University of California • Berkeley The Theodore P. Hill Collection of Early American Mathematics Books A SCHOOL ALGEBRA COMPLETE FLETCHER DUEELL, Ph.D. Mathematical Master in the Lawbencevillb School AND EDWARD R. ROBBINS, A.B. Mathematical Masteb in the William Penn Chaeteb School NEW YORK CHARLES E. MERRILL CO. Durell & Robbins' Mathematical Series Durell & Robbins — Elementary Practical Arithmetic. 201 pages, 12mo, cloth, 35 cents Durell & Robbins — Advanced Practical Arithmetic. 363 pages, 12mo, cloth, 65 cents Durell & Robbins — A Grammar School Al- gebra. 287 pages, 12mo, half leather, 80 cents Durell & Robbins — A School Algebra 374 pages, 12mo, half leather $1.00 Durell & Robbins^ A School Algebra Com- plete. 483 pages, 12mo, half leather, $1.12 Durell — Plane Geometry 341 pages, 12mo, half leather 75 cents Durell — Solid Geometry 2 1 3 pages, y2mo, hftlf leather 75 cents Durell — Plane ^hd Solid Geometry 514 pages, 12mo, half leather . . , . , $1.25 Durell — Plane Trigonometry and Tables 298 pages, 8vo, cloth $1.25 a Copyright 1897, by Charles E. Merrill Co. PREFACE. The principal object in writing this School Algebra has been to simplify principles and make them attractive, by showing more plainly, if possible, than has been done here- tofore, the practical or common-sense reason for each step or process. Thus, at the outset it is shown that new symbols are introduced into algebra not arbitrarily, but for the sake of definite advantages in representing numbers. The fundamental laws of algebra governing the use of sym- bols derive their importance in like manner from the econ- omies which they make possible in dealing with the symbols for numbers. Each successive process is taken up for the sake of the economy or new power which it gives as com- pared with previous processes. It is hoped that this treatment not only makes each prin- ciple clearer to the pupil, but also gives increased unity to the subject as a whole. It is also believed that this treatment of algebra is better adapted to the practical American spirit, and gives the study of the subject a larger educational value. While seeking to develop the theory of the subject in this manner, it has been deemed best to keep in close touch with the best current practice of teachers in other respects. For instance, the order of topics in text-books most used at pres- ent has been followed. 8 4 PREFACE. Great care has been taken in the selection and gradation of a large number of examples. It is hoped that they have been so graded that any example may be considered the last of a series of progressive steps, provided the teacher wishes to limit the work at any particular point. Frequent reviews have been provided for, especially in the all-important sub- jects of Factoring, Fractions, Exponents, and Radicals. This volume contains, besides the specified requirements in algebra for admission to the classical course of colleges, the more advanced subjects required by universities and scien- tific schools — to wit, Permutations and Combinations, Unde- termined Coefficients, The Binomial Theorem, Continued Fractions, and Logarithms. The authors will sincerely appreciate the courtesy, if their friends and fellow-teachers will kindly advise them of any discovered errors. FLETCHER DURELL, EDWARD R. ROBBINS. Lawrencevill,e, N. J., 1 December 23, 1897. / CONTENTS. CHAPTER I. PAGE ALGEBRAIC SYMBOLS 9 I. Symbols of Quantity 10 II. Symbols of Operation 12 III. Symbols of Relation 14 Algebraic Expressions 15 CHAPTER 11. METHODS OF USING ALGEBRAIC SYMBOLS .20 I. Laws for -f- and — Signs 20 II. Laws of Arrangement and Grouping 22 CHAPTER HI. ADDITION AND SUBTRACTION ! 26 Addition 26 Subtraction 29 Use of Parenthesis 32 CHAPTER IV. MULTIPLICATION 38 Multiplication of Monomials 38 Polynomial by a Monomial 40 Polynomial by a Polynomial 41 CHAPTER V. DIVISION 48 Division of Monomials 48 Polynomial by a Monomial 50 Polynomial by a Polynomial 51 5 6 CONTENTS. * CHAPTER VI. PAGE SIMPLE EQUATIONS 57 Solution of Problems 63 CHAPTER VII. ABBREVIATED MULTIPLICATION AND DIVISION .... 73 Abbreviated Multiplication 73 Abbreviated Division 81 CHAPTER VIII. FACTORING 86 CHAPTER IX. HIGHEST COMMON FACTOR AND LOWEST COMMON MULTIPLE ]04 CHAPTER X. FRACTIONS 117 General Principles 118 Transformations of Fractions 119 Processes with Fractions . . ..... . 126 CHAPTER XL FRACTIONAL AND LITERAL EQUATIONS 143 Problems 153 CHAPTER XII. SIMULTANEOUS EQUATIONS .163 CHAPTER XIII. PROBLEMS INVOLVING TWO OR MORE UNKNOWN QUANTITIES 178 CHAPTER XIV. INEQUALITIES ....!.... 188 CONTENTS. 7 CHAPTER XV. PAGX INVOLUTION AND EVOLUTION 192 Powers of Monomials 192 Powers of Binomials 194 Evolution .... 197 Square Root 199 Cube Root ....... 206 CHAPTER XVI. EXPONENTS 213 Fractional Exponents 214 Negative Exponents 216 Polynomials, etc. 223 CHAPTER XVIL RADICALS 226 Transformations of Radicals 227 Operations with Radicals . . ? r • •232 CHAPTER XVIII. IMAGINARIES 261 CHAPTER XIX. QUADRATIC EQUATIONS OF ONE UNKNOWN QUANTITY 259 Pure Quadratic Equations 259 Affected Quadratic Equations 261 Factorial Method of Solving Equations ........ 265 Equations in the Quadratic Form 266 Radical Equations . . . ; - . • • • 269 Other Methods of Solving Quadratic Equations .... 270 CHAPTER XX. SIMULTANEOUS QUADRATIC EQUATIONS ........ 278 Special Methods of Solving Simultaneous Quadratic Equations , 28X 8 CONTENTS. CHAPTER XXI. GENERAL PROPERTIES OF QUADRATIC EQUATIONS. . 289 Properties of ax^ + bx + c = 292 CHAPTER XXII. RATIO AND PROPORTION 295 Ratio . 295 Proportion 296 CHAPTER XXIII. INDETERMINATE EQUATIONS. VARIATION 304 Variation 307 Kinds of Elementary Variations 308 CHAPTER XXIV. ARITHMETICAL PROGRESSION 314 CHAPTER XXV. GEOMETRICAL AND HARMONICAL PROGRESSIONS ... 323 Harmonical Progression • • • 333 CHAPTER XXVI. PERMUTATIONS AND COMBINATIONS 336 CHAPTER XXVII. UNDETERMINED COEFFICIENTS 344 I. Expansion of a Fraction into a Series 347 II. Expansion of a Radical into a Series 349 III. Partial Fractions 350 IV. Reversion of Series 355 CHAPTER XXVIII. THE BINOMIAL THEOREM 358 CHAPTER XXIX. CONTINUED FRACTIONS 367 CHAPTER XXX. LOGARITHMS 376 CHAPTER XXXI. HISTORY OF ELEMENTARY ALGEBRA 393 CHAPTER XXXII. APPENDIX 403 SCHOOL ALGEBRA COMPLETE. CHAPTER I. ALGEBRAIC SYMBOLS. 1. First Source of New Power in Algebra. Let the fol- lowing problem be proposed for solution : James, John, and William together have 120 marbles. John has three times as many marbles as James, and William has twice as man}^ as John. How many marbles has each boy ? The solution of the problem is facilitated by the use of some symbol, as x, for one of the unknown numbers. Thus, Let X = number of marbles which James has, thenSx = " " " John has, 6x = " " " William has. Hence, x + 'Sx + Qx= " " " all have to- gether. But 120= " " " all have to- gether. Hence, x + Sx + 6x = 120 that is, lOc = 120 x — 12, number of marbles which James has, 3a: = 36, " " " John " 6a; = 72, " " " William" This solution of the problem illustrates the first new prin- ciple in algebra — viz. that a more extended use of symbols than is practised in arithmetic gives increased ease and power in the investigation of properties of number. In the above 10 ALGEBRA. example, a symbol being used for one unknown number, the other unknown numbers may be expressed in terms of this symbol ; then the relation of all the unknown numbers to the known number is expressed in a form which is readily reduced to another form so simple that from it the value of the first unknown, and afterward the values of the other unknown numbers are at once perceived. 2. Definition of Algebra. Hence, Algebra, in its first con- ception, is that branch of mathematics which treats of the properties of number (or quantity expressed by number) by the extended use of symbols. Algebra may be briefly described as generalized arithmetic. 3. Three Classes of Symbols. Three principal kinds of symbols are used in algebra: I. Symbols of Quantity. II. Symbols of Operation. III. Symbols of Relation. I. Symbols op Quantity. 4. Symbols for Known Quantities. Known quantities ar« represented in arithmetic by figures ; as, 2, 3, 27, etc. They are represented in the same way in algebra, but also in an- other more general way — viz. by the first letters of the alpha- bet ; as, a, 6, c, etc. The advantage in the use of letters to represent known numbers lies in the fact that a letter may stand for any known number, and a result be obtained by ihe use of letters which is true for all numbers. 5. Symbols for Unknown Quantities. Unknown quan^ titles in algebra are usually denoted by the hist letters of tlib alphabet ; as, x, y, z, w, v, etc. The advantage in the use of a distinct symbol for an un- known quantity is stated in Art. 1. ALGEBRAIC SYMBOLS. IV 6. Symbols for Groups of Similar Quantities. Groups of similar quantities are usually represented by groups of similar symbols; as, (1) By the same letter with different accents; for example, a', a", a!'\ etc., read " a prime," " a second," " a third," etc. (2) By the same letter with different subscript figures ; as, «!, «2, CL-ii etc., read " a sub-one," " a sub-two," etc. The advantages of these ways of representing groups of similar quantities are obvious. 7. Sign of Continuation. After a group of similar quan- tities a series of dots is often written to indicate that the group is continued indefinitely. Ex. tti, a2, as, This series of dots is called the sign of continuation, and reads " and so on." 8. Positive and Negative Quantity. Negative quantity is quantity exactly opposite in quality or condition to quan- tity taken as positive. If distance east of a certain point is taken as positive, dis- tance west of that point is called negative. If north latitude is positive, south latitude is negative. If temperature above zero is taken as positive, temperature below zero is negative. If in business matters a man's assets are his positive pos- sessions, his debts are negative quantity. Positive and negative quantity are distinguished by the signs + and — placed before them. Thus, $50 assets are denoted by +$50, and $30 debts by -$30. We de- note 12° above zero by + 12°, and 10° below zero by - 10°. The use of the symbols + and - for this purpose, as well as to indicate the operations of addition and subtrac- tion (see Arts. 9 and 10), will be justified later on (see Arts. 31,32). 12 ALGEBRA. n. Symbols of Operation. 9. The Sign of Addition is -f-, and is called "plus." Placed between two quantities, it indicates that the quan- tity after the plus sign is to be added to the quantity before it. Thus, a + & is read "a plus 6," and indicates that the quan- tit}^ b is to be added to the quantity a. 10. The Sign of Subtraction is — , and is called " minus." Placed between two quantities, it indicates that the second quantity is to be subtracted from the first. Thus, a — 6 is read " a minus 6," and means that b is to be subtracted from a. 11. The Sign of Multiplication is X, and reads "times" or " multiplied by," or simply " into." Placed between two quantities, it indicates that the one is to be multiplied by the other. Thus, aXb reads " a multiplied by 6," and means that a and b are to be multiplied together. The multiplication of literal quantities (and sometimes of arithmetical numbers) may also be indicated more simply by a dot placed between the quantities. The multiplication of literal quantities is indicated most simply of all by the omis- sion of any symbol between the quantities. Thus, instead of a X 6, we may WTite a-b, or ab. 12. Factors. The factors of a number are the numbers which multiplied together produce the given number. For example, the factors of 14 are 7 and 2 ; the factors of abc are a, b, and c. 13. Coefficients. In case a numerical factor occurs in a product, it is written first, and is called a coefficient. Hence, A Coefficient is a number prefixed to a given quantity to show how many times the given quantity is taken. ALGEBRAIC SYMBOLS. 13 For example, in bxy^ 5 is called the coefficient. When the coefficient is 1, the 1 is not written, but is understood. Thus, xy means \xy. When a number of factors are multiplied together, any one of them or the product of any number of them may be re- garded as the coefficient. Thus, in bahx^ bah is sometimes regarded as the coeffi- cient. 14. Powers and Exponents. A Power is the product of a number of equal factors. The expression for a power is abbreviated by the use of the exponent. An Exponent is a small figure or letter written above and to the right of a quantity to indicate how many times the quantity is taken as a factor. Thus, for xxxx, or four x's multiplied together, we write a;*, the exponent in this ca-se being 4. An exponent is thus in effect a symbol of operation. When the exponent is unity it is omitted. Thus, x is used instead of ic\ and means x to the first power. 15. The Sign of Division is ^, and reads " divided by." Placed between two quantities, it indicates that the quantity to the left of the sign is to be divided by the quantity to the right of it. Thus, a H- 6 means that a is to be divided by h. Division may also be indicated by placing the quantity to be divided above a horizontal line and the divisor below the line. Thus, for an- 6 we may write -• The expression - is often read " a over 6." h 16. The Radical Sign is ;/, and means that the root of the quantity following it is to be extracted. The degree of the root is indicated by a small figure placed above the rad- t4 ALGEBRA. ical sign. For the square root the figure or index of the root is omitted. Thus, Vd means "square root of 9." Va means " cube root of a." m. Symbols of Relation. 17. The Sign of Equality is =, and reads " equals " or " is equal to." When placed between two quantities it indicates that they are equal to each other. Thus, a = b means that a and b are equal quantities. 18. The Signs of Inequality are >, which reads "is greater than," and <, which reads " is less than." Thus, a > 6 means that a is greater than b. c 4-l) 2^ y_^ 6. 34a;-l)-ar'. * 5(a;^ + l)-3^ y" 3a;*' 11. (^ - 2a) (7a; + 11a) < ^^TtS* (a'— x' + l)' 12. • ["^^ ^^' ^ 5K - 3^)- (2/^ - 2n-z). Express in algebraic sj^mbols — 13. Five times a plus seven times h. 14. Six a square equals twice the quantity a minus h. 15. Four times the quantity a square minus nine h is less than the square of the quantity seven a plus h cube. 16. The product of x minus ten y square, and x cube plus y times 2, equals two a times x to the eighth power. 17. The quantity nine x plus two y divided by three z, is equal to nine x, plus two y divided by three z cube. 18. Five a cube, plus six ft square over the square of the quantity x minus two y cube, is greater than five times the quantity a cube plus h square, over the cube of the quantity X plus two y to the fourth power. 19. Four X minus a fraction whose numerator is x plus y square, and whose denominator is the square of the quantity X plus 2/, equals one, minus x square over y. 29. The Numerical Value of an algebraic expression is obtained by substituting for each letter in the expression the 2 18 ALGEBRA. value which it represents, and performing the operations in- dicated. Thus, if a = 1,6 = 2, c = 3. Ex. 1. Find the numerical value of lah — c^ 7a6 - c' = 7 X 1 X 2 - 3^ = 14-9 = 5, Ans. 96 Ex. 2. Find numerical value of - + bah'' - 7(a' + 26)' + 2>e. c We obtain = ^^^ + 5 X 1 X 2'^ - 7(1' + 2 X 2)M 3 X 3^ o = 6 -f 20 - 175 + 27 = - 122, Ans. EXERCISE 2. Find the numerical value of each of the following when = 5, 6 = 3; , c = l, x = Q: 1. 2a. 11. a + 2c. 21. 3 + 2(a;-a). 2. 3x. 12. 36 -a:. 22. 5x - 3(26 + c). 3. ax. 13. x' — ax. 23. 2(a;' - a') + 3ac. 4. 3ac. 14. 2a; -46c. 24. 7a:(5a - 4x) - ca:^ 6. h\ 15. a'-bx\ 25. 3a;(x - 3)' - 9a;. 6. d'c. 16. 3(a -h c). 26. (a; - 1) (a; - 3) + a;(a; - a). 7. ex. 17. x^ when a; = 3. 72. 3a;(a; + 1)* - 6a; - 7(a; + 2) when a; = 4. 73. Ta; + 5(4a; + 1) - 3a;(a; + 2)' when a; = 1. 74. ^7?{x + \y - (a; + 2)^^ (x - If when x = 2. 75. (3a; + yf - 2(a; - 2y) (a; + 2y) + 5(2a; - Syf when x = ^ t/ = 2. CHAPTER II. METHODS OF USING ALGEBRAIC SYMBOLS. 30. Second Source of New Power in Algebra. The second general source of new power in algebra lies in cer- tain standard ways or methods in which the symbols of algebra are used. These ways are termed the Laws of Al- gebra. There are two divisions of the primary laws of algebra : I. Laws for + and — Signs. II. Law^s for Grouping and Arrangement of Symbols OF Quantity. I. Laws for + and — Signs. 81. First Law for + and — taken together. As was explained in Arts. 8, 9, 10, the signs + and — are employed for two purposes — first, to express positive and negative quantity ; and second, to indicate the operations of addition and subtraction. We are able to put these signs to this double use because, as used in both of these ways, the signs are gov- erned by the same laws. Thus, if the distance to the right of be regarded as posi- tive, and therefore the distance to the left of as negative, —7—6—5—4 —8 —2 —1 +1 +2 +3 +4 -f 5 +6 +7 +8 XU I I \ \ \ I I I I \ \ \ I I \ I E' ^ K B A F ^ and a person walk from toward E a distance of 5 miles (to F)^ and then walk back toward W a distance of 3 miles (to A)^ the distance travelled by him may be expressed as 20 METHODS OF USING ALGEBRAIC SYMBOLS. 21 the sum of a positive quantity and a negative quantity; that is, (positive distance OF) + (negative distance FA)^ or, +5 + (-3) = 5-3 = 2. The position arrived at raay also be determined in another way — viz. by deducting (that is, using the operation of sub- traction) 3 miles from 5 miles. We obtain 5-(4-3)=5-3 = 2. Hence, we see that adding negative quantity is the same in effect as subtracting positive quantity; therefore, in the ex- pression 5-3 the minus sign used may be considered either a sign of the quality of 3, or as a sign of operation to be performed on 3. Hence, we are able to use the signs + and — to cover two meanings, as stated above. Whichever of these two meanings be assigned, we see that 4- (- 3) = - 3 ; also, - (+ 3) = - 3. Hence, Law 1. The signs + and — applied in succession to a quantity are equivalent to the single sign — . Or in symbols, + (— a) = — a ; —(+ a) = — a. 32. Second Law for + and — taken together. Again, if in the above illustration a person walk in the negative direction from 0—i. e. toward W—a distance of 4 miles to K, and then reverse his direction and go 2 miles, he will be at jB; or the distance travelled is expressed as -4-(-2) = -4+2 = -2; that is, the two minus signs in — (— 2) taken together give +. So also we see that deducting a certain sum from a man's debts is the same in effect as adding this sum to his assets ; 22 ALGEBRA. or, in general, that a double reversal of the quality of any quantity gives the original quality. Hence, Law 2. The sign — applied twice to a given positive quantity gives a -\- result. Or in symbols, — ( — a) = + a. It is also evident that + (+ a) = + a. By these laws any succession of + and — signs applied to a quantity can be at once reduced to a single + or — sign. Ex. -[+(-a)] = -[-a] = + a. These laws therefore enable us to use negative quantity with as great freedom as we use positive quantity, and hence are an important source of power. They also open the way to a free use of the second group of the laws of algebra. n. Laws of Arrangement and Grouping. 33. Formal Statement of Laws of Arrangement. The laws which govern the arrangement and grouping of the symbols for quantity in algebra are — A. The Commutative Law. 1. For Addition, a + b = b -\- a. 2. For Midtiplication, ah = ha. 3. For Division, a^hXc = aXc^b. B. The Associative Law. 1. For Addition, a + 6 + c = a + (6 + c) = (a + 6) + c. 2. For Mxdtiplication, ahc = a(hc) = (ah)c. C. The Distributive Law. 1. For Multiplication, a(h -\- c) — ah -+• ac. Hence, in- versely, ah + ac^ a(h + c). __,^. .. 6 + c6,c 2. For Division, = - H a a a 34. Meaning of Commutative Law for Addition. The meaning of these laws is best shown by examples. If it is METHODS OF USING ALGEBRAIC SYMBOLS 23 required to combine a group of 7 objects and another group of 5 objects into a single group, we may either count the 7 objects first, and then count on the 5 objects afterward, or the 5 objects first and the 7 afterward ; that is, groups of objects may be counted together into a single group in any order we please. The algebraic symbols representing different groups in like manner may be arranged in any order we please; that is, briefly, a-\rh = h -\- a. An example of the advantage in this quality of algebraic symbols is that similar terms in an expression may, by rear- ranging the terms, be brought together, and then counted into a single term by the use of the Distributive Law for Multipli- cation (inverse form). Thus, the terms of the algebraic expression, la^h - bxf + ^xy" + Za'h + Axf - 2a\ by the use of the Commutative Law may be arranged thus, la^h + ^o?h - 2a'6 - bxy^ + e>xif + Axy\ By the Distributive Law for Multiplication (inverse form) the first three terms may be combined into a single term, and the last three into another term, giving Wh + hxy\ Thus, by use of these laws 6 terms are reduced to 2 terms. The symbols used to represent number in arithmetic cannot be changed about in this manner. Thus, the number 234 cannot be written 324. If, however, we employ the + sign, the symbols used for the number may be put in a commutative form ; as, 234 = 200 + 30 + 4 = 30+ 4 + 200 = etc. The arithmetical form, 234, has the advantage of greater brevity than the algebraic form, 200 + 30 + 4, but the disadvantage of less flexibility. 35. Meaning of Commutative Law for Multiplication. 24 jiLGEBRA. To illustrate the Commutative Law for Multiplication, we recognize that if we have 15 objects, the number of the ob- jects is the same whether they be arranged in 5 rows of 3 objects each or 3 rows of 5 objects each — • • • • • • • • • • • • • e • So, in general, ah = ha. The advantages resulting from this property of algebraic symbols are illustrated by the fact that we are enabled by it to have a standard order for the arrangement of the literal factors in a term — viz. the alphabetical order. Thus, instead of writing Ic^xa^ or lax&. or lxac\ we write the literal factors in the alphabetical order, la&x. , When the factors in each term of an expression are thus arranged, it is much easier to recognize similar terms. 86. Meaning of the Distributive Law for Multiplioa- tion. Let it be required to reduce the expression, 6(a - 6 + c) + 2(a + 6 - c) + 3(a + 6 + c), to its simplest form. Applying the Distributive Law, the expression becomes ba - 56 + 5c + 2a + 26 - 2c + 3rt -H 36 + 3c. Using the Commutative Law, we obtitin 5a + 2a + 3a - 56 + 26 + 3S f 5c - 2c + 3c. Hence, by the Distributive Law, 10a -f 6c, METHODS OF USING ALGEBRAIC SYMBOLS. 25 In other cases the Distributive Law enables us to perform work, part by part, which would be difficult if not impossible in the undivided form. In general, therefore, these laws enable us to arrange and group the parts .of an algebraic expression to the best advan- tage according to the work to be done. They are, therefore, to be considered, from one standpoint, as economic methods which govern the use of algebraic symbols. It will be a useful exercise for the student to determine which of the fundamental laws for grouping and arranging algebraic symbols are used in the following illustrative ex- amples : Ex.1. 6(x + y) + S(x-y + z)-\-2(x + 2y-z), = 6x + Qy + Sx-Sy + Sz -\- 2x + 4y — 2z = 6x + Sx + 2x + Qy-Sy + 4ty + Sz-2z = llx + 7y + z . ^ ^ 12a'b' + 9a'b' + 6a'b' xLiX. Z. Ex.3. Bab = 12a:'b' 9n'b' Qa'b' Sab Sab Sab = 4a'b' + Sa'b + 2ab' =ab(4:ab+Sa + 2b) (2x + 32/) (3. * + 42/). ' = 2x(^ \x+ 4y)+Sy(S: c+ 42/) = Qx' + 8xy + dxy + 12y' = 6x' + 17xy + 12y' CHAPTER III. ADDITION AND SUBTRACTION. ADDITION. 37. Addition, in algebra, is the combination of several algebraic expressions, representing numbers, into a single equivalent expression. 38. Addition of Similar Terms. If the question be asked, How many books are 3 books + 7 books + 4 books ? the answer is, 14 books. In like manner, if the question be asked, How many a^6^'s are the answer is, 14a'6'. The simplification is obtained by the use of the Distribu- tive Law for Multiplication (Art. 36). Thus, similar terms are added by adding the coefficients of the terms and setting the result before the literal part common to the terms. If some of the similar terms are negative, the sum of the coefficients is taken, respect being had to their signs. The sum thus taken is called the algebraic sum of the coefficients. For example, add the similar terms Sa'x - 7a'x - Qa'x + lOa'x - a'x. The sum of the plus coefficients is + 18, the sum of the nega- tive coefficients is — 14, the algebraic sum of +18 — 14 is -f 4 ; hence, the sum of all the given similar terms is + Aa^x, 26 ADDITION. 27 39. Addition of Dissimilar Terms. If the terms to be added are dissimilar, the addition of them can be indicated only. Thus, h added to a gives a-\-h; also, a? — Sd^b added to Sa' - b' gives a' - Sa'b + 3a'- 61 Simplifications are possible only where there are similar terms. 40. General Method of Addition. The most convenient general method for addition is shown in the following ex- amples : Ex.1. Add 4a;' + 8a; 4- 2, 3a;' - 4a; - 3, -2x'-x-6. Arranging similar terms in the same column, and adding each column separately, we obtain 4a;' + 3x + 2 8a;'-4a;-3 ' . -2:g'- x-5 5x''-2x~6,Sum. Ex. 2. Add 2a' - 5a'6 + 4a6' + a'b\ 4a'6 + 2a' - ah' - Bab\ a'b-a' + 2ab\ Proceeding as in Ex. 1, 2a'-5a'6 + 4a6' + a'6' 2a' + 4a'6 - 3a6' -ab' -a'+ a'b + 2ab' 3a' + Sab' + a'b' - ab\ Sum, In the second column the algebraic sum of the coefficients is — 5 + 4 + 1, which = ; and as zero times a number is zero, the sum of the second column is zero, which need not be set down in the result. Hence, the general process for addition may be stated as follows : Arrange the terms to be added in columns, similar terms in the 28 ALGEBRA. same column; in each column take the sum of the + coefficients, and also the sum of the — coefficients; Subtract the less sum from the greater, prefix the sign of the greater, and annex the common letters with their exponents. 41. Collecting Terms. It is often required to add together the similar terms which occur in a single polynomial. This is called collecting terms. Ex. Simplify 2x + 7ab + 5- Sab + 2ab + Sx Collecting terms, we obtain 5a; -f 5 + 6a6. EXERCISE 3. Find the sum of- — 1. 2. 3. 4. 5. -11 4 8a; — X -7a; 6 - -10 -6a; -Sx 12a; 6. 7. 8. 9. 10. 2a - -x' 7xy a'b 7xy 5a Sx' -lOxy ba'b - 10a;y - -12a 5x' 2xy -Sa'b a^if 11. Sax, - - 2a,x, , bax, ax, - Sax. 12. 5x\ 12a:^ - 10:c^ x\ - IQa^, Sx\ -x\ 13. 7a*-'6^ - 12a'b\ -a'b\ -4a'b\ 5a'b\ Qa'b\ 14. 15. 16. Sx-2y 5a:^+ 7 a^ — ax-\- Aa^ 2x + Sy a;^-10 3a'^ + 2aa; — 5a;' X- y -7a;^+ 1 — a^— ax— x^ 17. a — 2b, Sa +. 46, a + 5b,—5a — b,a — 5b. 18. Sx" + y\ 2x^ - 1y\ - 4a:^ - 5y\ x' + Sy", - Sy\ 19. Sa^ - 5bif, 2ax^ + 46?/, 2by^ - 4aa;', 63/ - aar'. SUBTRACTION. 29 20. a^-xy + Sy\2x' + 2xy-2y\x'-\-y\Sx'-xy. 21. mn — 3n^ + m^, m^ + 2n^ — Zmn, m^ — n\ mn — 2m^ 22. x' + y' " 2s^ 3a;' -y'+ 2z\ ^ - 2x^ x' - z\ 23. 2x' - xy, Zxy - by\ Sif - ^x\ x' + 2y' - 2xy. 24. 7x-^y-i-bz — lOxy, 2y — Sz + l^xy — 4xz, 5z — Qx — 4xzy + 2xy, —i^y + 9z-{-7x — xz, ' 21xz — 16z + a: — 5xy. 25. x^ + 3xV -I- oxy' + 2/', ^'^ - Sx'y + 3a:2/' - 2/', 2x'y - 2a;2/' + 2/', x' - y\ x'y - Ax" - xy' - 2/^ y' + a;' - x^ 4~ ^^y. Collect similar terms in the following : 26. 2a; - 32/ - 5a: + 4z + 42/ + 2 — 2?/ — a; - 3z + 2a; — Sy. 27. Sxy — 5ax + Sy' — 2xy — 3a;' + 4aa; — 2y' + 3aa; — 2xy. 28. x — Sy + 2z + 22/ — 2x'— z— 3a; - 42 - 2a; + ;? + 2a;. 29. 2a; - 1 + 52/ - 2 + 3a; + 2 + 32/ - 3 - 2a; + 1 - a; - 32/. 80. 3a'6 - 2a'c + 3a' - 5a'b -a'- Sd'c + a'6 + 6a'c - 2a'. 31. 5a;^ - 3a; + 4 - 2a;' - 6x' + 4a; - 7 - a;' + a;^ + 3a;' - a; + 5 + 3x' - 6a; - a;' + 4a; - 2a;' + 2x. 32. 2a;"-5a;"' + 3a;'-a;" — 7a; + 3a;'-3 4- 2a;"* — 5a;'4-5 + 3a;"* SUBTRACTION. 42. Subtraction, in arithmetic, is the process of finding the difference hctween two numbers, and subtraction in alge- bra includes this work. But inasmuch as negative quantity is dealt with in algebra as well as positive quantity, the word difference takes a broader meaning, and we need a broader definition of subtraction which will cover both positive and negative quantity. If the quantity to be subtracted be named the Subtra- hend, and the quantity from which the subtrahend is taken be named the Minuend, and the result obtained be named the Difference, it is evident that for both positive and nega- 30 ALGEBRA. live quantity, the Difference added to the Subtrahend will give the Minuend. Hence, Subtraction, in algebra, is the process of finding a quan- tity which, added to a given quantity (the subtrahend), will produce another given quantity (the minuend). Thus, if we subtract Sab from 10a6, we obtain lab, for lab added to Sab (subtrahend) gives lOab (minuend). 43. Signs in Subtraction. From Art. 31 it is clear that subtracting a positive quantity is the same as adding a nega- tive quantity of the same absolute magnitude ; and from Art. 32, that subtracting a negative quantity is the same as adding a positive quantity of the same absolute magnitude. Hence, in subtraction, the most convenient way to govern the signs is to change (mentally) the signs of the '.^ terms in the subtrahend. -rr Thus, to subtract 46 from 76, we change 46 men- tally to —46, and add +lb and —46, and obtain the result 36. Again, to subtract — 26 from 76, we mentally change the sign of — 26, and add the result + 26 to + 76, and _J^ obtain +96. — ^ (Some concrete illustration of the reason for changing the sign of a negative term of the sub- trahend to plus in the process of subtraction should be fre- quently recalled by the student. For example, subtracting a $10 debt from a man's possessions is the same in effect as adding $10 to them.) 44. General Method for Subtraction. Accordingly, the most convenient general method in subtraction is to — Place the terms of the subtrahend under the terms of the minu- end, similar terms in the same column. Change the signs of the terms in the subtrahend mentally; pro- ceed as in addition. SVBTEACTlO^r. 31 Ex. 1. From 5a;'' -2x' + x~Z subtract 2x^-Sx'-x + 2. We obtain 5x' — 2x'+x — B 27^ -Sx"- x + 2 3x'+ x' + 2x-5,Diff'erence, since the coefficient of ^ is 5 — 2, or 3, of a;Ms — 2 + 3, or 1, etc. Ex. 2. Subtract 2a* - Sa'b - 6a'b' - 2ab' + 26* from a* + ba'b -6a'b'-Sab\ We obtain a* + 5a'b - 6d'b' - Sab^ 2a* - 3a^6 - Qa'b' - 2a6''' + 26* a* + 8a^6 - a6'-26* The coefficient of a^6* is — 6 + 6, or 0. The coefficient of 6* is - 2, or - 2. EXERCISE 4 1. 2. 3: From 7a6 bx x Take 3a6 9a: 2x 4. 5. 6. 5x -St^ -Ixy -Sx -4x^ Zxy 7. 8. 9. 10. From3a;'-4a: 3a;-9 2x^-5 5a;' + 4a;-3 Take 2x^4- x bx + 1 -ar' + 2 -7?-Zx + b 11. From 3a + 26- 3c- <^ take 2a-26 + c-2d 12. From 7 - 3a; + 231? take 15 — 4a; - bx\ 13. From a;^ - 2/' - 2^ + 8 take 2a;' + 2/' - 2^' + 10. 14. From bxy — 3a;z + byz + a;' take 4a:z — 2xy — o^. 15. From 2 - a; + a;' + a:* take 3 + a; - a;' - a;' - 2a;*._ IB. Subtract lOo^y + ZxY - 13a;?/' from T^y - a;^/' + 2a^y'. 17. Subtract 3 — 2a6 + 3ac — Acd from 5 — ac + Sec? — 5a(i. 18. Subtract 1 + a; - a;' + a;' - a;* from 2 - a; - a;' - a;' + a;^. 19. Subtract a + 26 — 3c + 4cZ from m + 26-fd — x-fa. 32 ALGEBRA. 20. Subtract Sx' - 2a:'^ + 5a; - 7 from Sa^ + 2x'-x-7. 21. Subtract — x' - 2x' + x' + 5 from y^ - x' + x'' — 2x + 5. • 22. Subtract Sx"* — 3a;" + a; — 3 from x"" + x"" — x"" + x — 1. IfA = x'-Sx';i-l,B = 2x' -5x-S,C=Bx^ + x' + Bx, find the values of — 23. ^ + J5 + C. 25. A-hB-0. 24. B-A + 0. 26. A -B+0. USE OP PARENTHESIS. 45- I. Removal of Parenthesis. Addition and subtrac- tion may be indicated briefly by the use of the parenthesis. Thus, the expression 2a + 36 - 5c + (3a - 26 + 3c) indicates that 3a — 26 + 3c is to be added to 2a + 36 — 5c. The process of addition thus indicated by the parenthesis may be performed in the usual way by placing similar terms in the same column, etc. But expressions like the above occur so frequently in algebra that it is found more conve- nient to simplify them simply by setting down the terms to be added in succession (omitting the parenthesis) and col- lecting similar terms. In accordance with this method we obtain, 2a + 36 - 5c + 3a - 26 + 3c = 5a + 6-2c. Similarly, the expression 2a + 36 - 5c - (3a - 26 + 3c) indicates that 3a — 26 + 3c is to be subtracted from 2a + 36 -5c. The most convenient way of making the subtraction is to USE OF PARENTHESIS. 33 change the signs of terms of the subtrahend (dropping the parenthesis which contains them) and to collect terms. Accordingly we obtain 2a + 36 - 5c - 3a + 26 - 3c = — a + 56 - 8c. Addition or subtraction performed in this way is called removing a 'parenthesis. The special rule to be observed in removing a parenthesis is that — When a parenthesis preceded hy a -{- sign is removed, the signs of the terms inclosed by the parenthesis remain unchanged. But — When a parenthesis preceded by a minus sign is removed, the signs of the terms inclosed by the parenthesis are changed, the + signs to — , and the — signs to +. 46. The Sign of the First Term within a Parenthesis is usually + understood, it being the custom to put a plus term first in an algebraic expression if possible. Owing to the absence of this + sign, the beginner frequently makes the mistake of using the sign of the parenthesis as the sign of the first term within it. This error may be obviated at first by writing out the sign of the first term in the parenthesis in full, till the fact of its existence is firmly realized. Thus, 5a - (+ 3a ~ 6) plainly reduces to 5a — 3a + 6, the — 3a being obtained by changing the + sign before 3a to — . This is equally true when the + sign is understood, cs in 5a — (3a — 6) = 5a - 3a + 6 = 2a + 6. In both cases the minus sign before the parenthesis belongs to the parenthesis, indicates subtraction, and disappears with the parenthesis. 47. Parenthesis within Parenthesis. Using the paren- thesis as a general name for the signs of aggregation, as brace, bracket, vinculum, it is evident that several parenthe- 34 ALGEBRA. ses may occur one within another in thie same algebraic expression. The best general method of removing several parentheses occurring thus, is as follows : . Remcyve the parentheses one at a time, beginning with the inner- most ; On removing a parenthesis preceded by a minus sign, change the sigiis of the terms inclosed by the parenthesis ; Collect the terms of the result. Ex. Simplify 5a; — 2/ — [4a; — 63/ + J — 3a: + ^ + 22 - (2x -=bx- 2/-[4a;-62/+ S-3a; + 2/ + 2z-2a: + zn = bx- 2/-[4a; — 62/ — 3a; + 2/ + 2z — 2a; + z] — bx— y— 4a; 4- % -\-Sx —y — 2z-{-2x — z = 6x + 42/- 3z. EXERCISE 5. Remove parentheses and collect similar terms : 1. 3a + (2a - 6). 7. x- [2a; + (x- 1)]. 2. 2a; - (a; - 1). 8. 5a; + (1 - [2 - 4a;]). 3. a; + (l-2a;). 9. 2- \1- (S- a) - a\. 4. 3a; - (1 + 3a;). 10. 2a; - [- a; - (a; - 1)]. 5. a;-(-a;-l). 11. 2y + ]- x- (2y -x)\. 6. x + 2y-(2x-y'). 12. a- J-a- (- a- l)i. 13. [a;' - (x'y - z") - z'] + (x'y - x'). 14. l-Sl-[l-(l--a;)-l]-lJ-a;. 15. a;-[-S-(-a;-l)-a;J-l]-l. 16. l-12 + [-3-(-4-5^^-7]S. 17. a—\a + lb — (ia + b + c — a + b + d)-c]\. 18. x-\2x^ + (Z^ - 3a; - [a; + x']) + [2a; - (x' + a;')]|. 19. X* - [4ar' - l^x' - (2a; + 2)] + 3a;] - [a;* + (3a;^ + 2a;* - 3a; -1)]. ^ USE OF PARENTHESIS. 35 20. X — x — y — } — a; — [— (a; — y) — (x + 2/) — «] — (« -yVs- 21. ~l-2x-l-(-2x-l)-2xl-l']-2x. 22. x — lx-{-(x-y) — \x + ^ — x)-2yl-\-y'] — y-\- x. 23. 25a: - [12 + S3x - 7 - (- 12a: - 6 + 15a;) - (3 + 2a;){] 4- 7 - (3x + 5) + (2a; - 3) + a; + 8. ^ 48. II. Insertion of Parenthesis. It is plain that the process of removing a parenthesis may he reversed ; that is, that terms may be inclosed in a parenthesis. Inverting the statements of Art. 45, Terms may be inclosed in a parenthesis preceded by the pltis sign, provided the signs of the terms remain unchanged; Terms may he inclosed in a parenthesis preceded by the minus sign, provided the signs of the terms be changed. Ex. a — b + c + d— e = a — b + (c + d — e), OT, = a — b — (^— c — d -\- e). EXERCISE 6. ♦ In each of the following insert a parenthesis, inclosing the last three terms ; each parenthesis to be preceded by a minus sign: a;^-4. 1. a^- - 3a:^ + 3a; -1. 4. l-a'-2 2. a- -6 + c + d . 6. x' + 4x-: 3. l + 2a- -a'- 1. 6. a'b'-2cd 7. 4a;*- - 9a;' + 12xy - -Ay\ 8. a'- -2a + l-9- -6x-x\ 9. x'- -4a;' + 4a;' + 4a;-4 — a;^ It is often useful to collect the coeflBlcients of a letter into a single coefficient. 36 ALGEBRA. Let it be required to collect the coefficients of x, 2/, and z in the expression, Sx — 4:y-\-5z — ax — by — cz — bx-{-ay + az. The complete coefficient of a; is (3 — a — 6) ; of 1/, (— 4 — 6 + a) or — (4 + 6 — a) ; of z, (5 — c + a). Hence, the same expression may be written, (3 - a - 6)a; - (4 + 6 - a)2/ + (5 - c + a)z. In like manner collect the coefficients of x, y, and z — 10. mx — ny -{- dz -^ 2x -{- nz — 4y. 11. x — y — 2z — ax -\-by — az—bx — ay+ cz. 12. —7x + 12y — 10z — 2ax-\-Sbz — cy + 2bx — Qdy. 13. abx — bey — cdz + acx — ady — acz — aby + adz. 14. by — Zacx — bcdz — Aabx — Scdy + 2cx — 4z — 5ax. Collect coefficients of x', x'^, and x — 15. Sx^ + X — 2x^ — ar' — 5 + a'^ — 2ax — C7? — cx^ — ex. 16. -x^-x- ax" + 3?-ax + bx' - ax' - Zbx - 2bx' -f Sa. 17. aV -ax -a- 6V - 26V + Sbx - aV - ex" + Sex - c. EXERCISE 7. SPECIAL REVIEW. Add- 1 . 2t* - Sar* - Zx^ + 2a; - 5, 2x^ - Sx* - 2x + 2x^ - 6, Zx^ + x* - Zx^ + 7 - a:, and 2 + Sar' + 2a;* - 4x - 2a;'. 2. 5a;^0 + Zx^yz — Zxy'^z — Zxyz"^, hxy'^z — Zx^yz — ^xyz, and locyz^ — xyz — x^yz + xy'^z. 3. 31/2 - 51/3 + 8, hV^ - 21/2 - 7, 31/3 - 41/2' - 2. 4. 2(a; + ^) - 3(a; + 0) + 2(y + 2;), 4(a; + 0) - 3(a; + 2/) - 5(y + 2;), and 4(a; + y) — (a; + 2) + 4(2/ + z). Subtract — 5. 2a6 - 36c + d from 1 - Zah - bo + X. 6. c - (? + a; - 10|2/ from 3a; - a + c. SPECIAL REVIEW, 27 7. 19a6 - c — 4x + Vy from \2ah - 3c + c' - Vy, 8. 3 - 2Vx + 5a; - a;^- a;* from 21/^ + a;» - 1. Fintl value of — 9. 3.-C -{X- 2y + 2{x + 1) (4 - a;) - Vbx + 1, when a; = 3. fd. 6a;2 - 3a:(a; -h |) + V^x' - 5a; + 2, when a: = 1. When a; = f . 1 < c>x- 2(4a;2 - 2a; - 5) + a;(a; + I) (5 - 2a;), when a; = 2. When x^ i. Simplify and collect — 12. 3a; - {- 2a; + [- 4a; - (a; - 2) - a;] - a;} - 1. 13. 9a; - {- 8a; - [7a; + (- 6a; + 1) - 5a;] - 4a;} - (3a; + 1) - 2a;. 14. X' - {y^ - a;2) - [(a;^ + z") - {{x" - z") + [y^ - z") - {x^ + z^)} - <1. Bracket coeflScients of like powers of x — 15. a;^ - ar'' + 2 - 3a;* - ax^ + ax^ - cx*^ - lax^ + 3ca;' - 2ca;* - 5a;'. IG. 1 - a; - a;2 - a;' + 2a - 2aa; + lax^ - 2ax^ - '6bx + Zbx'' + 36ar' + ex. 17. From the sum of a^ - lab + 36^ and 2a? - 66^ + 7a'^b\ take the sum of 4a262 _ 3^3 _,. ^a^ - 6^ and Sab - W + a^. 18. What must be added to a;^ — a; + 1 that the sum may be a?* ? That the sum may be 3a;? 15? 0? 1 9. What must be subtracted from 2x^ — 3a; + 1 that the remainder may be a;'? a;^ + 10? 7? a-a; + l? If 4 = 4a;' - Ix'y + 3a;^2 + ^3^ C = 3a;3 _ ^iy + ^y%^ -B = 4a:3 - x^y - xy"^ - ^y^ D = x^ - 2xy^ + y^. Find the values of — 20. A- B + C- D. 22. A -- (B + C) + D. 21. A-IB- {D+ C)l 23. B+ {A-[C- i)]}. 24. B- {-.A-i-B-(-C)-D]-C}-(C-B). CHAPTER IV. MULTIPLICATION. 49. Multiplication is the process of finding the result of taking one quantity as many times as there are units in another quantity. The Multiplicand is the quantity to be multiplied. The Multiplier is the quantity showing how many times the multiplicand is to be taken. The Product is the result of the multiplication. By def- inition of "factors" in Art. 11 it is seen that the multiplier and multiplicand are factors of the product. Thus, if X is the multiplicand and y the multiplier, the product is xy, and the factors of xy are x and y. MULTIPLICATION OP MONOMIALS. 50. Multiplication of Coefficients. To multiply 4a by 36, we evidently take the product of all the factors of the multi- plier and multiplicand, and thus get 4 X a X 3 X 6, or, rear- ranging factors as we are enabled to do by the Commutative Law, 4X3XaX6 = 12ah. Hence, in multiplying two monomials we multiply their coefficients together to produce the coefficient of the product. 51. Multiplication of Literal Factors or La^w of Expo- nents. To multiply d^ by a' : Since a' = a X a X a and a^^aXa .\a'Xa' = aXaXaXaXa = a\ ♦ MULTIPLICATION. 39 This may be expressed in the form a^ X a' = a' + ' = a^ or, in general, a"* X a" = a"* "•■ **, where m and n are positive whole numbers. Hence, in multiplying the literal factors of a monomial, we add the exponents of each letter that occurs in both multi- plier and multiplicand. Ex. A.a^hc' X U'h'x = 12a'h'&x. 52. La"W of Signs. The law of signs in multiplication follows directly from the general law of signs as stated in Art. 31. To proceed by way of illustration: (1) + $100 taken 5 times gives + $500, or, in general, + quantity taken a + number of times gives a vf result. (2) $100 of debts— that is, — $100, taken 5 times, gives - $500, or, in general, — quantity taken a + number of times, gives — quantity as a result. (3) $100 deducted 5 times, or $100 X — 5, gives as total amount of deduction — $500, or, in general, H- quantity taken a — number of times, gives — quantity as a result. (4) Deducting $100 of debts 5 times from a man's pos- sessions is the same as adding $500 to his assets ; that is, - $100 X - 5 = + $500, or, in general, — quantity taken a — number of times gives + quantity as a result. Thus, we see from (1) and (4) that either + X +, or — X — j gives -[-, 40 ALGEBRA. " and from (2) and (3), that either — X +, or + X — , gives — ; or, in brief, that in multiplication Like signs give plus, unlike signs give minus. 53. Multiplication of Monomials. Combining the results of Arts. 50, 51, 52, the process of multiplying one monomial by another may be expressed as follows: Multiply the coefficients together for a new coefficient; Annex the literal factors^ adding the exponents of each letter that occurs in both multiplier and multiplicand ; Determine the sign of the result by the rule that like signs give +, unlike signs give — . Ex. 1. Multiply da'bx^ by - 6a6y. The product is - SOa'iV^/l Ex. 2. Multiply Ba"" + ' by 2a" " \ Since n + S and n — 1, added, give 2n -f 2, the product is 10a'" + ' MULTIPLICATION OP A POLYNOMIAL BY A MONOMIAL. 64. Since, by the Distributive Law, Art. 33, a(b + c) = ab -\- ac, it follows that to multiply any polynomial by a monomial we proceed thus: Multiply each term of the multiplicand by the multiplier^ and set down the results as a new polynomial. Ex. Multiply 2a' - 5a'6 + 3a6'^ by - Zab\ 2a' - ba'b + 3a6^ -Ub' - 6a*6' + 15a'6' - 9a'6*, Product MULTIPLICATION. 41 EXERCISE 8. 1. 2. 3. 4. 5. 6. Multiply -5 -3a Zah 30a;y 4x -5x By 4 7. -2 - 8. -5 9. -1 -2x =_?? 10. 11. 12. Multiply 3ax -6x2/' 7ax - ba'h Q^ed -2x^2/2 By — Aax -"Ixf - -3a2/ -4ccf -Zed' -8x2/¥ 13. 14. 15. 16. 17. Multiply Wcdx^ -4x« 5a:Y 3a;Y-i ^2„^n-3 By - -zed 18. 3x'* 19. -7xY -xy+^ - x"2/" "" ' 20. 21. Multiply 2a + 3a; 3x- -22/ 40:^2/ ~ ^2/^ ' 7aa; — Aby By Ux -bxy 2a;2/ — 3a6a:2/ Multiply — 22. Sac' - 3m'n by ban. 26. Sx** + ' + Vx** by - 4a;. 23. m-m'- 3m' by - Im^n. 27. 2x'" - 5x"2/ by Zx^y"^. 24. 8x'2/ — bxy' — y^ by 3x2/. 28. ax"" — Iby"" by x^- 25. 2x" — 3x'* - ' by x\ 29. 5x' " ^ — 3x' - " by 4x^ 30. 2x'^ + '-3x" + ' — x" + ' — x"by 5x"-'. 31. x"2/ + 3x'* + y - Ax- + y by - 2x- - y. MULTIPLICATION OF A POLYNOMIAL BY A POLYNOMIAL. 65. Arranging the Terms of a Polynomial. The multi- plication of polynomials is greatly facilitated by arranging the terms in each polynomial according to the powers of some letter, the powers being taken either in the ascending or descending order. 42 ALGEBRA. In arranging the terms of a polynomial according to the ascending powers of a letter, the term containing the lowest power of the letter is placed first ; the term containing the next higher power of this letter is placed next, etc. Ex. dx^ -\- Z — X -\- x^ — 7x^j arranged according to the as- cending powers of x, becomes S-x + 5x''~7x^ + x\ In arranging the terms of an expression according to the descending powers of a letter, the term containing the high- est power of the letter is placed first ; the term containing the next higher power is placed next, etc. Ex. a* + 6* — 4a^6^ — 5a^6, arranged according to the descend' ing powers of a, becomes a'-5a'b-4:a'b' + b\ When arranging two polynomials for purposes of multipli- cation the same letter should be used, and the same order, either ascending or descending, in both polynomials. 66. Multiplication of Polynomials. The terms of each polynomial having been arranged, we proceed to multiply each term of the multiplicand by each term of the multi- plier, and take the sum of the results. The reason for this is made clear by taking two polynomials, a + b and c -f d, and forming their product by use of the Distributive Law: (a + b)(c + d)= a(c -\- d) -{■ b(c + d), by Distributive Law. = ac + ad -{- be -i- bd, by a second use of this law. We see that a similar result is obtained, no matter how many terms occur in each polynomial. Therefore, to multiply two polynomials Arrange the terms of the multiplier and multiplicand according to the ascending or descending powers of the same letter j MULTIPLICATION, 43 Multiply each term of the multiplicand by each term of the mul- tiplier ; Add the partial products thus formed, Ex. 1. Multiply 2x — Zy by Zx + by. The terms as given are arranged in order. The most convenient way of adding partial products is to set down similar terms in columns, thus : 2x - 32/ Sx + by Qx^- 9xy ) ^ . . + lQa^y-15.v' 3 products, Qx^ + xy— Iby"^, Product. Ex. 2. Multiply 2a; - a;' + 1 - 3a^ by 2a; + 3 — a;'. Arrange the terms in both polynomials according to the ascending powers of X. (Why is the ascending or'der chosen rather than the descending ?) 1 + 2X-SX''- 2^ S + 2x- x^ 3 + 6x- 9x^ - 3a;3 + 2x + 4a;2 - 6ar» - 2a;* - x'^ - 2x^ + 3a;* + a;^ 3 + 8a; - 6a;2 - Uxr^ + x^ + x^, Product. Let the student also multiply the two polynomials together with their terms in the order as first given, and hence discover the advantage of arranging the terms in order before multiplying. Ex. 3. Multiply Sab - 4b' + 2a' by - 2b'' + 3a' - 5ab. Arrange the terms in each polynomial according to the descending powers of a. 2a? + 3a6 - W 3a2- 5a6 - 26^ 6a* + 9a36 - l2aW - lQo?b - IbaW + 20a6' - 4a262 - 6a6' + 86* 6a* - a'b - Zla'b^ + l^ab^ H- 86*, ProdmL 44 ALGEBRA. Ex.4. Multiply a' + &' + c' + 2a6 — ac-6c by a + 6 + c. Arranging the terms according to powers of a, a^ + 2ab - ac + b^ - be + c^ a + b + c a^ + 2a^b - a^c + ab"^ - abc + ae^ 4 a^b + 2a62 - abc + b^ - b'^c + bc^ + d^e + lab c - a& + b'^c - 6c'' + c^ a^ + 3a26 + Safe^ + 6^ + c^, Product. Ex. 5. Multiply a;''" + a:''" + a;'" + 1 by a;™ 4- 1. a;3m + a:^"* + a;*" + 1 X"* + 1 a-4m _^ 2ar'"' 4- 2a;2'» + 2af' + 1, Prodmt. 67. Degree of a Term. Homogeneous Expressions. The degree of a term is determined by the number of literal fac- tors which the term contains ; hence, the degree of a term is also equal to the sum of the exponents of the literal factors. Ex. la^h(? is a term of the sixth degree^ since the sum of the exponents 3 + 1 + 2 = 6. A polynomial is said to be homogeneous when all its terms are of the same degree. Ex. ba^h — 6' + ah"^ is a homogeneous polynomial, since each of its terms is of the 3d degree. 58. Multiplication of Homogeneous Polynomials. If two monomials be multiplied together, the degree of the product must equal the sum of the degrees of the multiplier and multiplicand. For instance, in Ex. 3. Art. 56, the multiplicand and multi- plier are both homogeneous, and each is of the second degree, and their product is seen to be homogeneous and of the fourth degree. The fact that the product of two homogeneous expressions MULTIPLICATION. 45 must also be homogeneous affords a partial test of the accu- racy of the work. For if, for instance, in the above example, a term of the 5th degree, such as 5a^6^, had been obtained in the product, it would have been at once evident that a mis- take had been made in the work. The student should make use of this principle in testing the results obtained in exam- ples 8, 9, 10, 12, 14, 15, 20, 21, 22, 23, 24, 25, of the following exercise. EXERCISE 9. Multiply— 1. a;-4by 2a;+l. b. 1x^ -bf hy Ax^ -{-Zf-, 2. a; — 3 by 3a; + 2. <>. bxy + 6 by Qxy — 7. 8. 2a; + 5 by x - 7. 7. 4a' - h'c by 8aV + 2ah'c\ 4. 3x — Ay by 4x — Zy. 8. llx'y - Ixf by 3a;' + 2y\ 9. a' - a6 + 6' by a + h. 10. x^ -\-7?y + xy^ -{-'jfhyx — y. 11. 4a;' - 3a;' + 2a; - 1 by 2a; + 1. 12. 2a;' — Zxy + 2/ by 3a; - by. 13. a;' - 3a;' + 2a; - 1 by 2a;' + a; - 3. 14. Zx'^y — Axy^ — if by a;' — 2xy — y^. 15. x^ - 3a;'7/ f 3.r?y' - 2/' by 2;' - 2xy + if. 1(>. 4x' - 3a;' + 5a; - 2 by a;' +^3a; - 3. 17. .T* ~- 3a;' + 5 by a;' — a; — 4.^^ 18. ^ ~ Zxy -\- y^ by x^ — Sxy — y^, 19. a;'-7a;+2by a;'-7a;-2. 20. a'-ah + 6' by a' + a6 + h\ 21. 4a;' + dy' — Qxy by 4a;' + 9y' + ioxy. 2Z.X*- 7a;'2/' + 63;*,^ — y* by x' - 2a;?/' + f, 2S, x' - 6aa;' + 12a'a; - Sa' by - a;' - 4aa; - ia\ 2i, a' + 6' + a;' + 2ab - aa; - 6a; by a + 6 + a;. 25. a6 + cci + ac + 6c? hy ab -\- cd — ac — bd. 46 ALGEBRA. 26. a:'* + 2a;''-^ + 3:c~-'-2by X — 2. 27. a;** + ' - 3a:'* + 4a;" " ^ — Sa;*^ - ' by a:" + 2x'*-\ 28. x""-' — 2x^-' + 3a:"-' — 4a;"-^ 4- 5a:" by 2a:' + 3a: + 1. 59. Multiplication indicated by the Parenthesis. Sim- plifications. The parenthesis is useful in indicating multi- plications or combinations of multipHcations. Thus, (a — 6 + 2c)' means that a — b + 2c is to be multi- plied by itself. (a — 6 + 2cy means that a — b + 2c is to be taken as a factor three times and multiplied. To multiply out such a power is termed to expand the power. Again, (a — b) (a ■— 26) (a + b — c) means that the three factors, a — 6, a — 26, a + 6 — c, are all to be multiplied together. Also, (a — 2a:)' — (a + 2x) (a — 2a:) means that a + 2a: is to be multipHed by a — 2a:, and the product subtracted from the product of a — 2a: by itself. To simplify an expression in which multiplications are in- dicated in any of the above manners, means to perform the operations indicated and to collect terms. Ex. Simplify 3(a: - 2y) (x + 2y) - (a: - 2yy. 3(a: - 22/) (a: + 2i/) = 3a:' - 122/' (a: — 22/)' = a:' — 4xy + 4y^ Subtracting the second expression from the first, we obtain • 2a:' + 4xy - 16y\ EXERCISE 10. Simplify by removing parentheses and collecting terms : 1. a:' - a:(l + x). 4. (a: - 5)' - (a: + 5)'. 2. (x - 2) (2a: + 4). 5. 32: - 2a:(l + x + a:'). 3. 3a:(x - 2) - 2a:(a: - 3). 6. a: - (a: - 1) (a: + 2). MULTIPLICATION. 47 7. 3(a;-3)(x' + l) + 9. 8. (a-26)C3a + 46)~3a'. 9. (a + 26 - 3c) (a - 26 + 3c). ic. (a; - 2/ + z)'^ - xy'^ + ^3 + sr* . + Ixy"^ + "Ixyz - xH + xy"^ - 2xyz -V y^ + 2^ x^z — xz"^ — xyz xy"^ + xz' - xyz + y^ + !? xy-" + Tf + y'^z + xz' — xyz - y'^z •\- s? + xz' + yz' + ^ - xyz - y'^z - yz" - xyz - y^z - yz' Ex. 4. Divide a*" + ' - 4a'" + ' — 27a'" + ' + 42tt'" by oT + Sa" - 6a"^-^ ^m + 3 _ 4^m + 2 _ ^-j oT + ^ + 42a"' I g"* + Sa*"-^ - 6a**-' g"* + ^ + 3a*» + =^ - Bg*" + ^ a^ - Ta^* - Ta"* + 2 _ 2ia'» + i + 420"* - la"^ + 2 - 21a'» + 1 + 420** 64 ALGEBRA. Divide- EXERCISE 12. 1. 3a;^+7a: + 2by a; + 2. \ 2. Qx' + 7a; + 2 by 3x + 2. 3. Ux" + xy - 20y' by Sx + 4y, 4. 6x' - a;2/ - 122/' by 2:c - ^y. 6. 3a;'^ + a;-14by a:-2. 6. 6:c'' - Zlxy + 352/' by 2x - 7y. 7. 12a'' - llac - 36c' by 4a - 9c. 8. - 15x' + 59a; - 56 by 3a; - 7. 9. 44a;' -xy — By' by 11a; - Sy, ^. a' - W by a - 2Z>./ 13. 9a;' - 49 by 3a; + 7. 11. x'^ — fhyx — y. 14. 125 — 64ar* by 5 - 4x. 12. 27ar' + 8 by 3a; + 2. 15. 8aV + y^ by 2aa; + ^|| %. 2a;' - 9a;' + 11a; - 3 by 2a; - 3. 17. 353;^ + 47a;' + 13a; + 1 by 5a; + 1. 18. 6a' - 17a'a; + 14aa;^ - 3a;' by 2a - 3a;. 19. 42/* - 182/' + 222/' - 72/ + 5 by 22/ - 5. 20. c^ + c*a; + c'a;' + c'x^ 4- ex'' + a;^ by c + a;. 21.* 11a; - 8a;' + 5a;' - 20 + 2a;* by a; + 4. 22. 4a; + 6a;^ + 3a;'-lla;'-4by 3a;'-4. 23. -x?y- 11x2/' - 2a;'2/' + Qx'' - 6y' by 2a; - By, 24. 42/' + 6a;^ - ISx'y by Bx' - 2y. 25. X* — I62/* by a; — 2y. 27. x^ — y^ by x-^y. 26. x^ + Z2f by x + 2y. 28. 256a.-« - 2/' by 4a;' — y", 29. 9a; — 18a;' + 8a;* - 13a;' + 2 by 4a;' + a; — 2. 30. 10 - ar« - 27a;' + 12a;* -3a; by a; + 4a;' -2. * Arrange dividend in descending powers of x. DIVISION. 55 31. 22x' - 13r' 4- lOx' - 18x* + 5a; - 6 by x + Sx' - 2. 82. 14xy — 16a:'2/=' + Gx' + 2/' + 5^*?/ - Qxy'' by 3x^ + ^ - 2x2/. 33. Mh - 3a'6^ - a'h^ + 3a^ - 46^ by a' + 3a6 + 26^ 34. x' — / + 2^ — X2/2 — 2x^2 + 2yz^ hy x — y — z, 35. c^ + d^ + n^ — Scdn by c + d -f- n. 36. / - 2/ + 1 by 2/' — 22/ + 1. 37. 2x' + 1 — 3x* by 1 + 2x f x\ 38. 6xy - 62/V - 6xV - 13x^2^ - 5xy'z by Zxy + 2yz + 3x2. 39. x^ - 39x + 15 - 2x' by 3x' + 6x + x' + 15. 40. 4x'-9x* + 25-14x^-x'by 2x^-x-5 + 3a^. 41. 6x'" + ' - 13x"^" + 6x'» - ^ by 3x" + ^ - 2x^ 42. 12x*" + 13x'" — X" by 3x" + 1. 43. 4x** + ' + 5x" + ' — X" + ' - X" + x** - ^ by x' + 2x + 1. 44. 6x" + ' — 5x" — 6x"-' + 13x"-' — Gx'*-' by 2x'' - 3x + 2. 68. Division of Polynomials having Fractional Coeffi- cients. Polynomials having fractional coefficients are divided by the same methods that are used for those having integral coefficients. Ex. 1. Divide Ja' - ^a'b + ^ab'' - 16' by |a' - ^ab + ib\ ia' - la^b + ^ab' - ^b' \ W - \ab + \b\ Divisor. ia'-^a'b+ lab^ \a - lb, Qmtient. -ia26+ iab'-\b^ -Wb-\- ^ab'-^b^ Ex. 2. Divide 0.2x* - 0.01x^2/ - 0.44xy + xy' - 1.922/* by 0.5x' - 0.4x2/ 4- 1.22/1 0.2x* - O.Olx^y - 0.44xV + xy^ - 1-92^* | 0.5x^ - 0.4x.y + 1.2v' 0.2x* - 0.1 6x ^ ?/ + 0.48xV 0.4x2 + O.dxy - 1.6y^ + 0.15x^2/ - 0.92xV + xy^ •f O.lSx^.v - 0.12xV + OMxi/^ - 0.8xV + OMxy^ - 1.92?/* -^jxV_+ 0.64x.v^ ~ 1.92 i y* 66 ALGEBRA. EXERCISE 13. Multiply — rg.HxV+fx-i by|x-2. (^2H^'-2x + |byfx + |. J3i |x»-|x^ + |x-f byfa^ + f. ,4,; 1.2x^ + 1.5x + 6.4 by 2Ax - 3. ( 5. 3.6x3 _ 2.8x'^ + 7.2x - 0.32 by 1.5x + 0.25. , 6. 4.5x2 - 2.8x2/ + 5.62/' by 1.5x2 + 1.2xy + 2.42/^. , , Divide — 9. Jx* + iix'2/ + H^y + H-y' by |:«^ - \xy + 12/^ Q0>i 4.5x-^ - 7.1x^ - 0.4x + 0.24 by 2.5x + 0.5. ril^0.25x* - 1.8x^ + 3.24x2 - 12.25 by 0.5x2 _ ^<^^ _ 35 l5.'4.8x* + 0.18xV - Wf + 115.19x2/3-275.22/* by 1.6x^ - 2.5x2/ + 12.82^ Perform the operations indicated — 13. [1 -x3 + 22/(42/' + 3x)]^(l-x + 22/). 14. [1 - 2x2(x* - x^ + 1) - 3x*] - [1 - a:(2x2 + 1)]. 15. (1+ x« + x^O -^ [(1 - X + x') (1 - X* + x«)]. 16. [x^ + (4a6 - 62)x - (a - 26) {a? + S^')] ^ (x + 26 - a). 17. \x' + (3 - hy + (c - 36 - 2)x2 + (26 + 3c)x - 2c\ ^ (x2 + 3x - 2). 18. \x\y --z)^ yXz - x) + z\x -y)l-^ \xXy - 2) + y^z - x) + z\x-y\, 19. { (a^ - 3a6)x^ + (2a' + 4a6 + 362)x - (2a6 + 56') } -^ (ax - 6). CHAPTER VI. SIMPLE EQUATIONS. 69. An Equation is the statement of the equality of two dlgebraic expressions. An equation, therefore, consists of the sign of equality and an algebraic expression on each side of it. Ex. Sx-l=^2x + Z. 70. Members of an Equation. The algebraic expression to the left of the sign of equality is called the first memher of the equation ; the expression to the right of the sign of equal- ity is called the second memher. Thus, in the equation 3a; — 1 = 2x + 3, the first member is, 3a: — 1 ; the second member is, 2x + 3. The members of an equation are sometimes called sides of the equation. 71. Use of an Equation. An equation expresses the re- lation of at least one unknown quantity to certain given or known quantities, the object in the use of the equation being to determine the value of the unknown quantity in terms of the known. Thus, in the above example x represents the unknown quantity, and — 1, 2, 3 are known quantities. 72. A Numerical Equation is one in which all the known quantities are expressed as arithmetical numbers. Ex. 3a;- 1 = 2a; + 3. 57 58 ALGEBRA. 73. A Literal Equation is one in which at least some of the known quantities are represented by letters. Ex. ax + 26 = 3c — dx. a, h, c, and — d represent known quantities. 74. Degree of Equation having" One Unknown Quan- tity. If an equation contain but one unknown quantity, the degree of the equation (after the equation has been reduced to its simplest form) is determined by the exponent of the highest power of the unknown quantity in the equation. Exs. 2a; + 1 = 5x — 8 is an equation of the first degree. ax = b'^ + ex is of the first degree. 4x'-5x=20 " second" 'Sx'-x^ =6x + S " third " An equation of the first degree is also called a Simple Equation. 75. The Root of an equation is the number which, substi- tuted for the unknown quantity in the equation, satisfies the equation ; that is, reduces the two members of the equation to identical numbers. Ex. If in the equation, 3x — 1 =^ 2a; + 3, we substitute 4 in the place of x, we obtain 12 — 1 = 8 + 3 11 = 11, and the equation is satisfied. Hence, 4 is the root of the given equation. 76. The Solution of a simple equation is the process of finding the value of its root. In a simple equation the relation between the unknown quantity and certain known quantities is given, but in a more or less complex form, from which the value of the SIMPLE EQUATIONS. 59 unknown quantity in terms of the known is difficult to per- ceive. But since the algebraic symbols which represent these quantities may be rearranged and combined by the methods of Chapter II., as well as by the axioms of Art. 77, the com- plex relation first given can be reduced to a simple one, whence the value of x can be at once perceived. The processes of reducing an equation to its simplest form have been systematized, and take certain standard forms. 77. Axioms. Besides the principles given in Chapter II., which govern the use of algebraic symbols for quantity and enable us to use these symbols to the best advantage, there are other principles which are true of quantity in general, and therefore of algebraic quantity. These principles, -like those of Chapter II., are of great value in enabling us to use algebraic expressions to the best advantage. They are the so-called axioms : 1. Things equal to the same thing, or to equals, are equal to each other. 2. If equals he added to equals, the sums are equal. 3. If equals be subtracted from equals, the remainders are equaL 4. If equals be multiplied by equals, the products are equal. 5. If equals be divided by equals, the quotients are equal. 6. Like powers or like roots of equals are equal. 7. The whole is equal to the sum of its parts. 78. Application of Axioms to the Members of an Equa- tion. Since the two members of an equation are equal quan- tities, it follows from the axioms of Art. 77 that — The members of an equation may be increased, diminished, mul- tiplied, or divided by the same quantity, and the results will be Ex. 1. If 8a; = 24, dividing both members by 8 (Ax. 5), x = 3. 60 ALGEBRA. Ex. 2. If x^ + 3cc = x' H- 12, subtracting x^ from each member (Ax. 3), Zx - 12. Hence cc ^ 4 (Ax. 5). 79. Transposition of Terms. If we take the equation X -}-b = a, and we subtract b from each member (Ax. 3), x-\-b — b = a — b, or x = a — b. This result might have been obtained at once from cc + a = 6 in a mechanical way, by transferring -f- b from the left- hand member to the right-hand member, at the same time changing its sign to minus. So, if we take the equation, x — b=a, and add b to each member, we obtain x — b + b=--a + bj or x = a + b. This result also might have been obtained at once by trans- ferring — 6 to the opposite member, at the same time changing its sign. This process occurs so often in simplifying an equation that we abbreviate into the mechanical form and call it Transposition. Any term of an equation may be transposed from one side of an equation to the other, provided the sign of the term be changed. The main object of transposition of terms is to get all the terms containing the unknown quantity on the left-hand side of the equor tion, and all the known terms on the right-hand SIMPLE EQUATIONS. 61 I'fiis is the most important single step in simplifying an equation. This importance is indicated by the fact that the word algebra means transposition {al gebre, Arabic words meaning " the transposition "). 80. Clearing" of Parentheses. If an equation contain quantities in parentheses, it is necessary first of all to remove the parentheses by performing the operations indicated by them. 81. Changing Signs of all Terms. The signs of all the terms of an equation may be changed. This may be regarded either as the result of multiplying both members of the equa- tion by — 1, or as the result of transposing all the terms of the equation. Ex. For —ax = — b-\-c, write ax = b — c. 82. General Process of Solving a Simple Equation. This may now be stated as follows : Clear the equation of parentheses by performing the operations indicated by them; Transpose the unknown terms to the left-hand, side of the equa- tion^ the known terms to the right-hand side; Collect terms; Divide both members by the coefficient of the unknown quantity. Ex. 1. Solve the equation, 3x — 7 = 14 — Ax. Transposing the terms — 7 and — Ax^ 3a; + 4a; = 14 + 7 Collecting terms, Ix = 21 Dividing by 7, ^c = 3, Root. Verification. Substituting 3 for a; in 3x — 7 = 14 — Ax 3X3-7 = 14-4X3 9-7 = 14-12 .. ,- » -, - . . 2=2. 62 ALGEBRA. Ex. 2. Solve x(x -- 2) = x(ix + 4) - S(x - 3). (1) Removing parentheses, x^ — 2x = x'^ + 4a; — 3a; + 9 Transposing terms, x^ - x^ —2x — 4a; + 3x = 9 Collecting terms, — 3a; = 9 . . (2) Dividing by — 3, x= — S, Boot. Verification. Substituting — 3 for a; in equation (1), -3(-3-2) = -3(-3 + 4) -3(-3-3) -3(-6) = -3(l)-3(-6) 15--3 + 18 15 = 15. The value of the processes employed in the solution of an equation — viz. transposition, etc. — is realized when we com- pare the ease with which the value of x is perceived from equation (2), with the difficulty in assigning immediately in equation (1) a value for x which will satisfy that equation. EXERCISE 14. Solve the following equations : 1. 3a;=15. 9. — 7a; = 0. 2. 2a; = - 6. 10. 4a; - 5 = 1 + 3a;. 3. -13a; = 26. 11. 2a; -7 = 8 + 5a;. 4. _6a;=-12. 12. 2a; - 3(a;- 3) + 2 = 0. 5. 3a; = - 5. 13. 7(2 - 3a;) = 2(7 - 8a;). 6. — 2a; = ll. 14. x' - a;(a; + 5) = a; + 12. 7. - 5a; = _ 13. 15. 3 _ 2(Sx + 2) = 7. 8. 4a;= — 1. 16. 2a; - (a; + 5) = 4a;. 17. (x - 1) (a; + 3) = (x - 4) (a; + 2). 18. 3a; - 4a; + 10 + 5a; = 0. 19. a;(2a; + 1) - 2a;(a; + 3) = 7. SIMPLE EQUATIONS, , 63 20. 3(a: - 1) (a; + 1) = x(Sx + 4). 21. 4(x-sy = {2x + iy. 22. 8(2;-3)-(6-2a:)=2(a: + 2)-5(5-a;). 23. 5x - (3a; - 7) - {4 - 2x- (6a: -S)]= 10. 24. a; + 2-[a:-8-2S8-3(5-a;)-a;n=0. 25. 2(a; + 1) (2a; - 1) + 2|a; - (a; + 3) (2a; - 1)5 = - 32. 26. 2a;(a; -5)-\x' 4- (3a; - 2) (1 - a;) \ - (2a; - 4)1 27. (x + 1)^ - 2\(x - 1)^ ~ 3(a; + 2)'^S -= 3(a; + 4) (2a; - 4) ^(a;^-5). 28. 8a;'' + 13a; -2\x'- 3[(a; - 1) (3 + a;) - 2(a; + 2)^]S = 3. SOLUTION OF PROBLEMS. 83. The General Method of stating and solving will first be illustrated by an example similar to that given in Art. 1. James and John together have $18. If James has twice as many dollars as John, how many dollars has each boy? By solution of the problem is meant, of course, finding the value of the unknown quantity or quantities of the problem. The first thing to be done, therefore, is to de- termine what are the unknown quantities or numbers. In the given problem there are evidently two unknown quantities to be determined — first, the number of dollars which James has; second, the number of dollars which John has. The method of procedure is then to state the relation of the unknown quantities to the known quantities in the form of an equation, which is afterward solved. As a rule, when there are two unknown quantities in a problem, it is more convenient to represent the smaller of them by x. 64 ALGEBRA. Let X = number of dollars John has. Therefore, 2x= " James has. Hence, x-{-2x= " both have. But 18 also = " a Hence, by use of Axiom 1, Art. 77, .x + 2x = lS, 3:^-18, a: =^ 6, number of dollars John has. 2x-12, " James has. It is to be noticed particularly that we let x equal a definite number^ not a vague quantity. We do not let a: = the money which John has, nor X = what John has, but let X = number of dollars which John has. In solving problems the student will find it necessary to study each problem carefully by itself, as no rule or method can be found which will cover all cases. The following general directions will, however, be found of service: By study of the problem determine what the unknown quantity or quantities are whose values are to be obtained; Let X equal one of these expressed as a number; State all the other unknown quantities which are either to be determined or to be utilized in the process of the solution, in terms of x; Obtain an equation by the use of a principle, such as the whole is equal to tJce sum of its parts, or things equal to the same things are equal to ea,ch other ; Solve the equation, and find the value of each of the unknown quantities. SIMPLE EQUATIONS. 65 Ex. 1. James and John together have $24, and James has 8 dollars more than John ; how many dollars has each ? Let X = number of dollars which John has. Since James has 8 dollars more than John, a; + 8 ~ number of dollars which James has. .•.x + (a; + 8)= " " " both have. But 24- " " " " " Hence, by Axiom 1, a; -f a; + 8 = 24 2a: = 16 a; = 8, number of dollars which John has. a; + 8 -=16, " " " * James has. Ex. 2. The sum of two numbers is 36, and three times the greater number exceeds four times the less number by 10. Find the numbers. Let a: = the less number. If the sum of the numbers is 36 (as the line AB), and one part is x {AC), then the other part will be 36 - a; {CB). . * . 36 — a: = the greater number. 36 I -B 36 . * . 3(36 — x) = three times the greater number. 4a; = four " " less '' But (three times the greater number) — (four times the less) = 10. . • . 3(36 - a:) - 4a; = 10. Hence, 108 - 3a; - 4a; = 10 - 7a; = - 98 a; = 14, the less number. 36 — a; = 22, the greater number. Let the student verify these results. Problems may frequently be solved in more than one way. Thus, Ex. 1 above may be stated and solved as follows : 5 66 ALGEBRA. Let X = number of dollars which John has. Then 24 — a; = '' '' '' James has. Also, (number of James's dollars) — (number of John's dollars) = 8 ; that is, (24 - a;) - a; = 8 24 - a; - a: = 8 - 2a; = - 16 a; = 8, number of John's dollars. 24 - a; = 16, *' James's " EXERCISE 15. ORAL. 1. A has X marbles, and B has twice as many ; how many has B ? How many have both ? 2. There are 100 pupils in a school, of which x are boys ; how many are girls ? 3. If I have x dollars, and you have three dollars more than twice as many, how many have you? How many have we together? 4. Two boys together solved a examples : the one did x ; how many did the other solve? 5. The difierence between two numbers is 15, and the less is x ; what is the greater? What is their sum? 6. If n is a whole number, what is the next larger number? The next less? 7. Write three consecutive numbers, the least being x. Write them if the greatest is y. 8. John has x dollars, and James has seven dollars less than three times as many ; how many has James? 9. If I am x years old now, how old was I ten years ago? a years ago? How old will I be in c years ? 10. A man bought a horse for x dollars, and sold it so as to gain a dol- lars ; what did he receive for it? 11. A man sold a horse for $200, and lost x dollars ; what did the horse cost? 12. If a yard of cloth cost m dollars, what will x yards cost? 13. If a boy ride a miles an hour, how far will he ride in c hours? 14. A bicyclist rides x yards in y seconds ; how far will he ride in one second? In n seconds? SIMPLE EQUATIONS. 67 15. How many hours will it require to walk x miles at a miles an hour ? 16. A man has a dollars, and b quarters ; how many cents has he? 17. How many dimes in x dollars and y halves ? 18. I have x dollars in my purse and y dimes in my pocket ; if I give away fifty cents, how much have I remaining? 19. By how much does oO exceed x ? 20. Express the sum of the squares of two consecutive even numbers if the larger is x. 21. A gentleman is out x hours, of which he rides a hours at the rate of eight miles an hour, and walks the rest of his time at the rate of three miles an hour ; how far did he ride ? How far did he walk ? EXERCISE 16. 1. A boy has three times as many marbles as his brother, and together they have 48; how many has each? 2. A and*B pay $100 taxes; if A pays $22 more than B, what does each pay? 3. John solved a certain number of examples, and William did 12 less than twice as many ; both solved 96. How many did each solve ? 4. Three boys earned together $98; if the second earned $11 more than the first, and the third $28 less than the other two together, how many dollars did each earn ? 5. A man walked 15 miles, rode a certain distance in a coach, and then took a boat for twice as far as he had pre- viously traveled; altogether, he went 120 miles. How far did he ride by boat? 6. Find three consecutive numbers whose sum is 84. 7. The sum of two numbers is 92. and the larger is 3 less than four times the less; find the numbers. -8. The sum of three numbers is 50 : the first is twice the second, and the third is 16 less than three times the second ; find the numbers. 9. A farmer paid $94 for a horse and cow; what did each cost, if the horse cost $13 more than twice as much as the cow? G8 ALGEBRA. 10. Distribute $485 among A, B, and C so that B and C each get twice as much as A. 11. Divide the number 35 into two parts, such that three times the smaller shall be equal to twice the larger. 12. Find five consecutive numbers whose sum shall be 3 less than six times the least. 13. The difference between two numbers is 6, and if 3 be added to the larger the sum will be double the less; find the numbers. 14. Three men rent a store for $500: the first is to pay- twice as much as the second, and the second $60 more than the third; how much does each pay? 15. Divide $4500 among two sons and a daughter so that each son gets $150 less than twice the daughter's share. 16. A father is four times as old as his son, and the differ- ence of their ages is 24 years less than the sum ; how old is each? 17. A man is twice as old as his daughter, who is 5 years younger than her brother, and the combined ages of all three are 109 years ; what is the age of each ? 18. A father is now twice as old as his son ; 21 years ago he was three times as old. How old are they now ? 19. Find two numbers whose difference is 14, such that the greater exceeds twice the less by 3. 20. Find three consecutive odd numbers whose sum is 63. 21. The greater of two numbers is 5 more than the less, and five times the less exceedr. three times the greater by 3; find the numbers. 22. A man had five sons, to whom he gave $56, giving to each $5 less than twice the amount his next younger brother received ; what did each receive ? 23. It is required to divide 75 into two such parts that three times the greater exceeds seven times the less by 15. 24. The difference of the squares of two consecutive num- bers is 43; find the numbers. SIMPLE EQUATIONS. 69 25. The difference of the squares of two consecutive even numbers is 60; find the numbers. 26. The joint ages of father and son are 64 years : if the age of the son were doubled, he would still be four years younger than his father; find the age of each. 27. Two bicyclists, A and B, start respectively from New York and Philadelphia, 90 miles apart, and ride toward each other ; A rides 8 and B, 12 miles per hour. How long and how far will A ride before meeting B? 28. A man walks to the top of a mountain at the rate of 2 miles an hour, and back down at 4 miles an hour; if he is out 6 hours, how far is it to the top of the moun- tain? 29. How far into the country will a man go who rides out at the rate of 9 miles an hour, walks back at 6 miles an hour, and is gone 10 hours ? 30. A boy was engaged to work 50 days at 75 cents each day for the days he worked, and to forfeit 25 cents every day he was idle. On settlement he received $25.50 ; how many days did he work? 31. Five years ago A was twice as old as B, but 10 years ago he was three times as old as B; how old is each now? 32. A man is 30 years older than his son, and 10 years ago he was three times as old ; what is the age of each ? 33. Twenty yards of silk and 30 yards of cloth cost $99, and the silk cost three times as much per yard as the cloth ; how much did each cost per yard ? 34. A merchant paid a bill of $72 with dollar, two-dollar, and five-dollar bills, paying the same number of each ; how many of each did he use ? 35. How can $2.25 be paid in five- and ten-cent pieces so as to use the same number of each ? 36. How can $5.95 be paid in dimes and quarters, using the same number of each? 70 ALGEBRA. 87. A purse contains $10.50 in dollar bills and quarters, but there are twice as many quarters as bills ; how many are there of each ? 38. Twenty coins, dimes and half dollars, make together $8.80; how many are there of each? 39. A gentleman gave $525 to his son and daughter, so that for every dime the daughter received the son got a quarter ; how much did each receive? 40. A person was desirous of giving 80 cents apiece to some beggars, but found he had not money enough by 80 cents ; he gave them, therefore, 20 cents each, and had 30 cents remain- ing. Required the number of beggars. 41. A sum of money is divided among 8 persons, A, B, and C, so that A and B have $79, B and C have $70, and A and C $75 ; how much has each ? 42. A woman sold 12 new baskets for $3. For a part she got 20 cents each, and for the rest 32 cents each ; how many of each grade did she sell? 43. A certain flag-pole is 69 feet long, and has 12 feet of its length in water: the part in air is 3 feet more than five times the length of the part in earth; what is the length of the part above water? 44. B had $5 more than three times as much as A, but he gave A $9, and now he has a dollar less than twice A's sum ; how much did each have at first? 45. There is a fish whose tail weighs 9 pounds ; his head weighs as much as his tail and half his body ; and his body weighs as much as his head and his tail. What is the weight of the whole fish ? 46. A set out from a town, P, to walk to Q, 45 miles distant, an hour before B started from Q toward P. A walked at the rate of 4 miles an hour, but rested 2 hours on thfi way; B walked at the rate of 8 miles an hour. How many miles did each travel before they met? BEVIEW. 71 EXERCISE 17. REVIEW. Add— l.^x^ + ^xp - \y\ 2^2 _ 2,2 _ 2^^^ 2a;2 - xy - \y\ 2. \^ - x" ^ \x, Ix" - f a: 4- f , \x - \ - \7?, \7? + \x^. 3. l.Sx-' - Z.ixy + 0.6^2^ 3.2a;2 - S.l?/^ + 1.5^^^ j 7^.^ - 2/' - 0.7a:'. 4. 1.6 + 1.1x3 _ 2.2x2, hXx" - 0.7a; - 2.3a:3^ 1 + 0.2a;^ - 2.9a;' + 1.6a;. l.la;'^ + 2a;2/' - 1.5^/^. Subtract— h. ^- |a;2 + f a; - h\ from |x' -|a;2 + |a; + f. 6. 2.7a;3 - 0.4a;2^ - 1.3a;y + y. from 3a;3 - 1.: La;^ Multiply— 7. \x^- - f a; + 4 by %x + 2. 8. 1.5a; - 0.4^ by 2.4a; + 1.5?/. 9. fx' — ax -V \a^ by fx' + |aa; + \o^. 10. fa;' - fa; + 1 by fa;' + fa; - 1. 11. 3.2a' - 2.3a6 + 5.26' by 1.5a + 2.56. 12. 0.4a;' - \.%xy - 2.8^/' by 0.5a; - \.hy. 13. 4.5a;3^ - 1.2a;'?/' - hAxy^ by 0.4a;'^ - ^.hxy^, 14. 3.2a;' - 4.5a;^ + 1.8^' by 1.5a; - 3.5^. Divide — 15. 6a;3 _ 4^3 _ 2a;2 - I6a;'^ + 14a;^' + ^xy - 2^' by 3a; - 2^ - 1. 16. x'^ - 5a;V + 13a;*^ - 2a;V - l^a;'?/^ + 9?/' by a;' + 2a;y - Zy\ 17. 2 — a; by 1 + a; to five terms in the quotient. 18. 1 — a; + a;' by 1 + a; + a;' to five terms. 19. a;^ — 15 by a;' + a; — 1 to five terms. 20. 1 by 1 — 2a; — 3a;' to five terms. 21. -V-a^ - \x^y + 2a;?/' - -i^y^ by \x - \y. 22. -i^x'^ - Ix^y + if^V + \^y' by \x + \y. 23. 36a;' + i^' + i - 4a;?y - 6a; + \y by 6a; - I3/ - J. 24. 2.4a;3 _ o.l2a;'^ + 4.32^/=* by 1.5a; + 1.8y. 72 ALGEBRA. 25. 8.4a:* - l.ea:^ - lO.Sa:^ + 10.2a; - 3.9 by 2Ax'' + 1.6a; - 2.6. 26. 5a:* - 6.S5x^^ + SMxy^ - OM^/* by 2.5a: - 0.3^. Simplify — 27. 6z -h [43 - {8a: - {2z + 4a:) - 22a:} - 7a:] - [7a: + {140 - {4z - 5a:)}]. 28. a2(6 - c) - 62^a - c) + c2(a - 6) - (a - 6) (a - c) (6 - c). 29. 2a: - [ - 3(a: - ^) + {(a: - 2?/) - 2(.a: 4 3^/) -a:} - 2(^ - 2a:)]. 30. 6{a - 2[6 - 3(c + d)]} - 4{a - 3[5 - 4(c - d)]}. 31. l-2{-[- (a: -2/)] -a:} + 2{ - 2[- 1 - (a: -?/)]- 1}. 32. Sa'ia - b) - (36 - 2a) [a(26 - 3a) - {a + by] + 6(6 + a)^. Solve and verify — 33. (2a: -f- 1) (a; - 3) + 7 = a: - 2(a: - 4) (2 - x). 34. 7a: - 2(a; - 1) (2 - a:) - 17 = a:(3a: + 7) - (a: + 1)^. 35. 2a:2 - 3a: - 2(2a: + 1)^ + (2a; - 3) (3a: + 2) = 8. 36. 3a:2 „ ^5^ _ [4 _ (^^ _ i) ^2x - 3) - 7a:] + {x - 3)^ = 0. 37. 5a: + 1 - 2{2a: - 3[a: - (a: + 1) (a: + 3)] - 3(a: + 2f} = 0. 38. What is the dividend when the quotient is x^ + 2x^ + 7a: + 20, the remainder 62a: + 59, and the divisor a:^ — 2a: — 3 ? 39. What is the divisor if the quotient is x^ + 3x, the dividend a:^ — 8, and the remainder 9a: — 8 ? 40. If a: .= - f and 2/ = - f , find the value of (3a: - 2yy {9x'' + 4^') - Q[y - x) l/6a:^(a: + 2y' + ^). CHAPTER VII. CASES OF ABBREVIATED MULTIPLICATION AND DIVISION. ABBREVIATED MULTIPLICATION. 84. Value of Abbreviated Multiplication. In certain cases of multiplication, by observing the character of the expressions to be multiplied, it is possible to write out the product at once, without the labor of the actual multiplica- tion. Almost all the multiplication of binomials, and that of many trinomials, will fall under these cases^ and by the use of the abbreviated methods at least three-fourths of the labor of multiplication in them will be saved. The student should therefore master them as thoroughly as he has done the multiplication table in arithmetic. 85. I. Square of the Sum of Two Quantities. Let a + 6 be the sum of any two algebraic quantities. By actual multiplication, a-{-b u^ + ah - {-ah +6' a^ + 2ah + 6^*, Product^ Or, in brief, (a + by = a'-{- 2ab + 6^ which, stated in general language, is the rule— The square of the sum of two quantities equals the square of the first, plus twice the product of the first by the second, plus the square of the second. 74 ALGEBRA. Ex. (2a; + Syf = 4x' + 12xy + dy\ Product. Since the square of 2x is 4x'^, twice the product of 2x and Sy is 12a;y, and the square of Sy is %^ 86. II. Square of the Difference of Two Quantities. By actual multiplication, a — b a — b d — ab -ab +y a' - 2ab + h\ Product Or, in brief, (a - by =-a^- 2ab + b\ wliich, stated in general language, is the rule — (The square of the difference of two quantities equals the square of e first, minus twice the product of the first by the second, plus the square of the second.^^ Ex. (2x-Zy)'^^x^-l2xy^-'dy\ Product. 87. III. Product of the Sum and Difference of Two Quantities. By actual multiplication, a + b a — b ' a^ + ab -ab-b' a^ — b'\ Product. Or, in brief, (a -^b) (a — b) = a^ — b"^, which, stated in general language, is the rule — The product of the sum and difference of two quantiiies is the difference of their squares. Ex. (2x + Sy) (2a; - Sy) = Ax' - %', Product. ABBREVIATED MULTIPLICATION. 76 EXERCISE 18. Write by inspection the values of — 1. (n + 2/)'. 11. (x^z)(x~z\ 2. (c-xy. 12. (2/ -3) (2/ + 3). S.(2x~yy. 13. (3x-2/)(3x + 2/). 4. (Sx-2yy. 14. (7x + 4y)(7x-4y-). 6. (5a; + 1)'. 15. (x' -2) (x' + 2), 6. (x' + iy. 16. {aT^ - h'y) (ax' + b'y), 7. (x - 2/'')'. 17. (1 - lla^O (1 + liar'). 8. (l — 7yy. 18. (2x" + 52/™) (2x'* - 62/"*). ^ (3a:* + 5r')^ 19. (5a;" - 32/"z"*)l 10. (6xV-ll2/'zT 20j:4a;yz'" + 92/"*)'^ 88. Special Case under III. In applying any of the above abbreviated methods of multiplication we may have parentheses containing two' or more terms used as a single quantity. This is of especial ^importance in obtaining the product of a sum and difference. Ex. 1. Multiply x + (ia'h b) by a; - (a + 6). We have [a; + (a + 6)] [x - (a + 6)] = ar^ - (a + by, by III. = a;'-(a' + 2a6 + 60,by I. = a;' - a' - 2a6 — 6^ Product It is frequently necessary to re-group the terms of trino- mials in order that the multiplication may be performed by the above method. Ex. 2. Multiply x + y — zhyx — y-\-z. (x + y - z) (x -y + z) =--[x + (y - z)'][x - (y - z)] = x^-(2/-z)^byIIL = x'.-(y'-2yz + z'),hylL = x^ — 2/* + 22/z — z^j Produd, 76 ALGEBRA, EXERCISE 19. Write by inspection the product of — 1. [(a + 6) + 3][(a + />)-3]. 2. [2x-l-y-][2x-l + yl 8. [4-(a:+l)][4 + (^+l)]. 4. [a +(6 -2)] [a -(6 -2)]. 5. '(2x + Sy + 1) (2x -Sy- 1). ■ V 1^, (x^ + Sx-2) (x' + 3x + 2). ^.(4 --x-y)(4 + xi-y). h. (Sx' - 2:c + 1) (Sx' + 2x - 1). 9. (x^ - xy + 7/^) (x'^ +xy + y^). 10. (a' + a + l)(a'-a + l). 11. (2x'-Sx-5)(2x' + Zx-5). 12. (2a;'^ + 5xy - f) (2x^ - bxy - y^). 13. (aa:' -bx-{- 2c) (ax'^ + 6a; - 2c). 14. (x' ~\-xy — 2/') (a;2 -xy — 2/'). 15. (6ft^ - 3a - 2) (6a' + 3a - 2). 16. [(a + 6) - (c - 1)] [(a + 6) + (c - 1)]. 17. [(a:' -f 2/^) + (xy + 1)] [(^.^ + y') - (xy + 1)} ' 18. (x-y + z-l)(x + y^z + l). 19. (x'-2a;'-a;-2)(a;^H-2x' + a;-2). / Simplify — 20. (3a - 1)' + (2 - 3a) (2 + 3a). 21. (2x - 7y) (2x + 7y) - 4(x - 2yy + 132/(52/ - x), 22. (3a;' + 5)' + a;'(10 - 3a;) (10 + 3a;) - (5 + 13a;7. 23. (a-c + 1) (a + c-l)-(a-l)' + 2(c-l)l 24. (x + y — xy) (x — y — xy) + x'y -(x- y') (x + 2/'). 25. (x' - x-^) (x' + x-^) i- (^-t xr\- x') (x-x" i-S), ABBREVIATED MULTIPLICATION. 77 89. IV. Square of any Polynomial. By actual multiplication, a + b + c a + b -\- c + ab +b'-\-bc + ac -i-bc +c^ a'-h 2ab + 2ac + 6' + 2bc + c\ or, in brief, (a + 6 + c)^ = a'* + 6' + c' + 2ab + 2ac + 26c. In like manner we obtain ia-hb-{-c + dy = a' + b' + c' + (r + 2ab + 2ac + 2ad + 2bc + 2bd + 2cd. It is seen that each of these results consists (1) of the square of each term of the polynomial, and (2) of other terms formed by taking twice the product of each term into each term which follows it. It is also perceived that this method of forming the square of a polynomial will hold good no matter how many terms there are in the polyno- mial. For if we examine the above process of the multipli- cation of a + 6 + c by itself, we see that not only is each term multiplied by itself, but also a of the multiplicand is multi- plied by b of the multiplier ; and, vice versa, b of the multi- plicand is multiplied by a of the multiplier, and so for any other pair of terms, and that this list exhausts the partial products. Hence, in general, The square of any polynomial equals the sum of the squares of all the terms, together with twice the product of each term into each term which follows it. Ex. {a — 2b-{-c—3xy=^a^-\-ih^+c^-\-9x^—^ab-]-2ac — 6ax — 46c + 126a; — 6cx. 78, ALGEBRA. EXERCISE 20. Write by inspection the values of — 1. (2a; + 2/ + 1)'. 8. (2a} + 5a - 3)^ 2. {x-2y + 2z)\ 9. (x-y + z-l)\ 3. (3x - 22/ - 5)^ 10. (2a; + 3^/ - 4z - 5)^ JL (2a-6 + 3c)\ ll (3a;» - 4x^ + a: - 2)^ ^^a;-22/-3zy. 12. {2a-h')(2a + h'), 6. (4rr + 32/-l)^ 13. (2a'^>-56c7. 7. {x'-x + l)\ 14. (a-6 + 7)(a + 6-7) 15. (2a:' - 7a: + 11) (2a:' + 7a: + 11). 16. (5a:'-6x-3)l 17. (a:' + 3a: -5) (a:' -3a: -5). 18. (3a' - 5a& + h') (3a' - 5a6 - 6'). 19. (7x' - 5a: + 4) (7x' + 5a: - 4). 20. [(a' + 26') + (2a6 - 1)] [(a' + 26') - (2a6 - 1)]. 90. V. Product of Two Binomials of the Form x-\- a, Q, Product. Ex. 2. Multiply (x - 6a) (x — 5a). (- 6a) + (- 5a) = - 11a, (- 6a) X (- 5a) = + 30a' . • . (a; - 6a) (a; - 5a) =x''- llax + 30a^ In Case V., also, a parenthesis containing two or more terms may be used instead of a single quantity. In all cases the student should thoroughly acquire the ability to use a paren- thesis as a single quantity. By so doing many of the diffi- culties of algebra are at once overcome. Ex. 3. Multiply x + y-^Qhy x + y — 2. (.x + y + Q)h) (a - 3c). 20. (a — l)(a + ft). 23. (re -f 2?/ — 7) (:c + 2^/ + 2). 21. (a - 2x) (a + y). 24. (x^/'^' - 10) (xyV + a). 25. (a^6c — x^yz) (cfbc + bx^yz). 26. (a6 + ^c — ac) (a6 — 6c + ac). 27. [(a + 6) -3] [(a + 6) + 4]. 28. (m — 5 + 4n) (m — 5 — 8n). 91. VI. Product of Two Binomials whose Correspond- ing Terms are Similar. By actual multiplication, 2a -36 4a + 56 8a^ - 12a6 + 10 a6 - 156' W- idb-iw It is seen that the middle term of this product may be obtained directly from the two binomials by taking the algebraic sum of the cross products of their terms. Thus, (+ 2a) (+ 56) + (- 36) (+ 4a) = 10a6 - 12a6 = - 2ab. Hence, in general, The product of any two binomials of the given form consists of three terms: The first term is the product of the first terms of the binomials ; The third term is the product of the second terms of the binomials; The middle term is formed by taking the algebraic sum of the cross products of the terms of the binomials. Ex. Multiply (lOx + 7y) (Sx - lly). For the middle term, (lOx) (- lly) + (72/) (Sx) = - llOxy + 56xy = - 5ixy. . • . (10a; + 7y) (Sx - lly) = 80a;^ - b4.xy - 77y', Product. . ABBREVIATED DIVISION. 81 EXERCISE 22. Write by inspection the product of — 1. (2x + 3) (x + 2). __10. {x' - 3x2/ + Ay^.i^ % (2x + b){x- 2). n, (5x' - 7y) (5x' + ly). 3. (3x - 1) (a; - 2). 12. (lUy - 3) O^Y + 2). 4. (bx ^l)(x- 1). 13. (2a;' + ay) (Zt? + 2ay). 6. (a; + 3^) (32; - 8?/). 14. (a' - a + 7) (a' + a - 7). 6. (2a: - 72/) (3a; + lOi/).^ 15. (x' - 3x + 1) {x' - 3x - 1). 7. (3a' - 56) (3a' - 56). 16. (5x' - 9x - 8)1 8. (cr' — 22/2) (5a;' + 3?/2). 17. (Zx' -{- bxy — ^fy . 9. (3x - 2/')'. 18. (4a' - 36) (5a' + 46). 19. [(8a;=' -{-x)- (4a;' - 1)] [(8a;^ + a;) + (4x' - 1)]. ABBREVIATED DIVISION. 92. Value of Abbreviated Division. In certain cases much of the labor of division may be saved by performing the division operation in a typical case, noting the relation between the quantities divided and the quotient, and formu- lating this relation into a mechanical rule. ' 93. I. Division of the Difference of Two Squares. Either by actual division, or by inverting the relation of irt. 87, we obtain = a — bj and — = a-\-o, a+b a—b Hence, in general language, The difference of the squares of two quantities is divisible by the r,dm of the quantities, and also by the difference of the quantities, the quotients in the respective cases being the difference of the quan^ tities and the sxim of the quantities. 82 ALGEBRA. Ex. 1. -^ ^^ = 2x + 32/, Quotient ^ 2x — 3?/ Ex. 2. ^^ h \ = ^ - (a + 6), Qwo^ten^. a; H- (a + 6) EXERCISE 23. Write by inspection the quotient of — g (a^ + iy-gy (x + l) + a ^ a'-{h-2cf a-(b-2c) 5x-6y' ' 2x'-\-(iy' + l) 16a;^-4V ^^ (a-by-(c-iy ' 4x^ + 72/' ' * (a-6) + (c-l) \^ 5x^-2/* ' * l + (a + 6-c) 94. II. and III. Division of Sum or Difference of Two Cubes. By actual division we can obtain, ^3 0-7)3 „3 _ ^3 a-\-h a — b Hence, in general language, The sum of the cubes of two quantities is divisible by the sum of the quantities, and the quotient is the square of the first quan- tity, minus the product of the two quantities, plus the square of the second quantity ; also, . The difference of the cubes of two quantities is divisible by the ABBREVIATED DIVISION. 83 difference of the quantities, and the quotient is the square of the first quantity, plus the product of the two quantities, plus the square of the second. Ex. 1. §^1127/ ^ (2xy-(Sy)' ' ' 2x — 'iy 2x~By = (2xy + (2x)(Zy) + (&yy = 4x^ + 6xy + 9/, Quotient. = a' - 2ab + b'-3a + Sb + 9. EXERCISE 24. Write the quotient of— a + 2 c+(l-x) —.-^-1 * 2-(x + 7/).< ^ 27x^-64 ^^^ A^-(x2/-l)» 3a; — 4 ic^ — (xy — 1) l+8a;« 27a;^ + 125/ * l + 2a:^* • 3^:^ + 52/'' 6 125-^\ ^g (g- iy-a;\ 5-ar» ' (a-l)-a;'' ^ 27a^ + 2/" , j4 Sx^ + Ca^-iy * 3a= + 2/* ' 2a; + ar^-l * 7 ^1+1!. 15 8(a:- 1/^-2^ * cc'4-/ * 2(a:-2/)-z (g-iy-a^ ^ ^g 27(:r' + iy + 1252" (a-l)-a; ' * 3a;^ + 3 + 5«* 84 ALGEBRA. 95. IV., v., and VI. Division of Sum or Difference of any Two Like Powers. By actual division we can obtain, a* — 6* = a^ — a?h + ah^ — b^, Quotient. — a^-t- dh + ah^ + 6^, Quotient. But a* + 6* is not divisible by either a + 6 or a — 6 : ^' + ^' =a' -a'h + aW - ah' + h\ = a' + a'b-ha'b' + ab' + b\ a + b a'-b' a — b Hence, in like manner. The difference of any two like even powers of two quantities is divisible by the sum of the quantities, and also by their differ- ence ; The sum of two like odd powers of two quantities is divisible by the sum of the quantities ; The difference of any two like odd powers of two quantities is divisible by the difference of the quantities. For the quotient in all these cases — (1) The number of terms in a quotient equals the degree of the powers whose sum or difference is divided ; (2) The terms of each quotient are homogeneous (since the expo- nent of a decreases by 1 in each term, and that of h increases by 1 in each term). (3) If the divisor is a difference, the signs of the quotient are all plus; if the divisor is a sum, the signs of the quotient are alter- nately plus and minus. The last statement forms a general rule. for signs jof a quo- ABBREVIATED DIVISION. 85 tient in all the cases of abbreviated division, including I., II., and III. 2xi-y 2x + y = (2xy - (2xfy + (2a;)y - (2x)f 4- y' = 16a;* - ^xhj + AxY - ^xy" + y\ Quotient. EXERCISE 25. Write the quotient of — ^ a'-27b' ^ x'-l 1. TT-- 6. -• x — 1 x' + l' 81-^^ S + x"' x + y^ ^10 _ , .15 5. ^^-^. 10. V-r- Write all the exact binomial divisors for each of the fol- lowing : 11. 1 + Q4x\ 17. x'-y\ 12. dx' - 252/*. 18- *'° + 2/'"- 13. a;* -81. 19. l-^x-y)*. 14. a;^-32. 20. x' — y\ 15. l+x\ ^ 21. a;" -64/. 16. x'-27y\ 22. a;'^"-^/^ For what values of n \^ill — 23. a" — ^'^ be divisible by a — 6 ? By a + 6? 24. a" + 6" be divisible bya — &? Bya + 6? • a- 36 a* -166* a- 26 a^ + 1 x + 1 T>-24S x-3 x' + y' 3. — -• 8. x + 1 4. ^243. ,,_ CHAPTER VIII. FACTORING. 96. The Factors of an expression (see Art. 12) are the qiiiintities which, multiplied together, produce the given expression. Factoring is the process of separating an algebraic ex- pression into its factors. 97. Illustration of Value of Factoring. If a fraction, and it is known that and that 2x'' - ISx + 21 = (2x -7)(x- 3), for the original fraction we may write, (x-B)(x- 5) (x-Z)(2x-iy then cancel out the factor x — 3, common to both numerator and denominator, and obtain the simple fraction, x-b 2x-l' This is an illustration of the usefulness ojf" a knowledge of factoring in enabling us to simplify work and save labor. 98. A Prime Quantity in algebra is one which cannot be divided by any quantity except itself and unity. Exs. a, 6, a^ + h\ 17. 86 FACTORING. 87 99. Perfect Square and Perfect Cube. When an expres- sion is separable into two equal factors, the expression is called a perfect square, and each of the factors is called the square root of the expression. Ex. 9aV = ^ax' • 3ax^ . • . Sax^ is the square root of 9aV. Also, oi^ — 4x + 4:=(x — 2)(x— 2), and is therefore a perfect square, with a; — 2 for its square root. When an expression is separable into three equal factors the expression is called a perfect cube^ and each of the factors is called its cube root. Ex. 27a'xy = SaxY • SaxY • Zay^if. . ' . ^axY is the cube root of 27aV2/'. 100. Factors of Monomials. Since monomials are formed by simply indicated multiplications, the factors of a mono- mial are recognized by direct inspection. Thus, the factors of 7aV are 7, a, a, x, x, x. 101. Factors of Polynomials. When binomials or trino- mials are multiplied together, simplifications are often made in the product obtained by the addition of similar terms. Hence, given the simplified result, the problem of deter- mining the quantities multiplied (or factors) is made more difficult, and different cases must be carefully discriminated. CASE I. 102. A Polynomial all of whose Terms contain a Com- mon Factor. Ex. 1. Factor 3a;' + 6x. Each term contains the factor 3a;. Divide 3a;' + 6a; by 3a;, and the quotient is a; + 2. The factors are the divisor and quotient. . • . Sa;" + 6a; -= 3a;(x + 2), Factors, 88 ALGEBRA. Ex. 2. Factor \2xY - 16x/ + ^x\j\ 12x'y - 16x2/* + ^^Y = 4a:2/X3a: - 4^/ + 2rc2/'). Hence, in general, Divide all the terms of the polynomial by the common factor ; The factors will be the divisor and quotient. EXERCISE 26. Factor — 1. 2x'-{-5x'. 4. So:' -a. JT^^lSx' ~ 27x'y. 2. 7?- 2x. 6.^ la + Ua\ 8. x''-x^~ x\ 3. x" + X. \ U'x" - 15aV. 9. a\ - 2a'x\ 10. 3a' - ^ax + ^x\ :13,..a*6^c - a'6V + 2d'h'c\ 11. 2a: + 4a;' - Qx\ 14,. 2a:Y - 8x'y + Gx'Y- 12. lQa^6' - 35a'6l ^'J y 15. a"^6'c'" + lla"^6V" + \ CASE 11. 103. A Trinomial that is a Perfect Square. By Arts. 85 and 86 a trinomial is a perfect square when its first and last terms are perfect squares and positive, and the middle term is twice the product of the square roots of the end terms. The sign of the middle term determines whether the square root of the trinomial is a sum or a difference. Ex. 1. Factor 16a;' - 2Axy + %'. This trinomial satisfies the conditions for being the square of 4x — 3?/. . • . 16a;' — 2Axy + %' = (4a; — 3^/) (4a; - 3^/), Factors. Ex. 2. Factor a* + 4a'6 + 4^'. a* + 4a'6 + 46' = (a' + 26)', Factors. Hence, in general, to factor a trinomial that is a perfect square, Take the square roots of the first and last terms ^ and connect these by the sign of the middle term ; Take the result as a factor twice. FACTORING, * 89 ^ EXERCISE 27. Factor — 1. 4x^ + ^xy + y\ \^. 4x^ + 44a:y + 121a:/. 2. 16a'^ - 24a2/ + 9/. 11. 81a^6 + 126a^6^ + 49a6'. 3. 25x^ ~ lOx + 1. 12. My - 40aa:2/ + ^^x\j^ 4. a;"' - 20a;2/ + 1002/1 13. 2a;* ~ 8x' + 8x^ J|f= 49c + 286c'^ + Wc\ <4^0^^2/ + Bx* + 752/^ ^6. d'W- - 6a6c + 9cl 15. a^x + aa;' - 2aV. j^ a;?/' + 2x2/ + ^- 16. rc'^" + Ix^'y + 2/'- 8. 2m='n — 4m7i + 271. 17. {a-hj — Icia-h) ^ &. 9. a^ + 2a* + al 18. 9(a; + yj + 122(a; + 2/) + ^ °^C 19. 16(2a - 3)' - 16a6 + 246 + 6^ "^ 20. 2o(a; - yj - \%)xy{x -y) + 144a;Y. 21. a^ + 6^ +'c2 + 2a6 + 2ac + 26c. CASE III. 104. The Difference of Two Perfect Squares. From Art. 87, (a + 6) (a- 6) -a'- 6^ hence, a^ — 6^ = (a + 6) (a — 6). ^ But any algebraic quantities may be used instead of a and h. Hence, Ex. 1. Factor x^ — l^y\ x^ — IQy'^ = (a; + 4y) (x — Ay)^ Factors. In general, to factor the difference of two squares, Take the square root of each square ; The factors will be the sum of these roots and their difference. Ex.2. x'-y' = (x'-hy')(x'-y') = (ar* + 2/0 (a; + y) (x —y), Factors. * Apply Case I. first. 90 * ALGEBRA. 105. Special Cases under Case III. A. We may also factor by this case the difference of two squares when one or both of the given squares is a compound expression. Ex. 1. Factor (a + 2by - 4x\ (a + 2by - Ax' = [(a + 26) + 2a;] [(a + 26) - 2a;] = (a 4- 26 + 2a;) (a + 26 — 2a:), Factors. Ex. 2. Factor (3a; + 4yy - (2a; + Syy. (3x + 4yy - {2x + Syy = [(3a; + 4y) + (2a; + 37/)] [(3a; + 4y) - (2x + 32/)] = (3a; + 4y + 2x + Zy) (3a; + 4y-2x- Zy) = (6a; + 7y) (x + 2/)> Factors. Factor — EXERCISE 28. 1. x'-9. 2. 25-16a^ 8. a;y-362/. 9.* x^ — a;. 15. a«-4a;*. 16. 2a'6*-98a6. 3. 4a' -4961 10. a;«-25. 17. 144 -a;^'. 4. Sx'-12y\ 5. 100 -81m'. G. m — Mmn\ ^3a;^-75x/. 12. ,^2 -2a;*. ]^V-a:2/*. 14. a* -816*. 18. x^-\mf\ \^ x^-y\ ""20.*.,16aa;*-81a. 7^49a:*-l. 21. 225a;'" -y'. ' 22. x^ - /. 29. (x + 2yy - (3a; + 1)'. 23. x"- - f-z\ 30. 25(2a - 6)' - (a - 36)'. 24. {x + yy - 1. 31.* a;^y - yz^\ @. a;' - (3/ + 1)'. . 32. 81a;^^ - 16/. 26. {x - yy - 9. 33. x"^ - 144a;/2^ 27. 4(a; - yy - 25. 34. {a - by - 4(c + 1)1 28. l-36(a; + 22/)'. 35. 1 - 100(a;' - a; - 1)'. * May be resolved into four factors. FACTORING, '91 106. B. Grouping of Terms. It may be possible to group or rearrange the terms of a polynomial expression so as to produce the difierence of two squares. Ex. 1. Factor x^ — 4xy + 4y' — 9z\ x" - 4x2/ + 42/' - ^z' = (x' - 4xy + 4y') - 9^' = (x~2yy-9z' = [(x-2y) + Sz-]l(x-2y)-Bz'] = (a; — 22/ + 3z) {x — 2y~ 3z), Factors, Ex. 2. Factor 2xy — x' + a' - y\ 2xy -x' + a'-y' =.a' - (x' - 2xy + f) =^a:^-(x-yy \ = [« + (^ - 2/)] [a - (a; - :y)] = (a + a: «^ 2/) (<^ ~ ^ + 3/)> Factors. Ex. 3. Factor a' - ic'^ - 2/' + ^' + 2a6 + 2xy. a» - ic^ - 2/' + 6' + 2a6 + 2a;2/ = {a' + 2a6 + h') - {x^ - 2xy + y') = (a + by-{x-yy = (a-^b + x — y)(a + b — x + y'), Factors. Factor— 1. x'-ie(x-2yy. 2. 9(a-6y-25. 8. a'~2ah-{-b'-l. 4. 9x'' + 12xy^-4:y'-z\ 5. x'-^a'-y' -2ax 6. a' -^f- - a:' + 2ay. 7. a' -a:'- -2x'y-y\ 8. x" -f- -1-21/. EXERCISE : 29 • 9. l + 2xy -^-3/'. 10. e-a^- -6' + 2a&./l 11. a' + b'- -c^-2a6r^ ■z\ 12. 4.~x'- 4y^ — 4x2/. 13. 2a + b'- - a' - 1. 14. 2ab + a'. b' + l-x". 15. 2z' - 4z - 2z* + 2. 16. 9x'^ + 2/' -25z^-6a^. 92 ALGEBRA. 17. 2O2/Z + a;' — 42/' — 25z^ 18. 45x' — 20a;* - bf — 20icy 19. a^ + 2ah -\- b' - c' - 2cd - d\ 20. af + 4^/' - 92' - 1 - Axy - 6z. 21. ^a^ - 25a;' + 46' - 1 - 10a; - 12a6. 22. a' - 96V - 1 + 66a; - 10a6 -f 2561 23. 16a;^ - lQ>x'y - 16x - 48a:z - 36a;z' + Axy\ 107. C. The Addition and Subtraction of Some Quan- tity will sometimes transform an expression into a difference of two perfect squares. Ex. 1. a* + a'6' + 6* = a* + 2a'6' + 6* - a'6' = (a' + 6')'-a'6' = (a' 4- 6' + ah) (a' + 6' — a6), i^actors. Ex. 2. a;* — 7a;'2/' + 2/*. Add and subtract 9a;y. ic* — IxY + 2/* = a;* + 2a:y + 2/* — 9a;y =:(a;' + 2/^)^-9xy = (a)' + 2/' + 3j2/) (a:' + 2/' " 3^2/), i^actors. EXERCISE 30. Factor — 1. c* + cV + a;*. 7. 49a;* + 34a;V + 252*. 2. a;* + a;' + l. 8. 16a;*- 9a;' + 1. 3. 4a;* - 13a;' + 1. 9. 100a;* - 61a;^ + 9. 4. 4a* - 21a'6' + 96*. 10. 100a;* + 11a;' + 9. 6. 9a;* + 3a;'2/' + 42/*. 11. 225a*6* - 4a'6' + 4. 6. 49c* ~ We'd' + 2bd\ 12. 32a* + 26* - 56a'6'. 13. a* + 46*. 14. 1 + 64a;*. 15. xSf + 324. FAGTOBINQ. 93 EXERCISE 31. SPECIAL REVIEW. Factor- 1. 1 - 4a: + ^x\ / -^ 7j (/'!^ 1<^- 2«^* - 2a5. 2. 12^?/2 - 3x'\ y ^n, ^V.i'V^ ' fef49 - 140712 + iqOti*. 3. 9 + a;2 - 6a; - )/^. ,v^^ *I2: 4^2 + a;* - 4a;=' - 1. W , , , ^v a; - 2a:» + a;5. v.-_\T -^.^S^-^^'-'^S^ - ISa;^ + a;*: 6. a:* - 14^-2 ^ i '^^^' ^a:^ - 3 - 3a* + 6^ 7. 20a:V - 45^^ ^^ 16 ^^2 + ^^2^2 _ 4^2yj2^ 8. fa^ _ 52 _ ^2 + 26c.^ ^. .;, -N „. 17. 4a:y - (a;^ + y"" - \)\ 9. 32a:5 - 2a;3 + 2a;. , , ', .. /V.^ 18. ^a'x' + 256^ _ 20a6a;.i 19. 2a:^*^ ^xy"^ - 2a; + 4a;y. 20. 1 - 6a6 - 952 + I2a25 _ 4^* + 9^252^ 21. a'x'^ -b^ -y^ + 1- 2ax + 2by. 22. 9aW - 25 - IGa^o^s + 4^2 - I2a6a; - 40aa;. 23. (a;2 + ^2 _ 9)2 _ 4^222^ 24.. 49a;* + 66a;2^* + 25^8. CASE IV. 108. A Trinomial of the Form x^ + 6a? -f c. It was found in Art. 90 that on multiplying two binomials lixe a; + 3 and a; — 5, the product, x^ — 2x — 15, was formed by taking the algebraic sum of + 3 and — 5 to obtain the coeffi- cient of X, — viz. — 2, — and taking their product, — 15, to form the last term of the result. Hence, in undoing this work to find the factors of x^ — 2x — 15, the essential part of the pro- cess is to find two numbers which, added together, will give — 2, and multiplied together will give — 15. Ex. 1. Factor x' + llx + 30. The pairs of numbers whose product is 30 are, 30 and 1, 15 and 2, 10 and 3, 6 and 5. Of these, that pair wjtiose sum is also 11 is 6 and 5. Hence, x^ + 11a; + 30 = (a; + 6) (a; + 5), Factors. 94 ALGEBRA. * Ex. 2. Factor x" - 8fc + 7. ^ K It is necessary to find two numbers whose product is +7, and sum is —8. When the last term is positive, as in this example, the two required num- bers must be both positive or both negative, and since their sum is negative, they must be both negative. . ' . x^ - 8x + 7 = {x - 7) {x - 1), Factors. Ex. 3. Factor a;' — a; — 30. It is necessary to find two numbers whose product is —30, and sum is -1. Since the sign of the last term is minus, the two numbers must be one positive, the other negative ; and since their sum is — 1, the greater number must be negative. >, .. ^ x' - X- SO = (x- 6) {x + 5), Factors. ^^ Ex.4. Factor cc' + 3x3/ - lOy. Since 5^, — 2^, added give 3^, multiplied give — 10^', x^ + Sxy - 10^2 = (a; + 5^) {x - 1y), Factors. Hence, in general, to factor a trinomial of the form a? H- hx 4- c, Find two numbers whichy multiplied together, produce the third term of the tnnomial, and added together give the coefficient of the second term; X (or whatever takes the place of x), plus the one number ^ and x plus the other number ^ are the factors required. EXERCISE 32. Factor — 1. x' + 5a; + 6. 6. x' + x- 30. 2. a;' — a; - 6. 7. x' + Qxy - IG/. 3. a:' + a; - 6. 8. a:' - Qxy -IQf^ 4. a:' 4- 7x — 44. 9. x' + 8a; -f^l6.* 5. X* - Ux + 30. 10. a;' + 5x - 36. FACTORING. 95 11. x^-hx- 36. C ^ )ia;^ ~ 9a;' + 8. s.^ 12. a;* - bx^ - 36. 25. 2a - 14aa; - 60aa:'. 13. x^ + Zx- 28. 26. *>:e'' - 22a:' - 120a:. 14. a;' - 2a; - 48. 27. ■ 25a;y + b^y - 3(^2/*. 15. a:' - 8a; - 48. 28. 56«a;'2/ + 96ax2/ + ^a7?y. 16. a;' + 16a; + 48. ^29. 2a;« - 10a;' - 28a;'. 17. x' + 19a; + 48. J^^ + ^- 20a;. 18. a;' + 13a; - 48. 31. r' - 25x' + 144x. 19. x' - 22x - 48. 82. 3a;» - 51a;* + 48.. 20. a;'-49x + 48. 33. a;'" -a;" -56. 21. ar^ - 4a; - 96. 34. aW - llaic' - 26c*. Jg^ xY - 23xy + 132. 35^ Zaxf - 9ax'y - SOas^. ^x" - dax - 24a\ 36. 5a;' + SOs^y' - 35x3/*. . 37. x'-{-(a + b)xfab. 38. x" + (2a - 36)a; - 6a6. 39. a;' - (a + 26')a; + 2a6'. 40. a;' + (a + 26 + c)x + (a + 6) (6 + c). n^. i\: 7^ + (a + h)x + (a - c) (6 + c). 42. (a;-2/)'-3(a;-2/)-18. 43. 2(a;' + 2x)' - 14(ar^ + 2x) - 16.^ CASE V. 109. Trinomial of the Form ax^ + hx-\-c. From Art. 91 it is evident that the essential part of the process of factoring a trinomial of the form qt? -{-hx-\- c lies in determining two factors of the first term and two factors of the last term, such that the algebraic sum of the cross products of these factors equals the middle term of the tri- nomial. Ex. Factor 10a;' + 13x - 3. 96 ALGEBRA. The possible factors of the first term are lOx, x ; and 5x, 2a^ The possible factors of the third term are — 3, 1 ; and 3,-1. In order to determine which of these pairs taken together have the sum of their cross products equal to + 13x, it is convenient to arrange the pairs thus : 10a:, - 3 5a;, - 3 X, 1 2x, 1 Variations of these may be made mentally by transferring the minus sign from 3 to 1 ; and also by causing the 3 and the 1 to change places. It is found that the sum of- the cross products of 5a;, -1 . _ 2.; 3 ^^+^^^- Hence, lOa;'^ + 13a; - 3 = (5a; - 1) (2a; + 3), Factors. Hence, in general, to factor a trinomial of the form ax^ + bx + c, Separate the first term into two such factors, and the third term into two such factors, that the sum of their cross products equals the middle term of the trinomial ; As arranged for cross multiplication, the upper pair taken to- gether and the lower pair taken together form the two factors. 3tor — EXERCISE 33. ft. 2x^ + 33; 4- 1. 7. 6a;^ + 20a;' - 16x 2. 3a;'^-14a; + 8. ^ 8. 3a;' -4a; -4. 3., 2a;'^ + 5a; + 2. 9. 8a;' + 2x~15., 4. 3a;' + 10a; + 3. 10. 2a;' + x-m 5. 6a;' - 7a; — 5/^' 11. 12x'-5a;-2. 6. 23^^ + 53; -3. 12. 4a;' + llx-3, FACTORING, 97 "•^^^5a;' + 24x-5. 22. 12a:'-7a;2- 122». 9a:' - Ibx' - 6x. 23. 24x' + 104xy - 18ajy*. 16. 6x^2/ - 2x2/ - 42/. ^A ^ 24. 25a* + 9a'6' (^iTl 6a^-J6xJ^- 272^^ 25. IGx^-lO^y - 166*. cy-92/*. 17. 122;'^ + iC2/ — ^3p^ 26. 3a;'" - 8x"2/ - 82/'. 18. 422:' + 13x - 42. 27. 25a* - 41a'^6' + 166*. 19. 32a* + 4a6- 4561 28. 36a;* - 97xy + 361/*. 420. 4x* - ISx'^ + ^.4 29. 20 - 9x - 203:^. 21. 9x* - 148a:^ + 64. 30. 5 + 32a:y - 21af^^.^ 31. (a + 6)' + 5(a + 6) - 24. 32. 3(x-2/)' + 7(a;-2/>-6z^ 33. 3(a;* + 2xf - 5(x^ + 2a;) - 12. 34. 4a:(a:' + 3a;)' - d>x(o^ + 3x) - 32a;. 35. 2(a; + l)'-5(a;'-l)-3(a;-l)».- CASE VI. 110. Sum or Difference of Two Cubes. From Art. 94, .^^+Z =a'-ab + h\ a + b Hence, a' + 6» = (a + 6) (a' -a6 + 6') .... (1) In like manner, a' — 6' = (a — 6) (a' + a6 + 6') .... (2) But any algebraic expressions may be used instead of a and 6 in (1) and (2). Ex. 1. Factor 27a;' - Sy". 2l7?-S^ = (Zxf-(2y)\ Use 3a; for a and 2y for 6 in (2) above. 273;^ - 82/' = (3a; - 2y) {^x^ + Qxy + 42/'), Factwi. Ex.2. Factor a« + 86'. a« + 86' = (a')'4-(26»)» = (a' + 26')(a*-2aV + 460, 7 y 98 ALGEBRA. Ex. 3. Factor (a + hf - a:', (a + 6)' - a^ = [(a + 6) - a:] [(a + h)' + (a + 6)a; + a^]. Hence, in general, to factor the sum or difference of two cubes, Obtain the values of a and h in the given example^ and substitute these values in either the formula (1) or (2). 111. Sum or Difference of any Two like Odd Powers. Since the difference of two like odd powers is always divisible by the difference of their roots (see Art. 95), the factors of a" — 6", when n is odd, are the divisor, a — b, and the quotient. Ex. 1. a'-b' = {a- b) (a* + a'b + a'b' + ab' + b'). Since the sum of two like odd powers is divisible by the sum of the roots (see Art. 95), the factors of a" + 6**, when n is odd, are the divisor, a + 6, and the quotient. Ex.2. a^ + Z2^ = x'-^(2yy. = (ix + 2y) [x' - x\2y) + x\2yy - x(2yy + (2?/)^ -={x + 2y) [x* - 2x'y + 4xy - Sxy' + IQy'l Factors. 112. Sum or Difference of any Two Even Powers. The difference of two even powers is factored to best advan- tage by Case III. Ex.1. x'-y' = (x'-{-y')(x*-y*) = (a;* + 2/*)(^ + 2/')(a^-yO = (x' -\-y') (i^ + y')(x + y) (x-y). The sum of two even powers cannot in general be factored by elementary methods unless the expression may be re- garded as the sum or difference of two cubes (Art. 110), or other like odd powers. Ex.2. a^ + b' = (ay + (by = (a' + b') (a* - a'b' -h b*), Factors. But a' + 6', a* + 6*, a® + b^, cannot be factored by any ele- mentary method; and are therefore prime expressions. FACTORING. 99 EXERCISE 34. Factor — 1, m^—n\ 8. 1 — 1000a;3. 15. 250a;~2a;'^. 2. c^ + Sd\ 9. 21x^-\-a^x. 16. 8^« + ^/^ 3.27 — ^. 10. 512a:3 — ^6. 17. («+ ^,)3^ 1 4 a?-\-mcK 11. a + 343a*. 18. 125 + (2& — a)3. 5. a^3 — 125. 12.* a6 — ^6^ 19. s—(c + d)\ 6. 642/3 — 27. 13. a;i2_^6 20. {x—y)^—21x^. 7. a^fes + l. 14. a6_ 64^12 2I. 16xY—5^xz\ 22. ar5 + i/5 27. aii + a:ii. 32. 64— (a — 6)3. 23. x'^—y\ 28. a^ -f ^,9 33 8(a; — 22/)3 + l 24. t rt^ + m6. 29. 32a;5— 1. 34. a^^ — b^\ 25. 2^12 ^ yi2 30. ^11 _ 2,11. 35 ^10 _{_ JIO 26. a^— 1286^ 31. 243 — x^. 36. S2x^~a^\ CASE VII. 118. A Polynomial whose Terms may be grouped bo as to "be Divisible by a Binomial Divisor. Ex.1, ax — ay — hx-\-hy = {ax — ay) — (bx — by) = a{x — y)—h{x — y) = {a—h) (x — y) , Factors. Ex. 2. l-|-15a*--5a — 3a' = l — 3a' — 5a + 15a* X ={l — 3a')-5a{l — 3a') ={l — 3a^) (1 — 5a), Factors. Ex.3. «'+2/'+ar + 2/ = (a; + y) (x' — xy -[- y') -{- {x + y) =^{x+y) {o(^ — xy-}-y^-^l), Factors. Ex. 4. o' 4- 8a' — 4 =a' + 2a2 + «' — 4. = a2(aH-2) + (a + 2) (a — 2) = (a + 2) (a' + « — 2) = (a + 2) (a + 2) (a — 1), Zacfor*. *U8e Case III first. tSum of two cubes. 100 ALGEBRA, EXERCISE 35. Factor — 1. ax -^ ay -\- hx -\- by. 11. x^ -\-Sy — Sx — xy, 2. x^ — ax + cx — ac. 12. z' — z^ — z + 1. 3. 5xy — lOy — Sx + Q. 13. ab — by — a + y. 4. 3am — 4mn — Qay + Sny. 14. x^ — «* — 4x + 4. 6. a'x + Sax + aca; + Sex. 15. aV - 6'a:' - ay + by. 6. 3a''2/ + ^(^h — 5«^2/ - ^^^2/- 16. a;(x -f 4)' 4- 4(:r + 4). 7. a;* + x* + 2a;' -f 2x. ' 17. a^a + 3) - 3(a + 3). 8. 2x* - 2a;' - 2aV + 2a'a;. 18. 2(a;' - y') -(x- y). 9. 2/' 4- 2/' + 2/ + 1- 19. 4a;(a) - l)'^ + x — l. 10. ox' — 2a'a; — a; + 2a. 20. a;' — 1 + 2(a;' — 1). ^\ 21. 4a'-aV+a;'^-4. 22. Aa3i^ + Sax — Sa — 4aa^. 23. a(3a-a;y-6ax' + 2a;'. 24. a:'-8-7(a:-2). 25. 4(a;' + 27)-31a:-93. 26. (2a: + iy- (2x4-1) (3a; + 4). 27. (2a; - 3)» + 2a;' - 9x + 9. 28. a;» — 7a;-6. 29. a;» — 3a;' - 10a; + 24. 30. a;' - 8a;' + 17a; - 10. 31. 6a;' - 23x' + 16a; - 3. 114. General Principles in Factoring. Ii> order that the application of factoring may be as effective as possible, it is important to reduce each expression factored to its prime fac- tors. Hence it is important to use the different methods of factoring in such a way as to give prime factors as a result most readily. Hence, in factoring any given expression, it is useful to — FACTORING. 101 1. Observe, first of all, whether all the terms of the expres- sion have a common factor (Case I.) ; if so, remove it. 2. Determine which other case in Factoring can be used next to the best advantage. 8. If the expression comes under no case directly, try to discover its factors by rearranging its terms, or by adding and subtracting the same quantity to the given expression, or by separating one term into two terms. 4. Continue the process of factoring till each factor can be resolved no further. Ex. 1. Factor x' — dx\ This expression as it stands might be factored as the difiference of two squares (Case III.), but it is best to apply Case I. first. X^ -9x''-= x\x^ - 9) - = x\x + 3) (a; - 3), Factors. Ex.2. Factor a«-6^ This expression might be factored by dividing it by a + 6 or by a — 6, and taking the divisor and quotient as the factors, but it is factored to best advantage by the use of Case III., and afterward Case VI. a« - 6« = (a' + 6») (a^ - h^) = (a + 6) (a' - ab + b^) (a - 6) (a^ + a6 + 6'), Facton, Ex. 3. Factor a;' - Qx' + 6a; - 5. 7? -Qx"^ -^ Qx-b==Q^ -Qx"^ ^bx + x-h ^x{x-l){x-b)^{x-b) = (x - 5) [x{x - 1) + 1] = {x - b) {x"^ - X -\- 1), Factori, Ex. 4. Factor Gr'i/ — Sx^y — Ixf - \xY + 2x2/. ^7?y - ^x'^y - 2xy^ - ^x^y"^ + 2xy = 2xy{Zx^ - Ax-y"^ -'Ixy + 1) = 1xy{Ax^ - Ax -V\ — x^ - Ixy — y*) = 2a:^[(2a: - 1)'^ - {x-^yf-^ = 2xy{^x-\-y -1) {x—y - 1), Facton, 102 ALGEBRA. ■ l Ffictor— 1. Sx^ — Sx. %JLx' — 't^x'y -f '^xf, 8. a;^-lla: + 30. 4. \^ + hxy — 62/'. ^ 12a' - lah - "m^J U a:*- 1-2/^ + 22// T'40a' - 5. ^16x^-40^2/ + 252/^ l^^'^ + Zax — 3a - x. 10. 3x'-3a:. l^g* - 5a' + 1. j|*2x«-32. ^./ + 4a; — 45. 14. 4a;"' + 2a - a' - 1. ' ^"^IS^- 5ax^ — 5a. - ':i- liJ- 18x' - 3a:' - 36x. «* + 3xV + 4z*. 18. oV-9a;'-a' + 9^ 19. 110 -a: -a:'. 20.:; 3x' + 13x7/ - 3O2/'. '^; 7a - lan)\ |22.;6x' + 14a^ + 8. 23. a:* - {x - 2)'. 24. 3a + 3a*. 25. a'-a' + 2a-2. ^. 6x^-2x-4a;'. :^27. 1 - 23z' + z*. 28. 128-22/'. EXERCrSE 36. (gj 1 - «' - 6' - 2a6. ,30. 21a' -17a -30. 31. a;'' + 2/''. 32. 8a:^ + 7292». 38. 405jy-45a:*. /'Spa' - 4a=' + bd' - 20. 35.' (c + (^)»-l. 36. {x-yy-^2(x-y). 37. 24x' + 5x2/ - 36?/'. 38. x' - 2x\j - 4x2/ + 8/. 39. (a'-Qy~-n\ 40. z* + 2'+ 1. 41. (a'--^;'^-c7-46V. 42. 21a;'-40x2/ — 2I2/'. 43. 32 f n^ 44. 5a;' + 5x2/'. 45. m' + n\ 46. 2ax='-f ia2/-l 47. 1 -f a; — a;* - x\ 48. a;'-9-7(x-3)^ 49. 4a* - 37a' + 9. 50. x' - 64. 61. x' - 27 - 7(x - 3). 52. S2x^y-yz'\ 53. (x' + 2/"0*-16xy. 64. x' + xY-yV — z\ 66. aa;* — ax — x^y -\- y. 66. 4(a'-60-3(a + 6)l FACTORING. 103 67. a'' - 1. 59. 4a' - 96' - 1 - 66. 68. 4a' - 96' + 4a - 66. 60. Z^t? + 18x' - 40«. 61. (a;' ~ 1)' + (2x + 3) (a; - 1)'. 62. a'-6*-aV + 6V. 63. 3a;' - 27 + ax' - 9a. 64. 18a'6 + 86 - 27a'c - 12(;. 65. 3x' - 3a; + 4a;* - 4a;'. ^' 66. l-4a'6'c'-9a:'2/'z' + 12a6ca:2/2. 67. a'6ca; — amnpx + m^wpy — abcmy, 68. 4a; + 4an + a;' — 4a'-n' + 4. I 69. 2(ia^ - 8) + 7a;' - 17a; + 6. ^ 70. a* -46* + a' + 26'. 71. 4a;» — 19a; + 15. 72. 3a;'» + 7a;'-4. 73. 49a;' - 70a; + 25. 83. 45a;' + Sxy- 21y', 74. 2/' + 4a; - 1 — 4a;'. 84. aa;' + 5ax — 84a. 75. 49a;* - 22'a;'2/' + %*. 85. ISar' - 5a;' + 33x - 11. 76. 5xV - 5a;2/*. 86. a;'t/ - lOaj'i/'z' + 25xy'z\ 77. a;* + ar* -a; - 1. 87. a;* - 79a;' + 1. 78. 21a;' 4 2a; - 55. 88. a' - 9 + 96' - 6a6. 79. 18a;' + 62xy - Qy\ 89. a;'^ - 4a;* - lea;' + 64. 80. (a; + D' - x\ 90. (x' + 3)' - 64x*. 81. (1 - 2x)' - x\ 91. «* - 492/' + ^- 6«'- 82. ax' -cx + ax-c. 92. 60a;' + 119x - 60. 93. xY — 4x' + 4 - 2/' - 4xY + 4xy. 94. a'nx — bcrri'yz + acmxz - ahmny, 95. 5(a;^ + 27) - liar* - 46x - - 39. CHAPTER IX. HIGHEST COMMON FACTOR AND LOWEST COM- MON MULTIPLE. 115. Value of Highest Common Factor and Lowest Common Multiple. In the use of factors it is frequently important, in order to do required work most effectively and with least labor, to be able to find the factor of highest degree common to a number of given expressions, or to determine the expression of lowest degree which will contain exactly a number of given expressions. 116. A Common Factor of two or more algebraic expres- sions is an expression which divides each of the given ex- pressions without a remainder. The Highest Common Factor of two or more algebraic expressions is the product of all their prime common factors. This product will evidently be the factor highest in degree that will divide each of the original expressions without a remainder. Ex. 1. The H. C. F. of 4x\ 12a;', 16:x^y is ix". Ex. 2. The H. C. F, of 6xXx - y)\ Wxi^ - y') is Sx(x - y\ CASE L When the Highest Common Factor may be found di- rectly by Inspection. 117. H. C. F. of Monomials. Ex. Find H. C. F. of 60aV, 45ax\ OOaV^/. By arithmetic the H. C, F, of the coefficients, 60, 45, 90, is 15. 104 HIGHEST COMMON FACTOR. 105 a is common to all of the given expressions, and its least exponent in any of them is 1. X is common to all the expressions, and its least exponent in any of them is 2. . • . 15ax' is the H. C. F. In general, Take the highest common factor of the coefficients ; Annex the letters common to all of the expressions^ giving to each letter the least exponent which it has in any expression. 118. H. C. P. of Polynomials directly Factorable. Ex. 1. Find the H. C. F. of x' -Zt?, a?- 9a;, a;' - 6x + 9. x'-Z3? = i^(x-Z) ar'-9a; = a;(a;-h3)(a;-3) ic»-6x + 9 = («-3y. .•.H.C.F. = a;-3. Ex. 2. Find the H. C. F. of 6a:»y - 12ay 4- 6y» and Sic'y* + 9x2/* -122/*. GxV - 12x2/' + 62/' = 62/(x - yf 3xy + 9xy - 12y* = Zy\3? + 3x2/ - ^) = Sj/'Car + 4t/) (x - y) r.K.C.¥. = Sy(x-y). Ex.3. FindH.C.F. of 12a'b\ 8aXa - b)\ 16a'b\a + h)\ 4a« - 4a*b, H.C.F. = 4a'. In general, Separate each expression into its prime factors ; Multiply together the factors common to all the expressions^ taking lach common factor the least number of times it occurs in any one expression. 106 ALGEBRA. EXERCISE 37. Find the H. C. F. of— 1. 4a% 6ab\ (^a(a + 6), a' - b\ 2. 5x% 15xY. ^ix-yy, x'-y\ 3. abc\ Sa'bc'. IL 3^ — 3a:, x" - 9. 4. 2Aa'x\ 56aV. 12. 4a;=^ + 6x, Qx' + 9x. 5. Um\ 42am'. 13. a' - x\ a' - x\ 6. 2ixy, 4Sax\ 36a;. 14. a;^ + x, a^ ^ 1. 7. 34aV, Slaa:^. 15. xy — y, x^ — x. 8. aV2/, aV2/*z. 16. 4a' + 2a^ 4a' — a. 17. x' + x, x'' — l, x^ — x — 2. 18._a;' + a; - 12, a;^ - a; - 6, a;' - 6a: + 9. 19. 4a'a: — 4aar', 8aV-8ax*, 4aV(a — a;)l 20. 2a;' -2a:, 3a:* -3a:, ixi^x-iy. 2h Qx' + bxy - Ay\ ^x' + 4:xy - Zy\ 22. 3ar' - 5a:'* - 2x, 4a:' - 5x' - 6a:, a:' -4a;. 23. x' - 81, X* + 8x' - 9, 2a:* + 17a:' - 9. 24. b-d% Sb-a'b-2a% b'-a'b\ 25. 1 - a', 1 — a^ 3a + 3a' + 3a', 1 + a' + a*. 26. xy-hx-y-1, 14a:' + 10a; - 24, 3(a:'-l)\ CASE II. provided m is not a factor of A. m That is, one of two algebraic expressions may be multi- plied or divided by a quantity which is not a factor of the other expression without changing the H. C. F. of the ex- pressions. Ex. The H. C. F. of Sx, Qax, is the same as H. C. F. of 3a;, 12aa;, and of Sx, 6a;, the H. C. F. in all instances being 3a?. 108 ALGEBRA. II. The H. C. F. of a pair of expressions, A^ B^ is the same as H. C. F. of the pair A, B — rtiA. For any quantity which will divide both A and B will evidently divide B — mA. Conversely, any expression which will divide both A, and B — mA will also divide B. For any quantity which divides A will divide mA^ and, since it divides B — mAj must also divide B. Hence, the H. C. F. of one of these pairs of expressions is the H. C. F. of the other also. As applied in the method of finding the H. C. F. by the long division method, this principle amounts to this, that the H, C. F. of the divisor and dimdend is the same as the H. C. F. of the simpler pair of quantities, the divisor^ and dividend minus quotient X divisor; that is, of the divisor and remainder. Principle I. enables us to use other simplifications in the process of the work. 121. Examples Illustrating the Use of Principles I. and II. Ex. 1. Find H. C. F. of 4x' + Sa; - 10 and 4x' -h 7a;' - 3x -15. Divide the second expression by the first, 4a:' + 3x-10| 4a;» + 7a;^- 3a; - 15 | a; + 1 4ar' + 3a;' - 10a; 4x'-h 7a;-15 4a;' -h 3a;- 10 4x- 5 By Principle II., Art. 120, the H. C. F. of the two original expressions is the same as that of the simpler pair, 4a;' -f 3a; -10, 4a;-6. Proceeding with these, HIGHEST COMMON FACTOR. 109 4a;-5| 4x' + 3a; - 10 \x-\-2 8x-10 8x-10 Since 4a; — 5 divides the other expression, 4a;' -\-Zx — 10, exactly, it is the H. C. F. of the second pair, and hence of the original pair of expressions. .•.4a;-5 = H.C.P. Ex. 2. Find H. C. F. of a;' + 4a;' + 5a; 4- 2 and Sx* + 15x» 4- 12a;. The second of these expressions is divisible by 3a;, which is not a factor of the first expression ; hence, by Principle I., 3a; may be removed, and we proceed to find the H. C. F. of a;' + 4a;' + 5a; .+ 2 and x"^ + 5a; + 4. a;» + 5a; + 4 I 3:3 + 4a;' + 5a; + 2 I a; - 1 a;' + 5a;' + 4a; - a;' + a; + 2 - a;' - 5a; - 4 6a; + 6 We have now to find the H. C. F. of a;' + 5a; + 4 and 6a; + 6. But by Prin. I., Art. 120, the factor 6 may be dropped from 6a; + 6. a; + 1 I a;' + 5a; + 4 | a; -f 4 a;' 4- a; 4a; + 4 4a; + 4 . • . a; + 1 - H. C. P. Ex. 3. Find H. C. F. of 4ar» - 4a;' - 5a; + 3 and lOa:* - 19a; + 6. To render the first expression divisible by the second, by Principle I. we may multiply the first expreasion by 5, which is not a factor of the second «xpression, 110 ALOEBBA. 4a^- - 4a;'- 5x + 3 5 10a;' - 19a; + 6 20a;3- 20x3- - 20a;2 - - 38a;2 + 25x + 15 12x 2a: 18x2- 37x + 15 5 90x2- 185x + 75 9 90x2- 171x + 54 -7|- 14x + 21 2x- 3 J 10X2 - 10X2- - 19x -t- 6 -15x 5x-2 - 4x + 6 - 4x-h 6 . • . 2x - 3 - H. C. F. 122. Arrangement of Work. The following will be found a more compact and orderly method of arranging the work of finding the H. C. F. of two expressions (see Ex. 3, Art. 121) : lOx'-iGx + e Ax'- 5 - 4x^- 5x-f 3 20a;^ 20x' -20x^- 25a: + 15 -38x^+ 12a; 18x^- 37a; + 15 5 l(V-15x 90a;^ - 185a; + 75 90a;'^-171a; + 54 - 4x + 6 -71 -14a; + 21 - 4a; + 6 HC.F. = 2a;- 3 2a; 5a;-2 123. Removal of Simple Factors. It is important for the student to remember that if either one or both of the given polynomials whose H. C. F. is sought have simple fac- tors, these simple factors are to be removed at the outset, and their H. C. F. reserved to be multiplied into the H. C. F. of the remaining polynomial factors as found by the division method, HIGHEST COMMON FACTOR. Ill Ex. Find the H. C. F. of 6x* - 30a:' + 78x' - 54a; and 2x* -4a;* + 8x'-6xl 6x* - 30a:' + 78a;^ - 54x = 6x(a;' - 5a;' + 13a: - 9) 2x' - 42:' + 8x' - 6x' = 2a:Xa:» - 2x' + 4a; - 3). The H. C. F. of 6a: and 2a;' is 2a:. By the division method let the student determine the H.C.F. of a-'-5x^ + 13a:-9 and ar* - 2a:' -f 4a; - 3. This will be found to be a: — 1. Combining these results, the H. C. F. of the two original expressions is 2a:(a:-l). 124. The General Process of finding the H. C. F. of two expressions by the division method may now be stated as follows : Arrange the given expressions according to the descending powers of the same letters ; Remove simple factors of the given expressions^ reserving their H. C. F. as a factor of the entire H. 0. F. ; Use the expression of lower degree for divisor, or, if both are of the same degree, that whose first term has the smaller coeffi- cient ; Continue each division till the degree of the remainder is lower than the degree of the divisor ; Remove from each remainder each factor that is not a factor of both the given expressions; If the first term of a dividend is not exactly divisible by the first term of the divisor, multiply the dividend by such a number as will make the term divisible ; Continue the process by using each simplified remainder as a new divisor and the last divisor as a new dividend; The first divisor to divide its dividend exactly is the H. C. F. of the two original expressions. 112 ALGEBRA, EXERCISE 38. Find the H. C. F. of- X. 2x' — a: - 3 and 4x' - Ax" - 3a; + 5. 2. 6a;' -a: -12 and 6a;' - 13a;' - 6a; + 18. 3. a;' H- a;' + a; - 3, ar' - 3a;' + 5a; - 3. 4. 3a;» - 9a;' + 9a; - 3, 6a;' - 6a;' - 6a; + 6. 5. 6a;* - 5a;' + 6x' + 5a;, 2x* - 9ar' - 9x' - 2a;. 6. 3x' + x'-a; + 4, 3a;' + 7a;' + a; - 4. 7. 8a;' + 2a; - 3, 6a;» + 5x' - 2. 8. 3a;' + 7a;' -5a; + 3, 2a^ + 3x^ - 7a; + 6. 9. 2x* + a;' + 4x-3, 3a;* + 2a;' - 2a;' + 3a; - 2. 10. x*-a;'-a;' + 7a;-6, a;* + a;' - 5x' + 13a; - 6. 11. a;* + 3af^ + 9a;' + 12a; + 20, x^ -f 6a;* + 6a;' + 8a;' + 24a:L 12. 2ar' ■- 16a; + 6, 5a;' + 153;^ + 5a; + 15. 18. 2ar^ + a;* + 2af» - a;' - 1, 5a;* + 2af' + 33;* - 2x + 1. 14. 4ar^ - 10a;* + 10a;' - lOx' + 6a;, 4a;^ - 14x* + Sar* + lOx* -6a;. 15. 3x* + 2a;'y + 2a;y + bxy" - 2y\ 6x* + ar'y + 2xY + 2xy» 16. 3ar^ + 2a;* - 8a;» -Zt? + 4x, 3ar^ - 10a;* + 143;* - llx' + 4x. 17. 2a;* - 3a;' + 2a;' - 3x + 2, 3a;* - 4a;' + 5x' - 6a; -f 2. The H. C. F. of three or more expressions may be obtained by finding that of two of them ; then find the H. C. F. of this and another of the quantities ; the last H. C. F. thus obtained is the one required. 18. a;'-a;'-a;-2, a;» - 2a;' + 3x - 6, 23;* - 3x' - a; - 2. 19. 2a;' -3a;' -5a; -12, 3a;* - 73;* - 2a;' - 12a;, a;' - 9a;' + 27a; -27. 20. 2x* - 143;" + 12a;, 2x* + 6a;» - 32x' + 24x, 6a;* - 3Qx* + 42a;'-18x. LOWEST COMMON MULTIPLE. 113 LOWEST COMMON MULTIPLE. 125. A Common Multiple of two or more algebraic expres- sions is an expression which will contain each of them with- out a remainder. The Lowest Common Multiple of two or more algebraic expressions is the expression' of lowest degree which will con- tain them all without a remainder. Ex. 1. The lowest common multiple, or L. C. M., of 3a', Qa\ iax' is 12aV. Ex. 2. The L. C. M. of 3x and 4y is 12xy. CASE I. When the Lowest Common Multiple may be found directly by Inspection. 126. L. C. M. of Monomials. Take the L. C. M. of the coeffiQients ; Annex each literal factor that occurs in any of the given expressions j giving the letter the highest exponent which it has in any one expression. Ex. Find L. C. M. of 4aV, 5a3^, lOa^x^y. The L. C. M. of 4, 5, 10 is 20. The highest exponent of a is 2. " " " X is 5. " " " y is 1. .•.20aV2/ = L.C.M. 127. L. C. M. of Polynomials readily Factored. Ex. Find L. C. M. otx^ — Sx\ x^ — 9x, x^ — 6x-bd. x*-^x' = 7^ix-^) x'-9x =x(x-^^)(x-Z) x'-6x + d = (x-Sy .'.L.C.U. = xXx + Z)(x-Z')\ Hence, in general, 114 ALGEBRA. Separate each expression into its prime factors ; Take the product of all the different factors^ iising each factor the greatest number of times which it occurs in any one expression. EXERCISE 39. Find the L. C. M. of— 1. 3a^6, 2ab\ 8. 12a% lQab\ 2Aa'b\ 2. 6x'y, Sy'z. 9. 7a\ 2ab, Gb\ 21. 3. 12aV, 2dY. 10. 3x^ 8, Qx% \2xy\ 4. 16xy, 12xy. 11. 2x{x + \), x^-l. 6. 2ac, 36c, 4a&. 12. a" + ab, ab + b\ 6. Za% Aac\ 6b'c. 13. 7x\ 2x'-6x. 7. 42a;y, 2Sy'z\ 14. ar'-l, af'-l. 15. x' - y\ ^ - Sxy + 2y\ 16. 32:^- 3a;, 6a;' -122; + 6. 17. ax^Cx — yy, bxy(x^ — y^). 18. a;' - 3a; - 40, a;' - 9a; + 8. 19. 3a;'^ + 2a; - 8, ^^ + x- 12. 20. a' - 6^ a' - b\ a' + b\ 21. 6a;' + 6a;, 2a;' - 2a;', Sx" - 3. 22. a'b 4- ab', a'b - ab\ 3a' - W, 23. 2a;' + a;-l, 4a;' -1, 2a;' + 3a; + 1. 24. 32;^ - 3, 6a;' - 12'a; + 6, 2a;' + 2a;' + 2a;. 25. 12x' - 2x' - 140a;, 18a;' + 6a; - 180, 6a;' - 39a;' + 63a;. 26. l-a; + a;'-a;', 1 + a; + a;' + a;', 2a;-2a;'. 27. (a;-l)', 7a;2/'(a;' - 1)', Ux^yix + lf. 28. 18a;'-12a;' + 2x, 273;^- 3a;', 18x' - 24a;' + 6a;. 29. 36x*-81a;', 16a;^ - 48a;* + 36a;', 24a;* + 72a;' + 543;^. 30. a;' - 1 - 2a - a', a;' - 1 + a' + 2aa;, a;' + 1 - a' - 2a;. 31. (a;-l)(a; + 3)', (a; + 1)' (a; - 3), (a;'-l)', a;' -9. LOWEST COMMON MULTIPLE. 115 CASE II. Lowest Common Multiple Determined Indirectly by the Division Method. 128. If it be required to find the L. C. M. of expressions which cannot be factored readily, we proceed in general as in arithmetic when finding the L. C. M. of two large numbers ; that is, we first find the H. C. F. of the two numbers. Thus, to find the L. C. M. of 182, 299 by the division method, the G. C. D. is found to be 13. Then, since 182 = 13X14, 299 = 13X23, 13 1 13X14 , 13X23 14 , 23 . • . L. C. M. = 13 X 14 X 23 = 182 X 23. In brief, we find the G. d D. of the two numbers, divide one of the numbers by this G. C. D., and multiply the quo- tient by the other number. Similarly, to find the L. C. M. of two algebraic expressions which cannot be readily factored, we first find the H. C. F. of the two expressions by the division method. Thus, to find the L. C. M. of 4a;' + 3a; - 10 and 4ar' + loc" — 3a; — 15, we first find the H. C. F. by the division method ; this is 4a; — 5. Then 4a;' + 3a; - 10 = (4a; - 5) (a; + 2) 4ar* + 7x' - 3a; - 1 5 = (4a; - 5) (a;' 4- 3x + 3) . • . L. C. M. = (4a; - 5) (a; + 2) (a;' + 3a; + 3) = (4a;' + 3a;- 10) (a;" + 3a; + 3). Hence, in general, Find the H. C. F. of the two given expressions ; Divide one of the expressions by the H. C. F.y and multiply the quotient by the other. 116 ALGEBRA. EXERCISE 40. Find the L. C. M. of— 1. x' - 5a: — 2 and x'~x-\-Q. 2. Sx^-hx^-x^ 4, S7^^7x' + x- 4. 3. 6x' - Sx' - 9a: - 3, 6a;* + 9a:' + 9a:' + 3a:. 4. 6a:' - 3a:' - 10a: + 5, 8a:' - 4a:' + 20a: - 10. 5. 8x*-20a:'-14a:' + 5x + 3, 4a:'-3a:-l. 6. 12a:' -8a:' -27a: +18, 18a:' - 27x' - 8a: + 12. 7. 3a:'-2a:'-l, 4a:'-5a: + l. 8. 2a:' + 3a:' - a: + 2, 3a:' - x' - 9a: + 10. The L. C. M. of three or more expressions may be obtained by finding that of two of them ; then finding the L. C. M. of this result and the third, expression. The last L. C, M, thus obtained is the one required. 9. a:' - 7a: + 6, a:' 4- 7a:' - 36, a:' - 31a: + 30. 10. 4a:'-13x + 6, 4a:' - 4a:' - 5a: + 3, 3a:' + 7aJ'-4. Find the H. C. F. and L. C. M. of— 11. 20a'6'c', 35a'6'(i', 14a'6'c', lOa'6'c'cZ'. 12. 3a'5(a:' - 1)', 6a6'(5a:' + 3a: - 2)', 9(3i^ 4- 5a: + 2)». 13. a:* -f 2a:' + a:' - 4, a:* - a:' + 4a: - 4. 14. 4a:' - 3a: - 1, 2a:' - 3a:' + 1, Qx' - .t' -f 1. 15. 3a:' + 2a;' -7a: + 2, 4a:'-12a: + 8, *J^ -M2-.f:* - llaJ 4- 2. CHAPTER X. FRACTIONS. 129. Origin and Use of Fractions. It is sometimes neces- sary to indicate the division of one algebraic expression by another, but apart from this it is often useful to do so. For when a number of indicated quotients is combined in a pro- cess, cancellations and other simplifications are possible before making the final reduction, and in this way much labor is saved. 130. A Fraction is the quotient of two algebraic expres- sions indicated in the form -• . b This form of indicating a quotient is preferred, since it enables us readily to discriminate the parts of a fraction from the rest of an expression, and hence to compare the parts of difierent fractions to the best advantage. Thus, the fractional part of the expression is more readily x — 1 perceived in x' H 1- 5, than in cc^ + (x — 1) -^ (a; 4- 2) + 5. x + 2 131. The Numerator is the dividend part of the indicated quotient, or part above the line; the divisor, or part below the line, is called the Denominator. The numerator and denominator are called the Terrris of the fraction. 5x -\- 2 If (5a; + 2) -j- Sx^ be written as a fraction, we have — — — ; that is, the dividing line of a fraction takes the place of a parenthesis, and hence is in effect a vinculum. 132. An Integral Expression is one which does not con- tain a fraction; as, 3x^ — 2y, 117 118 ALGEBRA. 133. A Mixed Expression is one which is part integral, part fractional. 134. Sign of a Fraction. A fraction has its own sign, which is distinct from the sign of both numerator and de- nominator. It is written to the left of the dividing line of the fraction. GENERAL PRINCIPLES. 135. A. If the num,erator and denominator of a fraction be both midtiplied or both divided by the same quantii^j the value of the fraction is not changed. This principle is seen to be true at once, since the terms of a fraction are a dividend and a divisor. It is a useful exer- cise, however, to derive it from the fundamental laws of algebra (see Art. 33). CL -. = a-^b = a^bXm-^m = aXm^b-^m (Comm. Law.) aXm b ■ lib aXm bXm I- m = am b a b'^ m = a bm Similarly, and 136. B. Law of Signs. By the laws of signs for multi- plication and division (see Arts. 52, 64), a — a a — a a a a a 6—6 6 6 —b be —bXc —bX—c FRACTIONS. 119 Or, in general, The signs of any even numjjer of factors of the numerator and denominator of a fraction may he changed without changing the sign of the fraction. But if the signs of an odd number of factors be changed^ the sign of the fraction must be changed. TRANSFORMATIONS OP FRACTIONS. I. To Reduce a Fraction to its Lowest Terms. 137. A fraction is in its lowest terms when its numerator and denominator have no common factor. 138. Direct Reduction. When the terms of the fraction are monomials, or polynomials readily factored, Resolve the numerator and denominator into their prime factors, and cancel the factors common to both. Ex. 1. Reduce — - to its lowest terms. 4Sa'xy Divide both numerator and denominator by 12aV (see Art. 135). 36aV 3a Ex.2. * 48aV2/" Axy" 9ab-12b' 36(3a-46)_36 12a'-16a6 4a(3a-46) 4a The student should notice particularly that in reducing a fraction to its lowest terms it is allowable to cancel a factor which is common to both denominator and numerator, but that it is not allowable to cancel a term which is common unless this term be a factor. Thus, — reduces to -5 ac c 120 ALGEBRA. a ~\~ X but in ? a of the numerator will not cancel a of the a + 2/ denominator. This is a principle very frequently violated by beginners. 139. Finding H. C. F. of Numerator and Denominator by Division Method. When the numerator and denomi- nator of a fraction cannot be factored by inspection, Find the H. C. F. of the numerator and denominator by the method of Art. 12 Jf.^ and divide both numerator and denominator by their H. C. F. Ex. Simplify — ; ^ ^ 9a;'-22a:-8 The H. C. F. of the numerator and denominator i& ^und to be ^x'-Ax-2. Dividing both numerator and denominator by this> 6x^-110:^ + 2 2a:- 1 ^ ,^ — — = ) Result. 9ar'-22a:-8 3a; + 4 EXERCISE 41. Reduce to their simplest form — 1 ^^. 5 2a 8(^--l). ' 12aV' * 4a' -2a' * 12a: -12* 2 1?^. 6 ^^"^^ 10 i^-Zll)!. * 153:^3/^ ' 6aa:-12a?/ * \^(x-yy 3a'a; ' Ax-\'Ay a?h -{- ah" 6a'-9a'x* * Aax + Aay' * 2a'^6-2a6''' 12. 72a:VV JiZ"J(!_. 96a:2/V* ' (a; + 2/)'* " ^xSj-12xi^ L3 ^^ ~ ^^y 14 49a:' - QAf is^-dxy"' .' 14a:'-16a:'y' FJRACTIONS. 121 ^^ (^-yy(^ + yy 23. ^*~^^ 16. ^ . 24. — ^ 4x» - 2a;y - 1 2y' a;* + xy + y* ^^ Qx'-xy-2y\ ^ a:'-2x-l 67^-7xy + 2f ' ar'-2x* + l (a + 6)'-c' 18> + 1 9a;V-12v* * a'-(6+c)^* ,^ l-(a-a;)' 19. ^^ —' 27. „^ 4-(a + 6)» 20. ^- — er- 28. 27x* + 6xy - Sy' a^-Sx' + 17a: - 10 a;*-2x'-4u;'^ + lla;-6 2ay^ + (ix" - 9a St^-Sx-IS 3.r3 -f 4.7-2 _ a; + 6 2a:3 _^ 7^2 _^ 4^. _ 4- 2x^-llx'-9 (a-2)'-6» 2j m«-2ax-24a' 4a;'-2ax-6a' 22 ^-^ 30 . ' ic'y' + 2^2/' + 43/' * 4x* + llx* + 81 a;* — z' — 4 — 2x1/ — 4z + y* ol» • z^ — x' — 4 — 2yz —iz + y' n. To Reduce an Improper Fraction to an Integral OR Mixed Quantity. 140. An Improper Fraction is one in which the degree of the numerator equals or exceeds the degree of the denomi- nator. Since a fraction is an indicated division, to reduce an improper fraction to an integral or mixed expression, Divide the numerator by the denominator ; If there he a remainder^ write it over the denominator ^ and annex the remit to the quotient with the proper sign. 122 ALGEBRA. Ex. 1. Reduce — to an integral or mixed expres8ioi\ •- x^y — xy ^ .•.^-^» xy x^y + 2 X.2. Reduce 7? + X + 2 a:» + 4a:' 5 rB» + a;' + 2a; xy^ -y^ xy"^ 4- .v' x + y' x* + x + 2 BesvlU x + 3 3x^ - 2a: - 5 3x-' + 3a; + 6 - 5a; - 11 ;.^ a;» + 4a:'-5 ^^^3_ 5a: 4- 11 , j^^^ a:' + a; + 2 a;2 + a: + 2 When the remainder is made the numerator of a fraction with the minus sign before.it, as in this example, the signs of terms of the remainder must be changed, since the vinculum is in effect a parenthesis (see Art. 48). EXERCISE 42. Reduce each to a mixed quantity — ^ x' + Bxy-2f-l 6. 7. 8. 1. a;^-2x + 3 X 2. 4ar' + 6a: - 5 2x o lOftV -h5ax-7 — a bax 4. 7?-Zx' + x-\ x-\-y Zx' -13a; -28 a:^-3 7f- ar" - a; -h 2 - -a a;-l x' + 1 x'-x-l 2x* + 7 x' + x-^l x*-{-x''-x- -1 7^ + 2 9a» 3a' -26 T^-hx'-4x + 7 FBACTIONS. 14. 2a* a + 6 15. a^^-x^ + x*- -2x x'^1 16.* — i— . l+x 17. 1 l + a;-x» 18. \ 8 123 9. 10. 11. 12. ■iq x + 3 '" 2 + x-x' ni. To Reduce a Mixed Expression to a Fraction. 141. It is necessary simply to reverse the process of Art. 140 in order to reduce a mixed expression to a fraction. Hence, Multiply the integral expression by the denominator of the frac- tion, and add the numerator to the result, changing the signs of the terms of the nuw^erator if the fraction he preceded by the minus sign; Write the denominator under the result. fi 2 Ex. 1. Reduce a — 1 H to the fractional form. a-rS q- 1 + gLl-2 ^ g' + 2a - 3 + g - 2 ^ g^ + 3a - 5 j^^^^ g+3 g+3 a+3 Ex. 2. a: + y - ^-±Jl. x-y ^ (a; + y) {x - y) - (x^ + y^) x-y ^ aJ _ y2 _ a.2 _ yi ^ - 2.v' x-y x-y 2.V' y - X , Result * To three integral terms. 124 • ALOEBBA. EXERCISE 43. Reduce to a fraction — 1. a-H--. a 8. ^-^' + a 1. 2a 2. x + l-H— i— . x — 1 9. — ^ + a + 2. a — 1 3. ^ + x-l ^ ic — 1 ■ -t-i— ■ 4.4^-2 y-^. 2i+l 11. a; - a — ^^~^ + y, x + a 5. a 6+ 2*;. a + 26 12.1 (. .'+^;j. 6.- 1 '~^ • 7? + X-^l ''■''-H'-.li)] 7. a a: + l ''~-^. a-\-x ^^-'-f-^^/J- 15.^-{-.'-[x + l-^]} IV. To Reduce Fractions to Equivalent Fractions OF THE Lowest Common Denominator. 142. Since by Art. 135 we may multiply the numerator and denominator of a fr^ion by the same quantity without altering the value of the fraction, we can use the same pro- cess as in arithmetic for reducing fractions to their lowest common denominator. It is supposed at the outset that each fraction has been reduced to its lowest terms. FRACTIONS. 126 ( Find the lowest common multiple of the denominators of the given fractions ; Divide this common multiple by the denominator of each fraction ; Multiply each quotient by the corresponding numerator; the ^, results will form the new numerators ; \\jVrite the lowest common denominator under each new numerator. m-i- 2 3 5 Reduce - — > — — j - — - to equivalent fractions hav- Sax 4a'x Qaa^ ^ ing the lowest common denominator. The L. C. D. is 12a^x\ Dividing this by each of the denominators, the quotients are 4ax, 3a:, 2a. Multiplying each of these quotients by the corresponding numerator and setting the results over the common denominator, we obtain Sax 9x IQq Ua^'x^ ' 12a2a;2 ' Ua'x' ' Ex. 2. Reduce to their lowest common denominator > x — y X 1 x + y x^—y' The L. C. D. is x^ - y\ Dividing this by each denominator, the quotients 9.re x + y, X — y, 1. Multiplying each quotient by the corresponding numerator and setting ion denominator, we ol xy x^ — xy 1^ 7?.-y^' x'-y^' x'-y^ EXERCISE 44. Reduce to equivalent fractions having the lowest common denominator — 1. — , — • 4. 2x , 9' bx 6' 12a 56' 7 a lO' b 1 2a6» 2 ■' a'6 ' 1 2ac 46c 3a6 „....«, ^ 2 ^ 13 2 3^1 — > — » 2o, -• 3a' 4ax x 126 (5^c ab be ad bd cd ad be « 1 ^ o. ) ALGEBRA. 1 2 ^ a' — a a — 1 .. a: , 1 1 10. ) 1, -> 11. l-\- X X X-^7? X 1 x'-l x'-l N12. ^, ^, ^. VL^2 — 9 2a; + 3 > 13. m, 16. 17. 18. 19. a^b + a6' a^b - a6' 1 5^ 3x - 6 ' 2x 4- 4'x^ 21. ^^ .23. (C + l ^' 12, x^-^x-Q x^-^Ax + x-7? 3 + 3a; 2-2a; on 1 2 3 20. > » 4-a;' 2a; + x' 4-2a; a:-l_ -' «^ , 4, a6 111 ^--^2a;'4-3a;-2 a:' + 3a;+2 2a:' + a;-l PROCESSES WITH FRACTIONS. I. Addition and Subtraction op Fractions. 143. By the Distributive Law (Art. 33), inverting the order of the expressions, a b _ a + b c c c Hence, to add or subtract fractions, Reduce the fractions to their lowest common denominator; FRACTIONS. Add their numerators^ changing the signs of the any fraction preceded by the minus sign ; Set the sum over the common denominator; Reduce the sum to its lowest terms. Ex. 1. i. X x + 1 x + 2 -. _(x+l)(x^2)- 2x(x + 2)+x(x+l^ x{x + l)(x + 2) x'+Sx + 2~2x' -Ax + x' + x x(x+l){x-{-2) 2 x(x + l)(x-t2-) Ex. 2. - a a ,1,1 — 1 a^ — a a a a 1 a-1 1 a'~. a a o' — a' + a' + 1 + a(a-l) a-1 — a* 4- 2a' + a a(a~l) -a' + 2a + l a-1 ' Result. Collect— EXERCISE 45. 1. t X Sx' Ax _2 x 2. 2 3a Aax X ^ Za-b ■ '• 2a a-ib 36 3. 5 2ac 2 1 Sab be a + 26 ^- 2ab 6a-l 6a' 128 ALGEBRA. ^ 2a'x + 3 Sa + x ^^ 1 4ax* 6a; a — b a -f 6 ^ , , 4x-3 3x + 2 ,, ic-1 x + l 8. IH — • 11. 6 5 a; + l x-1 8 6 X 3x' 3a — 46 2a — 6 — c 15a — 4c ^^•""2 3"""^ 12 • 3x-l_x-6^+2 2x-4, 7 4 28 12 2x^-32 xz'-y'z y-Sxz" 2 * Sx'y 2xy' Qxh 3* ..a 6 o. 3x 2x , 10a; 16. -• 21. — —- - + a-b a + b x + 2 x-2 x'-A 17.^ ^. 22.-^-2' 2^' X — 3 x — 4 x' + x x' —X ' x-2 x + 2 ' &C-S Ix-Vl ei'-e (771-1/ m'-l 9-a' 3+a 3-a 2.3 7a; X . ^ X 20. - + 1 -• 25 ic-l x+l 2x-l 4x + 2 4a;*''-l 26.-^ + 2-^^-^^. a;'-l a; + l a;-l 27.-^—^.4- ^ a; + l x + 2 a; + 3 oo a; + 2 a;-3 , 2x4-5 28. _ „ ■ — — ; T "I 29. 2x' + x-l 4x^^-1 2x^ + 3x4-1 b ab ab^ a + b ~ (aT"67 ~ '(a~+W FRACTIONS. 129 30. 2x^-x-l 2x'-hx-3 4x^-h8a; + 3 a;"'* — 2/' 2a; 23/ 2xy 32. ^^~3/ _^ _li^J/_ . ' x-\-2y x^ —Ay^ ^ ~ % g — 6 , o _ g + 6 a^ — h^ a + h a — b _ 2 a; -3 a;' 35. — a; + 4 a;' -4x4-16 a;' + 64 2 2 1 a;' — 3x + 2 a;' — a;-2 x' — 1 6x 7 ' 26 2(x-3)' 3x4-9 4x^-36 38 i-\^^-r-J—i ^1_ll_^L.. ' X 1x4-1 Lx'-x4-l x4-lj i ar'4-1 l_| x»-6x-3 r_l ^_-]) 2 1 2a;^-2 Lx-1 x' + x4-lJ) x-2 Lx x^-3x + 2 U-l /J 144. Changing Signs of Factors. The process of reducing fractions to their lowest comm n denominator is frequently simplified by changing the sign of one or more of the factors of a denominator, at the same time making the necessary change in the sign of the fraction. It is to be remembered from Art. 136 that if the sign of an even number of factors be changed, the sign of the fraction is unchanged ; but if the sign of an odd number of factors be changed, the sign of the fraction is changed. 130 ALGEBRA. Ex. 1. Simplify - — - + a; -hi 1 — x The factors oi x^ — 1 are x -\- \, x — l. Hence, if the sign of the de- nominator, \ — X, be changed, it will become x — \, and be a factor of x^ — 1, But by Art. 136, if the sign of 1 - x be changed, the sign of the fraction in which it occurs must also be changed. Hence, we have x"^ X X _ x^ + x"^ - x ^ x^ + X ^ Zx^ ^ ^^^ a:'* - 1 a; + 1 a; - 1 x^ - 1 x' - 1 Where the differences of three letters occur as factors in the various denominators, it is useful to have some standard order for the letters in the factors. It is customary to reduce the factors so that the alphabetical order of the letters be preserved in each factor, except that the last letter be followed by the first. This is called the cyclic order. Thus, a — bf b — Cf c — a obey the cyclic order. Ex. 2. Simplify (a-6)(c-a) (a-b)(c-b) (c-b)(a-c) Changing c - b to b — c, and a — c to c - a where they occur, we obtain 1 I + 1 (a -b) (c -a) (a - 6) (6 - c) (6 - c) (c - a) ^ b — c — c + a + a — b (a - b){b - c) (c - a) 2«-2« -2 -Sum. {a - 6) (6 - c) (c - a) (a - 6) (6 - c) EXERCISE 46. Collect- 3^ +^+ 1 x'-l 1-x 1 + x 2. 2a^+_L,+ 2 a^ — b^ a + b b — a FRACTIONS. 131 ' x^ — 4y^ 2y -\- X '^y — x x-l 1 + a; 1— ic* 6 5 3a 4 -13a ' l + 2a l-2a 4a^-l' ' ^ — y" x-\-y y — x 8-8a 4a + 4. 8a'-8^ 9j + ^_+ 5. 1 3 a; x — 1 1—x^ x -{-1 x + x* 10. 1 1 1 (x-2)(3-a:) 10-7a;4-a;'^ (5-x)(aj--3) 2 3 4 11. z ;t-7; ^ - z ::rzz tt + (a - 3) (6 - 2) (a - 2) (2 - 6) (a - 2) (3 - a) + ^ (a - 3) (2 - 6) 5a 5a a l-^* T:^ ITT I 13. 6a-18 27-3a' 4a + 12 2b + a _ 26 — g _ 46a; — 2a' x + a a — x x^ — a^ ,, x + 1 2x~l , 2 14. — — -— + 6x-6 12a; + 12 3 - 3x' 12a; X , X . 1 — a; , 15x + 3 15. — + — 1 r 16.* 2a:-6 3a; + 9 6x 18a;-2ar» 7?-x-^ a;' + 4x + 3 15a; a:' + 5a; + 6 a;»-4a; + 3 O-a;** * Eeduce before adding. 132 ALGEBRA. 4a' -6' + a'-2a5 4-36« . a'-b' 2a' -Sab -2b' a' 1-x' , x'-O x'-4x + 3 3a6 + 26* 2x /T^f'^ 3(a; + 3) 19. ^^ + ^ ■ 5(x-3)'^ 5x^-45 X 4 — X 5a; + 6 20. 8a;-x'-15 b Ix 10 (a — 6) (a ~ c) + (6-c)(6-a) 6' + (c — a) (c — 6) 22. / {(x-b^{a-c) (6-c)(6-a) (c-a)(c-6) y 4-z z + a; x + i/ (2;-2/)(a; — z) {y — z){y — x) (z — x) (z — y) yz , zx _^ xy (x-y)(x — z) (y-z)(y-x) (z-x)(z-y) 1+Z . 1+m , 1+n '^ + + m) (^ — n) (m — n) (m — Z) (n — Q (n — m) n. Multiplication of Fractions. (I c 145. To find the product of any two fractions, - and - ^ we b a may proceed thus : ^X-^ = a^bXc-^d = aXc-^b^d (Art. 33) b a _ aXc , , aXc b bXd Hence, to multiply fractions. Multiply the numerators together for a new numerator^ and mvl- tiply the denominators together for a new denominator, canceling factors that are common to the two products. This reduces the multiplication of fractions to the multi- FRACTIONS. 133 plication of integral expressions, and enables us to use again 3ur knowledge of the latter process. ' M 126V 6a^2/ ^ 2 X 10 X 4a»6Vy» _ 2y*_ 6 X 12 X 6a*6V2/ ~ 9a ' Ex.2. ^±^X-^-^X ^"' cc* + xy'^ {x + J/)' = ^ + ^ X (a; + 2/)(a^-2/) w 4^ in. Division of Fractions. 146. To divide any fraction, - > by any other fraction, - > we d may proceed thus : Let a;=-) y = - ,* . bx = a, dy = c b d X ad a ^ , d , . X a c y be c y b d Hence, to divide one fraction by another, Invert the divisor and proceed as in multiplieation. This reduces division of fractions to the already-learned process of multiplication of fractions. = c bx _ a ' ' dy a b'^ c _a d 'd~b^c 134 ^ ALGEBRA, Ex. 1. Divide — - by x' — ^a' x — 2a a^ - Aa" 2a X ax -\- 2a* 2a ax + 2a' x — 2a ^ (x + 2a) (x - 2a) a(x + 2a) = 2, Quotient. Ex 2 ^-^ X ^'''~^^' • ^^~^^' ' xCx + l) aj' + x + l ' (x + 1)' ^ (x-l)(x' + x + l) (x'-l)(a:'-l) (x + l)' x(ix + l) x'-hx-^l (x-iy (x + iy ^ ,, = -^^ ^ J Result, z EXERCISE 47. Simplify — * 14a'c ISay*' * 2x + 2 ic'-xy' * 13z» ' 39z** ' 4a'- 1 ' 2a + 1 126 35a6 _5_^ ^ a;' - 9 . x-3 * 25a 48 76'' ' x' i- x ' x'-l 9a^ 28ax^ _ 21a^ (a-l)» x + l ' Sc'x 156V ' 106c»* 'a(a; + l)' (a-1)' .n ^ Af\Ji Q/*.2 1 492/" 40ar^ 9a;' -1 12a; -18 15x 2x(a;+l ) ^^ 2a;'-a;-l w 4a;'-l ' 2a;(2x-l) 5ar^ '2x' + x-l x^-l 13 Q^y — ax^y . a'y — 2aa;3/4-a;'2/ ^ aV -f a'a;'2/ a' -f ay FRACTIONS, 136 14. (a + 1)' (a + 1)' _ Z7? + x-2 ^ ear' 16. 4a:'-42;-3 2x'-a;-3 2a;' — a; — 6 , 2a:^ + x-3 2x' + a;-l * 2x' + 3x-2 2a;- 2 17. a;+ X \ x-lj ar'-fl 6x2/ ^ + y (p^—yy 2a^ * 4(a + 6)' 9(a-6)' " 8(a + 6)* 21 a:' + 2a;-3 ,, a:' + 2x-15 . ar» + Sx * * a:" 4- a; -12 a;' + 2a;-3 ' a:' + 4a;'* 22 6x^-4x3/' 30a: + 203/ ^V ' Adx" ~ 20y' 4xy x + y 23 ^^'-5a:-4 6a;' + a;-2 2a;' + 5a;-12 * 2x' + 7a;-4 4x'-4a;-3 9a;'-6a;-8 -^•)(-i)x('-5^)- 26. . ^ + ^ -X ^-^ X a:* + a;'3/ -f xi/' x^ — xy -{- y'^ \ a;-3// 27 a:^-(a-i y (g + xy-l . a + x-l * a'-(a; + l)' l-(a-x)' * a-x-1 23^2-6-a^^-6'-46-4_^^- 6-2-a a'-h6' + 2a6-4 6'-a' + 4a-4 136 ALGEBRA. 12x'-xy-20y' 27^ -]- bxy-12y' Gar* + 2Bx 'y + Ibxy^ ^ ' 12x'-Sxy-15y' 3x' -^ bxy - 12y' ix" + 21xy + 20y' ' \ab be acj \b c a J a'bV \a^ TT ax / L ax x J rw, -\-2n m — 2n~| ^ r m + 2n _ m — 2n ~| Lm — 2n m + 2nJ Lm — 2n m + 2nJ IV. Reduction of Complex Fractions. 147. A Complex Fraction is one having a fraction in its numerator or in its denominator, or in both. In simplifying any complex fraction it is important to write down the entire fraction at each step of the process. Ex. 1. Simplify a:X-^— = -^-, Remit. y y If the numerator and denominator of the complex fraction each contain fractions, the simplification is often effected most readily by multiplying both the numerator and denominator by the lowest common denominator of the fractions contained in them. Ex. 2. Simplify " V ^ . y % X FRACTIONS. 137 Multiply both numerator and denominator by xyz^ and obtain « z -f x'if H- yz^ Ex. 3. Simplify .+-i cc-2 A fraction of this form is called a continued fraction. In simplifying a continued fraction begin at the bottom, and reduce by alternate conversions of a mixed quantity into an improper fraction, and divisions of a numerator by a frac- tional denominator. Thus, 1 1_ 1 X H X -\ X -+ ._ 3 a; — 5 a;-5 a;-2 x-2 1 x-5 Simplify — x'-Ax-2 x' -4x-2 x-5 EXERCISE 48. ) Result, 4 1 . 1 X X X _ a; 1. 3. T' ^' a + 1 X 1 . , 1 14-- 1-- 1 + 2 re a-1 2-i ^-2c£ 6.1+ ^ « c 2. ^- 4.-^^ 1-- , 1 2cd a 138 ALGEBRA. 14. 7. 8. U yJ . 9. o =t-l -f 10. -lil- o — 1 11. 3 ^ " 1 + a 4 |a-l 12. a — 1 a 1 1 a — 1 a 1Q 13 2 X x' x' I-' ^-1+^ 15. 1 l + a 1-a 16. X 1 1 ~* l + a; x X \—x 1 + X X 17. 2^ + l_^ 3/ a; 18. e{a^hy-a^h^ a'bV io a a: ax ly. ^_a_2 1 a a; a aa; 20. -2). 2x-l ^ FRACTIONS. 22. 2a- 1 139 a-1 1 + a 23. 24. 25. 26. 27. 28. 29. l-jjah-cdf (ab-iy-c'd' (cd + iy-a'b' (ab + cdy-l 1 ]_ ^i^+ 2bc )• a 6 + c 1 a; — « + 1 X x + 2 l+ar« a: + : a;-2 l-a:» x^ 1 + 1 + 1-a; 1- 1 + x xf + T^y ' x^ -\r o^y -\- 7^y* (x^-yy M j_/i+iY i-/i-iY /i+i-^y a" \b cl b' \a cf \a c b) b' \a c] e \a b) [a c) b* 1-2 1+2 l-2a; l+2a; 4a-^ + a^)-i 14- 2a; ii^ + x + ix')-^ \-2x 140 ALGEBRA. EXERCISE 49. REVIEW. Reduce to their simplest form — 1 o^-^x + 2 . g 1 - iri - 3(1 - x)^ '■---- • l-|[l-2(l-a;)]' 7 2(^_li)_i(^_zi). ' 3(2; - I) ^{x + I) 8.3- 1 2x 3. x' + a;2- -3a; -2 x' - 2x'^ + 4a; - 3 x"- -5x-2 + 13a; - 9 Sx -4 3a; -2 2x + d 2a; + 1 Xj- -2n 2ri + a; . 3..- 3x a; + 1 4 •«- ~ *^" ' 4- "^'- ■ -^ . 9 2 — a; _ a; + 2 ^ 6a; . ' a; + 2n 2n - X * 1 - 2a; 2a; + 1 4a;'' - 1 |(|-a;^ + ia;-2) . ^(^ + 2;^ - V) Kfa;^ + fa; - 1) 10. ^-^f- — ^. »^2 2 ^^ x^ -^ xy — 2y^ 11 1 + 2 3 , 4a; - 3 _ a; - 1 a; - 2 a; - 3 (a;^ - a;) (a; - 2) a-- a + - a-^ a6 + - a a a' a 13 L/l--L\ +_A_ _6r 2(a-6) _:^) 2 U a - 6 / a + 6 2 \ a^ - 6^ / 1.1 1 14. 15. 9a;2 + 9a; + 2 1 - 9a;2 4 - 9a;2 2 - 9a; + 9a? 3 ^ 2_ ^ 1 a;2 - 3a; + 2 (a; - 1) (3 - x) (2 - x) {x - 3) 16. 2a; + y _ 1 ^_ ^ a^' . aJ + 2/ y — X y^ — x^ 17. (« + ? _ 2) (« + ^ + 2U (^ - ^y. \a; a / \x a I \x a} 18 i 1 + a; _ 1 + a;^ ]. ^ / L± ^' _ L±^ 1. FRACTIONS. 141 19. 20. 21. 22. 23. 24. 25. \ a / ^ ax I x-4- X x-2 -2 x-4 2a + 3 a; — 5 3a + 2 2a2 + a - 1 3a' + a - 2 6 (a + 1)2 2-7a + 6a2 (I -!)(!-)• 1 - ^ a;2 9 (^-f)(^^a) i'-t) o a 1 + 8a:' 1 9l 1 - 27a:' 2a; 1 + 2a; 1 + 3a; 3a; l-2a; 5x^ + 4 XX X x^ + 1 1 + 3a; 2+i 1 + ^ '>•« 27. 28. / ^-1 + 2/ \ /^ _ 2 + ^K ( ^' \ x\—^ ^— I \y^ xV \xy + y^) \ a^ - 2x^y + xy^ / \ a~b) i_6 + 6! 6> 1 + 6 + 6; 1.^^ 142 ALGEBRA. 1 + 29. 30. 31. 32. 33. 34. 35. x" ^-h X X + ('-9 l) + i \{x X — X + K^ + i) - fa; 1 . o _, „ . 7 a; + 1 + 3 X + X + 1 + 1 x-1 + X + S 1 + a' - a;' + 6a + 9 ^ a^ - x^ + Gx - 9 x-1 x^ — Sx + ax X- — a' — 6a M^a - i[2a 2a 3i - Y + |{f« - |[f 9 9 - a» - a;2 + 2aa; a^ - 3a |(3a + 2(a - 5))] + 2 + ^a}. i(4a-4(5a-3))]}. ax 1 + 1 + 1 + x 1 - 3a; 1 -3 1 + x i-Sx 1 + 1+x l-3x 36. 2 + ^ 1 + X 1 -3a; 1 + I 4 + 2 + 3 - 2a a-1 37. | 1 4- a n 1-a 1 + aJ 38. 39. 40. 1 + 1 n 1-1 a; / g + be y \a — 6c/ X a + be a -be -1 f — ^^^^1 1 x_ 1 + a J ^-1 + 1 ( g + be y _ g -f 6c , ^ g — 6c/ a — be a + l a-1 + 1 m)'^mr-~{-y CHAPTER XI. FRACTIONAL AND LITERAL EQUATIONS. 148. General Method of Solution. If an equation con- tain fractions, it is necessary to clear the equation of fractions before transposing terms and solving by the method given in Chapter VI. Ex. 1. Solve X -\ = 6 H 6 4 Multiply both members of the equation by 12, the L. C. D. of the fractions. 12a; + 2(x + 2) = 60 + S{x - 4) 12a; + 2a; + 4 = 60 + 3a; - 12 12a; + 2a; - 3a; = 60 - 4 - 12 11a; = 44 a; = 4, Root If a fraction be preceded by a minus sign, it is important to remember that the sign of each term of the numerator must be changed on clearing the equation of fractions. ^ ^ ^, a; + l 2x-5 lla; + 5 x-1^ Ex. 2. Solve — = — -— — • 2 5 10 3 Multiplying by 30, 15a; + 15 - 12a; + 30 = 33a; + 15 - 10a; + 130 16a; - 12a; - 33a; + 10a; = - 15 - 30 + 15 + 130 - 20a; = 100 « =- - 5, Boot. lis 144 ALGEBRA, Ex. 3. Solve --^ + ^^ - ^^ =0. l-\-x 1—x 1 — ic' Multiplying by the L. C. D., 1 - x^, 4(1 - a:) + (a; + 1)2 - a;2 + 3 = 4-4a; + a;2 + 2a; + l-a;2 + 3 = - 2a; = — 8 X = 4, -Koo^ Hence, in general, Clear the equation of fractions by multiplying each term hy the L. C. D. of all the fractions ; For a fraction preceded hy the minus sign the sign of each term of the numerator must be changed when the denominator is re- moved ; Complete the solution by the methods of Chapter VI. 149. Equations involving Decimal Fractions. If one or more of the coefficients of an equation is a decimal fraction, we may solve the equation by expressing the decimal frac- tions as common fractions. If all the coefficients are decimals or whole numbers, it is simpler to solve directly in decimals. Ex. Solve .3x-.14a; = .012x + .592. .32; - .14a; - .012a; = .592 .148a; = .592 a; = 4, Boot. EXERCISE 50. Solve — 3^ 5x_2^ 11 Sx 2x + 7 _ 4x-\-5 ^ *4 6~3 2* '5 3 "15* X 3a; 7a; _ 34 K^t^_,^_*|4'^ ' Z~J y~15* ' 2 3 5~4 "6* 2x-3 :MJ. ^ 5x + 2 i__?4.A = A_?:?. ' 4 6 ~ 12 ' 2a; cc 3a; ~ 4* 24* FRACTIONAL AND LITERAL EQUATIONS. 140 7 3 2- ■«-!^|- 16. f(x-l)=K^-2). "•M-- 17. i(l + a;)=i(2x + l). 12. - + 2 = 0. X 18. 3(|a;-l)(ia; + |)=a;», 10 ^-^ 2a; 4- 3 6-5a; 12 ^^- 3 6 -^- x-Z 6 -1^'^ + S^ 3 24 oi 3x-l a; + l 4a: + l 3(a:-l) ^ ^'' 7 6 21-4 ^- 22. 10--?^^-3a;-4i = 0. 4 2a: + 5 ^+li ,^_5£:zM 1 _^ cc — 1 23;-f3 , a; a; ^- ^"^^ + ^) + ^ - K2^ + 6) = ^^ ,. " ic + S a; + 7 , a; 4- 1 2a;-5 a; + 22 ^ , 27. — ' h = —• 7 6 2 10 70 10 146 ALGEBRA. 28 2a; + 3 5a; 4- 1 3a;+l ^1 6a;-l a;-f9 5 6 4 3 15 30 ' 29. |(5a; + 2) - |(7a; - 2) + i(3a; - 2) - a; - -• 80 y + 6 2y-18 2i/ + 3 ^ 3y + 4 ■ 11 3 4 ^ 12 " 32.0.5.-0.4. = 0.3. "- 1-5--1-6 3.5.-2.4 33. 1.5a; — 5 = a;. 34. 1.25a; + 1.9 = -1.125a;, 35. 0.6a; - 1.5 = 0.2 - 0.15a;. 39. , 2 + -^==^-M 1.2 3 9 1.2 0.8 9Q 3.2a; -3.4 0.6a; + 4 4.5 2.5 39. .0032a;- 0.1a; - 1 1.005 .0125a; 40, .0001 _ 1.00005 80a; 200.01 ''■ OA 42. a;-l d; + l 3 5* 43. 5 2x-l 8 3a; + 1 44. 2a; + 5 5a; + 3 _ 2a; + l 5a; +2 45. 2a;-5 2a; + 2 3a;-5 3a;-3 46. 6a;-5 3a:-3 8a; -7 4a; + 4 0.6a; + .045 5a; - 1.78 ^ ^^^5 X a;' - 5a; _ 2 ' 3 3a;-7~3' 48.-A_+-l-^-^^ 1-a; 1 + a; 1 — a;' 3-a; 34-a; 9- a;' 2a; 4-1 10 2a;-l 50. 2a;- 1 4a;' -1 2a; + 1 51.^+ ' ' x + 1 x-1 x + 2 62. FRACTIONAL AND LITERAL EQUATIONS, 147 x-1 ^ 3 l-3a; ar'-S x-2 x^-\-2x-^4' ^^ a^-x-^1 _ x' + aj + l 63. = 2x 64. x-1 x+1 a;-3 jx' + l ^ x + S 2x4-6 x^-d Sx-d 55. ^ + -^-..^ = .^-1- 66. x-1 x + 1 2x-2 3a;H-3 l-x» x + l cc' + T ^ 2 x-1 2x-3 4x^-9 2x + 3 6-4x* 67. — 2 — X = 3x-6 6x + 12 2x^-8 4 2 3 68. — + r : r + 3x' + 9x + 6 x'-3x x^-x-6 7?-2x-Z 4 3x' + 6x x + 1 ' x' + x-6 3x-x'-2 x=^ + 2x 60. 6x + 6 2x + 1 2x 2x^ + 5x + 3 2x^-x-l x'-f-2x 150. Special Methods. The work of solving an equation may frequently be diminished by using some special method or device adapted to the peculiarities of the given equation. 1st Special Method. If the denominators of some fractions are monomials, and of some are polynomials, it is best to make two steps of the process of clearing the equation of fractions, the first step being to remove the monomial denominators and simplify as far as possible before proceeding to the second step, which is to remove the remaining polynomial denomi- nators. 148 ALGEBRA. ^ , ^, 2a; + 81 13a;- 2 , a; 7x a; + 16 Ex. 1. Solve -^ h - = 9 17a; -32 3 12 36 Multiplying by 36, the L. C. D. of the monomial denominators, 8a; + 34 - 36(13a; - 2) ^ -^^^ = 21a; - a; - 16. 17x - 32 Transposing all terms except the fraction to right-hand side, 36(13a; - 2) ,_ 17a; - 32 Dividing by -2, M^^:^ = 25 234a; - 36 = 425a; - 800 191a; = 764 a; = 4, Root. 2d Special Method. Before clearing an equation of frac^ tions it is often best to combine some of the fractions into a single fraction. x — 1 x — 2 x — Z x~4: Ex. 2. Solve a; — 2 a; — 3 a; — 4 x — 5 In this equation it is best to combine the fractions in the left-hand mem- ber, and those in the right-hand member, each into a single fraction, before clearing of fractions. We obtain {x - 2) (a; - 3) {x - 4) (a; - 6) Clearing and solving, x ^ -' ,2t Solve — EXERCISE 51, 3a; -1 4x _ a; + 5 6 3a; + 2 ~ 2 ' 3-2a; x l-6a; ^ 2-3g ' 4 6 15-7a; 9 1.2. 13. FRACTIONAL AND LtTEttAL EQUATIONS. 3 2x-l X 14d 3. 21- 2a: + 4 6x + 13 2a; + 5 23 12 5 6 2ia;-3 4x-36 a; + 11 5-^a; 8 3a;-l 4x-7 7. 30 6a;-7 11a; + 5 a; + 4 15 a; + l 15 2a; ■ lla; + 5 16 3 7x 0. 15 + 8. 3 + 7a; + 11 3x-2| 7 10. 11. I 9 5 -4a; ^-n 2_ 12 |a; + ll 12a; -11 2a; - 1 ^ 199 30 ~ 'lO ' 3x 4a; + 9 12 -i 2x-li 60 4-a; 5 24 4 2a; + 3i 6 3x-l + iO 2r2x lf 3a;-l 3L9 2I 6 /2ix + l\ 3 Vila; + 6/ 2 11 O.la;- 100.01 ^ 10 1.001a; + .0002 .05 1 - 1.4a; _ 0.7(a; + 1) 2a;-5 7a; + 8 0.2a; = l-x. 19a; + 3 14. -1.6 54 2a; -0.2 = 0. 16. 17. 18. 19. 2 + a; 1.2a; 1.5 a; -0.25 0.125 1 l-0.5a; 1.5 0.4x+l 15. 0.4 2a;-3 0.3a; - 0.4 0.4x + 1 4a; + 0.5 "^ 2 0.4X-0.9 .06*-. 07 0.2a; -0.2 0.4a; - 1 5a; 0.5 20 .05 3x 1 0.2 1 a;-2 a;-3 a;-l a;-3 x-2 x-4 x-4 x-5 x-5 x-7 x-Q x-S 150 ALGEBRA, 20 x — A 8 a;-9 a:-5 x~Q 21.—^— 2 2 ^ 3x-2 2a;-3 2a; + 3 3a; + 2 ^„^ 2a; + l , 2a:4-9 2a; + 3 2:c + 7 x + 1 . a; + 5 a; + 2 a; + 4 4x-17 10a;-13 8a; - 30 5a;-4 _Q a;-4 2a;-3 2a;-7 a;-l ~ ' 151. Literal Equations are equations in which some or all of the known quantities are denoted by letters ; as, a, 6, c . . ., or m, n, ^ . . . The methods used in solving literal equations are the sama as those employed in numerical equations. Ex. 1. Solve a{x —a) = b(x -b). ax — a^ -= bx — b' ax -bx = a^ — b^ {a- b)x = a^-b^ X = a + b. Ex.2. Solve ^I1* = ^±A. x~c x-\-2c {a - 6) (a; + 2c) = (« + 6) (a; - c) ax + 2ac —bx — 2bc = ax + bx - ac — bo - 2bx = - Sac + be 26 EXERCISE 52. Solve — 1. 3x + 2a = a; + 8a. ^, bax — c = az — bc. 2. 9aa; — 36 = 2ax + 46. 4. aa; + 6 = 5a; + 26. * Transpose the second and third fractions. FRACTIONAL AND LITERAL EQUATIONS. 161 14. 5. Scx = a—(2b — a f ex). 6. 5x — 2ax^=S~b. ^^' a — 2x ~ a-\- 2x 7. 2ax~Sb = cx + 2d. 8. (x4-a)(x-6)=xl 9. ab(x + 1) = a" -j- b'x. 10. (a;-l)(a;-2) = (a;-a)'. ■'•i(!-')=ie-)- 12. (a — 5>= «'-(« + 5)a;. " 2a;-a a;-a" a a a 6 c 19. 2i:^* + *£:!-« + E^ = o. a6 be ac on ^^ r ^^ 3a'x + 6' 21. 22. 3a + 5 3a-6 9a,^-6' a; ^ _ 1 a; a a —b a + b b a^ — x b^ — x & — X _a^ _ 6^ e a b e a 23 5a'-7a; a5' + lOx ^ 10c' + 3a; 5(a-c) 6* 3a6 5ac ~ 66c 36 5c a + ■ X 1-? — a a 24. — X — a 1 X a a 1 X b 25. ■==■ ■ b c 26. a-\-bx a- hx 27. 36 + 2ax 6-2ax x — 1 X a _ o — 1 ^ g + 1 g 152 ALGEBRA. a-1 1 a; + l 1 + ^ a 31. a+ -^ = 1 + -^ 29. - = '— 1 + - 1-i a; f. a; a a I ^~T a 152. Problems involving Simple Equations containing Fractions. Ex. 1. A has 8| dollars more than | as much as B has, and together they have 56J dollars. How many has each? Let X = the number of dollars B has. Then fa; + 8| = " " A has, and f a; + 8f = " " both have. 5£ 35 _ 225. •'344* 20a: + 105 =-- 675 Hence, x = 28|, the number of dollars B has, and fa; + 8| = 27f, " " A has. Ex. 2. Divide the number 100 into two such parts that the eighth of the larger part exceeds the eleventh of the less part by 3. Let a; = the larger part. Then 100 — a; = the less part. - = the eighth of the larger. 8 ^^^-^ = the eleventh of the less, 11 X 8 ' 100 - a; o ~ 11 ^ 11a;- - 800 + 8a; = 264 19a; = 1064 X = 56, the greater part 100 - a; = 44, the less part. FRACTIONAL AND LITERAL EQUATIONS. 153 EXERCISE 53. 1. The sum of the third and fourth parts of a number is 14. Find the number. 2. Find that number whose fifth and sixth parts together are 16^. 3. What is that number whose third, fourth, and fifth parts are together 13 less than the number itself? 4. The difierence between the seventh and third parts of a number is 5 more than one ninth of the number. Find it. 5. There are two consecutive numbers such that one sev- enth of the greater exceeds one ninth of the less by one. Find them. 6. There are three consecutive numbers, such that if the first be divided by 6, the next by 7, and the largest by 8, the sum of the three quotients is \ more than ij- of the sum of the three numbers. Find them. 7. The difference of two numbers is 9, and -j^ of the less, increased by 3, is ^ of the. greater. Find the numbers. 8. A man left half his property to his wife, one fifth to his children, a twelfth to a friend, and the remainder, $2600, to a hospital. How much property had he ? 9. In a certain orchard there are apple, pear, and cherry trees : ten less than one half are apple, twelve more than one third are pear, and four more than an eighth are cherry trees. How many trees are there? 10. Find three consecutive numbers, such that if they are divided by 2, 3, and 4 respectively, the sum of the quotients will be the next higher number. 11. Divide $130 among A, B, and C, so that A receives J as much as B, and C, f as much as A and B together. 12. The sum of two numbers is 97, and if the greater be divided by the less, the quotient is 5 and the remainder L Find the numbers. Hint. The divisor multiplied by the quotient is equal to the dividend diminished by the remainder. 154 ALGEBRA. 13. Divide the number 100 into two such parts that the greater part will contain the less 3 times with a remainder of 16. 14. The difference between two numbers is 40, and the less is contained in the greater 3 times with a remainder of 12. Find the numbers. 15. Five years hence a boy will be f as old as he was 3 years ago. How old is he now? 16. A's age is f of B's age, and in 7 years he will be f as old as B. How old is each? 17. Eight years ago a father was 2>\ times as old as his son, and 1 year hence he will be 2^ times as old. How old is each now? 18. A tax of S5000 was paid by four men, A, B, C, and D, A paying f as much as B, C half as much as A and B to- gether, and D $400 less than A and B together. How much did each pay? 19. A man sold 4 acres more than f of his farm, and had 6 acres less than f of it left. How many acres had he ? 20. Find two consecutive numbers, such that f of the less exceeds f of the greater by \ of the greater. 21. If a boy can do a pieoe of work in 15 da3^s which a man can do in 9 days, how long would it take both working together ? Solution. Let Then But X = number of days both require. - = the part they both can do in 1 day. And xV + i Hence, xV + \ the boy the man both Or, X = 5f. Therefore, they both together require 5f days. These are called common-time examples. FRACTIONAL AND LITERAL EQUATIONS. 155 22. A can accomplish a piece of work in 6 days, and B can do the same in 8 days. How long will it take them together to do the work ? 23. A can spade a garden in 3 days, B in 4 days, and C in 6 days. How long will they require working together? 24. A and B can together mow a field in 4 days, but A alone could do it in 12 days. In how many days can B mow it? 25. If A, B, and C can together do a certain amount of work in 5^ days, which B alone could do in 24 days, or C in 16 days, how long would A require? 26. A and B together can dig a certain ditch in 1^ days ; A and C in 2 days, but A alone in 3 days. How many days would it take B and C together to dig it? 27. A and B in b\ days accomplish a piece of work which A and C can do in 6 days or B and C, in 7^ days. If they all work together, how many days will they require to do the same work ? 28. Two inflowing pipes ' can fill a cistern in 27 and 54 minutes respectively, and an outflowing pipe can empty it in 36 minutes. All pipes are open and the cistern is empty ; in how many minutes will it be full ? Hint. Since emptying is the opposite of filling, we may consider that a pipe which empties ^^ of a cistern in a minute will fill — 3*5 of it each minute. 29. A tank has four pipes attached, two filling and two emptying. The first two can fill it in 40 and 64 minutes respectively, and the other two can empty it in 48 and 72 minutes respectively. Tf the tank is empty and the pipes all open, in how many minutes will it be full? 30. A man labors 8 days upon a piece of work which he could complete in 4 more days, but he is then joined by a boy, and they finish it in 2|- days. In how many days could the boy do the entire task ? 31. At what time between 3 and 4 o'clock are the hands of a watch pointing in opposite directions ? 166 ALGEBRA. Solution. At 3 o'clock the minute-hand is 15 minute-spaces behind the hour-hand, and finally is 30 spaces in advance : therefore the minute-hand moves over 45 spaces more than the hour-hand. Let X = the number of spaces the minute-hand moves. Then a; -45= " " '' « '^ hour-hand " But the minute-hand moves 12 times as fast as the hour-hand ; hence, x = 12(a; - 45). Solving, x = 49xV. Thus the required time is 49x^r ^^^- P^st 3. 32. When are the hands of a clock pointing in opposite directions between 4 and 5? Between 1 and 2? 33. What is the time when the hands of a clock are together between 6 and 7 ? Between 10 and 11 ? 34. At what instants are the hands of a watch at right angles between 4 and 5 o'clock? Between 7 and 8? 35. A courier travels 5 miles an hour for 6 hours, when another follows him at the rate of 7 miles an hour. In how many hours will the second overtake the first ? Solution. Let x = the number of hours the second travels. Then a: + 6= " " '' " " first " b{x + 6) = " " " miles " " " 7x = " " " " " second " They travel equal distances, hence, 7a; = 5(a; + 6). Solving, a; = 15. Therefore the second courier requires 15 hours. 36. A courier who travels 5^ miles an hour was followed after 8 hours by another, who went 7^ miles an hour. In how many hours will the second overtake the first ? 37. A messenger rides for 6 hours at the rate of 13 miles in 2 hours, when he is followed by another at the rate of 8 miles an hour. How many miles will each travel before the first is overtaken ? 38. A train running 40 miles an hour left a station 45 min- utes before a second train running 45 miles an hour. In how many hours will the second train round the first ? FRACTIONAL AND LITERAL EQUATIONS. 157 39. An express train which runs 65 miles an hour leaves a station 4 hours after a freight traveling 11 miles an hour. How many miles from the station will the express round the freight? 40. A boy starts on a bicycle 2^ hours after his sister, who rode 8 miles an hour, and overtook her in 5 hours. How fast did he ride ? 41. A gentleman has 10 hours at his disposal. He walks out at the rate of S^ miles an hour and rides back 4J- miles an hour. How far may he go? 42. A and B start out at the same time from P and Q, re- spectively, 82 miles apart. A walked 7 miles in 2 hours, and B 10 miles in 3 hours. How far and how long did each walk before coming together, if they walked toward each otlier ? If A walked toward Q, and B in the same direction from Q? 43. A hare takes 7 leaps while a dog takes 5, and 5 of the dog's leaps are equal to 8 of the hare's. The hare has a start of 50 of her own leaps. How many leaps will the dog take to catch her? Solution. Let x = the number of leaps the dog takes. Then ix = '' " " hare takes tn same (tm«. Also, let w = " " feet in 1 leap of the hare. Then |n = " " " " dog. Hence a; x — = -^ = the number of feet in the whole distance. 5 5 And (^ + 50 ) X n = ^^ + 50n = the number of feet in the whole ^^ ^ ^ distance. Therefore §!^ = ZM + 50n ; or, a; = 250. 5 5 Thus the dog will take 250 leaps. 44. A hare is 50 leaps in advance of a hound, and takes 5 leaps to the hound's 3, but 2 of her leaps are equal to 1 of his. How many leaps must each take before the hare is caught? 158 ALGEBRA. 45. A greyhound pursues a fox which has a start of 60 leaps, and makes 3 leaps while the greyhound makes 2. Three of the dog's leaps are equivalent to 7 of the fox's. How many leaps does each take before the hound catches the fox? 46. A has a certain sum of money, from which he gives B $3 and \ of what remains ; he then gives C $6 and \ of what remains, and finds that he has given away half his money. How many dollars had A ? 47. A colonel in arranging his troops in a solid square found he required 51 men to complete it ; but on making each side contain one man less, had 32 men more than the square required. How many men had he? Hint. Let x = the number of men in each side of his first square. 48. A regiment is arranged in the form of a hollow square 15 men deep, containing 1800 men. How many men on the outer side of the square? 49. An officer arranged his troops in a rectangle, with 3 men more on a side than on an end ; if he should form a square each side of which is equal to one more than the width of the rectangle, he would have 44 men left. How many men had he? 50. If a bushel of oats is worth 40 cents and a bushel of corn is worth 55 cents, how many bushels of each grain must a miller use to produce a mixture of 100 bushels worth 48 cents a bushel? 61. A man has $5050 invested, some at 4%, and some at 5%. How much has he at each rate if the annual income is $220? 52. Divide the number 54 into 4 parts, such that the first increased by 2, the second diminished by 2, the third multi- plied by 2, and the fourth divided by 2, will all produce equal results. 53. Two laborers are hired at $3 and $4 a day each ; together they did 35 days' work, and each received the same sum. How many days was ^ach employed? FRACTIONAL AND LITERAL EQUATIONS. 159 64. Divide 180 into two such parts that if the less be sub- tracted from f of the greater, the remainder is ^ the difference of the two parts. 55. A boy bought some apples at the rate of 5 for 2 cents : he sold J of them at the rate of 2 for a cent, and the rest at f of a cent apiece ; he made 5 cents. How many apples did he have? 56. A lady in reading a book read the first day half the pages and 1 more ; the second day half the remainder and 1 more ; the third day half the rest of it and 1 page more, and still had 40 pages to read. How many pages were there in the book ? 57. Find two numbers whose difference is 12, such that if ^ the less be added to ^ the greater, the sum shall be equal to J the greater diminished by \ the less. 58. A man buys two pieces of cloth, one of which contains 3 yards more than the other. For the less piece he pays at the rate of $5 for 3 yards; for the other $1.50 per yard; he sells the whole at the rate X){ $9 for 5 yards and gains $36. How many yards were there in the less piece? 59. A and B together can do a piece of work in 2f days;' A and C in 2f days ; and B and C in 2^\ dajrs. How many days will each require to do the work alone ? 60. At what instants are the hands of a watch at right angles between 10 and 11 o'clock? 61. A fox is 100 leaps ahead of a dog in pursuit. The fox makes 3 leaps in the same time that the dog makes 2, but 3 of the dog are equal to 5 of the fox. How many leaps will each take before the fox is caught ? 62. A does y of a piece of work in 10 days, when he»receives the aid of B and C. They finish it in 3 days. If B could do the entire task in 30 days, in what time could C do it alone ? 63. A man has a hours at his disposal. How far may he ride in a coach which travels b miles an hour, and return home in time, walking c miles an hour? 64. Separate a into two parts such that the greater divided 160 ALGEBRA. by the less may give b for a quotient and c for a remainder. Prove your result. 65. The fore wheel of a carriage is a feet in circumference and the hind wheel is b feet. What is the distance passed over when the fore wheel has made c revolutions more than the hind wheel ? 158. Problems in Interest. The power of algebra as com- pared with arithmetic is well illustrated by the algebraic treatment of questions relating to interest. Let p = the number of dollars in the principal, r = rate of interest expressed decimally, t = time expressed in years, i = the interest* on the principal for the given time and rate, a == amount, i. e. the sum of the principal and interest. 154. I. Problems involving" Interest, Principal, Rate, Time. . The principal multiplied by the rate, that is, pr, gives the interest for one year, and prt gives the interest for t years. . • . i =prt (1) If interest, rate, and time be given, to find the principal, in equation (1) t, r, t, represent known quantities, and p the un- known quantity. Solving (1) for jp, 2' = 4- • • (2) rt In like manner, if interest, principal, and time are given, to find rate, solve (1) with reference to r, r = 4 (^) pt So also if interest, principal, and rate are given, t = - (4) pr FRACTIONAL AND LITERAL EQUATIONS. 161 Hence, in general, if any three of the four quantities, I, p, r, i, are given, the remaining quantity may be found by substituting for the three given quantities in equation (1), and solving the equation for the remaining unknown quan- tity, or by substitution in one of the formulas (2), (3), (4). Ex. 1. At what rate will $50 produce $20 interest in 6 years and 8 months ? We have given p = $50 i = 120 t = 6f years, to find r. Using formula (3), r = 20 _ 20 X 3 _ _3 ^ ^ 50 X 6f 50 X 20 50 r = 6%, Rale. 155. n. Problems involving Amount, Principal, Bate, and Time. ^ Since amount = principal + interest, a=p-\-prt . ,-',- . * i . . .(5) Any three of the four quantities, a, p, r, t, being given, the remaining quantity may be found by solving equa- tion (5). Th-. P-T^ (6) r = ^^ (7) pt t = ^^ (8) pr Thus all the problems of arithmetic relating to interest, principal, rate, time, and amount may be solved by means of equations (1) and (5). 11 162 ALGEBRA. Ex. 1. Find the principal that will amount to $335.30 in 8 years and 6 months at 5%. Here, given a = 335.30, t = 8^, r = .05, find p. TT • r 1 /c^ ^ 335.30 335.30 Uemg formula (6), P = J^^J^T:^ = ^m - $235.30, Principal. EXERCISE 54. 1. Find the interest of $650 for 4 years at 6 per cent. 2. Find the time in which $325 will produce $84.50 interest 1 5 per cent. 3. Find the rate at which $176 will yield $43.56 interest in 5 years 6 months. 4. What principal at 5 per cent, will produce $102.30 in 7 years and 9 months? 6. Find the time required for $123.45 to amount to $197.52 at 6 per cent. 6. Find the rate at which $75.60 will amount to $91.98 in 5 years. 7. At what rate will $15 amount to $16 in 10 months? • 8. What principal will produce a dollar a month at 3| per cent. ? 9. In what time will the interest on a sum of money equal the principal at 4 per cent. ? At 6 per cent. ? At 5 J per cent. ? 10. What principal in 2 years 4 months will amount to $609.30 at 5^ per cent.? 11. In what time will $76.80 amount to $80 at ^ per cent.? CHAPTER XII. SIMULTANEOUS EQUATIONS. 156. Simultaneous Equations are equations in which more than one unknown quantity is used, but in any set of which equations the same symbol for an unknown quan- tity stands for the same unknown number. Thus, in the group of three simultaneous equations, a; 4- 2/ + 2z = 13, 2x 4- 3/ - z - 3, X stands for the same unknown number in all of the three equations, y for another unknown number, and z for still another. 167. Independent Equations are those which cannot be derived one from the other. Thus, a; + 2/ = 10, and 2a; = 20 — 21/, are not independent equations, since by transposing 2y in the second equation and dividing it by 2, the second equation may be converted into the first. But Zx-2y-=b 158. Elimination is the process of combining two equa- tions containing two unknown quantities so as to form a single equation with only one unknown quantity ; or, in general, the process of combining several simultaneous equations so as to form equations one less in number and containing one less unknown quantity. V are independent equations. 164 ALGEBRA. 159. Value of Simultaneous Equations. In simulta- neous equations we have given the relations of a set of unknown quantities to known quantities in the shape of a group of equations. By the use of the methods of Chapters VI. and XI., and by elimination, we reduce this complex set of relations to more and more simple ones, till at last we arrive at relations so simple that the value of each unknown quantity can be perceived at once. 160. Methods of Elimination. There are three principal methods of elimination : I. Addition and Subtraction. II. Substitution. III. Comparison. These methods are presented to best advantage in conneo tion with illustrative examples. Ex. Solve \ ^ { 9x — 161. I. Elimination by Addition and Subtraction. \2x + by = 1b , (1) 42/ = 33 (2) Multiply equation (1) by 4, and (2) by 5, 48a; + 20y = 300 (3) 45a; - 202/ = 165 . (4) Add equations (3) and (4) 93a; = 465 Divide by 93, a; = 5, Boot • Substitute for x its value 5, in equation (1), 60 + 5^ = 75 . • . 2/ = 3, Boot. Since y was eliminated by adding equations (3) and (4), this process is called elimination by addition. The same example might have been solved by the method of subtraction. SIMULTANEOUS EQUATIONS. 165 Thus, multiply equation (1) by 3, and (2) by 4, 36a; + 15^ = 225 (6) Subtract (6) from (5), 36a; - IQy = 132 31y = = 93, y = = 3, and X = = 5. (6) It is important to select in all cases the smallest multipliers that will cause one of the unknown quantities to have the same coefficient in both equations. Thus, in the last solution given above, instead of multiplying equation (1) by 9, and (2) by 12, we divide these multipliers by their com- mon factor, 3, and get the smaller multipliers, 3 and 4. Hence, in general, Multiply the given equations by the smallest numbers that will cause one of the unknown quantities to have the same coefficient in both equations; If the equal coefficients have the same sign, subtract the equations; if they have unlike signs, add- the equations. EXERCISE 55. Solve by addition and subtraction — 1. Sx-2y = l. 6. Sx-2y = 4. x + y = 2. 5x — 4y=--7, 2. 2x-7y = 9. 7. 2y + x = 0. bx -+- Zy = 2. 4x + Qy =■■ — 3. S.ix + Sy = l. 8. 9x-Sy = 6. 2x-6y = S. 15x + 12y = 2. 4. 5x-Sy = l. 9. 4x-Qy + l=0. Zx-\-5y = 21. 5x-7y + l=0. 6. x + 5y=-3. 10. Sx + 5y = Q. 7x-^Sy = e. &y+2x = ll. 166 ALGEBRA. 14.^ + 2/^1. 11. 5ic-32/ = 36. , ^ ^ 7a; — 52/ = 56. 3a; , 5?/ ^ 2 8 ^_2/_ 3^^2^^7 4 9 * 4 ' 5 2" 13.^-^ = 3. 16.^^-^2/^-6. 3 5 9 ^ + ^ = 8. 5^_^ = _6. 5 2 4 6 162. II. Elimination by Substitution. Ex. Solve 5x + 2«/ = 36 (1) 2x-f 32/-43 (2) From (1), bx = 36-2^ .-. cr^^^-^^V (3) 5 In eq. (2) substitute for a; its valno given in (3), 2( ^« - ^11 ) + .V = 43 ^2^^ + 33, = 43 6 72 - 4?/ + 15^ = 215 ll.V = 143 ^ --= 13 Substitute for y in (3), a; = ^^ ~ ^- = 2 5 Hence, in general, 7n one 0/ ^/le ^iwn equations obtain the value of one of the unknown quantities in terms of the other unknown quantity ; Substitute this value in the other equation and solve. * Clear of fractions before eliminating. SIMULTANEOUS EQUATIONS. 167 EXERCISE : 66. Solve by substitution. 1. 3x-2/ = 2. 2a; + 52/ = 7. „. ^-4^-n 2. 3a;-42/ = l. Ax-by = l. hi"- 3. 2a: + 32/ = l. ■''T-r^^ 3a; + 42/ = 2. 4. 2/- 3a; =9. 5 10 .,.1-1 = 3,. 2a; + 72/ = — 6. 5. 3a; + 102/ = - L 2x + 72/ = -i roi-*- 6. 7a; + 82/ - 19. 6a; + 62/ = 13i. 7. 4a; +52/ = 10. 7a; + 32/ = 6. ... f -1^.,. 8. 6a;-52/ = 3. 5a;-62/ = 8. IS. 7.- 1?. 33. 3 9. 5a;- 1=42/. 72/+19 = 3a;. 6, +1^-9. i«.. 5+1^10. ...f=i.u 5-^'«- 32/ 2j; 8 3 * Clear of fractious before eliminating. 168 ALGEBRA. 163. III. Elimination by Comparison. Ex. Solve 2a:-32/ = 23 . . .' (1) 5x + 2y = 2d (2) From (1) 2x = 23 + Si/ (3) From (2) 6a; = 29 - 2^ (4) From (3) x = 23^^ (5) From (4) x = 29_=J^ (6) 5 Equate the two values of x in (5) and (6), 2 3 + 3.y _ 29 - 2.v 2 5 ' Hence, 115 +15^-58-4^ 1^ = - 57 y = — 3, Boot. 23 — 9 Substitute for y in (5), x = — - — = 7 Boot. Hence, in general, Select one unknown quantity, and find its value in terms of the other in each of the given equations ; Equate these two values^ and solve the resulting equation. EXERCISE 57. Solve by comparison — 1. bx — Zy = 2. 5. 3a; + 2^/ = 5. x + 2y = Z, 4a;-32/ = -l|. 2. y-2x = Z. 6. Zx-by = ld. Zy + x = 2. 4a: + 72/ = —2. 3. x-2y = &. 7. ^x-V\y = 5. 5x + 32/ = 4. ^x + \y = b. 4. 2a; + 7t/ + l-0. 8. \x-\y = l, 6a; + 82/-7 = 0. i^-i2/ = l. 9. !-!==- M-- 10. ^^y-^^. ?-«*• 11. 3 ^ 12 ^-2y = -|. SIMULTANEOUS EQUATIONS. 169 2 3 =c + 3 2y , 4 6" *• 3 4 2y^^+2j,^^^ o 2t 14. ^±^-L 2/-3 6 2a; + 5 y + x 164. Fractional Simultaneous Equations. = -9. Ex. Solve 2x + y ^b 2y + x 4' ~4 4 2x — y_ 2^/ — a; —J— -2/ ^ Transpose terms, J Clear the given equations of fractions, {4a: — 2a: — y = 5 — 2^ — a; 12 - 6a; + 3y = '[2y - 8^/ + 4a; 3a: + 2/ = 5 - 10a: - 2/ = - 12 - 7a: = - 7 a; = l y = 2. Hence, Simplify each of the given equations by the methods used (Chap' ter XL) for an equation of one unknown quantity ; Solve the resulting equations by one of the three methods of elimr ination. 170 ALGEBRA, EXERCISE 58. Solve— 3 4 • ' bx-Zy + Z 5* 3a; + 1 y^Q 2a;-y , x — 2y 4 3- 3 "^ 4 3/-2 _ . ^^-^2/ = 1. 2. a;-^-^=5. 7 12 4 _ ^ + 1Q ^3 7. 0.7a; - 0.02.V = 2. ^ 0.7a; + 0.022/ = 2.2. 3 4a; + 53/_^_ 8. 0.4a; - 0.31/ - 0.7. 40 0.7a; + 0.22/ = 0.5. 2a; — V — ^ + 2y = f ^ 2a; + 1.57/ - 10. o -, . o 0.3a; -0.052/ = 0.4. 5 7 10. 0.5a; + 4.52/ = 2.6. % + a ; 1.3a;^ 3.12/ = 1.6. 11 "^^~^* 11. 0.8a; -0.72/ = .005. 5.?^^ + 5^^ = 2: 2. = 32/. 5 11 ii±lZli_25_, ='2. 3:r = 0.42/ + 0.1. 4 3 ""^- .06y = 0.5a;-.02. 18. ^-^^ =^ + 13. , 2-3a: 102/ + 1 ^ 2a; + 3 5 "' .+3-i±^ SIMULTANEOUS EQUATIONS. 171 x + y 5 J 2__JL2 3_^ Hi If 16. (x--5)(3/ + 3)-(a;-l)(2/ + 2).: an/ + 2x = a;(2/ + 10) + 72i/. 5 3 4 2y + 6 4a;-hy + 6 3 8 j7^ 62/4-5 3a; + 5j- _92/-4 8 6x - 22/ 12 ' + 3 a; + 2/ _ ^y + 4 3x-2y~ 8 18 3a;-2 ^ 6a;-5 a; + 2/ + 6i 5 10 6a; + 2/ ' Sy-2 ^ 2y-5 S + 7x 12 8 103/-3a;' 165. Literal Equations. Ex. Solve ax + hy = c (1) o!x^h'y^d (2) Multiply (1) by a\ and (2) by a, aa'x + a'hy = a'e (3) aa^x + ah'y = a& (4) Subtract (4) from (3), {a'h — ab')y = a'c — ac^ • y = «^c — a(/ ' ' ^ a'b- ab^ Again, multiply (1) by 6^, (2) by 6, ab^x -f 66'y = b^G (5) a^bx + bb^y = bc^ (6) Subtract (6) from (5), {ab' - a'b)x = b'c - b& . b'c-bc\ ab'-a'b 172 ALGEBRA. It is to be observed that after finding the value of a:, it is best not to find the value of y^ as in numerical equations, by substituting the value of x in one of the original equations and reducing, but rather by taking both the original equations and eliminating anew. EXERCISE 59. Solve— 1. 3a: + 42/ = 2a. 11. (a + 1>— % = « + 2. 5a; + 62/ = 4a. (a — V)x + Zhy = 9a. 2. 2aa; + Zhy = 4a6. 12. (a + 6)a; + ci/ = 1. bax + 462/ = Sa6. ex + (a + 6)2/ = 1. 3. ax-\-hy = 1. I3. _^ [. _1_ = 2. a + 6 a — 6 a'a; + 6'2/ = 1. 4. x — y = 2n. mx--ny = m' 14. (a — b)x = (a — d)y. x — y = l. 5. 2bx -]- ay = 4b + a. . la , / , i.n ^ ^ (a — o)a; + (a + o)v ^ a6a; - laby = 46 + a. 15. ^ ~^:^^ ^ ■^* 6. ax~by = a^-\-b\ ax - 2by = a^ - 2b\ bx + ay = 2(a'^ + 6'). T. ax + by = c. a + 6 ' a— 6 16. ^z:A + lziA=._i. mx + ny^d. a; + 2a y-2b ^ a'-hb' 8. bx + ay = a-^b. a — b a + b a' — b^ ab(x-y)='a'-b\ ^^ rp-l y-a ^^ 9. c^x — d'y^c — d. * 6 - 1 6 — a C£Z(2(^a;-C2/)=2d^-c'. x-\-l ^ y-1 ^1 a;-hm ^ n ^ l-« ^ * 2/-n m' 18. (ic-l)(a+6)=a(2/+a+l> jc + 2/ = 2n. (2/+l)(a-6)=6(a;-6-l). SIMULTANEOUS EQUATIONS. 173 166. Three or More Simultaneous Equations. If three simultaneous equations, containing three unknown quanti- ties, be given, we may take any pair of the given equations and eliminate one of the unknown quantities; then take a different pair of the given equations, and eliminate the same unknown quantity. The result will be two equations with two unknown quantities, which may be solved by the meth- ods already given. In like manner, if we have n simultaneous equations, con- taining n unknown quantities, by taking different pairs of the n equations, we may eliminate one of the unknown quan- tities, leaving n — 1 equations, with n — 1 unknown quantities, and so on. '3a; + 42/-5z=32 (1) Ex. Solve ■ 4a; -52/ + 32=18 (2) bx-Zy-4.z = 2 (3) If we choose to eliminate z first, -multiply (1) by 3, and (2) by 5, 9a; + 12y - 150 = 96 (4) 20a; - 25^ + 150 = 90 (6) Add (4) and (5), 29a; - 13^ = 186 (6) Also multiply (2) by 4, (3) by 3, 16a; - 20y + 120 = 72 (7) 15a; - 9 y -12z = 6 (8) Add (7) and (8), 31a; - 29y = 78 (9) We now have the pair of simultaneous equations, r29a;-132/ = 186 l31a;-29^ = 78 Solving these, obtain a; = 10 Substitute for :p and y in equation (1), 30 + 32 - 53 = 32, = 6. 174 ALGEBRA. EXERCISE 60. Solve — 1. x + y + z-=6. 9. ix+.iy + lz = 2, ^x + 2y + z = 10. ix + iyi- |z = 9. 3x + 2/ + 3z = 14. ia; +^2/ + ^2 = ^^ 2. 3x-2/-2z = ll. 10. 2x + 2y-z = 2a. 4x-2y + z = -2. Sx-y-z = Ab. 6a:-2/ + 3z = -3. 5a; + 3^/ - 3z - 2(a + 6), 3. 5a;-6i/ + 2z-5. 2x 3^ _ 4z _ ^g^ 8x + 42/-5z = 5. '345 9a; + 52/ - 6z = 5. ^ _ §/ -f ??- = — 5. 4. 3x + 22/-z = 9. 6 8 4 7. + 52/ + 2z = 3. ?^_Z^ + ^ = _41. 2a;-72/ + 5z = 0. 2 5 10 12. x + y-{-2z = 2(a + 6). 5. 2x + 3y = 7. Sy + 4z = 9. x + z + 2y = 2(a + c). 6x + 6z = W. y + z + 2x = 2(b+c). 6. 2x + 4y + Bz = e, 13. a; + ^-z = 3-a-6. 62/ - 3x + 2z = 7. a; + z - 2/ = 3a - 6 - 1. 3a;-82/-7z = 6. 1/ 4-z-a; = 36-a-l. T. a;4-32/ + 3z = l. 14. 3x + 22/ = ya. 32;_5z = l. 6z-2a; = |6. 92/ + lOz + 3a; = 1. 52/ - 13z + a; = 0. 8. w + v-w = 4. 15. -x + y + z + v = a. u + v-x = l. x-y-^z + v = b. v-{-w + x = S. x-\ry-'z-{-v = c. ^_-^ + a; = 5. x-^y + z~v = d. SIMULTANEOUS EQUATIONS. 176 167. Use of — and — as Unknown Quantities. ac y Some equations which would otherwise be difficult of solution are readily solved by regarding - and - as the un- X y known quantities and eliminating them by the usual methods. Ex. 1 Solve ^5 + i^ = 49 X y ^ + - = 23. X y (1) (2) Multiply (1) by 7, and (2) by 5, Subtract (4) from (3), ^ + ^ = 343 X y 35 ^ 15 X 'y 76 y , 1 115 = 3 y y = i, Root, Substitute the value of ^ in (2), hence, X = If Boot. (3) (4) Ex. 2. Solve 2x 32/ ?_!- = ?? 4y 4 X (1) (2) When X and y in the denominators have coefficients, as in this example, it is usually best first to remove these coeffi- cients by multiplying each equation by the L. C. M. of the 176 ALGEBRA. coefficients of x and y in the denominators of that equation. Hence, Multiply (1) by 6, and (2) by 4, X .10^ y ' = 66 8 . X _ 1^ __ y = 29 Multiply (4) by 10, Add (3) and (5), 80 _ 10 - — - = = 290 . . . X y 89 — = X = 356 , ' .X = = h from (4), • y = -h EXERCISE 61. Dive — i.i+?=i. X y X y ?+5=i. X y — -?-3 X y 2.^ + ^ = 3. X y '■l-Ay^'- 5 + 5 = 2. X y -2+l = -5. 3x 2y 8.1-2=7. X y '■i-t''^- ? + ^=i. X y i-l-''^- 4.5 + 6^2. X y 8.i + Ul. X y n X y 1 1 = n. » y (3) (4) (5) SIMULTANEOUS EQUATIONS, 177 X y a 15. ^ y b a _ b — a X y a 2/ z lO. \- - = m^ 4- n. X y X y 16. Yx ¥y^Vz~^- 11. f + A=2. bx ay cc 2/ zz 6 g ^ g^ 4- 6\ A_A_1:^15 X y ab 2x 4y 3z ^L±l-i. ^-^ ^0^ 17 1_1_1 = 1 12. — ^ 1 = 2a. X y 13. 52/ — 3a; = 7a;2/. X y z a (^ , b , , 1111 ^ y y z X h 1111 15a: + 602/ = 16x2r. z x y c 14.^ + ^4-^ = 2. 18.^ + ^-^^Z. X y z X y .z 2 1,1^ g , c 6 4--=7. -4- =m. x y z X z y 3,2,5.. 6, eg -4 h- = 14. -4 =n. X y z y z X 19. 52/z 4- 6a:z — 3x2/ = ^xyz. Ayz — dxz -\- xy=^ 19xyz. yz — 12xz — 2xy = dxyz, 12 CHAPTER XIII. PROBLEMS INVOLVING TWO OR MORE UN- KNOWN QUANTITIES. 168. In the solution of problems we are sometimes obliged to employ more than one letter to represent the unknown quantities. We must always obtain from the conditions of the problem as many independent equations as there are letters thus involved. Ex. 1. Find two numbers such that the greater exceeds twice the less by 1, and twice the greater added to three times the less equals 23. Let X = the greater number, and y = the less. Then, x-'ly = \ \ 1 o , o , ^ OQ I f^®°^ ^^ conditions of the problem. Solving these equations by any of the common methods of elimination, one obtains a; = 7 ; ^ = 3. Hence 7 and 3 are the numbers required. Ex. 2. There is a fraction such that if 2 be added to both numerator and denominator, it becomes \ ; but if 7 be added to both numerator and denominator, it reduces to |. Find it. Let — represent the fraction. y Then, ^ + 2 2/ + 2 1 2' and ^ + 7_ 2/ + 7 2 3* Clearing these equations, and collecting like terms. 2x-y = -2 Zx-1y = -7 The solution shows x = Z and ^ = 8. Therefore | is the required fraction. 178 PROBLEMS. 179 Ex. 3. Two-fifths of A's age is 3 years less than | of B's age ; but | of A's age equals B's age 10 years ago. Let X = the No. of years in A's age. y = the No. of years in B's age. and y — 10 = B's age ten years ago. Then, |a: = |y - 3, ^x = y - 10. Ciearing and solving, a; = 45 and y = 35. Thtts, A is 45, and B, 35 years of age. EXERCISE 62. 1. Find two numbers whose sum is 23 and whose difference is 5. 2. Twice the difference of two numbers is 6, and \ their sum is Z\. What are the numbers? 3. Find two numbers such that twice the greater exceeds 5 times the less by 6 ; but the' sum of the greater and twice the less is 12. 4. If 1 be added to the numerator of a certain fraction, its value becomes \ ; but if 1 be subtracted from its denominator, its value is |. Find the fraction. 5. Two pounds of flour and five pounds of sugar cost 31 cents, and five pounds of flour and three pounds of sugar - 5 -2>-3 The First Member of an inequality is the expression on the left of the sign of inequality ; the Second Member is the expression on the right of this sign. 170. Two inequalities are said to be of the same kind, or to subsist in the same sense, when the greater member occu- pies the same relative position in each inequality ; that is, is the left-hand member in each, or the right-hand member. Hence, in inequalities of the same kind the signs of inequality point in the same direction, Thus, xy2x- 2x + \ ^ ^ 3 >^'' -3 -4 are of the same kind; «j but 2a>b- a 2 are of opposite kinds. 171. Properties of Inequalities. The following primary properties of inequalities are recognized as true : 188 INEQUALITIES. 189 (1) Adding and Subtracting Quantities. An inequality jvill be unchanged in kind if the same quantity be added to or subtracted from each member. Hence, (2) Terms Transposed. A term may be transposed from one member of an inequality to the other, provided its sign be changed. (3) Signs Changed. The signs of all the terms of an in- equality may be changed, provided the sign of the inequality be reversed. (4) Positive Multiplier. An inequality will be unchanged in kind if all its terms be multiplied or divided by the same positive number. (5) Raised to a Po"wer. An inequality will be unchanged in kind if both members be positive and both be raised to the same power. (6) Inequalities Combined. If the corresponding members of two inequalities of the same kind be added, the resulting in- equality will be of the same kind; but if the members of an inequality be siibtracted from the corresponding members of another inequality of the same kindj the resulting inequality will not always be of the same kind. 172. Application of Primary Principles. By use of these primary properties complicated relations of inequality may be reduced to simple relations, giving more or less definite results of value. Ex. 1. Given that x is an integer, determine its value from the inequalities. r 4a; - 7 < 2a; + 3 l3a; + l>13-a; f 2a; < 10 Transposmg terms, j 4a;>12 Dividing by coefficient of x in each inequality, x3 • . X = 4, ResuU. 1 190 ALGEBRA. Ex. 2. Prove that the sum of the squares of any two un- equal quantities is greater than twice their product. Let a be the greater of the two quantities, and b the lessr Then, a-b>0 r.(a-by>0 ,'. a'-2ab + b'>0 a'-i-b'>2ab. Ex. 3. Prove (a + 6) (6 + c) (a + c) >Sabc. The left-hand member when expanded becomes a{b^ + c') + 6(a2 + c^) + cia" + b^) + 2abc. But from Ex. 2, a(62 + d") > a{2bc) (1) bio" + c") > b{2ac) (2) c{a^ + 62) > c{2ab) (3) Also, 2a6c = 2abc (4) Adding (1), (2), (3), (4), (a + 6) (6 + c) (a +c) > Sabc. EXERCISE 64. Reduce — 2. (s-xy>(x-4y, ^ 3. 7ax + b>dax + 5b. 6. ^i:^>^Z:^- 4a; — 3 a; 3a; + 8 ^ a -f a; 64-a; 3 2 21 ' ' a-x 2b-x' g 4(x + 3) ^ 8a; + 37 7a;-29 9 "" 18 5a;-12* Find limits of x — 9. 3a;+i>2a; + 7. 10. 3(a; - 4) + 2 > 4(x - 3). ^-Kx + Q, 2(a; + l)<4(a;-l)-h3. INEQUALITIES. 191 11. What number is that whose fifth plus its sixth is greater than 6, while its third minus its eighth is less than 4? 12. A certain integer decreased by f of itself is greater than J of the number, increased by 5^ ; but if ^ of itself be added to the number, the sum is less than 20. Find the number. If the letters employed in each are positive and unequal, prove : 13. 3a'' + 6^^ > 2a(a + 6). ^^ ^^^-^2. 14. a'-h'>U'b-Zah\ ^^ ^ , T^ o ^ 17. a + 6>2]/a5. 15. a?-\-h^>a^b + ah\ is. a^ ^h^ + i^>ah^ ac^-hc. 19. 6a6c < a(6' -\-ah-\- c^ + cih" + 6c + a'). 20. ah{a + 6) + ac{a + c) + hc{h + c)< 2(a'' + 6' + c"). 21. a' + 6' + c* > 3a6c. CHAPTER XV. INVOLUTION AND EVOLUTION. INVOLUTION. 173. Involution is the operation of raising an expression to any required power. Since a power is the product of equal factors, involution is a species of multiplication. In this multiplication the fact that the quantities multiplied are equal leads to important abbreviations of the work. POWERS OF MONOMIALS. 174. Law of Exponents or Index Law. Since a^ = aXaXaj (a'f = (a X a X a) (a X a X a) (a X a X a) (aX aX a) In general, in raising a** to the m'* power, we have the factor a taken mXn times, or (a'*)« = a'"" L This law enables us to abbreviate the process of finding the power of a factor affected by an exponent into a mere multi- plication of exponents. Also, (aby = ahXabXab to n factors = (aXaXa to n factors) (bXbXb to n factors) by the Commutative Law for Multiplication. . • . (aby = arb'' 11. This law enables us to reduce the process of finding the 192 INVOLUTION. 193 power of a product to the simpler process of finding the power of each factor of the product. 175. Law of Signs. It is evident from the law of signs in multiplication that — (1) An even 'power of a quantity (whether plus or minus) is always positive. Exs. (-3)'' = 9, (-ahy = a*b\ (2) An odd power of a quantity has the same sign as the orig- inal quantity. Exs. (-ay = -a\ (+ay = a\ 176. Involution of Monomials in General. Hence, to raise a monomial to a required power, Raise the coefficient to the required power ; Multiply the exponent of each literal factor by the index of the required power ; Prefix the proper sign to the result. Ex. 1. Find the cube of Zx^y. (Sx'yy ^27xy. Ex.2. (i-2aby=-B2a'b'\ 177. Powers of Fractions. By a method similar to that used in Art. 174, it can be shown that \6'"/ ~ b^ Hence, to raise a fraction to a required power, liaise both numerator and denominator to the required power ^ and prefix the proper sign to the resulting fraction. 16a"x* Ex./-^y 6256y' 18 194 ALGEBRA. ' EXERCISE 65 , Write the square of— 1. la'b. 4. =^^. 5z' 6. - 13x»y. 2. -bxy\ 3. ixY. 6a6 '• ir * 102 8. -42/« + ». 9. 1. 8cd Write the cube of— 10. ^xy. 11. —2x\ 12. ^:^y\ 13. -SxY 7x'" Write the value of— 16. (7a6^c')l 17. (IWhJ. '•■(^J 21. (-2ar')'. 22. (-imO*. 18. (fx^2/)\ 20. (-JaV )*. 23. (3ia;'")^ 24.* (a' -25)1 25. (a + |)^ 26. (-x^-|2/)'. 29. 30. 31. 32. (a -26 + 3)'. (1-a + a^-ay (1 - 2x + la:^)'. 28. (|a:^ + |2/z)^ 33. (fx»-|x2/ + i2/T. POWERS OF BINOMIALS. 178. General Process. In obtaining a required power of a binomial, economies are possible still greater than those used in the involution of a monomial. It is sufficient in taking up the subject for the first time to * For this and the succeeding examples in this exercise see Art. 181. INVOLUTION. 195 obtain several powers of a binomial by actual multiplication, and, by comparing them, obtain a general method for writing out the power of any binomial. A formal proof of the method obtained is given later. (a + hy = a' + Sd'b + Sab' + b\ (a + by = a* + 4(f 6 -f Qa'b' + 4ab' + 6*. (a -f bf = a' + ba'b + lOa'6' + lOa'6' + 5a6* + b\ If b is negative, the terms containing odd powers of h will be negative ; that is, the second, fourth, sixth and all even terms will be negative. Comparing the results obtained, it is perceived that I. The Number of Terms equals the exponent of the power of the binomial, plus one. II. Exponents. The exponent of a in the first term equals the index of the required power, and diminishes by 1 in each succeeding term. The exponent of b in the second term is 1, and increases by 1 in each succeeding term. III. Coefficients. The coefficient of the first term is 1 ; of the second term it is the index of the required power. In each succeeding term the coefficient is found by multi- plying the coefficient of the preceding term by the exponent of a in that term, and dividing by the exponent of h increased by 1. IV. Signs of Terms. If the binomial is a difference, the signs of the even terms are minus ; otherwise the signs of all the terms are plus. Ex. (a + by = a' + 7a'b + 21a'b' + 35a*6' + 35a»6* + 21a'6'^ -\-7ab' + b\ To form the coefficient of the third term we have 196 ALGEBRA. The other coefficients are determined similarly. It is to be observed that the coefficients of the latter half of the expansion are the same as those of the first half in reverse order. 179. Binomials with Complex Terms. If the terms of the given binomial have coefficients or exponents other than unity, it is usually best to separate the process of writing out the required power into two steps. Ex. 1. Obtain the cube of 2x + 5y^. Since (a + bf = a" + Sa'b + Sab' + b\ substituting 2x for a, and 5y' for b, (2x + 5^)^ = (2x)' + 3(2a;)^ (by') + Z(2x) (SyJ + (ByJ = 8a;' + 60a;y + ISOc?/* + 125/. Ex. 2. (2a;' - \yy - (2xy - 4(2r')' {\y') + ^(2xy (\yy = i6x^'^ - 8xy + f xy - ixy + ,ji^2/'. 180. Application to Polynomials. By properly grouping its terms a polynomial may be put into the form of a bino- mial, and any power of the polynomial obtained by use of the above method for involution of a binomial. Ex. (a; + 22/ + Zzf = \(x + 2y) + 3z]' = (a; + 22/)' + 3(a; + 22/)'(3z)+3(a; + 22/)(32)= + (3z)' = «» -f Gx'^y + 12x2/' + 82/' + 9x'z H- 36a;2/z +362/'z + 27a;z' + bAyz' + 27z'. 181. Cases of Involution Previously Considered. In Arts. 85, 86, 89 important special cases of involution have, been considered, and should be here recalled: 1. The square of the sum or difference of two quantities; 2. The square of any polynomial. J II. For let v^'a = Xy yl) — y ; .•.a:" = a,...(l) y- = h...(2) But xV = (P^yT (by Art. 174) Substitute for a;« and 2/" fro- (1) and (2), «^ = (v^ai/F)« (3) Extract the n'* root of each member of (3), This reduces the process of finding the n'* root of a product to the simpler process of finding the root of each factor. 186. Law of Signs. From the law of signs for multipli- cation it follows that — (1) Any even root of a positive quantity may be either positive or negative. Ex. 1/9 = + 3, or - 3. It is convenient for the present to consider only positive roots of even powers. (2) No negative quantity can have an even root. Ex. The square root of — 4 is neither + 2 nor — 2^ since neither of these multiplied by itself will give — 4, (3) The odd root of a quantity has the same sign as the quantity itself. Ex. 1^=27^=- 3. SQUARE ROOT. 199 187. Entire Process. Hence, to extract a required root of any monomial, Extract the required root of the coefficient ; Divide the exponent of each letter by the index of the required root ; Prefix the proper sign to the result. EXERCISE 67. Write the square root of — 1. 9xy. 4. 16a;y. 6. |a;Y'. 2. 25a*. 36a^^ 121aV" 3. 144/". * 49a;'*** * 812/'" + ^ Write the cube root of— 27a"a;' iQ 8a;V n + 8 12. --— — • 14. 1000 10. -ia'6^ • 3432^9 * a;**"* Write the value of — 15. \'-bl2x\ 18. VM^^'^ 21 *I ^25P 16. 1^16^^. 19. V7=^. • \2/*" + «' 17. l^^^^^y: 20. V^^P- 22. l^-Aa^2/*- 23. l/25ar' + 20a; + 4. 25. l/a^6^ - f6a6c + 64c'. 24. l/9a;*-42a;-^ + 49. 26. Vl + 18xy + Sl^y. SQUARE ROOT. 188. Square Root of Polynomials. Our object is to dis- cover such a relation between the terms of a binomial (or in general of a polynomial) and the terms of its square (as, for instance, between a -\-h and its square, a^ + 2ah + 6^), that we 200 ALGEBRA. can state this relation in the inverse form as a general method for readily determining the square root of any polynomial which is a square. a" + lab + h'' The first term of the root, a, is the square root of the first term, d^^ of the square ex- 2a + 6 2a6 + IP" pression. The second term of the root, 6, 2a6 + IP' occurs in the second term, 2a6, of the square ^ expression, and may be obtained frcftn it by dividing by twice the first term, or la (called the trial divisor). If we take la and add h to it (giving 2a + 6, called the complete divisor), and multi- ply the sum by 6, we get 2a6 + 6^, which is the rest of the square expres- sion after a^ has been subtracted. This last step, therefore, furnishes a test of the accuracy of the work. Ex. Extract the square root of 16x^ — 2Axy + 92/'* 16x2 ^x-Zy - 24xy + 9y^ - lAxy + 9.^2 Taking the square root of the first term, \Qx^, we obtain Ax, which is placed to the right of the given expression as the first term of the root. Subtract the square of 4x from the given polynomial. Taking twice the first term of the root, 8x, as a trial divisor, and dividing it into the first term of the remainder, we obtain the second term of the root, — Zy. This is annexed to the first term of the root and also to the trial divisor to make the complete divisor, 8x — Zy. 189. Square Root to Three or More Terms. In squaring a trinomial, a + 6 + c, we may regard a-\-h as a single quan- tity, and denote it by a symbol, as ^, and obtain the square in the form p^ + 2pc + cl Evidently we may reverse this process, and extract a square root to three terms, by regarding two terms of the root when found as a single quantity. So a fourth term of a root, or any number of terms, may be found by regarding in each case the root already found as a single quantity. SQUARE ROOT. ^ 201 Ex. Extract the square root of x* — 6af' + 19x' - 30x + 25. ^ - 6^3 + 19a;2 - 30a; + 25 | a;' - 3a; + 5 a;* 2a;2 - 3a; I - 6a;^ + 19a;2 - fia;^ + 9a;2 2a;2 - 6a; + 5 + lOa;-'' - 30a; + 25 + ]0a;2 - 30a; + 25 The first two terms of the root, x^ - 3a;, are found as in the example in Art. 188. To continue the process, we consider the root already found, x^ — 3a;, as a single quantity, and multiply it by 2 to make it a trial divisor. Dividing the first term of the remainder, lOa;^, by the first term of the trial divisor, + 2a;'^, we obtain the next term of the root, + 5. The process is then continued as before. Hence, in general, to extract the square root of a poly- nomial, Arrange the terms according to the -powers of som» letter ; Extract the square root of the first term, set down the result as the first term of the root, and subtract its square from the given polynomial ; Take twice the root already found as a trial divisor, and divide it into the first term of the remainder ; Set down the quotient as the next term of the root, and also annex it to the trial divisor to form a complete divisor ; Multiply the complete divisor by the last term of the root, and subtract the product from the first remainder ; Continue the process till all terms of the root are found. EXERCISE 68. Find the square root of— 1. x' - 4a;' + 6a;'' - 4a; + 1. 2. l-2a-a' + 2a' + a\ 3. 9a;* - 12a:' + lOx' - 4a; + 1. 4. 25 + 30a;4-19a:' + 6a;' + a;*. • 202 • ALGEBRA, 6. n' — 4n^ + 4ri* + Qtv' - 12n' + 9. 6. ^x^ + 12x' + a:* - 24a:=' - 14x' + 12a; + 9. 7. 1 + IGm" - 407?i* + 10m - Sm'^ + 25m^ 8. 46?i' + 25n^ + 4n' + 25 - 44n' - 40n - 12n^ 9. 9a;' + 92/' + 24x^2/ + 24:X^ - S^y - 8xy - bO^^y", 10. m' + 9 + x' + 6m + 6a; + 2ma;. 11. 1 + 5x^ + 2a;* + a;' - 4a;' + 2a;-'' + 2x. 12. 28a;' - 47a;* + 49x' - 42a;' - 4a;' + 16a; + 4. 13. \x'' - bx + 25. 16. x* + 2a;' - a; + \. 14. |x' - 5a;2/ + ^y\ 17. >* - Ja' + -^a' - 4a + 36. 42/'' 2/ a;'' a; a a 19. ^x' - la;' + W^^' - 3a; + ^. 20. ^x*-|-ar' + |K--|a; + f|. 21. l + a-Aa'-K-|a* + t«' + «*'- r* r^ 1 ^'^ 1 ^^•T + - + 7 + i + & + 7 23. — -aa; + — -2 + — +—. x^ a' a; 4 Find to three terms the square root of — 24. 1 + 4a;. 27. a^ -\*4h. 30. a;" — 1. 25. 1 — 2a. 28. 9a' — 4a;. 31. a;' + 3. 26. a;' - 6. 29. 4a' - 6a6. 32. a' + 3a6 - 2h\ 190. Square Root of Arithmetical Numbers. The same general method as that used in Art. 188 can be used to extract the square root of arithmetical numbers. The details of the process, however, are somewhat different, owing to the fact that all the numbers which compose a given square number are given united as a single number. SQUARE ROOT. 203 Thus, (43)' = (40 + Zy = 1600 + 240 + 9 - 1849. Hence, given 1 849 to extract its square root, the square of the first num- ber, 1600, is not presented explicitly as it would be in an algebraic expres- sion, but must be determined indirectly. The first step is to mark off" the figures of the given number whose root is to be extracted into periods of two figures each, beginning at the decimal point, and then to determine the largest square number represented in the first period of figures at the left as a trial number. If the first figure of the root be in the tens' place, and therefore followed by one zero (as 4 in 40 above), its square will be followed by two zeros, as in 1600. If the first figure of the root had been in the hundreds' place, and therefore followed by two zero^, its square would have been followed by four zeros ; that is, there are two additional zeros in the square for each additional zero follow- ing the first figure of the root. Hence comes the significance of separating the given square number into periods of two figures each, and extracting the approximate square root of the left-hand period of figures. We will illustrate by an example, using the algebraic formula (a + by = a^ + 2ab + b"^, to show the essential identity of the arithmetical and jilgebraic processes. Ex. Extract the square root of 1849. Trial divisor, 2a = 80 b= 3 1849. 1 40 + 3 1600 249 249 Complete divisor, 2a + 6 = 83 This work may be put in the following abbreviated form : 1849. 1 43 16 83 I 249 I 249 191. Square Root of Decimal Numbers. If it be required to extract the square root of a decimal number, as 28.09. we may proceed thus : \100 1/100 10 It is better, however, to put this work into a different form 204 ALGEBRA. by marking off the given number into periods of two figures each, beginning at the decimal point and marking both to the right and left. If necessary annex a zero to complete the last period of figures to the right ; in such cases, however, the root cannot be exactly extracted. Ex. Extract the square root of 18.550249. 18.550249 I 4.307, Boot. 16 83 255 249 8607 60249 60249 192. Square Root of Comraon Fractions. If the de- nominator of the fraction whose square root is to be extracted is a perfect square, extract the root of the numerator and de- nominator separately and divide the one result by the other. ^ 1289 V2m 17 iliX. \ — — = — * ^324 1/324 18 If the denominator is not a perfect square, reduce the frac- tion to a decimal and extract the root of the decimal. Ex. Vi - T/0.66666666 + 0.66666666+ | 0.8164+^ 64 161 266 161 1626 10566 9756 1632 4 81066 65296 Hence, in general, to extract the square root of an arith- metical number, SQUARE ROOT. 205 Separate the number into periods cf two figures each, beginning at the decimal point ; Find the greatest square in the left-hand period, and set down its root as the first figure of the required root ; Square this figure, subtract the result from the left-hand period, and to the remainder bring down the next period ; Double the root already found for a trial divisor, divide it into the remainder {omitting last figure of the remainder), and annex the quotient obtained to the root and also to the trial divisor. Multiply the complete divisor by the figure of the root last found, and subtract the result from the remainder ; Proceed in like manner till all the periods of figures have been used. EXERCISE 69. Find the square root of — 1. 7225. 6. 337561. 11. 199.204996. 2. 2601. 7. 567009. 12. 10.30731025. 3. 8464. 8. 11573604. 18. 254046.2409. 4. 105625. 9. 36144144. 14. .0291419041. 5. 182329. 10. 8114.4064. 15. 1513689.763041 Find to four decimal places the square root of — 16. 7. 19. ^. '22., ^. 25. .049. 17. 11. 20. 2^. 23. 1|. 26. 1.0064. 18. 12.5. 21. 0.9. 24. ^. 27. 36^. Compute to three decimal places the value of — jT 28. \/2+VE. 32. l/2V7-f-3l/2. l 5(V^-T/^ _ 29. l/ 1/5-1. 33. 1/31/6-21/7. * 2 80. \/VW-V%. 34. J V5-1/2 36. j7, V^4-2l/5 ^ 81.1/31/3+1/5. 4 ^ a' 3a' + Zah + 6' + Sa'^ft + 8a62 + 6' Za? + 3a6 + b" + Za'b + 3a62 + 6' 206 ALGEBRA. CUBE ROOT. 193. Cube Root of Polynomials. Our object is to deter- mine such a relation between the terms of a binomial, or, in general, of a polynomial, and the terms of its cube (as between a + ft, and its cube, o? + 3a^6 + 3aft^ + 6^), that we may be able to state this relation in the inverse form as a general method for determining the cube root of any polynomial which is a perfect cube. a' + Zo}b + 3a62 + 53 [0^ + 5 The first term of the root, a, is the cube root of the first terra, a', of the cube expression. The second term of the root, 6, occurs in the second term of the cube expression, 3a^6, and may be obtained from it by dividing it by Za^ ; that is, by three times the square of the first term of the root (called the trial divisor). If we take the trial divisor, and add to it three times the product of the first term of the root by the second term, 3a6, and also the square of the second term of the root, h"^, we get 3a' + 3a6 4- 6^ (called the complete divisor) ; this multiplied by the second term of the root gives Za^b + Zab^ + 6^, the rest of the cube expression after a' has been subtracted. This last step, therefore, furnishes a test of the accuracy of the work. 194. Three or More Terms in the Root. In cubing a trinomial, a + 6 + c, we may regard a + b as a single quan- tity, and denote it by p, and obtain the cube in the form p^ + 3p'c 4- 3pc^ + (f. Evidently we may reverse this process, and extract a cube root to three terms, by regarding two terms of the root when found as a single quantity. So a fourth term or any number of terms of a root may be found by regarding, in each case, the root already found as a single quantity. We will now extract the cube root of a polynomial expres- sion indicating at each step the trial divisor and complete divisor. CUBE ROOT. 207- g fL o ^' ;2. OS g 1^ g + iS ^* I«-f 5 -< ^ + ►^ B 1 00 ^ za >^ C/, ^ S" fs II V .," ^ ^. CO " is: 5 i* I f5 tr a> o S (T o i_^ Q Q » 00 «• + 1? CO + %. CO 1 Oi I ^ f 1 1 1 OS 1 ^ I— »• at 1 CO 1 n. CO + ^ I—'' + t— > K ? + + fcO 1 ^i i 1 t— I to to 9^ Ci € + 1 1 208 ALGEBRA, EXERCISE 70. Find the cube root of — 1. a' + Qa'x + 12ax' + Sx\ 2. 27-27a + dd'-a\ 3. l-12a: + 48x^-64rl 4. a^ - Sa' - 3a* + Ha' + 6a' - 12a - 8. 6. a:« - 3x^ + 6a:* - 7a:' + 6a:' - 3a: + 1. 6. 1 - 9a: + 21a:' + 9a:' - 42a:* - 36x^ - 8a:«. 7.. 12a;* - 36a: + 64a:' - 6a:' - 8 + 1 17a:' - 144a:l 8. 95a' + 72a* - 72a' + 15a^ + 15a + a' - 1. 9. 114a:* - 171a:' - 27 - 135a: + 8a;« - 60a:^ + 55ar». 10. 8 + 27n' - 36?i - 81n^ + 907i' - 135n' + 13571*. 11. a:' 8 ' x' 4 -i 1 27* -i- 2y 22/ 3 18. 0?- • 3a;' + 6a: - 7 + 6_ X hi- 14. 1 + 3_ a a' a' 18 27 27 a* a' a' ■ 15. x'-V y 15a:* 2f 45a:' , 27_^_ 27 10i» 42/* + 2y ¥" 2/" 27ar^ 195. Cube Root of Arithmetical Numbers. The same general method as that used in Art. 194 can be used to extract the cube root of arithmetical numbers. As in square root, the process is slightly different from the algebraic one, owing to the fact that all the numbers which compose a given cube are given united or fused into a single number. - Thus, (42)=* = (40 + 2)=' = 4a'' + 3 X 402 X 2 + 3 X 40 X 22 + 2^ = 64000 + 9600 + 480 + 8 = 74088. Hence, given 74088, to extract its cube root, the cube of the first number, CUBE ROOT. 209 or 64000, is not given explicitly, as it would be in an algebraic expression, but must be determined indirectly. This is done by marking off the given cumber into periods or groups of three figures each, beginning at the deci- mal point, and then determining the largest cube represented in the first period of figures, and taking its cube root as a trial number for the first figure of the root. The reason for marking off" the given number into periods of three figures each may be briefly stated thus : If the first figure of the root be in the tens' place and therefore followed by one zero (as 40 above), its cube will be followed by three zeros (as 64000). If the first figure of the cube root be in the hundreds' place, and therefore followed by two zeros, its cube would be followed by six zeros. For every additional zero in the root there are three additional zeros in the cube. Hence arises the significance of separating the given number into periods of three figures each, and extracting the approximate cube root of the left-hand period. We will now illustrate the general process of extracting an arithmetical cube root, using the algebraic formula (a + 6)' = a? -\- Za^b + ^ab"^ + 6', to show the essential identity of the arithmetical and algebraic processes. Ex. Extract cube root of 74088. Trial Divisor, Complete Divisor, 74088 I 40 + 2 a3 = 403 = 64000 Za^= 3 X 40^ ■■= 4800 "TU08B Zah = 3 X 40 X 2 = 240 62 = 22 = 4 3a2 + Zab + 6^ = 5044 10088 m of work— 74088 [42 64 3 X 402 = 4800 10088 3 X 40 X 2 = 240 22= 4 5044 10088. 196. Cube Root of Decimal Numbers and Fractions. For a reason similar to that given in Art. 191 for square root of decimal numbers, in extracting the cube root of decimal numbers we mark off the decimal numbers into periods of three figures each, beginning at the decimal point, and sup^ 14 210 ALGEBRA. plying a sufficient number of zeros when the right-hand period is incomplete. Ex. 1. Extract the cube root of 130.323843. 130.323843 | 5.07 125 Trial Divisor = 3 x (500)^ = 750000 5323843 3 X 500 X 7 = 10500 49 Complete Divisor = 760549 5323843 Ex. 2. Extract the cube root of 3% to 4 decimal places. ^-0.416666666666+. 0.416666666 + | 0. 7469+ 343 3 X (70)2 _ 14700 3 X (70 X 4) = 840 42= 16 15556 3 X (740)2 _ 1642800 3 X (740 X 6) = 13320 62 = 36 73666 62224 11442666 9936936 1656156 3 X C7460)2 = 166954800 I 1505730666 I 1502593200 3137466 The first three figures of the root are found directly. The last figure is then found by division of the remainder, using three times the square of the root already found as a divisor. The number of figures of the root that may thus be found by division is two less than the number of figures already found. Hence, in general, to extract the cube root of an arithmetical number, Separate the number into periods of three figures each, beginning at the decimal point ; Find the greatest cube in the left-hand period, and set down its cube root as the first figure of the required root; CUBE BOOT. 211 Cube this figure, and subtract the result from the left-hand period, and annex the next period of figures to the remainder; Take three times the square of the root already found with zero annexed, as a trial divisor; divide the remaind^'r by it^ and set doivn the quotient as the next figure of the root; Complete the trial divisor by adding to it three times the product of the first figure of the root with zero annexed, midtiplied by the last figure, and the square of the last figure ; Multiply this complete divisor by the figure of the root last foundj and subtract the result from the remainder ; Proceed in like manner till all the periods have been used. EXERCISE 71. Find the cube root of— 1. 3375. 4. 43614208. 7. 344324.701729. 2. 753571. 6. 32891033664. 8. .000127263527. 3. 1906624. 6. 520688691.125. 9. 0.991026973. Find to three decimal places the cube root of — 10. 75. 12. 5.6. 14. 7^^. 16. ^. 18. 1^. 11. 6. 13. 3f. 15. 19f 17. y^. 19. 8^. Compute the value of — 20. Vb + 2V%. 21. I>'3l/10-2vl8. 22. v/3l70:8-2l/1.93'5. 197. Higher Roots Obtained by Successive Extractions. By the law of exponents the square of the square of any quantity gives the fourth power of the quantity. Hence, re- versing the process, the fourth root of a quantity is the square root of the square root of the quantity. Similarly, the sixth root of a quantity is the square root of the cube root of the quantity. The eighth, ninth, tenth roots of a quantity may be found by similar methods. 212 ALGEBRA. Ex. Extract the fourth root of 81a* + lOSa" + 54a' + 12a + 1. Obtain first the square root of the given expression, which is 9a' + 6a + 1. Extracting the square root of this, we obtain. 3a + 1, the fourth root of the original expression. -' EXERCISE 72. Find the fourth root of — 1. 130321. 2. 3418801. 3. 90. 4. 0.8. 6. 1 - 12ah + 54a'6' - 108a'6' + Sla'b\ 6. a:* - 2x' + ^x" -ix + ^, * 2/* y ^' 16a;* 8. 64x' - 56a;* + 16a;' + a;« + 16 - 323;^ + IQx' - Sx' + 64.'5. Find the sixth root of — 9. 7529536. 10. 1544804416. 11. 15. 12. x' 4- 1215a;' + 729 - 1458a; + 135a;* - 540a;' - 183;^. 13. 64a« - 192a* + ^-^ + -^-160 + 240a'. a' a* a* 14. 4096x^» - 3072a;^° + 960a;« - 160x« + 15a;* - fa;' + j^. CHAPTER XVI. EXPONENTS. 198. Positive Integral Exponents. Using a' as a brief symbol for aXaXa, and a"* as a brief symbol for aXa XaX a torn factors, we have already found the following laws to govern the use of positive integral expo- nents : I. a'"Xa"=:a"* + ». 11. — = ar-''A£ m>n. III. (a"*)" = a*"". IV. ^y^=fjm^ V. (ahy = orh\ 199. Fractional and Negative Exponents. Just as by using fractions as well as integers, and negative as well as positive quantity, the field of quantity and operation in al- gebra is greatly extended, some processes made simpler, and others more powerful, so by introducing fractional and neg- ative exponents we get like results. As fractional and negative exponents have no meaning belonging to them at the outset, it will be most advanta- geous to sui)pose that the first and fundamental Index Law, a™ X a" = a*" ^ '*, holds for fractional and negative exponents, and then inquire what meaning must be assigned to these exponents. We limit the fractional and negative exponents here treated to those whose terms are either positive or neg- ative integers, and commensurable; that is, expressible in terms of the unit of quantity used in the given problem. 213 214 ALGEBRA. Thus, exponents like 1/2, as in a^^ are not included in the iiscussion, though the student will find later that the same laws hold for these exponents. 200. I. Meaning of a Fractional Exponent. Since by Index Law I., it follows that a^ is one of the three equal factors which may 2 be considered as composing a^ ; that is, a^ is the cube root of al 2. Hence, in the exponent of a^, the numerator, 2, denotes the power of a to be taken, and the denominator, 3, denotes the root of this power to be extracted. So, in general, p p p p a^ X a'^ X a" X a'^ to q factors ?. + ?.+?.+ to 3 terms. = a« =--oF. Hence, in general, in a fractional exponent the numerator denotes the power of the base that is to be taken, and the denomina- tor denotes the root that is to be extracted. Ex. 1. 83 = 1^8^=1^63=4. Ex. 2. aixaixai = ai'i'i = aT2" Product Ex. 3. x^^'x^ = x^Xx^. = a;^ Product, Ex.4. 32*= 1^^ = 2^ = 64. EXPONENTS. 215 EXERCISE 73. Express with radical signs — 1. ai 4. 2aK 7. a^m*. 10. s 2. x^. 5. ax^. 8. 5x^2/^- 11. by\ 3. oi 6. 2a^6^. 9. 2c*ci* 12. n m ^my2n^ Express with fractional exponents — 13. V^. 14. 1/?. 16. aVx. 17. hv'y. 19. 1/5 iV. 20. 21^x^1/^. 22. 23. 4l^al/?'' 2i^STv'3' 15. 2l^P. 18. 2x1/^. 21. 1^5 V>^. SVTVW Find the value of— 24. 27^. 27. v' W" ' ■ 30. (-27)* 33. (-243)* 25. 25* 28. ^ 64\ 31. (-32)* 34. (^)*. 26. 16^. 29. (-8)*. 32. (-216)^ 36. (!i)* Simplify the following by performing the indicated opera- tions : 36. a^Xa^. 40. 2*a;* X 2^. 44. Vcc^^a?. 37. 2a X a*. 41. J Va\ 45. ^'2VT. 38. a'x2Xa%^. 42. 7l/av^a^. 46. a^y'^-x^Va. 39. 3x^ X a%3. 43. 2x'|>^^. 47. a;*|XF^ • 2x^. i^y-^ 2xiv^ V2^'l 2*7* a^ »/ a;- zrc* y a' V2-^'l 48. -—- X ——-' 49. -— V^ X i: ^^ iTc * i/3|>-5 1^3V5« 216 ALGEBRA. 201. n. Meaning- of the Exponent Zero, or of a\ By the Index Law I., a' X a"" = a' + "^ = or ==1 X or . • . a" = 1. Hence, aP is another symbol for unity. The student will realize the meaning of aP more readily thus, a™ By direct division, — = 1 ^ or By subtraction of exponents, — = o^ .-.byAx. 1, a' = l, 202. III. Meaning of a Negative Exponent. If n be an integer or a fraction, by the Index Law, a^Xa-^-a'^-^-a^^l a""" = — ,ora" = ^« Ex. 1. 4- = l=i- 4^ 16 Ex. 2. 8-4 = 1=1. ol 4 203. Transference of Factors in Terms of a Fraction. It follows from the meaning of a negative exponent that any factor may be transferred from the numerator to the denominator of a fraction, or vice versd, provided the sign of the exponent of the factor be changed. EXPONENTS. 217 Ex. 1. Transfer to the numerator the factors of the de- nominator of xy~^ — — = a6-'a;-'3/"^i Remit. xy~^ Ex. 2. Transfer factors in the terms of , so as to _ 2 ' xy ^z ~ ' make all exponents positive. 2a- 'h 2byh _ . = -^7—) Besult It will not be a difficult exercise for the student to prove a** Law II., — = a'"~", for fractional and negative exponents. EXERCISE 74. Transfer to the numerator all factors of the denominator — a a6' 3a 1- -T- 4. -f-. 7. — -. 10. ^f •J '■^- xz ' 4m-'n"^ xpress wit! I positive exponents^ 13. 7x-\ 16. 6a -^6'. 14. Sab-\ 17. 3a"^6-«. 1 2-^a;V 7 4 5ia;"^z" -i 11. 12. i _M (j!*x " 19. hd^x on 3«"'^ 1 20 15. a'6 ^. 18. a6-V2/ ^, cd 218 ALGEBRA. 21. J^. 23. 7'^-'^"* 25 ^ 3x-"2/-^ «^ ' * ' '^;>-^.;-^ 5-'cZ~n Sb-'y 3„.- 1 22. ^^. 24. ^!^I^. „„ BZIJ^^ Find the numerical value of — 27. 4~i 33.^ 39. (-125)"* 34. 3-^X44 40. -tl?^. 28. 27"*. 29. ^w 30. 1 5-' 31. 8-« 32. 1/S1-*. 35. 2-*8-* '-^> 36.1-.^ 41. (5J)-^. (!)■ •(-I)" -(s)'0 --' -iS^ Simplify the following by performing the indicated opera- tions, and reducing the results : 44. 2a^Xa-\ 45. a'x -^Xa- ^x-'. 46. ba- -'X2aV. 47. a*- j-a~'. 48. 4x- -'^2x- -3 49. a:2/- -'^xY -4 50. a*. .3a-i 01, C" ^(#-f-c- 'A 52. m "^n-mn'^. 53. 6a4a;"*-aV. 54. a"'4.2a^. 55. So; ~ ^2/ "^ 4a;^2/'. 56. 4x* ^ 3a:/. 57. 7a'a; ~ ^ -^ bz^y. EXPONENTS. 219 58. a*l^^.:.*l^^. Q^^^J^Ziy^. 59. X ^^y^^x%^. 7nWx ^ 6. — , 65, bU. _ 1 x ^Vx^ x'^Vy 66. 61. —.--—• 2 «' — :;:i xyy 67. — ^-^— Vx 62. a*6"^l^^^- T^'cl/5^ 2/"^^^. 63. , • 68. ^ ^ 3 |/a ^ a;j/2;^^^ 204. IV. {ary = a*"" for Fractional and Negative Bx^ ponents. It will now be found that using the meanings for fractional and negative exponents which have been deter- mined (Arts. 200, 202), Law III., {ary = a""» applies to theni also. First, when n is a positive fraction, -, the terms of the fraction, p and q, being positive integers. l{a-yr- = {ary Extracting the g'* root of both sides, p {a'^y - -a* Substitute tifor^, {a'^y- 9. = «»»". Second, when n is a negative integer or negative fraction ; as, — t. 220 ALGEBRA. rarY = (dr)-* = — = — = — ^a-^'^a"^. ^ («*") + ' or* It will not be a difficult exercise for the student to provn also Law V., {ahy = oJ'h''. First, when n is a positive fraction. Second, when n is a negative quantity. -3 -4 Ex. 1. Find the value of (4 2) ^ {^-i)~^ = 4.^ = Z% Result. Ex. 2. ^[^a'^y = (8a"^)* = 8*a"* - 4' ^^^^' Ex.3. /lg^V*= ^^"^^' ^gl!^ \ «i^' / 81-^6-4 let 27a^6^ ^ ,, = — - — ) Uemlt. 8 EXERCISE 75. Reduce to the simplest form — 1. {a')-\ 7. {a'b~^)~\ 13. (5a;~i)"* 2. (a; -=*)"*• 8. {x~^y^)~\ 14. (8a') "■^. 3. (a-')*. 9. (8-^* 15. (4a; -*)"*. 4. (x~*)*. 10. (64-0"*. 16. (9a-'a;^2/~')~*. 6. (c^)~^ 11. (9^) ~*. 17. (- 2a'a;~ V. 6. (a~*)"* 12. (3a-% IS. i~-bx-'y^-\ EXPONENTS. 221 19. (9x-'3/-»)"f 20. (a'Vo^"'. 21- (aVa^~^. 22. l^(a^^r^. 23. (aVS^^^"'. 26. (c'-a;-'-) 27 l2bVx\ * 28 1 82/V^~ 29. {.^p^ l/a^yj l-i-pf 30. 31. l>'8a-»6V? 32. l/x-^V't^^v'yi/^^. 33. l^a6-^c-^X V'SP?: 34. (xV^^"'Xi/5:"^^. 36. f-^^xi^f- 38. VlV(H)"*]. 39. [\(-U)"*] • 41. 8 ^ + 9^-2" +1 7». 42. _ 1 Va^ V7^ b ^i^ l^P^ -0 43. fa%~^»/a-=^5'T/?j. 222 ALGEBRA. 45. laJx-' X / aVx i 46. 47. m''a-5 + 4?= + -|-by2aA-t^ 1.. Va V(r Va Extract the square root of — 21, x^—^x^y^ ^-A.xy. 22. 9x1/"' + 12y-* + 4»-*. 23. a-^'-Wh^ + 10h~12ah^ + ^ah\ '. 24. a;"* + 8a;-^-2a;"^+16a;"^-8x'i + l. 25. 92; - ' - 30:c " % + 1 3x - y + 20x " V + 4a; " y. 26. 25a*6-»-10a%"^-49 + 10a"*6* + 25a"^6\ 2^^^_18v^^l%_6iy ^ ' a;* a;' x^ x ^ , 241/? 4x^ ^ /~i 4 28. 9ar^ - — ,— - + -r^ + 16^— + — !;=• y^ y 2S 2 9 ^^* 4a; 3^1 9]/^ y^ V" CHAPTER XVII. RADICALS. 207. Indicated Roots. The root of a quantity may be indicated in two ways : (1) By the use of a fractional exponent; as a^. (2) By the use of a radical sign ; as Va. For some purposes, one of these methods is better; for some, the other. 2 1 Thus, when we have a^ Xa^ Xa~^, where the quantities are alike except in their exponents, it is better to use frac- tional exponents to indicate roots ; but if we have 5VZ — 7l/27 + 8l/12, where exponents are alike, but coefficients and bases unlike, it is better to use the radical sign to indi- cate roots. In the preceding chapter we considered exponents ; we have now to investigate the properties of radicals. 208. A Radical is a root of a quantity indicated by the use of the radical sign. Exs. Vx, VT7. 209. Surds. An indicated root which may be exactly extracted is said to be Rational. Ex. 1/27, since the cube root of 27 is 3. An indicated root which cannot be exactly extracted is called a Surd. Exs. VB, v6. 210. The CoeflBcient of a radical is the number prefixed to the radical proper, to show how many times the radical is taken. Ex. The coefficient of 5l/3 is 5; of Qav'x is 6a. 226 RADICALS. 227 211. Entire Surds. If a surd have unity for its coefficient, it is said to be Entire. 212. The Degree of a radical is the number of the indicated root. Ex. Vx is a radical of the third degree. 213. Similar Radicals are those which have the same quantity under the radical sign and the same index. (The coefficients and signs of the radicals may be unlike ; hence, similar radicals must be alike in two respects, and may be unlike in two other respects.) Ex. .5t/3, — 4v^ are similar radicals. 214. Fundamental Principle. Since a radical and a quantity affected by a fractional exponent differ only in form, in investigating the properties of radicals we may use all the principles demonstrated concerning fractional exponents. Thus, since (aby = a'^i** is true, when n is a fraction, as i, (aby =a^b^ . • . i/'oS = i^a • y^. TRANSFORMATIONS OP RADICALS. 215. I. Simplification of a Quantity under Radical Sigm. If a factor of the quantity under the radical sign is a perfect power of the same degree with the radical, the root of this factor may be extracted and set outside as a factor of the coefficient. Ex. 1 Simplify 1^56. 1^=^WX7 =2^7, Result (Art. 214.) Ex.2. Simplify 6 l/18a'6V. 5 l/18a'6V = 5 VWWX^M= 15abc' V^Ec, Remit. Hence, in general, Separate the quantity under the radical sign into two factors^ one 228 ALGEBRA. qf which is the greatest perfect power of the same degree as the rad* ical ; Extract the required root of this factor^ and multiply the coefficien' of the radical by the result ; The other factor remains under the radical sign. 216. Quantity under Radical Sign a Fraction. To sim- plify in this case, Multiply both numerator and denominator of the fraction by such a quantity as will make the- denominator a perfect power of the same degree as the radical; Proceed as in Art. 215. Ex. 1. Simplify ]^^. 1^^ = v'^fXi = V'W = l^ift^Xl5 = J l^, Remit, Ex.2. Simplify ^-^^ /3^_ j'hax'' 2b ^ fWiM -^/ — 2 — X lOab = —VWab, Result. 366' 6& 217. Meaning of Simplification. By simplication radicals are reduced to their prime form, so that it is made easier to determine, for instance, whether a number of given radicals are similar or not. Thus, it is difficult to say whether 71^18, —5V12 are sim- ilar, but when the given radicals are put in the form 21 V2, — 30 1/2, it is easy to see that they are similar. Again, the radicals (a — l)-v/ and (a + 1) -y — -— , although unlike in present form, may be reduced not only to similar radicals, but to the same expression, Va^ — 1. The pupil should show this reduction for himself. MABICALS. 229 EXERCISE 77. Express in the simplest form — 1. 1/12. 11. ^2i. 21. 1/200^. 2. 1/18. 12. ;^^54. 22. |/Ii7^. 3. 1/27. 13. 1,^72. 23. -2i/^63?y^ 4. -1/2D. 14. -f|/108. 24. v' -81aV. 5. 21/24. 15. |>'i8. 25. i/aXx-y)'. 6. -31/28. 16. v*^28^^. 26. |/49ar'(a + 1/. 7. il/4i. 17. 1^250^^. pro-r;^ 27. 10, 12acn^ 8. J 1/45. 18. |/99a. >/ 25x* 9. 11/50. 19. 2^/4^^. 28. 3 /n2i^ mV^ 10. i^^iB. 20. rt|/8^^ ^ 9a» Simplify — ^- ^- 36. f#. *2- ^l^'?- ^3a 30. 2i/|. "- /45?5? 31. 3V^. St'KI^ ''-'VsxV' 38. 31^. "*• "V^- 33. VH. 39. 5a^. | I2(^^ , 34- V^- 40. -3i^|. ^5(x + y) -WIf- '■•-'=\'5^^- "■'-^ 230 ALGEBRA. EXERCISE 78. ORAL. Reduce by inspection — 1. VE. 6. VI 11. ^7t- 16. ^^' 2. Va^. 7. n. ''■ €• 17. 21/41-. 3. V'a^xK 8. VI 13. 3l/f. 18. |l/12i. 4. v'le^^. 9.JI- 10. l^f. 14. 2Vf. IK '« 19. l/2f. 5. V27xY- 20. 11/3^. 218. Making Entire Surds. It is sometimes desired to introduce the coefficient of a radical under the radical sign. This may be done by simply reversing the process of Art. 215. Ex. 1. Express 3 {^5 as an entire surd. 3i:^ = V^^'X5--=--l>'T35. Ex. 2. - 2l>^ = - 1^96 = 1^=^. Ex.3. -2v*3 = -|y48. EXERCISE 79. Express as entire surds — 1. 2T/S. 7. 2v^ 3.. 2- 3 1/5. g^ 2w.l/3m: 4. -21/5. ^ 6. -3i>^. 10. i>/6^ 6. -2v'^^=^ 11. fi/io: 12. 3m J 271 491 ^9m'^ 13. .,fc». 14. -^'^- 15. (x-l)V^. 17. "-Vs"— 2?: a + b -■<'"'>'a\ 6. V^z-^TO. 9. {TS^aVy. 2. ]^'^. 6. v^lOD^^. 10. v^9?^ 3. 1^ o^. 7. ^gaW^. 11. p^^'y'^. 4. ^W. 8. i^El^P. 12. 6|? 2|. 220. III. Reducing Radicals to the Same Index. Radi- cals of different degrees may be reduced to equivalent radicals of the same degree. Ex. 1. Reduce |/2 and y'S to equivalent radicals having the same index. v/2 = 2* = 2* = ir2' = i?'"B Ex. 2. Arrange in ascending order of magnitude |^5, y^Wy V2. We obtain Y/T2^, ^'81, y'M ; hence, the ascending order of magnitude is, ]/2j ^^3, |^5. 232 ALGEBRA. EXERCISE 81. Reduce to equivalent radicals of the same (lowest) degree — 1. VI'sLXid ^U. 7. ^% 1^9, ^5. 2. V 5 and ^% 8. Vd, ^/aF, ^oF. 3. |> B and ^'5' 9. ^Ta, ^2b, ^^^. 4. i/f and y^. 10. yx-{-y and y x — y. 6. 1^1^ and i^25. H- y^ and v^^. 6. T/6"and 1^200. 12. |/V^, yV, ]^cf. Which is greater— 13. l/3~or i^T? 17. i^ 10 or 2i^f? 14. 1^15 or 1/6 ? 18. T/2|orv^4j? 15. V^or I'/n ? 19. 3v' 6 or 2T/5f ? 16. v>^23 or 2^/2? 20. l/f or v'^S? Which is the greatest — 21. 1/B; 1^5; or i]^^^? 22. S^^^T^, 2l/6^ 2i^li}? OPERATIONS WITH RADICALS. I. Addition and Subtraction of Radicals. 221. The Addition of Similar Radicals is performed like the addition of similar terms, by taking the algebraic sum of their coefficients. The Addition of Dissimilar Radicals can only be indicated. Ex. 1. Add Vim - 2 1/50 + 1/72 - 1/18. 1/I2B -2V50 + 1/72 - l/IS = 8i/2 - 10i/2 + 6l/2 - 3l/2 = 1/2, Sum, RADICALS. 233 Ex. 2. 2l/| + il/6D + 1/15 + i/| = I Vl'5 + i Vio + 1/15 + i 1/15 Ex.3. vi28 + 2|^ r--3i.'/SI =5j^2"-V3', >^m. EXERCISE 82. Collect — 1. /IB + 1/8. 9. 2i>^lB9 - iMiB. 2. 1/50 - 1/32. 10. 1^ 24 + v'' 81 - iXBTS. 3. 21/27 + 1/75. 11. VI + 1/f. 4. 31/90-51/40. 12. 2l/f 4- 1/48. 6. 1/5 + 1/2D + 1/45. 13. %^4i 6. 4|^1B-2|^54. 7. 3i>H25- 4^^/135. 14. i^^^ + ii^l^. 8. 1^^162 + 3|;V48. 15. |v>^ j - f i^f 16.21/25^+31/4^-21/365^. 17. 3|> 2c + 3^^' 54^ - |^'2000c. 18. 1/12^ + 6 1/48"^ -6 1/3^^ 19. 2l/^"W- 3a 1/166? + 5cl/9^?5: 20. h^Ta + 1^^ 250^ - 26i>132^. 21. 1/2 + 1/18 - 1/5D + 1/IB2. 22. 1/73 -41/^3 + 2V108. 23. 6l/f-5l/24 + 12l/f: 24. 5 Vf- 12l/f + 61/60 - 30 1/^. 25. 3 1/5- 10 V"f + 2 1/45 - 5 V^. 26. 1/27 - 1/18 +1/3D0-1/1B2'+ 6 1/2-7 1/3; 234 ALGEBRA. 27. 21/63 - Syf - l/f + il/4o - 4 1/7: 28. l/2iB -\-V4E- 1/768 4- 9l/f + 1/75 - 3 i/'33f. 29. 21 >/f - 5 1/4 + 6 1/4J - 10 l/3i + ^0- Vn\. 30. 5a l/r2W^ - 36 l/27a' + 2 VHOO^OT - 40a6 >/p. n. Multiplication of Radicals. * ,, 222. Multiplication of Monomials. Since by the commutative law, = ac vbdj we have the general rule, Reduce the radicals if necessary to the same index; Multiply the coefficients together for a new coefficient ; Multiply the quantities under the radical sign together for a new quantity under the radical sign; Simplify the result. Ex. 1. Multiply Sl/e'by 2VW. 5 VWX 2 \/W= 10 |/I8 = 30 1/2; Product. Ex. 2. 5|/JX 2i/'/3"= 5],«/8 X 2^9 = 101^72, Product. 4 t7^, Product. 2 223. Multiplication of Polynomials. The Distributive Law applies here as in ordinary algebraic multiplication of polynomials; hence, Reduce each term of the mxdtiplier and multiplicand to its stm- plest form; BADWALS. 236 Multiply each term of the multiplicand by each term of the multiplier ; Simplify each term of the result, and collect. Ex. 1. Multiply 31/2'+ 5 1/3" by 3l/2'- 1/37 31/2 + 51/3 31/2 - 1/3 18 + ISl/S" - 31/6 - 15 3 + 121/6, Prodvxit. Multiply V6 - 21/12 + 5 T/S"by 3 l/T- 5 JV^ & - 21/12 + 51/3 = 1/6 - 4V3 + 5V'3 = = 1/3 + 1/5 1/3 + 1/^ 31/3 - 21/2 9 + 31/18 - 21/6 - 21/1^ =.9 + 91/^ - 21/^ - 41/3, ProducL EXERCISE 83. Multiply — 1. l/3^by 21/12. 11. l/a by ^a?F. 2. 3 1/5" by 1/13. 12. l/2 by |^3: 3. 2v' 4 by 31^6. 13. |^2^^by V^, 4. 1^24 by v''4. 14. v 9 by t/ST 6. 3 1/18 by 2 1/I2. 15. i^^F by ^/A. 6. 21^15 by 31^35. 16. vTS by l/l^. 7. 1 1/28 by f 1/35. ' 17. l^f by v'^. 8. il/fby|l/ff. . 18. l*/f by l^V. 9. 4n_byiin. 19 AW^AWl. 10. f v''' A by ||7f . * ^Z 3x-^^ A/ 4a6* '3S6 ALGEBRA, 20. V'SX Vji. 23. l^^X V^. 21. V^SX VT 24. V^ff X V'^X I^. 22. I^HI X V- W- 25. V'^ X V^H X V^360. 26. VS- V'6 + 2 l/ID by 2 ^27 27. 3l/5~- t/IO 4- 2l/r5 by 4l/5: 28. 4>/6~-3v/3> 31/2" by 2l/6: 29. i 1/f + i Vj- f Vjby 20 VT2. 30. 10 1/f - 5 Vf+ 14 1/ff by I V^. 31. 3+ 1/2 by 2-21/2" 32. 5 - 2V^ hj A i- BVK 33. 2 VT- 3 1/2^ by 4 1/3 + 5 1/2: 34. 3 VW+ 5 1/5^ by 5 1/3^- 3 VK 35. 4 1/2"- 3 l/3^by 3 1/2~+ 4 l/S: 36. 3 V5- 2 V2+ 1/3 by 3 1/5"- VW. 37. 1/2"+ 1/3"- 1/5" by V2- VW+ VK 38. 2 1/3"- 3 1/6 -4 1/15 by 2 1/3 + 31/6^+4 1/15. 39. 3 l/eO + 2 V5'- 3 l/6"by 2 1/5"+ 3 VW. 40. il/8"+ 1/S2 - 1/18 by SVS-iVS2 + 2l/l^. 41. 12l^-4l/f+4l/2l6by 6l/f-2l/f + 31/6: 42. V2x + 1/^=1 by VEx. 43. l/Bx + 1/^+T by y^TT. 44. l/x^^ - 3l/^Tl by 2VxT\. 45. a — Va — x-\- l/oTby 1/a — x + l/o: 46. 31/2^-51/^^=1 by 31/2^ + 51/^^^1. 47. VaTx — Va — x by Va-\-x + l/a — a;. 48. (21/2"+ 1/S") (31/^- l/S") (31/3""- l/2). 49. (2 1/^+2 + 3 1/2) (6a; - 5 VWV4.) (3 l/F+2 - 2 V^. RADICALS. 2¥^'^'2^X2^^'W = *^^^' ^''''^' EXERCISE 84. Divide — 1. VT by VB. 11. 4l/| by W^. • 2. 41/12 by 21/6. 12. 5l4"| by 21/^. 3. 121/15 by 41/5. 13. f l/JI by ^V^f. 4. 21/60 by 3l/5. 14. 21^3 by 3l/2. 6. 81/125 by lOl/lO. 15. |/55 by v* 36. 6. 31/405 by 9l/45. 16. |> 12 by ^^. 7. a» 1/^5^" by 2al/a'5: 17. ^"^ by v'^^S- 8. 41/18 by 51/82. 18. I/6J by x''^' 9. 3 1/iD by 5 1/28. 19. 3 l/ff by 2 l^ff |. 10. l/ifby >/f. 20. i^W by 2i^~ft. 238 ALGEBRA, 21. 5 1/35 --7 1/20 by 1/5. 22. 31/6 + 91/3 by 31/1 23. 12l/7-60V5by4VS. 24. 6vlD5 + 18l/40-45l/12by3l/m 25. 8 >/45 - 15 1/24 - 1/60 by 2 V/30. 26. 12 v'lB + 30 1^2T) + 42 y 30 by 2 K'lB. 27. 10v'i8~4l^60 + 5l^l00by 3v'3D. 225. Rationalizing a Monomial Denominator. If the denominator of a fraction be a surd, in order to make the denominator rational, Multiply both numerator and denominator by such a number as will make the denominator rational. 5 5 1^4 51^ ^"7i = 72^7^^W=^^^'^''"^'- One object in thus rationalizing the denominator of a frac- tion is to diminish the labor of finding the approximate value of the fraction. Thus, if we find the approximate 5 numerical value of — - directly, we must find the cube root of 2, and divide 5 by the decimal which we obtain. On the other hand, if we find the value of the equivalent expression, |v''4, we extract the cube root of 4, multiply by 5, and divide by 2. In the latter process we therefore avoid the tedious long division, and diminish the labor of the process by nearly one-half. 226. Rationalizing a Binomial or Trinomial Denomi- nator. If the denominator of a fraction be a binomial con- taining radicals of the second degree, since (l/a4- l^b)(V^-Vb) = a-b, RADICALS. 239 Multiply both numerator and denominator by the denominator^ with one of its signs changed; For a trinomial denominator repeat the process. 21/5 + 41/3 21/5 + 41/B SVB + V^ 31/5 -v^B 31/5 -i/B 31/5+1/3 42 + 141/15 42 + 141/15 3+1/15 =" 45-3 - 42 = 3 '^''^^^• ^^ 4 4 1+1/3-1/2 ' ' 1 + 1/3 + 1/2 1+1/3+1/2 1+1/3-1/2 2(1 + 1/3 - 1/2) 1-1/3 ^ ,^ /n ^ , = -^ ' X =2+1/2-1/6, Result 1+1/3 1-1/3 EXERCISE 85. Reduce to equivalent fractions having rational denomi- nators — 1 1-1/2 1^-1 1. 4. 7. '^ 3_ • V2 1/6 31^2 1/2 2+1/5 2v^^-3i?^i ^'21^3* ^*~2l^* * 5i?6 2 31/2-1/3 3-1/2 3. 6. 9. -• 3i/5 2l/6 3+1/2 5+1/3 21/13 + 31/ID ^^* 2^v¥* "^^* 41/3 + 31^2 31/3-21/2 SVa-4VB 11. ^ . 14. = 21/3 + 31/2 2l/a-3i/5 31/6-21/3 1/^1^1 + 3 12. 15. 4l/6~3l/B l/a; + l + 2 21/^ + 31/2+1/42 Vx' - 1 - l/x' + 1 Vx' - 1 + 1/x"^ + 1 Va+ Vb—Va + b 240 ALGEBRA, 2V2a-l + ZVa 21/^-31/2-1/52 16. =:^ 19. 3l/2a — l + 2l/a 2 + 1/B - 1/2 17. — . 20. 2-VQ+ 1/2 1/5-1/6+1 18. — z = 21. _ _ 1/6+1/5 + 1 Va-Vb+VaTh Find the approximate numerical value of— 3 1 1/3-1/2 22. 25. 28. 1/2 1/300 1/2+1^ 21/5 31/7 51/7-1 23. — -. 26. 29. 31/2 51/5 1/7 + 2 12 1 + 1/2 31/5-4 24. 27. 30 VI 2-1/5" 41/3-5 IV. Involution and Evolution op Radicals. 227. The process of raising a radical to a powei, or extract- ing a required root of a radical, is usually performed most readily by the use of fractional exponents. Ex. 1. Find the square of 3l^£ (3 V'x)' = (Sx^y = dx^- = 9 l^i». Ex.2. Extract the square root of 4a l/o^. (4a 1/^6^* = (4a • ah^)i== 4*a%^6* = 2a^6* = 2 lV&^ = 2a6 v'^, Result. Ex. 3. Extract the cube root of l^o^. (V¥F)^ = (a'b^)i = ab^ = aV¥ = abVl. This process might have been performed by extracting the cube root of a%^ as it stands under the radical sign ; thus, j/^y^^^ V^j^ = ab 1/6, RemiL RADICALS. 241 EXERCISE 86. Perform the operations indicated — 1. (l^m)*. 5. (1^^*. 9. (1^^«. 2. (I/?/. 6. {V^y\ 10. V/64aV8?. 3. (1/(7?)'. 7. (v/2?/. 11. (V- l^)«. 4. (VSB)'. 8. (}/~^\ 12. ^F^47M?^. V. Square Root of a Binomial Surd. 228. A Quadratic Surd is a surd of the second degree. Exs. 1/3; V^. A Binomial Surd is a binomial expression, at least one term of which contains a surd. Exs. V2 + bVW, or * a + 1/6. 229. A. The product of two dissimilar quadratic surds is a quadratic surd. Thus, l/2~XT/e'=T/l2-2l/3; or VoB X Vabc = ah Vc. Proof. If the surds are dissimilar, one of them must have under the radical sign a factor which the other has not. This factor must remain under the radical sign in the product. 230. B. The sum or difference of two dissimilar quadratic surds cannot equal a rational quantity. Proof. If Vadz VI can equal a rational quantity, c, squaring, a ± 2 Vab -\-b = (f, ± 2 Va6 = (^ — a — b. But Va6 is a surd by Art. 229 ; hence we have a surd equal to a rational quantity, which is impossible. 16 ^42. ALGEBRA. 231. C. If a+ Vb=x-\- Vy, then a=^x, b = y. Proof. If a + ]/6 =a: + l/^; transposing, V^ — Vy = x — a. If b does not equal y, we have the difference of two surd'" equal to a rational quantity, which is impossible ; hence, b = y, a~x. In like manner let the student show that if a — Vb = x— Vy, then a — x, b — y. 232. Extraction of Square Root of a Binomial Surd. If we expand (VW+VSy, we obtain 2 + 2l/6~+3, or 5 H- 2VW. Hence, the square root of the binomial surd 5 + 2VW is 1/2 -h V^. Hence, the square root of some bi- nomial surds may be extracted. To investigate a method of doing this, let a + Vb be a binomial surd, and let Vx -f Vy be its square root. •'■ Va + Vb = Vx + Vy (1) Square both sides, a + 1/15 = x + IV xy + y (2) .' . a = x^ y, Vh = 2Vxy. (See Art. 231.) , Hence, a-Vb = x + y- 2Vxy (3) Extract square root of (3), Va -VB ^Vx -Vy (4) Multiply (1) by (4), Va" -b = x-y (5) But a = x^ y (6) Adding (5) and (6) and dividing by 2, ^ _ a + Vd} - 6 ^ 2 Subtracting (6) from (5) and dividing by 2, ^ 2 2 2 RADICALS. 243 Hence, the square root of a binomial surd may always be extracted in form, but we get a result simpler than the orig- inal one only when d^ — 6 is a perfect square. Ex. Extract the square root of 5 + 2VK Let Vx + V^ = V 5 + 21/6 Then Vx-Vy = Vb- 21/6. {See (4) above.} Multiply! Qg, a; -y = 1/25 -24 .-. x-2/ = 1 But. x + y = 5 ,'. x = 3 .-. Vx + Vy = 1/3 + 1/2 Vb + 21/6" = 1/3 + 1/2, The required root 233. Square Root of a Binomial Surd by Inspection. By actual multiplication we may find, (1/2"+ v/5)' = 2 + 2T/10 + 5-7-r2l/m In the square, 7 + 2l/10, 7 is the sum of 2 and 5, 10 is the product of 2 and 5. Hence, in extracting the square root of 7 + 21/10, we are merely required to find two numbers such that their sum is 7, and their product 10 ; extract the square root of each, take the sum (or difference) of the roots. This may readily be done in all cases where the numbers involved are small. In general. Transform the surd term so that its coefficient shall be 2; Find two numbers such that their sum shall equal the ra- tional term, and their product equal the quantity under the rad- ical ; Extract the square root of each of these, and connect the results by the proper sign. 244 ALGEBRA. Ex. Find the square root of 18 + 81/51 18 + 81/5"= 18 + 21/80. The two numbers whose sum is 18 and product is 80 are 8 and 10. .1/18 + 81/5 = i/S'+l/lD = 21/2"+ i/IO. EXERCISE : 87. Find the square root of— 1. 17-121/2: 8. 77-241/10. 2. 23 + 4V15. 9. 87-361/5: 3. 35-121/6: 10. 14 + 31/3": 4. 9-61/2: 11. 8-J56-1/2: 6. 42 + 281/2: 12. 5i + 3l/H: 6. 73-121/^. 10a' + 9 + 6( 13. 4i-|l/3: 7. 26 + 41/3D. 14. 2m + 2l/m'' — 7i", 15. xVa^ + l. Find the fourth root of— 16. 28-161/3: 19. 193-1321/2: 17. 49 + 201/6: 20. H--I81/2: 18. 97-561/3: 21. y + yi/6: Find by inspection the square root of — 22. 3 + 21/2: 25. 23-6l/m 23. 9-2l/li. 26. 18 -121/2: 24. 21 + 12VW. 27. 7 + 41/3: 28. Prove that Va ±: VI cannot equal Vc, 29. Prove that Va cannot equal b + Vc. BADICALS. 246 VI. Solution op Equations containing Radicals. 234. Simple Equations containing Radicals. Ex. 1. Solve VxT7 - 1 = x. Transpose terms so that the radical shall be alone on one side of the equation. V¥T7 = x+l. Squaring, a;' + 7 = «' + 2x + l .-. 2x = Q a; = 3, Root Ex. 2. Solve V^T~B"+ Vx = 5. Transpose terms so that one radical shall be alone on one side of the equation. l/«TB"= d-Vx. Squaring, a; -}- 3 = 25 — 10 Vx + x .-.101/^=22 51/^=11. Squaring, 25a; = 121 In general, Transpose the terms of the given equation so that a single radical shall form one member of the equations- Raise both members of the equation to the power indicated by the index of this radical; Repeat the process if necessary. 235. Fractional Equations containing Radicals. If a radical occur in the denominator of a fraction, it is necessary to clear the equation of fractions, being careful to multiply correctly the radical expressions involved. 246 ALGEBRA. 2 Ex.1. Vx-V^^-^= y — -g - Vx — Q Multiply by Vx - 8, Vx^ - 8a: - a: + 8 = 2 Va;'^ - 8a; - a; - 6 a;2 - 8a; = a:^ - 12a; + 35 4a; = 36 a; = 9. V^4-3 '6Vx — 5 Ex. 2. l/x-2 31/^-13 Clearing of fractions, 3a; - 4Vx - 39 = 3a; - lll^i + 10 7l/i-49 Vx = 7 a; = 49. EXERCISE 88. Solve the following equations : 1. 1/^+1 = 3. 12. 2T/3F=^ = 3l/i'Tl. 2. 1/^+1 = 3. 13. SVx-l= Vx + 1, 3. 1/35-2 = 1. 14. VTTT6 = 8-1/^, 4. 5-l/2i = 3. 15. 1/^ = 3-1/^^=^. 5. 1 = 1/35^=3. 16. l/5^=n["5 = 15 - l/i. 6. V2x + d = 2. 17. I + 1/5 = l/-i^^. 7. 1 = 1^5-1. 18. f - 1/25= 1/25+^. 8. 1/2x^1 +1=4. 19. 1/5 + 1/5"+^ = 8. 9. 3 = 2-l>'35Tl. 20. 1/45+^ = 21/5^1 + 1. 10. a; - 1 = l/ST^. 21. 21/5- 1/45^=22 = l/2. 11. l/a; + l=l/13. 22. l/9a; + 35 = 7 1/5 - 3 V « 23. ^13 + ^V + 1/3+1/5 = 4. RADICALS. 247 25. l/25^^"29 - Vi^rnj = 3 1/i. 26. V^+ V1mTx = 2\/FTx. 28. ^^^+"^4- V^ = VT=R; 29. l/'9T2x = -— J:=r + |/2i. Vd + 2x SO. 2Vx-VW=^= ^ T/i^^'H 31. 3l/2STT-3i/2^^^ = ^ 32. ^-^' l^^+l l/a; + 3 l/a;-2 Vx + 2 Vx + 7 35. V^-1 l/a; + l 6v^-7 , 7l/i-26 34. — 5= — V^-1 7V^-21 l/^~a + Vx — == = a. Vx + a—Vx VWn + Vox 1 36. — ==^ — — = 1 + t;- x — a Vx— Va ^ _ 37. —= = . + 2 V^ Vx+Va ^ VWT2-Wx , 38. ^ -=4. VWT2 + 3V^ 248 ALGEBRA. 89. ^2^ + ^1^ + 1/^+1/^=1/5. 40. Vx + Va — V aS + cc" = l/a. a + 3 a - 3 2a 41. 1^ + 2 1^ — 2 a; — 4 42. J^- J? -1 = 1. 1,1 /I 43. - + - = \/- + v'^ + /XTT X b ^25 + V;i^ 44. Va:'^ + 4a; + 12 + Va;-^- 12a; -W=8. 45. l/4xM="6ar+T- l/4a;-^ + 2a; + 3 = 1. 46. l/9a;'^ + a; + 5- V 9a;^ + Va; + 6 - 1=0. 236. Summary of Principles relating to Radicals. Let the student form a summary of the principles relating to radicals, similar to tnat given in Art. 206 for exponents. Thus, Transformations of Radicals. I. Simplification of Quantity under the Radical Sign> 1. General Case. Ex. l/8a^= 2a i/2a6 2. Fraction under Radical. Ex. l/f = -J l/B. Etc., Etc. EXERCISE 89. REVIEW. 1. Write in the simplest form — 1 1 2 1 2 17 3 12 v%-i ' 2-1/5' 1/^-2' V5+V2' 1/7-1' 3V^-21/5' REVIEW. 249 Collect— 2. 2Vn - 1/75 + 1/18 - 1/2? + 1/5; 3. §1/45 - 30V^ + f 1/20 - 61/-^ + 1/500. 4. 4V'n7 - 31/75 - 6l/|^ + 18VJj - 24Vf. ' 36\l6a* a''\56''' a26\49a; 2a \ 20a;' Multiply — 6. 2 + 1/St - l/g" by" 2 + 1/3" + VE. 7. 1/a + l/oTl; - l/i by l/a - l/aT~a; - l/£ 8. ^l/f by 3l^^7; and 2l^|f by il^W- Divide — 9. 61/I2 + 31/8 - 6V30 + 4VI5 by 2V^. 10. f 1/70 - f 1^28 + 3VI05 by f 1/42. 11. a:^ _ a; -f 1 by a; + 1/a;- 1. Rationalize the denominator of— 12 2 + V^ . 14 6l/g - 41/g . * 3 - 1/^ ' 5l/r+ 31/^ j3 21/3 - 31/^ . jg 21/15 -f 8 ^ 81^3 - 6Vo _ 3 - 21/5 * 5 + Vl5 51/3 - Zi/S Find the numerical value of— 16. -^ • 18 21/5"- 31/g . V^ ' 31/5 - 41/2 17 V^-T^ . 19. 31/6 - 21/? . ' 21^7 + 1/2 * 41/2 + 5 Which is the greater — 20. 31/3 or 2VT? 22. 2l/6^ or 31^5? 21. 1/8 or 1^23? 23. V^ or l^'U? 350 ALOEBBA. Find the square root of — 24. 33 + 201/2. 26. 80 - 32V1S". 25. 35 - 12VE, 27. 107 + 121/77. Simplify 28.^^1 Vx -1 x + Vx a^+l'-i/^-i T + T/^i 1 + 1 29. ^f^- 30.{V|-2-Vf^}*- 1/1 -x» Solve— 31. xJ^^ + 2/V-^^^ - (3y' - *') V*?^ 32. 2 + l/x + 3 = 1/a; - 2 + 3. 33. i - l/^^r2 = 1/^. 1/5 3^» 34. Vx + Vx^^^ v^r^ gg 1/^ + 4 1/a; + 2 21/5 - 1 21/5 - 3 36. 21/35^F1 = 31^3^ + 5l/3F=TL l/3a; + 4 CHAPTER XVIII. IMAGINARY QUANTITIES. 237. An Imaginary Quantity is an indicated even root of a negative quantity. Exs. 1/^=^ i/^=T, V^H, The term " imaginary " is used because so long as we confine ourselves to plus quantity, and to its direct opposite, minus quantity, there is no number which multiplied by itself will give a negative number, as — 4, for instance. All the quan- tity considered hitherto that is plus or minus quantity, whether it be rational or irrational, is called real quantity. If we extend the realm of quantity considered, outside of plus quantity and its direct opposite, minus quantity, imag- inary numbers are as real as any others, as will be shown in the next article. A number part real and part imaginary is called a Com- plex Number. Exs. 3 + 2l/^n[, a + 61/^=^1. If an imaginary number exist by itself, it is called a Pure Imaginary. Thus, 3 V^^l is a pure imaginary. 238. Meaning of l/^=l. Let us consider the simplest imaginary, V—l^ and, by a geometrical illustration, try to discover how it has a mean- ing if we extend the realm of quantity outside of plus quan- tity and its direct opposite, minus quantity. If OA = + 1, and OA' be of the same length, but lying in the opposite direction from 0, OA' — — 1. Hence, we regard the operation of converting a plus quantity into neg- 251 252 ALGEBRA. ^v^ . f 1 -1 + 1 -V--1 -j ative quantity as equivalent to a rotation through an angle of 180°. If we divide this rotation into two equal rotations, each of these will be a rota- tion through 90°. But in seeking a square root of —1 we seek a B factor, l/— 1, which multiplied by itself will give —1. The result of a rotation of -f 1 through 90°, rotated again through 90°, gives -1. Hence, V— 1 must be equivalent (geometrically) to the result of ro- tating the plus unit of quantity through 90®. Hence, V- 1 on our figure will be represented by OB. Hence, it is easy to see, also, that B^ V^^l X V^^ = -1. We thus perceive that the introduction of imaginary quantity enlarges the field of quantity considered in algebra from mere quantity in a line to quantity in a plane. This gives a vast extension to power of algebraic processes and introduces many economies in them, as will be found by the student who pursues the study of mathematics extensively. In taking up the subject for the first time we consider only a few of the first properties of imaginaries, so called. 239. Fundamental Principle. We regard all the ordinary laws of algebra as applying to imaginaries, but, owing to the nature of a square root, one modification of these laws as ordinarily stated must be made. This fundamental new principle is that V — 1 X V^^^l = -1. Besides the above geometrical illustration (Art. 238), it is important to state formally the algebraic reason for this principle. The square of V — 1 must be such a number that when its square root is extracted we shall have the original quantity, If we use the law of signs in the most general form, ( V^=~l/ = V^l X V^^l = l/r= zt 1. IMAGINARY QUANTITIES. 253 Now, if we extract the square root of +1, we shall not have V^^^l. But if we extract the square root of — 1, we shall have V— 1. Hence, we must limit the product V — 1 X V^^^ to — 1. Likewise V^^X V~^^ = VaV^^l X VhV^^l or, in general, The product of two minus signs under a radical sign of the second degree is a minus sign outside of the radical. 240. Reductions to the Typical Form A -{- BV^~^1. It may be shown that any imaginary expression of any degree of complexity can be reduced to the form a + bV — 1. We will limit ourselves at this time to showing that any com- bination of the sum, difference, product, or quotient of im- aginary expressions of the second degree reduces to this form. First. Sum or difference of two complex numbers. a + 6l/^T ± (c + dV^^ = (a i e) + (6 ± d)V^^ = ^ + BV=~1 Second. Product of two complex numbers. (a + bV^^ (c + dl/^T) = ae-bd + {be + ad)V^=T. ^ A' + B'v^=n: Third. Quotient of two complex numbers. c + dV^ _ c + dl^^^T ^ a- h\^^^ a + 6V^^=T a + 6i/^=T: a - bV^^ go + bd + (ad - 6c)V- 1 a^ + b^ ac -f bd , ad - be ^y—^ = A'' + b'^i/^=t: Hence, any combination of the sum, difference, product, and quotient 254 ALGEBRA. of complex numbers may be reduced by successive steps to the typical form. This serves to illustrate the fundamental value of the imaginary unit, 241. Powers of V^ — 1. {V^=^y = - 1, . • . ^2 = - 1. {y^^f = {V^^^YV^^ = - V~l, ..e = -i. (See OJl^of figure {V^Ty = {V~-^iyV~i = + l, . • . i* = l. in Art. 238.) The symbol i is used for 1/— 1. Thus the first four powers of V- 1 are T/- 1, — 1, — V^HT, + 1 ; and for the higher powers, as the fifth, sixth, etc., these four results recur regularly. The same fact is plain from the figure in Art. 238. 242. Equational Properties of Imaginaries. I. If an im- aginary expression, ac + yV —\, equals zero, then x = 0,y = 0. Proof, x + y V — 1 = ,-. x^ + y'^O, which, since « and y are both real, can be true only when x = 0,y = Q. II. If two imaginary expressions he equal, the real part of one equals the real part of the other, and the imaginary part of the one equals the imaginary part of the other. Proof Let x + y V^^l =a + h V^^l. .'. x-a + (y-b)\/^=n.=0. . * . by I. of this Art., x — a=0, .' . x = a, y-b=0, .' . y = h. 243. Conjugate Imaginaries. If two complex numbers IMAGINARY QUANTITIES. 255 differ only in the sign of their imaginary part, they are called conjugate imaginaries. Exs. 3 — i/"=l and 3 -f- i/^T; a-\-b i/ — c and a—b -y/ — c. 244. Operations with Imaginaries. It follows from Art. 239 that, in performing operations with imaginaries, we Use all the ordinary laws of algebra^ with the exception of a limitation in use of signs, which may be mechanically stated as follows : The product of two minus signs under the radical sign of the second degree gives a minus sign outside the radical sign. But in dividing first indicate the division and after- wards rationalize the denominator. Ex.1. Addi/— 9, — 3 + 2i/— 1, 7 — 2i/=^16; v^^= 3 i/^^nr — 3+2 i/^^~l= — 3 + 2 i/"^^! 7_ 2 1/^^16= 7_8i/"=ri 4 — 3 i/— 1, Sum. Ex.2. Multiply2i/— 3 + 3i/— 6by3i/— 3— 5i/— 27 2i/^=^+3i/=6 3t/=^— 5i/^^ 6(-3)— 91/18 + 10 i/6 +15t/T2 18—27 i/2 + 10 i/ 6 + 30 >/ 3, Product, Ex.3. Divide — 2 ]/ 6 by i/— 2. — 2 i/'6 — 2 v/'6 y^=^ l/ — 2 i/- 2 i/— 2 — 2i/'=12 , ^ = — ^^72 — = 2i/— 3, Quotient, Ex. 4. Extract the square root of 1 + 4 1/ — 3. 1+4 1/— "3 = 1 + 2 1/^=12: The two numbers which multiplied together give — 12, and added together give 1, are 4 and — 3. . l/l + 4i/— 3 = 1/4 + 1/— 3 = 2 + 1/— 3, ^esttW. 256 ALGEBRA. EXERCISE 90. Reduce to the form aV^^l. 1. I/- 9: 3. v-im. 5. 2V~=i;. 2. 1/^-25". 4. - V^=^. 6. -ZV^=^. 7. -Ji/^=^SB: 8. 11/^=^324: Collect — 9. 7 V'^^ + 3 1/^^=^^- 10 i/^^a 10. 2V^T-ZV- 121 + 51/^^4. 11. l/'='400 + 2l/~ 900-51/- 144. 12. 1/^:^:2 - T/"~27'+ 21/^^+ 1/^=^75: 13. 5 V~^^- 3 V^=T+ 4 1/^^50"- 1/-200. 14. 2l/"=^^-3al/'^^T-r -V - 16a* - 1 1/ - 36al 15. a + 6l/^=T-6-al/"=T- l/'=^ + 2l/^=^^-a. 16. (a-26)l/'^=^~(2a + 6)l/^=T: Multiply — 17. 1/^=1: by 1/^^^ 22. y^=^by -31^^=^ 18. 2v"=^by 31/^=^ 23. 2 l/"=T4 by - 2 y^2r 19. V^=l> by -2V^=^b. 24. -bV^=2 by - 21^^=^: 20. - 1/^=^ by — l/"=T 25. — V^^ by Vy^=^. 21. - i/=T:2 by - T/=T8. 26. - aVY=^a by - l/(a - 1)' 27. V^=n. + V~=^hy l/^=T- 2 1/^^2: 28. 3l/=^- 2 1/'^=^2" by 21/^=^+31/"=^ 29. 2l/2-2l^=^by 3l/2 + 3l/'^^2: 30. ZV^^- 2l/^=^by 41/"=^+ SV^^^ IMAGINARY QUANTITIES. 257 81. l-3v^^^Sby 1+51/^^^^. 32. V=^-V^=^2+ V^^ by V^^+ V^=^+ 1/^=^ 38. x - 2 + 1/^=^ by a; - 2 - V^^^. 34. al/ — a + 61/^=^5 by al/^=^— 6l/^=T. 35. a; - 1 — V^=n. by a; - 1 + V~^l. 1-V^= ^ , 1 + V^=^ 36. X by a; 2 -^ 2 Divide — 37. -vlSby V^^. 39. - 6 l/^T5 by 2 1/"=^. 38. -V^^^T2 by -1/"=^. 40. 8V"=^ by -2v/a. 41. 2V=n^-AV'=lF+ lOl/BOby -2i/-'3. 42. a 1/"=^ - 2a l/^^- a VBc? by - a l/^=^. Express with rational denominators — 1 SV^=~5-V~^^ 43. 47. 3-l/^"2 2v^^^-3l/-2 2 - 1/=^ l/T="l + VT=^ 44. 48. 2 + V^=B 2l/r=^- 1/^"^=! 45. -— • 49. -^ ;=/-• 23/2+1/^=^ 2-3l/^=nL a + ftl^=n[ 3v/2 + 2l/"=^-l/^^ID 46. . 50. " a-bV^=A 31/2-21/^^ + 1/^=10 Find the square root of— 51. S-QV^e: 64. 121/ID-38. 52. 1-21/^=^6: 65. -29 -241/=^ 53. 12 1/==^- 6. 56. 7 + 40l/^=T. 17 258 ALGEBRA. 57. Find the value of, (v"=^)'; (-1/^=1)*; (V^^^l)*; (-l/-iy; (l/"-=l)-^ (-V'--l)-^ (1/"=^)* In expanding binomials containing imaginaries, the labor is perhaps less- ened by the substitution of a letter, as i, for the imaginary, V^^l, and after simplifying this derived expression, in i, return to the imaginary, V'-l. Fot example, let it be required to simplify the expression, 3(1/— 1 + 2)^ - (2l/~T - 1)2. Substitute i for V~=i:. 3(i + 2)2 - (2i - 1)2 = 3i2 + l2i + 12-4*2 + 4^-1 = - i2 + 16i + 11 = - {v^^y + 161/^^ + 11 = 12 + 161/^=^", Eesult Simplify — 68. (1/^1-1)' -(1/^=^-1^ + 2(1/^=1-1). 69. (1/=1 - 2) (3 V^l + 1) - (1/^=1 - By - (l/==^\ 60. ( V^l - 1)* + 3( 1/=~1 - 1)' + 4( l/'^I - 1)1 61. Prove that the sum and the product of any pair of con- jugate imaginaries are real. (See Art. 243.) 62. li x= ^~ ^^""^ , find the value of, 3x' - Ox + 7. 0£ 2 s?-6x' + 2x-l. 63. If a; = ^"^^^^^ ' find the value of, lOa;* -• 8x + 3. CHAPTER XIX. QUADRATIC EQUATIONS OF ONE UNKNOWN QUANTITY. 245. General Problem. The relation of the square of some unknown number (as well as of its first power) to known numbers may be given in the form of a more or less complex equation. It then is often required to reduce this complex relation to some simple relation from which the value of the unknown number may be at once recog- nized. 246. A Quadratic Equation of one unknown quantity is an equation containing Ihe second power of the unknown quantity, but no higher power. A Pure Quadratic Equation is one in which the second power of the unknown quantity occurs, but not the first power. Ex. 5x^-12=0. An Affected Quadratic Equation is one in which both the first and second powers of the unknown quantity occur. Ex. 3a;^- 7a; + 12 = 0. PURE QUADRATIC EQUATIONS. 247. Solution of Pure Quadratics. Since only the sec- ond power, x', of the unknown quantity occurs in a pure quadratic equation, Reduce the given equation to the form x^ = c; Extract the square root of both members. 25» 260 ALGEBRA, Ex.1. Solve — - — = —- — Clearing of fractions, 4x' — 48 = Sx'^ — 12 Hence, x" = 36 Extracting the square root of each member, a; = + 6, or — 6. That is, since the square of + 6 is 36, and also the square of — 6 is 36, X has two values, either of which satisfies the orig- inal equation. These two values of x are best written to- gether. Thus, a; — ± 6, Roots. Ex.2. Solve x'-b ~ x' — a ax'- -a' = bx'-b^ ax'- 6x^- ■a'-b' X — ^a + b --±Va + b, EXERCISE 91. Solve — 1. 5x^-80. g J L= 5 2. 3x^-5=a;' + 3. * 4x^ Sx' ~^' 9. 3. ix'-l=i-Sx\ 4. l-|x'=-x^-4f. 2x-l 2a; + 1 x" , ^ 10. ax'' + a' = 5a' — 3ax'. 5. — -4 + x' = 0. 8 ^ 11. ax' + c^b. ^ ^Z:5_|^-x\ ^2. ^ + 2a ^ x-2a ^,^,^ 5 X — 2a X + 2a U 6 x-c 3x ^_3^^^+5^3_ i3._2^+5x±2c^^^^^ QUADRATIC EQUATIONS. 261 14. (ax 4- by + (ax - bf = 10b\ 15. (x + a)(x-b) + (x-a) (x + 6) -2(a' + 6» + ab). 16. 3(2a:-5)(a; + l)-2(3a:4-2)(2a:-3)-a;-9. AFFECTED QUADRATIC EQUATIONS. 248. Completing the Square. An affected quadratic equa- tion may in every instance be reduced to the form x^ + px — q. An equation in this form may then be solved by a process called completing the square. This process consists in adding such a number to both members of the equation as will make the left-hand member a perfect square. The use of familiar elementary processes then gives the values of x. Thus, to solve x'' + Qx = 16, take half the coefficient of x (that is, 3), square it, and add the result (that is, 9) to both members of the original equa- tion. We obtain a:' + 6a; + 9 - 25. Extract the square root of both members, a; + 3 = ±5, Hence, a; = — 3 ± 5, That is, a; = - 3 + 5 = 2, Also, x= — 3 — 5= — 8, a Hence we have the general rule : By clearing the given equation of fractions and parentheses, transposing terms, and dividing by the coefficient of x^, reduce the given equation to the form x^ + px = q; Add the square of half the coefficient of x to each member of the equation; Extract the square root of each member; Solve the resulting simple equations. •262 ALGEBRA,^ Ex. 1. Solve Gx' - 14a: = 12. Dividing by 6, x^ — ^x = 2 Completing the square, a;' - |x + (|)' = 2 + ff = W* Extracting square root, a; — J = dz ^ ic = 3, or — I, i?oo^. Ex.2. Solve 3x^ = 2(1 + 2a:). Clearing, 3a:' = 2 + 4x Transposing, 3x' — 4x = 2 Hence, a:^ ~ |a^ = t x = i±VW = ^^^^ Roots, o EXERCISE 92. Solve — 1. x^ + 14a: = 32. 14. 35 - 2a:^ - 3ar. 2. x^ + lOx = 24. 15. 3a: + 77 = 2x\ 3. a:'-8a:-20 = 0. 16. 6a:' - a: - 35 = 0. 4. x'-5a: = 6. 17. 3x' + ia: = l|. 6. x^ + 11a: + 24 = 0. 18. 3a;' = ix + 2|. 6.3.^4-4x^7. I9.i = ^ + x'. 7. 5x'-6x = 8. 6 8. 2x^-5x--7. 20. ^^ + -^- = 2J. 9. 3x'^ + 7x-26. 2 x-1 10. 4x^-h8x-5 = 0. 21. ^-1 + — ^=0. ox a: ~r 2 11. 6^-5x-6 = 0. ^„ 3x + 5_, 2.-5 12. 2. + 3|a:' = 4. ^- ^ + 4 ~ x-2 ' 13. a:' + 5 = ^. 23. ^^^^ - -^±i = - i 3 x + S 2.-3 ' QUADRATIC EQUATIONS. 31. i(^ + l)-|(2«-l) = „„ 2x-l Zx-4 _ , 4a; -14 x + 1 x-1 l-x" x-B hx-1 2a; + 5 o4. _ _ + 3x-2 4-9a;=' 2 + 3a; „, 2a;-l l-3x x-1 , 35. = 4. a; + l a; + 2 x-1 36. a;' + 2a; = l. 40. lla;»- 12a; = -3. 37. 3x'-5x=-l. 41. 2a:' + 5a;=-4. 38. 9ar^-18a; + 4-0. 42. 3a;'-7a;=-5. 39. 5x' + 3a; = l. 43. 9x'-6a; + 5=0. 44. 3a;(a; + l)-(a;-2)(a; + 3) = 2H-(l-a;)'. 45. (x + l)(x-5)-2a:(a;-l) = l-(l-2a;)*. 46. ^^-i + ^^:^^2. x' + x+l a;' — a;+l N. B. The verification of equations having irrational roots is a profitable ercise. 249. Literal Quadratic Equations are solved by the same exercise 264 ALGEBRA. methods employed in solving quadratic equations with nu- merical coefficients. Ex. 1. Solve ^-^=% + a). 2 6 6^ Clearing, Sx'^ - ax = ax ■¥ o? Hence, ^x"^ — lax = a^ 3 3 ^-().(ff = if ^ Boots, 3 Ex. 2. Solve (a - 6) V - (a' - 6^^^ = - a6 ,2 _ «_±J'a; = ^^ (a - 6)^ a + b , a-h 2{a-b) 2{a-b) X = —^ , -^— , Roots. a — b a — b EXERCISE 93. Solve— 1. a:' + 4ax = 12a^ 10. 2aV + ahx = 156\ 2. a;'^ + 46a; = 216^ 11. x' - (a + l)a; = - a. 12. x' H = 4. x' + bahx = Q>a'b\ « 4^2 5. 6a:' -hx = 12b\ 13. x' + (2 - 3a)a; = Ga. 6. 3a;' + 4ccZa; = ISc'dl 14. 3aV + a(36 - 5)a; = 56. 7. 2aV + aa: = 3. 15. abx' + (a' + 6')a; + a6 = 0. 8. 7cV ~ lOaca; + 3a' = 0. 16. ax' - (a' — l)a; = a. a a' x' a' QUADRATIC EQUATIONS. 265 18. X — = a. 19. 4(x' - 1) = 6(4x - 6). X — a 20. Ca-{-b)x'-(a-b)x ^=0. a + b 21. abx" = -^ fxCa + 6) - — 1 • . a6 L abj x c a + b 23. a(a:' - b') + b(x' -b^-tc) + cx = 0. 24. (a + c>' — (2a + c)x + a = 0. FACTORIAL METHOD OP SOLVING EQUATIONS. 250. Factorial Method for Solving" Quadratic Equa- tions. If any factor of a product equals zero, the entire product equals zero. Hence, if a quadratic equation be reduced to the form ax^ + bx-{- c = 0, and the left-hand mem- ber be factored, and each factor be made equal to zero, the values of x thus obtained will satisfy the equation, and therefore be its roots. Ex. Solve a;' + 5a; — 24 = 0. Factoring, ' (ck + 8) (x -3) =0. Hence, letting a; + 8 = 0, and a; — 3 = 0, a; = — 8, a; = + 3, Roots. 251. Factorial Solution of Equations of Higher De- grees. Since the principle of Art. 250 applies to the prod- uct of any number of factors, equations of degree higher than the second may often be readily solved by this method. Ex.1. Solvea:(a;-l)(a: + 3)(a;-5)=0. The roots are a; = 0, 1, — 3, 5. 266 ALGEBRA, Ex.2. Solve x' + 1 -0. Factoring r, {x + l){x'- - X + 1) = -0 x+ 1 = = 0, gives X ==- 1, Root, Also, x' - X - f 1 = = Whence, x"- - X = = — 1 X - -I i \^^~ - 3, Roots. Ex. 3. Solve x\ +x'- 4(x' - 1) = 0. Factoring, x\x + 1) - 4(a;' - 1) = (a: + 1) {x" - 4a: + 4) = (a; + 1) (a; - 2) (a: - 2) = a;.- - 1, 2, 2, -Boote. EXERCISE 94. Solve by factoring — 1. a;' + 8x + 7=0. 11. a:*- 5x' + 4 = 0. 2. a;'-5a; = 84. 12. a;^-a;*-a; + 1 = 0. 3. Bx'' - X - 15. 13. (2x - 1) (Bx'^ - x - 2) = 4. Ba:^ + 7a;-90. 14. 3(x' - 1) - 2(a; + 1) = 6. 12x^ - 5a: = 3. 15. 5(a:^ - 4) = 3(3: - 2). 6. 3a:^ - lOa: + 3 = 0. 16. 7(a:* - 1 B) - 53a:(a;^ - 4) - 0. 7. 24a:^ = 2a; + 15. 17. 3x(ar^ - 1) + 2(a: - 1) - 0. 8. 3aV + lOax = 8. 18. a;' - 27 = 13a; - 39. 9. a:* = IB. 19. 2a:'' + 2x' = a: + 1. 10. a^ - 8. 20. 2a:' + Bx'' = 3a;' + 8x - 3. 21. Find the six roots of x" - 1 = 0. EQUATIONS IN THE QUADRATIC FORM. 252. Simple Unknown Quantity. An equation contain- ing but two powers of the unknown quantity, the index of one power being twice the index of the other power, is aa QUADRATIC EQUATIONS. 267 equation of the quadratic form. It may be solved by the methods already given for affected quadratic equations. Ex. 1. Solve x* - 5a;' =- - 4. Adding (f )' to both members will make the left-hand member a perfect square, giving X* - 5a;2 + (f )2 = f Hence, ic'' — f = =t f x^ = 4, or 1 x = ^2, ± J, Roots. This equation might also have been solved by the factorial method. Ex.2. So\ye2v'x-'-Sv'x-' = 2. Using fractional exponents, 2x~^-Sx~^ = 2 Whence x~^-ix~^ = l a:"^-() + T\ = M a;-^-f = ±f x-^ = 2, -1 Whence x^ = h -2 x = l - 8, Roots. 253. Compound Unknown Quantity. A polynomial may be used in the place of a single quantity as an unknown quantity. Ex.1. Solve(2a;-3)'-6(2a;-3)=7. Let 2a; — 3 = ^, and substitute. We obtain ^/^ - 6^ = 7 Whence 2/ = 7, - 1. Hence, 2a; - 3 = 7, also, 2a; - 3 = - 1 . • . x = 5, Root. a; = 1, Root. 268 ALGEBRA, Ex. 2. Solve V^rri2 + v'FTT^ = 6. 1 1 This equation may be written, {x + 12)^ + (a: + 12)^ = 6. Let {x + 12)^ = ^ ; then {x + 12)^ = y^ Hence, substituting, 2/^+^ = 6. Whence 2/ = 2, or — 3. . • . 1/a: + 12 = 2, Also, l/a; + 12 = - 3, a; + 12 = 16, a; + 12 = 81, X = 4, Boot. X = 69, BooU Ex. 3. Solve x\ - 7a; + Vx'-lx + l^^ 24. Add 18 to both sides, a:2 - 7a; + 18 + Vx' - 7a; + 18 = 42 Let Vx'^ - 7a; + 18 = ^ ; then y"^ + y = 42 y = 6, or - 7. Hence, l^a:'' - 7a; + 18 = 6, Also, 1/a;"^ - 7a; + 18 = - 7, a;2 - 7a; + 18 = 36, a;^ - 7a; + 18 = 49, a; = 9, - 2, Boots. x =-• 1(7 =1= VlTB), iJoote. EXERCISE 95. 10. 9a;"* + 4 = 13a;~*. n. Zv^-Wx = -2. 12.51^^=81^^ + 4. 13. 7l^^^-4l'/i^ = 3. 14. 3 l/2i- 21^2^ = 1. 15. (a;-iy + 4(a;-l)=21. 16. 2(a;^-3y-7(a;^-3) = 30. 17. 6(a;^ + l)=' + 13(a;'4-l)=-28. 18. 2l/2^^="3 + 5l^2a;-3 = 7. Solve 1— 1. %'- 17x^ + 16=0. 2. 4a:*- -13a;^ + 9 = 0. 3. 27a:« = 35a;'-8. 4. 3.*- -5a;i = 2. 5. 27a;' 4-19x^ = 8, 6. 3a;^ = = 4a;^ +4. 7. 2V^ = \/x-\-\. 8. 3a;" S-f5a;~* = 2. 9. 6x-' ^-x"* = 12. QUADRATIC EQUATIONS. 269 -d- !)■-(! -!)-■ 20. 5(4x + 1) - 27 VWTT= - 10. 21. 3(3x^ - 2x + 1) - 4 l/3x^-2a; + l = 15. 22. 2(2x^ + 3x - 4)^ - Z(2x' + 3x - 4) = - 1. 23. a^ + lx -3l/x-' + 7a;H-l = 17. 24. 6(a:' + a;) - 7l/3x(a; + 1) -2 = 8. 25. 3x^-7 + 3 l/3^^^=T6^T^r= 16a;. 26. 32;~^-7x*-4. 28. 16x^-22 = 33;** 27. 3x^ = 8x~*-10. 29. 2x''-l/x^-2a;-3 = 4a; + 9. 30. 5(2x' - l)i - 4 = f (2a;'^ - 1) - ^ 31. 3(x» + 1) ~ ^ -It 5 = 2(a;' + l)i KADICAL EQUATIONS. 254. Radical Equations resulting in Affected Quad- ratic Equations. If an equation be cleared of radicals by the methods given in Art. 235, the result is often a quadratic equation. Ex. Solve VWYVJ^ VxT2'-= VHdx + 16. Squaring, 3a; + 10 + 2l/(3a; + 10) (a; + 2) + a; + 2 = 10a; + 16 Hence, V{Zx + lGY{x~^T) = 3a; + 2 Squaring again, Sa;^ + 16a; + 20 = ^x^ + 12a; + 4 6a;^ - 4a; = 16 a; = 2, -f Substituting these values in the original equation, the only value that verifies is a; = 2, which is the root. The other value, a; = — f , is not a root of the original ecjuation, but is introduced by sijuaring in the pro- cess of clearing the equation of radical signs. It satisfies the equation, ^. -.A ,. yzxTiQ - yirr^ - vi^x + le. 270 ALGEBRA. EXERCISE 96. Solve — 1. x-l--= VSx - 5: 4. 3x - 2l/Bi = 6. 2. 2a; + 1 = VTx + 2. 5. V^SxTT - 2 l/2x = - 3. 3. a; - l/S^ = 6. 6. 2 + l/2xTT= VbxTT. 7. 1/30; + 7 = V^Tl + 2l/ic - 2. 8. VWTl = 2Vx- Vx^^^W. 9. 1/^'^=^ + Vx + 2d' = j/x~T7a\ 10. 1/a; + 2a + 2 Ka; — 2a = 3a - 1. 11. 3l/ar' + 17-2F'5r' + 41 + l/^n=0. 12. V'2x + 3-|l/r=^ = il/llx-33. 13. l/xT4 + VWTl - Vdx + 4 = 0. 4a; + 1 14. 2l/5x- t/2x-1 15 l/2a;-l 31/2^-5 9-21/^ 3+1/25 T/25-3 16. VxTJ-^ V^x — i = VdxTT- 17. V^T5"+ l/3x + 4 — l/12a: + l=0. 18. l/12x'^ - a; - 6 + l/12x^ + a;-6= l/24a;"^ - 12. x+vV^a' a;- l/a;-^-a'' ^/_,^ ^ 19. -_ ^-^=:^ = SV^^a\ X - Vx"^^^ x+Vx'- a' 20. l/4x 4- 3 + V2x + 3 =- V5x + 1 + 1/^+3. OTHER METHODS OP SOLVING QUADRATIC EQUATIONS. 255. I. Completing the Square when the CoelBficient of ac' is a Square Number or can be Readily made One. If in a simplified quadratic equation the coefficient of x^ is a QUADRATIC EQUATIONS. 271 square number, we may readily complete the square by di- viding the second term by twice the square root of the first term, and adding the square of the quotient thus obtained to both mem- bers of the equation. That this process gives a perfect square is readily seen from the fact that {ax + by = aV + 2a6a; + b\ Hence, given aV + 2abx, the term 6' with which to complete the square may be obtained by dividing 2ahx by 2aic, and squaring the quotient. Ex. 1. Solve 9x' + 4x = 5. To complete the square, take the square root of 9a:' (that is, 3a:), and divide 4a: by twice 3a: ; this gives as a quotient f . Add the square of this, %, to both members. . • . 9a;2 + 4a: + I = ^. Whence 3a; + f = =t | ' 3a; = I, or - 3 a; = f , or — 1, Roots. Ex.2. Solve 8a;' + 3a; = 26. To make the coefficient of x^ a square number, multiply both members of the equation by 2. . • . 16a:2 + 6a: = 52 Completing the square, ISa:' + 6a: + ^ = 52 + y*^ = -^^ 4a: + I = ± ^^ X = -V-> "~ 2, Roots. 256. II. Hindoo Method to Avoid Fractions in Com- pleting the Square. After simplifying the equation, multiply through by four times the coefficient of ac!^, and add to both sides the square of the coefficient of ac in the simplified equation. The reason for this process is evident, since if aa^ + bz=^o be multiplied by 4a, we obtain 4aV + 4a6a; = 4ac. 272 ALGEBRA. The addition of h"^ gives on the left-hand side 4aV + Aabx + 6', which is a perfect square. Ex. Solve Sx' - 2x = 8 by the Hindoo method. Multiply by 4 X 3, or 12, 362:2 _ 24a; .= 96. Add the square of the coeflScient of x in the original equation ; that is, (-2)2, or 4. 36a;2 - 24a; + 4 = 100 62: - 2 = ± 10 6x =12, - 8 X --= 2, - I, Roots. EXERCISE 97. Solve — 1. x' + Bx = 6. . 11. (x' + 3)' - 1{^ + 3) = 60. 2. 3x'-a; = 2. 12. 4a:-'^- lOla:"' + 25 = 0. 3. 6x^ + 52; = 4. 13. 6a:^-5a;^-6. 4. 1x^ + 11a: = 6. 14. 4a:^ + 4a:^ = 3. 6. 8a:'^-2x = 3. 15. 6 v'^^^ - 1 1 i^^r^ = 10. 6. 4a:' + 4a: - 35. 16. 3(a: - 2)=^ + 5(a: - 2) = 12. 7. 9x' - 3a: = 30. . _1 :^ 1 JL 8. 16a:* -40x^ + 9 = 0. *^ 2a: ~^ 2a 9. 6a:' -ax = 2a\ 18. (a - l)x' + (a + l)a: = - 2. 10. 4aV + 5aa: = 21. 19. (a' - 6')x' + (a' + i^x = aft. 20. 8a^a:' - (66' + 4a')a: = - Zah. 257. III. Use of Formula. Any quadratic equation can be reduced to the form ax' -\-hx-\- C—-0. Solving this equation by use of Art. 248, ^ = 2a "• QUADRATIC EQUATIONS. 273 By substituting in this result, as a formula, the values of a, 5, c in any given equation, the values of x may be at once obtained. Ex. Solve 6x' + 3a; — 2 = by use of the formula. Here a = 5, 6 = 3, c = — 2. Substituting for a, 6, c in the above formula, ^ - 3 ± 1/9 + 40 10 10 ~ ~ 1 ^, - 1, Boots. EXERCISE 98. Solve by the formula — 1. 2x^ + bx = l. 9. 2a;' + ax = 6a*. 2. 4a:^-3x = 7. 10. 126' = 3aV-5a6a;. 3. 6x' H- 7a; = 10. 11. aa;' = (H-a>-a. 4. 4a;'' -11a; = 3. 12. 2a;* -3x4 = 9. 6. 12x^ + 8a;-15=0. ' 13. 6a;-7T/x-20. 6. 6a;^ + 13a; + 6=0. 14. 8x-^ + 19x"'^ = 27. 7. 33a;'' - 17a; - 36. 15. 4x-*-73x-^ = -144. 8. 12a;^-a; = 6. 16. 3(a;'-l)' = 7(x'-l) + 6. 17. 1(23- — 3) — I- -2)' = K18-5x). 1ft ^ r a; l1 ^ . 18. — : — - ^ + d J {c-dy 19 2^ + ^ 7- ■X 7-3x •2(2a;-l) 2a; + 2 4 - 3a; 20. (a + 3)V- (a^- - 9)x = 3a. Find the approximate numerical values of x to four deci- mal places. 21. x'-4x + l=0. 23. 2x' + 3 = 10a;. 22. 9x'-12a;=-l. 24. 5x' + 2a;=2. 18 274 ALGEBRA. EXERCISE 99. REVIEW. Solve — 1. Gx^ + X = 1. 9x+- = -+-' 2. Sx"" + -\^-x = -2/. b X a 2. 10. V4x - 3 = 1 + 1/a; + 4. a:» - 16a; = 0. n. Sx ^ - 7a: 3- 5. SVx -\2x 2 = 5. 12. a;* - 27a: = 0. ^a:-l_^ x oi 13. 5a:-i + 6a: ^ = 11 0. + — Zg. X x-\ ^^ x-Z _ x±_Z ^ 6. 7 3 _^ 2a: + 1 _i, 'a:-4a; + 4^* * a: - 7 3a: ^ * 15. 2a:2 + 2a: + 1 = 0. 8. a:4- 1 _ g + 1 16. 5(a: + 2)* - 3(a: + 2)^ + 2. T/a: Vol 17. ZVxTJ - 5l^x~+l. = 2. 18.3-(l/^ L\=l/^ 1 — ^ Vx' Vx 19. 20-2(a: + ^y = 3(a: + ^)- 2Q abx^ + 1 = (Q^' + ^'^)^ . • a2_52 a2-62 21. -^4-^=1. VX^ 1^3 22. ^^ = 1+1+1. X -{■ a + b X a b 23 31/^ - 41/2 _ 21/^ - T/g' . ' 4Vx - 21/2 3l/3a: - 51^6 24. 4x' -71/2x2 + 3a;_ 2 = 19 - 6a:. 25 1 ^4a:-3 a;l 3 4 / ' la: + 1 a: - 1 / '^ 26. (x^ - 5xY - 8(a:2 - 5a:) = 84. 30. Sy^ = 36 + 41/^' _ 7^ 27. (a;2 + 6a:)" ~ 2(a:2 + 6a:) = 35. 31. 104 + UV^ = Si/^. 28. |n2 = 10 - ^n. 32. p'^ + a^ + y = lay + a + 6. 29. 9n* = 23n2 + 12. 33. a:^ - 1 = (1 - a:)l/2 - 4a;. QUADRATIC EQUATIONS. 276 EXERCISE 100. 1. Find two consecutive numbers the sum of whose squares is 61. 2. There are two consecutive numbers, such that if the larger be added to the square of the less the sum will be 57. Find the numbers. 3. There are two numbers whose difference is 3, and if twice the square of the larger be added to 3 times the smaller, the sum is 56. Find them. 4. Seven times a certain number is one less than the square of the number next larger than the original number. Find the number. 5. A gentleman is 3 years older than his brother, but twice the product of their ages is 17 more than 21 times the sum of their ages. How old is he? 6. If a train had traveled 6 miles an hour faster, it would have required 1 hour less to run 180 miles. How fast did it travel ? 7. Two numbers when added produce 5.7, and when multi- plied produce 8. What are they ? 8. What are the two parts of 18 whose product exceeds 8 times their difference by 1? 9. A gentleman distributed among some boys $9; if he had begun by giving each boy 5 cents more, 6 of them would have received nothing. How many boys were there? 10. A cistern is filled by two pipes in 18 minutes ; by the greater alone it can be filled in 15 minutes less than by the smaller. Find the time required to fill it by each. 11. A certam number of eggs cost a dollar, but if there had been 10 more eggs at the same price, they would have cost 6 cents a dozen less. What was the price of a dozen eggs? 12. One number is | of another, and their product, plus their sum, is 69. Find the numbers. 276 ALGEBRA. 13. Find two numbers whose product is 90 and quo- tient 21. 14. Find two numbers whose difference is 4 and the sum of whose squares is 170. 15. Find two numbers whose product is 42, such that if the larger be divided by the less, the quotient is 4 and the remainder 2. 49 [Let X and — represent the numbers.] 16. A number of boys bought a boat, each paying as many dollars as there were boys in the party; had there been 5 boys more, and each paid f as much as he did pay, they would have lacked $10 of the price of the boat. How many boys were there ? 17. A company of gentlemen agreed to buy a boat for $7200, but 3 of their number died, and each survivor was obliged to contribute $400 more than he otherwise would have done. How many men were there? 18. Divide the number 12 into two parts, such that the sum of the fractions obtained by dividing 12 by the parts shall be ff. 19. The length of a certain rectangle is twice its width, and it has the same area as another, 1^ times as wide, and shorter by 4^ feet. Find its length. 20. A rectangular lot is 8 rods long and 6 rods wide, and is surrounded by a drive of uniform width which occupies | as much area as the lot. Required the width of the drive. 21. A rectangular lot 20 by 15 rods is surrounded by a fence, within which is a drive occupying as much area as the rest of the lot. Find its width. 22. A number of two figures has the units' digit double the tens' digit, but the product of this number and the one ob- tained by inverting the order of the figures is 1008. Find the number. 23. A cistern can be filled by 2 pipes in 1 hour and 33f QUADRATIC EQUATIONS. 277 oiiniites, but the larger alone can fill it in 1 hoar and 40 min- utes less than the smaller one. Find the time required by the less. 24. The left-hand digit of a certain number of two figures is f of the right digit. If the product of this number and the number obtained by inverting the order of the digits be increased by twice the original number, the sum is 800. Find the number. 25. A man can row down a stream 16 miles and back in 10 hours. If the stream runs 3 miles an hour, find his rate of rowing in calm water. 26. Two trains run at uniform rates over the same 120 miles of rail ; one of them goes 5 miles an hour faster than the other, and takes 20 minutes less time to run this distance. Find the rate of the faster train. 27. A and B accomplish a certain task in a certain time, but if each were to do half the work, A would work 2^ days more, and B, 1^ days less than if they work together till the work is completed. Find the time required for each to do it. 28. If a carriage wheel 11 feet in circumference took -^ of a second less to revolve, the rate of the carriage would be 1 mile more per hour. At what rate is the carriage traveling ? Solve— a + b a 29. — ^'^— + a - X X a- X 30. 2cx2 + 2a\x + c) = ax{x + 5c). 31. a{h - c)x^ + 6(c - a)x + c{a - 6) = 0. 32. (4a^ - 962) ^^.2 + i) = ^xi^a" + 96'). 2. 33. a — h^-x ^ a + 6 a + 6 + a; 'a; + 6 34. d+ 46 a - 46 _ a; + 26 a; - 26 35. i^-(-«^> at b CHAPTER XX. SIMULTANEOUS QUADRATIC EQUATIONS. 258. The General Problem. If the relations of two un. known numbers to known numbers be given in the shape of two quadratic equations, the problem is to combine these relations so as to obtain simpler ones which will show directly the values of the unknown numbers. This can be done in certain special cases only, if we limit the work to methods already given for solving quadratic equations. 259. A Homogeneous Equation is one in which all the terms, containing an unknown quantity, are of the same degree. Thus, 3^V-5^2/' + / = 18 is homogeneous, and of the third degree. CASE I. 260. When One Equation is of the First Degree, the Other of the Second. Two simultaneous equations of the kind just specified may always he solved hy the method of substitution, ^ ^ , (2x-Sy = 2 (1) Ex. Solve ) J V y lx'-2xy=-7 (2) From(l), 2/ = ^^ (3) Substitute for y in (2), ^ 2x-2\ „ Hence, Sx' -Ax' + 4x = - 21 ic'-4a; = 21 Substitute for x in (3), 278 SIMULTANEOUS QUADRATIC EQUATIONS 279 It is to be observed thnt corresponding values of x and y must be used together. Thus, when x= —^,y must = — f, and not 4. Likewise the values, 7 and 4, go together. - EXERCISE 101. Find the values of x and y- - 1. 3a;^-22/^ = -5. 8. io; - iy = i. a; -|. 2/ - 3 = 0. {.x-yy = y--l. 2. x — 2y = S. 9. x'-3x2/ + 22/'=0. x' + 4y' = 17, 2x + 32/ = 7. 3. 2x'-\-xy = 2. 10. 42/' + 42/ = 4x-13. 3a; + 2/ = 3. . . 102/-2X-1. 4. rc^-32/^ = l. 11. 9^ — 62/ — 5 = 3ar. a: + 22/ = 4. 92/ + x + 5=0. 6. x-Sy = l. lxy-x^ = 12. 6. 2a; + 2/H-3-0. 3x^-72/^ = 5. 7. 2a; + 52/ = l. 3 2 9 12. --- = — y X xy 2x 10 32/ ^ 1 — =5. 2/ X2/ X 13.^-^-^ = 1. 2x 32/ 2x^ + 3x2/ = 9. 3:y + 4x = 6. 14. 3x- -by- 1=0. 2x' + 3x2/ -52/'-6x + 72/ = 4. 15. 4x^- -4x2/ = 2/^ + x + 32/-l. 4x -2- 52/ = 0. CASE II. 261. When both Equations are Homogeneous and of the Second Degree. Two simultaneous quadratic equations of this kind can always de solved by the substitution y — vx. 280 ALGEBRA. Ex. Solvfe x'-xy + y' = 21, 2/' — 2xy = — 15. Substitute y = vx, x^ — vx'^ + v^x^ = 21 (1) v^x' - 2vx^ = - 15 (2) From(l), -'-i^f^ (^) From(2), ^. = _=i|_ . (4) Equate the values of x^ in (3) and (4), 21 _ -15 1 - V -\- v'^ v^ — 2v Hence, 21^2 - 42i; = - 15 + 15v - 15i;» 36t;2 - 57v = - 15 12v2 - 19i; - - 5 . • . V = f, I Substitute for v its values in (3), ^2 ^ 21 21__ ^ 1 _ 5 4. 25' ^^ 1 _ 1 + 1 Hence, a; = =t 4, or ± 31^3 Since y = vx, multiply each value of x by the corresponding value of V, .•.y=(±4)|= i5, y = (i3l/3)^= il/3". EXERCISE 102. Find the values of x and 2/ — 1. x" + 82:2/ = 28. 5. 2x'-f = ie, xy-\-^f = ^. ^y + y' = 14. 2. 2x' + a:2/ = 15. 6. 3x' + f = 12. x'-y' = S. 5xy-4x':=ll. 3. x' + 3a;2/=7. 7. 22/' - 4x2/ -f 3x^ = 17. y^-\-xy = 6. 2/'-x'-16. 4. 2x^-32/^ = 6. 8. x' + xy + 2f = 74. 3x2/ -42/' = 2. 2x^ + 2x2/ + 2/' = 73. SIMULTANEOUS QUADRATIC EQUATIONS. 281 9. 22;' + Sxy + y' = 14. 11. x' + xyi- 2y' = U, '6x' + 2xy-4y' = d. 2x'' - xy + y' = IQ. 10. 4xy-x' = 5. 12. 2x^ - 7xy - 2y' = 5, 13x' - Slxy + Uy' = 2^. Sxy - x' + Qy' - 44. SPECIAL METHODS OP SOLVING SIMULTANEOUS QUADRATICS. 262. The methods of Cases I. and II. are the only genera] methods which can be used in solving all simultaneous quad- ratic equations of a given class. Besides these, however, there are certain special methods which enable us to solve import- ant particular examples. Examples which come directly under Cases I. and II. are often solved more advantageously by one of these special methods. The special methods apply with particular advantage to what are called symmetrical equations. 263. A Symmetrical Equation is one in which, if y be substituted for x, and x for y, the resulting equation is iden- tical with the original equation. Thus, each of the following is a symmetrical equation : X -\- y = 12, xy = &. 264. I. Addition and Subtraction Method (often in con- nection with multiplication and division). In this method the object is to find, first, the values ofoc-^y, and 00 - y, and tlien the values of x and y themselves. , _ , , . , (1) Ex.1. Solve [x + y = l I xy = V. xy = 12 (2) Here we have the value of x + y given, and the object is to find the value oi X — y. Square (1), a;» + 2a;y + y' = 49 (3) Multiply (2) by 4, 4a^ = 48 (4) 282 ALGEBRA. Subtract (4) from (3), x Extract square root of (5), Add{l) and (6), divide by 2, Subtract (6) from (1), divide by 2, 2 - 2xy 4- 2/^ = 1 . x-y =^ =t 1 rt'= 4 or 3 y 4 or 3 V 3 or 4J Boots. Ex. 2. Solve Divide (1) by (2), x" - xy ^- y" = 13 Square (2), x" + Ixy + ^/^ = 25 Subtract (3) from (4), Zxy = 12 Hence, xy == ^ . Subtract (5) from (3), x^ - 2xy + y"^ = ^ .♦. x-y= ±3 But Hence, Ex.3. Solve a; + 2/ = 5 (1 + ^=11. I ^ y Boots, Squaring (1), Subtracting (2) from (3), x' 2/ + — + A = 121 Subtracting (4) from (2), -I - A + 1 = i Hence, But, from (1), Hence, adding, c^ xy y' 2^ xy 1 xy y 1 y J_ y — = 60 X i + l-u t - 12, 10 3^ = i \] Boots, (5) (6) (1) (2) (3) (4) (5) (1) (2) (3) (4) (5) SIMULTANEOUS QUADRATIC EQUATIONS 283 EXERCISE 103. Find the values of x and y — I. x^y = U. X3. a^ + 2/» = 2a' + 6a. ^.y = 26. ic'-x?/ + 2/'^a' + 3. x + y = l, y ^ ^ S,x + y=~10. x + y = 5. ^y = '21, 15. ar* + 2/' =- 224. 4. a:^ + a:?/ -f 2/^^ --- 21. a:^ + a:^/' = 96. a; + 2/== - 1. 11 5. x'-xyi-y^ = S7. ^^' ? "^ 7 " ^^* x^ + xy + y^ = 79, l_fi=o 6. x' + 2/^ = 2i-. 2:2/ 3xv = 2i. 7. a: + 2/ + l=0. ^ ^ ^2/ + 3i=^0. . 17. - + -=3}. 8. x^ + 'if = d. ^ y" x + y^S. 1 + 1 = 2 9. a;' + 2/' = 37. x y ' x + y = l. 10. x3 + 2/^-218 ==0. 18. x» + 2/'--J:«^. a:' - X2/ + 7/ - 109. _ 11. x^+Sxy+y^= — 2f . « + 2/ =- i x''-a:2/ + 2/' = 12J. 12. xy-Qd^ = 0. 1^- a:* + a:y + y = 4^. a:^ + 2/' = a.-^ + 7a\ x' + a;2,' + y' = lh Solve also by the same method — 20. x' + y' = ^. 22. x' + 2/' ^ 5(a' + b'). x — y = A. y-x = a + Zh, 21. a:' - 2/' =- 98. 23. Z^ + Saty + 82/' - 13. x-y = 2. 5x^ + 3x2/ + 52/' = 27. 284 ALGEBRA. 265. II. Solution by the Substitutions, x = u + v and y = u — v, [0:^ + 2/^ = 242 (1) Ex. Solve 1 , o ro\ Substitute in (1) and (2), x = u + v, y = u-v From (1), 2w* + 20^3^2 + IOmv* = 242 (3) From (2), 2w = 2 (4) Divide (3) and (4) by 2, and substitute in (3) for u, i. e., w = 1, 1 + 10y2 + 5?;* = 121 Hence, v^ + 2v'' = 24 V = ± 2, ± 1/^=^ But w = 1 Hence, a; = w + v - 3, - 1, 1 ± l/^6| ^^^^ y = w-t;=-l, 3, 1 ^\^^Q) 266. III. Use of Compound Unknown Quantities. It is often expedient to consider some expression, as the sum, difference^ or product of the unknown qua7itities, as a single unhnoiun quantity, and find its value^ and hence the value of the unJcnown quantities themselves. Ex. Solve i.^-Vf = iy.-y (1) I xy = Q> (2) Add 2xy = 12 to (1). Then. x"^ + "tcy -\- y"^ =^ '^0 - x - y (3) Let X + y = V, Then from (3), v^ = 30 - v ^2 + V = 30 V = - 6, 5 Hence, x -^^ y = — % xy = 6. ,'.x= -3 ±1/3, y = _ 3 :p 1/3. also x + y = 5, xy = 6. .'. a; = 3, 2, y = 2, 3. SIMULTANEOUS QUADRATIC EQUATIONS. 205 EXERCISE 103 (A). Find the values of x and y. 1. x5 + 7/5=:244; x-f t/ = 4. x = 3, 1,2±3 i/^T7 y = l, 3, 2=F3 ^^X 2. a;2-ft/2 4.x4-y = 24; a;t/ = -]2. a; = 3, - 4, ib2i/3'; y=-4,3,=F2 v/ST 3. x-]-y-\-^x-\-y = Q; xy=3. x = l,3, 1(9 ±^69); y = 3,l,^{9 =f ^/W). 4. xV + a:y = 6;a: + 2y = -5. « = 1, - 6, - 4, - 1; l/ = -3,^-^-2. 5. a- 4-^ = 25; -/"E-f i/ y = a; - y. x = 9, 16; t/ = 16, 9. 6. y + i/a;-''- 9 = 6; i/x + 3 - •i/5"^=3= i/y. a: = 3, 5; y = 6, 2. 7. a:* + 2/* = y7. 10.* {x — yy—Z{x—y) =40, 8. a:'^ + 2/2 = a:-y+50. ^l' a^' + 2/' + a: + 5t/ = 6. 9. xY + 7xy = - 6. 12.t 'o:^ y^ {x^ + y^)= 70. 5a;2 + a:y = 4. a^i/^ 4_ a;i 4. y^= I7. EXERCISE 104. GENERAL EXERCISE. Find the values of x and y — 1. 2x-5y = 0. 6. a^ + y* = n. 6. a;' + 32/' = 28. 2-^ + 2/ = 2. ^ + xy + 2y^ = lQ. ?_L?=6 7,xy + 2x = 5. a? 2/ ' 2xy — y=-S. 3. 2a;'-a;2/ = 28. 8. Zx" -^ xy -^ y' = 16 4x 22/_34 3a;-62/ = l. 2a:-52/=-4. * There are eight roots for x and eight for y. t Coj^^ider^ first, tb»same time, two stations 240 miles apart ; each reached the station from which the other started, the one 3f hours, and the other If hours, after they met. Required their rates of running. 12. A crew rowing at f their usual rate took 32 bours to row down stream 48 miles and back to starting-place ; had they rowed at their usual rate it would have taken 18 hours for same circuit. Find their rate and that of the stream. 13. Two square plots contain together 610 square feet, but a third plot, which is a foot shorter than a side of the larger square, and a foot wider than the less, contains 280 square feet. What are the sides of the two squares ? 14. The fore wheel of a carriage makes 28 revolutions more than the hind wheel in going 560 yards, but if the circumfer- ence of each wheel were increased by 2 feet, the difference would be only 20 revolutions. What is the circumference of each wheel? 15. A number of foot-balls cost $100, but if they had cost $1 apiece less, I should have had as many more for the money as the number of dollars paid for each ball. Find the cost of each. 16. Find two fractions whose sum is equal to their product and the difference of whose squares is f of their product. CHAPTER XXI. GENERAL PROPERTIES Of QUADRATIC EQUA- TIONS. 267. Two General Forms of the Quadratic Equation. Any quadratic equation may be reduced to the general form ax' + bx + c^O ....... ^- I. Factoring, this becomes a(x' + -x-\--] = 0. \ a aj h c Dividing by a and denoting - by p, and - by g, we obtain (I CL x'' + px + q = II. When a, 6, c, or p and q are given, we can often infer at once, without the labor of solving the equation, important facts concerning the roots of an equation. Or if, on the other hand, the roots only of an equation be given, or some prop- erty of them, we can at once infer what the equation will be. PROPERTIES OIP x^ + px + q = O. 268. Relation of the Roots of x^+px + 4 = to the Coefficients p and q. Solving a;^ +_pa; + g = 0, and denoting its roots by n, ra, we obtain p Vf-4q 2 2 p Vf-4q '^"" 2 2 Adding, n + n =—P' Multiplying, rir, = q. 19 m 290 ALGEBRA. Hence, (1) The sum of the roots ofx^ + px + q — O equals —p^ or the coefficient of ac with the sign changed; (2) The prqcluct of the roots equals the known term q, Ex. In x' - 5a; + 6 = the roots are found to be 3, 2. The sum of these with the sign changed is — 5, the coeffi- cient of X ; the product is 6, the known term. This relation is used in the factorial method of solving quadratic equations (See Art. 250.) 269. Formation of a Quadratic Equation, the Roots Only being Given. If the two roots of a quadratic equation be given, the equation may at once be written out by the use of the rela- tion between the roots and coefficients determined in Art. 268. Ex. Form the quadratic equation whose roots are 5, and — 2. The sum of 5 and — 2 is + 3 ; hence, the coefficient of x in the required equation is —3. The product of 5 and — 2 is — 10, the third term ; hence, a;* — 3a; — 10 = is the required equation. This equation might have been formed also by subtracting each root from x, multiplying together the binomials thus formed, and letting the product = 0. Thus, (a;-5)(a; + 2)-0, or a;' -3a; -10 = 0. 270. Factoring a Quadratic Expression. Any quadratic expression may be factored by letting the given expression equal zero, and using the property stated in Art. 268. PROPERTIES OF QUADRATIC EQUATIONS. 291 Ex. 1. Factor a;' — 4aj + 2. Solving the equation, a;'' — 4a; + 2 = a; = 2 =tV2 .•.a:2-4a; + 2=(a;-2-V2)(a;-2 + 1/2), Fadon. Ex.2. Factor Sx'^ - 4x + 5. Take S{x' - fx + f ) = Solve a;2 - |a; + f - 0, Whence x= 2=^1/- 11 , 3 Hence, the factors of Sx^ — 4a; + 5 are EXERCISE 106. Find by inspection the sum and product of the roots in each of the following equations: 1. x' + ^x + 6=^0. 6. aV - ax + 2 = 0. 2. x'-x-\-7=0. 7. 5x-4x'^ = l. 3. x''-5x = 10. 8. S-Tx^llx". 4. 2a;'-6x-3 = 0. 9. 4x^-ax + x=^a\ 6. 6a;'-a; = l. 10. 1 - 2ca; - 2aa;' = 3«. Form the equations whose roots are — 11. 2, 3. 19. li -2*. „ 2±l/2 25. • 12. 2, -1. 20. 1 + a, l-«. 2 18. 3, —2. 21. a6, -a. 1 -h y^ZTl 14.-1,-5. ^ , 26. 15. i, 6. 22. - a b 2 6 a -2±:l/' 16. -1, -i 27. 17. - 2, - 1. 23. 1 + V^2, 1 - 1/2. 2 18. I, -|. 24. -3 ±1/3. 28. iadzcl/^^ 292 ALGEBRA. Factor — 29. 3a;' — lOc — 8. 84. x' + 14-6a;. 30. 24a;' + 2a; - 15. 35. 25a;' + 2 - 30a;. 31. a;' + 2x - 1. 36. 4a;' - 8a; + 7. 32. a;' — 4a; + 1. 37. 5a;' + 6a; + 7. 33. a;" - a; - 1. 38. 3x - 3a;' - 1. PROPERTIES OP ax" + boc + c =^ O, 271. Character of the Roots Inferred from the Coef- ficients. It is important to be able to infer at once from the nature of the given coefficients, a, 6, c, of an equation in the form ax' -f 6a; + c = 0, whether the roots of the equation be equal or unequal, real or imaginary, positive or negative. Solving ax' + 6x + c = 0, and denoting the roots by n, rj, we obtain - 6 4- Vb'-4ac -b- Vb^ - 4ac ri= 1 ra= 2a 2a From these expressions we infer that I. If b"^ — Aac !> 0, the roots are real and unequal. For if 6' — 4ac is a positive quantity (greater than zero), the radical 1^5' — 4ac is real and not imaginary, and since the fraction of which it is a numerator is added to to form 2a one root, and subtracted from ^ to form the other root, 2a the two roots are unequal. The roots are also rational or irrational, according as 6' — 4ac is or is not a perfect square. The roots are also rational if 6' — 4ac = 0. II. if 6' — iac = 0, the roots are real and equal, since each rootreduceato-l. - 2a PROPERTIES OF QUADRATIC EQUATIONS. 293 III. Jj h^ — 4:ac < 0, the two roots are imaginary. Since the character of the roots is thus determined by the value of b'^ — AaCj this expression is termed the discriminant of ax^ -f- 6x + c == 0. ]']x. 1. Determine the character of the roots of the equa- tion, 2x"' + 7x-15=0. We have a = 2, 6 = 7, c = — 15. . • . 6' - 4ac = 49 + 120 = 169. Hence, the roots are real, rational, and unequal. Ex.2. Of 9a:''-12x + 4 = 0. Here a = 9, 6 = - 12, c - 4 .'.b'- iac = 144 - 144 = 0. Hence, the roots are real and equal. Ex. 3. Of 3a;'-4a; + 2 = 0. ^ Here a = 3, 6=-4, c = 2 . • . 6^ - 4ac = 16 - 24 = - 8. Hence, the roots are imaginary. 272. Determining Coefficients so that the Roots shall satisfy a Given Condition. It is often possible so to deter- mine the coefficients of an equation that the roots shall satisfy a given condition. Ex. Find the value of m for which the equation (m — l)x' H- mx + 2m — 3 = shall have equal roots. By Art. 271, II., in order that the roots be equal, b"^ - ^ac = 0. In the given equation, a = m - 1, b = m, c = 2w — 3. .-. m2-4(m- l)(2m-3) =0 m^ - 8m^ + 20m - 12 = 7?7i2 _ 20m = - 12 m - 2, f . Proof. Substituting these values for m in the original equation, x' -j- 2x t 1 =0, a;2 - 6a: f 9 = 0, Ol each of which equations the roots are equal. 294 ALGEBRA. EXERaSE 107. Determine, wi43hout solution, the character of the roots in each equation. 1. x'-5x + Q = 0. 8. 2x'' + Sx = 5. 2. 3a:'-7x-2 = 0. 9.3x^-1=2;. 3. 4x'^4x- 1. 10. 6a;^ -{■^ = 10a;. 4. 3a:^ + 2a; + l=0. 11. x = ^^x' + 1). «. 2x'-5x + S = 0. 12. 35a; + 18 + 12x' =0. 6. 9x' + 12x + 4=0. 13. Jx' = 2a;-3. 7. 2a;^-f 5a; + 4=:0. 14. 7a;'' + l=5a;. Determine the value of m for which the roots of each equa- tion will be equal. 15. 2x'-2x-\-m = 0. ' 20. 2a;' — mx + 12^ = 0. 16. 2a:'+m + a;-0. 21. ISx" -\- Qx = m. 17. a:' + m = 3a;. 22. 4a;'^ + i = ma;. 18. -ma:'^ — 5a; + 2=0. 23. (m + l)a;'' + wa; -- 1. 19. 5x' + Sx-m = 0. 24. (?7i + l)a;' + 3m - 12a?. 25. (m + l)a;'4-(m-l)a: + m + l=0. 26. 2r/uc'^ + 3ma;-7 = 3a; — 2m — a;\ 27. If ri and r2 represent the roots of 3a;''' — 8a; + 5 = find without determining the actual roots, the values of: ri+r2; nrg; rf H-ri; n — r2; rl — rl; -8 I ^3. JL_1__L_. _i 1_, 1 r 1. 7^11-7^2, n ' r2» ri r2> rf ' rl 28. Find the values of the same expressions for the equation 2a;' — 9ic + 7 = 0. Also for the equation Gx^ — a; — 12=0. 29. Find the values of the same expressions for the equation 6 mb a : b has the same value as ma : mb, 291 296 ALGEBRA. PROPORTION. 277. A Proportion is an expression of the equality of two or more equal ratios. ^ a c , , JLx. - = -, or a\h = c'. d. b d 278. Terms of a Proportion. The four quantities used in a proportion are called its terms, or proportionals. The first and third terms are called the antecedents. The second and fourth terms are called the consequents. The first and last terms are called the extremes. The second and third terms are called the means. In a:b = c: d, d is called a fourth proportional to a, 6, and c. 279. A Continued Proportion is one in which each con- sequent and the next antecedent are the same. Thus, a:b = b:c = c:d~d:e. In the simple continued proportion a: b = b:c,b is called a mean proportional between a and c; c is called a third pro- portional to a and b. 280. Fundamental Property of Proportion. For alge- braic purposes the fundamental property of a proportion con- sisting of four quantities is, that The product of the means is equal to the product of the extremes. For, if a:b = c:d, XI, a c then - = -. Multiplying by bd, ad = be. In like manner, if a:b = b:c, b' = ac. .'.b = ■ Vac. This property enables us to convert a proportion into an equation, and to solve a given proportion by solving the equa- tion thus obtained. (See Art. 291, Ex. 1.) RATIO Al^D PROPORTION. 297 Before converting a given proportion. into an equation it is important, however, first to simplify the given proportion as far as possible. For this purpose we have the following tran?:* formations, which are possible in dealing with proportions: If four quantities are in proportion, they are in proportion by 281. I. Alternation ; that is, the first term is to the third as the second is to the fourth. For if a:b = c:dj a_c h~d Multiplying by -» ~ = ~.'> c c d .' . a:c = b b d Or, a-\-b;b=c-\-d;d. 298 ALGEBRA. 284. IV. Division ; that is, the difference of the first and sec- ond terms is to the second term a^ the difference of the third and fourth is to the fourth. Given, a:b=c:d, Then a c I'd' And i-H-^ That is, a—b c—d b - d ' Or, a — b:b = c — d:d. 285. V. 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. Given, a:b = c:d, Bycomposition(Art. 283), -^^=-^4^ (1) By division (Art. 284), if__[i _ ii_il ^g) 6 ' d a-b c-d b ~ d a + b c + d Divide (1) by (2), a—o c—d That is, a-{-b:a — b=c-\-d:c — d. 286. VI. Composition of Several Equal Ratios ; that is, in a series of equal ratios, the sum of all the antecedents is to the sum of all the consequents as any one antecedent is to its consequent. ^. a c e g Let each of the equal ratios equal r. mi. a c e a Then -=r, -=r, - = r, -=r. .'. a = br, c = dr, e=fr, g — hr. Adding the last series of equalities, hatio and puoportion. 299 a + c + e -h ^ = (6 -r d + / + A)r. • Q + c H- e + fl^ _a ' b + d-\-f+h ~^~b' .'. a-{-c-{-e + g:b + d+f+h-=a:b. 287. VII. Product of Corresponding Terms. In two or more proportions the products of the corresponding terms are in proportion. Given, a:b = c:dy j:k = l:m. Then a ^c e g j I b d f h k m Taking the product of corresponding members of these equations, oe; cgl bfk dhm ,' . aej : bfk = cgl : dhm. 288. VIII. Powers and Roots. In any proportion like powers or like roots of the terms are in proportion. Given, a:b =c:d. Then a _c b~d I I Hence, ^ = ^- Also,-=-., b- d- That is, ^n.jn^^n.^n^ And 11 11 a"" : b"" = c"" : d\ 289. IX. Cancellation of Factors of Terms. From Arts. 276 and 281 it is evident that if four quantities be in proportion, 300 ALGEBRA. and if the first two terms or the last two^ or the first and third, or second and fourth, be multiplied or divided by the same quantity, the resulting quantities are in proportion. Thus, if a'.b = c:d, Then ma :mb — nc: nd. And ma : pb =mc: pd. 290. Equal Products made into a Proportion ; that is, if the product of two quantities is equal to the product of two other quantities, either two may be made the means, and the other two the extremes of a proportion. For, if ad = be, Dividing hy bd, - = -. b d . * . a:b = c: d. This property is evidently the converse of the principle stated in Art. 280. 291. Application of these Principles. The use of propor- tion in solving algebraic problems and determining the prop- erties of algebraic quantities may be reduced essentially to the following: T. By taking the product of the means equal to the product oj the extremes, a proportion may be converted into an equation, and the proportion solved by solving the equation. Ex. 1. Find the value of x which satisfies the proportion, 4a; - 1 : a; -I- 1 = 3x + 1 : 2a; — 1. Taking the product of the means equal to the product of the extremes, (4a;- 1) (2x- 1) - (^ + 1) (3x + 1) .-. 5a;^-10a; = x = 0, 2. RATIO AND PROPORTION. 301 292. II. Before converting a 'proportion into an equation it is important to simplify the proportion, as far as possible, by use of the properties of a proportion, as Alternation, Composition, Division, etc. Ex.1. Solve x' — 2a; + 3:a;' + 2a;-3 = 2x'-a;-3:2x' + a; + 3. By Composition and Division, ^^ Divide by 1x\ 4a; - 6 2a; + 6 1 2 2a; - 3 a; + 3 .-. a; + 3 = 4a;-6 a;-3 The factor 2a;'^ divided out also gives the roots a; = 0, 0. VxTl + Vx^^^l 4x-l Ex. 2. Solve VxTl - Vx"=^l 2 By Composition and Division, ^ "^ 4a; + I 4a; -3 16a;' + 8a; + 1 1 16a;2-24a; + 9 X 16a;2 - 8a; + 5 Squaring, By Composition and Division: 1 16a; — 4 Hence, 16x' - 4a; = Ux^ - 8a; + 5 a; = |. 293. III. Given some proportion (or equality of several equal ratios), as a : b = c : d, a required proportion is often readily ft f» proved by taking - = - ~x (hence, a = hx, c — d-x), and sub- b d stituting for a and c in the required proportion. Ex. Given, a:b = c: d, Prove 2a' + Bab' : 2a' - Sab' = 2(f + 3cd' : 2c' - 3ccP. Let ^ = £ = x, .-. a = bx, G^ dx. o a Substitute in each ratio the values a= bx, c= dx. 2a^ + 3aP 2h^a^'+ 3b^x bh'{2x^ + 3) 2a;' + 3 I. II. 2a^ - 3rt6' ~ 2b^x^ - Sb^'x 6"^a;(2ar* - 3) 2x'' - 3' 2c^ + Scd^ ^ 2d'x^ + Wx ^ d'^ xj^x^ + 3 ) _^ 2x^ ^ 3 2c* - 'Scd' " la'x' - 3d'x d^x(2x' - 3) 2a!' - 3 2d^ 4- Sab"" 2c^ + Scd^ ' •• 2a^ - 3ab^ 2c' - 3cd?' since they are each ecjual to the same expression. a02 ALGEBRA. EXERCISE 108. Find the ratio of x to y — 1. lx-2>y = ^x + y. ^ Sx-2y o. — 2. 4:X — 5y:5x — Ay— f . 4x — Sy a 4. x^ + 62/^ = 5xy. Find a mean proportional between — 6. Sab' and 12a'. 6. 3^^ and 2|. 7. (a — x)' and (a + aj)*. 3x'-5a;-12 , 3x' 4- 4a; 8. and ^x' + 5x Sx''-4x--15' 21/6 + 51/3 9. ■ and 31/2-4 31/6-41/3 81/2 + 20 Find a fourth proportional to— 10. 2a, 36, 4ac. 12. h f , A. 11. x\ xy, ^x'. 13. a — 1, a, 1. Find a third proportional to — 14. x and 5. 16. (a + iy and a'- "1 15. 1^ and 7^. 1 ^ 1 . 17. a-- and --1. a a Solve the equations — 18. 2a: + 3 : 3x - 1 = 3x + 1 : 2a; + 1. 19. a; + 5:3-a; = 10 + 3x:a;-10. 20. 3a; + 5:5x + ll=7— x-.-Sx. 21. a;'-4:a;'-a; + 3 = a; + 2:2x + 3. 22. x' + 2x-l:2:' + 2a; + 5=:2x+l:2a;--5. 23. a;'-3x' + 5:x' + 3x'-5 = a;' + 2:a;'-2. 24. 2a;' - 8a;' - 3x + 1 : 2a;' - 10a;' + 3a; - 1 =a;' + 11 : x'-ll. 25. VWxl^:2VW=^==l/x=l:VxTT. 3 + V2xT'^ 4 + VxTT 6 - i/^x+'B 4 - l/x+1 RATIO AND PROPORTION. 303 ^ 3a + VAx - 3a' a + VxT^ 27. 6a — l/4x^ 3a^ 3a - VxT^ 28. 82/-6x:a; + 2/-l=5-3x:4-2/ = 7;4. Ui/ — 3:« — l=a + 2:l. If a : 6 = c : d, prove— 30. a^-.c'^ab'.dc. 31. a' : 6'-a* + c': 6»4- (P. 32. ac:6d = (a + cy:(6 + d)l 33. {a- cy :(h --ay = a^ + c' '.h' -{■ d^. 34. a : 6 = VaTTZ^ : V¥~+W, 35. 2a' + 3a6 : 3a6 - 46' = 2c' + Scd : Zed - 4(f . 36. a'-a6 + 6':^^^ ^ =c'-cci + d': -^ -- a e If a, b, c, d are in continued proportion, prove — 37. a:c-d = b':bd-cd. 38. a : c = a' + 6' + c' : 6' -f c' + d\ 39. a : d = a' + 26» + 3c' : b' + 2c' + 3cf . Prove that a : 6 = c : c?, it being given that — 40. (a + b) (c — d) + (6 + c) (d — a) = C(i - a6. 41. (a + 6-3c-3d)(2a-26-c + (i) = (2a + 26-<;-(0 (a-6-3c + 3c?). 42. Find two numb^s in the ratio of 2 to 5, such that when each is increased by 5 they shall be as 3 to 5. 43. Find two numbers, such that if 7 be added to each they will be in the ratio of 2 to 3 ; and if 2 be subtracted from each, they will be in the ratio of 1 to 3. 44. Separate 32 into two parts, such that the greater dimin- ished by 11 shall be to the less, increased by 5, as 4 to 9. 45. Separate 12 into two parts, such that their product shall be to the sum of their squares as 2 to 5. CHAPTER XXIII. INDETERMINATE EQUATIONS. VARIATION. 294. Indeterminate Equations. If a single equation con- taining two unknown quantities be given, this equation is called an indeterminate equation^ for the unknown quantities may have an indefinite number of different values which satisfy the ^quation. Thus, given 3a; + 22/ = 5. When r = 0, 2/ = f, x = l, 2/ = l,. x = 2, 2/=-i x = Z, 2/=-2, etc. In an indeterminate equation some limitation in the char- acter of the values of x and y may be imposed. Very fre- quently the values of x and y are limited to positive integers. Of the values obtained for x and y in the above equation, the only set that satisfies this condition is x = 1, y — 1. In like manner, if in a group of given simultaneous equa- tions the number of unknown quantities be greater than the number of the equations, the equations are said to be inde- terminate. The treatment of the subject here made wdll be limited to indeterminate equations of the first degree. 295. The Solution of Indeterminate Equations is best explained in connection with illustrative examples. 304 ' ' ■ INDETERMINATE EQUATIONS, 306 Ex. 1. Solve in positive integers 5z — 7y = 11. Divide through by 5, the smaller of the two coefficients. 5 5 Since x and y are integers, x- y -2 must be an integer. Hence, -^ must be an integer. 5 . 3 (2.y + 1) _ 6w + 3 , , • • ~^ — -^* or -»2— — , must be an mteger. o o (The particular multiplier 3 is used in this case so that on dividing the resulting numerator by the denominator 5, the coefficient of y in the re- mainder is unity.) -^- — ; hence, y + ^^— ' hence, 2L±_? must be an integer, o o 5 Let ^^^^=p. • .-.^ = 5^-3 (1) Substitute in the original equation for y, 5x - 35p + 21 = 11 .'. x = 7p-2 (2) In equations (1) and (2) p must have some integral value. If p = 1, then x = 5, y = 2. If p = 2, a; = 12, 2/ = 7. Etc. etc. It is seen that there are an indefinite number of positive integral values of X and ?/. Ex. 2. A number consists of two digits ; if the number be divided by the number formed by reversing the digits, the quotient is 2, and the remainder 2. Find the number. Let X = the tens' digit ^ y = the units' digit Then ^Q^ -^ y - ^ = 2 x + IQy 29 306 ALGEBRA, . • . 8a: - 19?/ - 2. Dividing by 8, a; - 2^/ - ^ - | ....-2^ = §^ . • . -^ must be an integer. Hence, -^-^ ^ > or -^ > and ^ must be int-egera. '8 8 8 ^ Let y-^=p 8 ^ \x= 19p- 14 The values of a: as digits in a number are limited to positive integers, lowest 0, highest 9. . • . a: = 5, 2/ = 2 is the only result allowable, . • . the number is 52. Ex. 3. In how many ways can the sum of $5.10 be paid with half-dollars, quarters, and dimes, the whole number of coins used being 20? Let X = number of half-dollars. y = number of quarter-dollars. z = number of dimes. '^'^ f^f + fo = **- Or, lOx + 5^ + 2^ = 102 (1) Also, a; -h ?/ + = 20 (2) Multiply (2) by 2, and subtract from (1), 8a: -h 3?/ = 62 r a: = 3p + 1. Solving, ■ y = 18 5p - %p. + 1. Let V- -0, then x=l, y = 18, z = \. P- = 1, a: = 4, y = 10, z = 6. P = = 2, a: = 7, y ■ = 2, z = 11. Any other values of p give negative results for one or more of th« quantities a:, ?/, z. Hence, tbere are thret ) ways of ' making the required payment. VABIATION. 307 EXERCISE 109. Solve in positive integers — 1. 7x 4- 42/ - 63. 6. lOx + 17y = 199. 2. 3x + lly = 31. 7. 5x~7y = 11. 3. 5x + 72/ = 82. 8. 132; - SOy = 61. 4. 7a; + 122/ -111. 9. 16a; - Ht/ = 26. 5. 15a; + 82/ - 101. 10. 13a; - 352/ - - 64. 11. Divide the number 107 into two such parts that one is divisible by 3, and the other by 8. 12. Divide 321 into two such parts that one is divisible by 9, and the other by 13. 18. Find two fractions whose denominators are 5 and 12 respectively, and whose sum is 4^V 14. A farmer sold a number of sheep and calves for $194 ; for each sheep he received $6, and for each calf $11. How many of each did he sell? 15. In how many ways can the sum of $5.80 be paid with dimes and quarters? 16. Find all possible ways of paying three dollars with five-, ten-, and twenty-five-cent pieces, so that half the coins used are five-cent pieces. 17. There is a number which, when divided by 17 gives a remainder of 6, and when divided by 23 gives a remainder of 21. Find it. How many such numbers are there? VARIATION. 296. Variables and Constants. A Variable is a quantity which has an indefinite number of different values. A Constant is a quantity which has a single fixed value. 297. Relation of Variables. Variations. One variable (called the function) may depend on another variable for its value in a definite manner. Thus, if a man be hired to work 308 ALGEBRA. for a certain sum per day, the number of dollars he will re- ceive as wages will vary as the number of days he works. Thus, if a; = number of dollars in his wages, t — number of days he works, X oc t. (The symbol a reads " varies as.") This expression is called a variation. This variation may also be expressed thus, X = mty where m denotes the number of dollars in one day's wages. Or, - = m. t Thus, if the ratio of two variables is always constant, their relation may be expressed in any one of three ways : (1) As a ratio. (2) As an equation. (3) As a variation. KINDS OF ELEMENTARY VARIATIONS. 298. I. Simple Direct Variations. The case considered m Art. 297, X ' X y = u + v. n .' . y = mx + - • ^ X 800. Fundamental Property of Variations. A variation may be converted into an equation by the use of a coefficient which is afterward to be determined, and the properties of variations de- rived and problems solved by the use of the properties of equations. 301. Elementary Properties of Variations. I. If X a 2/, and y cc z, then x cc z. For x — my, y—rvz, ,' . x = mnz. ,' . X ocz. 310 ALGEBRA. II. If a; a z, y oc z, then x±:y ca z and Vxy tjf^ 25, For X = mz, y = nz. . * . a; dz 2/ = (m zt n)z, And V^ = l/mz • nz = Vrm^ = z l/mn. Hence, x±y oz z. Vxy oc z. III. If a; a z, and y oc u, then xi/ a wz. For x = mz. y = nu. . * . iC2/ = mnuz. . • . a;2/ oc i(z. IV. If a; a 2/, then a;" oc 3/". For X = ray. . * . a;*' = m'*2/'*. . • . a;" oc 2/**. 302. Examples. Ex. 1. If a; varies inversely as 2/', and a; = 4, when y =» I, find a; when y-- = 2. Since xcc —, y' we have y^ Substitute x = = 4, 2/ = 1, in (1): , 4 = m. Substitute for m its value in (1), -^ (1) (2) Let 2/ = 2 in (2), then a; = 1, Result. Ex. 2. If 2/ equals the sum of two quantities, one of which VARIATION, 311 varies directly as a;', the other inversely as x; and y = 5 when a; = 1, 2/ = 1 when x = — 1, find y when ic = 2. Since 2/ = M + V, and u oc a;^, VOC —' X [Art. 299.] X (1) Substituting the given pairs of values for x and y in (1), 5 = m + n. 1 = m — n. . • . m = 3, n = 2. Substitute in (1) for m and n, 2/ = Sx^' + I (2) Let a; = 2 in (2), ^ = 13, Result. Ex. 3. The area of a circle varies as the square of its diam- eter. Find the diameter of a circle whose area shall be equiv- alent to the sum of the areas of two circles whose diameters are 6 and 8 inches respectively. Let A denote the area of a circle, and D the diameter. Then A ccB'', And A = ml)\ denote the areas of the two given circles by ^'' and A'^» Then ^/ = ryi x 6^ = 36m. ^// = 771 X 8^ = 6im, Adding, A^ -\- A^^, or A = 100m, Hence, since A = inD"^, and also 100m, mD"^ = 100m i)2 = 100 D = 10 Thus, the required diameter is 10. The student should review examples 1, 2, and 3 thoroughly, until he understands every step taken in their solution, before he undertakes a single example of the following exercise. 312 ALGEBRA. EXERCISE 110. 1. If X varies as 3/, and x is 10 when y is 2, find x when y is 3. 2. If a: ex 3/, and a: = 8 when 2/ = 6, find y when x = 3. 3. If a; + 1 a 2/ — 5, and a; =^ 2 when 2/ = 6, find a; when 2/ = 7. 4. If x^ oc 2/', and a; =^ 4 when 2/ = 2, find y when a; = 32. 5. If x^ a 2/'^ -f 8, and a; = f l/S when 2/ = 1? fi^^d 2/ when a; = 3. 6. If X varies inversely as 2/, and equals 2 when y is 4, find 2/ when a: = 5. 7. If a; varies inversely as 2/^ and is 6 when y is |, find y when a; = lj. 8. If X varies jointly as y and z, and is 6 when 2/ is 3 and 2 is 2, find a; when 2/ is 5 and z, 7. 9. If X varies jointly as 2/ and z, and equals 2 when y = \ and 2=1, find x when 2/ = 3, z = f. 10. a: varies directly as y and inversely as 2, and = 10 when 2/ = 15 and 2 = 6. Find y when a; = 16 and 2 = 2. 11. If 2a: — 32/ a 5a; + 92/, and when 2/ = — 2, a; = 4, find the equation connecting x and 2/. 12. If a;'-2a; + l cxy''-2y-\, and a; = | when y=-h find the equation between x and 2/- 13. One quantity varies directly as x and another varies inversely as x. If their sum is equal to 10 when a; = 2, and to — 2 when a; = — 1, find each quantity when a: = f. 14. Two quantities vary directly as x^ and inversely as x respectively. If their sum is 3^ when a; = 2, and — 3J when a; = 1, find the quantities in terms of x. 15. Given that w is equal to the sum of two quantities which vary as x and a;', respectively. If w = — 2 when a; = 2, and —5 when a;= — 1, what is w when x = |? 16. Given that w is equal to the sum of two quantities which vary as x and x^ respectively. liw=~2 when a; = — 1, and w = — ll^ when a; = — 2, what is the value of w when VARIATION. 313 17. Given that w is equal to the sum of three quantities, one of which is constant and the others vary directly as x* and inversely as x^ respectively. If w = S when x = ^jW = S when x= — Ij and w = 16f when x = — |, find the equation between w and x. 18. The distance fallen by a body from a position of rest varies as the square of the time during which it falls. If a body falls 144| feet in 3 seconds, how far will it fall in 8 seconds ? 19. The area of a circle varies as the square of its diam- eter. Find the diameter of a circle equivalent to two circles whose diameters are 5 and 12 inches respectively. 20. The volume of a sphere varies as the cube of its diam- eter. If three spheres whose diameters are respectively 6, 8, and 10 inches be formed into a single sphere, find its di- ameter. 21. The volume of a cone of revolution whose altitude is 7, and the radius of whose base is 3, is 66. Find the volume of a cone of revolution of altitude 6 and radius 5. Note. The volume of a cone of revolution (or cylinder of revolution) varies jointly as the altitude and the square of the radius of the base. 22. Find the altitude of a cone of revolution the radius of whose base is 7, and which is equivalent to two cones with altitudes 5 and 11 and radii 2 and 4 respectively. 23. 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 18 inches away, be removed to receive just J as much light? Interpret the two results. CHAPTER XXIV. ARITHMETICAL PROGRESSION. 303. A Series is a succession of terms formed according to some law. Exs. 1, 4, 9, 16, 25, l-x + x'-^-i-x'-, 2, 4, 8, 16, 32, 304. An Arithmetical Progression is a series each term of which is formed by adding a constant quantity, called the difference, to the preceding term. Thus, 1, 4, 7, 10, 13, is an arithmetical progression in which the difference is 3. Given an Arithmetical Progression (often denoted by A. P.), to determine the difference, from any term subtract the preceding term. Thus, in the A. P., |, - |, - 3, the difference = — |— f = — |. 305. Principal Quantities and Symbols Used. In an A. P. we are concerned with five quantities : 1. The ^rs^ term, denoted by a. 2. The common difference, denoted by d. 3. The last term, denoted by I. "" 4. The number of terms, denoted by n. 5. The sum of the terms, denoted by s. 306. Two Fundamental Formulas. Since in an A. P. each term is formed by adding the common difference, d, to the preceding term, the general form of an A. P. is — a, a + d, a + 2(i, a -f 3(i, -f 314 ARITHMETICAL PROGRESSION. 315 Hence, the coefficient of d in each term is one less than the number of the term. Thus, the 7th term is a + 6c?, 12th term is a + lie?, nth term is a + (n — l)d. Hence, l = a-i-(n-l)d (1) Also, s = a + (a + d) + (a-^2d)+ -\-(l-d) + l . .(2) Writing the terms of this series in reverse order, s='-l + (l-d) + il-2d)+ ^+(a + d) + a . .(3) Adding (2) and (3), 2s = (a + + (a + + (a + + + (a + H- (a + = n(a + 0. .•.s = |(a + ?) (4) If we substitute for I in (4) from (1), s-^[2a + (n-l)c?] (5) Hence, combining results, we have the two fundamental formulas for I and s, I. l = a + (n-l)d, II. s = ^(a + l) s=-[2a + (n-l)fq. Ex. 1. Find the 12th term and the sum of 12 terms of the A.P, 5,3,1,-1,-3, In this series a = 5, d = - 2, n = 12. From I., i = 5 + (12 - 1) (- 2) = 5 - 22 = - 17. From II., 8 = -»j«(5 - 17) = - 72, Sum. 316 ALGEBRA. Ex. 2. Find the sum of n terms of the A. P., a + b a — b a — 36 ■— — : — > 2 2 Here a = > d = — b, n = n. Substituting in the fundamental formula, s = - [2a + (n — l)(i], A «=|[a + 6 + (n-l)(-6)] = ^[a + (2 - n)6], ^um. EXERCISE 111. 1. Find the 8th term in the series 3, 7, 11, 2. Find the 9th term and the sum of 9 terms in 7, 3, — 1, 3. Find the 13th term and the sum of 13 terms in — 10, -13,-16, 4. Find the 20th and 28th terms in 5, -^, ^^, . . . . . 5. Find the 16th and 25th terms in — 13|, - 9, - 4| 6. Find the 7th and 10th terms and the sum of 10 terms in the series |, |, A, 7. Find the 18th term and the sum of 18 terms in the series 3, 2.4, 1.8, Find the sum of the series — 8. 3, 8, 13, .... to 8 terms. 9. — 4, — 7, — 10, . . . . to 6 terms. 10. 3, — 3, - 9, .... to 9 terms. 11. 21, 3f , 5, . . . . to 14 terms. 12- |j i? I, • • • • to 96 terms. 13. — i, i, f, . . . . to 38 terms. 14. — f , — f, — lij • • • • to 55 terms. 15. 3c, 1^, — 2c, .... to 6 terms. 16. 2a: — 3/, a; + 2/, 3^/, . . . . to r terms. 17. 5l/2-2v^, 4V/2-31/3, .... to 11 terms. 18. 3a ) 2a, a H- -, . . . .to 12 terms. ARITHMETICAL PROGRESSION. 317 307. Problem I. Given any three oj the five qiiantities^ a, d, If n, s, to find the other two. If we substitute for tbe three given quantities their values in the two fundamental formulas (I. and II., Art. 306), we shall have as a result two equations with two unknown quan- tities. The values of these unknown quantities may then be found by solving the two equations. Hence, by the use of these fundamental formulas, problems relating to A. P. are converted into problems relating to the solution of equations, processes already mastered. Ex. Given d = 2, 1=21, s = 121, find a, n. Substitute for d, I, sin Formulas I. and II., 21 = a+ (n-l)2 .(1) 121 = ^(«±^ (2) 2 . • . a + 2n = 23 (3) ari + 21n = 242 (4) Substitute for a in (4) from (3), «(23 - 2n) + 21n = 242 Whence n = 11 Hence, from (3), a = 1. 308. Problem 11. Given three of the five quantities, a, d, I, n, 8, to obtain a formula for one or both of the other two in terms of the three given quantities. Ex. Given d, I, s, obtain a formula for n. Since I = a -^ [n - \)d (1) s == ^[2a + (n - l)d] (2) Substitute for a from (1) in (2), 8 = ^[2^ - 2(n - l)d + (n - l)d] .' . 2s = 2ln- n{n - \)d Whence dn> - (d + 2l)n = - 2s c , . . d + 2^ ± V{ d + 2/)' -^"8^ . Solvmg for n, n= —z '■ • 318 ALGEBRA. No. Given. Requiked. Formulas. a, d, n a, d, s a + {n- l)d. _ irf i i/2ds + {a- idy. a. n s {n — V)d n 2 n, I, s d, n, s I- {n- \)d. \d ± V{\d + If - 2ds. n s _ (n — Vjd ^ n 2 10 11 12 a, d, n a, d, I a, n, I d, n, I = ln[2a + (n - l)d]. = ^+ « + ^^ - aV 2d ^{a + l). 8 s = \n\2l - (n - l)d]. 13 14 15 16 a, I, s n, I, s I — a ^ n-i 2(g - an) n[n — 1) 2s- I- a 2{nl - s) . n(ri — 1) 17 18 19 20 Of, c?, ? a, d, s a, ^, s d, ^, s I — a + 1. _ rf - 2a Jk l/(d - 2af + Sdg 2d 2s I + a 2^+ d=fcl/(2^ + d)'-'- 8d.9 . 2d ARITHMETICAL PROGRESSION. 319 EXERCISE 112. Find the first term and the sum of the series when— 1. d = 3, 1 = 40, 71 = 13. 2. d = l 1 = 1SI n = ZB. Find the first term and the common difierence when— 3. s = 275, ^ = 45, n = ll. 4. 5 = 4, /=-10, n = 8. 6. s=-246i, ^=-34^, 71 = 17. 6. s=-38i, l=-H, ri = 21. 7. s = 9, / = 2|, 71 = 9. 8. 5=-^, Z=-4, 71 = 47. Find 71 and d when — 9. a = -5,Z = 15,s = 105. 11. a=^,l= -^, $ = -2^. 10. a = 19,^ = -21,s = -21. 12. a= -3J, ^ = 9J, 5=48, Find a and 7i when — 13. ;=-8,d = -3,5 = -3. 15. l = 2,d=-i,s = 19{. 14. l=^l,d=ls=-20. 16. Z = -4,,i = _^,s = _i^. How many consecutive terms must be taken from — 17. 1, 1^, 2 . . . . to make 45? 18. }, i, J . . . . to make - 1 ? 19. J^, 2, f . . . . to make -20|? 20. I, j, 1 ... . to make 4.5? 309. Arithmetical Means. If it be required to insert a given number of arithmetical means between two given numbers, the solution of the problem is readily effected by means of Problem I. (Art. 307). The quantities in the A. P. which are given are seen to be a, l, n, and it is required to find d. 320 ALGEBRA. Ex. Insert 9 arithmetical means between 1 and 5. We have given a = l, 1—5, n = ll. Hence, we find ^ = f • The required means are therefore If, If, 2f , . . . In case but a single arithmetical mean is to be inserted between two quantities, a and b, this one mean is found mosi readily by use of the formula For if z denote the required mean, the A. P. is a, x, b. Hence, x — a — b — x 2x = a + b a-\-b EXERCISE 113. Insert — 1. Four arithmetical means between 7 and — 3. 2. Seven arithmetical means between 4 and 6. 8. Eight arithmetical means between | and 3. 4. Thirteen arithmetical means between ^ and — §. 6. Fifteen arithmetical means between — 4J and 9. 6. The arithmetical mean between 2f and — 5f . 7. The arithmetical mean between x + 1 and x — 1. 8. The A. M. between - and - • Between and b ^ x-\-y x—y 310. Miscellaneous Examples. Ex. 1. The 7th term of an A. P. is 5, and the 14th term it 9. Find the first term. ARITHMETICAL PBOGBESSION, 321 By the use of Formula I. (Art. 306), the 7th term is a + 6d, the 14th term is a + 13d. .'. a + ed'=5 (1) a + 13d = - 9 (2) Subtracting (1) from (2), 7d =-14 d= -2 Substitute for d in (1), a - 12 - 5 a = 17, Result. Ex. 2. The sum of five numbers in A. P. is 15, and the sum of the first and fourth numbers is 9. Find the num- bers. Denote the numbers by x + 2^, x^y, X, x-y, x- 2y. Add, 5a; = 15 (1) Also, {x + 2^/) + (a; - y) = 9 . • . 2a: + y = 9 (2) From (1) a; = 3 ; , hence, from (2), y = ^. Hence, the numbers are 9, 6, 3, 0, — 3, Result. Similarly, in dealing with four unknown quantities in A. P., we denote them by X + Sy, x + y, x-y, x - 3y. EXERCISE 114. Find the first two terms of the series wherein — 1. The 4th term is 11 and the 10th is 23. 2. The 6th term is — 3 and the 12th is - 12. 3. The 7th term is -^ and the 16th is 2J. 4. The fifth term is c — 35 and the liih is 36 —5c. 5. Find the sum of the first n odd numbers. 6. Find the sum of the first n numbers divisible by 7. 7. Which term in the series IJ, 1^,1^, . . . • is 18? 8. The first term of an arithmetical progression is 8; the 3d term is to the 7th as the 8th is to the 10th. Find the series. 21 322 ALGEBRA. 9. Find four numbers in A. P., sUch that the sum of the first two is 1, and the sum of the last two is — 19. 10. Find four numbers in A. P. whose sum is 16 and product is 105. 11. A man travels 2^ miles the first day, 2| the second, 3 the third, and so on ; at the end of his journey he finds that if he had traveled 6J miles every day he would have re- quired the same time. How many days was he walking? 12. The sum of 10 numbers in an A. P. is 145, and the sum of the fourth and ninth terms is 5 times the third term. Find the series. 13. If the 11th term is 7 and the 21st term is 8f, find the 41st term of the same A. P. 14. In an A. P. of 21 terms the sum of the last three terms is 23, and the sum of the middle three is 5. Find the series. 15. Required five numbers in A. P., such that the sum of the first, third, and fourth terms shall be 8, and the product of the second and fifth shall be —54. 16. The sum of five numbers in A. P. is 40, and the sum of their squares is 410. Find them. 17. The 14th term of an A. P. is 38; the 90th term is 152, and the last term is 218. Find the number of terms. 18. How many numbers of two figures are there divisible by 3? By 7 1 How many numbers of three figures are divisible by 6 ? By 9 ? 19. How many numbers of four figures are there divisible by 11 ? Find the sum of all the numbers of three figures divisible by 7. 20. If a body falls 16 A ft. the first second of its fall ; three times this distance the second ; five times the third, and so on, how far will it fall the 30th second ? How far will it have fallen during the 30 seconds ! 21. If a, 5, c,cZ are in A. P. prove: {l)thsita-\-d = h-{-c: (2) that ale, hJc, ck, dJc are also in A. P. ; and (3) that a-\-Jcj J) -{- k, c -{- Jc, d -\- k are in A. P. State this problem with- out the use of the symbols, a^h^c^ d, k. CHAPTER XXV. GEOMETRICAL AND HARMONICAL PROGRES- SIONS. 311. A Geometrical Progression is a series each term of which is formed by multiplying the preceding term by a con- stant quantity called the ratio. Thus, 1, 3, 9, 27, 81, is a geometrical progression (or G. P.) in which the ratio is 3. Given a geometrical progression, to determine the ratio: divide any term by the preceding term. Thus, in the G. P., - 3, |, - }, the ratio — -^ = — i« — o 312. Quantities and S3nnbols Used, a, ?, n, s are used, as in A. P. Besides these, r is used to denote the ratio. 313. Two Fundamental Formulas. Since in a G. P. each term is formed by multiplying the preceding telm by the common ratio, r, the general form of a G. P. is — a, ary ar', ar^y ar\ Hence, the exponent of r in each term is one less than the number of the term. Thus, The 10th term is aT^, ■ The 15th term is ar'\ The nth term, or l = af"* (1) 323 324 ALGEBRA. ' In deriving a formula for the sum, we know, also, s = a ^- ar ^- ar^ r\. _|- ^^.n- 1 . ^ , (2) Multiply (2) by r, rs ^ ar + ar^ -[- ar^ -{■ + af'^ + af . (3) Subtract (2) from v3^, rs — s = af — a. ar^ — a ' ^.^ ••• s^ T (^) r — 1 Multiply (1) by r, Substitute rl for ^r"* in (4), rl—a ,-. s== r (5) r — 1 Hence, collecting the results obtained in (1), (4), (5), we have the two fundamental formulas for I and s: I. l^ar^'-K 11. 5- - r — 1 W — a Ex. 1. Find the 8th term and sum of 8 terms of the G. P., 1, 3, 9, 27 . In this case, a = 1, r = 3, n = 8. From I., ^ = 1x3' = 2187. From II., s = ^ "" ^^^^ ~ ^ = 3280. 3-1 Ex. 2. Find the 10th term and the sum of 10 terms of the G.P., 4, -2, 1, -i Here a = 4, r = — ^, n = 10. Hence, I - 4{- If ^ - ^U = - jh- 8 = (-i)(-Th)-^ == 341. — J — 1 GEOMETRICAL PROGRESSION, 326 EXERCISE 115. 1. Find the sixth term in the series 2, 6, 18, 2. Find the 7th term in 3, 6, 12, 3. Find the 6th and the sum of 6 terms in 45, — 15, 5, 4. Find the 5th and the sum of 5 terms in 81, — 54, 5. Find the 7th and the sum of 7 terms in 1|, — |, 6. Find the 9th term in the series 2, 2l/2, 4, Find the sum of the series — 7. 3, - 6, 12, .... to 6 terms. 8. 27, - 18, 12, .... to 7 terms. > 9. —f, 1^, — 2, .... to 9 terms. 10. h-h-h,' • • • to 8 terms. 11. J 1, V^, .... to 8 terms. 12. V^, 1/6, 2V3, .... to 10 terms. 13. V^ — 1, 1, 1/2 + 1, .... to 6 terms. B14. Problem I. Given three of the five quantities, a, I, n, s r, to determine the other two. As in A. P., in the two fundamental formulas (I. and II., A-'t. 313) substitute for the three known quantities, and de- termine the other two quantities by solving the resulting equations. Ex. 1. Given a=-2, n = 7, l = -12S; find r, s. From I., - 128 = - 2r«. Hence, r« = 64, r = ± 2. Frora II, if r = + 2, s = ^(-m_-^{-^) = - 256 + 2 = - 254. I,,_2, ,^ (-2)(-128)-( -2 ) ^25^_3,^ HenotJ, there are two sets of answers ; viz., r = + 2, 8 = - 254. r = ~ 2, « = - 86. 326 ALGEBRA, No. Given. Required. Formulas. 9 10 11 12 13 14 15 16 17 18 19 20 a, r, s r, n, s a, n, s r, n, I r, I, s n, I, 8 a, r, n a, r, I a, n, I a, n, I a, n, s a, I, s a, r, I a, r, s a, I, 8 r, I, s a + (r — 1)8 r (r — l)sr^-* r " - 1 - ^)"-i - a{8- ay a -- a{i (r - 1)8 r^-1 ' rl— {r - l)s. i{s - ly a{7^- 1) Ir — a r-1 L n -^a» "-^r- (/•"- n - 1)1 -fa (r - l)r»*- ^a a a 8 — a 8-1 ' « .r»-' + — ^ =0. s- I s-l log ^ - log « _^ -|^ logr log [g + (r — 1 )s1 — log g logr log I- lo g g ^ J log (s - g) - log (s-l) log^- log[^r- (r- l)s1 ^ ^ logr GEOMETRICAL PROGRESSION, 327 Ex. 2. Given a-:f, r = -J, 8 = ^\; find I, n. From I., ^=3(_|)n_i (Ij Fromll, ^-i^ - ^^"1^2 1^ (2) Whence iff = (~y~^ Whence n = 6. Substitute for w in (1), I = — ^^^j. 315. Problem II. Given three of the five quantities^ a, I, n, Sf r, to obtain a formula for one or both of the other two in terms of the three given quantities. Ex. Given n, r, s, obtain a formula for I. Using l = ar''-^ (1) »=.75f (2) Solve (2) for a, a = rl - s{r - 1) . . . (3) Substitute in (1) for a from (3), I = rH — sr'^ - \r — 1). Hence, (r* - \)l = sr'^-^ir — 1) I = sr^-'^jr - 1) r«- 1 A complete table of all possible formulas for G. P. is given on the oppo- site page. These, the student should be required to derive for himself (except those for n). EXERCISE 116. Find the first term and the sum when — 1. n = 6,r = 3, Z = 486. 4. 7i = 8,r = -|,Z= -ffj. 2. 71 = 8, r = -2,/= -640. 6. n = 9, r= -3, /= -1215. 3. n = 7,r = f, / = if^. 6. n = 7, r =^1/6, / = 3. Find the ratio when — 7. a=-2, ^ = 2048, n = 6. 9. a = 2|f, Z= - Jf, n = 6. 8. a = 9,/ = 2^,5 = 23|. 10. a=-16i,^=^,5=-i2A. 328 ALGEBRA. Find the number of terms when — 11. a=^^,l= iV, ^ = i 13. a = 18, r = - §, s = 12f . 12. a = 3,Z=-96,s--63. 14. Z= -8,r= -2,s= -5^. How many consecutive terms must be taken from the series — 15. i,i,i, tomakefi^? 16. 15|, —61 2i .... to make 10|? 17. 5^, —8, 12, .... to make -221? 316. Geometrical Means. If it be required to insert a given number of geometrical means between two given numbers, the solution of the problem is readily effected by means of Problem I. (Art. 314). The quantities whicn are given are seen to be a, /, n, and it is required id find r. Ex. Insert 5 geometrical means between 3 and ^^. We have given a = 3, 1 = ^^, n = l, to find r. Solving by Problem I., r = \. Hence, the required geometrical means are, In case but one geometrical mean is to be inserted be- tween two given quantities, a and 6, this one mean is found most readily by using the formula Vah. For if x rep- resent the geometrical mean between a and 6, the series will be a, Xy b, X h Hence, - = - . a X ,' , x^ = ah. X = VaS, GEOMETRICAL PROGRESSION, 329 EXERCISE 117. Insert — 1. Three geometrical means between 8 and ^. 2. Three geometrical means between J and |. 3. Six geometrical means between ^V and — J/-. 4. Four geometrical means between — | and 3584. 6. Six geometrical means between 56 and — -^. 6. Five geometrical means between f and 12. Find the geometrical mean between — 7. 4i and |. 8. 3f and ^. 9. 51/2 + 1 and 51/2-1. 10. 31/5 + 2l/e and 3l/o- 21/3. 11. 28a'a;and 63aV" 12. %-^ and ^ ^ . c'v^ a: 1/^5- a' 8 13. Insert 6 geometrical means between — and lb Y^ 8 n^ 14. Insert 7 geometrical means between — and — n^ 2 317. Problem III. To find the limit of the sum of an infinite decreasing geometrical progression. If a line AB C D A I B 1 111 2 T "g- TS" be given of unit length, and one-half of it (AC) be taken, and then one-half of the remainder (CD), and one-half of the remainder, and so on, the sum of the parts taken will be i + i + i + A + A+ 330 ALGEBRA. This is an infinite decreasing G. P. in which r = ^. But the sum of all these parts must be less than 1, but approach closer and closer to 1 as a limit, the greater the number of parts taken. This is an illustration of the meaning of the limit of an infinite decreasing G. P. In general, to find the limit of an infinite decreasing G. P. we have the formula III. a — VL Formula II. of Art. 313 may be written, s — ■- Then, 1 — r as the number of terms increases, I approaches indefinitely to 0. rl .'.a-rl a — rl ^^ \—r 1 — r u 0. (( a-0 = a. a a 1-r Ex. Find the sum of 9, — 3, 1, —J, . . . . to infinity. Here a = 9, r=—\. 318. Repeating Decimals. By the use of the Formula III. of Art. 317, the value of repeating decimals may be determined. Ex. 1. Find the value of 0.373737 0.373737 = .37 + .0037 + .000037 + Here a = .37, r = .01. . .37 _ .37 _^ • •'~1-.01~.99~^- GEOMETRICAL PROGRESSION, 331 Ex. 2. Find the value of 3.1186186 3.1186186 - 3.1 + .0186 + .0000186 -f Setting aside 3.1, and treating the remaining terms as a G. P., a = .0186, r = .001. 1~.001 .999 ~^^^--sih' •'•3.1186186 =3A + rf|^ = 3^. EXERCISE 118. Find the sum to infinity of the series — l-2,i|, 6. A,A,^,... 2. 2,-1, i 7. 2if,~lJ,li, 3. - 9, 6, -4, 8. 6, SV2, 3, . . 4. l|,li,f, ' ^ 1 , 1 6.4J,-2i,li, 1/2-1 ' 1/2 + 1 10. il/2 + ii/3 + ii/2". ■■•■+('-^('-jj+ Find the values of— 12. 0.63. 13. 0.417. 14. 5.846. 15. 3.52424 18. 1.02727 16. 1.4037037 ..... 19. 1.027027 17. 3.215454 20. 0.30102102 819. Miscellaneous Problems. Ex. Find four members in G. P., such that the sum 332 alqebha, of the first and fourth is 56, and of the second and third is 24. Denote the required numbers by a, ar, ar\ ar^. Then * a + a?'^ - 56 ar + ar"^ = 24. Or, a(l + r') = 56 (1) aril + r) = 24 (2) Divide (1) by (2), ^ ~ ^ "^ ^' - 1. r Hence, 3 - 3r + Sr^ = 7r. 3r2 - lOr = - 3 r = 3, or |. And a = 2, or 54. Hence, the numbers are, 2, 6, 18, 54. Or, 54, 18, 6, 2. EXERCISE 119. Find the first two terms of the G. P. wherein^ 1. The 3d term is 2, and the 5th is 18. 2. The 4th term is f and the 9th is 48. 8. The 3d term is 5 and the 8th is ^K 4. The 5th term is 6 and the 11th is -A-. Determine the nature, whether Ar. or Geom., of each — ^' hhi 8. I, f , |- 6. i, i, i 9. 3i, 41, 7i T-iiiV 10. 7i5f,4T% 11. Divide 65 into 3 parts in geometrical progression, such that the sum of the first and third is 3^ times the second part. 12. There are 3 numbers in G. P. whose sum is 49, and tho sum of the first and second is to the sum of the first and third as 3 to 5. Find them. HARMONICAL PROGRESSION. 333 13. The sum of three numbers in G. P. is 21, and the sum of their reciprocals is -^. Find them. 14. Find four numbers in G. P., such that the sum of the first and third is 10, and of the second and fourth is 30. 15. Three numbers whose sum is 24 are in A. P., but if 3, 4, and 7 be added to them respectively, these sums will be in G. P. Find the numbers. 16. The sum of $225 was divided among four persons in such a manner that the shares were in G. P., and the differ- ence between the greatest and least was to the difference between the means as 7 is to 2. Find each share. 1 2 17. Find the sum of — , i/2 -^= . ... ad infinir 1/2-1' "^ ' 1/2 + 1 turn. 18. There are 4 numbers the first 3 of which are in G. P., and the last 3 are in A. P, ; the sum of the first and last is 14, and of the means is 12. Find them. 19. If the series f , ^ . .' . . be arithmetical, find the 102d term; if geometrical, find the sum to infinity. 20. Divide $369 among A, B, C, and D so that their shares may be in G. P., and the sum of A's and B's shares shall be $144. 21. The ages of three men, A, B, and C, are in G. P., and the sum of their years is 111 ; but | of A's age, -^ of B's, and ^ of C's form an A. P. Find their ages. 22. Insert between 2 and 9 two numbers, such that the first three of the four may be in A. P. and the last three in G. P. 23. Prove that the series l/2-l, 31/2-4, 2(5 1/2-7) .... is geometrical ; that its ratio is 2 — l/2 ; and that its sum to infinity is unity. HARMONICAL PROGRESSION. 320. An Harmonical Progression is a series the recipro- cals of whose terms form an arithmetical progression. Thus, 1, :j, 7, iVj is an harmonical progre^ion, sine©- the^?^biprocals of its terms, 334 ALQEBBA. 1, 4, 7, 10 form an arithmetical progression. The general form of an harmonical progression is — 1111 1 , } ~ 1 a a-\- d a-\r2d a-\-Zd a-\- {n — V)d 321. General Principle. Problems relating to harmonical progression are solved to best advantage by taking the recip- rocals of the terms of the given progression, and solving the A. P. thus formed. Ex. 1. Find the 15th term of the H. P., hhhii Taking the reciprocals of the terms of the given series, we obtain the A. P., 2, 5, 8, 11 In this a = 2, c2 =--^ 3, n = 15. .-. Z = 2 + l4X3=--44. Hence, the 15th term of the given H. P. is the reciprocal of 44 ; that is, 4^. Ex. 2. Insert 7 harmonic means between — 3 and 4. To solve this problem it is necessary to insert 7 arithmetical means be- tween — \ and \. Hence, « = — i> ^ ^ h ^ = 9« ,' . \= - \ + M, d = ^. Inserting arithmetical means, the A. P. is ri> ~ ij?j sVj ts> hh h - 3, - If, - V, - ff, - 24, 32, -V-, f f, 4. 322. Harmonic Mean. If but one harmonic mean is to be inserted between a and b, and this be denoted by if, then -— is the arithmetical mean between — and ^ : and by Art. 309, HARMONICAL PEOOBESSION. 336 1 a b a + b „ 2ab £1 = H 2 2ab a + b If the arithmetical, geometrical, and harmonic means be- tween a and b be denoted by A, (r, H^ respectively, we have A = — - — > G = VoBj H 2 ' a + b ,'. H=abX-^ = G'X^' a + b A .'. G' = AH and G=--VAB. Hence, the geometrical mean between two quantities is also the geometrical mean between their arithmetical and harmonic means, EXERCISE 120. Find the last term in the series— 1- I) T> 4) • • • • to 20 terms. 2. f , 2, 6, . . . .. to 18 terms. 3. -j%, -|, -1, .... to 27 terms. Insert— 4. Four harmonic means between J and |-. 5. Five harmonic means between f and ■^. 6. Seven harmonic means between — 3^ and — ^. Find the harmonic mean between — 7. 2| and 3. 8. 4i and - 3|. 9. ^ and ^. ^ ^ ^ x-\-l x-1 10. The first term of a H. P. is x, and the second term is y. Find the next two terms. 11. The arithmetical mean between two numbers is 15, and the H. M. is 14|. Find the numbers. 12. The fifth term of a H. P. is - 1, and the 14th, is f Find the series. 13. The G. M. between two numbers is 16, and the H. M. is 12|. Find the numbers. CHAPTER XXVI. PERMUTATIONS AND COMBINATIONS. 323. The Permutations of a group of objects are the differ- ent arrangements which can be made of the objects with respect to their order. Thus, the permutations of the three letters a, 6, c are, abc, acb, bac, boa, cab, cba. Taken two letters at a time, the per- mutations of a, 6, c are, a6, 6a, ac, ca, be, cb. 324. The Combinations of a group of objects are the dif- ferent collections which can be made from them without respect to order. Thus, the combinations which can be made from three let- ters, a, 6, c, taken two at a time, are, ab, ac, be. It is seen that ba and ab, for instance, are different permutations, but are the same combination. 325. The Number of Permutations in a Group of ft Ele- ments: To determine the number of permutations that can be made of a group of objects by actually writing out the permutations and counting them usually involves so much labor as to make the method impracticable. Instead of this, it is possible to establish a formula by which the number of permutations in a group of objects can be determined by a simple process of multiplication from the number of objects or elements in the group. Thus, to determine the number of permutations which can be made with a group of four objects, as a, 6, c, d, we conceive the permutations to be formed b}^ writing each of the four elements in the first place in turn ; after each of which the remaining three elements may be placed; after each of which the remaining two, and so on, 336 PERMUTATIONS AND COMBINATIONS. 337 Thus, we obtain [cd [hd Xdb {he d cb ad da ac ca M c < Whence we have all the permutations, abed, abde, etc.; as, 4 X 3 X 2 X 1, or 24 in number. Similarly, to form the number of permu- tations which can be made with a group of n elements, we may write each of the n ele- ments in the first place; after each of which the remaining n — 1 elements may be written; after each of which the re- maining 71 — 2 elements, and so on. Hence, the number of permutations formed from a group of n elements is nCn — l)(n-2) . . . . 8X2X1. If the permutations of four objects be formed, taking two objects at a time, in the first place each of ^the four objects may be written, after each of which the remaining three, which exhausts the number of per- mutations so formed. Hence, the permu- tations of four objects, taken two at a time, is 4 X 3. Similarly, the permutations of n objects taken 2 at a time is n(7i — 1) ; of n objects taken r at a time, is n(n-l)(n-2) (n-r + l); [r factors in all]. 326. Symbols Used. For the number of permutations of a group of n objects, taken r at a time, the symbol, „P^, is used. The product n(n — 1) .... 3 X 2 X 1 is called factorial w, and is abbreviated into the form, |n, or ?il. Hence, from the preceding Article, nP„ = n(7i-l)(n-2) .... 3X2Xl=|ri,ornI. nPr = n{n-l)(n-2) .... (n-r + l). 22 db ad da ab ba be cb ae ca ab ba 4X3X2X1 338 ALGEBRA, Ex. 1. In how many ways may the letters which form the word Baltimore be arranged ? Since there are 9 different letters in the word Baltimore, we have 9P9 = 9X8X7X6X5X4X3X2X1 = 362,880 Permutations. Ex. 2. How many of the permutations in Ex. 1 will begin with the letter I ? Since the letter I remains fixed in the first place, 8 letters are arranged in different orders. Hence, we have gPs = [8 = 40,320 Permutations. 327. The Number of Combinations in a Group of n Ele- ments. We denote the number of combinations which can be made with a group of n elements, taken r at a time, by „Cv. The permutations of a group of n objects, taken r at a time, might be formed by writing all the different combinations of the n objects taken r at a time, and then making all possible permutations (that is, arrangements) of the elements of each combination of r objects; viz. |r_ arrangements. Thus, for example, Or, in general, „p,=„ax|r p [r ^ n(n-l)(n-2) .... (n-r + 1) ^ " *■" r(r-l)(r-2) .... 1 Ex. How many combinations can be formed with the letters of the word Baltimore^ taken four at a time ? We have ^ 9X8X 7X6 ,^^^ ,. . •Ci = = 126 Combinations, 4X3X2X1 PERMUTATIONS AND COMBINATIONS. 339 828. Other Formulas for „c;. The formula for ^C^ may be put into another and often more convenient form by multi- plying both numerator and denominator by \n — r . n(n — 1) . . . . (n — r + 1) In — r In Thus, „a= '——T ^i-==^_. \r\n — r \r\ n — r It is also to be observed that for each different selection (or combination) taken from a group of objects a different group is left. Thus, if from a, 6, c, d^ e we take a, 6, cZ, we have left c, e ; if we take 6, c, e, we have left a, d, and so on. Hence, s^s^sC^s Or, in general, nCr = nCn - r- Ex. Determine the number of combinations of 20 letters taken 18 at a time. 20^18 — 20^2 ^ 20 X 19^ 1X2 = 190 Combinations, EXERCISE 121. Find the values of— 1. 6^4- 3. ^Ps' 5. 19C16. 7. jiPfi. 2. 20C9. 4. uO,. 6. uP*- 8. M. 9. How many different numbers of 5 different figures each can be formed from the nine significant digits? 10. From a company of 24 men, in how many ways can a committee of 10 be selected? 11. How many words of 5 different letters can be formed from our alphabet? 12. There are 13 points in a plane, no three of which are in the snme straight line. How many different triangles can be formed having three of the points for vertices ? 340 ALGEBRA. 13. How many different words of eight letters each can be formed from the letters in the word republic f 14. How many different numbers of 4 different figures can be formed from the ten digits 0, 1, 2, ? 15. From the letters in the word universal how many words can be formed, taking 6 at a time ? Taking 7 at a time ? 16. Five persons enter a car in which there are seven seats. In how many ways may they take their places ? 17. How many different throws can be made with 2 dice ? 18. How many different throws can be made with 3 dice ? 19. How many numbers is it possible to write by use of our 10 digits without repeating any digit ? SPECIAL CASES. 329. I. Elements of a Group repeated. It may happen that some of the elements in a group of objects are the same. Thus, if it be required to determine the number of permu tations which may be made of the letters in the word Chicago^ it is observed that in this word the letter c occurs twice, hence, all the permutations formed by interchanging the c's are identical. Hence, for the total number of permutatiopb we have \1 - — = 2520 Permutations. \i In general, if in a group of n elements, one set of elements, i in number, are identical, and another set, k in number, ar© identical, the total number of permutations in the entire group, taken all together, is \n \i\h Ex. How many permutations can be formed from the let* ters of United States, taken all together? We have 12 letters in all, but t used three times, s used twice, and e twice. PERMUTATIONS AND COMBINATIONS. 341 Hence, — ■ — — - = 19,958,400 Permutation. [£[2 1-? II. Different Groups taken Together. If combina- tions be formed from one group of elements, and other combina- tions from another group of elements, and these combinations be taken together in pairs, one from each group, it is evident that the total number of combinations will be the product of the number of combinations in one group by the number in the other group. For each combination in the first group may be joined separately to each combination of the second group. Ex. Out of 15 Republicans and 10 Democrats, how many different committees may be formed of 4 Republicans and 3 Democrats ? We have for the number of partial committees from the Republicans, . 15-X 1 4 X 13 X 12 ,_^ 4X3X2X1 From Democrats, 10X9X1^,20, '" ' 3X2X1 Hence, the number of entire committees will be 1365 X 120 = 163,800, Result. EXERCISE 122. Find the number of different permutations that can be made from each of the following words, taking all the let- ters: 1. Inning. 3. Upper House. 5. Independence. 2. Successes. 4. Mississippi. 6. Unconsciousness. 7. How many different arrangements can be made from dbYy when written in the expanded form? 342 ALGEBRA. 8. How many committees of 2 teachers and 3 boys can be selected from a school of 5 teachers and 25 boys? 9. From 9 red balls, 5 white balls, and 4 black balls, how many different combinations can be formed, each consisting of 5 red, 3 white, and 2 black balls ? 10. From 9 merchants, 14 lawyers, and 7 teachers, how many different companies can be formed, each consisting of 3 merchants, 5 lawyers, and 2 teachers? 11. How many different words, each consisting of 3 con- sonants and 1 vowel, can be formed from 12 consonants and 3 vowels? Hint. The number of possible combinations is i^Og x gCj. But each combination contains 4 letters, and may be arranged into [^ different per- mutations. 12. How many different words, each consisting of 4 con- sonants and 2 vowels, can be formed from 8 consonants and 4 vowels? 13. Find the number of different words of 3 consonants and 2 vowels each, that can be formed from an alphabet of 5 vowels and 21 consonants. 14. In how many ways can 2 ladies and 2 gentlemen be chosen to make a set at lawn tennis from a company of 6 ladies and 8 gentlemen? How many different words can be formed from the letters in the following words, using all the letters in each instance : 15. Volume J the second, fourth, and sixth letters being vow- els? 16. Numerical, the even places always to be occupied by vowels ? 17. Absolute, the first and last letters to be vowels? 18. Parallel, the first and last letters to be consonants ? 19. How many different quadrilaterals can be formed from 20 points in a plane, as the vertices, if no three are in the same straight line? FERMUTATIONS AND COMBINATIONS. 343 20. A man has 7 pairs of trousers, 5 vests, and 6 coats. In how many different costumes may he appear? 21. In how many different ways can a base-ball nine be arranged, the pitcher and catcher being always the same, but the others playing in any other position ? 22. How many different sums of money can be obtained from a cent, a five-cent piece, a dime, a quarter-dollar, and a half-dollar? 23. From 5 labials, 4 palatals, and 6 vowels how many words can be formed, each containing 2 vowels, 2 labials, and a palatal? 24. There are 12 persons in a coach at the time of an acci- dent at which 3 are killed, 4 are wounded, and 5 escape unin- jured. In how many ways might all this happen ? 25. The number of committees of three members each which can be formed from a certain number of boys, is to the number of similar committees possible if there were one more boy, as 10 to 11. Required the number of boys. 26. The number of words of four letters each which can be made from a certain number of letters, is 2V ^^ ^^e num- ber of words of five letters each which could be made if there were two more letters. Required the number of let- ters. Verify both results. 27. If„C3:2«a=--7:l5, findn. 28. If 20a = .0 a + 2, find, P4. 29. Show that „ + iC- = nC + nCr^X. 30. In how many ways can nine children form a ring, all facing the centre ? . CHAPTER XXVII. UNDETERMINED COEFFICIENTS. 331. Convergency of Series. Using the jesult obtained in Art. 317, viz., a s = - , 1 —r it is found that the infinite series i+i+i+i+ (1) approaches a limit, =2. Similarly, if we take the infinite series (2), we find that for any value of x between and 1, as -, where 6 < a, the series has a definite limit ; viz., 1 ^ a b a — h a But if in series (2) we let a; = 1, the value of the series is 1 + 1 + 1 + 1+ . . . . , which becomes greater without limit, the larger the number of terms which is taken. Similarly, if x have any value greater than 1 in series (2), the series has no limit to its value. Hence, in one kind of infinite series the sum approaches a fixed quantity as limit, while in the other kind the sum exceeds any assigned limit. A Convergent Series is one in which the sum of the first n terms approaches a certain fixed limit, no matter how great n may be. UNDETERMINED COEFFICIENTS. 345 A Divergent Series is one in which the sum of the first n terms can be made to exceed any assigned quantity by mak- ing n sufficiently large. 332. Use of Infinite Series. Infinite series are useful in representing complex algebraic expressions, since such series ordinarily consist of simple combinations of the functions, sum, difference, product, and power of quantities used. It is evident, however, that infinite series can be used in a valid way only when the series used is convergent. Identities. An identity is a statement of equality between two algebraic expressions, which is satisfied by all values whatever of the unknown quantity or quantities used in the expressions. Ex. a;'-4=-(a;-2)(a; + 2) is an identity, since it is satisfied by any value of a;, as 1, 2, 3,-1, etc.; whereas, for example, in 7a;-6 = 10-a;, X has but a single value ; viz., x = % The correct sign for an identity is — , but in the case of such identities as occur in the early part of the algebra, the custom prevails of using the equality mark for these identi- ties as well as for equations. The student, however, should form the habit of carefully discriminating between identities and equalities when the equality mark is thus used for both. In this chapter the identity mark will be used for identities, and the equality mark for equalities. 334. Fundamental Property of Identities. In the identity A + Bx-\-Cx' + X>ir' +....= A' + B'oc + C V -f- D'x^ + . . . . (each member being either finite or convergent)^ the coefficients of like powers of ac are equal; that is, A=A', B=B'f C='C'j etc, « 346 ALGEBRA. For, since the identity A + Bx + Cx'-^Dx' + ... . = J.' + ^'x + CV 4-i)V-t- ....(1) is true for all values of x, it is true when x — 0. Hence, from (1) A = A' (2) Subtracting (2) from (1), and dividing the result by x, B + Cx-\-Dx' + ....=B + C'x+D'x'+ . . . . (3) This identity is true for all values of x. , . • . letting a; = 0, B = B' (4) Subtracting (4) from (3), and proceeding as before, C=C\ D = iy, etc., etc. Ex. Find the condition that x^ + px + q is divisible by cc + a. li x' -\- px -\- q is divisible by a; + a, the quotient must be of the form x-\- k. Hence, x'^ + px + q = (x + a) (x -\- k), x^ + px -\- q^x^ + {a -\- k)x -\- ak. Hence, by the fundamental properties of identities, p = a + k (1) q = ak (2) Eliminating k from these equalities, q = a(p- a), which expresses the required condition. EXERCISE 123. Find the value of k that will make — 1. 8a:^ 4- kx — 91 exactly divisible by 2x + 7. 2. 6x' + x'^ -\- kx + 4: exactly divisible by 3a; — 4. 3. 3a;' + kx"" + 25 exactly divisible by a;' - 3a; + 5. 4. 2x* - ar* - 3a;' + lOx + k divisible by x' + a; - 2. . UNDETERMINED COEFFICIENTS. 347 Determine the conditions necessary that— 5. x'^-{-ax — h may be exactly divisible by x — c. 6. ax^ + 62: + c may be exactly divisible by a; + d 7. x^ + ax + h may be exactly divisible hy x^ — x-\-d. 8. a;* + ax^ + bx~ -\- ex ^- d may be divisible by x' + mx -f- n. Prove that if — 9. ax^ + hx' ^-cx + d is divisible by x^ — P, then ad = be. 10. ax' + 2hxy + 6?/' + 25ra: -\- 2fy + c is the square of a'x + h'y 4- c', then af = gh, bg =fh, and eh =fg. APPLICATIONS. I. Expansion of a Fraction into a Series. 335. General Method by Undetermined Coeflacients. A fraction may be expanded into a series by dividing the nu- merator by the denominator, but it is often more convenient to make the transformation by the use of undetermined coefficients. 1 — 2x' Ex. Expand — — into a series by the use of unde- ^ l + 2x-3x^ ^ termined coefficients. Let — 1 - 2x-^ ^^ + Bx^ Cx'' + Dx^+ (1) where A, B, C, D . . . . arc unknown numbers to be determined. C lear (1 ) of fractions, collecting the coefficients of x, x*, etc. by use of the vinculum in the vertical position instead of by use of parenthesis, thus, for {B + 2A)x use + ^1 ^• 1 - 2x2 £EE^ + 5| X + C a:^ + D x» + . . . . (2) + 241 +2B + 2C7 -ZA - 3P Since (2) is an identity, the coeflScients ot like powers of x are equal. 348 Hence, B+2A = 0, 1> 1 j/x>.£i..a.. r^ = i, 0, -2, whence - B=-2, = 5, 0, J i) = - 16, etc. D + 2C-SB = 0, etc. Substitute for ^, -B, C, i), . . . . in (1), — L=_2£i! — = 1 -2x + 5x^ - IQx^ + 1 + 2a; - Sx'' This result may also be obtained by division. , Series. 336. Special Cases. When the lowest power of x in the denominator is higher than the lowest power in the numer- ator, we may proceed in either of two ways, as illustrated by the following example: 1 + 2x Ex. Expand — — — -. into a series. 1 + 2a; 1 /I + 2a;> Let 1 + 2a; 1 -3a; Solving this identity, 1 + 2a; 1 / I + 2a; \ x'\l- Sx) Sx A + Bx + Cx^ + Dx^ + = 1 + 5a; + 15a;2 + 45a;' + Multiplying this result by — -> that is, by a;"'^, we obtain 1 + 2a; a;2-3a;3 = a;-2 + 5a;-i + 15 + 45a; + . The required series. Or, at the outset we might have divided the first term of the numerator, 1, by x^, the first term of the denominator, and determined a;~^ as the power of x occurring first in the required series, and let 1 + 2x Ax-^ + Bx-^ + C+ Dx + as usual. x^ - 3ar» and determined A, B, O, D, Again, if the numerator and denominator of the given frac- UNDETERMINED COEFFICIENTS. 349 tion contain only even powers of x, the process of expansion may be shortened by using only even powers in the assumed series. Thus, let ^ _ ^^^ ^''^^^ =A-^Bx'-\-W + Dz' + ..... EXERCISE 124. Expand into a series in ascending powers of x- 1. l + 2a; l + x 4. 3-a;' l + 2x' ^ 2-a;-4x' 2-i-x-7? 2. 2-3a; l-2a; 6. 1-x-x' 1+x^x^ ^ Z + X-27? 3. \-bx \^x-x' 6. x + Zx' l-2x-\-^x' 9. '-^ 2 + 6a;-a;» 10. 2-^^-^ 2^' 3 + x-2x' 1 11. 3-a:' + 4x» 2 + x^-a;» 12. l-2a: 14. 4 + 7a; l-2x'-cr:» X' + T? 2x + Zx^ 13. 2-x' 15. 2 + a; — 4x' 17. 1-^ + ^'. x' + x» 2ic» + X* x' + X* n. Expansion of a Radical into a Series. 337. Illustrative Example. Ex. Expand Vl + 2x into a series by use of undetermined coefficients. Let VI + 2x = A + Bx + Cx^ + Dx^ + Squaring both sides, using method of Art. 89 for the right member^ 1 + 2x = ^'^ + 2^51 X + ^'1 x"" + 2AD\ x^+ 2.401 + 2BC\ In this identity e(juate the coefficients of like powers of JC, ^. ^^ 350 ALGEBRA. 1 -^ 2x El -^ X - Ix^ + ^x^ + . . . . , Series. Another series may also be obtained by taking the negative value of A obtained from A"^ = 1, — viz. A = — 1, — and determining corresponding val- ues for B, C, D, etc. EXERCISE Expand into a series in ascending 125. powers of 5. 6. X — 1. VI - 2x. 3. V4:-'Sx- ^x\ + x. 2. Vl + Ax + 2x\ 4. Vd' — x\ -Zx-e^ m. Separating a Fraction into Partial Fractions. 338. General Method. If the degree of the numerator of a fraction be greater than the degree of the denominator, the fraction may be converted into a mixed number, the fraction obtained having the degree of the numerator less than the degree of the denominator. If we consider proper fractions only, such fractions, if their denominators can be factored, may be separated into partial fractions by the use of the properties of identities. It is evident that if in the original fraction the degree of the numerator is less than the degree of the denominator, the same must be true in each partial or component fraction. The problem before us is the inverse of that treated in Arts. 143 and 144. There, the several fractions were given to find their sum, but here, the sum is given to determine the constituent fractions. Tlie importance and use- fulness of the methods here presented will be more fully appreciated by the pupil whea be hag advanced to the Integral Calculus. UNDETERMINED COEFFICIENTS, 361 CASE I. Factors of the Denominator are of the First De- firree and Unequal. bx — 14 Ex. 1. Separate — — into partial fractions. a;^ — 6a; + 8 Let 5a: -14 ^_^4_ ^ _B_ a:2_6a;+8a;-2a;-4 ^ ^ Clearing of fractions, 5a; - 14 = ^(a; - 4) + ^(a; - 2) (2) Hence, 5a; - 14 =e (^ + B)x - 4A - 2B (3) Equating coelEcients of like powers of x, A + B = 6, 1 whence ^ = 2, -4/i -2E= -14,J and ^ = 3. Substituting for A and ^ in (1), 5a; - 14 _ 2 ^ 3 . a;2-6a; + 8~a;-2a;-4' The values of A and B may' also be obtained from (2) in another way, which usually involves less labor in this case. Thus, since (2) is an iden- tity, and therefore true for any value of x whatever, let a; = 2, then from (2) 10-14 = ^(2-4). Hence A = 2. Again in (2) let a; = 4, then ^ = 3. a;^ — a; — 3 Ex. 2. Separate - — into partial fractions. a:^ — 4a; a;''-a;-3^4^^ B . C x^ — 4x x X — 2 X + 2 Hence, a;^ - a; - 3 = ^(a; - 2) (a; + 2) + Bx{x + 2) + Gx{x - 2) . (1) In (1) let x= 0, then - 3 = ^(- 2) (+ 2), hence, A = \, In (1) let a; = 2, " B= - \, In (1) let a;= -2, ♦' C = f . Hence, ^'-^"zJ^A L_ + ? ' s^-4x ~~Ax 8(a;-2) 8(a; + 2) 352 ALGEBRA. EXERCISE 126. Separate into partial fractions — , x-12 ^ 2x-h24 . 2a;'-13a; + 9 6. — : --• 5.^ x{x-Z) x^-O 2x^-%x 2 — ?^_. 4 2a;' + 7a;-l x''-Q>x-l ' ^-x-2 ' a:(4x'-l)' 'ix'-x^-x --«*-- 2x^4- 4a; + 5 ax — 14a*' 8. ~ : I • 11. a:' + x — 2 cc^ — 3aa: — 4a" 2-llx-3x' 2x' + 5ax'* ISx^' - 18a:' + 4a; 2a:'H-aa;-a' ,^ a;' - 2x' + 16 ^^ 12(a;-l) ^^ 4a;- 14 lo. • 14. ;: • 15. 8 a;»-4a;' + 2a; 4x'^-20a; + 23 CASE IT. 340. Factors of the Denominator are of the First Degree and some of them Repeated. In this case some factors of the denominator will be of the form x"" or (ax + 6)". q~. -j If we consider the fraction > it is evidently sep- a;'(a; + 2) ^ ^ arable into ; 1 » the condition being that 0? a; + 2 the degree of the numerator in each partial fraction must be one less than the degree of the denominator. V\^ Aj^\Bx^C _A^ .^^.0 '.: , ar' ~ ^ ^ ar* X x^ of the latter being the more convenient form. * First reduce to a mixed number. UNDETERMINED COEFFICIENTS. 363 Hence, we write Sx~l _A .B ^C ^ D and find A, B, Q D as usual. Again, if the factor of the denominator which is a power be a binomial, it is similarly separable. Thus, J^^^.A^^^±0^±D ^A Bx'-ABx-^AB^-{C-VAB)x-¥D-\B ^ X (x~2y _A B ax-\-iy ~~ X x-2 (x-2)' __A B C'x-2C'-\-iy + 20' x ijc-2 {x-2y A , B , a ^ DC H ~ "T- T-. + ^ X x-2 {x-2y (x-2)* where A^ B, C\ Bf' are numbers to be determined. Ex. Separate into partial Iractions. 1 A ^_B_^ a . {X - If {x-Vl) x-1 {x-l}^ x + 1 Clearing of fractions, l = ^(a;2 - 1) + B{x + 1) + C{x - 1)> Hence, 1 = (^ + C)x^ + {B - 2C)x -A+B + a Hence, A + C = 0, ^ f A = - \. ^ - 2C = 0, I whence \ B = ^. -A + B + C^l, \ [C=\. • ' {x~ ly {x-i-l)" 4(« - 1) 2{x- 1}' 4{x + 1) 23 354 . ALGEBRA, EXERCISE 127. Separate into partial fractions — ^' xXx + 1) ^•(x+1)-* l-2x- IVar* " x\5x- 1) 8x' + 9x -35 ' (x-iyQ. r + 2)^ 0. ^'-' (x + 2y (a;-2)Xl-22;) ^ 3a;-20 ^ 28a;^-4a; 3a;'-4a;'' * (2a;-l/* 8a:* -81a: -54 10. 11. 12. a:\2a; + 3/ 5(9 + llx) (2a:^ + a:--3)^* 16a:' + 15a; -50 (a; -2)* xXx + 1) (2a: -5/ CASE III. 841. One or More Factors of the Denominator is of the Second Degree. It is necessary in each case to let the degree of the numer- ator of each partial fraction be one less than the degree of the denominator. X Ex. 1. Separate -r—-- into partial fractions. (x + 1) (a:' + 1) ^ Let ^ — ^ 4. Bx+ O . {x + l){x^ + 1) x + 1 x^ + 1 Hence, x^Ax^ + A + Bx^ + J5a: + Gc + 0. x = {A + B)x'' + {B + C)x + A^-G. Hence, A+B = Q>, ^ { A = - \. B+ C=\, \ whence \ B = \. ^ + c = 0, J yo = \. , X - 1 , a; + 1 ^ " * (x + l)(a;2 + 1) ~"2(a; + l) 2(a;'' + l)* UNDETERMINED COEFFICIENTS, 355 Ex. 2. Separate -— — - into partial fractions. ar* + 1 3 _ A ^ Bx^ a;' + l~a; + l x^ - x ^- \ Z = A{x'' -x + l)^ {Bx + C){x + 1) 3=(^ + £)x2 + {B + C- A)x + A + Q Hence, ^ + J5 == 0, 1 M = 1, B+ C- A^O, I whence ^ B = - I, ^ + C = 3, J [ = 2. . 3 _ 1 x-1 **a;3 + l~2;4-l aj-'-aj + l The following are other possible examples, with their methods of sep- aration : 3a; - 2 ^A^Bt^^Cx'^ + Dx^E x{x^ + 2} a; a;* +. 2 6a; + 7 _ A ^ Bx + C ^ Dx + E {x + 1) (a;2 +2)''' a: + 1 3:^ + 2 {x^ + 2)» It will be observed that the number of known letters used in the solution of all examples in Partial Fractions is the same as the degree of the De- nominator. EXERCISE 128. Separate into partial fractions — . 5a; + 4 „ 2x'-7a; + l , lOx + 2 a = -fa;~^^~i Ix2x3x4x 5\2/ \ ' " ) BINOMIAL THEOREM. 363 Ex. 4. Expand (1 + 2a; - Sx'Y by use of the binomial formula. (1+ 2a; - Zx^Y = [(1 + 2x) - Zx^y = (1 + 2xy - 3(1 + 2xY (3^2) + 3(1 + 2x) (3a;')' - (3x')» = 1 + 3(2a;) + 3(2a;)2 + {2xf - 9a;2(l + 4a; + 4a;') + 27a:*(l + 2a;) - 27a;* = 1 + 6a; + 3x2 - 28a;» - 9a;* + 54a;» - 27a;*. EXERCISE 130. Expand — 1. (a + 3)*. 5. (a:* - 2a:)^ 9. /x - ' - ^V- 2. (2a-x)^ 6. (xl/^ + l)«. 3. /l + ^y. 7. (x-t+v^y. 4. (3x^ - 21/')*. ^* \2y ^^) ' 12. ( V'^ - 1/^^/. ...(svi-y. "■(•^-Wi)'- 16. (a;' — a; + 2)'. 17. (2-3a: + a;7. 18. (2x^ + a;-3)*. 19. (a' + 2aa; - a;^*. 15. (3a " ^ 1/5 - 6 " * l>a)*. 20. (3x^ - 2x - 1)*, Find the— 21. Sixth term of (a - 2a;0". 22. Eighth term of (1 + xVy)''. 23. Seventh and eleventh terms of (x^ — 2/1^)^*. 24. Sixth and ninth terms of (|a'6 — 2 Paf. 25. Tenth and twelfth terms of fa: ^ + ^—z\ 26. Middle term of {Za'i-xVa)'\ l_ 364 ALGEBRA, X — I • 28. Term containing x^^ in Ix' j . fx —Y^ 29. Term containing x'* in j - + Vx^ J . (2\^' a;* 1 . 31. Term containing x in lyVx -\- \-] . FOR FRACTIONAL OR NEGATIVE EXPONENTS. 349. Examples. The binomial formula is true also when the exponent of the binomial, n, is a fraction or a negative number, provided the resulting series is convergent, though no simple elementar}^ proof in this case can be given. Ex. 1. Expand (1 + 3x)^ to 4 terms. Using formula (3), Art. 346, (1 + 3x)^ = 1 + i{Sx) + Mi zA)(3xy + Ki - 1) (I - 2) (3a;)» + 1x2 1x2x3 1+ f a; - |a;2 + ^^x^ + Ex. 2. Expand -^^ to 4 terms. Va-x _. -¥-i-i)(-i-2) ixy ^ 1x2x3 \a/ L 3a 9a' 81a= . J BINOMIAL THEOREM. 365 -i Ex. 3. Find 6th term of _ 1 9'** 'f We have [Art. 347], X = x^, a = - ^- — , n = - i, r = 6. o ... 6thterm^-^- t--i — J^^(^.)-i-» (_ S^'V 1.2.3.4-5 ^ ^ V 3 / 350. Extraction of Roots of Numbers by Use of the Binomial Formula. When a number whose root is to be extracted approximates an exact power corresponding to the index of the root, the required root may often be obtained readily by use of the binomial formula. Ex. Extract the cube root of 215 to five decimal places. 215 = 216 - 1 = 210(1 - 3j1^) = 6'(1 - ^h). 1^215-= 6(1 - A^)^ = 6[1 -h^h -^'{^h? - Mi^hy- ] = 6(1 - .001543 - .000002 - ) = 5.99073+, Boot EXERCISE 131. Expand to five terms — 1. (a + x)-\ 2. (x'-l)^. 3. (x + 3)*. 4. {l-\-2x'y. 11. (1^^-42/')*, 5. /,-2 A\-3 Q 1 (^ ^) • «• (1 + ^).' 6. 7. (x-' + xVy) ■^. 10. i/¥^^x. 12. (a ^+ Vax-'y\ 366 ALGEBRA. 13. ^ 16 1 14. Vx' -SVx I -I 2x y ^ 15. 1 r /v 9^/ ^ ^ 18 ^ ^ '' 3-' ' i 18. 1 3- ya-'+sx-^ ivy V Find in the simplest form— 19. Fifth term of (1 + x)^. 20. Eighth term of (1 + 2a:) ~ ^ 21. Tenth term of (a' — Sl/ai/A 22. Fifth term of 3-- 2a-Zv'x 23. Fifth and tenth terms of (x " * - 2 x/xf- 24. Sixth and eleventh terms of (x~^ + 3x~^)* 25. Seventh and thirteenth terms of ( V^ — — I I 2x1 26. The term containing a; ~ *° in [ o;^ — ' - 1^- (2 2 \4- 28. The term involving a;" in -3 Vx^ — aVx Find the approximate value of the following to five decimal places : 29. l/m 31. V'm. 33. 1/94. 35. 1>'260. 30. 1/65. 32. V128. 34. 1^15. 36. V'BB. CHAPTER XXIX. CONTINUED FRACTIONS. 351. A Continued Fraction is a fraction the denominator of which contains a fraction, the denominator of that fraction also containing a fraction, and so on, either for a finite or an infinite number of minor fractions. 3 Ex. 4-f "^ Continued fractions are usually limited to those in which each numerator is unity ;- as, 1 c + Continued fractions are more conveniently written and printed in the form, _1_ J 1_ a+ b-\- c+ 352. Integral and Converging Fractions. The simple fractions which compose a continued fraction are called In- tegral Fractions. Thus, in the above example, -» -> -> a c etc. are the integral fractions. The Converging Fractions are the continued fractions 367 S68 ALGEBRA. formed by taking one, two, three, etc. integral fractions at a time. Thus,, in the above evample, - is the first convergent, a is the second convergent, "+6 a+ ' is the third convergent, "-I etc., etc. 853. A Common Fraction made into a Continued Frac- tion. By a method essentially the same as that used in finding the greatest common divisor of two numbers, a com- mon fraction may be made into a continued fraction. For instance, 19^ 1 ^ 1 ^ 1 43 43- A 2 + 1 19 19 19 5 1 1 1 2 + -1-- 2 + -1— 2 + 4 1 3+? 3+^ 3+ 1 -i The process is more conveniently presented thus: 19)43(2 38 The quotients 2, 3, 1, 4, are the de- 5)19(3 nominators of the integral fractions 15 composing the continued fraction, the 4)5(1 numerators in each case being 1. 4 iH(4 CONTINUED FRACTIONS. 369 854. Computation of Converging Fractions. We shall now obtain a method of computing the values of the suc- cessive convergents of a continued fraction. Consider the continued fraction, J_ J 1_ J_ 1 J 1^ a+ b+ c-h....+p+ q+ r -\- s4- Ist convergent = - > a 2d convergent = ^ _^ 1 ab + 1 3d convergent = — ^ bo + 1 a + -i— ^^"^ + 1) + a 6 + 1 c An examination of the third convergent shows that it may be formed from the two preceding convergents. Thus, Num. of 3d conv. = (num. of '2d conv.) x (3d quot.) + (num. 1st conv.) Denom. of 3d conv. = (denom. of 2d conv.) x (3dquot.) + (denom. 1st conv ) In general, N. ofrth con. = [N. of (r — l)*^ con.] [rth quo.] + [N. of {r-2f^ con.] D. of rth con. = [D. of {r - 1}*^ con.] [rth quo.] + [D. o/ (r - Sf'^ con.] We shall now prove that if these laws hold for the formation of any con- vergent from the i)receding convergents, they will hold for the formation of the next succeeding convergent. Denote the convergents corresponding to the quotients p, q, r, s by ±—, hL, ±L, ^, and suppose the rth convergent to be formed accord- P' Q' R' S' ^^ ing to the law. Hence ^ = ^ + ^ (1) An examination of the general continued fraction given above shows that the sth convergent is formed from the rth by changing r into r + -' Making this substitution in (1), 24 370 A LQEBBA, s ^(^%-)^^ Qrs^Qr^Ps S' Q\r-v\)^P^ Q'rs + Q^ + P^s = {Qr + P)s+Q Rs + Q {Q'r + P')s+Q' E's + Q' Hence, if the law is true for the rth convergent, it is true for the next. But by actual reduction the law holds for the formation of the 3d convergent ; hence, by the general prin- ciple just proved it must hold for the 4th convergent ; hence, for the 5th, and so on. Ex. Find a series of converging fractions for ^f|. Forming the given fraction into a continued fraction, the quotients are, 6, 5, 4, 3, 2. Hence, convergents are, |, -^V, yW, /^, iri- The first and second convergents are readily determined from the contin- ued fraction. For the others, the following scheme may be found helpful : 3d^ j4xl+I = -2L; 4,1,^ f3x_21+l==_68 . ^^^^ 14x31 + 6 = 130 13x130 + 31=421 355. Convergents as Successive Approximations. The first, third, .... and all odd convergents are larger than the value of the entire continued fraction ; the second^ fourthj .... and all even convergents are smaller. For consider the continued fraction, o+ 6+ c+ d+ The first convergent, - ? is larger than the entire continued fraction, since the denominator, a, is smaller than the denom- inator of the entire continued fraction. The second convergent, — > is smaller than the contin- "+6 CONTINUED FMACTIONS. 371 ued fraction, since the denominator, 6, is too small ; hence, - is too large j hence, a + - is too large a denominator; hence, is too small. And so on alternately. The convergents, however, approach nearer and nearer the value of the original fraction. N. B. Should the original fraction be improper, the first convergent is an integer, and of course less than the value of the fraction. In the case of mixed numbers and improper fractions, therefore, the odd convergents are less and the even convergents greater^ than the value of the entire con- tinued fraction. 356. Degree of Approximation in Convergents. The dif- p Q .erence between two successive convergents, — and -;» can 1 ' P Q be shown to be j and therefore the difference between P'Q either of these convergents and the value of the entire con- tinued fraction is less than -^r^' P Q, The difference between the first two convergents is lb 1 . a ab + I a[ab + 1) We shall now prove that if this law holds for the difference between any pair of convergents, — and -^- » it will hold for the difference between the next pair, -^ > — • Let the symbol "^ be used to denote the difference between. T-t P^^9^=P9'-P'Q =^L. (I) 372 ALGEBRA. But ^^Q-^m^-R^Q^ Hence, if the law is true for the difference between any pair of converg- ents, — > ^ > it ia true for the difference between the next pair, ^ » ■n — • But by actual reduction it is true for the difference between the first B' ^ pair of convergents ; hence, by the general principle just proved it is true for the difference between the second pair ; and so on indefinitely. Ex. In the example of Art. 354, what is the error in using the third convergent, yVo^, instead of the value of the entire continued fraction, -g^ff ? The next convergent after yW is -^-^•, hence, the error is less than — - — -— • This is called the superior limit of the error. EXERCISE 132. Express as continued fractions — 1. fi. 2. «. 3. 5H. 4.^. Find the fourth convergent in — 5.3 + ~ 6.1 + -. 2 + -^—— 1 + 4 + -—- 3 + 1 + i 2 + i 7.1 + ::^ 2+ 1+ 2+ 1 + 3+ 2+ 1+ 3 + CONTINUED FRACTIONS. 373 Express the following as continued fractions, and find the fifth convergent in each : 9. li. 11. m- 13. Uh 15- tVoV. 17. 0.3029. 10. -\\\ 12. Iff. 14. 4H^. 16. 1.59. 18. 0.5678. " Determine the superior limit of the error in taking the fourth convergent for the continued fraction itself in each of the examples, 9-18 inclusive. 357. Surds Expressed as Continued Fractions. A quad- ratic surd may be converted into an infinite continued frac-* tion. Ex. Convert V^into a continued fraction. 2 is the greatest integer in 1/6. Hence, 1/6 = 2+ {V6 - 2) o_L V6- 2 T/6 + 2 2 = -^ + 7— X — n = 2 + 1 1/6 + 2 1/6 + 2 = 2 + ^ (1) 1/6 + 2 2 _ 2 4- 1 /Since 2 is the greatest integer in 2+ 1^6" -2 ( l/g+2 2 2 = 2+ ^ . 2+ T^6 - 2 ^ 1/6 + 2 2 1/6 + 2 «2+- 1 ) 2+-A 1/6 + 2 2+ 1 (2) 2+ L_ 4 + -^— 1/6 + 2 2 374 ALGEBRA. The last denominator fraction in (2), viz. > is the same as the last denominator fraction in (1) ; hence, continuing the process indefinitely, V6-2+J- -1- -1- J- 2-t- 4+ 2+ 4+ An infinite continued fraction in which the denominators repeat themselves periodically is called a Periodic Continued Fraction. 358. A Periodic Continued Fraction Expressed as the Root of an Equation. A periodic continued fraction may be expressed as the root of an equation. Thus, to express J_ J^ _1 1_ 2+ 3+ 24- 3+ .... , let X denote the value of the periodic continued fraction. 1 ,' . X — Clearing of fractions, 2a;' + 7x = 3 + aJ a;2 + 3a; = | ^ = K-3+ VT5). The sign + is used before the radical I^IS, since x can have positive value only. EXERCISE 133. Express each surd as a continued fraction — 1. V5. 4. yg. 7. 1/T9. 10. V^. 2. VZ. 5. VI. 8. 21/5. 11. 1/^. 3. V\Q. 6. 1/14. 1 9. 3 1/2. 12. ^33* 13. 3 + 1/23. 14. 1/15 + 3 ^^ 1/37 + 5 CONTINUED FRACTIONS. 375 Express each continued fraction as a surd — le. ^ ^ ^ ^ 1+ 3+ 1+ 3+ „. JL i X ^ 2+ 4+ 2+ 4+ ,8. -i- i A. J. 1+1+1+1+ 19.1 + JL J- -L J_ 2+ 3+ 2+ 3+ 20.3+^ X J. i_ 4+ 1+ 4+ 1+ 21.4 + ^ J- _i_ X A_ 3+ 1+ 2+ 3+ 1+ Express as a continued fraction the positive root of each equation — 22. x' — 2a; = 10. 23. a;'--4a: = 8. 24. 5a;' -7a; = 2. 25. Express each root of 3x' — 8x + 1 = as a continued fraction. 26. The circumference of any circle is 3.1415926 times its diameter. Required the series of fractions converging to this ratio. 27. The lunar month is approximately ^7.321661 days. Find a series of fractions converging to this quantity. 28. A solar year is 5 hours, 48 minutes, 49 seconds more than 365 days. Find a series of common fractions approxi- mating nearer and nearer the ratio of this excess to a day. CHAPTER XXX. LOGARITHMS. 859. The Logarithm of a number is the exponent of that power of another number taken as the base, which equals the given number. Thus, If 10 be the base, since 1000 = 10*, log 1000 = 3 ; if 8 be the base, since 4 = 8^, log 4 = f . if B'= N, log3 N=L This is read : log of N to the base B = l, 360. Source of Value in Logarithms. The source of new power in the use of logarithms may be illustrated by the multiplication of two numbers which are exact powers of 10, as 1000 and 100, by the use of exponents. Thus, Since 1000 ^lO*, and 100 = 10', 1000 X 100 - 10^ = 100,000, Product. In like manner, if 361 = lO'^-^"^' +, and 29 = 10'*«^*"+, we may multiply 361 by 29, by adding the exponents of the powers of 10 which equal these numbers, obtaining 10*°'^^^ "^, and then obtaining from a table -^f logarithms the number corresponding to this result, which will be the product. Thus, by the systematic use of exponents or logarithms, the process of multiplying one number by another is converted into the simpler process of adding two numbers (exponents). In like manner, by the use of logarithms, the process of 376 LOGARITHMS. 377 dividing one number by another is converted into the simpler process of subtracting one exponent (or log) from another; the process of involution is converted into the simpler process of multiplication; and evolution, into the simpler process of division. In the systematic use of these properties of exponents lies the source of new power in logarithms. 361. Systems of Logarithms. Any positive number ex- cept unity may be made the base of a system of loga- rithms. The base used is usually denoted by placing it as a small subscript to the word log. Thus, the logarithm of n in a system whose base is a is denoted by log^w. Two principal systems of logarithms are in use — 1. The Common (or Decimal) or Briggian system, in which the base is 10, used in numerical operations. 2. The Napierian system, in which the base is 2.7182818 -h, generally used in algebraic processes, as to demonstrate prop- erties of expressions by the use of logarithms. COMMON SYSTEM. 362. Characteristic and Mantissa. If a given number, as 361, be not an exact power of the base, its logarithm, as 2.55751 + for 361, consists of two parts, the whole number, called the Characteristic, and the decimal part, called the Mantissa. To obtain a rule for determining the characteristic of a given number (the base being 10), we have, 10000 = 10*, hence, log 10000 = 4. 1000 = 10^ " log 1000 = 3. Hence, any number between 1000 and 10000 has a log be-' tween 3 and 4 ; that is, it consists of 3 and a fraction. There- fore, every integral number consisting of 4 figures has 3 for a characteristic. 378 ALGEBRA. Similarly, since 100--10^ 10 = 10\ 1=10^ every number between 100 and 1000, and therefore containing 3 integral figures, has 2 for a characteristic; every number between 10 and 100 (that is, every number containing 2 inte- gral figures) has 1 for a characteristic; and every number between 1 and 10 (that is, every number containing 1 inte- gral figure) has for a characteristic. Hence, The characteristic of every integral or mixed number is one less than the number of figures to the left of the decimal point. 363. Characteristics of Decimal Numbers. Since 1 = lOO, .1- — -10-s 10 .01=— =^=lo-^ 100 10^ ,001 = =- — - - 10-', etc., etc., 1000 10* ' ' > the logarithm of every number between .1 and 1 (as, for in- stance, of .3) will lie between — 1 and 1 ; that is, will be — 1, plus a positive fraction ; also, the logarithm of every number between .01 and .1 (as of numbers like .0415) will lie between — 2 and — 1, and hence consist of — 2, plus a positive frac- tion ; and so on. Hence, The characteristic of a decimal number is negative, and is, nu- merically, one more than the number of zeros between the decirnal point and the first significant figure. The characteristic of a decimal number is written in two principal ways. Thus, log .0372 = 2.5705, the minus sign being placed over the characteristic 2, to show that it alone is negative, the mantissa being positive, LOGARITHMS. 379 We may also add and subtract 10 from the given log. Thus, log .0372 = 8.5705 - 10. In practice, we use the following rule for the characteristics of decimal fractions : Subtract the number of zeros between the decimal point and the first significant figure from 9, and annex — 10 after the mantissa. 864. Mantissas of numbers are computed by methods which are beyond the scope of this book. After being computed they are arranged in tables, and when needed are taken from the tables. The position of the decimal point in a number affects only the characteristic, not the mantissa. For example, 69.72 = ^^ ^ ^. Hence, if 6972 = 10'^*^^^ log ^^^ =: log - — ^ = log lO^-^^^''* = 1.84336. 5 10' ^10' In general, log 6972 = 3.84336 log 697.2 = 2.84336 log 69.72 = 1.84336 log 6.972 = 0.84336 log 0.6972 = 9.84336 -10 log .06972 = 8.84336 - 10 365. Direct Use of a Table of Logarithms ; that is, given a number, to find its logarithm from the table. We shall here insert a small table of logarithms, that the student may learn enough of their use to understand their algebraic properties. The thorough use of logarithms for purposes of computation is usually taken up in connection with the study of Trigo- nometry. In the given table (see pages 380, 381) the left- hand column is a column of numbers, and is headed N. The mantissa of each of these numbers is in the next column opposite. In the top row of each page are the figures 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. 380 ALGEBRA. N. 1 2 3 4 5 6 7 8 9 10 0000 0043 0086 0128 0170 0212 0253 0294 0334 0374 11 414 453 492 531 569 607 645 682 719 755 12 792 828 864 899 934 969 1004 1038 1072 11(16 13 1139 1173 1206 1239 1271 1303 335 367 399 430 U 4(51 492 523 553 584 614 644 673 703 732 15 1761 1790 1818 1847 1875 1903 1931 1959 1987 2014 16 2041 2068 2095 2122 2148 2175 2201 2227 2253 279 17 304 330 355 380 405 430 455 480 504 529 18 553 577 601 625 648 672 695 718 742 765 19 788 810 833 856 878 900 923 945 967 989 20 3010 3032 3054 3075 3096 3118 3139 3160 3181 3201 21 222 243 263 284 304 324 345 365 385 404 22 424 444 464 483 502 522 541 560 579 598 2:5 617 636 655 674 692 711 729 747 766 784 24 802 820 838 856 874 892 909 927 945 962 25 3979 3997 4014 4031 4048 4065 4082 4099 4116 4133 26 4150 4166 183 200 216 232 249 265 281 298 27 314 330 346 362 378 393 409 425 440 456 28 472 487 502 518 533 548 564 579 594 609 29 624 639 654 669 683 698 713 728 742 757 30 4771 4786 4800 4814 4829 4843 4857 4871 4886 4900 31 914 928 942 955 969 983 997 5011 5024 5038 32 5051 5065 5079 5092 5105 5119 5132 145 159 172 33 185 198 211 224 237 - 250 263 276 289 302 U 315 328 340 353 366 378 391 403 416 428 35 5441 5453 5465 5478 5490 5502 6514 5527 5539 5551 36 503 575 587 599 611 623 635 647 658 670 37 682 694 705 717 729 740 752 763 775 786 38 798 809 821 832 843 855 866 877 888 899 39 911 922 933 944 955 966 977 988 999 6010 40 6021 6031 6042 6053 6064 6075 6085 6096 6107 6117 41 128 138 149 160 170 180 191 201 212 222 42 232 243 253 263 274 284 294 304 314 325 43 335 345 355 365 375 385 395 405 415 425 44 435 444 454 464 474 484 493 503 513 522 45 6532 6542 6551 6561 6571 6580 6590 6599 6609 6618 46 G28 637 646 656 665 675 684 693 702 712 47 721 730 739 749 758 767 776 785 794 803 48 812 821 830 839 848 857 866 875 884 893 49 902 911 920 928 937 946 955 964 972 981 60 6990 6998 7007 7016 7024 7033 7042 7050 7059! 7067 61 7076 7084 093 101 110 118 126 135 143i 152 52 160 168 177 185 193 202 210 218 226 235 63 243 251 259 267 275 284 292 300 308 316 64 324 332 340 348 356 364 372 380 388 396 N. 1 2 3 4 5 6 7 8 9 LOGARITHMS. 38] N. 1 2 3 4 5 6 7 8 9 55 740-^ 7412 7419 7427 7435 7443 7451 7459 7466 7474 5663 669 675 681 686 74 692 693 704 710 716 722 727 733 739 745 75 8751 8756 8762 8768 '8774 8779 8785 8791 8797 8802 76 808 814 820 825 831 837 842 848 854 859 77 865 871 876 882 887 893 899 904 910 915 78 921 927 932 938 943 949 954 960 965 971 79 976 982 987 993 998 9004 9009 9015 9020 9025 80 9031 9036 9042 9047 9053 9058 9063 9069 9074 9079 81 085 090 096 101 106 112 117 122 128 133 82 138 143 149 154 159 165 170 175 180 186 83 191 196 201 206 212 217 222 227 232 238 84 243 248 253 258 263 269 274 279 284 289 85 9294 9299 9304 9309 9315 9320 9325 9330 9335 9340 86 345 350 355 360 365 370 375 380 385 390 87 395 400 405 410 415 420 425 430 435 440 88 445 450 455 460 465 469 474 479 484 489 89 494 499 504 509 513 518 523 528 533 533 90 9542 9547 9552 9557 9562 9566 9571 9576 9581 9586 91 590 595 600 605 609 614 619 624 628 633 92 638 643 647 652 657 661 666 671 675 680 93 685 689 694 699 703 708 713 717 722 727 94 731 736 741 745 750 754 759 763 768 773 95 9777 9782 9786 9791 9795 9800 9805 9800 9814 9818 96 823 827 832 886 811 845 850 854 859 863 97 868 872 877 881 886 890 894 899 903 908 98 912 917 921 926 930 934 939 943 948 952 99 956 961 965 969 974 978 983 987 991 996 N. 1 1 2 3 4 5 6 7 8 9 382 ALGEBRA. To obtain the mantissa for a number of three figures, as 364, we take 36 in the first column, and look along the row beginning with 36 till we come to the column headed 4. The mantissa thus obtained is .5611. If the number whose mantissa is sought contains four or five figures, obtain from the table the mantissa for the first three figures, and also that for the next higher number, and subtract. Multiply the differ- ence bettveen the two mantissas by the fourth {or fourth and fifth) figure expressed as a decimal, and add the result to the mantissa for the first three figures . Thus, to find the mantissa for 167.49 Mantissa for 168 = .2253 Mantissa for 167 =.2227 Difference =.0026 Since an increase of 1 in the number (from 167 to 168) makes an increase of .0026 in the mantissa, an increase of .49 of 1 in the number will make an increase of .49 of .0026 in the mantissa. But .0026 X .49 = .001274 or .0013— Hence .2227 13 Mantissa for 167.49= .2240 Hence, to obtain the logarithm of a given number, Determine the characteristic by Art. 362 or 363. Neglect the decimal point, and obtain from the table Cpp. 380, 381) the mantissa for the given figures. Exs. Log. 52.6=1.7210. Log. .00094 = 6.9731 — 10. Log. 167.49 = 2.2240. Log. .042308 = 8.6264 — 10. EXERCISE 134. Find the logarithms of the following numbers: 1. 37. 7. 175. 13. .0758. 19. 0.7788. 25. .08134. 2. 85. 8. 504. 14. 5780. 20. .04275. 26. .00-^32. 3. 15. 9. 32.9. 15. .00217. 21. 234.76. 27. 95032. 4. 6. 10. 4.75. 16. 1275. 22. 5.6107. 28. 91706. 5. 90. 11. .08. 17. 63.21. 23. 900.78. 29. 32.171. 6. 300. 12. 1.02. 18. 3.002. 24. 7781.4. 30. 328.07. 366. Inverse Use of a Table of Logarithms ; that is, given « logarithm to find the number corresponding to this logarithm, termed antilogarithm; From the table find the figures corresponding to the mantissa of the given logarithm; Use the characteristic of the given logarithm to fix the decimal point of the figures obtained. Ex. Find the antilogarithm of 1.5658. The figures corresponding to the mantissa, .5658, are 368. Since the characteristic is 1, there are 2 figures at the left of the decl?' jnal point. 'i . ^eIlce, the antUog. 1.5658=36.8. LOGARITHMS. 383 In case the given mantissa does not occur in the table, obtain from the table the next lower mantissa with the corresponding three figures of the antilogarithm. Subtract the tabular mantissa from the given man- tissa. Divide this difference by the difference between the tabidar mantissa and the next higher mantissa in the table. Annex the quotient to the three figures of the antilogarithm obtaiyied from the table. Ex. Find antilog 2.4237. 4237 does not occur in the table, and the next lower mantissa is .4232. The difference between .4232 and .4249 is .0017. Hence we have antilog 2.4237 = 265.29. 4232 11) 5.00 ( .29 For if a difference of 17 in the last two figures of the mantissa makes a difference of 1 in the third figure of the antilog, a difference of 5 in the mantissa will make a difference of in^ of 1 or .29 with respect to the third figure of the antilog. EXERCISE 135. Find the numbers corresponding to the following logarithms: 1. 1.6335. 7. 0.6117. 13. 0.4133. 19. 8.7727- -10 2. 2.8865. 8. 9.7973- -10. 14. 1.4900. 20. 2.4780. 3. 2.3729. 9. 7.9047- -10. 15. 3.8500. 21. 0.6173. 4. 0.5775. 10. 8.6314- -10. 16. 1.8904. 22. 1.9030. 5. 3.9243. 11. 7.7007- -10. 17. 2.4527. 23. 3.3922. 6. 1.8476. 12. 6.1004- -10. 18. 9.6402—10. 24. 9.7071- -10, 367. Properties of Logarithms. It has been shown (Arts. 199, 200) that— when 771 and n are commensurable. By the use of successive approximations approaching as closely as we please to limits, the same law may be shown to hold when m and n are incom- mensurable. It then follows that — 1. log o6 = log a + log &. 8. log a^ = _ploga. 2. log /^j = log a - log 6. 4. log^a = -^- 384 ALGEBRA J Proof — Let a = lO'". . * . log a =m. b = 10". . • . log b =n. a6 = 10"* + ". .'. log ab =--m + n =log a + log 5 . • (1) ^ = 10-^ b . • . log l-\ =m — n = log a — log b . •(2) aP^lO^rn^ . • . log a^ = pm = p log a . . . , .(3) ■^a = lo". .(4) The same properties may be proved in like manner for a system of logarithms with any other base than 10. 368. Properties Utilized for Purposes of Oomputation. I. To Multiply Numbers, Add their logarithms, and find the antilogarithm of the sum. This will be the product of the numbers, II. To Divide One Number by Another, Subtract the logarithm of the divisor from the logarithm of the dividend, and obtain the antilogarithm of the difference. This will be the quotient. III. To Raise a Number to a Required Power, Multiply the logarithm of the number by the index of the power. Find the antilogarithm of the product. IV. To Extract a Required Root of a Number, Divide the logarithm of the number by the index of the required root. Find the antilogarithm of the quotient. Ex. 1. Multiply 527 bv .083 by the use of logs. log 527 = 2.7218 log .083 = a9191 - 10 antilog 1.6409 = 53.7+, Product. Ex. 2. Compute the amount of $1 at 6% for 20 years, at compound interest. ' Amount= (1.06)20 log 1.06 .0253 20 antilog 0.5060 = $3.21 +, Amount, LOGARITHMS. 385 If the student will compute the value of (1.06)'^'' by continued multiplica- tion, and compare the labor involved with that in the above process by the use of logarithms, he will have a good illustration of the value of loga- rithms. Ex. 3. Extract approximately the 7th root of 15. log 15 = 1.1761 ■f log 15 = 0.1680 + antilog 0.1680 -f = 1.47 +, Root 369. Cologarithm. In operations involving division it is usual, instead of subtracting the logarithm of the divisor, to add its cologarithm. The cologarithm of a number is ob- tained by subtracting the logarithm of the number from 10 — 10. Adding it gives the same result as subtracting the logarithm itself from the logarithm of the dividend. The use of the cologarithm saves figures, and gives a more compact and orderly statement of, the work. The cologarithm may be taken directly from the table by use of the following rule: Subtract each figure of the given logarithm from 9 except the last significant figure, which subtract from 10. Ex. 1. Find colog of 36.4. log 36.4 = 1.5611 colog 36.4 = 8.4389 - 10. 8.4 X 32.4 Ex. 2. Compute by use of logarithms 21/576X3.78 log 8.4 = 0.9243 log 32.4 = 1.5105 colog 2 - 9.6990 - 10 colog 1/576 = 8.6198 - 10 colog 3.78 = 9.4 225-10 antilog 0.1761 = 1.5, Remit. 25 386 ALGEBRA. EXERCISE 136. Find by use of logarithms the approximate values of— 1.75X1.4. ^ 0.317 12. 0.48 -^ (- 1.79). 2. 9.8X3.5. * -0049* ^^ -9.91 3. 15.1 X .005. 8. 78.9 -45.7 4.831X0.25. 21 14.A:^> 1, a-« = — - — = 0. a* oo . • ..log„0=- — 00. But in any system whose base is less than unity, the loga- rithm of zero is plus infinity. For a"^ =■- 0, since a < 1. . • . loga ^ QO . • 375. Change of the Base of a System of Logarithms. Let a and h be the bases of two systems of logarithms, and n be the number. Then log,„ = |2g^ (1) Let a* = n; hence, a: = loga^, and fe'' = n; " y = \ogi,n, X Also, a" = 6^ ,'. a" =--h. .-. loga6 = -, or y y loga6 1 loga n loga6 390 ALQEBftA. Hence the logarithm of a number, n, in a system whose base is a, being given, the logarithm of the same number in a system with another base, 6, may be found by dividing the given logarithm by the logarithm of b in the system whose base is a. From the above relation (1) we may also prove logft a X loga h = \. For putting n = a in (1), , logo « 1 loga 6 loga 6 .-. logft a X loga 6 = 1. 876. Examples. Ex. 1. Find the logarithm of 0.7 to the base 5. ByArt.375,log5 0.7 = ^^^^ ^ 9.8451 - 10 0.6990 — 1549 = 06990^ ^ ~ ^'^^^^ ■^' ^^^^''^*^^^- Ex. 2. Find the logarithm of 243 to the base 9 without the use of the tables. Let log9 243 = a; ; then 9^ = 243. Hence, {Zy = ^\ or 3'* = 3^ Hence, 2a; = 5 a; == f , Logarithm, Ex. 3. Find the number of digits in 5^^ Log (5^«) - 16 log 5 = 16 X 0.6990 - 11.1840. Since the characteristic is 11, the number of digits in 5^^ is 12. LOGARITHMS. 391 EXERCISE 137. Find the approximate value of x in each of the following exponential equations : 1. 40- = 75. 6. 6^ + ^ = 17. 11. 20 = 5. 2. 20' = 100. 7. 3" -'• = 5. 12. 27'""' = 8. 3.5^ = 12. 8. 12* + *-45^ 13. 0.7'= = 0.3. 4. 7^ = 25. 9. 5^-»*=:2* + ^ 14. 13.18* = .0281. 6. 1.3^ = 7.2. 10. (0.4) -=" = 7. 15. 0.703^ = 1.09. Express in terms of log a> log 6, log c, and log x — ^ 16. loga'6». 21. log ^'^^. 17. logax^. 18. log 6'!^. 22. log vc 19. log V'a^ ' V^, ' 23. log a " 'h' Vc=^. 20. log \/-- 24. log bcVx Express the value of x in terms of the logs of the known quantities in each of the following equations : 25. a* = 76^ • 29. Bo' + ' = a'b'^ ^ \ 2Q, (f = a'-\ SO. Q(fb^'' = ll(a- by. 27. 13*+^ = a'6. 31. a'^-ft' = (2a- 1/'. 28. 7a' = 3^^ 32. 25a^^ = Va'-l}\ Find the values of — 33. logsSO. 86. log6 4.5. 39. logs 0.9. 34. logs 12. 37. logi2 8. 40. logo.sGS. 35. Iog9o25. 88. log4.t23. 41. log^.j>/3. 392 ALGEBRA. Find without the use of the tables the base when — 42. log, 9 = |. 44. log, 5 = 0.5. 46. log, 3f = - f . 43. log, 8 = |. 45. log, iV = - 0.8. 47. log,3 1/3 - 1.5. Find without the use of the tables — 48. logaie. 50. log9^. 52. Iog2.0625. 49. logs 4. 51. log^^ie. 53. logyi4T/2. Find the number of digits in — 64. (875)^«. 55. T\ 56. 9^. 67. 57". Show that — 68. (|i)^«'>100. 59. (|^y°^> 100000. 60. There are more than 300 zeros between the decimal point and the first significant figure in (0.5)^°^. In the following geometrical progressions find n : 61. Given a, r, I. 63. Given a, Z, s, ^2. Given a, r, s. 64. Given r, Z, s. 65. Given a = 2, r=:5, Z = 31,250. 66. Given a = -^, r = 3, s = 364|. 67. Given a = 0.375, Z = 96, s = 191|. 68. Find the amount of $1000 for 20 years at 5% compound interest, 09. Find the amount of $875 for 18 j^ears at 6% compound interest. 70. How long will it require for a sum of money to double itself at 5% compound interest? At 7 per centum? 71. If $1280 amounts to $37,770 in 50 years at compound interest, what is the rate per centum? rjHAPTER XXXI. HISTORYi^OF ELEMENTARY ALGEBRA. 377. Epochs JH \e Development of Algebra. Some knowledge of thif origin and development of the symbols and processes of Algebra is important to a right under- standing of them. The oldest mathematical treatise known is a papyrus roll, now in the British Museum, entitled " Directions for Attain- ing to the Knowledge of All Dark Things." It was written by a scribe named Ahmes at least 1700 b. c, and is a copy, the writer says, of a more ancient work, dating, say, 3000 b. c., or several centuries before jthe time of Moses. This papyrus roll contains, among other things, the beginnings of algebra as a science. Taking the epoch indicated by this work as the first, the principal epochs in the development of algebra are as follows: 1. Bg3^tian : 3000 B. C.-1500 B. O. 2. Greek (at Alexandria) : 200 A. D.-400 A. D. Princi- pal writer, Diophantus. 3. Hindoo (in India) : 500 A. D.-1200 A. D. 4. Arab : 800 A. D.-1200 A. D. 5. European : 1200 A. D.-. Leonardo of Pisa, an Italian, published a work in 1202 a. d. on the Arabic arithmetic, but containing an account also of the science of algebra as it then existed among the Arabs. From Italy the knowledge of algebra spread to France, Germany, and England, where its subsequent development took place. We will consider briefly the history of — 293 394 ALGEBRA. I. Algebraic Symbols. II. Ideas of Algebraic Quantity, "jt III. Algebraic Processes. It. \ I. History of Algebraic j /mbols. 378. Symbol for the Unknown Qua/.atity. 1. Egyptians (1700 b. c.) used the wr ^" hau " (expressed, of course, in hieroglyphics), meaning •"Lvlap." 2. Diophantus (Alexandria, 350 a. d. ?), ?', or ?° ; plu- ral, 9?. 3. Hindoos (500 a. D.-1200 a. d.), Sanscrit word for " color," or first letters of words for colors (as of " blue," " yellow," " white," etc.). 4. Arabs (800 a. D.-1200 a. d), Arabic word for "thing "or " root " (the term " root," as still used in algebra, originates here). 6. Italians (1500 a. d.). Radix, R, Rj. 6. Bombelli (Italy, 1572 a. d.), vL'. 7. Stifel (Germany, 1544), A, B, C, . . . , 8. Stevinus (Holland, 1586), 0. 9. Vieta (Prance, 1591), vowels A, E, /, 0, U. 10. Descartes (France, 1637), x, y, z, etc. 379. Symbols for Powers (of x at first). Exponents. 1. Diophantus, duva/j.i<;^ or d" (for sqiaare of the unknown quantity); xujSog^ or x" (for its cube). 2. Hindoos, initial letters of Sanscrit words for " square " and "cube." 3. Italians (1500 a. d.), "census" or "zensus" or "z" (for x') ; " cubus " or " c " (for x'). 4. Bombelli (1579), vl^, e^, ^ (for x, x\ x'). 5. Stevinus (1586), ©, ©, © (for x, x\ x'). 6. Vieta (1591), A, A quadrnius, A cubus (for x, x\ x^), 7. Harriot (England, 1631), a, aa, aaa. HISTORY OF ELEMENTARY ALGEBRA. 395 8. Herigone (France, 1634), a, a2, a3. 9. Descartes (France, 1637), x, x^, x^. Wallis (England, 1659) first justified the use of fractional and negative exponents, though their use had been suggested before by Stevinus (1586). Newton (England, 1676) first used a general exponent, as in x**, where n denotes any exponent, integral or fractional, positive or negative. 380. Symbols for EZnown Quantities. 1. Diophantus, /j^ovade? (i. e. monads), or /j.° . 2. Regiomontanus (Germany, 1430), letters of the al- phabet. 3. Italians, d, from '' dragma." 4. Bombelli, \^. 6. Stevinus, ©. 6. Vieta, consonants, B, Q, D, F, . ... 7. Descartes, a, b, c, d. Descartes possibly used the last letters of the alphabet, x, y, z, to denote unknown quantities because these letters are less used and less familiar than a, b, c, d, . . . . , which he accordingly used to denote known numbers. 381. Addition Sign. The following symbols were used : 1. Egyptians, pair of legs walking forward (to the left), j\., 2. Diophantus, juxtaposition (thus, ab, meant a + b). 3. Hindoos, juxtaposition (survives in Arabic arithmetic, as in 2f , which means 2 + f ). 4. Italians, " plus," then "p " (or e, or 0). 5. Germans (1489), -+-,+,+. 382. Subtraction Sign. 1. Egyptians, pair of legs walking backward (to the right), thus, ZV. ; or by a flight of arrows. 2. Diophantus, ^ (Greek letter i^ inverted). 3. Hindoos, by a dot over the subtracted quantity (thus, mn meant m — n). 396 ALGEBRA. 4. Italians, " minus," then M or m, or de. 6. Germans (1489), horizontal dash, — . The signs + and — were first printed in Johann Widman's Mercantile Arithmetic (1489). These signs probably originated in German warehouses, where they were used to indicate excess or deficiency in the weight of bales and chests of goods. Stifel (1544) was the first to use them systematically to indicate the operations of addition and subtraction. 383. Multiplication Sign. Multiplication at first was usu- ally expressed in general language. But— 1. Hindoos indicated multiplication by the syllable " 6/ia," from " bharita," meaning *' product," written after the factors. 2. Oughtred and Harriot (England, 1631) invented the present symbol, X. 3. Descartes (1637) used a dot between the factors (thus, a-b). 384. Division Sign. 1. Hindoos indicated division by placing the divisor under the dividend (no line between). Thus, a meant c^ d. 2. Arabs, by a straight line (thus, a — 6, or a I 6, or -7- ]• 3. Italians expressed the operation in general language. 4. Oughtred, by a dot between the dividend and divisor. 5. Pell (England, 1630), -^. 385. Equality Sign. 1. Egyptians, ^ . ^ i (also other more complicated symbols to indicate different kinds of equality). 2. Diophantus, general language or the symbol, K 3. Hindoos, by placing one side of an equation immediately under the other side. 4. Italians, se or a ; that is, the initial letters of " sequalis " (equal). This symbol was afterward modified into the form, X), and much used, even b}^ Descartes, long after the invention of the present symbol by Recordje. HISTORY OF ELEMENTARY ALGEBRA. 397 . 6. Recorde (England, 1540), =^. He says that he selected this symbol to denote equality because "than two equal straight lines no two things can be more equal." 386. Other Symbols used in Elementary Algebra. Inequality Signs (> <) were invented by Harriot (1631). Oughtred, at the same time, proposed |, | as signs of inequal- ity, but those suggested by Harriot were manifestly superior. Parenthesis, ( ), was invented by Girard (1629). The Vinculum had been previously suggested by Vieta (1591). Radical Sign. The Hindoos used the initial syllable of the word for square root, " Ka," from '• Karania," to indicate square root. Rudolf (Germany, 1525) suggested the symbol used at pres- sent ( ]/) (the initial letter, r, in the script form, of the word " radix," or root) to indicate square root, m/ to denote the 4th root, and mV to denote cube root. Girard (1633) denoted the 2d, 3d, 4th, etc. roots, as at pres- ent by 1^, y", 1^, etc. The sign for Infinity, oo , was invented by Wallis (1649). 387. Many other Algebraic Symbols have been invented in recent times, but these do not belong to elementary algebra. Other kinds of algebra have also been invented employing other systems of the symbols. 388. General Illustration of the Evolution of Algebraic Symbols. The following illustration will serve to show the principal steps in the evolution of the symbols of algebra : At the time of Diophantus the numbers 1, 2, 3, 4, . ... were denoted by letters of the Greek alphabet, with a dash over the letters used; as, a, "^, 7, . . . . In the algebra of Diophantus the coefficient occupies the last place in a term instead of the first as at present. 398 ALGEBRA. Beginning with Diophantus, the algebraic expression, x' + 5x — 4, would be expressed in symbols as follows : 3" a ?6 j/fx ix° d (Diophantus, 350 a. d.). Iz p.5 RmA (Italy, 1500 a. d.). lQ + bN-4: (Germany, 1575). lv!y p.5v^ mA^ (Bombelli, 1579). 1© +5© -4® (Stevinus, 1586). lAq + bA -4 (Vieta,4591). latt+5a-4 (Harriot, 1631). Ia2 + 5al - 4 (Herigone, 1634). 7? + bx — 4 (Descartes, 1637). 889. Three Stages in the Development of Algebraic Symbols. 1. Algebra without Symbols (called Rhetorical Alge- bra). In this primitive stage algebraic quantities and operations are expressed altogether in words, without the use of symbols. The Egyptian algebra and the earliest Hindoo, Arabian, and Italian algebras are of this sort. 2. Algebra in which the Symbols are Abbreviated Words (called Syncopated Algebra). For instance, "j9" is used for " plus." The algebra of Diophantus is mainly of this sort. Euro- pean algebra did not get beyond this stage till about 1600 a. d. 3. Symbolic Algebra. In its final or completed state al- gebra has a system of notation or symbols of its own, inde- pendent of ordinary language. Its operations are performed according to certain laws or rules, " independent of, and dis- tinct from, the laws of grammatical construction." Thus, to express addition in the three stages we have "plus," p, + ; to express subtraction, "minus," w, — ; to ex- press equality, " sequalis," ^, ^. Along with the development of algebraic symbolism there HISTORY OF ELEMENTARY ALGEBRA. 399 was a corresponding development of ideas of algebraic quan- tity and of algebraic processes. n. History op Algebraic Quantity. 390. The Kinds of Quantity considered in algebra are positive and negative ; particular (or numerical) and gen- eral; integral and fractional; rational and irrational; comHiensurable and incommensurable; constant and variable; real and imaginary. 391. Ahmes (1700 b. c.) in his treatise uses particular, pos- itive quantity, both integral and fractional (his fractions, how- ever, are usually limited to those which have unity for a numerator). That is, his algebra treats of quantities like 8 and I", but not like —3, or — f, or V2, or —a. 392. Diophantus (350 a. d.) used negative quantity, but only in a limited way ; that is, in connection with a larger positive quantity. Thus, he used 7—5, but not 5 — 7, or — 2. He did not use, nor apparently conceive of, negative quantity having an independent existence. 393. The Hindoos (500 a. D.-1200 a. d.) had a distinct idea of independent or absolute negative quantity, and used the minus sign both as a quality^sign and a sign of operation. They explained independent negative quantity much as is done to-day by the illustration of debts as compared with assets, and by the opposition in direction of two lines. Pythagoras (Greece, 520 b. c.) discovered irrational quantity, but the Hindoos were the first to use this in algebra. 394. The Arabs avoided the use of negative quantity as far as possible. This led them to make much use of the process of transposition in order to get rid of negative terms in an equation. Their name for algebra was " al gebr we'l mukabala," which means transposition and reduction. The Arabs used surd quantities freely, 400 ALGEBRA. 395. In Europe the free use of absolute negative quantity was restored. Vieta (1591) was principally instrumental in bringing into use general algebraic quantity (known quantities denoted by letters and not figures). Cardan (Italy, 1545) first discussed imaginary quantities, which he termed " sophistic " quantities. Euler (Germany, 1707-83) and Gauss (Germany, 1777- 1855) first put the use of these quantities on a scientific basis. Descartes (1637) introduced the systematic use of variable quantity as distinguished from constant quantity. ni. History of Algebraic Processes. 396. Solution of Equations. Ahmes solves many simple equations of the first degree^ of which the following is an example: " Heap its seventh, its whole equals nineteen. Find heap." In modern S3"mbols this is, Given - + a; =^ 19 ; find x. 7 The correct answer, 16f , is given by Ahmes. Hero (Alexandria, 120 b. c.) solved what is in effect the quadratic equation^ \\d' + ''-^d = s, where d is unknown, and s is known. Diophantus solved simple equations of one, and simulta- neous equations of two and three unknown quantities. He solved quadratic equations much as is done at present, com- pleting the square by the method given, in Art. 255. How- ever, in order to avoid the use of negative quantity as far as possible, he made three classes of quadratic equations, thus, ax' + bx = c, ax' -\-c =6x, ax' = 6a; + c. HISTORY OF ELEMENTARY ALGEBRA. 401 In solving quadratic equations he rejected negative and irrational answers. He also solved equations of the form ax"" — 6a;^ He was the first to investigate indeterminate equations, and solved many such equations of the first degree with two or three unknown quantities, and some of the second degree. The Hindoos first invented a general method of solving a quadratic equation (now known as the Hindoo method, see Art. 256). They also solved particular cases of higher de- grees, and gave a general method of solving indeterminate equations of the first degree. The Arabs took a step backward, for, in order to avoid the use of negative terms, they made six cases of quadratic equa- tions; viz.: ax^ = bxj ax"^ + 6a; = c, ax^ = Cj ax^ -{- c = 6'r, bx =c, , ax^ = hx-\- c. Accordingly, they had no general method of solving a quad- ratic equation. The Arabs also solved equations of the form ax^^ + hx^ = c, and obtained a geometrical solution of cubic equations of the form x" + pa; + g = 0. In Italy, Tartaglia (1500-59) discovered the general solution of the cubic equation, now known as Cardan's solution. Ferrari, a pupil of Cardan, discovered the solution of equations of the fourth degree. Vieta discovered many of the elementary properties of an equation of any degree; as, for instance, that the number of the roots of an equation equals the degree of the equation. 397. Other Processes. Methods for the Addition, Sub- traction, and Multiplication of polynomial expressions were given by Diophantus. Transposition was first used by Diophantus, though, as a process; it was first brought into prominence by the Arabs. 402 ALGEBRA. The Square and Cube Root of polynomial expressions ■were extracted by the Hindoos. The methods for using Radicals, including the extraction of the square root of binomial surds and rationalizing the denominators of fractions, were also invented by the Hindoos. The methods of using fractional and negative Exponents were determined by Wallis (1659) and Sir Isaac Newton. The three Progressions were first used by Pythagoras 569 B. C.-500 B. c.). Permutations and Combinations were investigated by Pascal and Fermat (France, 1654). The use of Undetermined CoeflQcients was introduced by Descartes. The Binomial Theorem was discovered by Newton (1655), and, as one of the most notable of his many discoveries, is said to have been engraved on liis monument in Westminster Abbey. Continued Fractions were first used by Cataldi (Italy, 1653), though none of their properties were demonstrated by him. Lord Brouncker (England, 1620-84) was the first to do this. Logarithms were invented by Lord Napier (Scotland, 1614) after a laborious search for means to diminish the work in- volved in numerical computations, and were improved by Briggs (England, 1617). The fundamental Laws of Algebra (the Associative, Com- mutative, and Distributive Laws ; see Arts. 33-36) were first clearly formulated by Peacock and Gregory (England, 1830- 45), though, of course, the existence of these laws had been implicitly assumed from the beginnings of the science. Students who desire to investigate the history of Algebra in more detail should read the second part of Fine^s Number System of Algebra, BaWs Short History oj Mathematics, and Cajori's History of Elementary Mathematics, CHAPTER XXXII APPENDIX. L PROCESSES ABBREVIATED BY USE OP DETACHED COEFFICIENTS. 398. Multiplication by Detached Coefficients. The proc- ess of multiplying one polynomial by another (see pp. 42-4) can often be much abbreviated, and the Hkelihood of error diminished, by detaching the coefficients of the terms of the polynomial, performing the multiplication with respect to them, and then supplying the proper powers of the letters in the product obtained. Thus, Ex. 1. Multiply 6x' — 5a;' - 4x — 3 by 6a;' + 5x — 4. Detaching coefficients, 6 -5- 4- 3 6 + 5- 4 36 -30 -24 -18 30 -25 -20- 15 - 24 + 20 + 16 + 12 36 + - 73 - 18 + 1+12 Annexing the powers of x, SQx^ + Oa:* - 7Sx^ - Y^x^ + a; + 12, or 36a;5 - nZx^ - ISa:^ + a; + 12, Product. Ex. 2. Multiply r' + Sa'x - 2a' by x^ - iax" + ^a\ 1+0+3-2 1-4+0+ 3 1+0+3- 2 -4-0-12 + 8 + 3 + + 9-6 1-4 + 3-11 + 8 + 9-6 Hence, afi - 4aa;^ +- ZaH*^ - lla^^ + Sa*^;^ + 9a^T - Qa^, Product, 403 404 ALGEBRA. EXERCISE 138. The pupil may work Exs. 12 to 23 of Exercise 9 by use of detached coefficients. As the method is especially advantageous in multiplying polynomials with fractional coefficients, Exs. 1 to 6 of Exer- cise 13 should also be worked by this method. 399. Division by use of Detached Coeflacients is per- formed similarly. Ex. Divide x' — 2xY + ^xf — 3^/* by x' + 2xy — j^. 1 + 0-2 + 8-3 1 1 + 2-1 1-2 + 3 1 + 2-1 -2-1+8 -2-4+2 3+6- 3 + 6- -3 -3 Hence, x'^ - 2xy + 3?/^ Quotient. 400. Synthetic Division. The above process may be fur- ther abbreviated as follows: 1+0-2 +8-3 |l|-2 + l , Divisor (with signs 2 + 1 of all the terms + 4 —2 except the first — 6 + 3 changed). Qibotient, 1 — 2 + 3 + + 0, Remainder. Hence, x^ — 2xy + Zy^, Quotient. This abbreviation is made possible by noticing that if we change the sign of each term of divisor (in the division in Art. 399), we can change the successive subtractions employed into successive additions. Further, as each successive term of the quotient is found by dividing only the first term of the remainder by the first term of the divisor, it is sufficient to add each column only as needed in order to determine the first term of each remainder, and hence the next term of the quotient. It is not necessary to multiply the first term of the divisor by each terra of the quotient, since these products are not used in determining the remainders which give the successive terms of the quotient. Thus^ in the above process, having determined the first term of the quo- APPENDIX. 405 tient, 1, we multiply — 2 + 1 by 1, and add the column _ « I the sum, — 2, divided by 1 gives — 2, the second term of the quotient. We now multiply - 2 + 1 by — 2, set down the product, + 4 — 2, in the proper -2 place, add the column + 1, and divide the sum, 3, by 1, etc. + 4 It is usually more convenient to set the divisor in a perpendicular column at the left. Thus, Ex. Divide 42^ - Qx'^y + Ax^y^ - llx'^y^ + y^ by ^x^ - Zxy - y\ 2 + 3 + 1 4-6 + 4-11 6 + 2 0+ + 9 + + 1 + 3 -3-1 + + 2+0+3- Hence, 2a;' + Sxy^ — y^, Quotient. EXERCISE 139. Solve Examples 16 to 33 of Exercise 12 by the use of detached coefficients and synthetic division. 401. H. C. P. and Evolution by Detached Coefficients. In finding the H. C. F. of polynomials by the method of Arts. 119-124, and Exercise 38, the work may be abbre- viated by the use of detached coefficients. In extracting the square and cube root of polynomials (see Arts. 189, 194, Exercises 68 and 70), work may be saved in the same way. It is to be carefully noted that the use of detached coef- ficients not only saves labor in all these cases, but has the further advantage of diminishing the probability of mistakes, since fewer symbols are operated with. II. BEMAINDER AND FACTOR THEOREMS. SYMMETRY. 402. Remainder Theorem. If any polynomial of the form Pix"" -\r PiQc'' ~ ^ + PiOf ~ ^ + . . . . -\-pnhe divided by x remainder will be pid^ + p-^d" -\- P,a^-' + a^ the tained by substituting a for ac in the original expression^. 406 ALGEBRA. Let the given expression be divided hj x — a till a remain- der is obtained which does not contain x, and denote the quotient by Q and the remainder by R. Then Pix"" -f-p2x''~'^ -i-p-.iX'''''^ + .... +pn^Q(x — d) -^ R. This is an identity and therefore true for all values of x. Let a: = rt, then 2>,rr+i?2a""'+P3a'*~'+ +Pn = Q(ia-ci) + R^R. .-. R=p,a''+p,a''-''+psa''-'' + .... + p„. Ex. Find the remainder when 2x^ + 3a;* — ox^ + 6x^ + 8a; — 9 is divided by x~-2. Substituting 2 for x in the given expression, 2 • 2^ + 3 • 2* - 5 • 2^ + 6 • 22 + 8 • 2 - 9, or 103, Remainder. 403. Factor Theorem. If any rational integral expression containing cc become equal to zero, when a is substituted for ac, then ac~a is a factor of the given expression. This follows directly from the remainder theorem when R = 0, or it can be proved as follows : Let E stand for the given expression. If E be divided hy x — a till a remainder is obtained in which x does not occur, denote the quotient by Q and the remainder by R. Then E~Q{x-a) + R. Let X = a, then = Q{0) + R (since E=0 when x = a). . • . i2 = 0, Hence, E = Q{x — a), or a: — a is a factor of E. This principle is frequently of value in factoring expressions* Ex. Factor 3a;' + 7x^-4. By trial we find that 3x^ + 7a;2 - 4 = 0, when x = - 1. . • . a; + 1 is a factor of 3a:' + la;^ - 4. By division. 3:i^ i- 7x^ - ^ = {x + 1) {Sx^ + 4a; - 4) = (a; + 1) {x + 2) (3a; - 2), Factors. APPENDIX. . 407 It is to be noted that the only numbers that need be tried as values of x are the factors of the last term of the given expression. This follows from the fact that the last term of the dividend must be divisible by the last term of the divisor. EXERCISE 140. Factor by use of the factor theorem. 1. a;2 — 4. 10. 2:z;2 + 7x — 15. 2. rc2— 3aj — 28, 11. 2x^ — x^—lx-^Q. 3. x^ — ^x^2. 12. 4x^ — 4x^—Ux — e. 4. a^2 — a\ 13. Sx^ + Sx^-^'Sx — 2. d. x^ — Sa\ 14. 2x^ + x^ — Ux^ + 5a; + 6. 6. a^—h^ + 3(a — h). 15. 6;r* — 13:z;3 — 45a:2 — 2^: + 24. 7. (a — 5)2 + 3(a — 6). 16.x^ — 2x^-{-l. S.a^ — ab\ 17. x^ — 6:^2 + 25. 9. a3 + 5a — 6. 18. ^4 — 28^2 _^ 3S;r — 90. 19. Prove that x*^ — y^ is always divisible by a? — y. 20. Prove that x^-^-y"^ is divisible hyx-\-y when n is odd. 21. Show that il — x)^is a factor of l — x — x''-\-x''+^. 22. Show that (x— 1)2 is a factor of na;«+i— (n + l)ic" + l. 404. Symmetrical Expressions. An expression is sym- metrical with respect to two letters when it is unaltered by an interchange of the letters. Exs. a + b, ah, a' + h\ a^ + ah + h\ are each a symmetrical expression with reference to a and h. Similarly an expression is symmetrical with respect to three or more letters, when it is unaltered by an interchange of any pair of them. Ex. a' + ^>' + c' — 3a6c, is symmetrical with respect to a, 6, c, since it is unaltered by substituting a for h and 6 for a ; a for c and c for a ; h for c and c for 6. Instead of complete symmetry, there are partial symmetries of different 408 ALGEBRA. kinds which an algebraic expression may have. Thus, an expression has eyclo-symmeb'y (see Art. 144, Ex. 1) with reference to a, b, and c, if it remains unchanged after a is changed to b, b to c, and c to a. Ex. ab^ + bc^ + ea^ has cyclo-symmetry. A symmetrical expression may often be denoted in an abbreviated way by writing only the typical terms of the expression with the Greek letter 2 before each one. Thus, for a^b + b^a write I^a^b ; for a^ + b"^ + c"^ + ab + be + ca write 2a^ + 2a6. Similarly for a product, as (a — b) {b — c) (c — a), write n{a — b). 405. Factoring' S3mametrical Expressions. The factor theorem (Art. 403) is frequently of use in factoring sym- metrical expressions. Ex. 1. Factor bc(b — c) + ca(c — a) + ah{a — b). When b = c, the given expression reduces to ca{c — a) + ac(a — c), or 0. Hence, . 6 — c is a factor of it. Similarly c — a and a — b are factors. . * . bcCb—c)+ca(c—a)+ab(a—b) = L(b—c)(c—a)(a~b). (1) Where L (since the right-hand member is of the same degree with the given expression) is some number to be determined, Since (1) is true for all values of a, b, c, Let a = 0, 6 = 1, c = 2. .-. 2(l-2)=:Z(l-2)(2)(-l). .-. X = -l, and the factors of the given expression are — (6 — c) (c — a) (a — 6). APPENDIX. 409 Ex. 2. Factor a(&+c— a)2 + 6(c+a — 6)2 + (•(« + &—c)^ . H-(6 + c— a) (c + a— W (a + 6— c). If we put a=0, the given expression reduces to zero. . * . rt, &, c, are factors of it. . * . a(6 + c — a)2 4-6(c+a— 6)2 + c(a + 6— c)2 +(6+c — a) {c-^a — l) (a+& — c)=Labc. Let a=l, &=1, c=l, then i=4. . • . Aabc are the factors required. Ex. 3. Factor (6' + c') (6 - c) + (c' + a') (c - a) + (a' + 6') (a -6). Evidently 6 — c, c — a, a — 6, are factors ; but their product is of the third degree only, while the given expression is of the fourth degree. Hence the given expression must contain another factor of the first degree, and since this factor is sym- metrical as well of the first degree, it must be of the form £(a + 6 + c). ,'. (b' + (f) (b-c) + ((f + a') (c - a) + (a' + 6') (a -6) = L{b - c) (c - a) (a + 6) (a + 6 + c). Hence, L = l. In factoring symmetrical expressions, it is also useful to remember (see Ex. 35 of Exercise 12) that aM- 6' + c' - Sabc = (a + b-\-c) (a^ + b' + c''-bc -ca-ab). Hence, for example, ar* - 82/' + 27 -f ISxy = r* + (- 2yy + 3' - Bx(- 2yX^) = (x-2y + SXx' + 4y' + d + Qy-Sx + 2xy). EXERCISE 141. Show that^ 1. (a + 6 + c)' - (6 + c - a)' - (c + a - 6)' - (a + 6 - cf = 24a6c. 2. a'(6-c) + 6'(c-a)+cXa-6) = -(6-c)(c-a)(a-6) (a + 6-i-c). 410 ALGEBRA. 3. a\b -c) + h\c - a) + c\a -h) = -(h- c)(c - a)ia - b), 4. (a; + 2/ + zy - (2/ + zy -(z + xy -{x + yy + X* + 2/* + 2* = ] 2xyz{x + 2/ 4- z). 6. h'c\h - c) + cWic -a) + a'b\a -b) = -(b-c)(c-a) (a — b) (be + ca + ab). 6. a^h" - c') + bX(^ - a') + c\a' — 6^ = - (6 + c) (c + a) (a + 6)(6-c)(c-a)(a-6). Factor— 7. a;X2/ - «) + 2/'(« - a;) + 2"(a; - 1/). 8. a(b - c)' 4- 6(c - a)» + cCa - by. 9. a(6 - c)' + 6(c - a)' + c where r and s are positive integers. s Then /(-) • f(-\ tosfactors=/(- + -+- + ....to s terms). • ■•[/(';)]"-/« (4) But since r is a positive integer, /('•) = (! + x)^. Substitute for/(r) in (4), (1 + xY = [f(^)T' Extract .sth root, {I + x)^ =f(^\' ' 412 ALGEBRA. This proves the binomial theorem for any positive fractional exponent. We can now prove that the theorem is also true for any negative expo- nent. In (3) let m = - n. .•./(-n)x/(n)=/(-n + n)=/(0) = l. •'^ ^ fin) {1 + xr ^ ^ .'.{l + x)--=f{-n) ^ ^ 1x2 IV. A MISCELLANEOUS EXERCISE. EXERCISE 142. Factor — 1. lOOOa;'— y'. 5. 132x -{- xy — xy\ 2. 36a:' — 13a;2 — 40a;. 6. 4a;* + 4a'' — a*— 4. 3. 36a;*— 289a;2+400. 7. a;^ — 9. 4. a;^ — 64a;. 8. a;^ + 27. 13. 15aa; — 5ai/ + 12&a; — 4&I/. 14. 7(p-l)»-27(p — 1) + 18. 15. a^ + Q — 4a;'^ — w'^ + 4na; — 6a. 16. 2aV4-(a'+3a)a; — a- — 3a — 2. 9. 4x — y\ 10. 11. a;2« -!/-«. J-8b-K 12. 25n^—y-\ 17. 3a;_8a;^— 35. 21. a*a;^-3a^ + 5a;^— 15. 18. 19. 20. 6a;3-a;"''-15. lOar^— 19a;* — 56. 12a;^ + 5a;« — 72. 22. 23. 24. 30-\-V'2x — 2x. 60 — 7-1/ 3a — 6a. 15x — 2-\/xy — 2^y, ltx = 2, y = — 3, z = — i, find the value of: 25. 3a;i/ — 2/(a; + 4^) — a;0(4i/ + 6a;) + 3?/^(a; + i/) {y-\-2z). 26. {x — y — lO^r) (2a; -\-3y + 62) — xyz (x' — y' — 20^) + UxY^^, 27. {y^ — z){x — 5)-{-{x'-\-2) {y — 20)-^{2x-{-5y—lO0)\ 28. x'^y + xy^ + xz"" + x'z + y^z + yz^ — 3yz{y^ —x — 1). 29. a^^yx-{.z^y ^ (^x -^ y^-y —]/^—{xz + xy)^ — {y -\- z)x' APPENDIX. 412 Find the H. C. F. and the L. C, M. of: ' 30. a?-\-ah^ ^-a^h^—y and a^ — ^V-aV — b', 31. a' + 6 -/ a — ah + i/¥ and a" -\- a ^~ab -\-ab^ — h i/X 32 . a^+ & i/"a — 26^ and a^ + «&*—& i/"^ — &^. 33. 3a2 + 7aM + a^h^ — V and 2a2— 13a6 — 2a^&^ + 3&'. Simplify: 34. (i/"^— 1) (3i/^+2)+a;^(2i/'^ — 5) — (2i/^ — 3)(3a;^ — 4). 85. + w , m — n m — n m-\-n ^__^ m -\- n m — n n m m — u m 36. \ x—\ x + x — \- x+l x + 1 37. 8"^ + 25^— (D-' + lS^ — (^) I h^l/ab 27a2b- ■^{x + D. 39. -/|-+i/|_-/i + ^36; 40. l/^-2l/j + 3l/6 ^ 1/2 — 21/3 — 3.1/6* Solve: 41. dx'' — 2x{x-{-5)—{x — l) (2a; + 3) = (5a; — 1) (a; + 7) — 6a;». 3a;— 2 7 a^ — 1 ^ 2 ' 2a;— 3 6 a; + 2 3* 43. 3j; + lli/ = 19| 11a; — 32/ = 9, 414 ALGEBRA. U. 2x — 6y — 2 = 5. 47. «a;-f (& + l)y = c. 3x + y—y=6, {b-{-i)x + ay=d, ^x — 9y-{-2 = 23. .. ^ K 10 ^®- ii + iy = -'^' 45. ^-52^ = 13. _4. , -£ „ ^ + i.= 4. ^^ + «^ = '- 46.4^-§ = -3. 49 a.-f5,-fc. = l. f bx-i- cy -\-az = l, dx -\- 4.y = — 6i. ex -\- ay ~{- b0 = 1. Write the following expressions with positive exponents and reduce the results to single fractions. 50. a +&-' + 2C&-2. ^^ 4(x — l)-^-^x-Hx-l). 51. (a — &)-i + («+&)-'. ' {x~l)-' — x-^ 52. a(a — &)-i — &(« + &)-'. _^^ , 1_-^ 55. a-' + Va-'b-^-i-h- 56. (a + &) (a — &)-i — (a — &) {a-\-b)-^ l-(a2H-6^)(a + 6) — 2 59 (a-^ + &-^-c-M' ■ a-'''&-na + &)' — c-2' 60. a(l -«-•'») (a + a-i)(l + a-i)-Ma' + l)-^ 62. 6(a + &)-i — a&(a + &)-2 — a&2(^4.j)_3^ 63. (& + — y^ = ^3. x-y = 3. «^-a-^r6 = *- 89.^^^- xy + y^ = 18. 3x^4y~^^- x — y = 5 y 2 u 1 y^ — xy — da; = — 1, 85. 3x'—lxy-\-y''—4:X=^ld^ 91. 2x'~lxy -\-^y= — ^^ 7xy — y^ = —15. 6x — 4y=15. 86. xhj'\'xy^=6. X 2y 3xY + 8xy = 3. ^2.^ + ^ = 1. 87.x^ + y^ = 5. ^4-^=2 ic4-y = 35. 2'^3 416 ALGEBRA. .,+i+L = 3i. ^«-Af+V!=- 94. a; + 2/ + v^ary = 14. VVy{x-Jry) = ^0. x-\-y = \Q. 95. i/^M^7 + y = 6. 99. V'^-V^^y = ll, 1/FT2V + ^' = 22. -i^^ - 2/ Vxy = 60. 96. i/Ff^ + V^^^ = 4. 100- y + l/^' — 1 = 2. I. Transformations of Physical Formulas. 101. Given v=at, find the value of t in terms of a and v 102. Given s == ^at^, find the value of t in terms of a and s. 103. Given s = — , find the value of v in terms of a and 5. 104. Given s = |a (2^ — 1), find ^ in terms of a and s. mv 105. Given /^ = , find each letter in terms of the other three. r 106. Given e = -^, find each letter in terms of the others. uw ^ 107. Given e=-?r— , find each letter in terms of the others. 2a 108. Given t=7:y —, find I and^, each in terms of the other letters. 109. Given 0=77, find each letter in terms of the others. li OS 110. Given R = — — , find each letter in terms of the others. 9 + s' 111. Given -7 = — -\ — •., find each letter in terms of the others. APPENDIX, 417 n. Transformations of Arithmetical Formulas. 112. Given i=prt, find each letter in terms of the other three. 113. Given a=p + prt, find each letter in terms of the other three. m. Transformations of Algebraic Formulas. 114. Consult pages 318 and 326. IV. Transformations of Geometrical Formulas. 115. Given A =^bh, find each letter in terms of the others. 116. Given A =^h(b + b'), find each letter in terms of the others. 117. Given C=27tR, find each letter in terms of the others. 118. GivenA=7^JR^ find each letter in terms of the others. 119. Given A =7tRL, find each letter in terms of the others. 120. Given A =47zR^, find each letter in terms of the others. 121. Given T =7zR{R + L), find each letter in terms of the others. 122. Given T =27zR(R + II) , find each letter in terms of the others. 123. Given V=7rR^H, find each letter in terms of the others. 124. Given V^^ttR^II, find each letter in terms of the others. 125. Given V=^7:R^, find each letter in terms of the others. 126. Given a+b+c=2s. show that a + b—c=2(s—c); a—b + c=2(s—b), etc. .«« ., r, xu X -. a' + b^-c^ 2(s-a)(s-b) , .t, * 127. Also show that 1 -^ = "^ ab ' ^* , a^-\-c^-b^ _ 2(s-a)(s-c) ^ 1 — ^ , etc. 2ac ac 128. Also show that 1+^^=^=?^^^; and that a2fc2-62 2s(s-b) H n =-^ > etc. 2ac ac CHAPTER XXXIII. GRAPHS. 407. Definitions. A variable is a quantity which has an indefinite number of different values. A function is a variable which depends on another variable for its value. Thus, the area of a circle is a function of the radius of the circle; the wages which a laborer receives is a function of the time that the man works. A graph is a diagram representing the relation between a function and the variable on which the function depends for its value. A function may depend for its value on more than one variable; thus the area of a rectangle depends on two quantities — the length of the rectangle and the breadth. The present treatment of graphs, how- ever, is hmited to functions which depend on a single variable. 408. Uses of Graphs. A graph is useful in showing at a glance the place where the function represented has the greatest or least value, where it is changing its value most rapidly, and in making clear similar properties of the func- tion. Graphs of algebraic equations are useful in making plain certain properties of such equations which are otherwise difficult to understand. A graph also often furnishes a rapid method of determining the root (or roots) of an equation. Copyright, 1906, by Fletcher Durell. 418 GRAPHS. 419 409. Framework of Reference. Axes are two straight lines perpendicular to each other which are used as an auxiliary framework in constructing graphs, as XX' and YY'. The x-axis, or axis of abscissas, is the hori- zontal axis, as XX'. The y-axis, or axis of ordinates, is the vertical axis, as FF'. The origin is the point in which the axes intersect, as the point 0. The ordinate of a point is the line drawn from the point parallel to the ^/-axis and terminated by the x-axis. The abscissa of a point is the part of the x-axis inter- cepted between the origin and the foot of the ordinate. Thus, the ordinate of the point P is AP, and the abscissa is OA. abscissa, the "x" of a point. Ordinates above the x-axis are taken as plus, those below, 420 ALGEBRA. as minus; abscissas to the right of the origin are plus, those to the left are minus. The co-ordinates of a point are the abscissa and the ordinate taken together. They are usually written to- gether in parenthesis with the abscissa first and a comma be- tween. Thus, the point (2, 4) is the point whose abscissa is 2 and ordinate 4, or the point P of the figure. Similarly, the point ( — 3, 2) is Q; (-2, -2) is R; and (1,-4) is S. The quadrants are the four parts into which the axes divide a plane. Thus, the points P, Q, R, and S lie in the jirstj second, thirds and fourth quadrants respectively. +Y Q X\ I I O Rl P --1(2,4) ■i — f— iZ, s (1, -4) F EXERCISE 143. Draw axes and locate each of the following points: 1. (3,2), (-1,3), (-2,-4), (4,-1). GRAPHS. 421 2. (2,§), (-3,-li), (5,-f), (-2,J). 3. (2, 0), (-3, 0),J0, 4), (0,-iMO, 0). _ 4. (l,\/2), (1,-V2) (Vs; 0) (V5,-3), (-iV5, 2\/2). 5. Construct the triangle whose vertices are (1, 1), (2,-2), (3, 2). 6. Construct the quadrilateral whose vertices are (2,-1), (-4,-3), (-3,5), (3,4). 7. Plot the points (0, 3), (1, 3), (2, 3), (5, 3), (-1, 3), (-2, 3), (-5,3). 8. Also (0, 0), (1, 0), (2, 0), (5, 0), (-1, 0), (-3, 0), (-5,0). 9. Also (0, 0), (0, 1), (0,2), (0, 3), (0, 5), (0, -1), (0, -3), (0, -5). 10. All points on the a:-axis have what ordinate? 11. All points on the i/-axis have v/hat abscissa? 12. Construct the rectangle whose vertices arc (1, 3), (6, 3), (1, -2), (6, -2)', and find its area. 13. Construct the rectangle whose vertices are ( — 3, 4), (4, 4), (- 3, -2), (4, -2), and find its area. 14. Construct the triangle whose vertices are (-3, -4), ( -1, 3), (2, -4), and find its area. 15. In which quadrant are the abscissa and ordinate both plus ? both minus ? In which quadrant is the ab- scissa minus and the ordinate plus ? In which is the abscissa plus and the ordinate minus ? G-RAPHS OF EQUATIONS OF THE FIRST DEGREE. 410. To construct the graph of an equation of the first degree containing two unknown quantities, as X and y, let x have a series of convenient values, as 0, 1, 2, 3, etc., —1, —2, —3, etc.; find the corresponding values of y; 422 ALGEBRA. locate the points thus determined and draw a line through these points. Ex. Construct the graph of the equation y = 2x—l. I,et 2-= 1 1 1 1 2 1 3 etc. | -1 | -2 etc. Theny= -1 1 | 3 5 | etc. | -3 — 5 etc. Construct the points (0, -1), (1, 1), (2, 3), (3,5), (-1, -3), (-2, -5), etc., and draw a line through them. The straight Hne AB is thus found to be the graph of y =2x—l. F t J t f t 1 ^ t 1 t 7 i_ 7 4- Y' 'X 411. Linear Equations. It will always be found that the graph of an equation of the first degree containing not more than two unknown quantities is a straight line. Hence A linear equation is an equation of the first degree. 412. Abbreviated Method of Constructing the Graph of a Linear Equation. Since a straight line is deter- GRAPHS. 423 mined by two points, in order to construct the graph of an equation of the first degree it is sufficient to construct any two points of the graph and to draw a straight line through them. The greater the distance between the points chosen, the more accurate the construction will be. It is usually advisable to test the result ob- tained by locating a third point and observing whether it falls upon the graph as constructed. If the given line does not pass through the origin, or near the origin on both axes, it is often convenient to construct the line by deter- mining the points where the line crosses the axes. Ex. 1. Graph3?/-2x = 6. When X =0, t/=2; when t/=0, x =S. Hence the graph passes through the points (0, 2) and (-3, 0), or CD is the required graph. lY t-H — I I iX Ex. 2. Graph 4x + 7i/ = 1. When x = 0, ?/ = I ; when y = 0, x = i. Hence the graph passes close to the origin on both axes. Hence find two points on the required graph at some distance from each other as by letting x = 0, and 9 and finding ij = \, —5. Let the pupil construct the figure, 424 ALQEBBA, EXERCISE 144. Graph the following (it is an advantage, if possible, to draw the graph line in red, the rest of the figure in black ink) : 1. y = x-\-2. 8. y^-x. 2. y = x-2. 9. i/ = 4. 3. 3a; + 2?/ = 6. 10. If x = 2, show that 4. y = 2x. whatever value y has, x al- 5. 4:X — 5y=l. ways = 2. Hence the graph X— 1 „ ofa; = 2isa line parallel to ^' 2 ~ ^* the X - axis. 7. x=3(y—l). 11. Graph x = 0; also y=0. 12. Show how to determine from an inspection of a linear equation whether its graph passes through the origin; near the origin on one axis; near the origin on both axes. 13. Graph 5x + Qy = l; also 6x-y = 12. 14. Obtain and state a short method of graphing a linear equation in which the term which does not contain x or y is missing, as 2y—Sx=0. Before graphing the following determine the best method of constructing each graph, and then graph: 15. x + 2y = 4:. 19. ^x + iy = i. 23. x~y = 5. 16. 2y = x. 20. x=~S. 24. y-\-2 = 0. 17. 5x~ey = l. 21. 5x + 4y = 0. 25. Sx-2y + \^=0. 18. yi-3x = 0. 22. 8x+3y = 2. 26. 4x = 12 27. Construct the triangle whose sides are the graphs of the equations, y — 2x + l=0, Sy — x~-7 = 0, y + 3x+ll =0. 28. Construct the quadrilateral whose sides are the graphs of the equations, x—2y-4: = 0, x+y = l, Sy — 5x-15 = 0, x+2y-4: = 0. 29. An equation of the form y = h, represents a line in what position ? One of the form x = a? GRAPHS. 425 413. Graphic Solution of Simultaneous Linear Equations. If we construct the graph of the equation x — y = S (the hne AB) and the graph of 3x-\-2y = 4 (the line CD), and measure the co-ordinates of their points of in- tersection, we find this point to be (2,-1). Y c> \ \ \ \ \ /B \ / \ / \ \ / \ &■ / 1) / S. / \ / \ / \ \ a' /> Y' {/r t/= 3 3x-l-2w=^4 by the ordinary algebraic method we find that x = 2 and 2/= —1. In general, the root^ of two simultaneous linear equations correspond to the co-ordinates of the point of inter- section of their graphs. For these co-ordinates are the only ones which satisfy both graphs, and their values are also the only values of x and y which satisfy both equations. Hence, to obtain the graphic solution of two simul- taneous equations, draw the graphs of the given equa- tions, and measure the co-ordinates of the point {or points) of intersection. 426 ALGEBRA. In certain cases (as when the values of x and y are not inte- gral, or when the graphs have already been constructed), the graphic solution of a pair of equations is more convenient than the algebraic solution. In making graphic solutions of equations whose roots are not integral, cross section paper in which each linear unit has been subdivided into five or ten equal parts should be used. Ex. Solve (1). ,. „ (5x-{-Sy = 2.. graphically I 2^_^J^g^__ ^^2). Constructing the graphs of equations (1) and (2) and measuring the ■0-92+ Uoote. co-ordinates of their point of intersection, -j J q ^ \ ^ ^ zkd <2i ^ U 414. Special Case. Simultaneous Linear Equations whose Graphs are Parallel Lines. Let the pupil con- {x-\-2y = 2 3^1 2^,^10 OBAPHS, 427 He will find that the graphs obtained are parallel straight lines. Let him now try to solve the same equations algebraically. He will find that when either x or y m eliminated, the other unknown quantity is eliminated also, and that it is therefore impossible to obtain a solu- tion. The reason why it is impossible to obtain a solution is made clear by the fact that the graphs, being parallel lines, cannot intersect; that is to say, there are no values of x and y which will satisfy both of these lines, or both equations, at the same time. 415. Graphic Solution of an Equation of the First Degree of One Unknown Quantity. By substituting for y in the first equation of the pair] ^~q~ ; the two equations reduce to a: — 3 = 0. Accordingly, the graphic solu- tion of an equation like x — 3 = can be obtained by combin- ing the graphs of y = x — 3 and y=0. In other words, the root of x — 3 = is represented graphically by the abscissa of the point where the graph of y=x—3 crosses the x-axis. EXERCISE 145. Solve each pair of the following equations both graphically and algebraically, and compare the results in each example: 1. (2x + Sy = 7, . {x + 7y+n = 0. \x-y=l, ^' \x-3y+l=0. 2 (y = Sx-4. (y = 3. ^- Xy=-2x+l. ^' \9x-5y^S, ^- Xx + y + Q = 0. ^' l2/ = 2x + 3. . (y = 2x, 8. Solve graphically, 2a;+ 3 = 0. \x+y=0. g Also3a:-5 = 0. 10. Discover and state the relation between the coefficients of two linear simultaneous equations whose graphs are par- allel lines. 428 ALGEBRA, 11. Solve graphically j Sx+by=7. Qx-\-2y=n. {Ox '\y = 5 fi _q _c* GRAPHS OF QUADRATIC AND HIGHER EQUATIONS. 416. To construct the graph of a quadratic equation of two unknown quantities, use the method of Art. 410. Sometimes, however, it will be found advantageous also to let X have fractional values as, J, i, yV; h ^tc. The ob- servant pupil will also find methods of abbreviating the work in certain cases. It will be found that the graph of a quadratic equation of ORAFES. 429 two unknown quantities is, in general, a curved line, and, in particular, either a circle, parabola, ellipse, or hyperbola. Ex. 1 . Construct the graph of 2/ = a:^ — 3x + 2. Let x= 1 12 3 1 4 1 f etc. -1 -2 1 etc. Then y = 2 2 6 -\ 1 etc. 6 12 j etc. The graph obtained is the curve ABC. A curve of this kind is called a Parabola. The path of a projectile, for instance that of a baseball when thrown or batted (resistance of the air being neglected), is an inverted parabola. Ex. 2. Construct the graph of 4a;2 — Q^/^ = 36. Let a;=0 1 2 3 4 5 6 etc. Theny= ±2\/-l ± ^V-2 ± fV-5 | ±1.7 ±2.6 ±3.4!etc. For negative values of x the values of y are the same as for the cor- responding positive values of x. Hence the graph is a curve of two branches, ABC and A'B'C, of the species known as the Hyperbola. Y a: ^n: ^v 'X- B' :z: 430 ALGEBRA. EXERCISE 146 Graph the following: 1. y = x^—\. 2. y = x^-2x-'i. 3. y=\x'. 4. y'^ = 4:X — x^. 6. 1/2 = 4a;. 7. a;2-2/2 = 9. 8. xy = 4:. 9. a: + a:?/=l. 10. a:2+(?/-4)2 = 5. 11. 92/2-a:2=-9. 12. y^ = 4:X + 4:. 13. a:2-a:?/ + T/2 = 25. 14. 2/2 = 4. 15. x2 — 4a; + 3 = (show that whatever the value of y, X always =1 or 3; hence the graph is two straight lines paral- lel with the 2/-axis) . 417. G-raphic Solutions of Simultaneous Equations Involving Quadratics. The method employed is stated in general in Art. 413. Y GRAPHS, 431 Ex. {t2 -L 7/2 _ 25 Constructing the graph of x^+y^ =25, we obtain the circle ABC, Constructing the graph of x+y =1, we obtain the straight hne FH. Measuring the co-ordinates of the points of intersection of the two graphs, the points are found to be (4, —3) and ( — 3, 4). These re- sults may be verified by solving the two given simultaneous equations algebraically. 418. Special Cases. Imaginary Roots. Let the pupil ( 'T»2 _|_ nj2 —— A construct the graphs of ^ t _ q • He will find that these (. x-j-y — o two graphs do not intersect. Let him then solve the given equations in the ordinary algebraic way. He will find that the roots are imaginary. If he treat the equa- ( a;2 r 2,2 _ 1 tions •] . 2 j_ Q 2 _ Q A ill the same way, he will obtain a simi- lar result. In general, imaginary roots of simultaneous equa- tions correspond to points of non-intersection of the graphs of the given equations. It should be remembered that in solving a pair of simultaneous equations, the number of values of x (and also of y) is equal to the sum of the degrees of the two equations. Hence, if two simultaneous equations are both of the second degree, their graphs should intersect in four points; and if their graphs are found to intersect in only two points for instance, the other two points must correspond to imaginary roots. The pupil may illustrate this by graphing and also solving algebraically j ^2^^2=5. EXERCISE 147. Solve both graphically and algebraically: ■ (0:2 + 2/2 = 25. o j 32/2-2x2 = 12. ^- \x-y=l. ^- \x^ + y^ = lQ. o jx+2/=-2. 4 (3x2 + 2/2=a ^' |an/=-3. *• iy=x+2. 432 ALGEBRA, . ^* (32/2 + 2;r2=14. ^' \x^ = dy-y^. . ja:2 + 2/2-10a;=0. « (4x2-92/2 = 36. ^* (2/-=2x. ^- (^^ + 2/^ = 1- ^ (0:2 + 2/2== 16. .^ Jx2 + 2/2 + x + 32/=18. '• (x2 + 4?/2-43. \xy-y=l2. 419. Graphic Solution of a Quadratic or Higher Equation of One Unknown Quantity. By substituting for y in the first equation, tbxC pair of equations i _ n reduces to ^2 — 3x+2 = 0. Accordingly, the graphic solution of an equation like x2 — 3x + 2 = is obtained by solving graphically the pair j ^~^ '*' In other words, the roots of a quadratic equation of one un- known quantity, ax^ + bx + c = 0, are represented graphically by the abscissas of the points where the graph of y = ax^-{-bx-\-c meets the x-axis. Ex. Solve graphically ^2 — 3a: + 2 = 0. The graph of y =x'^ — ^x-\-2 is the curved line ABC of the figure in Art. 416 (p. 428). This curve crosses the x-axis at the points (1, 0) and (2, 0). .'. a: =1, 2, Roots. The same results are obtained by solving the equation x^ — 3a:-|-2 =0 algebraically. This method of solution also applies to a cubic equation or to an equation of one unknown quantity of any degree. Thus to solve the equation x' — Sa:" + Sx — 2 = 0, graph the equa- tion y == x^ — 3x^ + 5x — 2. The abscissas of the points where this graph crosses the x-axi«5 have the same value as the roots of the given equation x^ - Sx^ -\- 5x - 2 = 0. 430. Special Cases. Let the pupil construct the graph of each of the following: 0RAPH8. 433 y=x'^-2x-l....{\) y = x^-2x+l....{2) 2/=x2-2a:+2....(3) It will be found that the graph of (1) crosses the x-axis at two points, and that x'^ — 2x—\ =0 therefore has two real and unequal roots; that the graph of (2) meets the a:-axis at only one point, and that accordingly the equation x'^ — 2x-\-\ =0 has two real and equal roots; and that the graph of (3) does not meet the x-axis at all, and that accordingly the equation rc^ — 2a;+2=0 has two imaginary roots. Writing the general equation of the second degree and one unknown, as ax''-\-hx-\-c =0, in (1) h^-^ac>0, in (2) h^-^ac =0, in (3) h^-4:acx + l.> 9. 4a; + ^. 18. '6a^ - Aax + x"^. 10. a + 26. 19. 2y' - Ay-" + y - 1. 11. a;'* + a;y + y\ 20. c* + c^a;^ + a;*. 12. 9a;2 - 6a; + 4. 21. 23^ - Zx"^ + 4a; - 5. 13. 3a; - 7. 22. 20? - x ^ \. 14. 25 + 20a; + 16a;2. 23. 3a;^ + 4a;2^ + bxy"^ + 2^. 15. ^a^x" - 2axy'' + y^. 24. 2a;* - 3a;2y - 2^^ 25. a;3 + 2x'V + 4a;y2 + 8^^ 26. a;* - 2a;V + 4a; V - 8a;^^ + 16^*. 27. x^ — a;*y + a;^^^ - a;^ + a;?/* — if". 28. 64.^^ + 16a;*^2 + 4^2^* + ^e^ 29. ^x" - 5a; - 1. 87. 2a;* - 4a;3 + Zx^ - 2a; + 1. 30. 3a;2 - a; - 5. 38. 2xy - 2xz - Zyz, 31. 2a;3 - 4a;2 - a; + 3. 39. x"- - 3a; + 1. 32. 2a;' + 3a;2^ - 4a;^2 4. ^3^ 4Q ^^z _ 3^2 + a; - 5. 33. 3a3 - ^a^h + 3a62 _ 26'. 41. 2a;" - 3a;" - \ 34. x'' + I/''' - z^ - xz ^ xy - yz. 42. 4ar»« + Zx^^ - a;". 35. c^ -I- d? -vn? - cd - en - dn. 43. 4a;" + ^ - 3a;" + a;" - ». 36. 2/* + 2i/2 + 3^2 ^. 2i^ 4- 1. 44. 3a;" - 1 + 2a;" - 2 - 3a;" - \ Exercise 13. 1. |a;3 - -V^ + |. 3. 2a;* - t|a;3 _ o^, _ 1. 2. |a;3 - |a;2 + f. 4. 2.88a;3 + 10.86a; - 19.2. 5. 5.4a;* - 3.3a;3 + 10. la;^ + 1.32a; - .08. 6. 6.75a;* + \.2^y + 15.84a;V + 13.443/*. 7. fa;2 - 4a; + |. 10. 1.8a;2 - 3.2a; + 0.48. 8. ^x" + fa; - 1. 11. 0.5a;2 - 1.8a; + 3.5. 9. |a;2 + ^xy + f^/^ 12. 3a;2 + 4.8a;?/ - 21.5^^ Exercise 14. 1. 5. 8. - i 15. - |. 22. 3. 2. -3. 9. 0. 16. - f. 23. 1. 3. -2. 10. 6. 17. - f. 24. 1. 4. 2. 11. - 5. 18. - f. 25. 6. 6. - |. 12. 11. 19. - h 26. 14. 6. - -^^. 13. 0. 20. - f . 27. -Y. 7. ^, 14. - 2. 21. f . 28. - 3. ANSWEBS. Exercise W. 1. 12 and 36 marbles. 2. A, $61 ; B, $39. 3. John, 36 ; William, 60 ex. 4. 1st, $26 ; 2d, $37 ; 3d, $35. 6. 80 miles. 6. 27, 28, 29. 7. 19, 73. 8. Ist, 22; 2d, 11; 3d, 17. 9. Horse, $67 ; cow, $27. 10. A, $97 ; B, $194 ; C, $194. 11. 14, 21. 12. 13, 14, 15, 16, 17. 13. 9, 15. 14. 1st, $280; 2d, $140; 3d, $80. 15. Daughter, $960 ; sons, each, $1770. 16. Father, 48 ; son, 12 years. 17. Father, 52 ; son, 31 ; daughter, 26 years. 18. Father, 84 ; son, 42 years. 20. 19, 21, 23. 19. 11, 25. 21. 9, 14. 22 Eldest, $21 ; rest, in order, $13, $9, $7, and $6. 23. 21, 54. 24. 21, 22. 25. 14, 16. 26. Father, 44 ; son, 20 years. 27. 4 1 hours ; 36 miles. 28. 8 miles. 29. 36 miles. 30. 38 days. 31.. A, 25 ; B, 15 years. 32. Father, 55 ; son, 25 yeais. 33. Silk, $3.30 ; cloth, $1.10. 34. 9. 35. 15 of each. 36. 17 of each. 37. 7 bills ; 14 quarters. 38. 17 halves ; 3 dimes. 39. Son, $375 ; daughter, $15(> 40. 11 beggars. 41. A, $42; B, $37; C, $33. 42. 5 @ 32 cts. ; 7 @ 20 cts. 43. 48 feet. 44. A, $21 ; B, $68. 45. 72 pounds. 46. A, 24 miles. I^y. Exercise 17. 6. 0.3a:3 - 0.7a:V + 3.3a:2/» 7. ^x^ - Ix + S. 8. 3.6a;2 -f 1.29.ry - 0.6^'. 9. fx* + la'^x' + fa*. 10. ^x* - Ux' + ilx - 1. 11. 4.8a' + 4.55a26 + 2.05a62 + 1361 12. 0.22r» - l.bx'^i/ + l.Sx^' + 4.2^'. 13. 1.82;V - 2.73:cV - l-^^^^ + 2.7a;V. 14. 4.S2^ - 17.95^2^ + 18.45ar?/2 - 6.3^'^ 15. 2x2 _ 4a;y + 2^2 IQ, z^ - 2x^y + 2a;y + 3xY - ^^y* - 8y*. i^iSf. 2. ^x^- -h 3. 4x'- - Z.by\ 4. 2.6 + 0.92r - -a:». 6. \2^- - i:r=' - \x-V Vi ANSWEES. 17. 2 - 3x + 3x^ - 3x^ + Sx*- 18. 1 -2x + 2x^ -2x* + 2x^ 19. a:* - a;' + 2a:2 - 3a; + 5 20. 1 + 2a; + 7a;2 + 20a;3 + 61a:* 21. |a;2 - 2a;^ + iy\ 26. 23^ - 2.5a:'-'^ - O.Sxy^ + 1. 22. far» - |a;V + h^y^ 27. 2z - x. 23. 6x- ^y - |. 28. 0. 24. 1.6a:=' - 2xy + 2Ay\ 29. 3a; + 7^. 25. 3.5a;2 - 3a; + 1.5. 30. 2a - 12c + 84(Z. 31. 3 + 4a; - 2^/. 34. - 3. 37. - 1. 32. 5d'b + 46'. 35. - 1. 38. x^ + a; - 1. 33. 2. 36. - 8. 39. x^ - 3. 40. 38. Exercise 18. 1. n' + 2ny + y^. 11. x^ - z\ 2. c^ - 2ca; + a;'. 12. ^2 _ 9. 3. \x^ - \xy + 3/2. 14. 49a;'^ - 16^'. 4. 9a;2 - 12a;2/ + ^y". 18. 4a;2« - 25^2«, Exercise 19. 1. d} + 2ah + 62 _ 9. 9. a;* + a^y ^. ^^i^ 3. 15 — 2x — x^. 11. 4a;* - 29a;2 + 25. 5. 4a:2 - 9^^ _ g^/ - 1. 16. a^ + 2a6 + 6^ - c' + 2c — 1. 6. a;* + 6a;3 + 9a;2 - 4. 17. a;* + 2/* - a;*^ - 1. Exercise 20. 1. 4x2 + ^2 ^ 1 ^ 4^^ + 4a- + 2^/. 2. a;'^ + 4^2 4. 422 _ /^xy + 4a;2; - ^yz. 3. 9a;2 + 4^2 + 25 - 12a;^ - 30a; + 20^. 4. 4a2 + 62 + 9c2 - 4a6 + 12ac - 66c. 9. x' + 2/2 - f02 + l -2a;^ + 2a;2; ~ 2a; - 2yz + 2y - tz. 20. a* + 46* + 4a6- -1. Exercise 21. 1. a;» + 7a; + 10. 5. a;2 - 6a; - 7. 2. a;2 - 8a; + 15. 20. a2 + (6 - l)a - 6. 3. a;2 - 3a; - 28. 23. a;2 + 4a;2/ + 4^2 _ 53. _ \^-\^ 4. ic' + 4a; - 32, 27, a2 + 2a6 + 62 + a + 6 - 12. ANSWERS. yii Exercise 22. 1. 2a;' + 7a; + 6. 3. Sx^ - 7a; + 2. 2. 2a;» + a; - 10. 4. 5x^ - 4x - U Exercise 23. 1. a + X. 9. a + 6 - 2c. 2. 3 + 2a;. 10. 2x^ - y"" - I. Exercise 24. 1. a} -2a + 4. 7. x* - a;^^^ 4. ^4^ 2. a;2 + a; + 1. 8. a^ _ 2gt + 1 + aa; - a; + a;*. 3. 9a;2 + 12a; + 16. 9. c^ - c + ca; + 1 - 2a; + x\ 4. 1 - 2a;"^ + 4a;*. 10. 4 + 2x + 2y + x"^ + 2xy + y'. 5. 25 + 5ar' + a*. 11. a;*^* + x^y^ - 2xy + 1. 6. 9a* - 3aV* + 2/®- . 12. 9a;* - 15a;V + 25^«. Exercise 25. 1. a' + Sab + 96^ 12. 3a; + 5^' and 3a; - 5y\ 2. a» + 2a26 + 4ab^ + 86». 13. a;^ + 9, a;^ - 9, a; + 3, a; - 3. 3. a;* - ar« + a;2 - a; + 1. 14. a; - 2. 4. a;* + 33^" + 9a;2 + 27a; + 8;. 15. 1 + x. 11. 1 + 4a;. 16. a;«— Sy', 17. x^ + ^, 7^ — y'^, ^ -^ y, ^ - y- 21. a;« + 8^^ a;* - 81/', a;* - 4.y2, x^ + 2^, a;^ - 2y. 23. Ist, all integral values ; 2d, all even integers. Exercise 26. 1. a;«(2a; + 5). 3. a:(a; + 1). 8. a;'(l -x- x^). 2. xix" - 2). 6. 7a(l + la"). 15. arb^&\l + lie). Exercise 27. 1. (2a; + yy. 3. (5a; - 1)^. 6. c(7 + 2bc)\ 2. (4a - Zy)\ 4. (a; - l()y)\ 17. {a - b - g)\ Exercise 28. 1. {x + 3) (a; - 3). 21. (15a;« + y) (15a;« - y). 2. (5 + 4a) (5 - 4a). 24. (a; + y + 1) (a; + y - 1). 3. (2a + 76) (2a - 76). 25. {x + y + 1) {x - y - 1). 4. 3(a; + 2y) {x - 2y). 29. (4a; + 2^+1) {2y - 2a; - 1). 6. (10 + 9m) (10 - 9m). 30. (11a - 86) (9a - 26). 9. xix"" + 1) (a; + 1) {x - 1). 31. y(a;«^*+2;8)(ar»3/H2;*)(a;3y-2?*). viii ANSWERS, Exercise 29. 1. (5a; - 8^) (8y - Sx). 6. {a + ^ + x) [a + ^ - x). 2. (3a -36 + 5) (3a - 36 - 5). 7. (a* + x^ + y) {a' - x^ - yi 3. (a - 6 + 1) (a - 6 - 1). 8. {x + y + 1) (x - y - 1). 4. (3a; + 2y + z) {3x + 2y - z). 9. {1 + x ~ 2/) {1 - x -¥ y). 6. {x — a -^^ y){x — a~ y). 10. (c + a - 6) (c — a + 6^ Exercise 30. 1. {c^ -V ex -\r x^) (c2 - ex + x"^). 2. [x^ + X + 1) {x' -x + 1). 3. (2a:2 + 3a; - 1) (2a;2 - 3a; - 1). 4. (2a2 - 3a6 - 36^) (2a2 + 3a6 - 36^). 6. (3a;=* + 3a:^ + 2y'') (3a;' - 3a;^ + 2y^). 13. (a^ + 2a6 + 26'^) {a" - 2ab + 26'^). Exercise 31. 1. (1 - 2a;)2. 4. a;(l + xf (1 - x)\ 2. Zx{2y + x) {2y - x). 9. 2a;(4a;2 + 3a; + 1) (ix-Sx r 1). 3. (3 - a: + ^) (3 - a; - y). 12. (a;^ - 2a; - 1) (a; - l)^. 17. {x-\-y + l){x +y-l){l-^- x-y){l-x+ y). Exercise 32. 1. {x + 8) (a; + 2). 31. x{x + 4) (a; + 3) [x - 4) (a;- 3). 2. {x + 2) {x - 3). 32. Seven factors. 3. {x + 3) (a; - 2). 37. {x + a) {x + 6). 4. {x + 11) {x - 4}. 38. {x + 2a) {x- 36). 5. {x - 5) (a; — 6). 39. (a; - a) (a; - 26^). . 12. {x" + 4) {x + 3) (a; - 3). 43. 2(a; + 1)^ {x + 4) (a; - 2). 24. {x' - 8) (a; + 1) {x - 1). Exercise 33. 1. (2a; + 1) {X + 1). 5. (3a; - 5) (2a; + 1). 2. (3a: - 2) (a; - 4). 6. (a; + 3) (2a; - 1). 3. (2a; + 1) (a; + 2). 7. 2a;(a; + 4) (3a; - 2). 4. (3a; + 1) (a; -f 3). 20. (2a; + 3)(a; + 1) (2a;-.3) (a;-l). 21. (3a; + 2) (a; + 4) (3a; - 2) (a; - 4). 24. {o? + 62 j (5a + 46) (5a - 46). 27. (5a + 46) (a + 6) (5a - 46) (a - 6). 31. (a + 6 + 8) (a + 6 - 3). 32. (3a; - Zy - 2z) {x - y ^ Zz). ANSWERS. bL 33. (3a:' + 6x + A) {x + 3) {x - 1). 35. 2(1 + 3a:) (2 - x), 34. 4x{x + 4){x + 2) {x + 1) (a;- 1). Exercise 34. 1. {m-n) (m' + wn + w'). 9- x{3x-j-a) (dx^ — Sax + a^). 2. {c+2d) {c'' — 2cd-i-Ad'). 10. (8a; — r) {Gix" + 8xy' + y*) , 3. (3 — a;) {9 + 3x-\-x^). 11. a(l + 7a) (1— 7a + 49a'). 12. {a-{-x) (a — x) {a^ — ax-^x'^) {a' -\- ax -{- x'^). 13. {x'-{-y) {x^-y) {x'-x'y -j-y^) (x'+x'y + y^). 17. (a -f & + 1) (a^ +2a& + &2 _ a _ & + 1). 18. (5 + 2& — a) (25 — 10& + 5a + 4&2— 4a& + an. 19. (2 — c — d) (4 + 2c + 2(2-f c2 + 2cd+d2). 22. {x -\- y) {x* — x^y-\-xhj^ — xy^-\-y*). 18. As tha 24th. 29. As the 23d. 34. As the 12th. 35. As the 22d, Exercise 35. 1. (a + 6) (a: -F ^). 18. {X - y) {2x + 2y- 1). 2. (a; - a){x + c)w 20. {x - 1) (a;2 + 3a: + 3). 3. (5^ - 3) (x - 2). 23. (3a - x) (3a + 2a:) (a - a;). 4. {m-2y) (3a ^ 4n). ' 24. {x - 2) (a: + 3) {x - 1). 6. a;(a + 3) (a + c). 25. {x + 3) (2a: - 5) (2a: - 1). 6. 2/(3a - 5n) (a + 6). 26. (2a: + 1) (4a: - 3) (a; + 1). 7. a:(a:2 + 2), (a; + 1). 27. (2a: - 3) (4a: - 3) (a: - 2). 8. 2a:(a: + a) (a:^ a) {x -1). 28. {x + 2) (a: + 1) [x - 3). 9. [y' + 1) (y + 1). 29. {x + 3) {x - 2) [x - 4). 16. {X + 4) (a: + 2)\ 30. {x - 2) (a: - 1) {x - 5). 17. (a + 3) (a^ - 3). 31. {X - 3) (2a; - 1) (3a; - l)c Exercise 36. 10. 3a:(a:2 + a: + 1) (a;^ - a: + 1) (a; + 1) (a; - 1). 11. (2a + 1) (a + 1) (2a - 1) (a - 1). 12. 2(a:2 + 2a; + 2) (a:^ - 2a; + 2) [x' + 2) {a^ - 2). 15. 5a(a:« + a;3 + 1) (a;* +'a: + 1) (a; - 1). 23. {x + 2) (a; - 1) {x^ - a: + 2). 34. (a' + 5) (a + 2) (a - 2). 35. (G + d - 1) {c^ + 2cd + (P + c + d + 1). 36. {x - y) {x-y-\- 2). 39. (a + 3) (a + 2) (a - 3) (a - 2). 41. (a + 6 + c) (a + 6 - c) (a - 6 + c) {a - 6 - e), 43. (2 + n) (16 - 8n + Ari^ - 2n' + n*). 46. ia(2a; + y) [ix^ - 2xy + y^). ANSWERS. 47. (1 + x') (1 + xy (1 - x). 48. &{x - 3) (4 - x). 51. {x + 1) (a; + 2) {x - 3). 63. (a:* + 62;^ + ^*) (a? + yf {x - y)\ 54. (a:2 + ^2 ^ ^^^ (3, + 2;) (a; - z). 66. (a + 6) (a - 76). 61. {X - ly {x + 2)2. 62. (a + 62) \a - 6^) (i _ a;) (i + a; + a:'). 66. (1 + 2a6c - Zxyz) (1 - 2a6c + Zxyz). 67. {dbc - 772W/)) {ax - my). 71. (2a; + 5) (2a; - 3) (a; - 1). 69. {x - 2) (a; + 5) (2a; + 1). 72. {x + 1) (a; + 2) (3a; - 2). Exercise 37. 1. 2a6. 7. 17aa;3. 13. a - a;. 19. 4aa;(a - x) 2. 5a;V. 8. aVy. 14. a; + 1. 20. x{x - 1). 3. a6c2. 9. a + 6. 15. a; -.1. 21. 2a; - y. 4. 8a2a;2. 10. x-y. 16. a(2a + 1). 22. a;(a; - 2). 5. 14m2. 11. a; - 3. 17. a; + 1. 23. X' + 9. 6. 12a;. 12. a;(2a; + 3). 18. a; - 3. 24. 6(1 - a?). 25. 1 + a + a2. 26. a; - 1. Exercise 38. 1. a; + 1. 6. 3a; + 4. 11. a;^ + 4. 16. a;(3a; - 4). 2. 2a; - 3. 7. 2a; - 1. 12. a; + 3. 17. a; - 1. 3. a; - 1. 8. a; + 3. 13. a;^ + a; + 1. 18. a; - 2. 4. 3(a;- 1)». 9. a;2 + a; - 1. 14. 2a^(2a; - 3). 19. a; - 3. 5. a;(2a; + 1). 10. a;2 - 2a; + 3. 15. 3a; - y. Exercise 39. 20. 2a:(a; - 1). 1. 6a262. 5. 12a6c. 9. 42a262. 2. 24a;»^3^. 6. I'ZdWcK 10. 24a;V. 3. 36a2a;2^2. 7. ^^y?y^z\ 11. 2a;(a;2 _ jj^ 4. 48a;3^^ 8. 48a26-^. 12. ah{a + 6). 13. 14a;2(a; - 3). 20. a« - 6«. 14. {7? - 1) (a; + 1). 21. 6a:2(a; + 1) (a; - 1). 15. {x' - y') {x - 2y). 22. 3a6(a + 6) (a - 6). 16. 6a;(a; + 1) (a; - 1)^. 23. (2a; + 1) (a; + 1) (2a; - 1). 17. abx'y{x + y) {x - yf. 24. 6a;(a;' - 1) (a; - 1). 18. {x + 5) (a; - 8) {x - 1). 25. 6a;(3a; + 10) (2a; - 7) (a; - 3). 19. [x + 2) (2a; + 3) (3a; - 4). 26. 2a;(l + a;'^) (1 + a;) (1 - ;|7), ANSWEBS, 2 27. lAx^y^ix + ly {x - ly. 28. Qx'iSx 4 1) (a; - 1) (3a; - 1)1 29. 3Ga;«(2a; + 3)' (2a; - S)^. 30. {x + a-¥ l){x- a- l){x + a- 1). 31. {X - ly {x + 1)2 {x + sy {x - 3). Exercise 40» 1. (a; + 2) {x^ - 2a; + 3) {x^ - 2x - 1). 2. (3a; + 4) (a;'-* - a; + 1) (a;^ + a; - 1). 3. 3a;(2a; + 1) (a;^ - a; - 1) {x^ + x + 1). 4. 2(2a; - 1) (3a;2 - 5) (2a;'^ + 5). 5. (2a; + 1)^ {x - S) {x - 1) (2a: - 1). 6. (4a;2 - 9) (9a;''' - 4). ' 7. (a; - 1) (3a;2 + a; + 1) (4a;2 + 4a; - 1). 8. {x + 2) i^x^ -7x + 5) (2a;2 — x +1). 9. {X + 3) (a; + 6) (x - 1) {x - 2) (a; - 5). 10. (X + 1) (a; + 2) (2a; - 1) (3a; - 2) (2a; - 3). 11. H. C. R, a'b^ ; L. C. M., HOa'S^c^d'. 12. H. C. F., 3(a; + 1)^ ; L. C. M., ISa'd^x-" - ly (3a; + 2)^ (5a; - 2)». 13. H. C. F., a;2 + a; - 2 ; L. Q. M., (a;^ + a; - 2) (a;" - a; + 2) (a;^ + + 2). 14. H. C. F., 2a; + 1 ; L. C. M., (2a; + 1)'' (a; - 1)^ (3a;''' - 2a; + 1). 15. H. C. F., a; + 2 ; L. C. M., 4(a; + 2) (3a; - 1)^ (a; - 1)^. Exercise 41. 1 2a. 3a;* 2.1?. dy X 2 -Sax Sxz 4y^ 1 2(a; - y) a + b 2{a - b) 2 2a - 1 3a; - 4y 6. A. 13. 1 - 10. 11. 12. x~y. x->r y 2(x + 1) 3 5 7. 2a 2a; + 3y 1, 14. 7a;-H8y . a 2x^ x+ a 15. 1 x-y 16. x + 2y , 2x + Zy 17. 2x + y, 2x-y 18. a + b — c a- b - c 19. 1 + a-x x + a-1 20. 2 + a + 6 2-a + 6 oi 3a; + 4a. Xll AUSWE UiS. 22.^-2. 24. x^ - y\ 26. 2^' + 3^^ 3a;2 + 2^2 23 a: + 3. a; + 2 25.^^1, x-\ 27. /-5 a;2 + x - 3 28. a(^x- -_3). ?,9 Sx' -2x + 3 Z{x- -2) 2x^ + 3a; - 2 3f J 1^ - Ax" \1x- 3 4x=* + 3x"^ - 18a; + 27 Exercise 42. 1. a;-2 + -- X 6.^a;2 ,,_.^... 2. 2x^ + 3- 5 2a; ' 7. a;2- -l + l-«. a; - 1 3. 2a'x' + 1 7 + a 5aa; 8. a;2 - -a; + 2 -^(-= a;2 + a; 1) . -1 4. x-^-4x + 5- - ^ ' 16. 1 a:+ 1 X + a;2 - -^- . 1 + a; 6. x + 2y- V + 1 . X + 2/ 17. 1 - X 1 ^x^ ^^' ' 2a;* ^ ^"^ 1 + a; -a;» Exercise 43. 1. a^ - a + 1, a 6. ^-^ . a;2 + a; + 1 11. ^i±^. a; + a 2. x-" x-1 ^ a2 _ a.2 + ^ + 1 a + a; • 12. -^ • 1 + a; 3. x^ -2x X- 1 o 1 - 2a + a^ ^' 2a ' 13. ^-1. x + 1 4. 2a; + 1 a-1 14. ^'-*. x' + l 6. a2 + ab a + 26 10. «' + i- a - 2 16. 1 + 2^-^. 1 + a; Exercise 44. 15. ^ • a; - 1 1. 4a; 15a; , 18' 18 2 *^4« ^' 106 76 10a ' 106' 106 ANSWERS. xui a 4b lab g 1 3a; -3 . 1 2a2 - 2a 3a 9. a? — a o? — a ^ o? — a 11 a;^ + a;'' + a; x + l ' {x+ l)(ar''-l)' (a; + 1) (a;3 _ 1) 12 a; a;(2a; - 3) 4a ;''- 9 * a;(4a;'' - 9)' a;(4a;2 - 9) ' a;(4a;2 - 9 * 2a; + 4 15a; - 30 18 18. 6(a;2-4)' 6(a;2 - 4) ' 6(a;2 - 4) 21. {x + 1)^ 12(a; + l)(a;-2)(a; + 3) (a;-l)(a;-2) . (a; + l)(a;-2)(a;+3)' (a; + 1) (a;-2) (a; + 3)' (a; + l) (a;- 2) (a;+3) 22 2 (a - 6)^ abja^ - h'^Y (a + by 4(a^ - 6') _ ^6_ . (a^ - bY " (a' - 6')' ' (a' - &')'' («' " 6')' ' («' " 6')' l.ii 6a; 8a; - 9 + 12a 12aa; 156 -4c- 6a 6a6c 4a;- -7 4a; 2a^ -\- ab - -362 6ab 3a=» + b 6a^b jg 3w2 -f 1 20. 21. (m + 1) (m - 1)2 x^ + 2a; - 1 Exercise 45. 7 9 -f- 10aa;2 ' , 12aa;2 ,o 25a -206. *^- 12 Q 2a; -f- 3 ^' 30 ' 14. 8x + 65 . 21 9.3a;+l. 24 6xYz -^^• Ifi «' + »'. "■r^- 17. 1.-.. . 7a; - x" - 12 12. 7-«^. -A- 24. 1 + a . a;'- 1 a;" 25. 9-a» 1 8a;»-2 4a; -1 '^ - 4 26. a;»-l 22 2^' + 3a;-l . a;(ar»-l) ^7 a;' -K 5a; -HO 23. 0. * (a; + 1) (a; 4- 2) (a; -h 3) xiv ANSWERS. M^jL?) 30. {2x + 1) (2a; - 1) {x + 1) (2a; + 1) (2a; + 3) (a; - 1) 29 _^L_. 31 5a;'. y - 3.y ^ . ' (a + 6)3 * a;(a;2 - y^) 32. 0. 3^ 44 - 9a; . 3^3:2 + qq^. _ 9 33 a:' + 4a: - 13 . ^ + ^^ ^(a:^ - 9) (a: - 3) 2(a:'^-l) 36 3a: + 2 38—1—- 34. 0. • {x' - 1) (a: - 2) * a:(a:» + 1) 39. — ^^— • 40. ^ + 1 a:^ — 1 x^ — X Exercise 46. 1.^^. 4. * 7.0. 1-x^ a-Sb a'-b^ Zxy a^-b^ 8. 1^ 2-^i-^- 5-0. 8(1 -a^) 3. _^^_. 6. ^~^^' -' 9. 4^2 _ 3.2 1 - 4a'^ 1 - a:3 10. ^^l-^: .. 14. -7 (a: - 2) (a: - 3) (a: - 5) 12a:(a: + 1) 11. 5-46 j5 :c (a - 3) (a - 2) (6 - 2) 3(a:2 - 9) j2 7ft' + 19a . 16 ^' ~ 15^ ~ 1^ 12(ft2 - 9) {tP- - 9) (a; - 1) 13. 0. 17. 0. jg^ 17x^j^i2x+_39. 20.0. 21.1. 22.0. 23.1. 15(a:» - 9) Exercise 47. 10. a-1 a{x + 1) 2a: + 3 1 26»^. g 3(a: + 1) . * Zacy ' x{2x — 1) 2. 9^. 7 ?a?. 11 4x^ ' X ' S{3x - 1) 3. i. g ab . j2 (2a: + 1)' , 4. 1. ' 2a - 1 ' (a; + 1)» 5^ 5.v^g . g a.-' + 2a: - 3 . ^3 a 4- a; 7 'a: ' x\a-x)' 14. a' + a + 1 a 15. (3a: -2)'. (2a: - 3)2 16. x'-\ x^-1 17. 2 x+l 18. x a+ 1 2a- 1. a- 1 4a:' + 2a; + 1 ANSWERS, XV 20. -^ • 27. ^ "*" ^~ ^ . 3a6 a — x + 1 21. 1. 28. 1. 23. 1. 24. 1. x^ -x-1 a:' -x + S x^ -Ax + 9 2(1 - 5a:) 3 2(1 - 5a:) g 3(4 - 3a;) * (2«+3)(2a; + l)* * 4(2 - a;) 29. X. a + b + o 22 • ' X + y 30 «'c + «6' + 60* 31. 1. 32. i 26 a x^-xy + y^ 25. ^ ^^ ^^ + 4^^ 19- 1- 26. x + y. ' 4mn Exercise 48. 1. 2(2 - a: ) . ^^ «-l . jg a;-a + l . a; ' a + I ' x + a — 1 2 ^ - n 2a -1 20. -i 2^+1 a ' 21. 0. 3. a: + 1. 12. (a + 1)2. 22. a - 1. 4. ^ ' ^ + 1 23. ^^ - e^ + 1 ^^'-^(^TZ) ab-cd-l ^ ~ ■ ^ 14. a + a:. '^^- ~- 26c 15. «(^--.tJl . o. 1 6.-^. ^^--.Tl^- 25. X - 4a; ' + 2a; + 1 . jg _J__. 26. 2a;. 2a; ' 2a:2-l 27. (^ - y) 8.1, 17. ^±i^. ^ a:' a; - y 28. - 1. §. . 18 <^<^ + ^^ ~ <^^ . 29 1 , ' X ' ac + bo + ab * 1 + 2a; Exercise 49. 1 (^ ^ 1)' . 4 8na; . y 14a; ' in" - x^' ' 9a:2 - 4 2 ^ -^^o . 5 ^{2x_.z_S), 6 2(3a: - 1) 2-z 9. a :vi ^JVSfTT^iJ^'. ' 10. *- 13. ^' . 2a(6^ - a») "•x(2. 9 -a:)(; J. \%x + 3 x-Z) •(9x^-l)(4-' 9a;') ''■a'\i ^^•(T 4 -x){x-1) (0^-3) 17. 1. 13 1 + 2^ 21. 22. • 4(1 -a'). 25. 26. _^. ^■' 27. hx. X x^^y\ 30. ^r^-> 2a; -r • X{1 + x^) 20. 1. 1 23. 24. : -f- a + 6. 29. a-6 a + h a-b X x + 1 32. «. a; 33. ia. 34. 6 - 3» 35. X. 37. - f . QQ M 2 + 52^2 ^^' d^ + Zb^c^ OR 5a- 1. 1* 38. - X. Exercise 50. 40.^. a - 1. 6. 13. V- 25. - 2. 37. tV 49. 2. 2. 2. 14. |. 26. f. 38. 5. 50. f . 3. 3. 15. -i 27. 0. 39. 2825. 51. - i 4. -2. 16. i- 28. 1. 40. .00025 . 52. - ii. 5. 10. 17. -tV 29. 2. 41. -.04. 53. 0. 6. 2. 18. ¥• 30. 5. 42. 4. 54.^. 7. 2. 19. -¥. 31. - 5. 43. 13. 55. -5. 8. - 1. 20. -2. 32. 3. 44. - A. 56. - 7. 9. 5. 21. 5. 33. 10. 45. f f 57. - 3. 10. i 22. ¥• 34. - 0.8. 46. H. 58. -If 11. - 9. 23. -f. 35. H. 47. - 7. 12. - f. 24. -2. 36. - 5. Exercise 51. 48. 4. 1 -f 6. -\' 11. - 23. 16. 8. 21. 0. 2. -3. 7. -i 12. - 0.5. 17. 0.4. 22. - 3. 3. -if 8. -2. 13. - 0.1. 18. I 23. f 4.12. 9. -f. 14. - 0.1. 19. 5. ft. - 9. 10, - f 15. if 20. 7. ANSWERS. XVii Exercise 52. 1. 3a. g a- b « a. jo a* - 6 * . 2c * 6 ' g» + 6' 2- 5* 6.1^. 10.^^2. 14.0. ^-2« ^«7' 15.17a. 4. -A^. 8. -«^. 12. «. 17. ^. a-b a-b 2 3 18. «6cd 22. gc» 27. ^1 - 2a - a*), ab + 6c + ac ' gc - g6 + be 28. - a. 19. 1. 2^- ^- 29 ^^ . 2a -36 22. ac» ac - ab + be 23. 0. 24. g3 + g 3a2-l 25. c^. Oft ab 20. ^-^;- ^«' - A 30 36g6 2a + 6 orr ^2 • 2262 _ 15^2 2j a6(a — 6) ofi g6 o^ a^ - a" - a + 1 • a»-2a62-6' * * 2{b^ - a')' ' a Exercise 53, 1. 24. 4. 63. 7. 33 and 42. 2. 45. 5. 27 ; 28. 8. $12,000. 3. 60. 6. 48 ; 49 ; 50. 9. 144 trees. 10. 26 ; 27 ; 28. 15. 13 years. 11. A, $32 ; B, $48 ; C, $50. jg f A, 21 years. 12. 16 and 81. * *-B, 35 years. 13. 21 and 79. jy f Father, 48 years. 14. 14 and 54. ' \ Son, 20 years. 18. A, $960 ; B, $1200; C, $1080 ; D, $1760. 19. 70 acres. 23. \\ days. 26. 2f days. 20. 44 and 45. 24. 6 days. 27. 4 days. 22. 3f days. 25. 12 days. 28. 36 min. 29. 169tV min. 33 f 32^^ min. past 6. 30. 20 days. * ^ 54y«x min. past 10. 32 / ^^ri min- P^* ^- 34 / ^tt and 38,2^ min. past 4. t 38^T min. past 1. ' *- 21 j»y and 54^^ min. past 7. 36. 24 hours. 38. 6 hours. 40. 12 mi. an hr. 37. 208 miles. 39. 55 miles. 41. 19^ miles. xviii ANSWERS. 42. Ist, each VZ hrs. 54. 108 and 72. A, 42 mi.; B, 40 mi. 55. 30 apples. 2d, each 492 hrs. 56. 334 pages. A, 1722 mi. ; B, 1640 mi. 57. 4 and 16. 44. Hound, 150 ; hare, 250 leaps. 58. 81 yards. 45. Hound, 72 ; hare, 108 leaps. 59. A, 4 days ; B, 5 days ; C, 6 days. 46. $63. 60. 5t\ and 38y\ min. past 10. 47. 1713 men. 61. Dog, 600; fox, 900 leaps. 48. 45 men. 62. 19^^ days. 49. 2160 men. go r46|bu. oats. 1 53i bu. com. gj f $3250 at 4%. I $1800 at 5%. 52. 10 ; 14 ; 6 ; 24. 53. First, 20 days; sec, 15 daj^s. " b -a Exercise 54, 1. $156. 4. $264. 7. 8%. 10. $540. 2. 5^ yrs. 5. 10 yrs. 8. $333}. 11. 8 months. 3. 4^%. 6. 4|%. 9. 25 yrs.; 16 yrs. 8 mos. Exercise 55. 1. a; = 1. 5. a; = 2. 9. x = |. 13. x = 15. y = i. y^-^' y = \' y = 10. 2. a; = 1. 6. a; = 1. 10. a; = - \. 14. x = 3. y = -\. y = -h 2/ = 2. ^ = -4. Z. x = l 7. x = -S. 11. a; = 3. 15. x - 10. 63. b + c miles. 6<1 ab + ( G a - c 6 + 1 ' 6 + 1 65. abc feet. y = -h y = H' y = -7. y = -io. x = 2. 8. a; = |. 12. x - 8. 16. x = 12. 2/ = 3. y = -h y = 9' y = is. Exercise 56. 1. a; = 1. 5. a; = - 2. 9. a; = - 3. 13. a; = f y = l. y = h .V = - 4. ^ = - 4. 2. a; = - 1. 6. a; = 3. 10. a; = 12. 14. x = l y = -i. y = -h y = 12. y = -h 3. a; = 2. 7. x = 0. 11. a; = 6. 15. a; = 4. y=--i. y=2. 2/ = 20. y = -s. 4. a; = - 3. 8. a; = - 2. 12. X = 15. 16. a; = - 21. y-0. y = -3. y = 10. y = - 40. 10. X = = 7i y = = - 2J. 11. X = = i y- = i. 12. X = = 4. y = = 5. ANSWERS. XIX Exercise 57. 1. a; = 1. 4. rr = 3. 7. x = 6. y = 1. 2/ = - 1. y = 6. 2. X = - 1. 5. a; = |. 8. a; = 12. y = 1. y = f. 2/ = 12. 3. x - 2. 6. a; = 3. 9. x = 12. y = - 2. 2/ = - 2. y = 35. 13. a; = 0. 2/ = 3. Exercise 58. 1. X = 5. 5. a; = 3. 9. a; = 2. 14. x = 18. y = 12. 2/ = 1. y = 4. ^ = 12. 2. a; = 5. 6. a; = 1. 10. a; = - 0.2. 15. a: = 9. y = 2. 2/ = 1. y = 0.6. .y = - 1. 3. X = i 7. a; = 3. 11. x = .015. 16. x = 17. 2/ = i. 3/ = 5. ^ = .01. y = 6. 4. a: = 7. 8. a; = 1. 13. a; = 2. 17. a; = 2. y = 5. y = -l. ^ = -3. y == - l^ 18. a; = - 2. y=-3. Exercise 59. 1. a; = 2a. 6. x = a + 26. 10. a; = w - m. y = - a. y = 2a-b. y = n -\^ m. 2. X = — b. „ en r- bd 11. a; = 3. = 2a. * an- bm ,„ _ 2a + 1 b' - b ,, _ «d-_cm , ^ 6 ^- ^ ab'-a'b an-bm yi, x ^ a-vb -^ G y- «-« -^ ab' - a '6* «- = !• V ^ ■ "" a + 6 + c 4. X = m + n. ., = *. 13. X = a ^ b. y = m — n. a y = a-b. 5.x 26 + 1. b 9.. = 1- c 14. a- d b-d -"-^^- -i- -H- 15. a; = a. 16. a; = - a. 17. X = b. 18. a; = a + 1. y-6. y -6. y = a. y = 6 - 1. XX . \ Exercise 60. 1. x = 1. 3. a: = 3. 5. X = 1^. 7. a; = - 3. y-2. 2/ = 4. y = li y = 3i z = 3. = 7. z-=lh z= -2. 2. a: = 2. 4. a; = 3. 6. a; = 2. 8. u = 2. y = 3. 2/ = -2. y = 3|. v = 3. z- - 4. 2= -4. 2 = -4. w; = 1. x = 4. 9. x=- 12. 10. a; = a + 6. 11. x = 6. 2/ = 18. y=- a-b. y = 40. = - 24. 2 = 2a. 2 = 20. 12. a; = - a + 6 + c. 14 a; = ^^ ~ 2& . y = a ~ b + c. ' 6 2; = a + 6 — c. 2a + 36 13. a; = a - 6 + 1. 6 y = — a + 6 + 1. g^ g + 6 . = a + 6-l. 6 Exercise 61. 1. a; = ^. 4. a; = — ^. 1. x = \. 9. a; =* a. y = -\. y = i 2/ = -i y=~-ck 2.x=-l, 5.a. = f. 8.:r = -2^. 10.. = 1- 3/ = 1 2/ = f • * 1 + n' m 3. a: = f 6. a; = i. ^ _ 2n . 1. 11. a: = ?. 17. a: " ^60 . 6 6 + c 2ac a 12. a: = 1. 2 = y = l. 13. ^ = |. 13 ^ _ 2a 14. a: = l: 2/ = -i; = i .V = r^ \b.x=2;y=-\',z==l. 16. x = i;y = i;2; = i. w+n 19. a; = i; y = -|; ;s = l. ANSWERS, XXI Exercise 62. 1. 9 and 14. 2. 9 and 12. 3. 2 and 8. 4. f 6. Flour, 3 cts. ; sugar, 5 cts. 13. A, 91 years ; B, 30 years. 6. f . 14. 84 and 60. 7. 49. 15. A, 4| miles ; B, 4 miles. 8. Man, $3 ; boy, $2. 16. 12 boys, $60. 9. y*j. 17. Length, 8 in. ; breadth, 6 in. 10. 57 pe?r trees ; 43 apple trees. 18. A, in 24 days ; B, in 48 days. 11. Sheep, $4 ; calf, $7. 19. A, $70 ; B, $110. 12. A, $660 ; B, $480. 20. 480 miles. 21. j\. 24. f 27. 24. 22. 11 and 36. 25. 23. 28. 64. 23. i^ and |. 26. 56 and 65. 29. 253. 30. 151. 37. i and f. 31. Silk, $1.80 ; satin, $1.50. 38. A, $26 ; B, $14 ; C, $8. 32. 16 ; 20 ; 24. 39. A, $70 ; B, $50 ; C, $90. 33. 15 gals, from 1st; 6 gals, from 2nd. 40. 8 men ; 6 women ; 10 children. 34. $600 ; eldest, $200. 41. 6 doz. at 30 cts.; 3 doz. at 40 cts. 35. A, $40 ; B, $50 ; C, $80. 42. 12 yards by 8 yards. 36. 8 dollars ; 40 halves ; 36 quarters. 43. 15 ft. by 6 ft. 44. Fore wheel, 5 yards ; hind wheel, 6 yards. 46. 6f miles an hour. 48. 1^ mi. an hour. 47. 32 miles ; 5 mi. an hour. 49. 24 bu. from 1st ; 16 bu. from 2nd. 50. A, 60 yds. a min. ; B, 80 yds. a min. 61. A, 27 mi. ; 3 mi. an hr. B, 30 mi. ; 5 mi. an hr. 62. A, 9 ; B, 12 ; C, 8 hrs. 64. A, 4f yds. a sec. ; B, 4} yds. a sec. 55. A, 5^ min. ; B, 5^ min. 57. C helped 6 days. A, in 45 days. 66. A, 4 1 min. ; B, 4^f min. 58. A, $35 ; B, $26 ; C, $20. Exercise 63. 2. H. C. F. = 2a; - 3. 6. ^' "^ ^ "^ ^ 3. iV. x{x-l)*' ' x'-l -x-1 7 ^(^^ - ^^) *• A- * 6(2a; - 3) 9. a ' + x* a} 10. X = -2. 11. X = -f 12. X = -8. 13. X = 4. xxil ANSWERS. 17. X 14. a: = f. lb. x=\. 16. a: == 7. « - & 2^ = |. ■ y = \' y = 10. y = —^ a-\^ b Exercise 64. 1. a; < 1. b. X > 6. 9. a; > 6 and < 7. 2. a; > 3J. 6. a; < ^. 10. a; < 2 and > \\ 3 a:>^. 7 a: < — «^ • ^^^ 1^, 18 or 19. ^- "^ > a ^- "^ < '^^r^Tb 12. 13. 4. a; > 2. 8. a; < 6. Exercise Q^, 10. 27ar'^'. 25. a' + 3^ + ^9^. 11. - 8a;«. 26. a:* + 3a;V + f^^ 18. \\xY' 27. ^ - 1 + ^• 33. fa;* - 2x^y -^ f fa: V - \xy^ + j\y*. . Exercise 66. 1. a» - 3a26 + Sab^ - bK 3. 1 - 4a; + 6a:2 - 43:3 + a:*. 2. x^ + Sx^ + 3a; + 1. 4. a^ - 6a^ + 12a - 8. 6. 16 + 32x2 + 2ix^ + 8a;« + a^, 6. a* - 10a*b + 40a362 _ 30^253 ^ gOaft* - 326*. 7. a;'' + 15a;* + 90a;3 + 270a;2 + 405a; + 243. 8. a« - 6a*6 + UaW - 86». 9. 32c5 - 80c*c?2 ^ 30^3^^* _ 40c2(i« + lOcrfs _ (^lo. 10. a* - 12a'62 + 54a26* - lOSaft^ + 8168. 11. 343 - 441a;2 + 189a;* - 27a;«. 12. a;*2/8 + 8a;''^« + 24a;2^* + Z^xy"^ + 16. 32 13. 8 + 12a; ^ 6a;' a a^ -+^3- a^ 14. 243 405c*^ ^ 2 136c* 2 45c6 4 ^ 15c8 16 15. 1 - 2c2 3c* b 262 263 C8 166* 16. 32a;« + 5c* ^ ' 16a;* 5c« ^ 4a;3 5c2 ^ 2a;2 2a; ANSWERS. ZXlll 17. a* + Zx^-b3^ + Sx-l. 18. 2^-9x^ -^ 24a:* - 9x^ - 24x^ - 9x - 1. 19. a8 + 4a^c + iOa^c^ + IQa^o^ + 19a*c* + IGa'c^ + lOa^c* + 4ac' + c«. 20. a:» - 3a:V + Sx'^z + '6xif - &xyz + ^xz" - y^ -\- ^y'^z - Zyz^ + 7^, 21. 8a:« - 12a:* + 42x* - 37 ar» + GSx^ - 27a: + 27. 22. 1 + 4a; + 2a:^ - Sar* - 5a:* + Sa:^ + 2a;6 - 4a:^ + x^. 1. Zxy\ 2. 5a». 3. 122/«. Exercise 67. 8. 3a:^ 9. hy^. 10. \ab''. 15. - 8a:». 16. 2^». 17. - Ix^y, Exercise 68. 1. a;2 - 2a: + 1. 2. 1 3. 3a:2 - 2a: + 1. 4. 5 + 3a; + x*. 5. ri' - 2n' + 3. 6. 2x^ + 3a:' - 2a; - 3. 13. ^a; - 5. 16. a:' + X - ^. 17. ^d' - ia + 6. 18. ^ + 3+^- 19. |a;2 - fa: + f. 24. 1 + 2a: - 2a:2 25. 1 - a - ia' . 26. a: - ^ - A _9^ 2a:3 27.a+26_2^' Exercise 69. 1. 85. 4. 325. 5. 427. 10. 90.08. 11. 14.114. 14. 0.17071. 16. 2.6457 . 17. 3.3166 18. 3.5355 19. 1.8257 20. 1.4529 21. 0.9486 22. 2.5819 23. 1.2747 24. 0.3415 25. 0.2213 26. 1.0031 6.0075 1.9318 1.1117 1.3687 27. 28. 29. 30. Exercise 70. 1. a + 2a;. 2. 3 - a. 3. 1 - 4a;. 4. a' - a - 2. 5. a:' - a: + 1. 6. 1 - 3a: - 2a:'. 7. 4a:' - 3a; - 2. 8. a' + 5a - 1. 9. 2a:' - 5a; - 3. 10. 2 - 3n + 3n'. 11.^ h 12 ^ - -%. 2y 3x 13. a; - 1 + - - X 14. 1 + - - -^ a or 15. a;' + — 2^^ XXIV AmWEHS, Exercise 71. 1. 16. 3. 124. 5. 3204. 7. 70.09. 2. 91. 4. 352. 6. 804.6. 8. 0.0503. 9. 0.997. 13. 1.542 . • 17. 0.2147 . . . . 10. 4.217 14. 1.953 . .... 18. 1.021 11. 1.817 15. 2.704 . .... 19. 2.0033 . . . . 12. 1.775 16. 0.3968 20. 2.901 21. 1.730 22. 0.0535 Exercise 72. 1. 19. 7. 2?- _^. 11. 1.5704 . . . . 2. 43. y 2x 12. 2; -3. 3. 3.08006 .... 4. 0.9457 8. 2 + 2a: - x\ 13. 2a- i. 5. 1 - '6ab. 9. 14. a ^.x-l. 10. 34. 14. 4a:' -i. Exercise 73. 4. 21^0: 6. 2a}^y¥. 30. 81. 31. 64. 38. a^x\ 44. a^. 7. Va^m^. 32. 36. 39. 3a%. 45. 1^2ir. 24. 9. 33. - 27. 40. 4x\ 46. a^y. 25. 125. 34. if. ij- 47. 2x\ 26. 8. 35. -W-. 41. a^. 27. 16. 28. 128. ^* 32 36. ak 37. 2ai 42. 7ai 48. 2«^^\ c 29. 4. 43. 2a:i 49. ,Vir. Exercise 74. 7. 3 X 2-^ac-\ 1 OK 750 « 34. f. 43. i 13.^. 25. ♦ ca:V 35. i 44. 2a. x^ 36. 216 50. 3ai 51. c'dl 14.-^. 27. i 28. i. 37. f ^. 38. - f 20. ^^f. a?e 29. 32. 30. 25. 31. h 39. ^h. 40. - -i I- 52. m\ 24IJ5M. 32. ^i^. 41. tW 63.^- x^ a^x 33. 54. 42. 1. ANSWERS. XXY 62. «^. 7aio • i I 66. -^ • 68. X ="* . 6a ^. j,^ Exercise 75. 12. A. 26.^. 31.2^ 40.0^. 13. o* c^* ^f 41. 27. x^. 27.-^^- 32.1. 42.4 25 i9!i^l 33. Va. ^ 125a; 34. Va. jQ ^i ,, 1 2 ^4. Va. 43. ai5c« • 4a^ 28.^. o. A. .. a» 15. J 44. ^. 9x^ ^y * 1/5 24. 1 ^ ^1^ 36. ^' «¥ y^ ' 37. ,j^. «^^»V 4^- it. c^& 25.^. 30. -i^. 38. (f)l ^^-"fed' 2/^ «* 39. -l^i 47. xyz, 48. 8a:3 - 36 + 54a: - 3 - 27a; - ». 49. a:2 - 8a;'^^^' + 24a;^ - 32a;^ + 16a;t 60. 243a;^ + 810a;''^' + 1080a;6 + 720a;"^' + 240a:'''^ + 32a; ^. 61. T^a; - 2 - a; ~ '^ + 6a; " ^ - 16a; ^ + 16a;. 62. 16a; - 2 + 160a; ~ '^ + 600a; " ^ + 1000a; " ^ + 625a; " i 63. a;2 - 4a; ^2/ ~ ^ + 6a;y - * - 4a;^y ~ ^ + y - 2. 64. 1 + 4a;~%"^ + -^a;-'^"^ + -¥/^~^^~' + f^a;-V~^ + fta;"^^"^ + A^a;-^-*. 66.4. 56.9. 57.^. 68. f 59. - 32r 60. 1. 61. -^ • 62. ^ V^ • (~ 3)' XXvi ANSWERS. Exercise 76. 1. 2a^ - a + 9. 3. 9x + bx^y^ + Qy, 2. a + 1. 4. 4x^-1 + 12a;" i 8. ex^ - *lx^ - 19a;^ + 52; + 9x^ - 2x^, 9. 2- 4a"M + 2a~%^ 10. 5 - 3a;~y + 12a;"^^^ + 4x^y~^. 11. bx^ - 3a;^ + 1. 12. 4a; - ^ - 3^ - 1 - 2a:3^-« 13. x"^- 2a:~^ +3a;"^- 1. 18. x^y - 1 - ^x^y ~ ^ + 2 - 4x~ ^y^. 19. 3a~^- 2a"M + 4a~^a;-a;i 20. 2a^ - 3a" tV _ a ~ TJ • 23. a ~ ^ - 26^ + Sah. 21. x^ - 2x^y^. 28. 3a;^ - ixy~^ - 2x~^y'K 29. ix'^y- ^x~^ + Sy-K Exercise 77. 1. 21/3. 4. - 2V5^. 29. }l/6. 32. |1/3D. 2. 31/2. 5. 41/6. 30. 1/TO. 33. t'^t/6. 3. 31/5. 6. - 61/T. 31. ^Vm. Exercise 79. 34. :r-^l4a. 4a; 1. l/I^. 2. 1/35. 3. 1^432. 4. -1/20. 18. -Vx-i Exercise 80. 1. Va. 2. l/a. 5. V^. 10. f 3^^^ Exercise 81. 1. 1^345"; l^m: 4. l^i; l^f. 14. l^B. 2. l^m-; i>27. 7. 1^64; i^Sl ; 1^125: 18. 1^?^. 3. 1^27 ; 1^25. 13. V^. 21. 1/3. ANSWERS. XXVll 1. 5T/2; 2. v5. 6. 21^. 7. 31^5. 11. il/6. 12. 51/g. 13. 2Vax. 14. fl^20. 15. yi^i^e. 16. 4VB. 17. 2l^2c. 18. 0. Exercise 82. 19. dacVE. 20. - 66T^2a. 21. 8K2. 22. - 191/3. 23. - 21/6. 24. 71/io. 61/5; 25. 0. 26. 61/J 27. 51/T. 28. - 51/3. 29. 121/6 + 101/5". 30. a6l/3a. Exercise 83. 1. 12. 2. 15V/3. 11. al^aU^. 12. 1^72. 13. a:1^864a^. 14. 31^24. 3. 121/3. 4. 21^3. 15. 16. 17. 18. 5. 361/6. 6. 301/21. 1^3356. 61^. y 1y 1^1^. 20. ^1^288: 26. 21/& - 41/3 + 8V5. 29. 201/2 + 30 - 81/15. 31. 2 - 41/2. 32. 2 + 71/3. 33. - 6 - 21/6^. 34. 16VIS - 30. 35. 71/6 - 12. 36. 42 ^6V'I0 + 21/5'. 37. 21/15 - 6. 38. - 282 - 721/10. 19. V 7. 71/5. 8. tV 6|3a5ai2~ 9. fl^3. 10. ^1^5. 22. i^f. 23. !^^ 21 24. 25. ^. 39. 301/6 + 541/5^ - 34. 40. 2. 41. 96 - 161/5. 42. a;l/6 + l/3a;'^ - 3a;. 43. l/3a:-' + 3a; + a; + 1. 44. 21/a;'* - 1 - ^x - ^, 46. 25 - 7a;. 47. 2a;. 48. 251/3". 49. 36a:2 - 50a; - 100. 1. 3. 2. 21/f. 3. 31/3. 10. il/6. Exercise 84. 14. ^1^72. 16. 3. 16. l^f. 17. 1^^*. 21. 51/7-14. 22. 1/2 + 3. 23. 1/21 - 51/B. 24. 21/7 + 41/6 - 6VH 1. iv^. 2. il/6. 3. 2Vi5 15 21/5 6 Exercise 85. 21/7 + 1/35 6. 6. 14 2l/g - V\ 4 1^20 - 1^? 6 l^T2-vlB 9. 11 5 6Vl 10. 13 + 71/5. xxviii ANSWERS. ' 13T/g-30 12 1A±J^. 13 Yl±}^. 6 '23 * 5 14 6 a + V^g?) - 126 . 13 5 - T/30 - 51/6 + 6t/5'. 4a - 96 5 x-S 15 rg + T/^Tl - 5 . 19 21/3 - 1/^21. 1^ 6a- 6 + 5l/2a^- a. 20. Vx^ - ] [-a;». •*^"- ' " ,^ n 14a- 9 17. VS + V2. 21. H«? + W-Vab b 22. 2.12132. 25. 0.057735. 28. 0.10104. 23. 1.05409. 26. 0.709929. 29. 2.63224. 24. 4.53556. 27. 9.00996. 30. 0.62034. Exercise 86. 1. w'. 4. 6. 7. 8x^\ 10. 4al/25:. Exercise 87. 1. 21/2 - 3. 8. 31/5 - 41/2. 15. a -f 31/a*^ + 2. 1/3 + 21/5. 9. 21/15 - 31/3. 16. 1/3 - 1. 3. 3V3 - 21/2. 10. ^1/2 + |V/6. 17. 1/3 + 1/2. 4. VE- 1/3. 11. |l/3 - |l/6. 18. 2 - 1/3. 5. 1/14 + 21/7. 12. f + 1/3'. 19. 3-1/2. 6. 31/5-21/7. 13. 14. 2 - il/3. 20. 2 - il/2. 7. 2V'5 + 1/6. 1/m + n + l/m -n '. 21. 1 + \V^. 22. 1 + V% Exercise 88. 1. 8. 5. 2. 9. -|. 13. 1. 17. i 2. 4. 6. -i 10. - 1. 14. 9. 18. \. 3. 3. 7. 8. 11. 12. 15. -V-. 19. %. 4. 2. 8. 14. 12. -2/. 16. 64. 20. ^. 21. 18. 28. 29. 30. 31. 1. 8. 1. 35 ^^- ly 41. a', f) 42. f. 22. 5. 23. 1. 24. 9. 25. 9. 35. ^ 36. 4a(l I + 26 («-*)'. 2a -6 32. 33. 9. 37. 16a. 38. I .o 20a> ^^' 25 -4a- 27. |. 34. 64. 39. i. 44. - 3. 45. ^. 46. - 6. - ANSWERS. Exercise 89. XXIX 2. VS. 3. 1/5. 4. VS. 5. -^1/5E 4a 6. 2 + 4VS. 1. - 2Vax. 8. f V"3D ; fi^^IOS: 9. 3V2 + 1/3 - 31/5 + VlD. 10. 1^1/15 - 1/6 + 21/m 54 - 381/2 . 16. 2.88675. 17. 0.18362. 15. 4 + Vl5. 18. 0.218286. X — 11. 12. 13. X -Vx - 1. 9 + 51/? . 6 21/3 1/2 14 7 4 + VTd. 25. 31/5 - 21/2. 26. 41/5 - 41/2. 27. 31/7 + 21/n. 28. 29. 33. I. 34. 1/a; V-. 35. 25. Exercise 90. 19. 0.36452. 22. 31^. 20. 21/7. 23. l^Ti. 21. l^'gg. 24. 5 + 21/2, 30. I (a; - Vx' - 36). 31. 21/^2^^. 32. 6. 36. 4. 37. f. 9. 5l/=T. 10. 9l/^=T. 11. 20V^=T^. 12. 6V^^. 13. 81/"=^. 14. 0. 15. al/^^-6. 16. - (a + 36)1/^=T. 17.-1/2. 18. - 18. X&. 10. 41. -?/6 + 2V^vSl/' 48.^ + ^^^ - 7. -61/6^. 91/2. 281/6. - 101/ro. (y - a;)l/"=T. 26. a(a-J)2l/- 1 27. 3 + 1/2. 28.-6-5 1/6. 29. 24. 30. - 19 - 21/35. T^. 42. - l-4l/"="3 31. 46 + 21/^. 32. - 2 - 21/3. 33. a;2 - 4a; + 7. 34. 6' - a\ 35. x^-2x + 2. 36. x2 - a; + 1. 37. 1/^=^. 38. 1/6^. 39. - 31/5. 40. - AaV^^, 46. 47. 11 fl' - 6^ + 2a6l/^^ aM- 62 24 + 71/10 48. liSV^HI. 53: iV^ + 31/^=^. 54. 21/^^3 - 31/^^^. 55. 4 - 31/- 6. 14 3 - V~=^. VS - V^^. 60. 2 - 21/"=^. 62. |i/^:T:+4;fl/"=T-i. 63. - V-. XXX ANSWERS. Exercise 91. 1. i4. 6. if. ,. -.J^EI. iA -.26 11. A=J^E^. 14. ±: 2. i2. 7. ±5. Af a a 3. ±^. 8. ±1. 12. i3a. 4. ±2. 9. ±f. j3 ^e. 1^- =^(« + &)- 6. ± |. 10. ± a. ' ■ 2 ' 16. ± 1. Exercise 92. 1. 2, - 16. 10. h - |. 19. h - |. 28. 1, - -^. 2. 2, - 12. 11. I, - f . 20. 2, 5. 29. ± 1. 3. - 2, 10. 12. - I, |. 21. 3, - |. 30. 3, - |. 4. 6, - 1. 13. 3, |. 22. 3, - 1. 31. 5, - ^. 5. - 3, - 8. 14. 5, - |. 23. 3, - f . 32. 3, - f. 6. 1, - |. 16. 7, - -1^. 24. 3, - I 33. 2, - 5. 7. 2, - |. 16. f, - |. 25. 4, - |. 34. i, |. 8. - 1, 1. 17. f, - f 26. 1, - -V-. 35. 5, - f. 9. 2, - -V-. 18. 1, - f . 27. - 2, - ^. 36. - 1 ± V^. gg -3=fcl^29 . ^j - 5 i i/:r-7 43. o, 5 =t 1/13 ^'' 6 oo 3il/5" 38. ^ ■ , 1 ± 2V- 1 40. g'^'^^ 42. ^ * '^^IT^ 1. 2a, -ea. 2. 36, -76. 3. 2c, -5c. 4. ab, -6a6. n 36 '• 2 ' 46 3 3 » -3( a 3. 2a Q a o. — > c 3a. 7c* 9.1, a _6. a 10_ 4 }^ 42. 7 -^ ^ ■ 11 6 44. - 1, - 3. 45. lifVe. 46. il^^^T. Exercise 93. 10. ^, - ^. 18. a + 1, a - 1. 2a a 19.^^2, b + 2, 2 2 11. a, 1. 12. A, _36. ^ 5 2a 2a ^0. ^^^, - ^-^. 13. 3a, - 2. 1^ & 5 21. -— -» -— -• 14. > — - • a6^ a^6 a 3a 15 6, a. 22. «±-^ -^. 15. --> -^ c a + 6 16.a,-l. 23.-6,«^fA_-. a a + 6 17. a, --^- 24. 1, -^• a+1 ate ANSWEBS. XXXI Exercise 94. 1.-1,-7. 8 A, _i. 14.-1, f. 2. 12, - 7. ' Sa a 15. 2, - ^. 3. f, - f. 9. =1=2, =t2l/^^. 16. =1=2, 7, f 4. V-, - !• 10. 2, - 1 =1= 1/"=^. j7 J - 3 =1= 1/- 15 5. f, -i 11. =tl, ±2. * ' 6 * 6. 3, f 12. 1, ± 1, ± V^^. 18. 1, 3, - 4. _ 7. I, -|. 13. i -i |. 19. -1, ±^l/2. 20. W,- 3. 21.^1,L±^,Z.1A1^. Exercise 95. 1. ±1, i4. 2. ±1, if. 3. 1, I, .=J-±i^:Z3, and " ^ ^3^"^ ' 4. 16, ^V 7. 1, ^V 10. 1, W. 13. 1, (- 1)1 5. 1, f 8. 27, - i. 11. 1, i^%. 14. i jI^. 6. 8, - ^. 9. t, T^5. 12. 8, - xb. 15. 4, - 6. 16. =t:3, ±^1/2. 23 1 -8 - 7 =^ 3l/^ . 2 18. A ^H-. „. . , 8 --t 21/37 17. ±iV^3, =tfl/"=^. 25 5 i 19. 2, 6, - 2 i 21/"=T. ■ ' " 3 20. 6, - ^^V. ^^- ^' ^'^• _ 27. i|v/6, ±81/^:T. 21. 2, - I, 3^J^^ . 28. - ^V, f 1^1^. ^ 29. I, -f, l=tV^ 00 1 _ 5 - 3 ^ 31/^ . 30. ±1, =t^V^310. ' ^' 4 31. =1=21/2, ^\V^^. Exercise 96. 1. 3, 2. 3. 3,* 12. 5. 8, ^^* 7. 3, - -»/.* 2. 1, - \. 4. 6, f* 6. 9, - f * 8. 4, - f 9. 2a^ - 2M. 11. 2, -1, ^^^' -l±l/^. lO.a^ + 1, 81«!^1. 12. 3, ¥. '9 13. 5, - tV 14. 0, 5. 15. 8, ^V , 16. i, - H- 17. 4, - yV 18. ±1, =t|. 19. ±a, ±2a2. 20. 2, f . * Will not satisfy the equation as it stands. xxii ANSWERS. Exercise 97. 1.1, -6. 3. i - f . 5. f , - - I 7. 2, - |. 2. 1, -|. 4. f , - 2. 6. 1, - -1. 8. ii, ^|. .M a ' -2* 13. \h if. 14. i, - -¥• 18. ^ » 1. 1 - a -^ a 15. tI^, - ¥• 16. Y, - 1. 19. ."^ ' \- 6 - a a + b 11. ±3, 12. i2. i2l/-2 • ''' «' 2I • 20. « , ^^ . 26 4a Exercise 98. • 1. 1, 2. -] -I 3. - 2, |. 5. 4. 3, - \. 6. 9 3a, 2 -2a. 11. a, ^ • a 14. 1, f. 15. i|, ±i. 10.36, a 46 . 3a' 12. 27, - -2/. 13. -^^, V-. 16. ±2, iii/s: 17. 6, -V-. 18. c{c + d) d(c + d) 21. 3.7320, 0.2680. c-d ' d-c 22. 1.41202, - 0.07868. 19. 20. 3, f. /TT 4- 5i /^ - 3 r 4- R 23. 24. 4.67945, 0.82055. 0.46332, - 0.86332. Exercise 99. 1. I, - i 2. -»/, - f . 3. 1, 16. 11. ^v, - -V-. 4. 0, i4. 5. 9, ^f. 6. 3, - 2. 12. 0, 3, 3i 3l/^=^ 13. 1, T?,V 14. hS - 3. 15. ^l^i^ZI. 2 16. -1, -3, -2± iV^^^IO. 17. 15, - If 18. 4, 4. 0, A. , 1. 9.^ - a a, 6 ^'a 10. 3, i. 19. - 1, - 3, 5 ± 1/- 23 4 20. ^ + 6 a ' a-6 6 21. 16 1, 1. 22. - a, - 6. 23. 2, 648. 24. 3, -!, - 3 il/43 4 - 25. - 2,f Exercise 100. J1.JV.A.11 1. 5, 6. 2. 7, 8. 3. 2, 5. 4. 5. 5. 23 years. 6. 30 mi. an hr. 14. 7, 11. 22. 24. 7. 2i 3i. 15. 14, 3. 23. 4 hrs., 10 min. 8. 5, 13. 16. 10 boys. 24. 32. 9. 36 boys. 17. 9 men. 25. 5 mi. an hr. 10. 30 and 45 min. 18. 4i, 7i 26. 45 mi. an hr. 11. 30 cents. 19. 18 feet. 27. A, 20 days. 12. 6, 9. 20. 1 rod. B, 12 days. 13. 6, 15. 21. 2i rods. Exercise 101. 28. 9 mi. an hr. 1. a; = 1, - 13. 6. X = - 2, - f |. 11. a; = 1, 10. y = 2, 16. 2. a; = 4, - 1. y = h- iz' 7. a; = 3, - -1^. y = - f , - f . 12. a: = 1, -V. ^ = i - 2. 3. a; = 1, 2. 2/ = - 1, H. 8. a; = - 3, 7. y = - 3, -^/. 13. x = Z,\. y = 0, - 3. 4. a: = 2, - 26. y- -1, 4. 9.^a: = |, 2. y = - 2, 1. 14. a: = 2, - f y = i, 15. 6. a; = 4, - f . 2/ = 1, 1. 10. a; = 7, 12. y = \- f f 15. a; = 3, Jf . y = 1, - if. Exercise 102. y = 2, 3. 1. a; = ±4, ^14. 5. a: = =i=5, =±=61/2. 8. a: = ±8, =t3. y = il, ±4. 2. a; = ± 3, =t f v^. .V = =f1, TiV^. 3. a: = ±], i|l/-2. 2/ = ±2, ^fl/-2. 4. a; = ±3, ifVlTO. y = ±2, ±ii/ro. y = ±2, =1=71/2. 6. a; = =i: 1, ± Jl_. 1/91 y = ±3, i-^. 1/91 7. a; = =4=3, =tf. y = i5, =tV-. y-=F5, ±5. 9. a; =±5, i -i^ Vh\ 2/ = =f3, i-^ V51 10. a; = ±l, i2. y = ii, ±|. 11. a; = ±2, =4=1/2. 12. a; = ±l, =bl4f. y = =t 4, ± 31/2. y = T3, ±3f Exercise 103. 1. a: = 9, 4. 3. a: = - 3, - 7. 5. a; = =±=7, ±3. y = 4, 9. y = - 7, - 3. y = ±3, ±7. 2. a; = 4, - 3. 4. a; = 4, - 5. 6.a: = ii =t|. y = - 3, 4. y = - 5, 4. y-^h =ti. :xxiv ^JV^STTEIJ^: 7. ar = f , - |. 12. x = = =t2a, ±3a. 17. a: = f, 2. y=-h I y = : =t 3a, i 2a. y = 2, f 8. 2; = 2, 1. 13. x = = a + 1, a - 1. 18. x = i,~l y = 1, 2. y = a - 1, a + 1. y = -h f 9. a; = 4, - 3. 14. a; = :2, 3. 19. a: = ±|, =^f y = - 3, 4. y = 3, 2. y = Tl, ±f. 10. a; = 7, - 5. 15. a; = = 6, 2. 20. a; = f , |. y = - 5, 7. y = = 2, 6. y = - 1, - i 11. a: = if, Tf. 16. a; = = ±i ±i. 21. a; = 5, - 3. ^ = =ff, if. y - ^=ti ±i. 2/ = 3, - 5. 22. x = a - 26, - 2a - 6. 23. a; = ± 1 i 1/2: y = 2a + 6, 26 - a. 2/ = ± 1 =p V^. Exercise 104. 1. a; = ± 5. 4. a; = 2, -¥• 7. a: == 1, |. 2/ = i2. y = h - ¥. y = 3, 2. 2. 2; = 1, i b. x = 6, -5. 8. a; = ±1, =b2. y = f , f . y- -5, 6. y = i3, il. 3. a:=i4, ±|V^. 6. a; = ±4, ±1. 9. a; = 3, ^. y = ±l=F|v/2. y = =f2, =f3. 2^ = 2, f J. 10. a: = 2, - f . y = 3, - H. 17. a; = 2, 3 11. a: = 5, 13. y = l, 2 -3iT/33 9 2/ = 3, i 12.^-1,1,7=^^^. 18. a; = 3, - 2, - 2 ± 1/^. 3/ = - 2, 3, - 2 ^ 1/5: 19. a: = ±4, ±fl/35. 13. a: = 2, 1, 6, - f. y = il, i^i/S5. 2/ = 3, 1, - 10, - f. 14.a:--5, -1, 7^^-^^. 20. a; = 2, , 3±l/-65 2 * y - - 2, - 6, ^ 2 1/-23 y = h S=FV-6b. 2 15. 2: = f , |. 2 21. a; = 3, l±3l/-3 " ^' 2 • y = - i - f 16.x = i 1,-3^^^-33. y == - 2, 3, l-i3l^E?. y-|,j, -'1^/33. 22. a; = 3, - 1, 1 ± 1/- 10. 18 y - - 1, 3, 1 =F V^Tt). ANSWERS. XXXV 23. « « 1, 4, > y = - 4, - 1, 24. a; = ±2, =tl, ^2V^, iv/^T. y = ±l, i2, =fV-l, =f2l/^n[. 25. jc = 3, - 2, 1, - 3. 26. a; = 3, 4, - 2 i V^. ^ = 2, - 3, 3, - 1. 2/ = 4, 3, - 2 :,= l/S: 27. a; = ± 2a, ± 26. 29. a; = 5, /y. 31. a; = 3, 4. y = i6, ±a. y=_i, 1^. 2/ =-1,0. 28. X = i f 30. a; = 3a, - 9a. 32. a; = J. - 7 y = i f 2/ = 2a, - 7a. y = |, I 33. ar = - 3, - 4, 6 ± V^. 37. a; = 1, V, 2, 5. y = - 4, - 3, 6 =F 1/43. 2 5 1 10 14 y = — • ' - 34. a; = - > - • a 7a a a a a L, _i. 38. x = 64, 1. 6 6 35. a; = 6a, ^ 6 y = 1, 64. 36 356 ' 39. a; = 16, 9, ^l y = % 16, (--»#. ^=2' 4 a y = a + 6,«(6^. 2/ = M, 0. 41. X = ±4, ±2, =tl/^=^/ ± I/- V^. y = i2, ±4, il/'^=^/TV~^ 42. a: = 8, 2. 43. a: = 4, 3, 6, 2. y = 2, 8. y = f, 2, 1, 3. Exercise 106. 1. 3, 7. 6. 15 by 18 rods. 9. 18c. @ $12. > 2. 3, 8. 6. 28. or 16c. @ $14. i 3. 7 by 12 ft. 7. 9 by 18 in. 10. f . 4 7 by 11 rods. 8. 40 yds. 11. 40 and 60 mi. 12. Bow 6 mi.; stream, 2 mi. an hour. 13. 21 and 13 ft. 14. 10 and 12 ft 15. $6. 16. f and f. XXXVl ANSWERS. Exercise 106. 11. a;» - 5a; + 6 = 0. 25. 2a;2 - 4a; + 1 = 0. 23. a;» - 2a; - 1 = 0. 26. 2ar^ - 2a; + 1 = 0. 24. a;' + 6a; + 6 = 0. 28. 4a;2 + a^ + 4c^b = 4aa;. Exercise 107. 1. Real; uneq. 15. i 19. - ¥• 23. - 2. 2. Real; uneq. 16. i. 20. i 10. 24. 3, - 4. 3. Real; eq. 17. |. 21. - I 25. - 3, - |. 4. Imag. 18. V-- 22. i f 26. - 1, V. Exercise I08. If 13. a 23. 0, 1, - f. 2.|. a-1 24. 0, 5, 20. 3 3a -26. • 4a - 36 14 25. X 25. 26. 0,5. 3, - 1. 4. 3; 2. 15 36 27. 3aK 5. 6a'6. 6. 2f. 7. a' - x". g 3a. + 4. 3a; + 5 16. 17. oo. (a-l)«. a-1 . a{a + 1) 28. 29. a;=-3;y=-4. \ ' a + 1 h = a + l,l. 18. 2, -|. 42. 4 and 10. 9. 11/3. 19. 5, -4. 43. 5 and 11. 10. 66c. 20. - 7, - -V-. 44. 13 and 19. 11. 3y. 21. ±3, -2. 45. 4 and 8. 12. f 22. 0, -4. Exercise 109. 1. a; = 1, 5. 3. a; = 15, 8, 1. 5. a; = a y = 14, 7. S/ = h 6, 11. y = 7. 2. a; = 3. • 4. a; = 9. 6. a; = 8. y = 2. y = 4. 2/ =7. 7. a; = 5, 12, 19, etc. 11. 99, 75, 51, 27, 3. y = 2, 7, 12, etc. 8, 32, 56, 80, 104. 8. a; = 7, 37, 67, etc 12. 243, 126, 9. y = 1, 14, 27, etc. 78, 195, 312. 9. a; = 3, 14, 25, etc. 13. V, -\S |, f . y = 2, 18, 34, etc. j%, |i, f |, f f. 10. X = 22, 57, etc. 14. Sheep, 3, 14, 25. ff = 10, 23, etc. Calves, 16, 10, 4. 16. 12 ways.* 16. 11 ways.* 17. 159; an indefinite No. * Including zero yalues. ANSWEBS. MXVU Exercise llO. 1. 15. 5. ±2. 9. 4. 13. 9, - f . 2. 2\. 6. f 10. 8. j4 3^, _5. 3. 5. 7. ^V2. 11. a; = - 2y. " ^ ' a? 4. 8. 8. 35. 12. z^ -2x + 2^y^ -2y. 15. ^. 18. 10291 ft. 21. 157}. 16. ^. 19. 13 in. 22. 4. 17. «; = f - 4ar» + -\ • 20. 12 in. 23. 9( ± VG - 2). Exercise 111. 1. 31. 5. 54, 94^. 9. - 69. 2. - 25. 6. i, 0. 10. - 189. «=-81. 8 = 3|. 11. 148|. 3. - 46. 7. - 7.2. 12. - 300. s= - 364. » = - 37.8. 13. 573^ 4. _ 23^ - 13. 8. 164. 14. - 165. 15. (^"^ -56)6c . 17.-771/3. 4 Exercise 112. 1. a = 4 ; s = 286. 9. n = 21 ; d = 1. 2. a = - 5f ; s = 209. 10. n = 21 ; d = - 2. 3. a = 5 ; d = 4. 11. n = 18 ; d = - ^Jj* 4. a = 11 ; d = - 3. 12. n = 16 ; d = f. b. a = bl, d= -2^. 13. a = 7 ; n = 6. 6. a = f ; d = - i. 14. a = - 5 ; n = 10. 7. a = - I ; rf = tV 15. a = 1 ; n = 7. 8. a = 3f;d=-i 16. a = -i; n = 16. 17. 12. 18. 8. 19. 10. 20. 4 or 9. Exercise 113. 1. d =» - 2. 3. d = f 5. d = f . 7. a:. _ 2, d = i. 4. d = - xV 6. - Hi. 8- ^-y* Exercise 114. 1. 5, 7. 5. n\ 8. 8, 7i 2. 4i 3. g 7n(n + 1) . 9. 3, - 2, - 7, - 11 3. - -V, - ¥• ' 2 10. 1, 3, 5, 7, 4. 5c - 7&, 4c - e&. 7. X02d t«rm. 11, 24d, xxxviii ^ivsTrj5;i25'. 12. 1, 4, 7 . 13. 12. 14 . ~ 5, - 41 .. fll, 6, 1, -4,-9. 21. Exercise 115. 1. 486. 4. 16. 6. 32. 10. mi 2. 192. 8 = 55. 7. - 63. 11. -V(3 + 1/5). 3. - i^. 5. j\. 8. -\V-. 12. 31(1/6 + 1/S^) s = 33if. Exercise 116. 13. 211/^ + 28. 1. 2. 4. 45. 7. - 4. 13. 5. 8 = 728. 8 = -%W. 8. I 14. 6. 2. 5. 5. - ^. 9. - f. 15. 5. » = - 425. 8 = - H¥^- 10. - I 16. 4. 8. 16. 6. f. 11. 5. 17. 6. ,-ii^. 65 + 191/6^ 12. 6. Exercise 117. 1. r = i 3. r = - 2. 5. r = - ^ 7. =t f . 2. r = |. 4. r=-4. 6. r = 1/2. 8. ±5. Exercise 118. 1. 3. 6. i 12. ^j. 17. 3t\^V 2. f 7. H. 13. H|. 18. Iji^. 3. - ^. 8. 6(2 + 1/2). 14. 5xVt. 19- lyV 4. ^. 9. K31/2 + 4). 15. 3Hi 20. |Mf 6. W. 11. a. 16. Un. Exercise 119. 1. f, f 13. 3, 6, 12. jg r2, 4, 8, 12. 2. A, f 14. 1, 3, 9, 27. ■ ^-¥-, ¥, I I 3. ¥, V- .5 8 11 19- -24|, f. 4. 96, 48 "^^^ i 15 8 l' ^0. $64, 80, 100, 125. 11. 5, 15, 46. ' ' • 21. 27, 36, 48 yrs. 12 /7, 14,28. 16.115,30,60,120. r 2, 4, 6, 9. • 1 63, - 21, 7. 17. 21/2 + 3. ^ 2, i, - f , 9. Exercise 120. 6. - Jy^, - H, etc. jQ a;j/ ^ xy . 7. ff ' 2x -:i/ Sx-2y 8. - 36. 11. 12 and 18. 9 ^Lui. 12. -i, -i -i «« + 1 13. 8 and 32. 1. A. 2. -A- 3. tS. 4. 1. 1, h !• 5. f , 1, A, et 12.1-A__JL_+ 4 X 7^ a; + 1 (2a; - 5)^ ANSWEBS, xli Bxercise 128. . 3a; -i- 5 _ 3^ _ J5_. ' x^ + I X ' X* %-^ ----■* . 5 ^-^ a;-6 . 1 a; 2a; -5. a;=' + 2 3 3a;- 4 a: -2 a;* + 2a; + 4 10a;- 7 4 Z.—±}^±^-L 3 6. a;' + a; + l a;»-a; + l a; + 5 . 2a; - 15 3(a;» + a; + 1) 3(a; - 1) a;' - a; + 3 (a;* - a; + 3)' 2a; + 3 2a; - 2 1 3(2a;'^ - 3) (2a;'' - 3)» 3a; 2a; -1 a; + 2 3 1 a;2 + 1 (a;» + 1)' 2a; 2(a; - 1) Exercise 129. 1. « =• y - 2/' + y' - 2/* 5. a; = y - ^y» + j2/» - |y*. . 2. a; = y + 3^2 + 13y» + 67^/* 6. a; = 2^ + l^/' + ^y' + ffty*- 3. a; = y - 2?/2 + 5^/' - 14^ 7. a; = \y - W ^ i^y^ - ^^jV^ - 4. a; = y + i2/' + i3/=«+^V2/* »• ^ = y-iy' + ihy'-h^y' -- ^3 15 315 10. a; = - My - 1) + 1(2/ - ir - ^(2/ - 1)' 11. a; = (2/ - 1) - K2/ - 1)' + 1(2/ - 1)' - i(2/ - 1)* Exercise 130. 1. a* + 12a' + 54a= + 108a + 81. 2. 32a5 - 80a*a; + SOa'a;' - 40aV + lOaa;* - x^, 4. 81a;2 _ 216a: V + 216a;i/* - 96a; V + 162/®- 5. a;^ - lOx^ + 40a;' - 80a;" + 80a;'^^' - 32a;*. 6. a^y^ + Sx''y^ + 2Sx^y^ -f 56a;5^^ + 70a; V + 56a;V^ + 28a;'2/+8a;2/^ + l. 7. a;""^^* + 7a; ~ ^ + 21a; " "^ + 35a;~ ^ + 35 + 21a;^ + 7a;^ + x^ 8. ^^MZL _ ^5^a;ly -i + ^x*y - ' - ^x^y ~^ + |a;V - ^^y^- xlii ANSWEBS. 10. 243a^a;~^ - 405a'a;-« + 270a^x~^ - 90aa;-' + 15aK~^ - 1, 11. 16a:« + 32a;"'^V + 24a:^y^ + 8a; V + ^^M- 12. a* - 6aS^ + 15a~^ - 20a~^ + 15a~'^^ - 6a~^^' + a-». 13. a^x^ - 12a^x^ + 54a "V - 108a ~V^^'" + 81a- V. 14. 64a*a; - ^ 4- biea^^x ~ ^ + 2160a ~^x~^ + 4320a " ^x^ + 4860a ~ ^^x^ + 2916a ""^'a;'^' + 729a -»a:». 15. Sla-W- 108a-26-2 + 54a-i6-«- 126-^" + a6-i*. 16. a^-Zx^ + 9x*- IZx^ + ISx^ - 12a; + 8. 18. 16x» + 32a:^ - 72a:« - 136a;* + 145a;* + 204a;' - 162a;2 - 108a; + 81. 20. 81x8 _ 216a;' 4- 108a;« + 120a;5 _ 74^^* _ 403^3 ^ I2a;2 + 8a; + 1. 21. - 14,784aV». 24. - 198a"*3V. 26. - 61,236aTV. 3 2 27. — 1320a;^. 22. 1716a;Vi 7920a^-6*. 28^ 1365a*a;i8. 23. 3003a;"y«. 25. i-%^^-5.x-^yK ^9. ^f f^x^*. J 30. - 212,640. 1001a; ^yio. mi^x^yn. 31. 24,310a;2/". Exercise 131. 1. a-»- 2a-3a; + 3a-*a;2 - 4a-5a;3 + 5a-«a;* 2. ar» - fa; + fa;-* + ^^x-^ + jh^'^ • 3. a;^+ 3a;-3_|a.-f ^ f |a; ~ ^ - f §f a; ~ ^ 4. 1 - 2a;' + 4a;* - 8a;< + 16a;8 5. a;-« + 3a;~ *^' + 6a;-9 + lOa:""''^^" + 15a;-" 6. a;^ + |a;~^- fa;~^ + ffa;"^-^H^~^ 7. x^ - \xS^ + \x^y - ^^x-^y^ + A\a;'^V 8. 1 - 7a; + 28a;2 - 84a;3 + 210a;* 9. a^ - \a-^x - j\a-^x^ - jha"'^'^ - ^iha''^x* 10. a-a-^x- a-H^ - !« -s^ _ 1^0^-113.4 11. a;^ - 14a: V + "^Oa;^ - 140a; V + 70a; ~ ^2/" 12. a ^ - 4a^» ~ .^ + lOa'a;-' - 20a'a; ~ ^ + 35a'^^a;-« ANSWERS. xliii 13. a: " ^ + ^ax ~ ^ + f a'a; ~ ^ + ||a»a; "" ^^' + if f a*a; ~ "^ 14. a; - 1 + f a; ~ ^ + -V-^^ - * + W"^ ~ ^^' + -W/a; - ^ 15. a^ - a^x^ + 2ay - Ya^^ + -%^-a"^^'x^ 16. x~^ - x^y^ - ^x^y - ^x'^^y^ - f aj'^V' 17. 1 + ix^y ~ ^ + '\*x^y - ^ + Vt^^ ~ ^ + If f ^^^ - ' . . . 18. x^y ■*" - 5a;% ^ + V-a; ^y"^ - fa; ^^ * - |a; ^"^/^ . 19. ^^jx^. 24. Ya^ ; - 55^V^a;i9. 20. - ^l2^a;^ 21. iof ^^55 • * Exercise 134. 4. 0.7782. 15. 7.3365 - 10. 18. 0.4774. 5. 1.9542. 16. 3.1055. 19. 9.8914 - 10. 11. 8.9031 - 10. 17. 1.8008. 20. 8.6309 - 10. Exercise 135. 1. 43. 4. 3.78. 9. .00803. 18. 0.4367, 2. 770. 8. 0.627. 17. 283.6, 19. .05925. Exercise 136. 1. 105. 17. 1.324. 33. 3.936. 49. - 2.158. 2. 34.3. 18. 43.36. 34. 1.946. 50. .02864. 3. .0755. 19. - 4.08. 35. 0.459. 51. - 2.483. 4. 207.71. 20. 0.2785. 36. 18.02. 52. 0.873. 5. 4.082. 21. 0.4287. 37. 14.44. 53. 1.792. 6. 0.5036. 22. 12.16. 38. 5.624. 54. .01567. 7. 64.7. 23. 37.82. 39. - 9.365. 55. - 3.908. 8. - 0.7995. 24. 162.5. 40. 0.3933. 56. 7.672. 9. 0.04775. 25. 0.7518. 41. 1.403. 57. 0.8686. 10. 147.1. 26. 0.2526. 42. 2.052. 58. - 0.4704. 11. - 2.773. 27. 4.359. 43. 0.1755. 59. - 0.1606. 12. - 0.2681. 28. 1.487. 44. 22.58. 60. 0.2415. 13. 0.2168. 29. 1.502. 45. 1.19. 61. .0725. 14. 1.427. 30. 11.86. 46. - 1.162. 62. 3.076. 15. 2.407. 31. 0.6633. 47. 3.271. 63. 1.805. 16. 0.3016. 32. 2.571. 48. 0.1424. 64. 0.7876. Ivi ANSWERS. Exercise 137. 1. 1.17. 5. 7.52. 9. 1.206. 13. 3.37. 2. 1.54. 6. 0.76. 10. 2.12. 14. - 1.38. 3. 1.54. 7. 3.47. 11. 0.537. 15. - 0.24. 4. 1.65. 8. 1.88. 12. 0.81. 16. 21oga + 31og6, 17. log a + 5 log X. 18. 3 log 6 + ^ log X. 20. idog a + log a;- 3 log 6}. 22. ilog a + 1 log x-\o^h -\ log c. 25. ^og7 . 28. ^og3-log7 4- 21og6 . log a — log 6 log a 26. log^ . 29. ^og CT + log 6 - l og 5 . log a — log c * log a - 2 log 6 27. log6-logl3 . 30. logll-log6 log 13 - log a 2 log 6 + log c - log (a — 6) log (g -f 6) + log (ct — 6) . 31og{2a-l) I \ log fee + 6) + I log (ct - 6) - log 25 1 '. I log a 33. 3.56. 38. 2.03. 43. 16. 48. 4. 34. 1.19. 39. -.065. 44. 25. 49. f. 35. 0.71. 40. - 3.46. 45. 32. 50. - f. 36. 0.84. 41. 1.58. 46. %. 51. - f. 87. 0.83. 42. 27. 47. 3. 52. - 4. 61-64. See page 326, formulas 17-20. 65. 7. 67. 9. 69. $2497. 66. 8. 68. $2654. 70. 14.2 and 10.24 yrs. Exercise 142. 25 0. 28. -271. 35 _^!jt^' 37. —6f. 26.44i. 29.25. ' ^"^" " 38.-%- 2a;2 ft 2 27. i. 34. -x-^iw/x- 14. ^^- 2^r^2- 39^ |^e; ^Q 27 i/6 + 99 1/2"— 48 i/3 — 176 * 94 u i 11 iiiiiiiiii iiiiii iii^