 ft., 64 ft. 34. A, 9 ; B, 5. 35. 467. 86. Express train, 45 miles an hour ; slow train, 27 miles an hour. 37. A, $72 ; B, $81 ; C, $63 ; D, $ 180. 38. First, 38; second, 18. 40. Rate of crew in still water, ^^^ + ^""^^ miles an hour ; of cun*ent, ^»^=^ miles an hour. ^'^'^ 2mn 41. Going, 10, I miles an hour ; returning, 4V miles an hour. 42. 78. 43. 369. 44. 75 ft., 54 ft. 45. $ 375, at 4 per cent. 46. ^^ - ^'^ dollars, at ^^^C^ " ^> per cent. 47. A, 15 ; B, 21. m — n bm — an 48. $ 2000, at 6 per cent. 18 ALGEBRA. 60. Rate before accident, 36 miles an hour ; distance to B from point of detention, 90 miles. 51. 647. 62. A, $6; B, $12; C, $8; D, $20. 53. A, .$13 ; B, $7 ; C, .$4. 54. Fore-wheel, 9 feet ; hind-wheel, 15 feet. 65. A, days; B, ; days: C, days. mn-{-np—'mp mp + np — mn mn + mp — np 56. A, 8 ; B, 12 ; C, 24. 67. First, $15000 at 4^ per cent; second, $18000 at 3i per cent; third, $ 13000 at 5^ per cent. 68. A, ^ hours ; B, -^^ hours. 6-f-c — a a — b 69. Rate of crew in still water, 9 miles an hour ; of current, 5 miles an hour. 60. Principal, $5000; time, 3 years. 61. A, $55; B, $19; C, $7. 62. 12, each paid $3. 63. Express train, 40 miles an hour ; slow train, 25 miles an hour. 64. A, 18 ; B, 16. 65. 3 quarter-dollars, 8 dimes, 9 half-dimes. 66. 30 of 3^ per cent stock, 20 of 4 per cent stock. 67. A, 8 ; B, 7. § 184 ; pages 168, 169. 3. a;<3. 4. x>i 5. x<-. 6. x>8. 7. x<^' 3 2 ^ 8. x>a-6. 9. a;3, y<2. 11. x>6 and <9. 12. 7. 13. 18 or 19. 14. 38, 39, or 40. § 187 ; page 172. 4. x* + 4x3 4-6x2-f 4x + 1. 6. 4^4 - 4a3 + 17 a2 ~ 8a + 16. 7. 25a;* - 30x3 - a:-2 + 6x + 1. 8. 9x'^+2ix^+2Sx^+mx+i. 9. 36 n6 + 12n4-60wHn2-10 71+25. 11. a* - 8 a8& + 22 a262 _ 24 «63 4. 9 54. 12. 4 x4 + 12 x^y + 13x2y2 + 6 xy^ + y^, 13. x6 + 12x5 + 36x4 - 14x3 - 84x2 + 49. 14. 16 a^ - 40 a^x^ + a^x^ + 30 a''-x^ + 9 x^^. 17. x6-2x5-x4 + 6x3-3x2-4x + 4. 18. a6 + 4a5_2a*-20a3-7a2 + 24a+16. 19. 4x6 - 20x5 + 41 x^ - 52 x3 + 46x2 - 24x -|- 9. ANSWERS. 19 § 188 ; page 173. 4. x8 + 6x2 + 12x + 8. 8. 210a3_i08a26 + 18a62-.63. 5. 27a3-27a2 + 9a - 1. 9. 125x3 + 150x22/+60x|/2+8y3. 6. m=5-12m2n+48mw2-64n3. 10. 64m3-144 w2w3+108mn6-27n». 7. x6 + 15x* + 75x2 ^ 126. 11. 27 x^ - 135x5+ 225 x*- 125x3. 12. 64 xi2 + 240 x^yz^ + 300 xl^yH^ + 125 yH^. 13. 8x3-84x5 + 294x^-343x9. 14. 125 ai8 + 450 a^'^h^ + 540 a^h^^ + 216 ¥^. 16. a3 + 63 - c3 + 3a26 - 3a2c + 362^ -Zh^c -\-Zd^a + 3c26 _ 6a6c. 17. x5 + 3x5 + 6x4+7x3 + 6x2 + 3x + l. 18. x8-y3+8^-3x2|/+6x25; + 32/2x+6i/22 + 12 02^-12 22y_i2iry0. 19. a® - 9 a^ + 24 a* - 9 a3 - 24 a2 - 9 a - 1. 20. 8x6 + 12x5 - 30x4 - 35x3 + 45x2 + 27x - 27. 21. 27 - 108 X + 171 x2 - 136 x3 + 57 x* - 12 x^ + x«. § 193 ; page 176. 24. 56. 25. 135. 26. 252. 27. 432. 28. 588. 29. 24. 30. 105 a6c. 31. 462. 32. 45. 33. 12. 34. 6. 35. i^6. 36, 28. 37. a3 + 4 a2 + ^ - 6. § 195 ; pages 178, 179. 3. 4. 5. 6. 7. 2x2 + x + 1. l-3a + a2._ .3x2_4x-2. 2x2 + 5x-7. a — h — c. 10. 3x + 11. 7m2 12. 3a2- 13. 5x2- 5y-4^. — nin — 4 ??2. -5a + 4. - 2 xy - 3 y2. 16. m + 4-1. m 17. 1 - X + x2 - x\ 18. x3-4x2-2x-.'^ 19. X y ^y\ 8. 9. 2 a3 + 3 a2 - 1. x3-2xa2 + 5rt3. 14. 4m2 15. 3a2- + mx2 -2a6 -3x*. -5 62. 2 2x 21. 2a3 + 3a26 + 4a62_5 63. 26. 1+a -f-f-- 22. a^ ah 62 2 3 4* 27. 1-^. 2 x2 a;3 8 16 "*' 23. 3 x3 - 2 x2y - a;y2 + 4^3. 28. J 3a 2 ; 9 a2 27 a' 8 16 24. 4 X 2 x2 3 a a2 29. x + §- X - ^ + 2^ + .... 2x3^2x5 25. l + 2x-2x2 + 4.x8+ .... 30. 2a-- 2 6 62 6'* la 16 a3 64 a5 20 ALGEBRA. § 199 ; pages 182, 183. 1. 65. 10. 3581. 20. 3.6055. 30. .8660 2. 148. 11. 274.9. 21. 6.9282. 31. .7453 3. 713. 12. .4027. 22. 8.0436. 32. 1.148. 4. 8.07. 13. 51.04. 23. .44721. 33. .7071. 6. .396. 14. .07906. 24. .23664. 34. .7745. 6. .254. 15. 9.318. 25. .62449. 35. .9354. 7. 62.9. n. 2.6457. 26. .094868. 36. .6373. 8. 9.82. 18. 2.8284. 27. .027202. 37. 1.035. 9. .0567. 19. 3.1622. 39. 28. .6085. 2.9265. 38. 1.258. § 201 ; pages 185, 186. 7. x'^-2x-l. 11. a2 - 3 a - 2. 14. x2 + 2 xy + 4 y^. 8. 2a2 + 3a + l. 12. 2x'^-bx-\-2. 15. ^-1+1 3 X 9. Sy^ + y- 2. 13. Sa^-2ah-\- b^. § 206 ; pages 189, 190. 1. 27. 6. 9.5. 11. .0481. 16. 1.442. 21. .7413. 2. 53. 7. .608. 12. 92.4. 17. 1.912. 22. .7631. 3. 3.9. 8. 3.59. 13. 7.63. 18. 2.087. 23. .7368. 4. .85. 9. 806. 14. 697. 19. .2714. 5. 136. 10. 57.2. 15. .1048. 20. .8549. § 207 ; page 190. 1. Sa + 2b'^. 2. \-Zx-x\ Z, 2a'^-a- 2. 4. x^ + y2. 5. a -2. 6. 21.4. 7. .46. § 217 ; pages 195, 196. 8. wi 9. A^ 10. 6w"i 11. 7a~^i 12. Qah^. 15. ax~'^ 17. a-h. 18. 8a;-2 + 27. 19. 8a-2-18a-i-47-15a. 20. x-^-\Q 1112 7 fi _4 _5 4 3 _1 21. ar^ + x^ys+y'J. 22. m^n-^— 4w^n ^-\-Qmn 3— 4 m'n'^+w'n ^ 23. a-%-^ - 3 a-^b-' + a-'6-9. 24. 2 m~^ + 4 mThr^ + 18 n-* 25. 4a^6-2-17a^?)2+16a-f66. 26. ]^m^x~'^ -20m^x^+2m~'^x, ANSWERS. 21 § 218 ; pages 196, 197. 1. nk 9. 3a:"K 11. a* - a^ft^ +&^. bK 6. 2-a;'s\ -^ -L^ 13. a;2 _ 2 -f x-\ 14. a^ - 2 a^ + 1 _3 _1 _1 a ^' + a -^ +a * + 1. ^^•-^ 16. m-2-2?>ri + l-2wi. 17. 3a;V+a;2y+a-. 19. aV5-2-3a"26i 2 3 _3 4 2 3 20. m^x ^ + 2m^ + w^x^. s a;^. wi' § 220 ; page 198. 11. a^ 13. c' 12. a;"^. 14. a" 16. m~^. m 16. X". 2. 125. 243. 256. 27. § 221 : page 198. «.i. 8. 128. 9. 49. 11. - 128. 12. 32. 13. 625. 15. -1024. 16. 81. 17. -I § 223 ; page3 199, 200. 2a*+l-5a"^. 9. 3a;"3-2x"5-f 1. 10. Jb-^-iab-^-Sah-\ + 2x^-3x1 17. a 18. tt^^-sn, 19, a;"'-'. X 2(x -1. 21. X""*. 22. a'. 23. -• 24. z^'^"*. 25. ^i^- 8 1 - rt ' + yh x^ -y- x" + 2. 27. a" + 1 + a 30. 2 xy x^ + y^ 28. 31. 2(x + y) . a;-2/-(x - y) 2a + 16a^ftf a - 86 12. 13. VUab^. V5 xy^ § 228 ; page 202. 14. V2ahn. 15. V3 m^n^ 16. \/2x3m2. 17. v'^^s^ 22 ALGEBRA. § 229 ; pages 202, 203. 19. 6 abWS ab'^ + 2 a^b. 22. (x-\-S)V6x. 20. 3x?/v^5xV-4 2/*. 21. (a - 2 6) Va + 2 &. 27. 12 V6. 29. 42\/2. 28. 5\/l05. 30. 75 Vs. 35. 12^. 23. (^a-2b)VSab. 24. (x - 3) \/x2 + 7 a; + 10. 31. 28\/42. 33. 7\/l2. 32. 5\/9. 34. 14^28. 36. 315a6>/l5a&. 2. i\/6. 3. fV5. 12. ^v^. 16. — \/42a. 6a 19. |^V3^. § 230 ; pages 203, 204. 4. i\/T5 6. j\\/^5. 8. -|Vl2. 5. |V2. 7. |\/3i. 9. aV2. 13. |\/10. 17. 10X2 20. J-V98«2. 7a 14. iv/8t, V30^. Va^ - b\ X + 2 10. iv^. 11. 1^5. 15. ^\/l8. 18. ^y/W^. 6cd3 21. -Lv/20^. 4y V2x. 14. vT 15. § 231 ; page 204 Vx2"^=n. 16. \ a - b a + b 17. J(£.^ >! .X2 + 1 V2. 3/ § 233 ; pages 205, 206. 3. 7\/3. 4. 4\/2. 5. -2V5. 6. 5^2. 7. 3v/3. 9. 5\/3. 10. \/7-2Vn. 11. V-v/2. 12. |V0. 13. 1^6 14. 0. 15. -i^VTO. 16. 2^9-3^5. 17. -i>/l5 18. -a2&2V2^. 19. lOm^v'Im^^. 20. (5 a - 4x2) V2 a^ - 3x 21. fVIi. 22. v^. 23. 6^3-2^6. 24. -Sv^B 25. 7V2-5\/5. 26. 4x\/6^. 27. 2 62 VTo^ - 3 a ^76 29. ^§\/30-fVlO. 30. (7x-l)V5x 31. 13?/\/3. 32. ^ 28. ^V3-V6 a — b -J a?- - bK ANSWERS. . 23 § 234 ; page 207. 2. \^, V^. 3. v^, 'v/l4>v^l75. 16. '-(^253 > V3 > v/T5. 17. V3 > Vo > v/?. § 235 ; pages 208 to 210. 4. 12. 5. 6 a. 6. 6V7. 7. 5V3U. 8. 110. 9. 10 ay/^Tbc. 10. 12. 11. 3v'35. 12. Gv/55. 13. |Vr5. 14. 3v^. 15. 2^^. 16. P>x^^. 17. 2\/486. 18. v'SOO^^^. 19. 5v^5. 20. 2 6v^l6a56cs. 21. 3v^. 22. 2v^27. 23. 3v^. 24. i/6. 45. 61 + 24 V5. 46. 37 - 20\/3. 47. 168 - 96\/3. 48. 665 + 70V70. 49. 5 a - 4 + 2\/iU^-Sa. 50. 13a; + 5y- 12 Vx^ - y\ 51. -31. 52. 28. 53. 4 - 21 a;. 54. 2 6. 65. 3-46^. § 236; page 211. 3. 2V3. 4. |V5. 5. ^>/7. 6. 3^3. 7. h 8. i\/225. 3 9. 3. 10. J-v'T62^. 11. 2V'2. 12. \/2. 13. a/-^- 3 a >'5c 14. ^^-. 15. I 16. AVI5. 17. ^>/ J^. 18. ,^3^^^. 19. vWoF. 20. ^v^TeO. 21. a/-- 22. ^^^- 23. ^. '2 ' 2y 24. ^. 25. J- ^18^. 26. ^^^8. 24 ALGEBRA. § 237 ; page 212. 6. 18\/2. 8. av^. 10. 3v^. 12. ^Om^x^STi. 7. S2 a'^b^Vab. 9. 5\/2^. 11. 2v^. 14. 2?/v'a^. § 238; page 213. 8. \^. 10. \/l&2xy^ 11. \^. 13. v^. 14. /3+ 10 9 22VT5-85 ^^ l _ Vi~I~^ 2 ' * 5 " ' a " g a + b^ + 2h y/a -^ a; — 4 — Va; — 2 -- x + Vx'^ — y'^ a — b'^ X — 4 y g a; + y-2\/xy ^^ & -2a-2\/q2_qft ^^ Vg* - x^ - a^ X — y b x^ 7 9 + 4V3 -2 1 +4a;\/l -4x-^ -„ I4x - 10 + llV^^^^ 3 * • 8a;2-l ' 5X-13 § 241; page 215. 2. .949. 4. .535. 6. -4.560. 8. .268. 10. -.330. 3. 2.224. 5. .684. 7. 4.442. 9. 13.354 §247; page 218. 3. V7+2. 8. 2\/7-V2. 13. 3\/5-l. 18. 2\/5-\/l5. 4. 3-2\/2. 9. 3-V3. 14. 5+VTo. 19. SVS+VlO. 6. 4\/3+l. 10. V6+\/5. 15. 5V'2+VG. 20. 6\/2-V3. 6. 2V3+\/7. 11. 4+>/T0. 16. 3\/3-2V2. 21. V^+T+V^^. 7. 2\/6-2. 12. vTl-3. 17. 4\/2-\/5. 22. Vo^-Vft. ANSWERS. 25 § 251; page 219. 2. 9V^^. 3. WS v^^. 4. \/2 \/^. 6. (a; + 1/ + i2) V^n. 7. 0. 8. SV^HT. 10. y/lV^. 11. (l-a:)V^^. 5. bV^^. 4. 10. 13. 16.' 20. 23. 27. 31. 34. § 252; pages 220, 221. -14. 6. 12 a2. 6. -2>/l5. 7. -Va6. 8. 18. 9. -60. 26-7V^. 11. 66-33 V^^. 12. - 61 + 18VI5. - 8a + 18 ft. 14. - xyzV^^. 15. 48\/2 V^H". -8V30-17>/T5. 17.2. 18.480. 19. - v^lO. -2 + 2>/^^. 21. -74-40>/3. 22. 11 -8\/^^. - 30 + 12\/6. 24. x2 + y 25. 61. 26. - 9a + 4 6. ^ 1 _ v^^ - 50. 28. 73 + 40\/3 23 _ 10 - 9\/^3. 12 32. 51 -20\/l5 33 2. § 253; page 222. 6. -V^. 6. 3>/^n". '•a/^- 8. Va 30. V-1. 33. -2 + 2V'3T. 9. V5. 11. V3. 10. -Vs. 12. V2. 6. 2 3' 16. 25 36* 1 6* 9. 9 20 10. -2. 11. 2. 5 12. 13. H. § 254; page 223. 5a 15. 4. 25 16. 17. 18. 144 16* 19. - ). 1. 21. 4 4 62 10 g 3 25 26. 7a 8 ' ab 27. 28. 9a 16* 64 5a 4 * 71 120* 31. 6. 26 ALGEBRA. § 256 ; page 225. 3. ±3. 7. ±5. 11. ±6. 15. diSv" -1. 19. ±\ 4. 4 8. ±T- ^'^• -H- --!• 20. ±(«-6). 5. ±\/3. 9. 4-- "• ±4. n. ±8. 21. ±i>/l5. 6. ±6. 10. ± 2. 14. § 259 ±2. 18. ±1. ; pagre 227. - =^1- 3. 1, -7. 7. 5, - 6. u.1,-5. 15. 6,-1 4. 8, -4. -•!• n. . |. 18 1 ^. 5. -2, - 9. 9.1,-1, "•II- 6. 10, 3. 10. -|, -5. 5 260 ; page 229. • 3. I'-- . -1, -. ■ --l-l- in 1 1 4. o 5 4* '•II- - -\ -\ 16. i, -1. 9 3 5. ''!• -hT\^ ^--\-\ 6. 1 2' 4* 10. ^, -|. u. . -I § 262 ; pages 230, 231. 3.10, -3. 7. -1. -|. 11. -f, -\ 15.5,?. S,,, -|. 12. I -|. 16. 3, I 9.1, I 13.1,-4. n.|,-f. ^»-l-i- ^*--l-^- "--i'-l- 4. ^'-1- 5. y- -- 6. 1 3 4' 2' ANSWERS. 2 § 263; pagres 231 to 233. 1. «'-!• 10. 3, 4 5' 20. -4, -7. 30. 'i'- 2. 1- 11. 1, 7 18 21. 1 1 ~9' ~8' 31. -1, -2. 3. 3 1 12. 2, 1 3* 22. -3, -4. 32. ^'1 4. -!• -- 13. 3, 1 7* 23. 1 1 2 I 33. -1, -3. 5. !■-■ 14. 1, 2 3* 24. 4, -1. 34. ^'-t- 6. U, _3. o 15. 2, - 1. 25. -3, -4. 35. 1 2 6' 7' 7. ^'1 16. 26 , 2. 26. ^'k 36. 8 18 5' ~r 8. 5, -6. 17. 6, -3. 27. ?•-• 37. ^-• 9. 0±V3. 18. 119^ 7. 28. 2 23 ' 12* 38. !•-■ 19. 3, -13. 29. ^' -f r § 264 ; pages 234, 235. 3. 2a ±3 6. 4. 1, -2m -1. 5. a, -1. 6. - b, -a 7. 771^ m3. 8. -, --. 9. 3a-f5, -a + 7. 10. 1, -^ c a rt — 11. _«^, _6. 12. -«J:V«^^^. 13. a + 6, «_±-^ a + 6 a 2 14. 2 a, -a. 16. a, -^^^tl. le. a - 2 ?>, -85 2a ,- a — 6 a + 26 m r ^ 13 a -« 2 — a o^ 1 17. — - — » — 18. 5a, — — 19. — - — , — 2a-l A , o o 2 ' 20. (3a + 6)2, -(3a-6)-2. 21. ^^±^, ^^. 22. a-2 6, -2a + b a-b a+6 23. 3 a, -^. 24. a- -c, -6 + c. 25. 42 a, 2 a. 26. ^r, 2 m. 3 27. 3 a, -4 a. 28. 1, ^-«. a - 6 a^ + l a^-l — a2_i' ^2^.1 on a-b c 31. a + 6, 2«^ a+ b 3„ , 6 + c-2a oU. , • c a — ^^- ^' c + a-26 28 ALGEBRA. § 265 ; page 236. 3. I -2. T. 1, -|. U. -I, -5. 15. -I 1. 4. 9, -4. .. 4, -|. 12. 1, -|. 16. -!, |. 5. -6, -8. 9. 3, -. 13. -, -. 4 5 5 6. 2, ?. 10. -i. -^. 14. -^, -I- 5 14* 2 3 4 § 267 ; pages 237, 238. 6. -7, 4. 6. 5, 9. 7. -8, -3. 8. 12, -6. 9. 0, ^. 5 10. 0, -8. 11. 0, ±?. 13. 0, ^, -?. 15. 3, -^, -4. 4 4 3 2 16. 0, 2drV2l. 17. ±3a, ?, -a. 18. -1, ^ =^ ^^^-^. 2 2 19. 3, :z3±3VH3. 20. ±3, ±?V3i. 21. i«, -2ad:2aV-3 2 2 2 3 3 22. -| 5 ± 5 V^Ts 23. i2, liV^S, _ 1 ± V^s. 24. 1, ±\/^. 25. 0, |. 26. ±5, ^. 27. ± -, - -• 2 5 2 2 28. -f, ±3\/^2. 29. 0, i^. 30. 0, 4a -4. 4 5 § 268 ; pages 239 to 242. 3. 21, at $6 per barrel. 4. 11 and 7. 5. 9 and 16 : or, -^ and -^. 2 2 6. 3, 4, 5. 7. 16 and 4 ; or, 25 and - 5. 8. 6 and 2. 9. 1, 2, 3, 4 ; or, 5, 6, 7, 8. 10. 18, at $6 per barrel. 11. 21. 12. $40. 13.4,5,6. 14. 6 miles an hour. 15.300. 16. 27 and 36 miles an hour. 17. 18 rods, 12 rods. 18. 20 cents. 19. $ 75 or $ 25. 20. 9 miles an hour^ 21. Area of picture, 25 sq. in. ; width of frame, 4 in. 22. Fore-wheel, 12 ft. ; hind-wheel, 16 ft. 23. Larger, 6 hours ; smaller, 10 hours. 24. 22. 25. $ 3000. 26. 5. 27. 5. 28. 8. 29. 4. 30. 12100 and 1225 sq. ft. ; or, 8836 and 4489 sq. ft. M. 6. 32. 136 or 68 miles. 33. 72 miles. 34. 80, at $ 60 each. ANSWERS. 29 § 270 ; paeres 244, 245. 4. ±3, ±2V3. 5. ^, 'f 6. 4, /2; or, x = -2\/3, y = ±2\/2. 6. X = 2 a - 6, y = ± (2 & + a) ; or, x=-2a + b, y = ±(2b+a). 30 ALGEBRA. § 277 ; pages 250, 251. Note. — In this, and the three following sections, the answers are arranged in the order in which they are to be taken ; thus, in Ex. 2, the value x = 2 is to be taken with y = 3, and x = 10 with y = — IS. 2. x = 2, 10. 7. X = 6, 1. 12. .=-3.-f. . = .>. 3. y = S,- 13. x = 6, -9. y=-9, 6. y 8. X y = 1, 6. = a + l, -a. = a, — a — 1. 13. 4. x = S, -7. y = 7, -8. a; = 10, -3. y = 17, 4. 9. X = 8,-3. 14. 5. 10. X y 11. X = a + 6, a - 6. = a - 6, a + 6. = 5, -3. 15. . = 3.-?I. 6. x=2, -5. t/ = 5, -2. y =-•!• . = 12, -f § 278 ; page 253. 4. a: = 8, 6. 9. X = 5, 2. 14. , x = 8, -2. y = 6, 8. y=-2, -5. y=-2, 8. 5. ic=l, -10. 10. x = -l, -6. 15. x = 6, -9. 2/ =-10, 1. |/=-6, -1. y = 9, - 6. 6. x = 4, -3. 11. x = 6, -7. 16. x = 4, 17. y = 3, -4. y=-7, 5. 2/ =-17, -4. 7. a; = 5, -9. 12. x = 2, -16. 17. x = ±7, il3. y = 9, -5. y = 16, -2. ?/=Tl3, T7. 8. x=±Q, ±2. 13. X = 4, 20. 18. x = 2, -7. y=±2, ±6. 19. 2/ =-20, -4. a; = _6, -25. y = 25, 6. 2/ =-7, 2. § 279 ; page 254, x = ±4:, ±|\/2. 3. x = ±2, ±fV2. ?/=il, TfV2. !/=T5, T|V2. ANSWERS. 31 4. x = ±3, ±4\/3. y=±6, T5V3. 5. x=±4, ±1. 6. x=±6, ±i3tv/::r3. y=T4, ±-V^V^=^. 7. a; =±4, ± fV7. y=i3, T|V7. 8. x=±2, ± j\v/ - 13 . 2/=Tl, ±3-VV-13. 9. a; = ±5, ± fV - 10 . y=±l, if^V^n^. 10. x = ±l, ±fv^. y=T7, ±fV77. 11. a; =±2, ±T%V3. 2/ =±5, TffVS. 6. X y 7. X y 10. X y 13. a: y 16. X: y- 19. X y 23. X y 26. X y 28. X = ±4, i^V^Ts. = ±3, T^^^. = 6, -4. = ±4, ±iV46. § 280 ; pagres 257, 258. 6. X = 4, - 3, - 1 ± >/l3. y = 3, - 4, 1 ± Vl3. 8. x = ±l, ±^. 9. x = 3, 6. 11. x = 8, 11. y=-ll, -8. 4,2,8,--. 14. x = 2, -4. 16. x = 2, '22 12. X = 3, 9. y = 9, 3. 3, 8, - 6, 16. y = 4, y=-l,2, 2 ±2, ±V^. 17. x=r3, -6, -, --. 18. x=-5, - 55 = :fl, ±2V^. = a±l. = « T 1. y . 9 9 '' ~4' i' y= -6, -^. 7 20. ^=\\' 21. x=.6, -6 1 1 4' 3* 22. x = a±h y = ± 3, ± 3. y = aTh. :2a-3, 1^. 24. x=±3, ±1. 25. x=3a+2,2a-3. lo :3a-2, 126 a -169 26 = ±2, ±Hv/-31. = T2, i}fV-31. b±y/- 151 2 y=±l, ±3. 2/=2a'-3,3a+2. • 27. x = ±(2a-6), ±(a-26). y=±C«-26), ±(2a-6). = 3,2, x = 2, 10 5±Vl93 :-2, -3,ZLi±^^EMI. 3 4 63 ± 3\/l93 ■ 4 21, 32 ALGEBRA. 30. x=27, -8. 31. x=2, -1. 32. x=a + l, -a. 33. x=2, 12. y=8, -27. y=-l,2. y = a, -a-1. y=-3, -i-- 72 34. J 4 o y=||,o. ^ = 2' |- ^ = 26, -6. 37. x=2, -1, l±:^Illl. 38. a;=0,2, ±V2. 39. x=±3, ±V-7. y = l, -2, Ill±^3dl. ^=0, 2, 2T V2. i/=2, 6. 40. a;=±l,±2. 41. x=3, -1, -1, -2. 42. x=3,4, -6±>/i3. y=±|±| y=l, -3,2, 1. 2/=-4, -3,6±Vi3. 43. a; = -2, -4. 44. x = 3, - 1, 2, - 3. 45. x = 2, 1. y=_4, -2. y=-],3, -3, 2. y = 1, 2. -^-•-V^— '•-!-=-• -i- § 281 ; pages 258 to 260. 1. 6, ± 4 ; or, - 6, ± 4. 2. ± 5, ± 3 ; or, ± SV^ T SV^^. 3. 18 rods, 9 rods. 4. 7, 5 ; or, - .5, - 7. 6. 5, 2. 6. Cow, l$70 ; sheep, $40. 7. 32 or 23. 8. 9, 4. 9. - or :=^. - 8 22 10. 24 in., 16 in. 11. Rate of crew in still water 6 miles an hour, of stream 3 miles an hour; or, rate of crew in still water -'/ miles an hour, of stream | miles an hour. 12. Length 30 rods, width 12 rods ; or, length 60 rods, width 6 rods. 13. 60 ; A gives to each ^ 3. 14. A, 6 hours ; B, 3 hours ; C, 2 hours. 15. Length 32 rods, width 30 rods. 16. 6 and 4; - 4 and - 6 ; or, l±^ and -^±/^K 17. A's rate of walking, 3 miles an hour ; distance 12 miles. 18. A, 4 hours; B, 8 hours ; C, 12 hours. 19. 1 and 3 ; or, 2 + Zy/^^ and 2 - Zy/'^. ANSWERS. 33 § 283; pagres 262, 263. 2. a:2- 15x + 54 = 0. 8. 8x2+17x = 0. 3. a;2 + x-6 = 0. 9. SGx^ + 77x + 40 = 0. 4. 3x2- ic -2 = 0. 10. a;2 + (2 6-3a)x4-2a2-5a6-3 62=o. 5. 2 x"2 + 19 X + 44 = 0. 11. 052 _ 2 ax + a2 - 9 77i- = 0. 6. 30x2 -31x + 5 = 0. 12. x2-6x-89 = 0. 7. 28x2-x-15 = 0. 13. 4x2 + 4xVa + a - 6 = 0. § 285; pages 264, 265. 6. (3x-2)(x + 3). 19. (9-4x)(5 + 3x). 7. (5x + 8)(x + 2). 20. (7-2x)(6 + 5x). 8. (2x-3)(3x-l). 21. (6x-5)(4x- 1). 9. (3x-4)(5x + 2). 22. (4x + 5)(2 x + 7). 10. (5-3x)(4 + x). 23. (3x-4y)(7x + 62/). 11. (5-3x)(7 + 2x). 24. (7 x - 5ab)(x -\- Oab). 12. (6-x)(2 + 6x). 26. (x - 3y + 4)(x + 4?/ + 3). 13. (x-7a)(3x + 4a). 27. (x - 2?/ - l)(x + ?/ + 2). 14. (3x-7m)(2x-3w). 28. (x -2y + 4)(x + 2y - 1). 15. (7x + 2)(2x + 3). 29. (2x - ?/ + 3)(x + 4y - 1). 16. (3x-2)(6x-l). 30. (a-26-2)(3a4-6-l). 17. (1 - 4x)(5 + X). 31. (3y - 2 - x)(3y - 3 + 4x). 18. (9x4-2)(2x + 3). 32. (2x - 5y- z){3x + Sy-\-2zy § 286; page 266. 4. (2x + 5)(2x + 3). 5. (3x - 2)(3x - 4). 6. (4x + 7)(4x - 3). 7. (x+l + 2\/3)(x + l-2>/3). 11. (5X + 3+ V3)(5x + 3-\/3). 8. (2x+H-V2)(2x+l-\/2). 12. (2\/2-2 + 3x)(2v/2 + 2-3x). 9. (6x + 6)(Cx- 1). 13. (7x + C)(7x + 2). 10. (x + 2)(4x-3). 14. (l+8x)(5-2x). § 287; page 267. 4. (x2 + 2x + 3)(x2-2x + 3). 5. (x2 + 3x - 5)(x2 - 3x - 6). 6. (2a2 + 3a6 + 462)(2a2-3a6 + 462). 7. (3x2+ 4x?/- 2?/2)(3x2-4xy-2y^). 8. (4 m"' + 3 mn + n'^){i m2 - Smn + n^). 9. (2a2 + 6a-7)(2a2-5a-7). 84 ALGEBRA. 10. (3a;2 + a;Vl3 + 3)(3a;2-a;\/l3 + 3). ^ 11. (2»i2 + mV5-2)(2m2-??iV5-2). 12. (x2 + 2 a; V2 + 4) (x'^ - 2 xV2 + 4). 13. (_x^ + xy/3-l){x^-xVS-l). 14. (3a2 + 5ax-d^2)(-3^2_5^^_5^2). 15. (4 a2 4- am + 6 m2) (4 a2 - am + 6 w2). 16. (6x2 + X - 2)(6x2 - a; - 2). 17. (5 m2 + 2 mx + 4 x2) (5 ^2 - 2 mx + 4 x2;. 18. (4x2 + 2x?/-72/2)(4x2-2x?/- 72/2). - 19. (6a2-f 2a?>\/2-5 62)(6a2-2a&V2-6 62). § 288; page 268. 2. V3±V32, -V3±v:ri. 5 1 JrV^ -^^^^^. 2 2 3. V3±V6, -V3±V6. 6. ^^1±2^, -V3J:Vl5. 4. ±1, ±i. ,. 3vl±JivE2^ § 299; page 277. 3. x = 9, y =1; x = 6, ?/ = 3; x = 3, y = 5. 4. X = 4, ?/ = 13 ; X = 8, y = 6. 5. x 6. X = 4, y = 122 ; X = 13, y = 91 ; x = 22, ?/ = 60 7. X = 3, 2/ = 50 ; X = 10, y = 26 ; x = 17, y = 2. 9. X = 3, ?/ =z 59 ; X = 13, ?/ = 16. 10. X = 78, ?/ = 4 ; X = 59, J/ = 12 ; X = 40, ?/ = 20 ; X = 21, y = 28 ; x = 2, y = 36. II. x = 2, y=l, z = S. 12. X = 2, ?/ = 30, = 3 ; x = 9, y = IS, z = 48 ; x = 16,y = e,z = 93. 13. x=2, y = l. 14. x=5, i/=2. 15. x=8, y=6. 16. x = 3, y=ll. 17. X = 7, ?/ = 1. 18. X = 9, ?/ = 4. 19. Either 2 and 8, or 6 and 3, twenty-five and twenty-cent pieces. 20. Either 1 and 17, 3 and 12, 5 and 7, or 7 and 2, fifty and twenty- 19 2 10 7 1' 12 cent pieces. 21. Either — and -, — and -, or - and — • 9 5 9 5 9 5 22. Either 1, 18, and 1; 4, 10, and 6 ; or 7, 2, and 11, half-dollars, quarter-dollars, and dimes. 23. 5 pigs, 10 sheep, 15 calves. 24. Either 17, 2, and 8 ; or 3, 11, and 25, quarter-dollars, twenty- cent pieces, and dimes. -3V2 2 ±3V- ■2 2 3, y = )0; x = 8. X 5. :.31, y = = 3, y :29. = 2. ANSWERS. 36 § 322 ; pages 285, 286. 4. 8. 5. 30. 6. -• 7. If. 8. ^^. 9. 10^. 10. a: - 3. 32 a + o 11. 2a- 1. 12. -1, ^- 13. 5, 22, -4. 14. ^. 11 15. rK = ±a26, ?/ = ±a?)2. 16. 32, 18. 17. 25, 11. 18. 31, 17. 19. 6, 8. 23. 3 : 4. 24. a : - 6. 25. 1 or - 15. 29. 5 : 4. 30. 3:4. 31. 3, 9, 27. § 332 ; pages 289, 290. 3. 72. 4. y = lz\ 5. J. 6. ^. 7. | 8. -18. 9. ]- o V o 4 10. 579 ft. 11. — , - — . 12. 7. 13. 16. 14. — • 15. 12 in. 4 3x 2 16. 3. 17. 5. 18. 9 in. 19. 15(>/3-l) in, 20. y=3 + 5x-4a;3. § 337; page 292. 2. ; = 69, ;S' = 432. 3. Z = -77, >S = -C30. 4. Z = 36, 6^ = - 264. 5. ^^_60^ ^^_561. e ^^m ^,^793. 4 4 4 4 7. i=m, ^=mi. 8. Z = -21, ^ = -165. 6 6 4 9. Z = -— , 6^ = -Zii. 10. Z = 34a+19&, ^=162 a + 63 6. 5 10 11 ;_ 17?/-8a; ^_ 80y-35a; ~ 2 ' 2 ' § 338; pages 294, 295. 4. a = l, S=biO. 5. a = 7, I = -69. 6. d = 3, ^=552. 7. cZ = - 5, Z = - 95. 8. d = -, n = 35. 9. a=-, d = -—' 4 5 15 10. Z = -, n = 16. 11. n = 22, S = -- 12. a = -3, Z = 5. 12 2 13. a = --, n = 9. 14. « = §,y-w(n-l)d ^^ 2^+n(n-l)d 2n ' 2 n 24 w, = ^-^ + ^ s = G + <^)(^-« + ^) f? ' 2 c? 25. « = ;-(n-l)(^, .S' = ^[2?-(n-l)d]. 26. a = ^-^^',d = ?M:::^. w n{n — 1) 2 2S-a-i a+! 7. d=- 5 4' § 340; page 296. ^■h- 2. X2 4-49. 3 4«2 4.i 4 a-2 - 1 OQ cZ ± V(2 ? + cZ)2 - 8 (Z6' ^ _2 I -\- d ^ V( 2l + d)^ - SdS 29. a = ^ ,n- — § 339 ; page 296. § 341; pages 297 to 299. 3. 6050. 4. 250500. 6. -50. 6. 10, 2, -6, -14. 7. 840. 8. 65x + 52|/. 9. 3, 5, 7, 9. 10. 100. 11. 44550. 12. 31. 13. ^. 14. -6,-2, 2, 6, 10 ; or, 21, ^, ^, - ^, - ^. 15. «wMl^. 16, 124. 17. 17. 18. 30. 19. 5. 20. 15. m + 1 21. -3, 7, 17; or, _3, -^, -^. 22. 579. 5 5 § 345; page 301. 3. Z=2ie7, ;S'=3280. 4. Z = ^, >S'=^. 5. Z=-1250, ,9= -1042. 6. ^ = 2048, *-4094. T. ! = - J^, « = -||. ANSWERS. 37 8. Z=-1280, S = -^^' 9. 1 = —, S = -^^' 2 625 625 10. ^ = -243, ^^_463. ,, ;=21 781. 64' 192 128' 384 12. Z = 768, ^ = 2457. 4 § 346; pages 302, 303. 3. a = 1,^ = 511. 4. a = 3, Z=-. 6. r = - 4, ,5 = 1638. 6. n = 10, S=—' 7. a=-, 1 = -- 256 2' 2048 8. r = ?, ^.191Il;or,r = -§, S = '^. 9. r = l, n = 9." 2 384 2 384 2 10. lz=- J-, n = 6. 11. a = 3, w = 7. 12. r = -, n = 8. 324 2 13. ;^ «4-(r-l)>y , j4 r = ^^^. 15. a = rl-ir-l)S. r S-l ^ ^ 16. a = -L,S= ^(r;-l). n.a = ir-^)S m-i^r-DS, yn-l jrn-l(^y _ J) r« — 1 »«" — 1 »• -(-:y ' '^"TT — X" § 347 ; pagre 304. 2. ?. . 4.-1 6. li. 8. -^. 2 6 5 40 3. ^. 6. -5. 7. L2. 9. A. 6 55 21 348; page 305. 25 ^ 581 ^ 107 « 2284 2475* 2. 1-. 3. A. 4. 25. 5. 581, 107. - 11 27 36 990 925 § 349 ; page 305. 2. r=3. 3. r=-2. 4. r=±2. b.r=±~. 6. r=-4. 7. r=±-. 'Z o § 350 ; page 306. 1. 2^. 2. 1. 3. rt2 _ 52. 4. ^Ltll. X — '2y 38 ALGEBRA. §351; pages 306, 307. 2. -4. 3. 4, 12, 36, 108. 4. 5, -10, 20; or, -5, -10, -20. 5. $4118. 6. 32 ft. 7. - ^^. 8. (a'"b)"^. 9. -3,4, 11; or, 13, 4, -5. 10. A, $108; B, $144; C, $192; D,|256. 11. - 4, 1, 6, 36 ; or, 8, 1, - 6, 36. 12. 3. iQ A a Q ^v 76 190 475 13. 4, b, 9 or, — , , • ' ' ' ' 39 39' 39 § 355 ; page 309. --i • -i, 5. 1. 6. -A. 61 17 7. 1 ». . f . 10, -10, 10 2 1^ --, -2, --, - 10 ' 9* - -1 - 4 -^j ~ -4, 2, 4,1, A, 2. '52 11 7 4 3 12 2 12 3 4 6 12 1 .11. — , 5 ~5' "^' ■ 5' ~35' 10' 15' 25' 55' 5* .1. 4. - -I 12. 1-^^ X 13. xy 2x-y xy xy 14. 5 and —3. Sx-2y 4x-Sy § 360 ; page 314. 10. ai^ + 5 a^b^c + 10 a%^c^ + 10 a'^h^c^ + 5 a^&i^c* + b^h^. 11. X^2m^(J ^I0m^3n_|_ I5 x8"'2/6n + 20 a;'5'»2/9''+ 15 a;4'»|/12n + 6 iC^'^J/l^w -f ^/18«. 12. 16 «4 _ 32 a-3 + 24 a2 - 8 a + 1. 13. x5+ 10a;*4-40x3 + 80a;2 4.80x + 32. 14. «4 - 12 a36 + 54 a25-2 _ iqb a63 + 81 &*. 15. 1 + 12 m2 + 60 m* + 160 m^ + 240 jji^ + 192 m^o + 64 «ii2. 17. a;4 -f 5 x'=" + 10 o;^ + 10 x~^ + 5 a;""^" + x-^ 18. a^ - 14 a*^ + 84 J - 280 a^ + 560 a^ - 672 a + 448 a^ - 128. 19. 243 + 405 x3 + 270 x^ + 90 iK^ + 15 x^^ + x^^ 20. ?>r^ + 6 w"r^ + 15 9W~3 + 20 TO* + 15 rn^' + 6 w^^ + mK 21. 256 a^ - 256 a 'x-^ + 96 a^x^ - 16 a^x + xl , 22. X-IO - f X-8 y^ + li> X-6^8 _ |0 ^-4^12 _^ _5_. a.-2yl6 _ _1 ^ 2,20. 23. to12 + 20 m^-^ + 150 m^x-"^ + 500 w'^x-^ + 625 x-12. ANSWERS. 39 24. 16 a^ + 16 a- + C a"^' 4 a~^ + ^^ a-^. 25. x^' - 7 aj's^y"^ + 21 a;2i/"^ - 35 x^y'^^ + 35 x^y-i - 21 x^?/"^ 2 _3 _7 + lx''>y 2 - y ■*. 26. 16 a~* - 32 a-^h^ + 24 qTH - 8 a~3-62 + b'K / _15 3 -?^ '^ _3 9 _3 J ?. / 27. 3^2 ^ * — A" ^"^wi' + I x ^m"^ — I X ^wi^ + t ic ''w* ^ — m^. \ 28. ac + 16 a^ + 96 a'^'" + 256 a^ + 256 ai 29. «3 _ 18 a V^ + 135 a'^x'^ - 640 a V* + 1215 ax~'^' - 1458 ah~''^ + 729X-8. 30. 32 «'^ - 240 a'^b + 720 a^b'^ - 1080 a'-^ft^ 4- 8 10 «&* - 243 b^. 31. a 'ft's + 7 a'ft"^ + 21 a^6-i + 35 ah~^' + 35 a'h^ + 21 a'^ft + 7 a~^6^ H- a"^6i 32. 81 m2n-2 - 216 mu"! + 216 - 96 m-^n + 16 m-^n^. 34. 1 -4a; + 10x2- 16a;3+ 19x*- 16x5+ 10 x^ _ 4 a;7 + a:8. 35. x8 + 4x7 + 14x« + 28x6 4- 49x4 + 56x3 + 56x2 + 32X + 16. 36. 1+ 12x + 50x2 + 72x3 -21x4- 72x5 + 50x8 - 12x7 + x^. 37. x8- 8x7 + 12x6 + 40x6 -74x4- 120x3 + 108x2 + 216 x + 81. 38. 1 + 5x+5x2-10x3-15x4+ 11x5+ 15x6-10 x7-5x8+5x9-xio. 39. xio - 5x9 + 20x8 - 50x7 + 105 x^ - 161x5 + 210x4 - 200x3 + 160x2 -80x + 32. § 362 ; page 316. 2. 56a5x3. 7. Vis'a"^^*- 12. j%\\a^h-^ 3. 165 m3. 8. -220xi5y-3. 13. 42240 a'^x's". 4. 126a564. 9. 5005 a6»'+9''. 14. 21840 ^jAV^. 5. -11440x9. 10. - 219648 x-6i/i 15. - -^002 ^-'iy-i-^ 6. 495w8n24. 11. 61236 rt""j\25. 16. -i^^a^xu^ § 371; page 323. 3. 1 +4x-4x2 + 4x3-4xt+ .... 4. 3 + 10x + 40x-+ 160x'' + 640x4+ .... 5.2 + 13x2 + 39x4+117x6 + 351x8+.... 6. 2 X - I x3 + -2^t x5 - 3S x7 + -IjV ^^ . 7. 1 + x + x'J + 2x4 + 5x5+ .... 40 ALGEBKA. 8. 2x-7x2 + 38a;3_204a:4 + 1096a;5 . 9. ix-2 + |x-i + |f + w^ + ili^' + •.•. . 10. i-ix-Y^-'-ffa:^ + Ma^'+-- 11. l-2x + a;2 + 2x3-3x4+ -. 12. 2 + 9x4-23x2 + 47x'5 + 73a:t+ .... 13. x-3 + 5x-2 + 20X-1 + 106 + 570x + .... 14. 3x-2 + 14x-i + 39 + lOlx + 264x2 + .... 15. ^x2-2x3 + fx*-|x5 + |x6+.... ID. 3 + eX— 2yX — TT^ 2¥3^ — •••. 17. fX-1- fX+^x2-h 3x3- |X4+ .... § 372 ; page 324. 2. 1 +2x-2x2 + 4x3- 10x*+ .... 3. l_fx-^/-x2-Jj-V-x3--3ji^/-x*-.... 4. 1 +X-X2 + X3- fx4+ .... 5. l-ix-fx^-J^jX^-rVsX*-.... 6. l+x-x2 + fx5-J3'>xi + .... 7. 1 -ix + fx2 + i!x3 + ^13X^4- -. § 374 ; page 325. 4. J- ? 6. » 1 1 8 4 5- 4. 2x + 3 2x-3 6. 8 7 7. 2x + 3 3x-2 0. .-IN 3x 3(5x-6) X x+5 x-6 4a 3a g 10 3 x + 5a X — a 2 — 5x 4 -f x 1 10. 1_^_+ 2 1 2(2x-l) 2(4x-3) 3x + 2 x x-2 x + 3 x-3 § 376 ; page 327. 9 e_ 2 1 2x-3 (2x-3)2 6(5x+2) (6x+2)2 5(5x+2)3 1 g 4 „ 1 4 3 x + 5 (X + 6)2 (X + 5)3' ^13 5 3x-l (3x-l)2 (3x-l)3 x+2 (x + 2)2 (x+2)3 (x + 2)4 6._2_+_i2 _1 92,5 4 2x-3 (2x-3)2 (2x-3)3 3(3x-2) 3(3x-2)2 3(3x-2)* ANSWERS. 41 § 377 ; pagre 328. 2 2___5 8_. 6J+2__3 i_. X ic-3 (x-3)2 X x2 x-l (x-l)« 3 3 4 2 ^ 5 g 1 3 5 'xx2x3x + 4 'xx+1 (x + 2)2* 4. --^_ + -^^ I 7.-1 L_+ ^. 3x-l 2x + 3 (2x + 3)-^ 4x4-1 2(2x-3) 2(2x-3)« § 378 ; page 329. 2.3x-2+-«---A_-. 4.x-l-l-l+4- + ^!_. x + 2 3x-l X x2 x8 x+l 3. 2 ^ + — -• 5. x4-2+?--l ?— + 2 X - 2 (X - 2)8 X x^ X - 1 (X - 1)2 6. x2 + 3-3..1+2_.^_. X x2 x3 x+3 § 379 ; page 330. 2 -2,-. I S a; - 1 , g 3 _ 1 X - 1 ' x + i a;2-x + r ' x + 1 x-1 x^+l' 3 5 ^ 2x + 3 g _J 3X4-.1 3 X + 1 X- - X + 3* ■ 2 X - 3 4 x2 + 6 X + 9* 4 4 _ x-3 ^ . 5x + n _ 3x-4 ■ 2 X - 5 x2 + 2* ' x^-^x+l x2 - X 4- 1' § 380 ; pagr6 331. 2. x = y + ?/2 + y3^y4+ .... 6. a; = y+ i ?/2 4. i ^3+ i y4_|. .... 3. x = .v+i2/2+^2/»+ Ay^+-. 7. x = 2y-2t/2 4-4y3_|y4+.... 4. X = y-2y2 4.5y8_i4y4+.... g. .« = y _ yS 4. ^6 _ ^,7 ^ .... 6. x = 2/ + 3y2 + 13y3 + 67t/44--. 9. a- = .v-i y3 + ^^y6_^Y^y74..... § 383 ; page 336. 7. J -\ a~^x - /j a"^x2 - ^1^ a~'^''ot^ - ^^l-g a~'^'x* . 9. a-6 + 6 a-76 + 21 a-»b^ + 56 a-^fts + 126 a-^%*+ .... 10. x^ - 5x!/ + V-a^V - t a;"V _ 5 3.-1,/*+ .... 11. m8 - f w^w"^ + Jg2 + ^io^c^-^t ^ 7103.3-^3+ .... _9 _3 2 34 96 158 ^i^oT'x^ _1 3 23 1 ff "27)6 1365 a;ii. -192a:7yl tV-t a~%8. § 384 ; pagre 337. 7. ^Wx\ 8. - _4J) 9. - 2002 a:-i6m«. 10. ^p wr'^8^n--8. 12. Y#-«""'^'ft-^ 13. -l-4_8^ 14. 220x-iV2>-6 15. _2 3 11* 3 2V6 8 ^ ^-:»^«. § 385 ; page 338. 5.09902. 4. 2.08008. 9.89949. 5. 2.97182. ; § 397 ; page 342. 1.5441. 7. 2.1003. 12. 2.5104. 1.6990. 8. 2.2922. 13. 2.5774. 1.6282. 9. 2.3892. 14. 2.6074. 1.8751. 10. 2.3222. 15. 2.9421. 1.6020. 11. 2.7960. 16. 2.8363. 6. 2.03055. 7. 1.96100. 17. 3.0512. 18. 3.4192. 19. 3.7814. 20. 4.0794. 21. 4.2006. 2. .5229. 3. .2431. 4. 1.1549. § 309: 5. 1.6532. 6. .2589. 7. 2.3522. page 343. 8. .2831. 9. .7939. 10. 2.1303. 11. 1.4592. 12. 1.3468. 13. 2.0424. § 402 ; page 344, 3. 3. .3397. 5. .7525. 7. 7.7205. 4. 4.19-10. 6. .6338. 8. .4824. 10. .286.3. 1.0460. ANSWERS. 4S 11. .3943. 12. .0682. 13. .1165. 14. .0939. 2. 0.4471 3. 1.0491. 4. 9.7993-10. 15. .4042. 16. .6250. 17. .4978. 18. .2542. 20. .0495. 21. .0366, 22. .7007. 23. .8752. 24. .0794. -^ 25. .4248. 26. .1341. 27. .1807. § 406 ; page 346. 6. 1.5104. 10. 6.5353 - 10. 14. 3.2646. 7. 7.5741-10. 11. 9.9421-10. 16. 0.1151. 8. 3.8293. 12. 0.4134. 16. 0.7335. 5. 8.9912 - 10. 9. 8.5932 - 10. 13. 2.4383. § 411 ; pagre 350. 6. 3.0286. 9. 7.8605-10. 12. 2.4032. 15. 7.8108-10. 7. 1.9189. 10. 0.8923. 13. 9.9632-10. 16. 8.1332-10. 8. 9.9830-10. 11. 6.5783-10. 14. 3.6099. 17. 0.6059. § 413 ; page 351. 4. 64.26. 7. .8143. 10. .09215. 13. .5061. 5. 2273. 8. .004897. 11. 64.23. 14. 366.8. 6. 461.2. 9. 7.488. 12. .003856. 15. 17008. 16. .0001994, § 418; pages 355, , 356. 1. 189.7. 15. -1.167. 29. .6682. 45. 2.627. 2. 8.243. 16. -.002893. 30. .6458. 46. 2.527. 3. - 1933. 17. 3692. 31. .1377. 47. -.8378. 4. .3091. 18. .2777. 32. -.3702. 48. 1.033. 5. .002976. 19. - 16893. 35. 30.12. 49. .2984. 6. -.01213. 20. .001233. 36. 2.487. 50. .3697. 7. 6.359. 21. 316.2. 37. 1.056. 61. .7945. 8. .03018. 22. .7652. 38. .0006777. 52. .9348. 9. - 6.853. 23. 243.9. 39. .007105. 53. 179.6. 10. 311.9. 24. .00001085 40. .8335. 64. 1.883. 11. .2239. 25. 2.236. 41. .5428. 55. .0001931, 12. -.009544. 26. 1.149. 42. - 36.03. 56. -.09954. 13. .1261. 27. - 1.276. 43. - 11.11. 67. .1711. 14. .02367. 28. 1.778. 44. .9432. 58. - 74.88. 44 ALGEBRA. S. .28301. § 419 ; page 357. 4. -2.172. 5. 1.155. 6. -.1766. 51oge 3 lag a log a — 2 log b log n — 4 log m 9. 10. 4, - 1. 11. ^^ log^-loga _^i, 12. ^^lo ^[(r-l)>y+a]-log« , log r log r II. n = log ? — log g log(6^-a)-log(>S-0 4-1. 14. ^^log?-lo g[W-(r-l)^j ^^^ logr • §420; pai^e 358. 2. 3.701. 3. -.06552. 4. -2.761. 5. 2.389. 6. -.3763. 7. .3731. 9. 4. 10. -• 11. --• 12. -• 3 3 5 OF THE UNIVERSITY OVERDUE. LD2l-l00m-7,'40 (6936s)