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BT JAMES LOUDOI^T, M.A. •PROFESSOE OF MATHEMATICS AND NATURAL PHILOSOPHY, UKIVERSITY COLLEGE, TORONTO. iFJLiaT I, TORONTO: COPP, CLARK & CO., 47 FRONT STREET EAST. 1878. Entered aooording to the Act of the Parliament of Canada, in the year one thousand eight hundred aud seventy-three, by Adam, Stivknbor & Ck>., 1q the Office of the Miuister of Agriculture. ^0^^ CONTENTS. PAOll. Introduction 1 Addition and Subtraction 6 The Signs + and - 16 Brackets 18 Multiplication 23 Division 36 Homer's Method of Division 47 Miscellaneous Theorems 62 Involution 61 Evolution 66 The Gkeatest Common Measure 73 The Least Common Multiple 87 Fractions 89 Surds 102 Equations containing one Unknown Quantity Ill Transposition of Terms 114 Clearing of Fractions 117 Clearing of Surds 121 Problems 125 Quadratic Equations containing one Unknown Quantity 133 Equations of a Higher Order than Quadratics 140 Problems 1 46 Properties of Quadratic Equations 162 Simultaneous Equations of the First Degree containing two Unknown Quantities 157 Method of Substitution 160 Method of Comparison 161 CONTENTS. PAQfc. Method of Cross Multiplication 162 Method of Arbitrary Multiplier 166 Simultaneous Equations of the First Degree containing three Unknown Quantities 166 Problems 170 Simultaneous Equations of Higher Orders containing two Unknown Quantities 177 Simultaneous Equations of Higher Orders containing three Unknown Quantities 181 Problems 187 Inequalities 192 Katio and Proportion 196 Variation 209 Arithmetical Progression 214 Geometrical Progression 218 Harmonical Progression 223 Determinants 225 Answers ^ ., 247 PREFACE. In writing the following Treatise the author has had primarily in view the development in natural order of the general laws that govern the operations of real quantities in Algebra. The various quantitative symbols have ac- cordingly been introduced at as early a stage as possible, so as to bring them at the same time under the general rules. In the remaining parts of the book the treatment of the subjects will l>e found to agree, in the main, with the works of standard authors. Instead, however, of placing at the end a set of general examples, as was originally intended, the author was induced, from a con- sideration of the simplicity and importance of the subject, to insert a chapter on the method of Determinants. In commencing the study of such a treatise as the following, the pupil should of course know something of the subject. To supply this preliminary knowledge the author intends to prepare forthwith a small treatise for the use of beginners. Should the present work bo received with favour by teachers a second pnrt will be brought out, in which the higher parts of the subject will be treated. The author would feel obliged for any suggestions or corrections that may be communicated to him. Toronto, June, 1873. THE ELEMENTS OF ALGEBRA. INTRODUCTION. 1. In the ordinary processes of Arithmetic the quantities involved are expressed by means of the nine digits and zero, and combinations of these ; but the operations performed are not indicated by the results obtained, although they may be sometimes sug- gested by the arrangement of the various parts of the process. We can always tell, for instance, when the operation of Division has been performed, by the arrangement of its various parts, but the con- nection between the several figures of the quotient and the corresponding steps of the process can only be exhibited by a lengthy explanation in words. In Algebra, on the other hand, quantities, and the operations on quantities, are denoted by means of symbols and signs, the use of which enables us to exhibit operations, either simple or complex, on any quantities whatsoever without the assistance of words. The results of these operations on general symbols do not exhibit the individual values of the quantities which are the subject of investigation. They only indicate the operations which ought to be performed on the given quantities to obtain the value sought. If, however, .the given quantities are expressed by 2 ELEifENTS OF ALGEBRA, nambers, the resalts of the Algebraical will cor- respond to those of the Arithmetical operations, which are thus included in the more general pro- cesses of Algebra. 2. Any object or result of an Algebraic operation is called an Algebraic quantity or expression, 8. There are two distinct and independent ele- ments in Algebraic quantities, the designative and the qtiantitative : the former of which is denoted by means of signs and combinations of signs, and the latter by means of symbols and combinations of symbols. 4. The usual symbols are (1) figures, as in Arith- metic; (2) letters of alphabets, either alone, or marked with accents, dashes, suffixes, or other marks, as a, h, X, y, o, A a', Xu a^, ^', &c. 5. Quantities are said to be positive or negative according as their designative elements are denoted by the signs + or — , or the equivalents of these written immediately before their quantitative sym- bols. Thus +8, +a?, +y are positive quantities whose quantitative elements are denoted by 8, a?, and y; —6, —a, — « are negative quantities whose quantitative values a .e 5, a, and z. 6. Ilie sign + is called the plus sign, and the sign — the minvs sign, 7. Positive and negative quantities are also called real quantities. ELEMENTS OF ALGEBRA 8 8. Quantities which are not real are called imagi" narpf or imposaible quantities. The manner of de- noting such quantities will only be treated of inci- dentally in the following Treatise. Unless otherwise stated, therefore, all quantities will be supposed real. 9. The symbol = stands for is, or are, equal to, and is written between the quantities whose equality jit is desired to express. Thus a = 8 denotes that the value of a is 8. 10. The symbol > stands for is, or are, greater than, and the symbol < for is, or are, less than, 11. The symbol .*. means hence, or therefore, and ihe symho\\* since OT because, 12. The following are some of the laws by which symbols are combined to denote the quantitative element of Algebraic quantities : (1.) The product of any quantitative symbols is repre- sented by writing them down in a horizontal line one after another, in any order, with or without the multi- plication sign X , or the dot . , between them. Thus ab, ba, a.b, b.a, axhj bxa all denote the pro- duct of a and b ; abc, a.b,c, axbxc, the product of a, b, and c; and so on. Figure symbols are written first in order ; thus 8a, habyQahc. "When there are more figure symbols than one the multiplication sign x only must be used between them. Thus the product of 6 and 6a is denoted by 6 x 6a, and not by 5.6a, or 56a, whose values are Jive decimal, six times a, and fifty-six times a, respectively. ▲ 2 ELEMENTS OF ALOEBRA. (2 ) The symbol a« stands for aa, a* for aaa and gene- rally a» for aa X with a written n times. (8.) The quotient of one quantity a by another b is denoted by the symbol a-^b, or the fractional form ^. U.) The product of -a fraction £^7^ quantity c is denoted by y ' and the product of two fractions ^ and -^is denoted by ,-^. ,..2 ,„.6 ^2 ,«.2a Thus the product of - and 8 is - ; — 8y take 2x—Sy-\-x. 8. From a+6— c take b—o-{-d. 4. From a— 6+<'+<^ t^ke —h-^-d—e, 25. The results of operations in Addition and Sub- traction may also be represented by combinations of signs and symbols, according to the following Laws : Law L +(+«)=+«. +(-«)=-«, -(+«)=-«, Thus the sum of a and -f &> which is a-\-b, may also be written a+(+i) if +(+ft) = +6; the sum of a and — 6, which is a — b, may be written a-f (— 6), if +(—6)=— 6; the difference between a and +6, which is a-b, may be written a— (+6), if —(+&)= —jj and the difference between a and —b, which is a-)- 6, may be written a— (—6), if — ( — 6) = -|-6.. 26. The bracket ( ) is usually omitted in the case of a mononomial. Thus + (— o) is written + —a. 27. Law n. + (+a+6) = + +a++6=+a+6, -(+a-6)=~-+a 6=_a^.6, and so on, the sign outside the bracket being said to operate distributively on all the signs within. Thus the sum of a and -ft+c, which is a— 6+c, may also be written a+{-b-\-c) if J^(-b^c)= -6-fc; and the difference between a and +b-c, which is o-6+c, may be written a-(+6-c) if _(4.6-c)= -b-\-c. So also .the sum and difference of a and — 6-|-c— d may be written a+(-fc4-c-d), and a-(-fc-|-c-(/), respectively. ELEMENTS OF ALGEBRA. 28. In the above combinations signs other than those written immediately before symbols are called signs of operation^ the sign + being the sign of operation in the case of Addition, and the sign — in the case of Subtraction ; and all those quantities whose terms are affected by such a sign are enclosed within brackets ( ), { }, [ ]. 29. The bracket is sometimes replaced by a vin- culum drawn above the whole quantity. Thus a — \-h—c\B equivalent to a— (+6 — c). 80. Also the sign of the first term of the quantity within the bracket or under the vinculum, if it is positive, is generally omitted. Thus a-(6-c)=a-(+6-c). Examples. 1. Enclose within a bracket preceded by + the second and third terms of 2a— 8&+c> Here2a-86+c=2a+-86++c=2a+(-86+c). 2. Enclose within a bracket preceded by — the second and third terms oi2x—y-\-z. Here 2x—y-{-z = 2x — [-y z=2x — (-\-y--z) = 2a? -{y-z). Exercises, III. 1. Enclose within a bracket preceded (1) by +, (2) by—, the second and third terms of a-\-b-\-Ct a-\-b—Ct a—b-\-c, a—b—Cf a — b-^C'—d, 10 ELEMENTS OF ALQEBBA. 2. Enclose within brackets preceded 1^ — the first and second) and third and fourth terms of aj-y4-«-l, oj+y— «— 1. 81. Law ni. +a6= 4-a (+6)= -«(-&)• -.a&=+a(-6)=-a(+6), + (+a6+«c) = +a(+6+c), +(— a6+ac) = +a( — 6+c), ~(+a6— ac)=— «(+*— c), + (a6 — oc — ad) = +a(6— c— d), and BO on, the symbol outside the bracket being said to operate as a multiplier on each symbol within. Thus 6aj- 6y = 6 {x—y) ; a — 86 — 8c = a — (86 + 8c) = a-8(6+c); a-i6 + ic = a-(i6-ic) = a-i(6-c); 2a?— a'+fl6=2a?— (a*— a6)=2a?— a(a~6). EXEBOISES, IV. Enclose the second and third terms of the following quantities within a bracket preceded by the symuol common to those terms : 1. 2a—xy—xz. 2. a^—l^+bc. 8. a-2bc-\-2c-d. 4. x-iy-^z. 82. Law rV. 1+1=2, 2+1=8, 8+1=4, and so on. 83. From this and the preceding Laws it follows that when the symbols are figures, Algebraical Addition includes Arithmetical Addition and Subtraction of whole numbers and fractions if quantities to be added are denoted by +, and quantities to be subtracted by-. ELEMENTS OF ALGEBRA, 11 ■■I .1 Thus the Arithmetioal operations 7 6 _8 2 8 16 _4 _s y 11 5 4 may be denoted by the Algebraical operations 6+2+4=11, 8-8=6, 7+8-11=4. So also i+i+i=i(l+l + l) = + |, i-|=^^-^, = (9-8)tV=t^- The following examples, however, of Algebraical Addition have no corresponding place in Aritluuetic : — -1-2-8= -(1+2+8)= -+6= -6, +4_6=+4-4-l=-l, |-| = H-tS = (24-26)5>5=-,V 84. From the preceding Laws it follows that a+a=(l + l) a=2a, —a—a—a= — (l + l + l)a= —8a, 7a— 5a= (7 — 5) a=2a, i«+J«+i«=i(l+l+l)«=f«, fa - f a = T^a - T»»ja = (8 - 9) tS« = -■ i^«» and so on. 85. Hence the sum of any number of like quantities is a like quantity whose coefficient is the Arithmetical difference between the sums of the positive and nega- tive coefficients, and has the sign of the numerically greater sum. If all the coefficients have the same sign the sum will have that sign, and be equal numerically to their sum. 86. The operations of Addition and Subtraction may now be conducted by arranging the expressions IS ELEMENTS OF ALGEBRA. under each other so that like terms shall stand in the same column, and proceeding as in the following 2a+Sb a— b Examples. a^+Sx+2 a-\-b— c 2a~b-\-2c a-b-\-Sc 2aa^—dbx-\-c -Sax^- bx-\-2c 2aa?-\-2bx-^c a^—2bx—c 8«+26 8aj" +8 4a-6+4c In the first example the coefficients in the first column are 2 and 1, and as they have the same sign their sum is 8; in the second column the coefficients are +8 and — 1, and as they have different signs their numerical difference 2 is taken preceded by +, the sign of the numerically greater coefficient 8. la^—dxy —x -2a^+4a^+6i/" -y &x'--6xy-\-3if+iix^ fa;"-2a;+i - a^-\-x -2 2a^-x +i lQ^-2x-i In the first column of the last Example the sum of the positive coefficients is 2f , and the numerical dif- ference between this and the negative coefficient 1 is j, to which the sign of 2i is prefixed. Exercises, V. , Add together 1. 6+Jc», a«+6»-icS 6»-a»+Jc«. ^. a ^ ,c ^ c a c ah 2~"8"^4* 2""8'^4' 2"8'^i* 87. In Art. 24 it was shown that the dififerenee between one quantity and another was the sum of the former, and the latter with its sign or signs changed. Hence the process of Subtraction can be conducted as in the following Examples. 1. From 7a +146 take 5a— 66. Here the second quantity with its signs changed is — 5a+66, which is to be added to 7a+146. 7a+146 -6a+66 2a +206 It is usual, however, to put down the result without actually changing the sign, but merely supposing it done, as in the following examples : — 2. From 8a?-2y+« take 2a?-5y+8«. 8a;--2y+« 2ir-6y+3« «+8y— 2« 14 ELlSMmm OT ALGEBRA. Here the first term of the difference is ^—2x=sXt the second term is — 2y+6y=+8y, and the third term is +«— 85f=— 2^. 8. Froma" + 8a"6 + 8ai» + 6' subtract a»-8a«6 + 8a6«-i«. a»+8a«64-8a6«+6» a»-8a»6+8a6"-6» +6a"6 +26" 4. From 2«»-a?4-6 subtract -a!»+a;»+8. -a^+aj" +8 (c»+a;»-a?+2 6. From ia-fi+^c subtract a+i6+c Exercises, VL 1. From 2a -56 subtract a +26. 2. From 8x-73/+4« subtract 2a?- 8y+2;j. 8. From a?— y+« subtract —x—y—z 4. From -2a- 86+ 2c subtract 2a+6-2c. 5. From4a^-8a!»-2a;»-7aj+9subtractaj*-2a!»-2as^ +7a?-9. 6. From - 6a«+ 86«+ c« - 7a6 + 86c - 8ca subtract -7a»-86»+6c«+9a6-46c-5ca. 7. From ia+6 subtract ia-6. 8. From a+6 subtract ia-ib. 9. From ia^-fa?+i subtract -^a^+jar-f, 10. From iy«—^zx+iaey subtract iy«— i«a?— |ajy. ELEMBNTS OF ALOEBRA. 16 8ca subtract THE SIGNS + AND -. 88. In the preceding Articles it will be observed that the signs + and — are employed to denote (1) the designative elements of quantities, (2) the operations of Addition and Subtraction. They are also used in combination in a twofold sense to denote the operations of Addition and Subtraction of quan- tities, and the designative elements of the quantities on which those operations are to be performed. These various uses of the signs mil be illustrated in the following Articles : — 89. Let the sign + denote that a quantity is addi- tive , or to he added; then the sign — will denote that a quantity is attbtractive, or to be subtracted. Thus if 40 and 50 are to be added to 100 the 40 and 50 and the operation of Addition may be denoted by +40, +50 the sum of which is +90. That is, the result will be the same if 90 be added to 100. If 40 is to be subtracted from, and 50 added to 100, the 40 and 50 and the operations of Subtraction and Addition may be be denoted by --40, +50, the Alge- braic sum of which is +10. That is, the result will be the same if 10 be added to 100. If 40 is to be added to and 50 subtracted from 100, the 40 and 50 and the operations of Addition and Sub- traction may be denoted by +40, —50, the Algebraic sum of which is —10. That is, the result will be the same if 10 be subtracted &om 100. 40. Again, suppose a man to walk along a straight road (running east and west) from ^ to B and then to C, so that AC is Ma distance from the starting- point A, 16 ELEMENTS OF ALGEBRA. Here if the + sign denote east the — sign will denote west. If he walk from A to B east 10 miles, this distance and direction may be denoted by + 10. If now he walk from B to C east 4 miles, this distance and direction may be denoted by 4-4; and the distance AC will be denoted by +10 +4 or +14; that is, C will be 14 miles east of A, If AB be 10 miles east, and BC 4 miles west, BO will be denoted by -4, and AC will be +10 -4 or +6 ; that is, (7 will be 6 miles east of A. If AB be 10 miles east, and BO 16 miles west, BG will be denoted by -16, and ^0 by +10-16 or -6; that is, C mil be 5 miles west of A. 41. If + written before a sum denote that it is due to a person, then — will denote that it is dv£ by him. Thus if 60 dollars be due to and 60 dollars due by him, the former may be denoted by +60 and the latter by —60, and the Algebraic sum of these, which is +10, will denote that 10 dollars are due to him. Again, if 60 dollars be due to, and 60 dollars due by him, the Algebraic sum of these, which is —10, wiU denote that 10 dollars are du+7, +5 >-6, -2, >-6,0 > -4, be- cause the difference between the quantities is in each case positive. Again, +6 < +7, —8 < —8, —2 < 0, because the differences are negative, being —2, —6,-2, re- spectively. Hence it follows that such a series of quantities as -4, -2,-1,0,1,2,4, is in ascending order of magnitude, whilst the quan- tities 6,8,2,0,-1, -4, -7, are arranged in descending order of magnitude. BRACKETS. 46. The use made of brackets in Art. 27 may be extended, so as to enable us to collect within a bracket various parts of an expansion in which a bracket is already employed. When it is necessary to use more brackets than one they are generally made of different shapes. The following examples will illustrate the mode of introducing additional brackets. Examples. 1. Since the sum of -6 and -c+«i= -ft+ (-c+rf) the sum of a aiid -6+(_c+d) may be written ELEMENTS OF ALGEBRA, 19 e the mode of So also the difference between a and —6+ (— c+d) may be written 2. The quantity a —b +c +d —e may be written in the equivalent forms a — 1 + 6 —c —d -\-e} a -{+6 -{c +d -e)} a — j+6 —(c — —d +«)}. Here the signs of the terms within the bracket first introduped must be changed on account of the sign - {. ' In the same manner the signs of the terms with- in ( ) are changed on account of the sign — ( ; and the signs of the terms under the vinculum on account of the — sign preceding it. 8. The quantity a —h +c ■\-d —e may also be ex- pressed in the equivalent forms a +{ -h^c +rf -4 a +{ -(h-c) -{-d +«)}. Here the introduction of the bracket preceded by -h does not affect the signs. Exercises, VII. Enclose within a bracket preceded by — all but the first term of 1. a —b -\-(c —d) 2. a+b -{c-d) 8. a-b -(c+d) 4. a —{b—c)-\-d 5. a -{-(b — c)—d 6. a —(b—c)--d 7. a -6— {-c+(c?-e)}. 77. Conversely, an expression may be freed from brackets beginning with the inside pair by removing B 2 io mumiVTa OF ALQ^EBTtA. them in succession, the signs of the terms within any bracket being retained, or changed, according as the sign immediately preceding is + or — . Examples. 1. a-{-b-(c-d)} ^a-{-b-c+d} = a-\-b-\-c—d. 2. 2a-86-{-8&+(2a-c)}=2a-8i>-{-8H2a-c} =2a-86+86-2a+c —c. 8. 2x-[dy-{-2z-(2x-2z-{-Sy)y} ^2x-iSy-{-2z-(2x-2z-Sy)}] =2a?- [dy-{-2z-2x-{-2z-\-Qy)] -2x- [Sy-\-2z+2x-2z-3y] ^2x-Sy-2z-2x+2z+Qy =0. 48. If the brackets be removed successively, com- mencing with the outside ones, the sign immediately preceding any bracket will not affect the signs of quantities within the other brackets. Thus the preceding example may be treated as follows : 2a?- [8y- { -2z-(2x-2z+Sy)}] =2a?-8y+ { -2«- (2a? -2^+3^)} = 2x-Sy-2z- (2a? - 2z+dy) = 2x-dy-2z-2x-\-2z-\-8y = 2a? - 81/ - 2« - 2a?+ 2«+ 8y =0. 49. Expressions may also be freed from brackets lyy retaining the several combinations of signs till all the brackets are removed, and then replacing each combination by its equivalent + or — sign according ^0 the law, ELUMENTS OF ALGEBRA, n A combination of the signs +, — is equivalent to — , or + , according as it contains an odd number of — signs or not. For example, H |-a=-| a = — a \- a = a = -^ a '\ \. a = -\ a = -\--{-a=:+a» Thus a - { -b+(-c-d)} =a- { -b+ -C+ -d] =a b — I — c — I — d =a-\'b-\-c-\'d, 50. Hence the brackets may be removed simultane- ously by the law that each sign of operation affects all succeeding terms as far as its accompanying bracket extends. For example, a— [—b—(-\-c—d-\-e)] — a b |-c d — = a +i +c —d —e. --+« EXEBCISES, YIII. Simplify the following expressions by removing tho brackets and collecting like terms : 1. a}-{-y-z) -(z+x) 2. (a-b) - (36-1) S. {a+o}) -(b-x) - (a-b) 4. 2x - {-Sy-(-2x+4:z)} 5. {a-{x-y)}-(2a-\-x-\-y) 6. 8a - {b-\-{2a-b)-{a-b)} , , fj, 2a-b-{-(c-d)-{-2a+b+d)} 8. 6a-[46-{-4a-(6a-46)}] 9. 16-{5-2a;-[l-(3-a?)]} 10. a^b-[-c+d-{ -a- {b-d^}] 11. [a-66-{a-(6c-2c-6-46) + 2a-(a-26+c)}]. 51. In the same manner as the use made of brackets in Art 27 was extended in the preceding Articles ELEMENTS OF ALGEBRA, may the use made of brackets in Art. 81 be ex- tended. For example, the quantity ah—acd-^-ace may be written in the equivalent forms a{h—cd-\-ce}y a [h — c {d—e)]. So also the quantity ^—^x—^xy-\-^xyz may be written in the equivalent forms 8 \l—x—xy-\-iicyz\ 8 [l^x{\^y-yz]] 8 [l^x{l+y{l-z)}], 52. In such cases, the effect of 'ach symbol of operation extends as a multiplier of the terms after it as far as its accompanying bracket. Thus in the last example 8 is a multiplier of every term as far as ] . 53. Conversely, such quantities as the last may be freed from brackets by the law, that each sign and symbol of operation affect all terms as far as their accompanying bracket extends. Thus if the brackets be removed successively, com- mencing with the inside ones, 4[l-a»{-l-2y(l-^)}]=4[l-2a;{-l-2y+2i/^}] =4 \l-\-'2ix-^4txy—4iXyz\ = 4 + 8a?4- IQxy - IQxyz, The brackets may also be removed simultaneously, as in the following example : a [l~a;{-l + 22/(-l+^)}] =2 2a?- + -2a?x22/- + -|-2a;x2y« =2 '{■2x-\-ixy—4xyz, ELEMENTS OF ALGEBRA. 88 Lrt. 81 be ex- d-^ace may be f8ajy« may be EXEBCISES, IX. Simplify the expressions • 1 {a-\-b)x-j- {a — b)y. 2. a-2(9a+b)-\-d(2a-b). 8. S[a-2{b-(c-^d)]]. • 4. 8a- [2a-2 {a- (a-l)}+2]. 6. a-2(8a+i)-8{6+2(a-6)}. 6. 4a- [2a-{26 (a?+2/)-26 (a?-?/)}]. 7. a-2 [6+8 {a-2 (6-c) +26-8 (a-&+2c)}3. 8. a~8(6-c)-i{(a-6)-4(6-c -i a-6)}. 54. The use of brackets may be still further ex- tended to express such a quantity as a{c + d) + b (c+d) in the equivalent form {c+d) {a + h). In the same manner a (-c+d) -6 (-c+d) = {-c+d) (a-b). (a—b) X —{a—b)y —{a — b)z = (a — 6) (x—y—z), xy-9x+2y-Q:=x(y-2) +2 (y-8). = (2/-3) (^+2). {a+\)x+{a+l)y-a-l = {a+l)x+(a+l)y-{a+l) = (a+l) (x+y-1). MULTIPLICATION. 55. The product of any quantities is represented by enclosing each of them in a bracket, and writing them together in a horizontal line in any order with or without the dot . , or Multiplication sign x be- tween them. Thus (+a) ( — 6) denotes the product of +a and —6; ia— 6J (— c) denotes the product of a— 6 and -c; a—b){c—d) deaotes the product of a— 6 andc — d; and {a—b) {c—d)(e—J) denotes the product of a— 6, c—df and e—f. M ELBXSNT8 OF ALGEBRA. 56. Each of the quantities so enclosed in a bracket is called 2k factor of the pifodnct. Thus +a, —6, and^+c are factors of (+a)(— 6) 57. When a factor is a mononomial it is called a simple factor ; otherwise, a compound factor. Thus —a is a simple factor, and 6— c a compound factor of ( — a) (6 — c). When the factor written first is a simple factor, its bracket is generally omitted. Thus(-a)(-6) = -a(-6),(+a?)(y-«) =+a?(y-«). 58. From the preceding mode of representing a prodact, it appears that it is equivalent to the sum of certain terms which can be obtained by laws the con- Terse of those laid down in previous Articles for expressing sums in various equivalent forms, Multi- plication being thus another form of Algebraical Addition. 59. It) is convenient to make three cases in Multi- plication, namely, L The Multiplication of simple factors. n. The Multiplication of a simple and a compound factor. m. The Multiplication of compound factors. We shaU take these three cases in order. 60. I. The product of two simple factors is obtained by a law the converse of that given in Art. 81, namely, +a (+6) = + +oi = +a6, -f-a ( — 6) = ■\ ah = —aft, —a (-f-6) = \.ah — —ab, —a (-6) = ab = +a6. XLEMnirTS OF ALGSBRA. ed in a bracket Here it will be observed that the sign of the pro- duct may always be obtained from the signs of its two factors by the law that like signs product + , and unlike signs — . Thus the product of —2a? and — 8yis + 6a?y, the pro- duct of +4fl? and — 8y is — 12ic"y. 61. Again, since +a (— 5) = — a6, it follows that 4-a (—6) (+c) = — aft (+c) = —abc. In the same manner it may be shown that -a (+6) (-c) (-\-d) = ■\-ahcd, and generally that the product of any number of factors is obtained by finding the product of the first and second, and multiplying this result by the third, and so on. 62. In addition to the modes given in Art. 12, for representing the quantitative elements of quanti- ties we shall now use the symbol a", where n may be positive or negative, integral or fractional. 63. The symbol a* is called the nth power of a, or a to the power of n, and this power is said to be of the nth degree ; also the symbol n is called the index, or exponent of the power. Thus a* is called a to the power of 2, or a squared ; tfi is caled a to the power of 8, or a cubed ; whilst a*, a""*, a*, a® are called a to the power of 4, a to the power of —8, a to the power of i, and a to the power of — |, respectively. Also the index of a^ is 5, and of a?""» is -i. The meanings of such symbols will be explained farther on. ae ELEMENTS OF ALGEBRA. 64. PowerSi and the products of powers of a quan- tity are represented according to the following laws, called Index Laws, I. a"* «♦* = «•*+*, II. (a*?*)" = a«», III. (ab)* = a-J", IV. r^ (-V = - b) b* m and n being any real quantities whatsoever, including zero. Thus, by Law I. we have a-^.a*' = «-«+* = a*, By Law II. it follows that (a»)» = a", (a*)» =ai (a«)* = a*, (a"*)* = a"*. By Law m. 66. The meaning of such symbols as a^, a*, cf can now be established. For by the second Index Law, or a is a quantity whose square is a; that is, in Arith- metical language, a* is the square root of «, or a* = V a. Again, (a*)» = a*, or a* is a quantity whose cube is a"; that is, a* is the cube root of a', or a*= >^a*. ELBMBNT8 OF ALGEBRA. 27 In like manner, since (a*)** =«"», it may be shown that a^ = V a"* ; so that, m and n being positive in- L V^ tegers, the symbols a» , a* may be called in Arithmetical language the nth root of a, and the nth root of the mih. power of a respectively. Thus 2^ is called 2 to the power of i, or the cube root of 2 ; and 8^ is called 8 to the power of |, or the 5th root of the 4th power of 8. 66. By the first Index Law it follows that a°a~' =a'-'=a°. The meaning of such a result may be established as follows : — m /,o ar.a a ;"»+« = a"», Also a ,m Xl = a m Q a , a ftt can 1 ■ i Index Law, 1 But the product I Therefore at is, in Arith- 1 ' ev, or 1 Thus «-!=-, a-8 a ^hose cube is 1 68. The quanti >r a*= ^a\ 1 Thus i is the rei Therefore a* = 1. ThvLB(a^Y = aP=l; (y^y=tf>=l; 2a:».ar* = 2a^ = 2, 4a?*x8aj~* = 12a5» = 12. 67. The meanings to be assigned in accordance with the Index Laws to such symbols as a-^ a"*, a"*, can now be established as follows : By Law I. we have a«.a-«=ra«=l. of a*» and — is also 1. a" «-"=. a" 1-^1 _4 1 a" a t a' dr. ty - is called the reciprocal of a. a iprocal of 8, and | is the reciprocal SLEMENT3 OF ALaSBRL Examples. 1. -Ir (-8y) (+«)= \-6a!yz=:-{'6xyz. 2. -a«(+a«) (-«)=_ + _««+•+»=+«•. 8. -ix-V(-i^2r')(-i^)= *'H^»+» Exercises, X. Find the products of 1. —la, +46, — c. 2. +6iP, — Tas", — 2aj". 8- -iy«, +i«^, +ajy. 4. +2a?-*y, - Sasy-*, 2a!y. 6. 2a*6*, 6a*6* 6. 4a?V» 8fl7~V» 7. 2a*6*, -a*6*, a6*. 69. n. The product of two factors, one of which is a simple factor, is obtained by a law the converse of that given in Art. 31, namely, +a (-\-b -\-c\ = + -\-ab + -}-ac — -\-ab +«c •j-a { — b -j-c) = -| ab + -}-«c = — o* -j-oc, —a { — b +c) = ab \-ac = +a6 — oc. Hence the product of two factors, one of which is a simple factor, is the sum of the products of the simple factor and each term of the other factor. Thus, —2a (a? — y -\-z) = — 2aa? -\-2ay -^az; -8j; (2a;«- 2a; -1) = -Qa? +6a;" +8a?. The work may be arranged as follows : 2x" -2a? -1 -8a? -ea.*" +6«» H-Sa?. EXEBCISES, XI. Multiply 1. 2a^-8a?-lby -4a?. 2. 6a?» +a? -8 by ^a?*. 8. 3a^ -4y» +52» by 2a;»y. 4. 2a;* -a;* +2 by 8a;** 5. a^ _«y +y« by ary-i. 6. ia?"*-ia;"'*+l by 8a;*. SLSMENTS OF ALGBBRA. 29 70. III. The product of two compound factors is obtained from the law fa-^b) (c-\-d) =(a-{-b)c -\-(a'\-h)d ssac+bc -\-ad'\-bdf — a-\-b){c — d) = ( — a-\-b)c — ( — a-\-b)d = —ac-\'bc-\-ad —bdf and so on, which is the converse of the law given in Art. 64. Hence the product of two compound factors is the sum of the products of one factor and each term of the other, and may be obtained as in the following Examples. a^ -X -2 X +8 oj" —a^ —2a? H-8a;»-8a?-6 «»+2a;«-6a?-6. Here a^ —a^ —2x is the product of a? — « — 2 and x, and 8a^ —Sa? — 6 is the product of a;* —a? —2 and +8, and the sum of these is ^ -f 2a;* —6a; —6) the required product. a? H-iff^+l X -.r^+l ar» +x^ +x —a?* —a? —a?" +x-\-x^+l «* +a? +1. 5aj"- 2aj + 7 a? + 3 - 2a;-* 6a;8__ 2ar»+ 7a; H-16a;a- 6a;+21 -10a;+ 4 -14a;-' 6a;»+18«»-9a;+26— 14a;-».' 80 ELEMEVT8 OF ALGEBRA, EXEBGIBSS, Xn. Find the products of 1. 2aj"+aj+l and 4a?- 8. 2. aj"+8aj-l and aj»-8a;+l. 8. a^—xy+y^ and x-\-y. 4. a*-a«ft»+6* and rt*+rt«i»4-ft*. 6. a^-s^y-\-a^\/-x\/-\-y*' and a;+y. 6. l+4a?-10a;»andl-6aj+8a!". 7. aj«-7a!«+5j?+l and 2a:»-4a?4-l. 8. a;«-2a:"+8a?-4 and 4aj»4-8a5«+2j?+I. 9. a^+y*— a?i/+a7+i/— 1 anda?+y— 1. 10. a?+2y — 8« and a?— 2y+8«. 11. a'+i^+c"— 6c — ca— aft and a-f 64-0. 12. a«-2a6+6»+c«anda«+2a6+6?-c*. 13. a?*+3/*anda?*-2/*. 14. a?*+y and x^—y^. 16. a*+aM+&* and a*-fr*. 16. a^-ah^-\-h^ and a* +6*. 17. a?+a?*+2 and a?+a?*-2. 18. a^+2+aj-» and a^-2^x'\ 19. aj*+a!»+l and aj-*-aj-«+l. 20. a-*+a-*+l anda-*-l. 21. a^-2+a-^ and «*-«-*. 22. a+a*6*-a;Vanda+aV+a?V- 8^ 1 1 8 11 28. xi—ccy^-\-x^y-yi and a;+a?y-f y. 24. a?+^y— 2 and fa?-y+|. 26. ^a?-i+a?-» and ^a;4-i-aj-». 26. aj»P+a;PyP+i/»p and aP-yp, 27. a«»-2a'»-ia?+8a"^ar» and a*+2a"-%-8an-*a^. - - L !L 28. «»+2a?V+8/Panda«-2a?y+8^. HMB— SLEMEirrS OF ALOEBRA. 81 71. When two factors contain integral powers only of the same symbol the successive coefficients in their product may be obtained by the following method, called the method oi Detached Coefficients, which con- sists in leaving out the letters in the ordinary process, and writing the coefficients only of the successive powers. Whenever a power is wanting, therefore, its place must be supplied by a zero. Ex.1. a^-8a:*+8a> -1 ic"-2a?4-l 0^' 8a?* -f- 8aj»- 2aj*+ 6a^- 6a!»+2aj + aj»- 8a!"-f8a?-l This product can be obtained by the above method, as follows : — 1_8+ 8-1 1-2+ 1 1-84. 8- 1 «a+ 6- 6+2 4 1, 8+3-1 l_64.10-10+5-l. Here we have the coefficients of the successive powers in the product commencing with a?. Ex. 2. Find the product of 5a^-2a;+l and ba^-[- 2a?— 1. Here the coefficient of a?^ is wanting in each factor, and its place must be supplied by 0. 5+0- 2+1 6+0+ 2-1 25+0-10+5 +0+ 0-0+0 +10+0-4+2 -6-.0-t-2-l 25+0+0+0-4+4-1. Therefore the product is 25a!«-4a^+4«-l. S) 32 MjIIMENTS of IZGEBBA, EXEBOISES, Xm. Apply the method of Detached OoefGicients to find the product of 1. a!»+2a;+l anda?-l. 2. 2a^+8a!"+4a?+5anda!'-2a?+l 8. 6a;"-a?4-l and 6aj"+a?-l. 4. 6aj"-8a;»+l and 4a^+^+l. 6. 7aj»-(k»i»+2andaB»+2aj»-7a?+l. 72. The product of two like factors is called the square of either. Thus (a^b-^c) {a—h+c) is the square of a—6+c, and may he written (a — 6 -f c)*. When the two factors are the same, since the product of like terms in the multiplier and multiplicand is the square of either term, and the product of unlike terms in the multiplier and multiplicand is repeated in the process, it follows that the square of any quantity is the sum of the squares of each term and twice the product of every two terms. Fox example, find the square of a— 6+c. a —6 +c a —b +c a*—ab •{•ac —ah +^— ^ +ac — Jc+o* a«~2a6+2ac+6»-26c4-c«. Here the products of the like terms a, a; —6, —ft; +c, +c are the squares of a, —6, +c, respectively; and the product of any two unlike terms, as —6, +c is repeated, giving twice the product of those two, that is, —26c. ELEMENTS OF ALGEBRA. 33 Examples. 1. (a+6-c)«=a"+6«+(-c)'»+2a6+2a(-c)+26(-c) =a+6''+c»+2a6-2ac-26c. In taking the products of the terms, two and two, it will he found of advantage to take in order the product of the first term, and each term that follows it, then the product of the second term, and each term that follows it, and so on, if there be more terms than three, 2. (2a;»- 8a? - 4)«=:4a^+9ir»+16+2 (2aj") (-3a?) +2(2a?«)(-4)4-2(-8a?)(-4) = 4a?*+ 9a?+ 16 - 12a?» - 16a;'»+ 24a? s= 4a?*- 12a?» - 7a!*+ 24a?+ 16. 8. (iP-l-a?-*)«=a!»+l+ar«+2a?(-l) + 2a?(-ari) +2(-l) (-ar^) := a?"+l+aj-«-2a?-2+2a?;;i as a?" - 2a? - 1 + 2ari+a?-». 4. (a>*-i + aj'~*)«ata? + J + a?-»+2a?^(-i) + 2«* (+«-*)+2(-i)(+^-*) ^a?+i+ari-a?^+2-a?'^ B=aj-»*+f-a?"*+ar*. ExEBasEs, XIV. Find the values of 1. (l+a?+a?»)«. 2. (l-a?+a^)* 8. (l-a?-.a^)» 4. (l-8a?+2a?^. 6. {a^-^x+^f, 6. (a?«-2+ar»)*. 7. (a?^-a?*+l)* 8. (ia?*-ia?^-l)». 9. (a-^b+c-dy, 10. (2+8a?+4a?«)»-(2-8a;+4a?>)«. 11. (a+6+c+d!)«-(a-6+c-ci)*. ELEMENTS OF ALGEBRA. 73. Since by actual multiplication (a+6)(a-6)=a«-6>, (a 4T+ c) (^W- c) = (a + *)" - c", and so on, we can hence write down at once the product of two factors, one of which is the sum, and the other the difference of the same two quantities. Examples. 1. (ar+3y)(ar-8y)=(2ar)«-(8y)»=4a!»-.93^. . 8. (aj+y4-2«)(aJ+y-2»)=(a?TH-2«)(a?+y-2«) . = a? + y* + 2ajy — 4«*. Here the factors are arranged so that one is the saniy and the other the difference of a;+y and 2z. 4. (2a+6-8)(2a-6+8) = (2a+6-8)(2a-6-8) = (2a)«-(6-8)» =4a>_(i»+9__66) Here the factors are arranged so that one is the sum and the other the difference of 2a and 6—8. 6. (ir»-a^+a?-l)(a;»-a!«-a?+l) = (V-aj")"-(a?-l)» r=a5«+a^-2«»-ir>- 1 +ar =a^-2a;»+a5*-a!»+2a?-l. 6. (^^-a!-i) (a*+a?-l+a?-»)=(;/-a;_l+a?-») (a^+a?-l+a;-») =aj*-(a^+l+a?-»-arH-2~2aj-») =0?* - «»+ 2af - 8+ 2»-* - »-•• ELEMENTS OF ALGEBRA. 8S Exercises, XV. Employ the preceding method to find the product of 1. a+86 and a—Sb. 8. J+b^ and a*-6*. 6. 4a*+66^and4a*-56^* 6. ^a^+lb and ^aJ'-^b. 2. a«+6«anda»-6». 4. a* +5* and a* -6*. _ t:^ c and - — -i- 8. 2a?+yH-« and 2x-\-y—z, 9. 6«— 86+c and 6a+86— c. 10. 2/i» - 86» - 4c* and 2aH 86*+ 4A 11. x—y-\-z—l anda?— y— «4-l« 12. 2a?-82/+«-l and 2x-8y+z-\-l, 18. 2a«-86+c-4dand2a»+86-c+4(«. 74. Since by actual multiplication (x+y) (a^-xy+y^)=afi+f^, {x-y) {oi^+xy-\-'f)=a?-fy we can write down at once the product of any two factors one of which is the sum of two quantities, and the other the sum of the squares less the pro- duct of the same two quantities. Examples. 1. (2a+b) (4a*-2a6+i") = (2a)»+6'=8a»+^. Here the second factor is the sum of the squares of 2a and b less their product. 2. (x-Sy) (a^-j-Sxy-\-9y^)=af^-(Syy=a^-21i/', Here the second factor is the sum of the squares of a and — % less their product. 8. (a? + a?-») (a^-H-a;-«) = (aj + a?-i) (a» + x-*-l) o2 88 ELEMENTS OF ALGEBRA, Exercises, XYI. -Find the product of 1. 2a;+3i/ and ix'-Qxy+df. 5. a'-iandrt^+a^i+J* 4. J+b^ and a-ah^+b. «. 2a?*-8y*and4a?+6a;*2/*+9y. 6. a^+y and o^P-x^^f+yK 7. ax-^+a-^x and a^x-^-l-\-a-*sfi. DIVISION. 75. Any factor of a product is said to be the quotient of tlie product divided by the other factor. Thus +2y is the quotient of — 8a; (+2y), or — 6ajy divided by — 8aj; 2a?— 1 is the quotient of (2a?— .1) (a?+2), or 2a!"+8a?-2, divided by a?4-2. 76. The quotient of one quantity divided by another is denoted by enclosing them in brackets and writing the sign -;- between them, Dr by writing the second below the first with a line between them. Thus the quotient of —a divided by ■\-bi& denoted by (-a) -7- (+6), or by ~; the quotient of 2aj*— 8aj+l "T b divided by a? — 1 is denoted by (2a? — 8a?+ 1) -^ (« — 1), or , 2aj"-8a;4-l by i — ; and so on. " X — J. 77. The first or upper quantity is called the Divi- dend, and the second or lower )=-«' ("^)(-'0=+a, it follows that —a a —a a ^ +'' « —b b -i-b b —6 h Hence the Rule of Signs is the same in Division as in Multiplication. Thus -f4.T 4a7 — ar* X ,.2 -2a; 2a? 32/2' -3 """^ 3* are mono- 81. When the Dividend and Divisor involve powers of the same quantity, the quotient can be obtained according to the Law m — = tf ni-n m and n being any real quantities whatsoever. This Law follows from the first Index Law, thus- a'*~^a'»=a'"; a"* • • _ — I* • a" 88 ELEMENTS OF ALGEBRA. a' a' a* a' Thus -„ =«"-' = «, -=«*-*= rt-2; -r=rt-^"=rt" — /.s^o a -8 a* a* 1. 8. 4. +x -xy^ xhz\ Examples. ■ — - = — 2.r. a? +a''6'' = _ei»-66*-». , Exercises, XVII. Divide 1. -2a!»by +07. 2. +6a^y'by -2.^^. 8. — a;^y by x^y, 4. +a?^«' by —x-hjz^, 5. 2a* by -a~*. 6. -5a;-y by +a?-V. 7. a*6^c* by a^feM. 82 that . II. Since (4-|-|)(+6) = +a-c, it follows "+6 "^fe-ft- In the same manner it may be shown that '^ ^ e e " e e^ and generally that the quotient of a polynomial by a mononomial is obtained by dividing each term of the former by the latter. BLEMENTB OF ALOEBRA. 89 Examples. 1. = -4a?+2. 2. —a? ax s=af+aa?— a*. EZEBOISES, XYIII. Divide 1. 48a;*-16a5»+24a!«by8aj". 2. 16aa!»-266a5V+85cajy»by -5a?. 8. mhia^—mr^a^ by mnajy. 4. a"6a;^+«— a6'a^+'»» by abaf^-^*, 6. 6a^-i-6a?-y by -a;-^-». 6. mai^+*i^-na^-^*—a^fhya^, 88. m. Since (a+5) (c+-i.fa?-« Here the third term of the quotient x-^ is the quo- tient of 1 or a?" by x, 8. Divide 20^ + 5^*-|^-JJ^*+?^+^a.*-^ by 2x^^y+i 2 5 (x^+^x-l 2a? —-x -4- - I2ar + -x — -zor x A — x4- -x 2 ^6/2 2 40 ^6 ^2 6 2aj»-^a!«+ ic* 2 6 8 4 96 4 , 8 -a;*— — aj*+_aj 2 40 ^6^ 84 8 £ , 8 -a?* — -a?*+_a? 2 8 ^6 -2aj* -2a;* 2 2 4. Divide 2a»-6a»&+13a»J>-6a6»-8a<6 by 2a-86. Here we shall arrange the dividend and divisor ac- cording to descending powers of a. JCLEMENTS OF ALGEBRA. 43 2a - db) 2a» - dba* - 6i«»+ 18iV - 6i»« {a* - 86rt>-f 26«a 2rt»-8&rt* . ■ 4lM-mi 6. Divide a* - ab" - flc»- 2ia»+26*+ 2k-»+8m8-8ci» -8c* by «+ 8c -26. Here the dividend will be arranged according to powers of a, the coefficients of a' and of a being col- lected within brackets. a+ 8c-26)a*4- (8c - 2h)a'^ - (c»H- fc'»)a - 8c*+ 2^" - 86«c+'2/>* - (c» + 6»)rt - 8c*+ 26c« - 8fe8c + 2h * -(c»+6>-8c*+26c''-86''c+£&« i)ivide Exercises, XIX. 1. 2-8a;+iB»by 2+a?. 2. 10a:»4-7a!»-8aj+6 by 2a?+8. a. aj»+l bya?+l. 4. 2a!»-ar»+8a;-9by 2a?-8. 6. 6««+14a;»-4a?+24 by 2a;+6. 6. 7aJ»-24a!»+58a;-21 by 7a?-8. 7. afi-1 by a?-l. 8. a;*-6a;»+lla;»-12a?+6byrB»-ai?+8. 9. a;*-18a;"+86bya!"+5a7+6. 10. l-x-Sa^-a:^hyl'^2x-\-a^, 11. a^-\-x*-2hy i€*-\-2a^-\-± 12. .T8+2a^H-8a;*+ar"+l by a?*-2aj'+8a;2-2a;+l. 18. ia!»-4a?*+^a^-^a^--^a?+27by ia;»-a?+8. 14. a;"4-a?~' by Dy x-^-cc'-^. 15. 2aj«-8af-8+7a?-*-8a;-«by 2a?+l-8a?-». 16. y'-y-'by y-y-\ 44 FlUMENTS OF ALOEBE^. 17. a:«-2+^-"bya?-2+a;-' 18. a!»+64bya?+4a;* + 8. a^+y-9a;-16aJ*-4 by a?+4a:*+4. 19. 20. 21. rt'-ft" by a»-2a't + 2H9i*by aHSaft+at'. 27. a:»-8a?y-y-l by ar-y-l. 28. a^- (a+6+c)a!"+ (6c+ca+ai) ar-a6c by a;*' (a-\-b) x-\-ab. 29. a^+y»+3"-8an/«bya;+2/+«'- 80. ar^-y by a;*-t/'» 81. «»-)'>'' by a* -6*. 82. x^-y^hy ai^-y^. 88. ar-y by a;*-y* i.-i 84. aH2a6-"*+96-*by«-2fl^6"*+86- 85. ;r3-2a;^i/ ^-{-2xy -x^y * by -aj^i/'^bva?*-!/"* 86. i^r^ 2/ 1+ A *+6 « by a*+an *+ft 87. x^-2Jx^-\-(^ hyx^-2a^x^-^a. 88. a? -4.r*2/ * + 6a;*y *-4a;V +2/ ,i -f^.-i ,^..-4 by a;^-2.»^2/ ^+y ^. 86. When there is no exact quotient the process of the preceding Article will give at any stage a quantity which, multiplied into the divisor, and added to the last difference, will be equal to the dividend. J!LIIl£S]lf}V 07 ALGEBRA. 48 X-' - 2/" ?+n. - abc by x"- \h >i -y -i -* For example, a».^2;a»"-6a>-f.7f'2» 2aj"-4a? --2jj +7. Here2a?(aj-2) -2Lj»+7=2a!"-6*+7. If the process be oontinued, the next term of the qnotient being —2, and the next difference 8, it follows that (2a; - 2) (aj - 2) + 8 = 2a;* - 6a? + 7. 87. The statement of the preceding Article is true, no matter what the aiTangemeiUt of the terms of the dividend and divisor is. Thus, in dividing a;*-8x»+4a>»- 7a? +6 by a^-2x-^d, we may proceed as follows : - 2a?+a!» - a; - 8aj*+ a?* - 7aj+ 4a5"+ 6 f" Jaj*+ ia^ -8a5»+^a?* -fa;" -ia?*-7a7+-L7aj*+6 -ia;*+ia;»-faj» - ia;» + K + -!Jar» - 7«J^+ 6 If we stop at this stage of the process it vnll follow that (fa!*+iaj») (-2a?+a?*-'8)-iaj*+|aj"+3Ja!"-7a>4-6 =a?* - 3a;"+4ar»- 7iC+6. 88. When the dividend and divisor are arrangeci according to descending positive powers of the same letter, it is always possible to continue the process of t the process of I division until the leading term of the last difference is of stage a quantity ■ a lower degree than the leading term of the divisor. ad added to the I The last difference in such cases is called the idend. I Remainder. 4B ELEMENTS OF ALGEBRA Examples. 1. Find the quatient and remainder when lOaj'+Taj* -8a?+14 is divided by 2a?+3. 2ar+3;i0aj»+7ar»-8a?+14(6aj«-4a?4-2 ■Sop 4fl?+14 4aj+6 9 Henoe the quotient =5a^— 4;i;+2, and remainder =8. 2. Find the quotient and remainder when a^+px-^-q is divided by a?— a. a^ — ax {a-\-p)x-\-q (a+p)x-a(a-\-p) Hence the quotient = a; + « + />. and remainder =fl'(a+i>)+g. EXEBCISES, XX. Find the quotient and remainder in dividing 1. 4a^-4a^+8a?+2by2a?+l. 2. aj»+a« by x-{-a, 8. d^-a?bya?+«. 4. 2a?»-2a;*+9a^ by 2a^4-a?+l. 5. 2a;»+2aj*+5;»»byaJ»+a!»+a?+l. 6. 2a?*+8aJ'y4.8a;»y+8a;2/8^y4_,.2y5by2aj+y. 7. «»+3aaj3 by aj»4-5aa?+a«. ELEMENTS OF ALGEBRA. 4Br d remainder HORNER'S METHOD OF DIVISION. 89. When the dividend and divisor contain integral powers only of the same symbol, and the coefficient of the first term of the divisor is unity, the quotient may be found by the following process, which is the inverse of Multiplication by Detached Coefficients. By multiplication the product of Qa^^x-\-2 and a^—2a;+ 8 is obtained as follows: 8-1+ 1-2+ 2 8 8-1+ 2 - . -6+ 2-4 + 9-8+6 8-7+18-7+6. Therefo e the product is 8iB* - 7aj"+ 13ic» - 7a?+ 6. Here, since the sum of the third, fourth, and fifth horizontal rows of coefficients is equal to the sixth, it follows that the third row is equal to the sum of the fourth and fifth with their signs changed, and the sixth. Thus 8-7+18-7+6 +6- 2+4 - 9+8-6 8-1+ 2. Consequently the quotient of 8aj*— 7ir»+18a:*— 7a?+6 divided by «*- 2^+8 may be obtained as follows. 3-7+18-7+6 +2 +6- 2+4 -8 - 9+8-6 8-1+5 m 48 ELEMENTS OW ALQEBRA. In forming the product above it will be noticed that the diagonal columns -6 +2 -4 +9 -3 +6 are the products of —2+8 and 8,-1, 4-2, respectively. Therefore in performing the inverse process of finding the quotient ilie diagonal columns +0 -2 +4 -9 +8 -6 are the products of +2 —8 and 3, —1, +2, respec- tively. +2,-8 are the coefficients of the second and third terms of the divisor with their signs changed, and are written vertically, each horizontally opposi'^ the products of which it is a factor. Ex. 2. Multiply 2ic>-8aj+2 by a^-2a?+4, and divide the product by oi"— 2a? +4. 2-8+ 2 1-2+ 4 2-8+ 2 -4+ 6- 4 + 8-12+8 +2 -4 2-7+16-16+8 +4- 6+ 4 - 8+12-8 2-8+ 'J Here in the inverse process +2,-4 the coefficients of the second and third terms of the divisor, with their signs changed, are written in a vertical line, and the products of these and the successive coefficients of the quotient are written in diagonal columns. The first co- efficient of the quotient is the same as the first coefficient of the dividend, the coefficient of the divisor being unity. Ex. 8. Multiply Sa^-2as+b by a5»-2«"-l and divide the product by «" - 2«* - 1. ELEMENTS OF ALGEBRA. 49 8+0-2+6 l_2-i-0-l 8+0-2+6 _6-0+4-10 . +0+0- 0+0 ^8- 0+2-5 3-6-2+6-10+2-5 +2 +6+0-4+10 -0 -0-0+ 0-0 +1 +3+ 0-2+6 3+0-2+6+ 0+0+0 Here zeros are supplied where any terms are wanting. The process of division exhibited in the above ex- amples, as the inverse of multiplication, is called Horner's Method of Division. The following additional examples will exhibit the mode of conducting the vari- ous steps in the process. Ex. 4. Divide 2a:» + 7aj* + 20a5» + 80ar»+ 34a?+ 85 by 2+7+20+80+84+35 -2 -4- 6- 8-14 -5 -.10-15-20-85 2+3+ 4+ 7+0+0 Therefore the quotient is 2a^+3ar»+4a;+7. Here the coefficients of the dividend are written in a horizontal line, and the coefficients of the divisor, ex- cept the first, with their signs changed, in a vertical line. The first coefficient 2 in the dividend is repeated below as the first coefficient in the quotient, and the products of this 2 and —2, —5, give the first diagonal column to the right of the divisor, namely, -4 : ^>] 10 60 ELEMENTS OF ALGESRA. The sum of the second vertical column to the right of the divisor gives +3, the second coefficient in the quotient, and the products of this +3 and —2, —6, give the second diagonal column, . -6 -16 The sum of the third vertical column gives +4, the third coefficient in the quotient, and the products of this +4 and —2, —5, constitute the third diagonal column, -8 -20 The process is continued in this manner and ceases when the last product, (in the above case —35,) falls vertically below the last term of the dividend. The successive coefficients of the quotient are thus found, and as the first terms of the dividend and divisor are of the fifth and second order, respectively, the first term of the quotient must be of the third order. Therefore the quotient is 2a^+3ar»+4a?+7. If the dividend in the last example had been 2aj^+ 7a;*+ 20a^+ 30ar»+ 33a;+ 39, that is, 2iB*+7a;*+20a^+80ar»-|-34j?+35-a;+4, and the divisor the same as before, the remainder would evidently be — a?+4. The sucessive coefficients — 1, +4, of this remainder will therefore be the sums of the last two vertical columns in the following, the process being conducted as before. 2 5 2+74-20+30+834-89 -4- 6- 8-14 -10-15-20-35 2+3+ 4+ 7- 1+ 4 ELEMENTS OW ALGEBRA. n In such a case as this the last multiplier for forming the diagonal columns will be the last coefficient of the quotient, and the sums of the vertical columns to the right of this multiplier will be the successive coefficients of the remainder. The order of any term in the re- mainder is the same as that term of the dividend in the same vertical column. Thus in the last example — 1 is the coefficient of x, as is also -{-83 in the dividend. It is usual to draw a vertical line as above between the coefficients of the quotient and those of the re- mainder. Ex. 6. Divide lOx^-la^-ldx" by a? -8. Here the places of the coefficients of the two terms wanting in the dividend must be supplied by zeros. +8 10- 7-19+ + 80+69+150 10+23+60+150 + +450 +450 Therefore, quotient =10a^-\-23x^-\-50x-\-U0y and re- mainder =450. Ex. 6. Divide 5a;»- 18aj»-8ar»+20a;-5byay'+2ar»-3. In this case the place of the term wanting in the divisor is supplied by a zero. -2 +0 + 8 5+ 0-18 -10+20 + 5-10+ 2 - 8+20-5 - 4 - 0+0 + 15-80+6 + 8-10+1 Here the operation of forming the coefficients of the quotient ceases as soon as the last product +6 falls below the last term of the dividend. A vertical line is then drawn after +2 the last multiplier, and the sums of the vertical columns to the right of this line will be the successive coefficients of the remainder. d2 62 ELEMENTS OF ALGEBRA. Therefore, quotient = 5x^ — 10a? + 2, remainder = 8a?-10a?+l. E2u:b6is£s, XXI. Divide according to Hprner's Method 1. Ba^-^-Sx-dOhy x-3, 2. a^-\-Sa^+Sx-\-l by a?+l. '8. af^-5x*+10a^-10a^-{-5x-lhy a^-'2x-\-l. 4. 20a^^20a^-\-57x*+Safi-7a^-2x-\-6 byar»-a7+8. 5. 17a;*-15a?*-804bya?-.2. 6. a^+x^-\-x'^-\-2£d^-x*-x'-2x-l hyx^+a^+x+l, 7. iaf^-7x*+25i»^-15a^+Sx-\-10hy a^-x-^5, 8. lla?*-2a;»+14a;-539bya!3-7. 9. 6a;*-7a?8+15ar»-12a?-86byar»+a?+8. 10. 4ar*-18a:«-12ar»+18a?bya7-5. 11. 7a^+19a;*-5ar»bya?+8. 12. 7a;*-8ar»+lbya!»-2a;+8. 18. lOx^+Sx^-dOa^-Ux'+lOxi-l by a^-9. 14. 5a!«-15a^+20a?+42bya;'-2dr»+8. 15. 8a;*-400a^+176a; by afi+lx'-S.x+l, 16. 10iP^»+10a;«+10a;8-100bya;'+a^-a?+l, 17. Sa^+5x*-daf'+7a^--\-S by a;3-2a?. 18. 9a?i3+4a,-«-27a;»+lbya;8+2a;*+l. MISCELLANEOUS THEOKEMS. 90. Any quantity is said to be a function of the symbol or symbols involved in it ^ Thus 5a?, a^-l, 5a^-x-^l, af^-f-x+y are func- tions of Xi the latter being also a function of y. ELEMENTS OF ALGEBRA. 53 91. A function of any quantity x is denoted by enclosing x ia & bracket, and writing before it any one of the symbols f, ^y '^, &c., either alone or affected with accents, dashes, or subscripts. Thus f(x)y f^(x), f{x\ /a (a?), 0(a;), ^{x) denote dif- ferent quantities involving x, 92. It must be carefully noted in this notation that such symbols as /, y employed before the brackets do not denote numbers, but are merely short forms for expressing the words quantity y expression, or function involving. Such a symbol, therefore, as f{y) must not be confounded with the product of /andy. 93a If a certain quantity involving x be denoted by f{x)y then the value of the same quantity when x=a is denoted by /(a). Thus, if /(a?) =a:»-5ar»+10, it will follow that J{a) =a»-6aH10, J{h) =6»-66«+10, /(ca) =c«-5c*+10, /(2a?) = 8^.-8 -20a!2+ 10, /(I) =1-6+10=6, /(-8)= -27-45+10= -62, m =10, and so on. Examples. 1. If /(a?)=a^-2a7+3 and ^(a;)=2«»-8ai"-5, find the value of /(l)+^(2). Here/(l) = l-2+8=2, ^(2) = 16-12-5=-l. Therefore /(l)+^(2)=2-l = l. 64 ELEMENTS OF AlGSJiRA. 2. If/(ur) = 6..'«-2.i-y+j/«, fiml/Cv). Here/(y) will be the value 6.j'»-2.ri/-f i/* when for ^r we write y. Therefore f{y) = 5./» - 2y* + i/ = 4.v«. 8. If /,(.«•)=. i:»-2.n/-. A ami /,(y) = »/»+2i/.u-u^ find the value of /,(.f) -f^ (»/). /•,(.r)=.,J-f-2.^•«-.l•« = 2,l^ >,0/)=.v«-2,/»-./'=-2y«. Therefore /, (x) -t\ (//) = 2.t'» + 2»/«. Exercises, XXII. 1. If /(.f)=j^-2.i-»-f 6.i'-10, liml/(2). 2. If/(rt) = 6a«+6rt-20, fiud/(-8). 8. If/(y)=V-8»/»-l,find/(iJ). 4. If /(.f) = 5.t'»-8a-+10, and 0(.v)=8»/*- 6»/- 15, find/(-l)-^(-2). 6. If /;(rt)=a«-fi"-(^ and /„(/>) =/>«-c«-rt», find 6. If /(.r) = 2.(//,»? 4^) a will vanish, hecauHo ono of iis factors l)0(5omoH zero, and 11 will remain tho same, booauso it does not con- tain .r. Hence /©- and therefore tho romaindor is the value of the dividond when w has the value which makes the divisor vanish. Examples. 1 . Find the romaindor when .^•' — Qx^ -f 2a; — 7 is divided by a?— 2. Here af'-Sji'-{-2x^7r=:Q(x-2)-\-IU and since this is true for all values of x it will be true when x=2, in which case wo havo 2»-8x2»+2x2-7=0+i2 .•.7J=8-12+4-7 = -7. 2. Find the remainder when 8a;"+12a!'— 4aj— 5 is divided by 2a? +8. Here f{x) = 80.'"+ 12a!*— 4a? - 6, and a;= — ^ makes tho divisor vanish. .M2=/(-f) = 8(-i)-+12(-t)»-4(-f)-6 = -27+27+6-5 =1. A [mi M ELEMENTS OF ALGEBRA. 8. Find the remainder when af*—a* is divided by d7+a, n being an integer. Here /2=(— «)" — ««, the first term of which is —a", or +«", according as n is odd or even; therefore i2= —2a", or 0, according as n is odd or even. Thus x—a^!i^—(^y!j^—(^^3^—(ff have remainders — 2a, — 2a', —2a", —2a', respectively, when divided by x-^a\ whilst a^—c^f x^^a^f a^—a^ are exactly divisible by a?+a. 4. Show that {b-\-c—a) (c+a—h) (a+6— c)+8a&c is exactly divisible oy a+6+c. Here the dividend may be denoted by /(a), and since — ft — c when substituted for a makes the divisor vanish, i?=i=/(-6-c)=(J4.c+6+c)(c-6-c-6)(-&-c+6-c) +8k(-6-c) =2(6+c)(-26)(-2c)-86c(&+c) =86c(6+c)-86c(6+c) =0. EXEBOISES, XXIII. Find without actual division the remainder after dividing 1. 6a;*-6a;»+7iP-8 by x-\-l. 2. 6aj»-6a!"+10a;-100bya?-8. 8. 18aj»-27aj+40 by 8a?-4. 4. 8aj»-16aj«-12a?-10by2a?+6. 5. a? -\-a^ hy x-\-a, 6. aj'-a*by a?-a. 7. a;»+a» by ar-a. 8. aj»-(a+l)ar+2a-l by a?-a+l. 9. Prove that (6-c)a»+(c-a)6»4.(a-5) c» is exactly divisible by a+6+c. 10. Prove that a«(6» + c»-a«) + 6»(c« + a«-ia)+ca («* + 6* - c») is exactly divisible by a-|- 6+ c. ELEMENTS OF ALOEDlU. By 96. The value of any quantity /(a?) when x=a may be determined by finding by actual division the re- mainder when /(a?) is divided by x— a. For the value required is/ (a), and this by Art. 95 i^ the remainder when/ (x) is divided by x—a, Thusthevalueofar"— 7a?+9whena?=6is6'— 7x5+9; but the remainder when ar^ — 7a;+9 is divided by a?— 5 is also 5"— 7 X 6+9. Hence the value required may be determined by finding the remainder by actual division. Examples. 1. Find the value of a?"— 4a; +8 when a? =8. Here the value required will be the same as the remainder when a;"— 4a? +3 is divided by a?— 8. 8 1+0-4 +8+9 I 1+8+6 + 8 + 15 + 18 Therefore 18 is the value required. 2. Find the value of Sas"— 4aj*+8a;'— 4a;"+a?+4 when a;=— 4. -4 6_ 4+ 8- 4+ 1 _20+96-896+1600 + 4 -6404 6-24+99-400+1601 | -6400 Therefore — 6400 is the value required. 8. Find the value of a;«-102aH»+100a?*+102a;8-99aj> -201a? when a? =101. 101 1-102+100+102- 99-201 +101- 101 - 101 + 101 + 202 1- + + 101 1- 1- 1+ 1+ 2+ 1 I +101 Therefore 101 is the value required. m 88 ELEMENTS OF ALQEBTtJL Exercises, XXIV. Find the value of 1 . 7a:* - 1 1 J?" + a; - 60, when a? = 2. 2. 6ir" - 2ur> + 10a; - 28, when x = 4. 8. a;"-19A'"-121, when a? =-8. 4. a:»-98a^-98a;»-100a:»+98a?+100, when a?=99. 5. 2a;» + 401a;* - 199a;» + 899a;* - 602ar + 211, when .i = -201. 6. 10a;*-1109a;»-109a;«-212a;-llll, when a;= 111. 7. a;"— 4a;"— 2a;', when a;=— 4. 8. 72a;'— 48a;, when x=—\. 9. 1000a;* -81a;, when a;=0.t 97* Although the arrangement of the terms or factors of a quantity is arbitrary and does not affect its value, it will be found useful in some cases to prefer one arrangement to another. Whenever, for example, the parts of an expression are analogous to each other, corresponding letters involved should be arranged in the order of the alphabet, the last of the letters being followed in order by the first. Thus if a, h, c be the letters, a will be followed by 6, h by c, and c by «, in the same manner as if they were arranged on the circumference of a circle. 98. Quantities arranged according to this law are said to be written in alphabetical circular order. The following are examples of its application. Ex. 1. a+b-i-c; ^a-b-c; a«+i«+c»; a;*+y*+A ELEMENTa OF ALGEBRA, BO It will be observed that correeponding parts have like signs. P Q V Ex. 2. aa-\-by-{-cz ; a*x-\-l^-\-<^z; --f -+-. Ex 8. abc il-hlh bc-\-ca-\-ab. In the latter form it will be observed that the first factors of the three terms are b, c, a, and the second factors Ct a, b, and that the first term does not contain a, the second does not contain b, and the third does not contain c. Ex. 4. The quantity a*(b-c) -6''(rt-c) -c» (6-a) may be arranged in the more symmetrical form a" {b — c) -\-l^{c—a) -\-c^ i^ — b) in which the corresponding parts follow each other in alphabetical circular order. Ex. 6. a(b-cy-^b(c-ay-\-c(a-bf. Ex. 6. (bz-cyy-{-(cx-azy-{-{ay-bxy. Here the first part does not involve a or a?, the second part does not involve b or y, and the third part does contain c or z. 99. When we know the letters involved in a quan- tity which can be arranged according to the above method, we can infer one part from another, and therefore may write the first part only and leave the rest to be inferred. Thus the quantities in Exs. 4 and 6 may be written a'(6— c)+&c., and a(6— c)''+ &c., respectively. Thus also — (6"— c")+&c. may stand for y^ s eo ELEMENTS OF ALGEBRA. 100. In the same manner as a" is used to sym- bolise the product of n a% +" and -" may be employed to denote combinations of n + signs, and n - signs, respectively. Thus, (-«)»=-»«»=: -a»; (—r)«= -» a;»= +a^; and, generally, if n be a positive integer, so that 2n is an even integer and 27i+l an odd integer, Thus also we may write +»+»=+"= + ; —«—*=—' = — ; — »— *= —8= + ; and, generally, j_m j_n-- _L.m+n • 101. By assuming this Law to hold for all values of m and n, positive or negative, integral or fractional, we would be thus enabled to extend our notation for representing the designative elements of quantities by the introduction of such signs as +^, — ^, &c. We would be thus led to the mode of representing imaginary quantities, the discussion of whose pro- perties is beyond the scope of this Treatise. 102. Also, if the Law hold for zero values of m and n, it will follow that +»= -f- =-^ For +<>-}-8=+«= + ; and -|-.+3= + ; therefore +<>= + . Again, -o_8^_83^_. and +-'=-; therefore —"=: + . Thus (+a?)«=+?aj»= 4-1; (-a?y=-?aj»=-|-l. ELEMENTS OF ALGEBRA. 61 INVOLUTION. 103. The process by which the power of a poly- nomial is expressed as the sum of a series of mono- nomials is called Involution, and the power so expressed is said to be developed, or expanded. 104. In the following Articles we shall give the method of effecting the development or expansion when the index of the power is 2, 3, 4, —1, or —2. 105. The expansion of the square of any quantity is most easily obtained by the method already given in Art. 72. The two following modes, however, of arranging the results given by that method will be found useful. 106. t. In this method the various parts of the expansion will be written in horizontal rows. In the first row occurs the square of the first term of the given quantity whose square is to be developed. In the second row the product of twice the first term added to the second and the second. In the third row the product of twice the first term added to twice the second term added to the third and the third. And so on. Examples. 1. (a+6)«= a> +(2a+i) 6=&c. 2. (6-c)= 6« +(26-c)(-c)=&0. 8. (a+6+c)«= a* +(2a+6)6. +(2a+26+c)c=&o. m 62 ELEMENTS OF ALGEBRA. 4. (a-6-c)»= a« + (2«^-6)(-6) + (2a-2&-c)(-c)=&e. 5. (a+6+c+rf)2=: «« 4-(2a+6)6 +(2a+26+c)c. + (2a + 26 + 2c 4- «?) c?= &o. From this mode of arrangement is deduced the inverse process for extracting the square roots of quantities. ExEROISEd, XXV. Develop 1. (.r-2/)«. 2. (2a?-8i/)». 8. (l-x-\-yf 4. (2a— 36 — c)**. 6. (yz-{-iS£C-\-£cy). 6. (2a?-l+a?-i)». 7. (a^+a-aa?-!)". 8. (a?-2/+;?-l)». 9. (aa-.6+c2-.d)>. 10. (2a;»-3a?+l-;»-i)». 107. II. The second method of arranging the development will be evident from the following Examples. 1. (a+6)«=a>+2a6+6». 2. (a+6+c)»?=a''+2a(6+c)+6»+c»+26c. 8. (a+6+c+d)»= aH 2a (6+c+d)+6»+ c»+ +8)» 4. (a;»-2)». 6. (2;r+8y7. 6. (l-a+a«)«. 7. {x-l+x^^y. 8. (a!»+2-a;-»)». 9. {l-xi-y^zf, 10. lbc+ca-\-abf. 111. The results given by the Rule of Ari. 109 may also be arranged according to the method ol the following examples, in which the various parts are arranged in horizontal rows. 1. (a+fc)»= o» 2. (a+b+cy= aS -a +(8 a+b +8 a+6.c4c«)c=&c. ELEMENTS OF ALGEBRA. 65 8. (a+fc+c+d)»= a» +(3a»+8a6+6")6 + (8 a+6+c* + 8 «+6+c. d-^^)d = &c. 4. (2a!»-aj+8)»= (2a^)» +(3.2;r2"^»+9.2a!"-a;+9)(3)=&c. From this mode of arrangement is deduced the inverse process for extracting the cube root of a quantity. Exercises, XXVIL Develop 1. (a;+y)» 2. {w-yf 8. (a-J^. 6. (l+aj-a^f. 112. The fourth power of a polynomial is found l)y squaring its square. Thus (l+flj)*={(l+a?)*}'={l+2a?+a;»}« = 1 -f 4a?+6a?»+4a;»+a?*. r*.l Develop 1. (1-a)*. 4. (a^-b+cy. Exercises, XXVIII. 2. (a?-y)*. 8. (l-a?+a:y. 113. The development of the negative integral powers —1, —2 of polynomials may be effected by expressing them as the reciprocals of the correspond- ing positive powers. Examples. 1. Develop (l+aj)-i. Here(l+a?)-»ssz— , and if 1 bo diyided by 1+a? the mi m m I ''^'il'h mfi A- ill 6ft ELEMEHrrS OF ALGEBRA. quotient will be l—ar+aj'— a;"+&o., and the remainder "will depend on the stage at which we stop dividing. .The quotient 1— aj+aj"— a^+^C' is then the develop- ment of (1+a?)-*; but it must be carefully observed that such developments do not express the exact values of the powers unless the remainders be taken into account. Thus in the present example if we stop at —a;' in the quotient, we shall have (l+a;)-» = l-a?+a?»-a^+ ,r—- . If we stop at -{-x^ in the quotient, we shall have and so on. 2. Develop (l-a?)-a. Here (1 -ar)-«= ^_^-^ =1 + 20? +'8a5»+4aJ»+&o. by actual division. K we stop at 4a;' in the develop-^ ment, the remainder will be ^ai^—^a^. EXEBCISES, XXIX. Develop to four terms 1. (l-ar)-i. 2. (l+a^)-». 4. 6. l+2a;+3a;^ 1+a? 8. (l+a?4-a;»)-». l+a?+a;* 6. (l+xf ' EVOLUTION. 114. The process by which the root of a polyno- miai is expressed as the sum of a series of monono- mials is called Evolution, and the root when so expressed is said to be extracted, 116. In the case of the square and cube roots the extraction can be effected by processes the inverse ELEMENTS OF ALGEBRA. QT of those given in Involution for the development of the squares and cubes of polynomials. We shall consider in the first instance the method of extracting the square root of a quantity. 116. Since the square of a quantity is equal to the square of the same quantity with its sign or signs changed, it will follow that there will be two square roots, one being derived from the other by a change of signs. Thus since (+«)''= {-af=a\ it follows that the square root of «" is -\-a, or —a. These two values may be represented by the symbol ±a, the two signs being combined. Again, smce {a-bf = {-a-\-bf = a^-2ab-\-l^, it fol- lows that the square root of a'' — 2a6+6" is a — b, or -a+b. In the following articles we shall only determine the root whose first term is positive. 117. In the process of Art. 106, the successive terms of the quantity to be squared occur as factors of the several products written in the horizontal rows, and therefore, in extracting the square root of such a development, the successive terms of the square root may be found by arranging the given development in the lorm of such successive pro- ducts. Thusa«+2a64-6'»=a» + (2a+6)6 from which arrangement we infer that the square root ofa24-2ai+6»isa-l-6. b2 ELEMENTS OF ALGEBRA, Ex. 2. 4-12i;»+9^=2» +(4-ai!") (-ar«). Therefore the square root of 4 — 12a!'+9ar* is 2—Ba^, Ex. 8. a>-2a6+6"-2ac+26c+c" = a» +(2a-.6)(-6) +(2a-26-c) (-c) Therefore a— 6— c is the square root of a*— 2a6+&o. 118. The preceding process may be more con- veniently replaced by the following equivalent one in which the successive terms of tlie square root are written as they are obtained to the right of the quan- tity whose square root is required. iiX. 1. a«+2a6+6' (a+'J a* 2a+6)2a6+i» 2a&+6» Here the first term a is the square root of the first term of the given quantity from which its square a* is subtracted leaving 2a6-4-6*. The second term b is obtained by dividing the first term ^ab of this remainder by 2a, double the term already found ; b is then added to 2a forming the complete divisor 2a-\-bi and this multiplied by b gives 2ai+6" Ex.2. 4-12a;»+9a?*(2-8«" 4 4-8a!») -12a^+9a?* -12j!'-f9a? ELEMENTS OF ALGEBRA. G9 Ex. 8. a«-2a5+i'-2ac+26c+c« (a-6-o a* -2a6+6» 2a-26-c) -2ac+26c+c« -2ac+26c+c» Here the first two terms a and —b are found as be- fore. The third term — c is obtained by dividing —2ac by 2rt, the second divisor being formed by doubling a— by the part of the root already found, and adding to it —c. The third term — c is then multiplied into the second divisor 2a — 26 -- c giving the product -- 2ac In the same manner a third divisor would be formed by doubling the three terms of the square root pre- viously found and adding the fourth term ; and so on. It will be observed that the successive terms of the root and the successive divisors correspond to the factors of the development of the square, when arranged according to the method of Art. 106. Ex.4. a;*-6x»+13a;"-12a?-t-4 (a;a-3a?4-2 2j!»_8a?)-6a!»+13a^ 2ir»-6a?+2) 4»"-12a?+4 4a;'-12a7-f4 Ex. 6. ia?*-aj^-«?-f 4a?"*+4i»-« (ia3«-a?*-2a?~i ^ a^-2a?*-2a?-i)-2a?-f4a?-*-t-4a?-" -2aJ+4a;~*+4a?-» Ai )m 70 ELEMENTS OF AlOFTiRA. 119. In the foregoing examples the process of extraction terminates, and the quantity whose root is extracted is said to be a perfect square. When the same process is applied to a quantity which is not a perfect square, a result is obtained, the square of ivhich, added to the last difference, is equal to the given quantity. For example, let the process be applied to find the square root of l+o?. 1 If the process be stopped at this stage, it will follow that (i4.^a?-ia;^)»+^^'»-oV'*^=l+^. 120. Care must be taken to arrange the given quantity according to ascending or descending powers of some one letter ; otherwise the process may not terminate, although the quantity be a perfect square. For example, in finding the square root of 4a?*+4a!"+ 1 let the quantity be arranged in the order 4a;»+4a?*+l. 4a^ -afi+l ELEMENTS OF ALGEBRA. n By proceeding in this manner we wonld get a scries of terms in the root, the square of which added to the last difference would be equal to 4a?*+4a;"4-l. Thus if we stop at the second term of the root we have (2aj+aj»)«-a5«+l = 4;»*+4ir»4-l. Exercises, XXX. Extract the square root of 1. 4a7+4aaj'+fl?, 8. ir—px-\- — . 7. 4aj-4a?*+l. 2. 4a«-12a6+9i^. 4. a*4-2a''+8a'+2a+L 6. 4»-*+12a;-»+9a;-*, 8. a'*+4a?+2-4a?-»+^"'- 9. 9a*-12a6-|-24a*+12aj*+a:"-86a:*+86a?*+27a?-64a;*+27 8*-" - 6i»* + 4a>*; - 6a;* + 1 2a;* + iB* - 6a;* + 12a;* -8a;* 8x'«-12a;*+12j>*+9a;-18a;*+9;9a;»-86a;*+86u;*-|-27a;- 64a;*+27 9a^-86a;*+86*+27iP- 64ar*+27. It will be observed that the successive terms of the root and the successive divisors correspond to the factors of the development of the cube, when arranged according to the method of Art. 111. Exercises, XXXI. Extract the cube root of 1. l+6a;+12a;"+8a;". 2. 27a;" - 54a;" + 86a? -8. 8. 8a" - 84a"a;+ 294aa;» - 848a;". 4. 27a;"4-54a;+86a;-»+8a;-". 6. 8d;"-.12a;'-fl8a;*-13a;"+9a;»-8a;4-l. 6. 27a;"-54aa;»+63aV-44a"a;»+21aV-6a"a;4-a«. THE GREATEST COMMON MEASURE. 123. A quantity is said to be of so many dimen^ sions, in any letter, as are indicated by the index of the highest power of that letter involved in it. Thus a^y is of 2 dimensions in x, and of 1 dimension in y; a;"— 2a;+8 is of 8 dimensions in a?; a^—o- — l is of I dimensions in a;; x'-^y^-^a^x'^y ib of —1 dimension in X, and of 2 dimensions in y, 124. By the phrase total dimenaiom is meant tho 1 n niEWENTfi OF AiavmnA* groatost 8um of iho indicos of the powers involvtid in any ono torm. Thus tho total dimonaions of ifV+^'V/J^^'^^ ^* "-"d tlio tlio total dimonsions of iV^ff+iV~hf aro 2. 126. Wlion tho total diraonsionH of cncli torm of a quantity are tho same, tho (|iuintity in Htiid iohohotno- Thus tlio quantit ioa .«" - a'y + 8//", a?* -f B.^"*/ + *«?•/ — fhmf +»/*» are homogouoous, 126. Any quantity can bo rondorod liomopfonoouB in form by introducing proper poworH of somosyinbol, as ijy whose numorical value is 1, as factors of tho several terms. Thus «'— 3.r4-2 and .t;"— 8.»j'-f B-^— 1 ^i^^y bo writtc^n in tho horac)f,'oiio()us forms a:'— 0. (looinoirioiil (iiiuonHiou. AKaiii, tlio quaiiiiiy rt^-} ft.r— H, if .»•* bo iwkow to ro|»ioHoni an aroa, will liavo no moaninjLf tinloHM f fi.r inul —II mIho roproKonfc aroan, iliai in, unloHM (>lu« iorniM { (i,v and — M liavo ilio isanio (]oon)(«tvi(Mil (iinuMmionH iiH.r"; und tiio qtuintitv ii*"-— JU'I IJ, if .»'" 1)0 iiikon to roprowiMil. a volunio, will have no nioanin^j; iujU^hh -~l\,v luul ) "2 aJHo roproHcnl. VolunioH, tlmi iH, nnl(»SH ilu> lornm - Jl.r and 4 *I Imvo tiho nanio (loonjolrical «linu>nHi»)nH aK.r", naniolv, lon|.:lli, broadlli, anHM. 'I'lio pn'oodin^ plniiHonlofty v^horo ilio dinionnionH aro 1, 2, or W in Uion ovtondod to (j[uaiitiiiuH of any diniouHionu whatHouvur. I'iO. A wholv (WptntHtim or (jmintitjf Ih ono wliich iuvolvoH powoi'H with poHiiivo iudiooH (including zoro), and iuiogral ooollloicMitH only. ThuH -Tm'", ii'«-2.r } n, (b-" -Tm-' I 7.i'"l().«'* I n aro wholo oxproHHionH, an aro alno all ponitivo and nogativo intof^orH, ninoo tlioy ntay l)o (M)nKidorod au coollununtti of tlio aoro powor of any Hynibol. IHO. Whon t-wo or nioro wliolo cx|)r(^HHi'onH two multiplied toj^othor, <»a(d» iH Haid to ho a iniutHurc of tho product, and tho product Ih said to bo u rnuUiple of each factor. TluiH rt, .r, and ir-f 1 aro inoaHurcH of 0.u"-f H.— 8 — x—S, Here 8x-\-9 is the difference of the given quantities, and therefore a multiple of their G. G. M. Also since 8a;+9 contains 8 as a factor, its other factor a;-f 8 must be a multiple of the G. 0. M., which contains no simple factor because the fr^-^^^ quantities do not. The G. 0. M. will therefore be tko Jie as the G. CM. of x-\-S and a^-{-2x-^S; and as the former is a measure of the latter the G.O. M. will be a; +8 itself. 2. FindtheG.C.M.ofar"+2a?+landa;»+2a^+2a?+l. a^+2x+l)a^-\-2a^-\-2x-}-l(x a i-lJx'-^2x-\-l(x-{-l a^+x a +1 06 +1 SLBIUSNTB OF ALQBBRA, 81 Here «+l is the difference of the eedbnd quantity, and X timen the first; and is therefore a multiple of their G. 0. M. The G. C. M. of the given quantities is therefore the G.O.M. oi x+l andaj«+2a;+l, that is, 8. Eind the G.C.M. of 2a;»— 7aj— 2 and 2«*— «— 6. a»"— a;-6;2a?»-7a?-2fa:^ +1 2a;»-aj«-6aj 0^- -X -*2 2 -2a?-4 - x—6 - x-\-2 a?-2;2aj» -jj-6f2«+8 8a?— 6 8»-6 O.0.M.s=:«-2. Here we first find a/^— a?— 2 a multiple of the G. G. M. of less dimensions than 2a;°— 7a; — 2; so that the G.O.M. required is the G.C.M. of 2a;'— a;— 6 and a;"— a?— 2, which is the same as the G.C.M. of 2a;"— «— 6 and 2(a;*— a?— 2). The partial quotients x, +1 of 2a;»— 7x— 2 and 2a;«— 2a;— 4 divided by 2a;''- a;— 6 are s'jpantted by a comma to distinguish them from parts of an ordmary quotient. At the next step, we find — «;4-2 a multiple of the G. C. M. of less dimen- sions than 2^;"— 2,r— 4; so that the G.C.M required is the G.C.M. of 23;*— a;— 6 and — a;+2, which is the same as the G.C.M. of 20;*— a;— 6 and a;— 2, that is, »— 2, because 2a;'— a;— 6 is a multiple of a;— 2. 4. Find the G.C.M. of 7a;'-2da;+6 and 5a;>-18a;« ^^:l 1 "MlJ * M , :u'--'..'Ll ELEWENTB OF ALGEBRA, 7 7a^-28iP+6;85a?"-126a!*+77a?-42f6a>, U 86aj"-116«*+80a? -11« +47aJ-42 - 7 77««-829a?+294 77a?«-258a?+ 66 -76;- 76a;+228 ar— 8;7aj*— 28a?+ 6/7»— 2 7a!»-21a> — a»+6 G.C.M.=a?-8. Here the second given quantity is multiplied by 7 in order to make its first term a multiple of the first term of 7a;'— 23a;+6, and the process is conducted as before. 6. Find the G.O.M. of Ux'S5a^-\-iaT'-12x+6 and 4a:»-10a?»+6a?-15. 4a!»-10aj»+6a?-15;28«*-70a;»+ 8a?»- 24a;+10r7iP 2ar*-70«»+42a;»— 106a: -l;-34a;«+ 81a?+10 • 84a;»- 81«-10 ELEMENTS OF ALOEBRA. 88 4aj"- 10dj"+ Qx- 16 17 84«"-81a>-10;oa»»-.17{)^"+ 1()2u7-255(^2;p, 4 68aJ»--lG2j;«- 20u> - 8jj"+ 122;t— 266 - 17 180ur'»-2074aHh4886 ISOu-a- 824j?- 40 -876;-1760aj+4876 2a:-5;84A'»-8U— 10('17«+2 84u;'~85;i; G.C.M.=ar-6. 4.1;- 10 4a?-10 Here 4.1?'— 10a?'+6a?— 15 is multiplied in the second step by 17, in order to make its first term a multiple of the first term of 84d;"-81a?-10. 148. The process of the preceding examples will frequently enable us to find the G.C.M. of polyno- mials involving powers of different letters, as in the following Example. Find the G.C.M. of 2^Ha'y~8y« and 8a;»-4a^+J/'. 2 6a^+ 9xy- df ^llyJ-lUyi-lly' w—yj2x^-\-xy'-Qy'^(2x-^dy 2a^ — 2xy G.C.M.=»- ?/. Sxy — By" dxy—Sy* v9 mk m t f^^ ii(%, il- t' : 'ji.-,! 'i V-l: 84 ELEMENTS OF ALGEBRA. Here the suppression of the factor -rlly cannot affect the G. G. M., which contains no simple factor. 144. When the preceding method fails we must introduce, when necessary, and suppress, when possible, polynomial factors, provided they are not common to the given quantities. The reasoning is analogous to that of Art. 141. Example. FindtheG. CM. of 2aaj»-(a-2)a;-l and 26a;»- (6+2) a?+l. a 2aa^-(a— 2)a?— l; 2a6a;«— a (6+2) a?+a ^6 « 2a6ar»— 6(a— 2)a?— 6 — (a+6) /-~2(a+6)a;+~(a+6) 2a?— i;2aar»— (a-2)a;— 1 (aa? + l 2aa;'— 00? G.c.M.=:a»-i. 2a?--l 2a?-l Here the suppression of the factor —(a +6) which is evidently not a measure of either of the given quanti- tie:^, will not afifect their G. G. M. ExEBOisES, XXXm. Tind the G. C. M. of 1. a;"— 1 and a;*— a?— 2, 2. a;»+2a?-8 and a;»+5a?+6. 8. a;*— 1 and a?'— az-'+a?— 1. 4. a?»-6a?«+lla?-6 and a?»+4a?»+a?-6. 6. 2a?«+a?-3 and 8a?»-4a?+l. 6. a;*— 1 and aT^+a?"— a?— 1. 7. 8a?»-22a?+82andaj»-lla?«+82a?-28. 5. 2a.'«+28a?+45 and »^-8a?+9»* - 27.' ELEMENTS OF ALGEBRA. 8tf 9. a:«-9a;»+29as*-89a?+18and4aj«-27a;*+56»-88. 10. 8aj»-88aJ-f-119anda:»-19aj>+119a?-246. 11. 9aj»+68a5»-9aj-18 and a;«+llaj+80. 12. a»»+ir"-8aj+5 and 7jr»-12a?+5. 14. 20jr*+a^-land26aJ*+5. "-ic-l. 15. 2a;»+9a;'+4aj-15 and 4a^+8a?2+a»+20. 16. ic*+5aj»+6 and a;*+40a?+89. 17. 8a^-20a^+ 15^+2 and a:*-4a?+8. 18. aj*-a*anda^-6a^-a'a;+a'6. 19. a'"+aj'y+ajy+2/' and cc*—y\ 20. a^+aj'y'+a^y+y'anda?*— y*. 21. 6a!«+26an/+88i/»and7ar»H-19a??/-6y«. . 22. 8a:*-a?V-22/*and2A-»+8a;»//-2a?y»-8t/». 145. in. The G. CM. of two polynomials, in- volving simple factors, is the product of the G. 0. M. pf the simple factors and the G. C. M. of the polyno- mial factors. Examples. 1. Find the G. C. M. of 8ar»+12a?H-4 and 6a;«-6a;— 12. Here the given quantities are equivalent to 4 (2aj«4-8a?+l) and 6 la^-x-^). The G. C. M. of 4 and 6 is 2, and theG. C. M. of 2a;»+8a?+l and ar»— a?— 2 is x-\-l. Therefore the G. C. M. required is 2 (a?+l). 2. Find the G. C. M. g{ 20a^+104a^i/"+182ajy and 42ar'y»+ 114a;y — 86^^. The first quantity is equal to ia^ (5a^+26i»y+88y') and the second quantity to 6xi^ (7aj*4-19a?y— 6j^). I ■'■u '4/'"* ':ill f ■km Mm mm IMAGE EVALUATION TEST TARGET (MT-3) &c ^0 /.. .f^ ,v ^ ^ Mr ^ 1.0 I.I m 12.2 us ■a |» 120 IL25 i 1.4 III 1.6 Photographic Sciences Corporation 23 WEST MAIN STREET WEBSTER, N.Y. 14S80 (716) 872-4503 A^ A^ \ \ fv % '<«> (S^ ^^ ^'^f4 '^ V Kp t/. 1^ c\ \ >1* o'^ ELEMENTS OF ALGEBRA, The G. G. M. of the simple factors is 2a^, and of the compound factors isx+3y. Therefore the G. 0. M. required is 2xy (a?+8y). EXEBOISES, XXXIV. Find the G. 0. M. of 1. 24a?-40ar+16 and 24aj»-24a!«-6a?+6. 2. 6a^-{-12af'y*-{-12a^f+6xy* and 9a^+9sfiy*-[-9xi^, 8. 4«V+2a»a;»y - 6a»a^ and 9a«* - dax*y - 8aa:»/ 4. 2a«6« + 3a»6» - 8a*6* - 2a»6» + a«6«-a6' and 2(fb +8a%*+a»i«+4a*6*-a»6»+a?6^. 146. The G. 0. M. of three quantities is the G. C. M. of any one and the G. C. M. of the other two. Thus whatever measure is common to the three quantities A, B, 0, must plainly he common to A and theG.O.M. ofB, 0. Examples. 1. Find the G. C. M. of 4an/V, 6a^««, and Qa^h, The G. C. M. of Axyh* and Qa^yz^ is 2xyz* ; and the G. C. M. of 2xyz^ and Sx^y^z is 2xyz. Therefore the G. C. M. required is 2xyz, 2. Find the G. 0. M. of a?*-l, a?*+2aJ«-8, and 2a;*+2aj»+8a?+8. TheG.O.M. ofa;*-l and a;*+2a:«-3 is a^-1; and the G. 0. M. of aj«-l and 2aj*+2ir8+8a;+3 is a?+l, th« G. C. M. required. ELFMENTB OF ALGEBRA* 87 EXEBOISES, XXXV. , I^dtheG.O.M. of 1. ir»+a?-80, aj*+lla?+80; anda!"-»-42. 2. 6aj"-aj-2, 21a^-17aj+2, and 15aj*+5aJ-10. 8. 12a»+7a6-10&», 16a«+2ai-86», and 15a«+6a6 -10ft«. 4. 10aj«-80a?+20, 16aj«-76aj+90,and6a!»-21a?+18. 6. aj'4-8a!'y+3a?^+y', a^^+aj^y+ayH-/* and a^-^-a^y 6. a?*-9ic»+29aj»-89x+18, 4a!»-27«»+ 68^-89, and af».8a5»+19a?-12. THE LEAST COMMON MULTIPLE. 147* The L. 0. M. of two or more quantities is the common multiple of lowest dimensions and least numerical coefficient or coefficients. 148. To find the L. C. M. of any two quantities A andB. I If they contain no common measure except unity, their L. CM. is plainly their product. Thus the L. C. M. of 4a and 56 is 20a6. n. If they have a G. C. M., let it be C; so that A=aO and B=&0, where a and h are two whole expres- sions having no common measure. Then the L. C. M. of A and B is the L. 0. M. of aO and 60, that is abO. Anda60=«B=A.B. Mm III m 88 ELEMENTS OF ALGEBRA, Therefore the L. G. M. of two quantities is equal to one of them multiplied by the quotient of the other divided by their G. 0, M. 149. In the case of two mononomials the L. C. M. may be obtained by multiplying the L. C. M. of the numerical coefficients by the greatest power or powers of the several letters involved in the given qnantities* Examples. 1. The L. 0. M. of Qx'yz' and Axyh is 12a^i^, 2. Find the L. 0. M. of ar«- 1 and ar*+2aj«-8. The G. C. M. of these quantities is a^—l; and the quotient of a?*— 1 divided by a;*— 1 is aj*+l. Therefore the L. 0. M. is (a?'+l) (aj*+2a!»-8). a Find the L. 0. M. of 4a;»-4a^-8y» and 6aj*-6^. The G. 0. M. is 2 (a?+y)» and the quotient of 6a;»— 62/* divided by 2 (a?+y) is 3 {as—y). Therefore the L. C. M. is 8 (a? -y) (4aj*-.4»y-8y«). EXEB0ISES» XXXYI> Find the L. a M. of 1. 12a»J»(; and 9fl6V. ^. 21a!»y*« and ISoajV. 8. 6 {x'y-sDy*) and 16 («"-2/*). 4. 16iB«-20a?-6and2aj»-.6a;*+6a?-8. 6. a;»— a;"+a?— 1 anda!»— 1. 6. 12aj»-a?-l and Qx'-Bx-^l 7. 8«"-6a7+2 and 4aj"-4a^-a;4-l. ELEMENTS OF ALGEBBl, E& m 150. The L. 0. M. of three quantities is the L. CM. of any one, and the L. C. M. of the remaining two. Thus the L. G. M. of the three quantities A, B, C, is the L. C. M. of one of them, as A, and the L. G. M. of B,G. The L. G. M. of four quantities is the L. G. M. of any one, and the L. G. M. of the remaining three. Example. Find the L. G. M. oiaf^-l, af^-ly and aj*-a»*+l. The L. G. M. of a;»-l and a5»-l is (a^-1) (a!»»-[a?+l); and the L.G.M. of (a^-l)(a^-^a!+l) and a;*-2a«+l is (a^+x+1) (a;*-2aj»+l). r Ml ,'V v:^ H t nit- a k \ I "la '4\ EXEBOISES, XXXV 11. Find the L.G.M. of 1. 2iB^% 6a?y;2*, and 8a?*t/a". 2. aj»+a?~30, ir"+lla;+80; anda!*-a?-42. 8. a;*-l, 7ar»+6a?-2, and Ta?*- 5a? -2. 4. 12a«+7a6-10^», 15^« + 2a6-86«, and 15a« + 5a6 -106>. 5. a;«-4, aj»+8, 2a!»+8a?-2, and 2aj*-8a;-2. 6. 21aj»+ Ban/ - 41/', 49a;"- 42/«, 21a;* - 20an/+4y*, and 49a;*-28an/+4^. FKACTIONS. 151. When one quantity is not exactly divisihle by another, the quotient is represented by ^ting them in the form of a fraction. % * I I'jT 90 SLEMSNTa OF ALGEJBRA, Thus the quotient of +o divided by -6 is — , —b and the quotient of a^^3x-\-l divided by «"— 8 is 152. Hencoi conversely, the product of such a fraction and its denominator is equal to its nume- rator. Thus (±-:)(-6)=+<.. and (^^+iy^.s) -aj«-8aj+l. 153. A fraction is not altered in value by multiply- ing or dividing the numerator and denominator by the same quantity. For, m being any quantity, we have, by the previous Article, ^^ (±^)(-w6)=(±|)(-6)w=+ma. That is, ^^^ is equivalent, as a factor, to •-mb -b* ■\-ma +^ * * —mb ~" —b ' And so for other fractions. 154. A fraction which involves powers with negative indices or fractional coefficients in the terms of the numerator or denominator, can always be reduced to one whose numerator and denominator are whole expressions by multiplying the numerator and deno- minator by a proper quantity. Thus, . ."",^ , — 5 is reduced to the equivalent 4ar — i^ar— ^a;—* ELEMENTS OF ALOEBRA. 91 fr«^°° 24^-a»'-2 ' ''y ""^«Ply^8 numerator and denominator by 6a!'. In like manner, by using the multiplier pqx^t ^ ^, , is reduced to f^. ^^ . pxy—qx*y~^ p'qixn^'-pq^ • Exercises, XXXVm. Beduoe to fractions whose numerators and denomi- nators are whole expressions X y 1. 4. aj« 2. a 8. ^aj«-2a;4-| ax-\-hy' *" a?*— 2aj"— 3aj-** oa? — 6a?-i+l 5. a~^a!"— 6a?~*— 1 155. A fraction is said to be in lowest terms when its numerator and denominator contain no common measure. A fraction, therefore, is reduced to lowest terms by dividing its numerator and denominator by their G.C.M. Examples. 1, Beduce a'ft^ to lowest terms. Here ab is the G.C.M. of a^l^ and a'S-oJ'; therefore ah the reduced fraction is a- 2. Eeduce ^J^±^fJZ^ to lowest terms, a!"— 1 The G. C. M. of numerator and denominator id 8a?-2 fl^+»+l; therefore the reduced fraction is r 0?— 1 I'll If ■J* ill. +j()^2^j^ _ 2x (a!-\-y)+2y (xr-y) 2aj«-j-4an/-2y« -r »*— ^ I:. Jl'f M SLEMSNT3 OF ALGEBRA. a. Simplify «-^^+^+i The L. 0. M. of the denomixiators is ab (a*+V), Therefore o_a'+2afej_ft^ a'(fl^+y)-q6(a*+2a6)+y(a«+y) "* a6(a"+6") 8. Simplify _iL_,- ^^ ^ ** "^ ^ •^(a-6)(a-c) (6-c)(6-a) (c-a)(c-6) The L. G. M. of the denominators is (b—c) (c-^a) (a— 6), and the multipliers for the several fractions are, therefore, — (6— c), ^(c—a), —(a— fc), respectively* Theitefore (a-6) (a-c) (b-c) (6-a) {c'-a)(c-'b) —a (b'-c)+ b (c—a)+c(a—b) (b—c) (c^a) (a—b) 2ac-2ab -2a B. 10 1. , * 12 2a a>- (6— c) (c— a) (a—b) (c—a) (a— 6) ExEBOISES, XLI. Simplify the expressions 1. ^-J?. 2. iL-i.. 8. 8^ 2a; 46c 6cd as-2 x-3 ^ 23?+l _ 4a?+5 ' 8a;-f 2 6^+4* 6. .i-+: » 2~8y 2+8y SlUT^NfB OIP AlOMBRA. 97 6. B. 7. .+ : !+«+«• 1— aj+^jB* 9. I 1 10. -^+-^+ JL. 11. 0^+1 ^+^ ^+8 2 8 _ 8 12. 14. 16. 2a :+ ip 18.^-- x" x^ a»^^* a+b a-h 1-x {1- xf (l-a?) 8a— 45 2a— i— c , 15a— 4c 12 a +i« H--^- a"— i* a+6 -a— 6 16. -4 2 ,l+d> 1- o; a? 1— a? 1+a? 17. 18. 19. ^0. 21. 6- c , a^+l s,c»— 1 i»»+l (a-i) (a-c) (a?-a) ' (6-c) (6-a) (x~h) (c^d) (c— () (a?— c) a u '•> »l iii 1;'! 98 ELEMENTS OF ALGEBRA. 161. Since a whole expression may be considered as a fraction whose denominator is unity, the snm or difference of a whole expression and a fraction may be expressed as a fraction ; and, conversely, a fraction may sometimes be expressed as the sum of a whole expression and a fraction. 1. 2a^-±. X 1 Examples. 2aj» l_ 2a;»-l X X 2. 8a- 1 + 8. .iJ^-l- 8a-l 1 9a« 8a+l 1 8a+l 8a+l x—1 a^—l x—1 cd^—x x'-^l 1 a^+1 ««+l . ab4-c ah . c „ , c ■ x^—x-\-l a?"— a?+l Here the numerator 2^:^— a;— 2 is the renudnder when 2a;*— a?*+2a?— 8 is divided by a?"— aj+1. EXEBOISES, XLII. Find the fractions equivalent to 1 ^ , y a o I 1 I ^ 1. _-f£— 2. 2. ax-\- — + — . bx ca^ 8. X" y X a^-1 x—y 4. ar-^1- 8 a?+l 6. 8ar+2-T£±6. 6. a^4-^y+2/»-^^. 2a?+3 «-y ELEMENTS OF ALGEBRA. 99 Beduce to the sum of a whole expression and a fraction !• — ;r • 0« ^ — • «7* • 2a?y 2f aj«-8 10. 2fla- a+1* * a»+l 12 ^•^'-Ba^+a^'+l a;»»+l mm mi m m u m 4 162. To find the prodtict of two or more fractions. Let there be two fractions -— , ^^. B D A C Then since _..B= A, and ~-.D8bO» the product of B D A C the four quantities—, B,— ■, D is equal to AC, that is B 1) 4- w- BD=AC. B D Therefore^. £=.^^ B D BD In like manner the product of three or more fractions may be found. Hence the product of any number of fractions is a fraction whose numerator is the product of their nume- rators, and denominator the product of their denomi» nators. 163. The case of the product of a fraction and a whole expression is included in the preceding Article o2 mm «^ '' f 100 ELEMENTS OF AiaEBRA. by considering the whole expression as a fraction, whose denominator is unity. Examples. m a-{-b a* _a* (a-\-b)_a*-\-db a a~b a{a—b) a—b 8 (%-^^^^\^ -l_ 8a!'--8-2a^+4 a? -1 ' \ a;"-f/d?>+l d^-1 ' ira+l _^+l a?-l_ 1 ExEBOisES, XTiTTT. Find the product of 1. 8. l-x* 1+x' a*-b*: a-b aa-2a6+ft«* o«+a6* 00^ 18ca«/* a*— ax 9bf* 1^* a«-^' a'— a;" * ac+6ca?' 4. 2w"n m*+mn m' ;'—»** 6mn* 6. 7. 8. m'—wiw+n' m*—n* m"— 8mn(w— n)— n'* m^+n"* (a?-y)(a^-a;y+y') (a?+y) C^+a^y+y*) i»»+2a!'i/+2a?i/'+2/' a?— y aj"— / * a?+y * Simplify ^■''-i-^«'+^^-e«). 06 ^"' i2(^^"2(a+6)"*"a«-W 26 ' ELEMENTa OF ALGEBRA, m 164. Since the quotient of one fraction divided by Another is the fraction which, multiplied by the latter, is equal to the former, it follows that the quotient of <^ne fraction divided by another is the product of the former, and the reciprocal of the latter. -,, AOADAD, ADCA ^^^« B"^ D^B-O-^BO'^^^^^'^BCD^B- Hence also to divide a fraction by a whole expres* sion we must multiply its denominator by the latter. Thus:^-C=^-^=A1=^ 1. 8 x-y x-\-y Examples. x-y' 8 8(a?-y)* 2 ^-y' ^.^ ^-y _ a^-y* xy _ xy(x-\-y) s^+y^ ' xy x^-\-y^'x—y a;*+2/* (a?— 2/) a +y) ii«i#>i i ;te: m 'M :«\if »! i^"' It:' ■il'^lll 108 ELBMENTB OF ALGEBRA. Divide EXBBOISES, XTjIV. 1. a nB—y i,„a?+y a by a*b ai» a ^- a^-y' ^^ a^-^ab by x-y {^-{-ff -^ a^+y^' 4. (a^bf by o'-ft' a—b b—c a^- 6a? y a^_6a?4-6 * Simplify W— t/" ar^+w^i ' la?— w aj+v) 7. ■y a^+f) ' (x-y x-\-y)' ^- (a« + o6 + 6« "^ a* - aft 4- 6») * la'' + a* + 6? ? + o6 + 6« ' a*-a6 + 6«r a'— a64- SURDS. 165. Those roots of quantities of which the nu- merical values cannot be exactly expressed are called irrational quantities or surds Thus, for example, V2, -^6, ^7 are surds. 166. When a surd is combined by Addition, Sub- traction, Multiplication, or Division with a whole number or a fraction, evidently the exact value of the combination cannot be expressed. Such quantities are also called surds. Thus, for example, 3+ v'2, 5- Vs, 7^4, i^lO are surds. ELEMENTS OF ALGEBRA, 108 167. The quantity under the radical sign is called the haae* Thus 2 is the base of ^2, and 5 of '^ 6. 168. The number which expreaees the root to be extracted is called the radical index. Thus 2 is the radical index of Vs, and 3 is the xadical index of 6 ^^4. 169. A surd whose radical index is 2 is called a quadratic surd, and one whose radical index is 3 a cMc surd. 170. Similar surds are those which have, or can be made to have, the same base and the same radical index. Thus 6'^2 and 8 "^ 2 are similar surds; whereas V 4 and V i are dissimilar surds. 171. Since a=a'*= V^a", it follows that any rational quantity can be expressed in the /orm of a surd. Thus 2= ^28= ^8 and 5= V25: 172. When the radical index of a surd is the pro- duct of two or more integers, the radical sign and index may be equivalently replaced by a combination of radical signs and indices. Thus ^a=a^=(#= V ^= V V«. In like manner '\/ a ma y be writt en in the equiva- lent forms V^ V ^* \f V ^»- Thus 101 ELEMEWrS OF ALOISBRA. 1 \ 173. Since aV'6=a6*=(<*"^)" = V^a"6, it follows that a rational multiplier of a surd may be brought under the radical sign by multiplying the base by the rational factor raised to the power indicated hj the radical index* Thus 6/2= V6«>c2= V5O; 2^i=^4^^x^=^9^ 174. Conyersely, a surd whose radical sign is n can be reduced to another whenever the base is a multiple of a complete nth power. Thus V48=V^4*x8"=4V^; V'eOrs V'2«xl6=» VI6; ^40=^^x6=2/^5.. m 175. Since >«C^6» ^ ^^126; *'3=''y8i=5^8n , 176. Conversely, a surd whose radical index is mn can be reduced to one- whose radical index is n, if th& base of the former is an exact nth power. Thus ^9=^/8; ^»='^2»"=V2; ^16= ^2*" = V2l 177. A surd is said to be in its simplest form when the base is neither a power indicated by any measure of the radical index, nor a multiple of a power indi^ cated by the radical index. Thus ^6 is in its simplest form because 6 is neither an exact cube nor a multiple of an. exact cube; ^Itt ELEMENTS OF ALGEBRA. lOS is in its simplest form because 10 is neither an exact square, cube, or sixth power, nor a multiple of an exact sixth power. Exercises, XLY. 1. Bring the rational factor under the radical sign in 8^2,2^7, 8^6. 2. Simplify ^48, '^lOaT 8. Reduce Vs, Vs to surds whose radical index Z&6. 4. Reduce ^^2, ^^3, '^5^6 to surds whose radical index is 9. 6. Simplify ^9, '^1257 178. The Algebraic sum of any number of surds is expressed in its simplest form by simplifying the several terms> and collecting similar surds under one term. 179. Surds are said to be simple or eompotmd according as they contain one or more terms. Examples. 1. 6V8+V1O8 =5V8+6V8=(64-6)V8 =11^3 2. V6+8^6i-2'^16=V6+9'^2-4^2=V'5 Exercises, XLVI. Simplify 1. 8V2+4V8-V82. 2. Vi2+V75-V48 m .-..m ^1 m m / :ii ■ ■■, i'%L 106 ELEMENTS OF ALGEBRA. 8. 2^4+5^^82-^1087 4. ^108"+ 2^676. 6. 8^64-2^16. 6. -v/48-V8-|^72r+2^27 180. To find the product of. two simple snrds ^a and Vft. Let I be the L. 0. M. of m and n, so that l=:mp=nq. Then VI Vb =' hn^g. hi ={dP.h9) i= 4^^^ 181. The products of simple surds are also some- times expressed as follows : 11 11 In like manner ^a ^ h ^"c may be written J^a \Jb V'cT Thus, V2.^8">C^6=\/^ V^~VT Conwsely, ^a s^iT^c^ ^- ^l ^1 Examples. 1. y''* *s/b= \fab. Heret 2. Va 8. •# f- I nil m tnumima of Aiamiti. quantities, x the unknown; and in a!'+pa?+3=0, p and q are known, as unknown. 192. Quantities whicli, on heing substituted for the unknown, reduce the equation to an identity are said to satisfy the equation, and are called its roots. Thus 5 is a root of 2^7—85=7, because 6 when sub- stituted for X reduces the eq^uation to the identity 10— 3=*7. So 2 and 3 are the roots of a^+6=5a?, because when either is substituted for x the equation is satisfied. 193. The determination of the roots is called the «oZt££io9t of the equation. 194. An equation is isaid to le redticed to its simplest form when its members consist of a series of mononomials involving positive integral powers only of the unknown. Thus a?+l=a?* is reduced, by squaring its sides, to its simplest form a^4-2a' -f-il=a?; and a?— a?-^=2 is reduced by multiplying its sides by « to its simplest forma;^ — l=£2a?. 195. Equations when reduced to their simplest forms are classified according to their order or degree. 196. The right-hand members of the standard forms are generally made 0, and the term independent of X is called the absolute term. Thus - 8 is the absolute term of a^+ 4a? - 8 = 0. 197. Simple Equations, or those of the first degree^ are those in which the highest power of the unknown quantity is the first. Their general form is ax-j-bssO* ELUmiNTS OF ALGEBBA. U8 198. Quadratic Equations, or those of the second «legree, are those in which the highest power of the unknown quantity is the second. Their general form is 199. Equations of the third and fourth degrees are called Cubic and Biquadratic EquaUonSf respectively, their general forms heing And, generally, if the highest power of the unknown is the nth, the equation is said to be of the nth degree. 200. The coefficient of the highest power of x can always be made unity by dividing both sides of the equation by the coefficient of that power; so that the general forms of simple, quadratic, cubic, &c., equa- tions may be written a?+j»=0, a^-\-px-\-q=:0, (if-\-px^-\-qx-\-r=:0, &C. 201. It is proved in works on the Theory ofEqua' tions that the number of the roots of an equation is equal to its degree; so that a simple equation has one root, a quadratic two roots, a cubic three, and so on. 202. In order to solve an equation it is generally necessary to reduce it by one or more of the following processes: Transposition of Terms, Clearing of Fractions, Glearing of Surds. 11 itf' '/'SB !'('■ 'I'M .; "1: ■ 'Ilk U4 ELEMENTS OF ALGEBRA. These operations will be illustrated by applying ihem in order to the solution of simple equations. TRANSPOSITION OF TERMS. 203. If an equation contains neither fractions nor surds, it may be solved by transposition of terms» which consists in taking the unknown quantities to one side of the equation and the known to the other side, the sign of the quantity which is so transposed being changed. Thus if the equation is ax-\-b=cx-\-df by adding —6 to each side we get ax-\-b — b=cx-\-d-~bf that is «a?=ca?+d— 6. and so any quantity may be transposed from one side to the other by changing its sign. Therefore that is aa;—cx=d-'b, _ (a— c) xsnd—b ; and the equation is solved by dividing each side by a— c, the coefficient of x. d-b a— c Therefore x-. Thus -J^ is the root, and the equation is solved. a—c Examples. 1. Solve the equation 6;v+10=15. ELEMENTS OF ALGEBRA. 116 Here 4-10 is to be taken to the right-hand side of the equation, which is then reduced to the form 6a?=16-10=6; divide by 5, the coefficient of Xy and we get a?=l. 2. Solve7a?-f2=llf4a:. Here +2 is to be taken to the right-hand side and ^\x to the left-hand side ; hence we get, at one step, 2; * 7a?-4a?=ll that is 8a?=9; therefore a?=8 8. Solve 2(8^-l)-6(2a?-fl)=20-7a?. Here the left-hand side must be reduced to a series of mononomials by removing the brackets ; thus 6a?-2-10a?-6 = 20-7a? ; transpose —2, —6 to the right-hand side, and —Ix to the left ; therefore 6a? - 10a? + 7aj = 20 + 2 + 6 ; that is 8a?=27; therefore a? =9. 4. Solve a? (a?-8)+2 (l-a?)=a!»-8a?-6. Bemove the brackets ; thus ar'-8a?+2-2a;=a?'>-8a?-5 : strike out a?* and — 8a?, which are common to both sides; thus 2 --2a? =—5 transpose — 2a?=— 6 — 2 that is -2a?=-7 divide by —2; therefore a?=8i. B 2 116 ELEMENTS OF ALGEBRA. 6. Solve (.V - 1) (ur+1) - 6;ir»+ 8ii? = (8 + 2;»)) (6 - 2w) -20. Porform tlio multiplioatioiiR indioatod ; tlms iB>_l_5j,4i+8u;=15+4;i7-4a^--20. Transpose, .'c«-5.t'«+ar-4.r+4.j?"=16-20+l ; that is, —0?=— 4; therefore a? =4. 6. Solve a {x—a)-\-b (;»— />)+.i^=(iP— a) (a;— 6). Perform the multiplications indicated ; thu strike out ;»", which is common to botli sides, and trans- pose ; thus ax-\-bjo-^a^-\-bx=:a*-{-b*-^ab; collect coefficients of a;, 2(a+6)i»=rt«+6«+a6; divide by 2 (a+6) ; therefore a;=^+*!±f. ^ 2 (a+o) ' Exercises, L. 1. 8a?-7=a?+8. 2. 8a?4-9=5aj+18. 5. 16aj-10=41-a?. 4. 5-19a;=2+lla7. 6. 21aj+7=4 (a?~8)+8aj+61. 6. 8(2~a;)+7(l-2a?)=:87-28ay. 7. 8(a?-6)-6(a?-8)=21a>-46. 8. 2(8. 0. Or- 10. 2.f(i +10. 11. (X- —a?. 12. (.1?+ 18. (1- 14. rtaj= 16. ax-\- 10. X {x 17. (.i* — 18. (rt — 19. (x+ 20. (X- miEMENTa OF ALnEimA 117 8. 2 (8.»-l)-17-:B (.it+2)-7 (8.H-1). 0. 0«-l)(aj-2)4-(^-l)0f-»)-=2(.w-2)(.r-H). 10. 2.t' (2jj+7) - 4 (;w -f a) - 11 = (1 - iv) (8 - 4.*') + 2x 11. (ir ~ 1) (o) - 2) + (a? - 2) (flJ - 0)-2 {x-l)(iK - 8) 12. (.'»-|-2)»+7(8aj-l)=0(2;«+4)+(a;-2)M-8. 18. (1-2jj) (l-8j?)-28 = (0a;+l) (.i'-l)-8a;. 14. (UV=:hx-\-C, 10. X {x — a)-\-itt (in—h) — 2 (x—a) (x—b), 17. (x-a) (a; -/>) = («? -«--/>)". 18. (rt — a;) (h — x) = (j)-\-x) (q-^x). 19. (.^+2tt) (d?-rt)'' = (a;H-2/>) {x-b)\ ^0. {x-aY {x-{-a-2b) = (x-by {x^2a+b). CLEARING OF FRACTIONS. 204. If an equation contains fractions, it may be reduced to a form capable of solution by transposi- tion, by multiplying each side of the equation by the L. C. M. of all the denominators of the fractions. y »., 118 ELEMENTS OF ALGEBRA. Examples. 1. Solve ^-^=^+1. 2 8 6 Here 80 is the L. C. M. Multiply each side by 80 ; thus 15a?-10a?=6aj+80; therefore — a?=80; therefore aj=— 80 2. solve ^+?^8^6^1_9; Zoo Multiply each side by 24, the L. CM. of 2, 8, 8; thus 12(a?-l)+8(2a;+3)=8(6a?+19); the solution of which is a; =4^. 8. Solve ?-A+l= 1+8. X *Ix ox bx Multiply each side by 60?, the L. C. M. of x, 2x, QXf 6x ; thus 80-9+2=7+480?; whence x=^. 4. Solve K+ ' x-1 x-\-4: 2a? -2 8a; +12* Multiply by 6 (a?-l) (a?+4), the L. C. M. of the dfir nominators ; thus 18 (a?+4)+12 (a?~l)=21 (a?+4)+8 (a?-l) ; whence a? =16* ELEMENTS OP ALGEBRA. 119 5. Solve b _l + 6 00 x-\-a x-\-b Multiply by x (x+a) (a?+6) ; thus (x-\-a) (a?+6)+ bx {x+b)=x (1+6) (x+a) ; whence, on reducing, we obtain {b^-\-b — ab)x=—ab; therefore a?= —r«^ =_Z^ = _^_. U'-^b-ab b+l-a a-6-1 EXEBGISES, LL 6^7 8 2. ?4--=3-?^. 8 6 5 ■n« :?^ Ili''' a ^-8=74-^. 9 12 4. ?+^±l=a:-2. 2^ 7 6. x—l_x—2_xS ~^ "8 4~' 6 8a?+l _ 4a;-l _ 2-a? 2fl?-5 * 18 5 2*8' 7. I (^-8)+| (^-9)-^ (^-11) =7-|(a?-17). s i^ 8. 8ar-7_ 8a?-14 4a;+2 4a;-18' HI I 120 ELEMENTS OF ALGEBRA. 9. 1?+JL=E X 12a? 24 10. a?+l-^^±?=2. a?+2 11 7'a>+lg _ a?+8 _28 , ar 21 4a;+10 70 8' a c b d ' 2a;+l "8^ x ' 15. 8 a?— 1 a?— 2 a?— 8 16. x—a.x— 6_a*+ ^ a a6 17. 8a?~l _ 4a?-2 _l 2a?- 1 8a?- 2 6* 18. 2 :+-^= 6 2a;-8 a?-2 8a;+2 19. 8+a? ^ 2-^a? _ l+a? _|^ 8— a; 2— a? 1— a? 20. a^—a* a~-x 2a? a bx X 21. a— 6 a?— a a?— 6 a^^ab ELEMENTS OF ALGEBRA. 121 22 a!-~a _iff-{-a^ 2ax 28. (-f±^Y=^±fL. \2^+l/ 4a?+a 24. /"^-A'-^-g^ \a?-6/ a?-2& CLEARING OF SURDS. 205. If an equation contains one surd, the radical sign may be removed by bringing the surd to one side, and the remaining terms to the other, and then raising both sides to the power ' indicated by the radical index. The equation, if of the first degree, can then be solved by the previous methods. U Ai:;^ III i Examples. 1. Solve \/iB"_6a;+8-a?=2. Transpose, Va;"— 5a;+8=a;+2; square both sides, a;"— 5ii?4-3=iir'+4a;+4; whence a?=— ^. 2. Solve ^a;»+8ar»-5-a?=l. Transpose, ^a^+8a^-6=a;+l; cube both sides, a:»4- 8a;»- 5 = aj»+ 8a;»+ 3a7+ 1 ; whence a;=— 2. 206. If the equation contains two quadratic or two cubic surds it may be reduced to an equation contain- 11' .*"a iiiii 122 BLUMENT8 OF ALOEBBA. ing but one surd (i.) by bringing one surd to one side, and all the other terms to the other side of the equation, and then raising both sides to the power indicated by the radical index; or (ii.) by bringing the two surds to one side, and all the other terms to the other side of the equation, and then raising both sides to the power indicated by the radical index. Examples. 1. Solve V2xH-8- V2x^S=:2. • . Employing the first method, we transpose — Via^S to the right-hand side ; thus V2x+S=^ V'2a?-8+2; square both sides, 2a?-f3=2aj-8+4+4 V^x-d; that is, 2=4^2^37 whence a?=l|. 2. Solve V2x-l+V2a:-j-i=:5. Employing the second method, we square both sides as they are ; thus 2a?-l+2a;+4+2 \/4a^+6aJ-4=26; transpose and divide by 2, v'4ar»+6a?-4 = ll-2a?; whence a;=2^. 8. Clear of surds, ^l-x-\- ^8i^=S. Cube both sides by the formula of Art. 110 ; thus l_aj+8+aj+8>v/(l-a;)(8+a;){>v^ri^+\^ 8+^ = 27} substitute for v'l-"^4. \/8+a? its value 8; thus 9v'8-7a?--a!»=18; BLEMENTa OF ALGBBBA. US divide by 9, cube, 8-.7a?-x"=8; a quadratic equation. 207. If an equation contains three quadratic or three cubic surds, it may be reduced to an equation containing but one surd by bringing two surds to one side, and the remaining surd to the other, and then raising both sides to the power indicated by the radi- cal index. Examples. 1. Clear of surds ^a?+4+ V2^+9= ^8a;+26. Square both sides, a?+ 4 + 2a;+ 9 + 2 ^2^+17^+86"= 8i»-|- 26 ; transpose and divide by 2, ^2^+17^+86"= 6; square, 2a!»+17a?+86=86 ; that is, 2a^-|-17a?=0, a quadratic equation. 2. Clear of surds 4^a+ ^b+ >^c=0. Transpose, ya+^b=: — ^c; eabe both sides, substitute for ^a+^b its value —^c, a+6-8'^a6c"=~c; transpose, a-\-b-{-c=S^abc; cube both sides, {a+b+cY=21abc, m i 1^ i '' l:m !l ■*: 'n 194 ELBMEma OF JLLOEBRA. Exercises, LII. Solve or simplify 1. v'4a?» 4- 8a? -T6 = 2a;+ 2. 2. v'^+2*+8=a?-6. 8. V'a?«+8a?+4 + 10=a?. 4. V. 6. -V^ ^+4 +^^+16 = 11. . 6. V^^36-'/^I^=l. 7. V^^^S^ ^^+6= v'4a?-6. 8. i^^T8+V^a;+8=2v/a?. 9. V^a?+6+ v'a;-3=2\/ar. 10. V^a;»+aa?+6«+c=a?. 11. V 007— V^6i»=a— 6. /- / 2 12. Va?+Va?+1=— p===. ^a?+l 13. -Z^+^^+S -7==. 14. a/^- "V^a- V'^+^= -/^ ^12^+1+ Vi2^ *"• T7s= — 7 = 1+ ^ ■ VSaJ-t-l 2 ELEMENTS OF ALGEBRA, 126 18. a3«+aV^«""^^«=:d?{a+l/i^"IT"}. PROBLEMS. 208. Whon a question is assigned for solution the unknown quantity or number is generally involved in the various conditions which are proposed for its determination. The expression of these conditions in Algebraical language leads to an equation, the solu- tion of which will be the solution of the question. 209. In some cases, although there are more unknowns than one, they are related to each other in such a manner that when one is determined the others become immediately known. In such cases the unknowns can be expressed in terms of one unknown. • Thus, if the sum of two unknowns is equal to 8 we may denote one of them by x, and the other by 8— a:; if the greater of two unknowns exceeds the less by 3, the former may be denoted by x, and the latter by a? — 3 ; if the product of two unknowns is equal to 12, one 12 of them may be denoted by a?, and the other by — ; if there be two numbers, of which one exceeds 4 times the other by 7, the former may be denoted by 4a?4-7, and the latter by a?. In like manner, if there are three unknowns, of which the first exceeds the second by 3, and the second exceeds the third by 5, the first may be denoted by a?, the second by a?— 3, and the third by a?— 8; if there are three unknowns, which are to each other as the numbers If 8, 6, Uiey may be denoted by x, 8a;, Bx, '. ,!' ELEMENTS OF ALGEBRA. 210. The foUomng examples will illustrate the method of solving problems by means of simple equations of one unknown. 1. What is the height of a house wall in which a window 6 feet high has under it i, and above it i of the whole height ? Let the height sought =a; feet. Then under the window there are ^x feet, and above it ^x feet ; x=S6. 2. How may a debt of £5 be paid with 29 coins, some of them croWns, and the rest florins ? Let there be a? crowns; then there are 29— a? florins. The value of the x crowns is 5x shillings, and the value of the 29— a? florins is 2(29— a;) shillings; .-. 6a;+2(29-ar) = 100; ^ a^=14, 29-aj=16. Thus a debt of £5 can be paid in the required way only with 14 crowns and 15 florins. 8. If A can perform a given work in 120 days, and B in 80 days, in how many days will A and B, working together, be able to perform it ? Let the whol') work done be denoted byti;, and the re- quired number of days by x; then amount of work done by A in one day = m; 120' w tt work done by B in one day = ^jr ; w work done by A and B in one day ss - ; 9 ELEMENTa OF ALGEBRA. 127 therefore - = as w w ^ 120 "^80* divide by w, - = 120 "^ 80 ' a?=48. , Thus A and B, working together, can do the work in 48 days. 4. A number consists of two digits, the first of which is greater than the second by unity, and the sum of the digits is one-sixth of the number itself. Let X denote the second digit, then x+1 will denote the first. The number is, therefore, 10(a;+l)+a;r=llaj-flO; and the sum of the digits is 2a;+l ; ^, - lla?+10 „ . ^ therefore — ^ — = 2a? + 1 ; o • aj=4, a?+l = 5. Thus the number is 54. 5. One hundredweight (112 lbs.) of bronze contains by weight 70 per cent, of copper, and 80 per cent, of tin; with how much copper must it be melted in order to contain 84 per cent, of copper ? Let the amount in lbs. of copper be denoted by sc; 70 then in the cwt. there will be zr^ . 112, or 78*4 lbs. of 80 copper, and j^ • 112, or 83*6 lbs. of tin. Therefore the whole amount of copper in mixture will be 78'4+a;, and this is to be 84 per cent, of the mixture, which weighs 112+ a? lbs.; 84 therefore 78-4+a;= — (112+a;); a?=98. Thus 98 lbs. of copper must be added to the cwt. w 11 I J "II ,'li- ■-M m i; '*/p| ' Itil I i im ULEMENTS OF ALGEBRA, 6. To find at what time between h and h+1 o'clock the minute-hand is m minute divisions before the hour- hand. Let X denote the number of minute divisions between the mark h and the position of the hour-hand; then the number of minute divisions between the mark h and the minute-hand will be m+a?, and between the mark 12 and the minute-hand 5h-\-m-{-x. Therefore the number of minute divisions passed over by the minute-hand since h o'clock is 5h-\-m-^Xj and the number of minute divisions passed over in the same time by the hour-hand is x; but, in the same time, the minute-hand passes over 12 minute divisions for the hour-hand's one; theref ore 5 A + wt + a? = 1 2a? ; 5h-\-m X=i 11 therefore the required time is 5h-\ -^ |-m, or 12 Y^ (5^-fm) minutes past A. Thus the time between 2 and 8 o'clock when m is 15, that is, when the hour and minute-hands are at right- 12 angles to each other, is ry (10+15), or 27x^ minutes past 2. ExEBGISfiS, Lm. 1. The Bum of two numbers is 10, and their differ- ence 4; find the greater nuinber. 2. Divide 80 into two parts, such that one may be two-thirds of the other. ELEMmrS OF ALGEBRA. 129 8. The difference of two numbers is 3, and their product exceeds the square of the less by 12 ; find them. 4. Find the number to which, if its third part be added, the sum will exceed its half by 5. 6. The denominator of a certain fraction is one less than the numerator, and twice the fraction added to three timep its reciprocal makes 5; find the fraction. 6. Divide £34 4s. into two parts, such that the number of crowns in. the one may equal th"e number of shillings in the other. 7. A and B sat down to play. A had seven shillings more than B, but, after losing ten shillings, finds that he has only half as much as B. How much money had A and B originally ? 8. A person invests two-thirds of his property at 4 per cent., one-fourth at 8 per cent., and the remainder at 2 per cent.; his income is £430; what is his pro- 9. If B gave half of his money to A he would have only a quarter as much as A; but if A gave B £60 he would have only Imlf as mu€h as B. How much have A and B, respectively ? 10. A and B set out at the same time to meet each othel*. A, travelling 5^ miles an hour, meets B travel- ling only 3^ miles an hour, 3 miles beyond a midway station ; what is the distance of the points from which they started? , , 11. How much wine at 15s. a gallon must be mixed with 20 gallons o^ wine at £1 a gallon to make a mix- ture worth 17s. a gallon ? 12. A mixture is made of a gallons at p shillings, b gallons at q shillings, and c gallons at r shilhngs; ^hat will be the value per gallon of the mixture ? 18. A cistern is supplied from two taps, by one of ^hich it is filled in 45 minutes, and by the other in i p i ! 'if 130 ELEMENTS OF ALGEBRA, 76 minutes; in what time will it be filled by both togetlier ? 14. A starts on a journey 20 minute^? before B ; A walks at the rate of 4 miles an hour, and B at the rate of 4^ miles an hour; at what distance along the road will B overtake A ? 15. A, who walks at the rate of 8f miles per hour, starts 18 minutes before B; at what rate per hour must B walk to overtake A at the ninth mile-stone ? IG. A and B start to run to a flag-staff 450 yards off, and back. A returning, meets B 80 yards from the flag-staff, and arrives at the starting-point half-a- minute before B; how long did A take to run the whole distance ? 17. Divide 24 into two parts, such that their sum shall be to their difference as 3 to 2. IB. Divide 80 into two parts, such that their sum shall be to the difference of their squares as 1 to 6. 19. A number consists of two digits, the first of which is less than the second by 2, and if the difference of the squares of the digits be subtracted jfrom the number itself the remainder is 19 ; find the number. 20. A bill of £100 was paid with 202 coins, consist- ing of crown pieces and half-guineas; how many of each were used ? 21. What is the first time after 7 o'clock when the hour and minute hands of a watch are exactly op- posite ? 22. A watch gains as much as a clock loses, and 1798 hours by the clock are equivalent to 1802 hours by the watch ; find the error in each per hour. 28. At what times will the hour and miimte hands of a clock be together during 12 hours ? 24. A hare pursued by a greyhound is 60 leaps in ELEMENTS OF ALGEBRA. 131 advance, and makes 9 leaps while the hound makes 6, but 3 of the hound's are equal to 7 of the hare's. How many leaps must the hound take to catch the hare ? 25. A steamboat which can travel at the rate of a miles an hour, in still water, goes from one station to another with the current in t hours, and goes back in t' hours ; find the velocity of the current in miles per hour. ^ 26. In the previous question, if the distance between the stations be 19^ miles, the time of going down the river 1 hour 18 minutes, and up the river 2 hours 10 minutes, calculate the velocity of the current, and the rate of the steamer in still water. 27. A certain number of sovereigns, shillings, and sixpences together amounts to £8 6s. 6d., and the amount of the shiUings is a guinea less than that of the sovereigns, and a guinea and a half more than that of the sixpences ; find the number of each coin. 28. Two minutes after a railway train has left a station. A, where it had stopped 7 minutes, it meets the express, which set out from a station, B, when the former was 28 miles on the other side of A; the express travels at double the rate of the other, and performs the journey from B to A in an hour and a half: find the rates at which the trains travel. 29. The circumference of the fore wheel of a carriage is 10 feet, and that of the hind wheel 12 feet ; the former Jias made 100 more revolutions than the latteri how many times has the hind wheel revolved ? 80. The epitaph of Diophantus, the celebrated mathe- matician, states that he passed the sixth part of his life in childhood, and the twelfth part in the state of youth ; that, after an interval of 5 years more than one-seventh of his life, he had a son who died when he had attained to half the age of his father, and that the father sur- vived the son four yet^s. Find, from these data, the age of Diophantus. i2 '!'■■■<(:] 182 ELEMENTS OF ALGEBRA. 81. A number consists of two digits, of which the first exceeds the second by 4 ; and when the digits are reversed in order, a number is obtained which is four- sevenths of the former. Find the number. 82. A steamer makes a journey of 2,568 miles in 9 days ; for 3 days she is retarded by winds and cur- rents at the rate of 3 miles an hour ; for 4 days she is helped at the rate of 2 miles an hour ; and for the remainder of the time her speed . is due solely to her steaming power. What is her rate in still wind and water ? 83. The hour is between 2 and 8 o'clock, and the minute hand is in advance of the hour hand by 14^ minute spaces of the dial. What o'clock is it ? 84. A man at his death leaves £5,850 to be divided among his family, 'which consis''jS of 3 sons, 4 daughters, and his widow. Twice the widow's share is to be equal to the share of a son and a daughter, and the share of two sons is to be equal to that of three daughters. Find each person's share. 35. A quantity of leaden shot is shaken on a sieve ; twice as many grains go through as are left behind ; what remains is shaken on another sieve, when three times as many go through as are left ; what remains ir, shaken on a third sieve, when four times as many pass through as are left. The number of grains which is finally left is 100. Find the number of grains of each size. 36. In one specimen of gunpowder there is n per 'cent, of nitre, s of sulphur, and c of charcoal ; in another n', s', and c' of these ingredients, respectively. If IV lbs. of the first be mixed with w' lbs. of the second specimen, what will be the per-centages of each material in the mixture ? 37. Gun-metal is composed of 90 per cent, of copper and 10 per cent, of tin. Speculum metal contains ^7 per cent, of copper and 38 of tin. How many cwt. ^112 lbs.) of the latter should be melted with 8 owt. of ELEMENTS OF ALGEBRA. 133 the former in order to make an alloy in wliich there is three times as much copper as tin ? 88. A garrison of 500 men is provisioned for 60 days. On the 14th day they lose 80 men in a sortie ; on the 84th day they lose, by the explosion of a mine, 2G men and 2,000 rations ; after a week they receive a reinforcement which enables them, by reducing the rations one-third; to prolong the defence until the Tlst day, when they are relieved. What was the number of men in the reinforcement ? 89. One half of a population can read ; of the re- mainder, 42 per cent, can read and write ; of the re- mainder again, 16 per cent, can read, write, and cipher, while 248,600 can neither read, write, nor cipher. "What is the population ? 40. A person possessed of £5,222, invested a part of his property in 6 per cent, stock, which he bought at 105, and the rest in 3 per cent, consols, at 96. How much did he invest in each kind of stock, if his annual income amounts to JE191 16s. 8d. ? QUADRATIC EQUATIONS CONTAINING ONE UNKNOWN. 211. The general form of a quadratic equation is in which p and q are supposed to be known, and x an unknown quantity, whose value is to be expressed in terms of ^ and q» 212. Quadratic equations are called adjected or pure according as the term involving the first power of the unknown quantity does or does not appear. Thus .r^- 2^7+ 8=0, 6j72-6a?=0, are adfected quad- ratios ; 2a^— 6=0, a;»^+&=0, are pure quadratics. ii* .^ 134 ELEMtJNTS OF ALGEBRA. 218. The solution of a quadratic equation, whose right-hand memher is zero, can always he immediately effected if the left-hand member is in the form of the product of two factors, each involving the first power of the unknown. For if the product of two factors be zero, one or other factor must vanish. Thus, if (iP— 1) (a?— 2) = 0, it follows that either a?— 1=0, and therefore a?=l; or a?— 2=0, and there- fore a; =2. Thus the roots are 1 and 2. Examples. 1. Solve a!"-2aj=0. Herea?(ar-2)=0; therefore, either x=0 ; or a?— 2=0, that is x=2. Thus the roots are and 2. 2. Solve (2a? -3) (3a?-f-l)=0. Here we have 2a?— 8=0 ; and therefore a?=| ; or 8a? -f 1 =0 ; and therefore a?= — J. Thus the roots are | and — i. ... 8. Solve (ax+b) {cx+d)=0. vr Here aa?+6=0 ; and therefore a?= — -; a or ca;-{-d=0; and therefore a?=— -. e Thus the roots are — - and — — a c 214. The substitution of each root gives rise in general to two different identities. ELEMENTS OF ALGEBRA. VM Thus, in the last example, if — - be substituted in a the equation for a?, the identity will be {-a}--^b\ (-c}-{-d\=0; but if — - be substituted for a?, the identity will be c /-a.--\-b\ {-c.i-\-d\ = 0, 215. A pure quadratic, as x^—a'^z=:0, may also be immediately solved by transposing and extracting the square root. Thus, x^=a\ Here, on extracting the square root, we get +a;=+a; — a;=+a; or, ^x=—a ; amongst which equations, it will be observed, the first and last are equivalent, as are also the second and third. Hence the four equations may be combined into the two or, x=—a ; And the solution may be written in the form 216. If the given quadratic be ax^-\- 6=0, we have ^ 6 h If now in — -, a and h have opposite signs, the right- a hand side — - will be positive, and x will be either a Y 13C ELEMENTS OF ALGEBRA. 1 positive or negative. If, however, a and b have the same sign, x can neither be positive nor negative, since the square of any positive or negative quantity is posi- tive. In such a case x is said to be imaginary and the two imaginary roots are written in the form Conversely, it will follow that the square of + ^ — , or -V-^ will be -1 ^f a a • Thus the square of 4- -/ — 1, or — y' — 1 is —1. 217. The solution of an adfected quadratic, as is effected as follows : transpose, a^-^px=—q; add |- , the square of one-half the coefficient of a?, to both sides. 4 4 The left-hand side is now a complete square, and the equation may be written (.+iy=p'-4.. Extract the square root, , P . 1 / ^+2==^2^"-42; transpose ar=-| ± | v'^2_4j. Thus the roots are - | + -- Vp^-^, and- 2 -2^/^*-4g. ELEMENTS OF ALGEBRA. 187 The preceding method is called the Italian method, having been used by the Italians, who first intro- duced a knowledge of Algebra into Europe. 218. If the given equation be of the form it may be reduced to the standard form by dividing both sides by a, thus a!»+ -x-\--=0, a a which equation may be solved by the Italian method. 219. The equation aa!^+bx-^c = may also bo solved by the following, called the Hindoo method : transpose, aa^-\-bx=—c; multiply each side by 4flf, 4:0^0^ ■\-iabx=—4iac; ' j add b* to each side, ia^x^-{-4abx-{-b^=P-iac. The left-hand side is now the square of 2ax-\-b, and the equation may be written Extract the square root 2ax+b=±Vb^-4^ac; transpose and divide by 2a, ^= ^ 2a -• Examples. 1. Solve a^-+-9=0. _ .;• Transpose ar* = — 9 ; extract the square root, x=± V~9= ±QV — 1. Thus the roots +8 v^^, — 8 V-O. are imaginary. V' i 188 ELEMENTS OF ALGEBRA. 2. Solve a!"-f 4a+8=0. Transpose a^+4aT=— 8; add to each side 4, which is the square of oiie-half the coefilicieut of d?, extract the square root, therefore, ir=— 2±1=— 1, or —8. Thus the roots are — 1 and — 8. 8. Solve a?=l+-. a? Clear of fractions and transpose, add to each side i, the square of one-half the co- efficient of Xf extract the square root, 1 . V5 therefore mi. XI- X 1+V6 ,l--/6 . Thus the roots are — ^ — ^^^ — o — 4. Solve 6iB"-f 5a?— 21=0 by the Hindoo method. Transpose, 6a;^+5a;=21; multiply each side by 4 X 6=24, 144a^+120a?=504; add 5^=25 to each side, 144a^H- 120^7+ 25 = 529; extract the square root, 12a? + 5=±23; ,, , -5±23 8 7 therefore x= — j^ — — a* ^^ ^ R' ELEMENTS OF ALOEBRA. 139 8 7 Thus the roots are ., and — _ , 5. Solve 2.P-1 8ur-|-4 27+1 "'" 3x-4 20 ii* Clear of fractions, 12.f"4-100u?= -108; divide by 4, 8^-»+25a;= -42; multiply each side by 4 x 8 = 12, add 25*=G25 to each side, 86.1^* -f800u;+ 625 = 121; extract the square root, 6u.'+25=±ll; -25±11 therefore x= 6 = -2j, or -6, 6. Solve 8a; = 41 10-8d?* Clear of fractions and transpose, 9ar»-80a;=-41; multiply each side by 4x9 = 86, and add 900, 824a!»- 1080^+900= -1476+900= -576; extract the square root, 18a?-80=±V"r676 = ±24V'-l; 5±4 V ^T" therefore 0? = 8 Exercises, LIY. Solve 1. (a?-l)(a;-2)=0. 8. (a;+8)(a;-5) = 0. 5. (6ar-l)(2a:+8) = 0. 7. 2.ta-8 = 0. 9. iB*-aj?+16=0. 2. 2.r2_7.r=0. 4. (2.f-5)(^+8)=0, 6. 5w{7u--S) = 0. 8. 8ar»+12 = 0. 10. ar»+9a?+14=0. 4, ' * « '£11 140 ELEMENTS OF ALGEBRA. 11. a^-a;-12 = 0. 13. 6ur»-5a'+l=0. 15. 15x'+7x-4:=^0. 17. a;+--s=0. 12. ar»+a;--20=0. 14. 80x'2-a;-l=0. 16. 35^+81a;+6=0. 18 ^lZ^j_^±?_?? ■ if+4"*"a;-2 - 10 19. 20. 21. 22. 23. + T + A'-l a?+l 8 5.i-2 _ 3.1-4-10 = z + r + a;+a-|-^ ^ a OOJ-fl + + rta?— 4 "16" = 1. ax-\-l ax—1 Vn^—i* ar+1 x—1 24. _L4 _ nc — l x-{-l = 2a. .i^' u 25. '/4a;+17+^^a?+l = 4. 26. v/5"d;4- v'2"a;+2= 'v/:»~+2'. 27. «c.i^ — bcx + fl^^ — 6c? = 0. 28. 29. 2j;-1 2a?- 3 + 3^-2 3a;+2 ' 2j:+1 6j^-x-2 a^+5^- 2.g+l =x^+Sx+2. 80. 2V8j?+7 = 9~'/2j;-3. 81. ^6a;+l+ ^x+4:+Vi'^+1^2. 220. Equations of a higher order than the second can be solved either partially or wholly by the aid of quadratics when they can be thrown into one of the following forms : — ELEMENTS OF ATQElRA. 141 I. (cx-\-d) (aix^-\-bx-^c)=0. II. (aa^+bxi-c) {a'x^+b'x-\-c')-Q, III. iv*-{-aii^+bx^+ax-^l=0. IV. iaay^+bx+cY^+p{ax^-\-bx+c)^-\-q=:0. 221. I. The equation {cx-\-d)(ax^+bx+c)=0 is satisfied by equating either factor of the left-hand side to zero. We thus get a simple and a quadratic equation, whose roots will be the three roots pf the given cubic. Examples. 1. Solve a^-5a^+6x=0. This equation may be thrown into the form therefore, either a? =0, or ar*— 5a; -|- 6=0. The roots of the latter equation are 2, 3. Hence the three roots of the cubic are 0, 2, 8. 2. Solve a!»-l = 0. This equation may be written (a;-l)(a;"+a;+l)=0; and, therefore, a?— 1=0, x=l; " ora!»+a:+l=0, -l±v^-3 x= 2 Hence the three roots are 1, 2 222. n. The biquadratic {ax'^-^bx + c){a'a^+h'x +c')=0 can be fully solved by equating each U2 ELEMENTS OF ALGEBRA. dratic factor of the left-hand side to zero. We thus get two quadratic equations, each of which has two roots. Examples. 1. Solve {ar^-l){ar'-^x-6) = 0. Here we have ar»— 1=0, and therefore .x'=±l; or 3^-\-x—6—0, and therefore x=2, or —3. Hence the four roots are 1^ —1, 2, —3. 223. III. The biquadratic a;*+o^+[)x2-|-a.r+l=0 may be solved as follows : Divide both sides by or, ,U., . i,. > X ar * 1 / 1\ ' ■■'■' or, a^+ ^ +a yx+ -j+6=0. ,, . /| Add 2 to each side, V 0'+5)'+''(^+5)+*=2- Let a?-f- - =y; hence y^-\-ay-\-b = 2; from which two values can be found for y, that is, for x+ . If these vail ^ be called p and q we have thus two quadratics, 1 J 1 the roots of which will be the roots of the given biquadratic. ELEMENTS OF ALGEBRA, 148 Example. Solve 12a,-*-104^-8+209x2-104a?+12=0. By dividing both sides by 12 the equation will be reduced to the required form; thus, , 26 , , 209 ^ 26 . , ^ ^- •;i 1 ^ ^ 26 , 209 26 1 . 1 dmde by a?, a?- _ ^+ _ _ _ ._+_ =0; add 2 to each side, / ^ly 26 , / i\ , 269 „ write 2/ for a;+ - , • ^ 8 ^^ 12 ' 87 6 from which we obtain 2/ =--, or - . XT , 1 87 ,15 Hence a?4- - = — - , or x4- - = ^ • X 6 X 2 The roots of the former are 6, ^, and of the latter 2, i. Henco the four solutions of the given equation are i, ^, 2, 6. 224. IV. The equation (aa;=+6a;+c)»«+2)(aa;24.6a: +c)''+g=0 is reduced by writing y for {a£-\-hx-\-cY to the form the solution of which gives two values of y, d and e, suppose ; thus (aa^-\-bx+c)'^=:dt or, (flW3"+6a;+c)"=:«; I (Hi ;i ll 144 ELEMENTS OF AIQESBA. extract the nth root of both sides, or, aa^-^-bx-^c— V e; two quadratics which may be solved by the usuaj method. It is to be observed, however, that if n be an integer, there will be really 2n quadratics instead of 2. For it is proved in the Theory of Equations that the quantity 'y/d has M values, and therefore there will be n quad- ratics corresponding. In the following examples, how,ever, we shall generally assume ^d to have but one value. Examples. 1. Solve ic«+19^-»- 216=0. Here, writing y for a^, we get 2/5^19y_216=0; from which we obtain yt=—217, or 8. Thus a^=-27,OTa^==S; whence a;=3— 8, or a;=2. !* 2. Solve a;"+24=12 v/^4^. This equation can be written in the fona " (a^+4)_l2V'^4:4=-20; add 86 to each side, (a:2_f.4)-l2 v^^^»+4+86=ie; extract the square root, - ' whence, . V^+T = 10, or 2 ; ' = square, aj^+4==100, or 4; a:"=96, or 0. Hence the four roots are 0, 0, 4^6^ —4^6. 8. Soh This e< thas, add ( 1' 24. ( extract (b Bdtf^ 1. ^■f 9. 4^4 8. a^-j. 4. One remaining B. (a;- 6. (a^- 7. a^- 8. 2x-\ 9. 12a 10. 18a? 11. Onel theremaiJ XLEWBKTB OF ALOBBRA. 145 »• Solve i<:-±i)Vv(:-±i)=i2. This equation may be solved by finding -^ first ; Z^+iy. 7 /^+l\ /7 Y_ 625. Vaj-1/ *^ 12 U-a/ "^ \24/ " m' extract (he square root, whenoe ^+1 7 25^ «-l"'"24~*24* aj:=s —7, or ^. EzEBCISESy LV* Solvd 1. aj*+2a?-24sc0. 2. af4-»^-72=a, 4. Onerootof 0^— ll^>40t:i=0 is 2: find the xemaining roots. 07 o ^ 6. («*+a;-2)«-^18 (a^+»'-.2)+88«a 7. a^-2a?+6Va?-2a?+6=ll. 8. 2a;4-17=9^2»--l. 9. 12a;*-91a^+194aj»-91aJ+12=0. 10. 18aJ*"-171aj»+406a;«-171a;+18=0. 11. One root of o^~21a^+ 148a; -816-0 is 5: find the remaining roots. .fij'ifl 148 ELEMENTS OF ALQEBBA. 12. One root of a^ -120^+ 41 x-QO=zO is 4 : find the remaining roots. 18. V^8a;»+2a;+4=6a!»-|.4a?-622. 14. v'5iB»+6a?-2=16a;»+18a;-80. 15. ^^^+22-^^^+8=1. a?+V2^^ 4 16. 17. a,+ Vg-a? 7 18. V8a;"+ai;+14+ V8aj»-8a?+14=8ar. 19. V8«»+8a;+14+ V8a!"-a»+14= V40a!»+24. PBOBLEMS LEADING TO QUADRATIC EQUATIONS, 225. The conditions of a problem are sometimes such that their expression in Algebraical language leads to a quadratic equation. The roots of this quadratic may both satisfy the conditions of the given problem, as in the following example : Find two numbers such that their sum shall be 16, and the sum of their squares 118. Lata; denote one number; then 15— a; will denote the other. Hence, by the conditions of the question, a;«+(15_a?)»=118; whence a?=7, orS; and therefore 16— a?=8, or 7. Thus the two misibers are 7 and 8. ELEMENTS OF ALOEBRA. 147 ind the 226. In many cases, however, both roots will not satisfy the conditions of the problem. Whenever, for example, the unknown quantity is denoted by +Xf and one of the roots is negative, this root will be incompatible with the conditions of the given problem. In the following example the unknown time is denoted by + X, and therefore the number of gallons that flow through the two cocks in the same du'ection must have the same sign. )metimes language } of this the given 01 be 15, in denote uestioD, A cistern can be filled in 56 minutes by two cocks flowing together. If they flow separately, it will take one of these cocks an hour and six minutes longer to fill the cistern than the other. In what time will the cistern be filled by each ? Let the cistern be filled by one cock in x minutes ; then it will be filled by the other in a; +66 minutes. Also, let g be the number of gallons in the cistern ; then the amounts in gallons that flow through the first and second cocks in one minute wiU be ^ and — £ — , a? a;+66 respectively ; and — ^, — I- will be the amounts that X a?+66 flow through in 56 minutes. m 56a a?+66 66 Hence 5^+ X X a;+66 whence a? =88, or —42. The negative root must be rejected as being incon- sistent with the conditions of the question ; for if «=— 42, it would follow that the number of gallons E 2 ■ IB m§ S «■ re 14B ELEMENTS OF ALQEBRA. lowing through the two cocks in one minute would be — ^ and +|!-, respectively. 42 24 227. The exisoence of a positive and a negative root may often be explained by the fact that the latter is the solution of an allied problem which has the Bame quadratic statement as the given one. In order, however, that the expression of the conditions of these two problems may lead to the same quadratic statement, the unknowns must be so related to each other that they may properly be denoted by +a; and —a;. In the two following examples the unknowns, the number of oxen bottght, and the number sold, may properly be denoted by symbols with opposite signs. V Ex. 1. A person bought a number of oxen for £120, and found that if he had bought 8 more with the same money, he would have paid £2 less for each. How many oxen did he buy ? 120 Let +07 denote the number bought ; then — is the ^rice paid for each, and is positive ; if he had purchased <8 more, the price of each would have been 120 Therefore, by the conditions of the problem, 120_ 120 __g X a?+8 ifhich reduces to a?+8a?=180 (1) Ex. 2. A person sold a number of oxen for £120, and if he haid sold 8 fewer for the same money, he SLBMENTS OF ALGEBRA. 14B "wonld have reoeived JS2 more for each. How many oxen did he sell ? Let —X denote the number sold; then the price received for each =- — = — -— , a negative quantity. — X X If 8 fewer had been sold, the number disposed of would s= — 07+8, and the price of each would then 120 120 -aj+8 «-3 Therefore, by the conditions of the problem, remember- ing that money received is in this case negative, we have -l??-2=-— which reduces to that is, a?-8a;=180, (-.a.)»-|-3(_a?)=180 • • • • (2). If now we write yfor the unknowns of both problems, that is, if we write y for -f-a; in (1) and for —x in (2), those equations reduce to the same form t/*+8i/=180 .... (8) and therefore the two problems have the same quad- ratic statement (8), the roots of which are 12 and — 16, the former of which is the solution of Ex. 1, and the latter of Ex. 2. Exercises, LYI. 1. Find two numbers such that their sum may be 14, and the sum of their squares 100. 2. Find two numbers such that their sum may be 10« and the sum of their cubes 280. 1^ ■'1. i •m dp ' 1,1. ':■! i M ELEMENTS OF ALGEBRA. 8. A Bum of money, amounting k) £10 16s., was divided equally amongst a certain number of persons ; if there had been three more, each would have received one shilling less. Find the number of persons. 4. The difference of two numbers is 4^, and their product 28. Find the numbers. 6. The product of two numbers is 86, and the differ- ence of their squares 65 ; find the numbers. 6. A gentleman bought a horse for a certain sum, and having resold it for £119, found that he had gained as much per cent, as the horse cost him. What was the prime cost of the hone ? 7. A, working alone, can perform a piece of work in IG days less than £ takes to perform it alone ; both together can perform the work in 12 days : how long does it take A to do it alone ? 8. A and B, working together, can perform a piece of work in 10 days ; after working together for 4 days, A is taken ill, and £ finishes the work in 8 days more than A would have taken to do the whole : in what time would each of them do it separately ? 9. Find two numbers whose sum is 100, and the difference of their square roots 2. 10. The height of a certain triangle is 4 inches less than the base ; if the base be increased 6 inches and the height lessened as much, the area is diminished by one-eighth part : find the base of the tri igle ? 11. A rectangular field is an acre in extent, and its perimeter is 308 yards : what are the lengths of its sides ? 12. A boat's crew can row in still water at the rate of 6 miles an-hour. They enter a current, and row ELEMENTS OF ALGEBRA. 151 witn it for a distance of 6 miles ; on coming back they find that it takes them two hours longer to make the same distance against the current than tlic time it took them when rowing with it. At what rate did the current run? 18. Two vessels, one of which sails faster than the other by 2 miles an-hour, start at the same time upon voyages of 1162 and 720 miles, respectively; the slower vessel reaches its destination one day before the other : how many miles did the faster vessel sail ? 14. A number is composed of two digits, the first of which exceeds the second by unity, and the number itself falls short of Hie sum of the squares of its digits by 26. What is the number ? 15. A number is composed of two digits, the first of which exceeds the second by 2. The sum of the squares of the number, and of that which is obtained by reversing the digits, is 4084. What is the number? 16. A merchant sells two casks of wine for £76 6s. ; one holds' 6 gallons more than the other, and the price of each wine is in shillings the number of gallons in the cask which contains it. How many gallons are there in each cask ? 17. A person drew a quantity of wine from a full vessel which held 81 gallons, and then filled up the vessel with water. He then drew from the mixture as much as he before drew of pure wine ; and it was found that 64 gallons of pure wine remained. Find how much he drew each time. 18. In order to resist cavalry, a battalion is usually formed into a hollow square, the men being four deep, but a single company is usually formed into a solid square. If the hollow of the square of a battalion, consisting of seven equal companies, is nine times as ■''■fi !? ''i where ^and— | are the roots of aj"4-|ar -t=0. 285. Conversely, if the roots be given, the equa* tion may be formed. *•■ wi "■-' ■»'ri'v^-4g of a;«+j9a:+g=0 is shewn, as follows, to depend on the nature of the quantity V J)*— 4g. 238. I. If2)«-4g=0,or2)2:=4g, it follows that a=6 = — ^, Bind therefore the roots are equal, as is other- wise evident, since the left-hand memher of the equa- 239. n. If2)«-4g he positive, that is, if l)«>4?, the roots will he real and different, a being the sum, and h the difference, of the same two real quantities -|, \ V p*-4gr. This will always he the case if the absolute term be negative. For example, let the absolute term be — r, then the quantity under the radical sign is j3''+4r, which is positive, and therefore both roots are real. Thus, for example, the roots of ar'4-5x'+2=0 are real and different, because 6" > 4 x 2 ; the roots of a^— 7a?4-8=0 are real and different, because (—7)" >4x8; andtherootsof ar"+7a;-l=0,a;2_iOa?-8=0, are seen by inspection to be real and different, be- cause their absolute terms are negative. 240. in. If 2)2-4^ be negative, that is, if 'f 243. The preceding tests for determining the nature of the roots of a quadratic may sometimes be employed to find the maximum and minimum values of a function oix. . 244. A function of x is said to attain its maxi- mum or minimum value when by assigning gradu- ally increasing or diminishing values to x, the function ceases to increase, and begins to diminish, or vice versd. Thus, the quantity aj» — 2a?-f6 = aj* — 2a?+l-f4 es(a;'-l)*-|-4, has a minimum value 4 corresponding to ELEMSirrs OjP Algebra, 167 x=il: for all other real values of x render (a;— 1)* posi« tive, and, therefore, (a?— 1)"+4 greater than 4. 245. In like manner it follows that the quantity x'^+px+q={x+^Y-^ ^""^ , has a minimum value ^ 4 4g— j) ^ corresponding to the value «= — ^. 4 si EXEBOISES, LViii. Find the minimum value of 1. a^-Bx-\-5. 2. a;»+4a?+8. 8. aj»-|aj+J, 4. 50a;»+40a?+9. 6. 72a5«-63a?+64. SIMULTANEOUS EQUATIONS OF THE FIRST DEGREE CONTAINING TWO UNKNOWN QUANTITIES. 246. An equation containing two unknowns is said to be of the degree indicated by the highest total dimensions of the unknowns involved in any term. Thus 2x—8y=5iB an equation of the first degree in X and y ; a;"— y'=2, a^=8, a?+2iry=8 are equations of the second degree ; and so on. 247. If there be only one such equation between X and y the number of solution^ will be infinite, for the equation will be satisfied hy giving any value to 'III .■•>■*! in "n.;' .*^- ••■!,«« 16B ELEMENTS OF ALGEBRA, 'I ■ one unknown, when the other unknown will haye one or more corresponding values. Thus in the equation 2a;— 8y=5, if we give y the value 1, aj=4 ; if 2/=3, a?=7; if a?=3, y==^; and so on ; 80 that the equation is satisfied by the sets of values 4, 1 ; 7, 8 ; 8, i ; &c. Again, the equation a^—y^—2 is satisfied by giving any value to one unknown, when the other unknown has two corresponding values. Thus, ify=:2, a;=±2; ify=l, a?=±V^8. 248 If there be two such independent equations in X and y at the same time, they are called simulta- neous equations, the same values of the unknowns being taken in both equations. In such a case the number of solutions is finite. Thus, if we have the simultaneous equations 2x—Sy = 5, 3.i!+y=13, it will be found that the only set of values which will satisfy these equations is a; = 4, Again, the equations 2x—y—4y «"— 2a^+8i/'=9, the former of which is of the first, and the latter of the second degree, will be found to have the two solutions a?=3, y—2; a?=_, y=— — ; either of which sets of values, and no others, will satisfy the equations. 249. And, generally, it may be stated (as is proved in works on the Theory of Eqimtions) that if the two equations in x and y be of the mth and nth degrees, respectively, the number of solutions may be equal to> but not greater than, mn. Thus the number of solutions of the equations ipy-t-y»=6, a:"— 8ajy4-2^=:8, which are of the second ELEMENTS OF ALGEBRA. U9 and third degrees, respectively, cannot be greater than 2x8, or 6. 250. It must be carefully observed that the number of solutions will not be finite unless the two simulta- neous equations be independent. If, for instance, one equation follows necessarily from the other, the two are really equivalent to one, and therefore the number of solutions will be infinite. For example, the equations 2a; — 81/ = 1 , 8a? — 1 2i/ = 4 , have an infinite number of solutions, for the second equation is deduced by multiplying the sides of the first by 4, and therefore there is only one independent equation. Again, the equations a;+2y-l=0, have an infinite number of solutions, for the second is deduced from the first by multiplying its sides by 2a?-2/+3. 251. The solution of such simultaneous equations is effected by deducing from them two other equations involving each one unknown quantity. This process is called elimination, and may be conducted according to one of the following methods : I. Substitution. n. Comparison. III. Cross Multiplication. rV. Arbitrary Multiplier. These methods will be illustrated, in the first instancey in solving simple equations of two un- knowns. K.^ uj ' I'L {♦»» '•W 'fi'ij 108 SLUWSNTS OF ALGESnA, ,13 "'i! METHOD OF •SUBSTITUTION. 252. This method consists in finding from one equation the yaluo of one unknown in terms of the other, and substituting the value so found in the second equation^ which is thence reduced to a simple equa- tion in one unknown. For convenience of reference the given equations and others which arise in the process of solution aia numbered (1), (2), (3), &c. Example. Solve a?+y=8 (1), 2a?+2/«4 (2). Prom (1) we find y=8-a? (8). Substituting this value of y in (2) we obtain This value of x substituted in (8) gives y aSI, Hence the solution is a; = 1 , ^ = 2. ExEsasEs, LIX. 1. 8a?-4y=2, lx^2y^l.^ 2. 6a?+2y=:29, 6a;-.2y=26. : 8. 8a?+2y=0, 8a?-.42/=18. 4. 6aj+4=2y, 4ir+y=28. y 6. 2jj-8y=0, 4a?,-|=H. 6. 6a;-5y=l, 7a;-4y=8i. 7. 2d;+6ys2i, 6a;+2ys2j^ ELEMENTS OE ALGEBRA. ICI 8. ~ +62/=13, 2x+ ^^ :=33. ^' -3" + "2"=^' 2 +"9"=^* 10. Ax+5y=A0(x-y), 2a?+5y=li. 11. 5a?-ay=0, - + | =2» a 12. f+f=i, 1+^=1. a a METHOD OF COMPARISON. 253. This method consists in finding from each of the proposed equations the value of one and the same unknown quantity in terms of the other, and equating the values bo found. Example. Solve 7x-3y=ld. (1). 4a?+72/=37 (2). Prom (1) we obtain .= M (8); and from (2) y== ~ - (4). Eqjiating these values of y we get a simple equation ino^i 7a?-19 _ 87-4^. 3 "" 7 ' whence a?=4. Substitute this value of a? in (3) or (i), and we get y=3. TJms the solution is a?=4, y=d, ' " , L i-^f'i • ' 'V, . ' i. < "^^ tfl. <* 'ft *% in « 162 ELEMUNTS OF ALGEBRA. EXEBOISES, LX. 1. 6j?4.11y=146, lla;+5y=110. 2. 8a?-2i/=19, a»+8y=48. 8. 7a?-2i/=14+|, 7y-ar=82+|. 3a7+5y 6a?~8y a>+l _ 2 *• 20 "^ 8 -^''^H^-g* o. ^c— — - — . a b 11 ^^ 2 6 ' 8(3a?+4)=10y-15. Q a?+l_a?— 1 a?+a _ i»+^ j • y+«~" y+b* 2/+1 "y-l * METHOD OF CROSS MULTIPLrCATION. 254. This method consists in multiplying the given equations (reduced to their simplest forms) by such quantities as will render the coefficients of the same unknown numerically equal. By adding or subtract- ing the equations so found we obtain a simple equation in one unknown. Examples. 1. Solve 7a?-9t/=29 (1), 13a;+4y=:116 (2>. Multiply (1) by 4 and (2) by 9; thus "^ 28aj-86y=116 (8), 117i»+86y=1044 (4), ELEMENTS OF ALGEBRA. 1^ by adding (8) and (4) we eliminate y; 146ar=1160, a?=:8. To eliminate a?, multiply (1) by 18 and (2) by 7, 91a?- 1172/= 877 (6). 91a;+2tey =812 (6); subtract (6) from (6), r , 146t/=486, • Thus the roots are 8, 8. 2. Solve Ux'+4y^5i (1), 's^^I .^^ - 18a?-62/=61 (2). ^ -"^ - Here, in order to eliminate y, we may multiply (1) by 8, and (2) by 2, 8 and 2 bearing the same proportion to each othe? as 6 and 4; thus 45aj+122/=162 (3), = ' 26a?-122/=122 (4)* Add (8) and (4), ' ..ii..-.:.' .: 71a?=284, ^ a?=4. To eliminate a?, multiply (1) by 18, and (2) by 16, and subtract the equations so found, 142y=-213, ,,. ■..:. ...^/ ..--V^. • ^~* 2 117 ^- ®^^®- 4^ ~ % =e-..-..-".(i)» Nuui: i ^-i-?? r2^ by first finding*, -. X y 1.2 ■'|i i,;V ■'//'IS i 164 ELEMENTS OF ALGEBRA. Here, in order to eliminate y, it is only necessary to divide (2) by 8, 8x Qy IQ* And subtract from this (1), J^ _ J_ _ 23 __ 7 8a; 4u? - is 6 * 8 To eliminate x divide (2) by 4, and ict from (1) ; 2 whence we obtain y= — - . • 255. Instead of using either of the proposed equa- tions we may use any equation derived from them which has smaller numerical coefficients, as in the following Example. Solve 16a;-15i/ = -88 (1), 12a;-132/=-46 (2). Subtracting (2) from (1), we obtain ' 4a;-2i/=8 or, 2a?- 2/=4 (3). Equation (3) may be used instead of either of the proposed equations, as (1); thus we have to determine .X and y from the equations 12a?-13i/=:-46, 2x—y — 4. Eliminating as before we obtain a; =7, 2/ =10. Exercises, LXI. 1. 6a?+2i/=12, 3ar+5y=ll. fi. 8a?-7y=ll, 5a?-8i/=ll. ELEMENTS OF ALQEBBA, 165 U 8. 9x-7y=7, 4aj+8y=60. *• 4 2~'12+6~ 5. 82a;+81i/=43, 28jj-89i/=1. 6. 96j?+75>/ = 102, 92a?+80^ = 101. 7. a;-\-y=a-\-bi bx-^ay=2ab, 8. iv-\-y=sCj ax—hy — c{a — h), 9. a(a?+2/) + 6(«-2/) = l. a{^-y)-\'h{x-{-y)-=l, 10. a(a;— «)+6(i/ — 6) = 0, fl(a?— y— a)-{-6(a;+2/— 6) = 0. METHOD OF ARBITEARY MULTIPLIER. 256. This method consists in multiplying either of the proposed equations by an arbitrary multiplier m, then adding the other equation to the equation so found, equating to zero the coefficient of y and solving for Xf or the coefficient of x and solving for y ; the value of m corresponding to each case being found £rom the coefficient which is equated to zero. Example. Solve 4a?+5y =7 (1), 8a;-10j/=19 (2). Multiply (2) by m, and add to (1) ; thus (4+8)n)a;+(5-10m)2/=7+19m .(8). Equation (8) is true for all values of m, and therefore for that value which makes 5 — 10m=0, or m= s; in which case (8) becomes a;=8. . hi .'I , 1. * 1 186 ELEMENTS OF ALGEBRA, Equation (8) is also true for that value of m which 4 makes 4+8m=0, or m= — -, in which case (8) becomes y=-i. ; EXEBCISES, LXII. ^^.:.ti .V 1. 16a?+28y=58, lla;-7y=16. , , ' . 2. 7a?+8y=:18, 21a?-.6y=114. , . ^ ', 8. 8a?+8i/=5, 10a?+9y=8. 4. 2a?+y=i, 18a;+22i/=-28. 'v> < '.' l.l 1 6. aa?+6y=c, mx-^ny=p. 7. ra?4.|»=«y+n, ^"^^ -*** 8. CD—y n p _ g n-\-y m—x* q—x p-^-y n SIMULTANEOUS EQUATIONS OF THE FIRST DEGREE CONTAINING THREE UNKNOWNS. 257. When three independent equations conttiining three unknowns are given, the number of sets of solutions depends on the degree of each equation, and (as is proved in the Theory of Equations) may be equal to but not greater than the product of the numbers expressing the degrees of the three equa- tions. Thus, if each equation be of the first degree, there will be only one solution ; if the first equation be of the first degree, the second equation of the second degree ); ELEUlBNTa OF ALOEBRA. im and the third equation of the third degree, the number of solutions may be equal to, but not greater than, 1x2x8, or 6; and, generally, if the first equation be of the mth, the second equation of the nth, and the third equation of the pih degree, the number of solu- tions may equal, but not exceed, mnp. If the three equations be not independent, that is, if one of them be deducible from the other two, or if two of them be deducible from the remaining one, it will follow, as in the case of two unknowns, Art. 250, that the number of solutions will be infinite. 4 Vj.*' ' 'm I! 258. If the three given equations are of the first degroe, the values of the unknowns may be found by eliminating one of them between the first and second equations, and also between the first and third, or' second and third; whence we shall obtain two equa- tions involving only two unknowns, which may be found by previous methods. The value of the third may be found by substitution. Example. Solve 2aj+8t/+4;?=20 (1), 4a;-8t/-2;:=-8 (2), 8a?+4yH-5«=26 (8). Here z is eliminated from (1) and (2) by adding (1) to twice (2); thus, 10a;-8y=4 ...........(4). Also « is eliminated from (2) and (8) by adding five times (2) to twice (8); thus 26a?-72/=12 (5). Erom (4) and (5) we obtain a;=l, y=2. ^ 0" ■ 1'L m M fi >iii 1% 168 ELEMENTS OF ALQEBRA. Substitute these values in any one of the proposed equations, and we obtain ;s== 8. Thus the solution is a; =1, 2/= 2, ^=8. 259. By the following method of Cross Multipli- cation we can eliminate two of the unknowns at the same time by multiplying the given equations by cer- tain quantities, and adding the derived equations. Let the given equations be «i'»+%+Ci3;+tii=0 (1), aja?+%+Ca«4- ^+2/~*=c* 12. ?+|+f=l, ?+»+f=l, f+?^+f=l. .- PROBLEMS LEADINa TO SIMPLE EQUA- TIONS, WITH TWO OR MORE UNKNOWN QUANTITIES. 261. When there are two or more unknowns in a problem to be determined, and they are not imme- ELEMENTS OF ALOEBBA. 171 diately expressible in terms of one unknown, as in the cases referred to in Art. 209, they must be re- presented bjr distinct sj'mbols. The conditions of the problem being then translated into Algebraical language will lead to a corresponding number of in- dependent simultaneous equations, the solution of which will give the yalues sought. EXAUPI4ES. 1. What fraction is that which becomes, when its numerator is increased by 7, equal to |, and when its denominator is increased by 10, equal to ^^ ? Let cc denote the numerator, and y the denominator of the uaction. K the numerator x be increased by 7, the denominator remaining unchanged, the fraction becomes ^"^ ; and if the denominator y be increased y by 10, the numerator remaining unchanged, the frac- tion becomes X problem, y+10 Hence, by the conditions of the a?+7_3 y » ■7:' J* ■7^ > y+10 2 the solution of which is a; =11, 1/= 12. ft The fraction is, therefore _. 2. There is a number consisting of two digits, which exceeds five times the digit in the unit's place by 6 ; L\ I [ t y. J,. : , .,.1,- iipi 1-' .*■• >: W'M ii $m *f- '""''4' f"'i : ' ■ '■ I'll'; ■'€■■( ■ - > ':|" .:,:' -Willi V,^ ■jivri'i'. ^'- -V^M 172 ELEMENTS OF ALOEBEA. I' I and if 27 be added to it, the sum will b.e expressed by the same digits in an inverted order. Let X be the digit in the place of tens, and y the digit in the place of units. By the first condition of the problem we have lOa:+y = 5y+G; and by the second condition, 10A'+2/+27 = 10//+.r. From these two equations wo find a;=8, 2/=6. Therefore the number sought is 80. 8. A and B can do, a piece of work in G days ; A and C can do it in 9 days, and A, B, C can do 8 times the same work in 45 days. In what times can they do it separately ? Let X, y, and z denote the numbers of days in which A, B, and C, respectively, can do the work w. Then, since — , —, ^ are the amounts of work done X y ' z by them separately in one day, we have by the condi- tions of the problem W .IV _w ~^ y~~^ w .w w + TT* X z 9 X y z i5 ELEMENTS OF ALGEBBA. 173 These equations become, on dividing by w, -+-=J- (1) X y b 5+r9 ^^^ -+-+-=il • • • • • (3) X y z 4lO By subtracting (1) and (2) from (3), we find .- = . , 00 1 1 and- — — , tlie reciprocals of wliicb give ;2=90, and y 15 y=15. The value of x is then found, by substitution, to be 10. Exercises, LXIV. 1. Find two numbers, such that the sum of the first and half the second shall be 19, and the sum of the second and half the first shall be 20. 2. A number composed of two digits is equal to seven times i^s unit's figure \ and, if the digits be re- versed, its value is increased by 18. Required the Dumber. 8. If the numerator of a certain fraction be increased by 2, its value becomes one-half ; but if the numerator be diminished by 2, the value of the fraction is one- fourth. What is the fraction ? 4. Three numbers are required whose sum shall bo 22 : the sum of the first and third to exceed twice the second by unity ; and four times the first, three times the second, and twice the third added together, to make 69. ■lii .1:.... "««?, %lt:: ■■4> ."jjiwi'^'- ■''m ikSI;' ■■'M - - , .pi' . 174 ELEMENTS OP ALGEBRA. 6. Find two numbers such that twice the first and three times the second shall together make 18 ; and, if double the second be taken from five times the first, 7 shall remain. 6. A cistern supplied by two taps is filled by both together in 50 minutes ; after both had been open to- gether for 85 minutes, one is closed, and the cistern is found to be full 27 minutes afterwards. In what time would each tap fill the cistern ? 7. A man being asked what money he replied that he had three sorts of coin, namely, half- crowns, shillings, and sixpences ; the shiUings and six- pences together amounted to 409 pieces, the shillings and half-crowns to 1,254 pieces ; but if 42 was sub- tracted from the sum of the haK-crowns and sixpences, there would remain 1,103 pieces. What sum did the man possess in all ? ^^ . ^ 8. A courier travelled a distance of 240 miles in four days, diminishing his rate of travelling each day alike ; during the first two days he travelled 186 miles. What number of miles did he travel on each day respectively ? 9. A and B can do a piece of work in 2 days ; A -^rl can do four times as much in 9 days; A, B, ai t C can do eleven times as much in 18 days. In how many days can each do it separately ? ^v 1 ■, -i ' "j,f y f, ... . ,^' 5T 10. The ten's digit of a number is less by 2 than the unit's digit ; and, if the digits are inverted, the new number is to the former as 7 to 4. find the digits. 11. The sum of three numbers is(p+l) (q-\'l)n; the sum of +he two larger is equal to jtiiimes the smallest, and the sum of the two smaller to q times the largest^ Find the number^. ELEMENTS OF ALGEBRA. 176 12. At an examination there were 17 candidates, of whom Bome were passed, some sent back in one subject, and the rest rejected. If one less had been rejected, and one less sent back, the number of those passed would have been twice those rejected, and five times those sent back. How many of each class were there ? 18. There are two numbers in the proportion 8:5; if 10 be added to the first, and subtracted from the second, the proportion is reversed. Find the numbers. ,14. Find three numbers, such, that one-half the first, one-third the second, and one-fourth the third, together shall be equal to 62 ; one-third of the first, one-fourth of the second, and one-fifth of the third equal to 47 ; and, finally, one-fourth of the first, one-fifth of the second, and one-sixth of the third equal to 88. 15. A man has two casks, each containing a quantity of wine. From the first he pours into the second as much as is already there; from the second he pours back into the first as much as is already there ; and, finally, from the first he pours into the second as much as is already there: he then finds there are 80 quarts in each. How many were in each origin- ally? .' 16. A number consists of two digits whose sum is 12, and such that, if the digits be reversed in order, the number produced will be less by 86. Find the number. 17. A, B, and C sat down to play, and found when they stopped that they each had the same amount of money. When they commenced, A had £11 8s. less than B and C together ; he now finds he has lost £21 4s., that B and have won ^618 14s. and £2 IDs., respectively. With what money did each begin, and with how much did they conclude the game? .. ■■. Id " I '.I : ;| ^ i e IW ELEMHin^ OP ALGEBRA* 18. A cistern can be filled from three different cocks. If the first and second run together, it will be filled in 83^ minutes ; if the first and third run together, in 86 minutes ; if the second and third run together, in 40ii minutes. How long will it take for each separately, and for all running together ? 19. A silversmith has two alloys of silver ; he melts 10 ounces of the first with 6 ounces of the second, and produces an alloy, the fineness of which is 687^ per mille ; again, he melts 7i ounces of the first with 1^ ounces of the second, and produces an alloy which is 625 p^r mille fine. T/hat is the fineness of each kind ? . 20. Three towns. A, B, and C, are at the angles of a triangle. From A to C, through B, the distance is 82 miles ; from B to A through C, is 97 miles ; and from C to B, through A, is 89 miles. Find tho direct distances through the towns. 21. The diameter of a five-franc piece is 87 millir metres, and of a two-franc piece is 27 millimkres. Thirty pieces laid in contact in a straight line measure one m^tre exactly. How many of each kind are there ? 22. A man invested £2,000 in cottages; a certain number were then bnrnt down ; but, in consequence of their having been insured, the loss in each cottage was only 10 per cent, on its cost pwce, and was also found to be at the rate of £10 on each cottage bought ; some time after the remainder were sold 20 per cent, dearer than they had been bought,' and the total gain was £100. How many cottages were originally bought, And how many were burnt ? ULISMENTS OT ALGEBRA. Ill SIMULTANEOUS EQUATIONS, WHEREOF ONE IS OF THE FIRST DEGREE AND THE OTHER OF THE SECOND DEGREE IN TWO UNKNOWNS. 262. Let thfi general equation of the second de- gree be «wj*-f 6a;y+cy'-f- do? -fey +/==0. .»...(!), end of the first degree «'a;H-%=c'... (2). In order to solve these equations we substitute in fl) for one unknown, as t/, its value in terms of x derived from (2); (1) thus becomes a quadratic in a;, and has two roots. By substituting these values of ^ in (2) we get two corresponding values for y» ExAMPLfi. Solve 8a3"-6a:y+42/«-2aj-2y+2=0 (1), 6a; — 2i/=10 (2), From (2) we find y=*3a;--5 * (3); substituting this value for y in (1), we obtain 2la;»-98x+112=0» 8 the roots of which are § and 2. Substituting these values for x in (8), we find the corresponding values of ^ to be 3 and 1. g The required solutions are, therefore, a?= -, 2/= 3; and 47:^2, y&cl. 263. For certain forms of the general equation of the second degree in x and y the preceding method 1 -^ 1 fii I 'rim ^ii r4i, . :■:■.* ■i'.M'.> 178 ELEMENTS OF ALGEBRA. may be advantageously replaced by others. The yariuus artifices are exhibited in the following Examples. 1. Solve a?+y=14 (1), a^=lS (2). In this case, squaring (1) and subtracting 4 times (2), we get a;* — 2a^ + 2/" = 144 ; extract the square root a'-y=±12 (8). Combining (1) and (8) by addition and. subtraction we obtain, if +12 be taken in (3), a?=13, y=l; and, if — 12 be taken, x=l, y=18. The same method will answer when the first member of the first equation is a;— y, or mx-\-ny. In the latter case (2) is multiplied by 4mn, and then subtracted from the square of (1). 2. Solve 8a;»-22a^+15i/»=5 (1), 2a?-8y=l (2). Here (1) is equivalent to (2a?-8y)(4a?-5y) = 5 (8); substituting in (3) for 2x—Sy its value from (2), (3) becomes 4iX—5y=5 (4). From (2) and (4) we find ar= 5, y= 3. Tliere is, therefore, only one solution in this case. That this must be so may also be seen by substituting in (1) for one unknown its value from (2), when (1) is reduced to a simple equation in one unknown. ELEMENTS OF ALGEBRA. 179 8. Solve 4a!»+12jry+9i/''=64 (1), Sx-y=l (2). Here the left hand side of (1) is a perfect square; therefore 2i7-f3j/=±8 (3). Combining (2) and (8), we find a;=l, w = 2; or 6 26 4. Solve o?-2xyi-if=d (1), 2x-y = 4: (2). From these two equations we may derive an equa- tion whose first member shall be a perfect square, as follows : Multiply (1) by m, and add to square of (2); thus (m+4)a^-2(w+2)a?y+(8wi+l)2/2=9w+16...(3). In order that the left-hand member of (3) may be a perfect square we must have, by Art. 242, (jw-f 2)^=(m+4) (Sm+l), 9 or, m=-^. 9 If, therefore, (1) be multiplied by — ^, and added to the square of (2), we will get the left-hand member of which is a perfect square ; therefore x—5y~±7 (4). IS 10 Combining (2) and (4) we find ii*=-— , y= — — ora?=8, y=2. m2 rMm M: ■■, ajy=?. 28. aa!-by=Pt xy=q, SIMULTANEOUS EQUATIONS, SOME OF WHICH ARE OF THE SECOND DEGREE m THREE UNKNOWNS. 264. The following examples will illustrate the methods of proceeding to solve equations of this dass. 1. Solve «+2y+8«=14 (1), * 2xy-['Sxz=lS (2), y«=6 (8). Here (2) may be written (2y+8«)a?=18. If now we write u for 2yH-8«, (1) and (2) may be written x-{-u=14kf a?M=18; from which we find x=l, w=18; or a?=18, w=l. Choosing the former solution we have from (2) and (8), 2y+Sz^lS (4), y«=6 (6). Square (4), subtract 24 times (5), and extract the square root, 2y-8«=±5 (6). ■''mil m -0m SI m 188 ELEMENTS OF ALGEBRA. Q Combining (4) and (6) we find y=2, «=8; or y=- , 4 9 4 «= 5. Thus we have the solutions 1, 2, 8; 1, ^, ^; and by taking a;=13, m=1, we would get two other solutions. 2. Solve «"+2/»+s«=61 (1), xy-]-xz=SQ (2), y«=18 (3). Add the sum of (1) and twice (2) to twice (8); thus, a?+2/+«=±18 (4). If now u be written for y+z^ (4) and (2) may ba .written a?u=86; and the solution is effected, as in Ex. 1. 8. Solve a!»+t/»+2«=110 (1), a? 4-y +^ = 18 (2), y;2= 80 (8). Add (1) to twice (8); a^+{y-^zy=nO (4). If now we write u {or y-\-z, (4) and (2) may be written a^+M« -170 (5), a?+M =18 ........... .(6). From the square of (6) subtract (5), 2;rM=1.^4; and the solution follows as bftfore. BLEMmTB OF ALGEBRA. 188 4. Solve yz=a* (1), • gx=b^ (2), ^=c» (8). Multiply together (1), (2), and (8) ; thus, xyz=Jc:^bc (4). Dividing (4) by (1), (2), (3), we obtain the two solutions he ca ab he ica ah T' T' T' "'~^' ""y^ ""T* Exercises, LXVI. 1. a^+2/*+«»=14, x+yi-z=Q, y«=-6. 2. d^-\-f+z^=S6h a?H-«/+«=10, a;y+a?«=16. 8, 0^=12, y«=20, zx—15. 4. a;'-t-^+«'=21, yz-\-zx+xy='-Qf a?+y— «=— 5. 5. a!»+3/»4-2'=60, jp+y+»=li!, «y+a?«=27. 6. a;'+/+»*=90, a;+y+«=6, a?«=28. X y z xyz « 8. 2y«=a(y+»), 2«a?=6(KH-a'), 2xy—€{x-\-y). 9. aj(y+«) = 2a, y(;s+a;) = 26, «(a;+y)=2c. 10. x=ayZf y—hzx, z=cxy, 265. Equations which do not come under the head of those already discussed can sometimes be solved by previous metliorlM, if special artifices are adopted to transform the equations into simpler forms. Some of these artifices are exhibited in the following ex* JSf'''"!-S fIdW ■Wm 'I 194 BIEMEKT8 OF ALOEBRA. amples. The successful application, however, of such artifices can only be acquired by long experience; 'and the student is recommended not to spend too much time at such exercises* 1. Solve 6a»+8aJ5f=26 (1), 8j(»+2a^= 7 (2). Whenever, as in this case, all the terms involving the unknowns are of the second degree, assume y=vx: (1) and (2) then become a!^(5 +8v)=26 (8), «* (8»»+2v)= 7 (4); | dividing (8) by (4), and clearing of fractions, ire obtab 78v*+81v-86=0, 1 8fi the roots of which are - and — — -. 2 89 Substituting these values successively in (8), or (4). we get x== ±2, and therefore y= + 1 ; or a?= ±?| V 16, and tiierefore y= ±Z V 16* lo 9 2. Solve 18yV ^+8==6aj»+20y..:......(l). y Here (2) is homogeneous and of the seeond order« In such a case the ratio of a; to y can always be deter« mined. Thus let ys^rxt and substitute in (2). tLmnnma of algebra. 185 Therefore f*-10r+16=0, r=8, or 2. y=:8a;, or y==2a). Substitute either of these values in (1) and a qua- dratic in X results. 8. Solve 0*4.2^=: 188. .(1). x-^y=i 7 (2). Assume xssu+t, y=su— v ; (1) and (2) then become 2u»+6m«^=138 (8), n Substitute for u in (8) its value i, and we obtain 2 -4 Whence the values of x and y may be found. ^^. /■,'* ■'"■tiri , ''9''i**<**A*«^I^MMlM Distern. In what time would the first cook alone fill) and in what time would the second alone empty the full cistern ? INEQUALITIES. 267. From the definition laid down in Art. 45 we infer that one quantity a is greater or less than another b, according as the difference a^^h is positiye or negative. Thus, a*+6">2a6, because the difference 6 we infer 6a>6bt and i^>ib. . 270. If a ^6, then ma^mh, if m be negative. Thus if a<6, then— 4«> —46, and — |a> — 16. 271. If a^6> then a+c^h-hc, where c is positive, or negative* - • > ^ •! : , Thus if a > 6, then a »- 6 >0, and conversely. Hence quantities may be transposed from one side to the other by changing signs, i).s in equations. 272. l(a>bf then - t according as a and b have a>o ELEMENTS OF ALGEBRA. 103 like or unlike signs. For the difference ~—-=- - ah ah ^ which since 6— a is negative, is negative or positive according as ah is positive or negative, that is, accord- ing as a and h have like or unlike signs. 273. In like manner it may he shown that if a<& , then -^r according as a and h have like or unlike a^o signs. 274. If «i>6i, a2>&2» &c., then «!+%+ &c. >6i + 68+&c. ^ For the difference 26c, (j"+a*>2m, and a»+ft*>2a&, we have 2 (a"4-6"+c')>2 (6c+ca+a6) ; and therefore 276. If a ^ 6, then a.**"*"* ^ 6"*+S where n is a posi- tive whole numher; for the members of the latter inequality have the same signs as those of the former. ^ Thus from a > ft we infer that a" >f^t a^> V, &o. We cannot, however, infer that a*>I^. For example, although 4>— 6, it is not true that 4*>( — 6)'; and •lihough ~2> -4, it is not true that (-2)*>(^4)>. **■ '','<■■ ¥■'''■ I'll Pi F;^ d '; m I ■?*ii 194 ELEMENTS OF ALQEBRA, 276. If (i">fc", then a cannot lie between —6 and h\ and if a*9, a cannot lie between —8 and +8; if fl'< 10, a must lie between —4 and +4, that is a> -4 and < +4. 277. By the aid of this proposition, and the con- dition for the existence of real roots in a quadratic, we can find the limits to the real values of a fraction "*'+''*+"., as in the following ExampleB! of the form jpx"+(l-y)(8-42/), ^^ or, (2y-l)> -V 4 and 8 = Hence 2y 2. Find a positive number such that the sum of it and its reciprocal shall be a minimum. Let X be the number; then a?-f--=y is to be a SB minimum. Therefore, in the equation a^—yx+l=0, we must have ^^4; and, therefore, the minimum value of y is 2, and the value of a corresponding is 1. ELEMENTB OF ALOEBRA. 105 8. If a, 6, c bo poeitive, prove — -f — -f- ^_ a b c Since 6'+c">2k, we have in this case, dividing by be, \^l>^ w. similarly, - 4. - >2 (2), d c h\>'--- («)• Now multiply (1) by n, (2) by /;, and (8) by c, and wo obtain, by addition, .v » be , ca , ab therefore _ ^_ -j_ __>«+/>+ c. « c EXEBGISES, LXIX. 1 . Prove that 4 < ic" - 8a? + 20. 2. Prove that y + - > 2, if a and b have like signs. 8. Find the values between which x must not he- in order that 4a;"+4a?— 1 may be positive. 4. Divide a line a into two parts, such that the rectangle under the parts shall be a maximum. 6. Divide a line a into two parts, such that the sum of the squares on the parts shall be a minimum. 6. Prove that ~ -'-r hes between 8 and 7, . ar+a;+l o 7. Prove that :—- cannot lie between 1 and 25. x—\\} n2 I " , '.1 ^„Wi l\ m^ w t •i* 196 ELEMENTS OF ALGEBRA. 8. Prove that -'-:r7i t^— cannot lie between 1 and 2. 12.1^-8 rt" I)' ia-by 9. Prove that — | — - — cannot lie between . X a—x a and 2x—l 10. Prove that ^-^ — ^ — :. can have no value between 2ar»-2a?-5 -r and 1. 11. Prove that the greatest value which a. - ; - a^+a^ admits of is — . — 12. If rt, 6, c be positive, prove RA.TIO AND PROPORTION. 278. The relation which one quantity a bears to another h, as measured by the fraction t , is called the I'atio of these quantities, and is expressed either AS a fraction r , or by the notation a : h (read a to b), 8 Thus the ratio of 3 to 4 is expressed by - , or 8 : 4. 279. The quantities which constitute b ratio are . 283. A ratio of greater inequality is diminished or increased, and of less inequality increased or dimin- ished, according as each term of the ratio is increased or diminished by the same quantity. a Let T he the given ratio. If x be added to each term a-\-x < a we get the- ratio j-r;- > which is ^ -- according as ab-\-bx ^ ab-\-ax, that is, as > bx ^ ax. Now (1) if a>b, then bxax, if X is negative. In the former case the ratio is diminished, and in the latter increased. (2) If a < 6, hx < ax, if X is negative, and bx > ax, if x is positive. In the former case the ratio is diminished, and in the latter increased. i '•iriii SS'; i 'I ' m W '111, 198 ELEMENTS OF ALQEBEA, 4 6 Thus if 2 be added to each term of ^ , we get -= , and the ratio is diminished; and if 1 be taken from each o term we get ^ , and the ratio is increased. etc 284. The ratio v^ is said to be compounded of the ratios r , ;i»* the ratio j-r. is said to be compounded of ct c e the ratios r , ^, 5 ; and so, generally, any ratio is said to be compounded of the ratios expressed by the fractions whose product expresses the given ratio. Thus the ratio compounded of - and ^ is -- . 285. The ratios ^, y-^; fitj^t &c., are called the duplicate, sub-duplicate ; triplicate, sub-tnplicate, &c., ratios, respectively, of ^ . 2 4 Thus the dupUcate ratio of 3 is ^ ; and the sub-trip- 8 2 licate ratio of ^" is - . V 286. "When two ratios are equal they constitute a proportion; and the four terms are said to be pro- portionals, ft /> Thus ii a : b :: c : dy or - — -. , the four quantities a a, 6, c, d are said to be proportionals, the equality between the ratios being expressed by ::, or =. 287. In this case a and d are called the extremes, 6 and c the means. ELEMSNTB OP ALOISBRA. 109 288. From the proportion t = j » we can dednce certain other proportions, as follows : (i.) Since ad=hc, we obtain on dividing by ci, l-\ (1)- (ii.) Since be=:adt we obtain on dividing by ab, ^=f (2). (iii.) By adding 1 to each ratio of the given propor- tion we get (iv.) By subtracting 1 from each ratio of the pro- portion we get b ^ d *' a — b c—d ^^» »7 . or^ — - •— J ••••««••• •t«^*i» (v.) By dividing (8) by (4) we obtain a-\-b c-^d yjj* a—b c—d In Geometrical language (1), (2), (8), (4), (5) are said to be proportions which follow from the given proportion by taking its constituents r., h, c, d alter- nandOf invertendOf componendOfdividendOf cow/gonendo et dividendOf respectively. '. 'i 1. '.■ 1 Sp'^ *'h :ii ^!!! m 8p0 KLSMKNTS'Or AhOVBRA. a c 2B9. If . =i j , aiul/((r, />), <^(rt, b) donoto any two homo(jfonoouH functiouR of n (limoiiHioiiH iii a, b, thon /(a, h) ^f(r, d) 0("' /') " 0(r, d)' For lot ,- .=.»•; ihou a -.-.-- h.r, c=d.v. d Substitutiug thoso valuoH for a and r in /(a, h), Ac, wo obtain /{a, b)^-^/{h% /)) = /;•»/*(.«•, 1), Hinco ovory torm of /(/m?, /») iH of »j dinionHionH in h. In liko manner wo j^ot VyOi,/>)=/>'»V.(.r, 1); j\i\d)=dy\.v, 1); ^^(.n d)=:dy(c, l). 0((i.,/>) /^-yil-r, 1) 0(.r,l)' and /(^'O_{c,dy ■-■■■■ - ■■] , ■ ;.r- , ''''-. ' . . ,, , Examples. ^* ^^b- d' ^^^"^ '^ib^sW - 2cd-S^' Here the terms of the last proportion are homoge- neous functions of two dimensions. Let ~z=z -=x, and substitute bx for a, and dx for c. 6 a Then and Therefore a^^ab - 6« _ Ir^a^+bKv - />» _ a^+ar-l ^ 2a6-3i* ~ 2b''x-Sb^ ~ 2a;-8 ' 2cd-8t? ~ ~"25";r^8a 5r»-8r^/» rt _ c fur ~ »ah^ _ Or-" - 8r^/" ft ~ d ' ^'^^^ 2a''/>+76» "" 2(A/+7^8' 8. If = , prove , -' = „ '^/a. 4. If^ = 5,prove^^^^, = ^^,_^^. 290. If ^'^=^^=^=...=.% then each of these »1 «2 «» «« : :I ratios is equal to 1 »h*? Hr wjjftj 4- wig/v^ 4- . . . 4- m„i;; ) where m^, w,, w?,, &o., are any positive or negative quantities. Forlet ''L=:^a=&o.,=a?. Then a^ = biX^ a^z=b^x, &c. Therefore aj =6?.}", a5=6^j;", &c. ; and mia!'=m^bp\ ?n,a;=m36Saf», &c. fji \£i^'.r' i'fr" illl-'* 20B ELEMENTS OF ALQEBRA. Adding the last set of equalities, we obtain therefore and, therefore, extracting the nth root, we get Examples. 1 . If - = ^, each of these ratios is equal to | ^^ V' h a \lr-\-ar/ Here mi=ma=:l, and 71=2. 2. If -=-=71 each of these ratios is equal to h d J ab-—cd-\-ef' Here iiii=a, m,= — c, »ng=«, and n=l. to 8. If _-^= y ■= ■ ^ -, ^jich of these ratios is equal y+« «+a? x+y 2a?+2y+2« 2 Here mi=m,=»n,=l, and n=l. 4. If^J^ b-\-o c-\-a a+h , prove a=b=c. SLFMENTS OF ALOEBRA. Mf We have, as in Ex. 8, each ratio=^ ; so that 2a=fc+c (1), 26 = c+o (2), 2c=a+6 (8). Bnbtraoting (2) from (1) we get 2a— 26=6— a, and therefore a=b. Similarly a=c. Hence a=b—c, 6. If»!.+f^'=^«!r^=!^^_lf. prove each be ca ab ratio equal to 1. By taking the first and second ratios together, we have each ratio equal to y4-c»-a'+.r=« , prove that o^ — yz y*—zx z* — xy In this case each of the given ratios is equal to a^jc-\-bhi-\-(?z X (3^-yz)-\-y {f-zx)-\- z {z^-xy • ••••••• • t X I) and also to rt«+6«+c» j^-{-]^-{-z^-yz-zx- xy or (g'+fe'+c') ( ^+y±f)___ (2). I nit;' ■ m "::'-'r ;-i'i ■■■'*i "t'J ! ■'■* ii '».!;' Ill ti''/.5 HO 1, ',t?' It If '''if*!, 1 III Si 4 Ill Mi:w I i 808 KLEWFIVT**? OF AlOJUBRA. Thoroforo, combining (1) and (2), each of tho given ratios is oqual to the dononiinator of wliich vaniHlioB idoutioally ; tlioro* fore tlio uumuiator also vaui»ho8. 'li 1 m^ EXKR0I8E8, LXXII. hx — cy _ C.V — ax _ ay — hx a prove that ax -{-by 1. If 2. If _^±^,=^ J+^ , = ^;+% prove that 82a H-85ft+27c=:0. |i;!ij: ! 8. If ?±„J= '+« = «+« , provo that 8«+8J a— 6 2(6--c) 8(c— a) f! 4, If 1 = I = ! , prove that-+ a (y—z) b (x—x) c (aj-y) a y + -=0. c ' : ^ I ■J. 6. K __?__==__?__=__?._, prove that a (y+ «) * («+ a?) c (x +y) -(y-«)+^-(«-^)+-(a:-y)=0. O 6. If ^ = y ■ , Umb+nc-^la) m (ftc+2a— mi) jSLnumrs of alqebra. aoo : -■ ., prove that ft (la-^-mb—nc) ^(i«6-nc)+A(nc-//i)+* (/fl~m/>)=0. 7. If r A that laojny^ mz) my (Ix-^ nx) nz (mx— ly)* w fHy nz prove VARIATION. 292. One quantity is said to vary directly as another when it bears a constant ratio to that other. Thus w varies directly aa y if x ". y is constant, that iB,i[x=skyf where A is a constant quantity. This is also expressed by the sign of variation, oc , written between the quantities, as a; oc ^. Ex. The area A of a triangle is one-half the product of the base b into the height A, or A=^bh; therefore, if b be kept constant, and h varied, A oc A. So that, if h be doubled, A will be doubled also. 298. One quantity is said to vary inversely as another when it bears a constant ratio to the reciprocal of that other. Thus « varies inversely My it x : - is constant, or k '■■WW N Mill i!i,«.il '.':li%i*k|' ! IS; i«;. m m ■ :. .1 ■ ■■/ lit ! ! '''1ii!! 'icii £10 ELEMENTS OF ALGEBRA. This is also expressed thus : a? oc - (read x varies in- versly as y). Ex. If the area A of a triangle be kept constant, bh will be constant, and therefore b will vary inversely as h; so that, if b be doubled, for instance, h will be reduced one-half. 294. One quantity is said to vary jointly as two others when it varies directly as their product. Thus, if a? varies jointly as y and z, xcc yz, or x=kyz, k being constant. 295. If a quantity x varies directly as one y, and inversely as another z, the variation is expressed thus : y ky a: a -, ora;=-^ , z z where A; is constant. 296. When one quantity varies as two others in such a manner that when either is kept constant it varies as the other; then when both vary it will vary as the two jointly. Let X vary as y when z is constant, and as z when y is constant. (1). z being constant, let a, b be the corresponding ralues of x and y; therefore a b .(1). (2). y being constant and equal to 6, let f, d be cor- responding values of x and z ; therefore " : .-W . - a vanes in- ELEMENTS OF ALGEBRA. 311 Hence, dividing (1) by (2) we get X yz c ' '" and therefore x'''i m u 212 ELEMENTS OF ALQEBBA. For let a?=»ty, a?=n«; therefore ' a^=mnyz, or x= y mn . ^ yz. Therefore x a '^yz^ since y mn is constant. 8. A varies as 6 and C jointly; and A=l when B = l and 0=1; find the value of A when 3=2 and = 2. Here A=wBC, where m is some constant. To determine m write for A, B, their corresponding Talues 1, 1, 1; therefore m = l, and A=BC. Hence if 3 = 2, and 0=2, A=4. 4. If £70 pay 10 men for 36 days' work, for how many days will £120 pay 80 men ? Let x^ yy z denote the wages, number of men, and days, respectively. Then since the amount of wages varies jointly as the number of men and days, we have x=kyZf where A; is some constant. Substitute for x, y, z their corresponding values 70, 10, 85; therefore 70=^x10x85, or, k=l. Hence x=^yz, and therefoie if a? =120, and 2^=80, 120= ^^.80«, or«=20. 6. The value of diamonds varies as the square ot their weight, and the square of the value of rubies varies as the cubes of their weights. A diamond weighing a carats is worth m times a ruby weighing b carats, and both together are worth £p. Find the value of a diamond and of a ruby each in terms of its weight w (Carats. ELEMENTS OF ALGEBRA. 218 Let the value in £ of a diamond of a carats be Aa', and the value of a ruby of b carats be kbi, h and k being certain constants to be determined. By the conditions of the question, a%=mkbif ha^+kbi-p; two equations in h and A;, from which we find mp , p h= , k=: Therefore the value in Jg of a diamond of w caratd mpiv^ and of a ruby of w carats (»i+l)a«' pivi (m+l)br Exercises, LXXIII. 1. If a? a 2/, and a?=2 when 2/=l, find the value of X when y=2. 2. 3a?+5y varies as 5x+dy, and a;=5 when y=2» find the ratio x : y. 8. X varies as y and « jointly; and x=8 when y=2 and z=2; find the value of yz when a; =10. 4. If y^/+« xa;+2/-«, and. t'^+/-i-^'»ocar»+/-;sS show that X X z and yacz. ARITHMETICAL PROGRESSION. 297. Quantities are said to be in Arithmetical Progression when they increase or decrease by a common diflerence. The common diflference is the difference between any term after the first and the term immediately preceding. < Thus the series iS) 5, 8, 11....... 20,18,10,14, ' a, a-\-d, a-\-2(l, a-{-S(l, , are in Arithmetical Progression, the common differences being 8, —2, and d, respectively. 298. The terms between the first and last are called Arithmetical mea/ns* ELEMENTS OF ALGEBRA, 216 299. If the first term be denoted by a, the common difference by dy the last term by l, and the number of terms by w, then the series of terms will be rt, a-\-d, rt-f 2(/, rt-|-8 1, 8, 5, • Here a =1, d=^; therefore, by formula (8), . f={2+2(n-l)^2=«^* 2. Insert n Arithmetical means between the given numbers a and b. Let d denote the common differonoe. As there are n-|^2 numbers in the series, the kst one 6 being the n-f 2tb, ^yo hare, by formula (2), 6=a+(n+2-l)d; therefore rf=^=^, and hence the means can be found. If 0=9, ft=24, andn=5, d will be 2i. Therefore the means in this case are 111, 14, IGJ, 19, 21*. 3. Find the number of terms in a series of which tho first term is 10, the common difference 6, and the sum 176. Substituting the given values izr formulas (1) and (2), we obtain .(10+ On =860, ?=10+6(n-l). Eliminating I from these equations, we get n»+8n-70=0, the roots of which are 7 and — 10. From the nature of the ease the positive root 7 mast be taken. ELEMENTS OF ALOEBHA, 217 Exercises, LXXIV. 1. Sum the series 2J+2{^+8i + to 18 terms. 2. Sum the series ff+H-i + to 8 terms. 8. Sum the series 10-f 7-f4+ to 10 terms, 4. Sum the aeries 2J+1J+1+ to 6 terrids. 8 29 2 6. Sumthe series _+-^- 4--^ + to 6 terms. U VO DO 0k ct Lv. ' n — l.n — 2,n — 8, . , 6. Sum the series +- + +...to n terms. n n n 7. Insert 4 Arithmetic means between 5| and 6|. 8. The sum of a series in A. P. whose first term is 2 and last term 42 is 198 ; find the common difference and the number of terms. 9. How many terms of the series 17+15+18+... amount to 66 ? 10. Find what number of terms of the series 6+9 +12+... will amount to 105. 11. Between the numbers 8 and 27 as extremes, find a series of Arithmetic means whose sum shall be 76. 12. For boring an Artesian well 500 feet deep the cost is 28. 8d. for the first foot, and a halfpenny in addition for each foot following ; what is the cost for boring the last foot, and how much for the whole well ? !l'i 'f: \f'^' it ••• a, aVf ar^f ai^ ELEMENTS OF ALGEBRA. 219 are in Geometrical ProgreHsion, the common ratios being 8, i, and r, respectively. 303. The terms between the first and the last are called Geometrical means. 304. If the first term be denoted by a, the common ratio by r, the last term by I, and the number of terms by w, the series of terms will be a, aVf ai^, ar" the index of r in any term being one less than the number of the term ; therefore, for the last, or nth, term we have Z=ar''-i (1). #15. To find the sum of a given number of quan- titiei in Geometrical Progression in terms of (1) a, r, n ', (^2) a, l, r. Let s denote the required sum ; then 8=:a-{-ar-\-ar^-{- -\-ar^~^. Also rs= ar-f-ar'-i- +ar''"~*+tt»", by multiplying the first equation by r. Therefore, by subtraction, we obtain (r— 1) 8 = ar^—at /•■ ^ —' (It- « • • ... ...( iB )« f.i' "t m (. ■ I :. <:'. 'I ,''''H, f up ]}, ■MP' ■■t> Jio Ml tIK ii-,.; m m 111! m ■\W ■m Mr m 220 ELEMENXa OF ALGEBRA. Since from (1) we have lr==ar*f we get, on substi- tuting this value in (2), Ir—a 8 = r-1 .(8). The equations (1) and (8) are independent, and involve the five quantities a, I, n, r, s ; therefore iif any three of these quantities be given, the remaining two can be determined from equations (1) and (8). 806. If r be less than unity, r" decreases as n in- creases, and becomes indefinitely small as n becomes indefinitely large. This is generally expressed by saying that r"=0 if w=oc (infinity); by which it is meant that ?* approaches zero as n approaches oc . In this case the value of s in (2) continually approxi. mates without limit to the value which is called 1 — r the sum of the infinite series, the value of r" being neglected. Hence, r being less than unity, the sum of the nfinite series is given by the formula 9 SSS _ ••••••••• I ^ )• 1 — r ^ Examples. 1. Find the sum of 6 terms of the series 1, 8, 9, 27,... Here a=l, r=8, n=6; therefore from formula (2) we have ft»— 1 8-1 ELEMENTS OF ALGEBRA. 221 2. Insert 3 Geometrical means between 2 and 82. Here the series is to consist of 6 terms, the first of which is 2 and the last 32 ; therefore if r denote the common ratio, the value of the 5th term is 2r*=:32, .•.r=:2. Therefore the means are 4, 8, and 16. 8. Find the sum of the series l+i+i+i+... to infinity. Here a=l, r=si ; therefore, by formula (4), =2; 1-* that is, the greater the number of terms in the series, the more nearly does its sum approach 2 ; and if n be indefinitely large, the difference between the required sum and 2 is less than any assignable quantity. 4. Eeduce the circulating decimal '8242424... to a finite vulgar fraction. Here -82424... =-i+?l-+?l_+... 10 10» 10» The terms after the first belong to an mfinite series 1 24 of which the common ratio « .— , and sum = ^* M ■'ill;- 1 I' -1 i !• .,:|, id:, V\ V,;: ■-i .»^m' ■' ('6 .1, , ■ 'fill' ami ii'tii,' 1 1' i„'n .'i!^ Ii'ii< vm .;: I'lf "I':'. :Mi 822 ELEMENTS OF ALGEBRA. Therefore the value required = — -}-_=?_. ^ 10^990 990 Such examples are better solved as follows : Let s= -32424... therefore 105=3-2424... 1000s =324-2424.. subtracting the last two equalities we get 9905=321. Therefore «=??!. 990 Exercises, LXXV. 1. Sum the series 4—2+1 — ... to 6 terms and to infinity. 2. Sum the series f — J+f— ... to 6 terms and to infinity. 8. Sum the series 9+3+1+.. . to 5 terms and to infinity. 4. Sum the series |-(f)^+l-... to 5 terms. 6. Sum the series (^)"'*+l + (^)*+... to infinity. 6. Insert three Geometric means between 2 and 162. 7. There are firve terms in G. P. ; the snm of the ELEMENTS OF ALGEBRA. 223 even terms is 4J, and of the odd terms 8t\. Find the series. 8. There are seven terms in G. P. such that the sum of the first six terms is 157i, and of the last six 315. Find the numbers. W 9. "What is the infinite Geometric series of which the sum is 2>- and second term — ^ ? 10. If a and b are respectively the Arithmetic and Geometric means between two numbers, find the numbers in terms of a and b. 11. Eeduce 'BlTSeTSe... to an equivalent vulgar fraction. 12. Eeduce '3526464... to an equivalent vulgar fraction. ;'f£' ■M.it HARMONICAL PROGRESSION. 307. Three quantities a, h, c, are said to be in Harmonical Progression when a : cy,a^b :b—c. 308. Any number of quantities are said to be in Harmonical Progression when every three consecu- tive ones are in H. P "'\[ WUmi*' 309. The quantities between the first and last terms of a series in H. F. are called Harmonical means. 310. The reciprocals of quantities in H. P. are in A. P. S24 ELEMHNT8 OF ALGEBRA. Let a, bf cheia. H. P. ; therefore a, tt'^b c fc— c* or, ab—ac^ac—bCf divide by ahc. c b b €t and therefore -,r-,- are in A. P. ab G Examples. 1. Find the Harmonic mean between a and b. Here we must first insert an Arithmetical meao between _ and -. If this be called x, wo have a b a 6 or x 2ab ' Hence the required mean ss-i OB a-\-b . If a, 0, c are m H. P., prove — ^ . Since a, i, « are in H. P., we have and therefore c 6— c a— ft ft — c SLEMENTS OF ALGEBRA. How, each of these latter ratios h eqttal to f f a+c god ^^J—-— — , which are formed by adding and snb- a—c Iraoting the antecedents and consequents ; therefore fl— o_rt+c--25' EXBBOISEB, LXXYI. , ' 1. Find two Harmonic means between 8 an^ 4. 2.* Insert four Harmonic means between 2 and 12. 8. The Harmonic mean between two quantities is f of the Geometric, and the Arithmetic mean is 6 ; find the quantities. 4. If a, &, y making forms (1), lumber of er of un- mknowns B is equal te all the efficients; ELEMSNT8 OF ALOEBRA. 227 and, when the number of equations is greater than the number of unknowns, we can eliminate aU the unknowns in more ways than one, and so obtain several inde* pendent relations among the coefi&cients. We shall apply the method of determinants to the consideration of I. One and two equations in two unknowns. n. Two and three equations in three unknowns. HI. Three and four equations in four unknowns. 81S. I. Let there be one equation in x, y, then r = — ; and therefore - = — ^* (1). By putting y=l we deduce from (1) the solution of the simple equation in one unknown 0^0;+ ^i=0, 314. Again, let there be two equations in two un- knownSf aix+biy=0 (1), fi^ Cia,-l|!'' ■I " '! •! I l'*- r.u. /."(vli !•! "H), •I'll, III w m it 11 I BLEUENTB OF'ALaEBBA. Here z=sl, and —a, 6, c may be considered the oo- effioiente of unity; therefore the eliminant required is 2, 1, -a, 1, -1, b z^O. 2, -8, c Expanding this we obtain 2 that is, 2 (-e+8&)~(«-8a)+2 (6-a)=0« or a+86-8c=0. -1, ft -8, c — 1,-a -8, e +2 -1, ft =0, 6. Eliminate a, ft, and c from ^ In like manner, by eliminating y and u, and ^, «» we ]Kroye each of the preceding ratios equal to ft «i> hy d\ ss Oj, 61, Ci «s, 6a, *8> ^'s «8» *8» ^8 Hence when we have three equations (1), (2), (8), we can determine the ratios xiyvz : u from the formulas X y z w 6„ Cj, rfi 6«, Cv da «1, <•!, <^l «2> <^3> ''a «8> ^8, C?» t*l, 61, dl «j, 6a, dj ff», 6„ d, — «fl» 6a, Cj a„ 6„ c :fH,i»^l ■' 820. if we put t(=:l, we derive from the last for- nmlas the solution of the equations Oja? + 61^ + Ci« + d, s= 0, «r»+ 692/ + Ca* + da = 0, «l»+%+C8«+<^=0f »8 BLEMENTB OF ALQEBBA, in the form X=s t &0* «!» hv Ci «1, t>v c* «8. bv 6„ Cj, d. »"* &„ c„ d^ 64, C4, ^4 64, C4, ^4 64, C4, (^4 bit c„ <^ respectively, and add. The coefficients of y, «, t<, yanish identically, and we ohtain ( 6„ c„ d^ ^ 64, C4, ^4 — &G. >-d;sO» and therefore XLEMENT8 OF ALGEBRA, -«4 hv Ctt ^ 6„ c„ et, ho Co ^i K Ci» ^ 6„ c„ d, 5j, c„ d, -0, 64, Co d^ =0 (6), +«» 61, c„ di 6„ c„ d, 64, C4, (^4 wbicli is tlie eUminant of (1), (2), (8), (4), and is nsaally written, in the determinant notation, Oj, ftj, Ci, dj 0,, 6„ c„ d, «8. t»» c,, ^4 =0 (6), {he left-hand member of the last equation being the determinant of which Oj, 6^, c^, dj, &c., are eomtituenu, and Oi &| Cg ^4 &c., elements* Thns, according to the preceding definition, the Talueof 'M\ wm 1, 1, 1, 1 2, 4, 1, 1 4, 1, 2, 6 2, 4, 2, 8 'N IB 4,1,1 1,1,1 1, 1, 1, " 1,1,1 1,2,6 -2 1,2,6 +4 4, 1, 1, -2 4,1,1 4,2,0 4,2,8 4, 2, 8, 1,2,6 the Talae of which will be found to be -15. i.i'li^i JBLBMBirtB OF ALOBBBA. Xn like mumer, we have Expa 6, -10, 11, -82, -85, 84, 11, 12, -11, 2 1, d, 8, He: -85, 84, 12, -11, 2 5, 8,0 +82 -10, 12, - 5, 11,0 11,2 8,0 -10,11,0 -85,84,0 - 5, 8,0 -10 - -86 12 ,11, ,84, -11, 2 +11 the value of whioh will be found to be 8100. 2. I Examples. 1. Find the ratios x:y:»:u from 2aj+8y— 4«— 10i*s=0, 8a;— 4y+2«— 5u=0, 4a?— 2y+8«— 21tts=0. By formula (4) Art. 819 we have m y 8, -4, - -4, 2, - -2, 8, --1 tt 10 5 21 • — 2, 8. 4, -4, -10 2,-6 8, -21 2, 8, -4 8, -4, 2 4, -2. 8 2, 8, -10 8, -4, -^ f Her< -2, 6, 3, - lo; 8 1, Exp] Ther -10 ELEMENTS OF ALQEBBA. 241 Expanding the determinants we get ^ - y . 296 286 ~T77 "59* Hence the required ratios are 295:236:177:69, or 6 : 4 : 3:1. 2. Solve the equations 10a?— 2y+4«— 10=0. 8a?+6i/+8;s— 20=0, ^-j-3t/-2«-21=0. Here u=l, and therefore, as in Ex. 1, X 2, 4, -10 5, 8, -20 — 3, -2, -21 10, 8, 1, 3, •2, •10 ■20 ■21 10, -2, -10 8, 6, -20 1, 8, -21 1 [ 10, -2, 4 8, 6. 3 1, 8, -2 Expanding the determinants we get 676 Therefore 768 -576 192 576 x= 192 8, i; III'' ii' ]! , 'I" n 11 ''illliil!! m if m\ '!■!'■ I. ' ' 248 ELEMENTS OF ALGEBRA. y= «=- 768^ 192" -576 192 4, = -8. 8. If CD _ y _ , finda:6: mb-\-nc — la nc-{-la—mb la-\-mh — nc e in terms of ar, i/, Zy i, m, w. Let each of the given ratios be equal to \ ; then the given equalities may be written — la-\'mh-{-nc-'~=Ot A ia+wfe— wc— -=0. \ Here we may consider a, 6, c, - to take the place of \ Xy y, «, M in the standard equations ; therefore by (4) Art. 319 we have a my w, —a? -^ w, —a; -2, m, —a? n, — w, -2/ —2 2, n, -y — n, —;? 2, w, — « 1 X -I. ?», n I. - -m, n I. m, — n Expanding the first three determinants we get inda:6: hen the place of by (4) —a? -y — z ELEMENTS OF ALOEBRA. 24S a — 2wm {y+z)~ —2nl {z+a))~ —^llm {.r-^if) and therefore the required ratios are win {y-\-z) : nl (z-^x) : Im {x-{-y). 4. Eliminate x, y, z, u from the equations ax-}-hy-\-z-\-u=:0, x-{-ay-\-bz-\-u=0, ii!-\-y-\-('Z-\-bu=0, hx-\-y-\-z-\-au=0. By formula (6) Art. 821 the eliminant of these equations is a, h, 1, 1 1, a, by 1 1, 1, a, b by 1, 1, a =0, which on being expanded becomes a*-2a2 (1 + 26) +4a (l+fe2)-46+2J«-6*=0. 5. Eliminate a, b, c, d from ax-{-by-\-cz—d=0, ax'+b7/+cz'-d=0, ax"+by"+c!^'-d=0, ax"'+by"'+cz"'-d==0. q2 .!;'. 1 !■■! :, i 111 .;.!'' ill' III' ' : ' M mm ,1 ll !!• |!;l ii'i'il M 244 ELEMENTS OF ALGEBRA, a?, ?/» «. -1 a^\ ?/'. ^', -1 *•", ?/". ^",-1 a;'", /', «"'-! Here a, b, c. d take the place of .r, y, 5f, u in the standard equations, and therefore the constituents of the deterniinant will be the constituents of a, b, c, d in the proposed equations. Hence the eUminant is =0, which may be expanded as before. 6. Find the condition for the co-existence of the equations ax-\-by-\-c=:a'x-\- b'y-\-c'=a"x + b"y + c"=a"'x + b"'y H-c'". Let the common value of these quantities be \; then the equations may be put in the form ax-\-by-\-c—\=0^ • a'x^b'y-{-c' -\-0, a"x+b"y+c"-\=Oy o"'a;+6"'y+c"'-\=0. Here 1 and \ take the place of z and u in the standard equations; and therefore the constituents of the determinant will be the coefficients oix^ ^, 1, \ in the last equations. ELTIMENTS OF ALQEERA. 246 Hence the elimiuant is a, 6, c, -1 «', b', c\ -1 «", y\ c", -1 a"', h"\ c"\ -1 = 0, vhich on being expanded gives the required condition. 1. Evaluate 1.1,1, 4 2,4,1, 8 4,1,2,13 2,4,2,11 2. Prove Exercises, LXXIX. 5,-10, 11, -10,-11, 12, 4 11, 12,-11, 2 0. 4. 2.-6 7,-2, 0,61 -2, 6,-2,21 0,-2, 6,8 6. 2, 8, 4 (6+c)^ a" a" (c+6)« t'S 8. Prove 0, c, 6, d c, 0, rt, e 6, a, 0, / d, e, /, cS (a+6)» = 2a6c(a+6+c)» = a?da 4- &V 4- c^/ - 2a6ffe - Ucef- 2adcf, 4. Prove 0, 1, 1, 1 1, 0, z\ f 1, z\ 0, x" 1, 2/=, ^S 0, X, y, z a-, 0, z, y ?/, «, 0, a; «, 2/, X, 6. Find the ratios x :y : z lu from the equations 4a;-8i/+«+17M=0, 9a;H-72/-65(+84M=0, lla;-2y+7«-8M=0. II ■rl,.i' • 1 ' ';%.;':! > -I '■'i.il'^ ih i ' iil> :^Mi, i' ii'lii I' 't I )■ I II' 246 ELEMENTS. OF ALGEBRA. 6. Solve the equations x* — 1/4-«=3, 8^'-{-y— « = 18, ar — y — «=— 1. 7. Solve the equations 8. If bz-\-cij=^cx-{-az=ay-\-bx~\y prove x :y i z w equal to a(a — 6 — c) : i(6 — c — rt) : c{c — a — h) : — 2fli(?. 9. If x=hy-\-cz-{-dUj y—ax-\-cz-\-dUi z=ax-\-by-\~dUf u=ax-\-hy-\-cZf abed prove ^-— 4- rrr + Ti — 1" i~r:i =i« 1+a l+t' 1 + c l+» 1. tU ANSWERS. I. 2. a+6. 8. .T-f-6+c. 1. a — c. 2. —z. Ss a—d. 4. xy. 4. a+c+tf. •%. ' ''til ' III. 1. a4-64-c = a+(i+c) = rt — ( — 6.— c). a+6 — c = rt+(/> — c) = rt — ( — i+c). a — 6+c = rt + ( — 6+c)=a — (6 — c). a — 6 — c = a+( — 6 — c) = rt — (6+c). a— 6+c — d=a+( — 6+c) — d=« — (6 — c)— d 2. aJ-2/+«-l=-(-a;+2/)-(-«+l). — irH-2/+;2+l=-(a;-2/)-(-s-l). x+y-z-\=-{-x-y)-(z+\), IV. 1. 2a-a?(2/+«). 2. a''- 6(6- c). 3. a -2c (6-1) - V;s^ 4. -12y-» 6. 10a6. 6. 12a;i/. 7. -^a^b. XI. 1. -ac»+12ar2+4a;. 2. S.i-^+ia.-'-laj^. 8. 6a?*y - Sx't/' -\- 10x'yz\ 4. 6j;^ - 3x^ + 6a?*. XII. 1. 8a^-2ar»+a?~3. 2. a?*-9.rH6^-l- 3- ^+2/** 4. a8+a*M+68. 6. a,-»+/. 6. l-2a;-81ar^+72a;» -80a;*. 7. 2«»-18a;*+39ar»-25a;'^+a;+l. 8. 4x«-6a*+8a;*-10.i-3-8ar^-5a?-4. 10. ar»-42/«+122/2-952. - 11. a»+68+c'-3a6c. 12. a*-2«''iH&*+4a6c"-c*. 3 3 13. a!-y. 14. a;^~t/2"- 15. a-b. 16. a^+R 17. a?+2a;i4-a;-4. 18. a^-2+a;-*. 19. a?*+l +a?-*. 20. a-*-l. 21. a''-3a^+3a'"^-a-». 22. a»+2A*+«&-^V» '■?^i I'll! i 'ii: M II pij' w ' '; ,ii it ''■'i'lH ■..It, ■''■' l! W 250 ELEMENTS OF ALGEBRA. A3 a. ft 24. J.i-2-i^^-3a;-i2/''+VV-8. 26. i^'H^'*«^-i+§^-'-ic-^ 26. ju^P-y'P, 27. «™+"— 4a"*+"-V+6a'»+"^V-9a™+"-*iC*. 28. aj^+2a?^'*+%*P. xm. 1. a!»+ar»-a;-l. 2. 2a!«-a;*-6a?+5. . 3. 36d^-ar'+2;»-l. 4. 24a;»+4a;*-3i:«+2ar'+l. 6. 21aj8+14a;'-49a?«-8a^-10a;*+41a^-ar»-14a?+2. XIV. 1. l + 2a?+3ar»+2it'8+a;*. 2. l-2^+3ic«-2aj8+a?*. 8. l--2a?-a!»+2a^+a;*. 4. l-6a;+13ar»-12a;8+4x*. 6. a?*-a;»+VV-2j;+4. ■ 6. a?* - 4ar»+6 - 4a;-'» +a7-^ 7. «" -2ar»+2a?^+.r -2a;*+l. 8. ix-lx^-\-lx^-x^+^x^+l, 9. a»+6Hc'+<^-2a6+2ac-2ad~26c+2M-2(M2. 10. 24a;+48a;». 11. 4a6+4ai+46c+4ci. ELEMENTS OF ALGEBRA. 251 XV. 1. a?-9b\ 2. a*-b\ 3. J-b^. 4. a-b. 5. 16a*-256*. 6. ^Va^-sV*'. 7. ^2—^4. 8. 4x^-i-y^+ixy-z\ 9. 25rt2-96H6ic-c2. 10. 4a*-96*-246»c2-16c*. 11. a^-^xy+if-z'-^-^z-l. 12. 4a'2+9»/H«'-12a,'?/+4a?«-6?/«-l. 13. Aa'-db^-c^-lQ(P-]-Qbc-2ibd+8ca. XVI. 1. 8a:»+272/». 2. a^-Sy\ 4. a^6^. 5. 8a;^-272/l 3. a«-6». 6. a?P+^9, 7. a^x-^+a-V. XVII. 1. -2j;. 2. -3;»i/2. 8. -X. 6. -2a. 6. Ill -5a7/. 7. aH^"c'«> 4. —ah/z. 1. 6a;^ -2aj+8. 8. mx^y — nxf. 6. -5j:8+62/». ; XVIII. 2. -8rta;H5ftaTt/-7c2/». 4. rta'"' — ia'". 6. mx'^—mJ^ — 1, m ■Ii;l •'.il ■■;:|,;'tii .'I'ii''''^ ji! I ; r ■i|l!i|:l||: ill! 252 ELEMENTS OF ALGEBRA. • ■ ■ XiX. 1. l-2x-{-a^. 2. 5ar^-4x-\-2. 8. a^-x+1. 4. a-2+ar4-3. 6. Sx'-9>x-\-4. 6. a-^-Sar+V. 7. .^•*4-a;»+a!2+.^•^-l. 8. ar»-2a;+2. 9. .-"-Sj^'+G. 10. l-3a;+2^-a.'8. 11. ar^-l. 12. .i^-\-2a^-^\iar' 4-2^+1. 13. ='j^-6ar»+i-^'+9. 14. a^-l-^x-\ 16. a?- 2 + .r-\' * 16. y*+y2+l+2/-H.'r*. 17. a;+2+a;-i. 18. a?-4ari+8. 19. a: -3x^-1. 20. A-+2ar?/+82/». 1 21. fl»+2tt''6+2rt6H6'. 22. a2_3/,34.i. 23. aj'-nix -\-n. 24. rt+6+c. 25. ar'-mx+nv'-n. 26. a2-2a6+36». 27. .iH^(y+l)+2/'-2/+l. 28. jc-c. 29. a;2+2/2+52_^.y_^^_y^^ 80. a; +a?y i-x y -\-y . 81. a'+a^r+a^ft'+rtft+flV+ft'. 82. a;+a?V+2/. 33. a;^+a;*y* + 7/*. 84. a+2a 6~*+86-». 35. a;^-ary"*+a;^2/~'' 36. a' -a b '-f ft"*. 87. x-{-2Jx^ -\-Sax^ +2Jx^ -\-a^ 38. a-^-2a;^y"^+2/~^* ELKMENTS OF ALGEBRA. 268 XX. 1. 2a!='-3a7+y, -J. 2. x-a,2a\ 8. a?-ax+a\ -2a\ 4. ar»-a?+4, -Sx-i. 5. 2ar>+8, -5jr^-Qx-d. 7. a? -2a, 9rt^j;4-2a'». 1. 5ar»+lla;+30. 2. ar*+2a;+l. 8. a?-^!i^-\-^x-l 4. 20a;*-3a,H2. 5. 17x*+19A-3+38jr»+7C.r+152. 6. a?Ha^-«-l. 7. 4/ -3a;24-2.r+2. 8. lU'2-2a;+77. 9. 5ar2-12a;+12, 12a?-72. 10. 4ar»+2a;2-2a'+8, 15. 11. 7a;*-2a'3+A-2-3a'+9, -27. 12. 7a;«+14a?+4, -84X--11. 18. lOA-'+Sa.-S- 1,1007+ 10. 14. Saj'^-f-lOar+S, -5ar»-10a?+27. 15. 8a--56, 16ar^+56. 16. lOo^, 10a?«-100. 17. 3^+11j;H19.^'+45, 90a,— 8. 18. 9a;* - 18a?, 4a;«+ 18a;+ 1. it! !•• I SR4 niftMKNTS OP" Ar,ap:nnA. XXfT. I. a. «. 7. n. i. 4. I p. /)V,,i». n. -^2*. xxrir. I. I. a. 47. n. HO. 4. -205. r>. o. «. 0. 7. *2«l^ a 1. XXIV. 1. 20. 2. mm. a. -nar*. 4. i. f>. lo. o. -i. 7. -800. 8. 7^ U. ~8. XXV. 1. .r«-2.r»/-f\v°. 2. 4.r»-12.i7/4-0»A «. l-2.r4-2»/-2.n/-f-.r''-f.'/'. •1. .la»-12r(/»-fO/>«--4./r4-(>/'C + rV 5. t/':" + 2.f7/:« + c'.r'' + a.JV/'c 4- 2.r V 4- A'* « 4.*^_4.r4-r>-2.r~»4-.r-''. 7. .>'*-!- 2^1.1'" 4- "" - 2 4- iH2rtV-2k«4- c*-2a''(i 4- 2hd-2c^d 10. 4.p*-12j^4-18a?''-10.r4-7-2ir-Ha?-'. ELT}MT!NrS OF ALOEBRA. 255 ''\r ' -^2«. (I. 0. --I. XXVI. 1. t/"-! nfz-{ a//:" f .•». 2. 8a" - 80rt2/;+ C4rt/>'' - 276". n. H/r"f mU* I 5/l"r» -f r''a=' -f V 4- 'daWc^+Qa%c''+ba%c^ 4- »«(!4- Ofi»/A' 4- ««"/>'(-•'. XXVII. 1. a'»4-8u^»/4-8.«7yH.'A 2. ar'-SA+Sa^y-t/'. 4 . a-" - 8 A-^y 4- Sxf - if - ^i.iH 4- ^xyz - difz 4- Sxz^ - Si/s^ li- - 4-3»/J*-s°. 6. l4-3j?-5a''4-3^'-a;^ xxvm. 1. l-4a4-6a^-4rt»4-«* 10 256 ELEMENTS OF ALGEBRA. 2. a;*-4a.->»/+6ar»//»-4a.yH-2/*. 4. a^- ia% + 4aV + Ga*b^ + 6«*c» - 1 2a*bc - 4a^lr + 4(f :• + 12tWc - Vlu^bc" - Uc^ - 4i3(;+ 66V4-6*+c*. XXIX. 1. l+a;+ar»+ir»+ &c. 2. l-ar»+a?*-a;°+ &c. 8. l—x+a^—oc*-^ &o. 4. l+a?+a'=+2ar»+ &c. 5. l+a?+2ar»-2a;'+ &c. • 1. 6. l-a;+2ar'-3a;'+ &c. 6. . 9. i XXX - 18. i 1. 2^^-{-a. 2. 2a -86. 8. aj-f 17. i ■ 4. rtHrt + l. 6. K+t'>"- 6. 2a-2+8.r-». 21. ^ 7. 2x^-1, 8. a?+2-a;-^ 9. 3rt-26+4(7. ■ 10. 6jr»-3a;4-l. 11. ar'+4 + 4a?-^ 12. a^-p. 1. 2 13. a;*-a;-*+a?""^ 14. ai^+px^-^q, 15. 5a*+2a4^ -86*. i • 4: al ELEMENTS OF ALGEBRA. 267 XXXI. 1. l4-2a?. 2. 8aj-2. 8. 2a-7aj. 4. 8»+2a?-^ 6. 2aj».T-«+l, 6. 8a?-2aaj+a«. '1 31' ii , ■' r xxxn. 1. 4aa5*. 2. Ba^;. 8» SicV^". 4. tiJuv. 6. 4a*. 6. an/MV. xxxm. 1. x+l. 6. a?— 1. 2. a?+8. 6. a;«-l. 8. aj-1. 4. «— 1. 7. «-2. 8. aj+9. 9. ar»-4a;+8. 10. a?-?. 11. a?+6. 12. a?-l. 18, ar»-.2a;+l. 14. 5aj«-l. 15. 2a;+6. 16. aj+1. 17. a?-l. 18. a^-a\ 19. a^+y, 20. a^+y". 21. a?+8y. 22. ar»-3/». XXXIV. 1. 2 (aj-1). 2. 8a; (a;»+a^+2r'). 8. ax(x-y). 4: ad(2a»-l-8a»&-a6H6')- B 266 ELEMENTS OF ALGEBRA. XXXV. 1. a?+6. 2. 8a:-2. 8. 8a -2b. 4. a;-2. 5. x+y, 6. a;— 8. XXXVI. 1. Sea'^'c*. 2. 105aa;V3. 8. 480^ (a^- 2/*). 4. 2(4a?+l) (2a,-«-5ar»+5a7-8). 6. (a:~l)(aj+l)(ar»+l). 6. (2a:-l) (3ar-l) (4a;+l). 7. (a;-l) (2a;-l) (2a?+l) (8a?-2). XXXVII. 1. 24ar»2/V. 2. (^+6) (ar+5) (a;-6) (a^-7). 3. (a?+l) (a?-l) (707+2) (7a? -2). 4. 5 (8a-26) (4rt+56) (6a+46) (a+6). 5. (aj»-4) (4ar»-l) (a!»-2ar+4). 6. (7a?-2i/)2 (7aj+2i/) (8a?-2i/) (8(f+22/). 1. ahc XXXVIII. 4. 1. ay^—hx hx—mi .___•- « c^hx-\-ahhj 8a^-12a;«+4ar 6a*-T2^^^S' ELEMENTS OF ALOEBliA. 250 i. d?»-l a!^-\-x^-\-x 5. a^j^^a bx^-\ - x^ — ab — ax XXXIX. ax , 2 - ^1 8a;-7 a— 2 ^27. 4. '^^S, ^^27, '^125. 5. '^S, ^^6. XLVI. 1. 7V2. 2. 8 Vs. 8.9^^4. 4.86^8. 5. 5^2. 6. Vs. XLVII. 1. 6\/i5. 2. 6^28. 8. ^'^ . 4. 10^^432. ELEMENTS OF ALGEBRA. 263 6. '^n- 6. ^15-3^2- 7.1. 8. 2V2-2V*3 +2. 9. 2+ '^72-'^ 12- JJ/ 2187. 10. \^¥.~¥\ v^2^»r3», 4^8, -^32. XLVIII. 1. ^9. 2. ^4^5. 8. Ve. 4. ^4. 6' ^3 6. i;5/2^ XLIX. 'II ; ■;■'! ii ::ti| 'tl I; jif !"■(! 3. '432. 1. . 5\/7 2^25 ^Q* 6 2. V7+V8 3.^(1 + 2 V3)(3+V2)- 4. 8-\/5 5. (3- Viy g ( 4+\/2)(4-\/3 ) 2 ' * 13 7. (\/8+A/i2)(V3+V2). 8. (2+V3- V5)(2V3-1 ) 22 INI 1' ' I 11' ifii 264 ELEMENTS OF ALGEBRA. Q (a- V2- VS) (4+8 \/2) ^' — i — • '12. .f. 2(1-a/6+V8)(8+2V5) 16. 11. (^2+^6- V?) VlO 20. 12, 10 L. 1. 5. 2. 8. 8. 8. 4. ■^, 6. 8. 6. 2^. 7. 2. 8. 1. 9. 2i. 10. 8. 11. -2 12. 2. 13. 7. 14. c n — h 2ab r. 15. 16. — -7- a— a—m a-\-b I 17. a+6 18. ab—pq «+*+JP + ? 2(a«+a&+6a) 19- S^^W 20. i(a+6> LI. 1. 105. 2. 8i. 8. 72. 4. 6, 5. 11. 6. 4. 7. 17. 8. 7. 9. 10. 10. 5. 11. 5. ELEMENTS OF ALGEBRA. 2«S r ad '12. Tc 13. a— b 0+6* 16. --+y- 17. 2. 14. 1. 15. |. 18. fg. 19. f. 20. b-a. 21. 2«& a+6 22. a' b —a 23. 2rt-3 4^3^' 24. 2ab ai-b LH. 1. -4. 2. V. 3. 4^j. 4. 4 J. 5. 21. 6. 5. 7. 61. 8. U- 9. 4. 10. 2cTa' 9a 11. (Va+Vfc)". 12.*. 13. f 14. ig- 15. |g|. 16. 25a^^-50:c+9=0. 17. x'^lQ^, 18. r-i i 1 I 5 1,1 11. 5. Lni. 1. 7. 2. 12, 18. 3. 4, 7. 4. 6. 5. |. 6. Je28 10s., £5 148. 7. 28s., 16s. 8. ^£12,000. :i,i !:' ( i ' ■II ■ rf :■■! i!il 2fi6 ELEMENTS OF ALGEBRA. 9. 1 12. £112 10s.,de75. ap 4- hq + cr 10. 13. 13^ miles. 281 min. 11. 14. 80 gallons. 12 mncb. 15. 4f miles. 16. 3^ min. 17. 20,4. 18. 18, la. 19. 35. 20. 22 crowns, 180 half-guineas. . 21. 5t\ min. past 7. 22. 4 seconds. 23. At ^;^' hours, where vi lies between 1 and 11, inclusive 24. 72 leaps, while the hare takes 108. t'-t 25. a'TTT'.' 26. 3 miles, 12 miles. 27. 55 sixpences, 59 shiUings, 4 sovereigns. 28. 21 miles per hour, 42 miles per hour.- 29. 600. 80. 84. 81. 84. 32. 12 miles per hour. 33. 26-^^ min. past 2. 34. Son's share £900, daughter's £600, widow's £750. 85. 4000, 1500, 400. nw-\-n'iv' siv-\-s'v)' civ-\-c^v' W-f-W W-\-W W-^W ELEMENTS OF ALGEBRA. 2G7 87. 15 cwt. 38. 90 men. 89. One milhon. 40. £1750 and £3472. Hi ti LIV. 1. 1, 2. 2. 0, 31. 3. -3, 5. 4. 2^, -8. 5. h-h 6. 0, |. 7. 2, -2. 8. 2V-1, -2V-l7 9. 3, 5. 10. -2, -7. 11. 4, -6. 12. 4, -5. 13. h i. 14. i, -J. 15. i, -|. 16. -f, -^ 17. 2, h i8- 6, -18. 19. 3, -f. 20. 6, 0. bill; 21. -a, -6. 22.^(1±V2): 23.^Z-:i±- 24. liVl + a-^ a 25. 8, -^- 26. 0, -10. a c 10 11 i I i 80. 14, 2-48. 31. I (7± V~55). ^•v.*,. 268 ELEMENTS OF ALGEBRA. LV. 1. ±2, ± V36. 2. 4, 3^3. 3. -1. -^^"^ 4.4,5. 6. -2,i,li^l?. 6. 2,-3, p -liSVs 7. 1, 1, l±2Vl6- 8. 18i, 6. 9. 3, 4, i, i. 10. 6, 3, J, i. 11. 7, 9. 12. 8, 6. 13. 10,-V»-i±JA/363l 14. 1, -V.-|±AV49r 15. 6,-80. 16. ±^. 17. 6, -Hi. 18. ±1. 19. ±l,±V-|i. 1. 8, 6. 2. 6, 4. 8. 24. 4. 8, 8^. 5. 4, 9. 6. £70. 7. 20 days. 8. 15 and 30 days. 9. 64, 36. 10. 24 in. 11. 44 yards, 110 yards. 12. 4 miles an hour. 13. 12, or 8. 14. 87. 15. 53. 16. 25, 30. 17. 9. 18. 64. ELEMENTS OF ALGEBRA. 269 ^i! LVII. 1. (aj-2) (a;-9). 2. (x+4) {x-ld). 8. (2x-l) (3a;+4). 4. (7+5a;) (l-2a;). • 6. (ll + lOa?) (3-44 6. (5-i9a;) (13-1U-). LVIII. 11 1 ** 2. 4. 3. ^ TU* 4. 1 6. 5O5',. LIX. 1. 10,7. 2. 6,2. 8. 2,-3. 4. 4,12. 5. 3, 2. 6. 3i, 4. 7. i, |. 8. 19, 2, 1- if; 4,9. ab ah 9. 6, 12. 10. i, i. 11. a, b. 12. ^:j:j»^q:5' LX. 1. 5, 11. 2. 11, 7. 8. 4. 6. 4. 5, 7. a7o ELEMENTS OF ALGEBRA, 8. a -f- /> '» + '' 9. c {c — h) c {c -^ a) rt_t-2' a^b-'Z "• a{a-b) 6(6-0) LXI. 1. 2, 1. 2. 4, 8. 8. 8jV. Oa- 4. 8, 2. n. h h c. 3, ?. 7. «, 6. a ^^_^^» ^_j_^-' 0. , -r, 0. 10. J. bm — a?/ w?^ -- ir^ (ni-ny m {t-r)-n (< + r) m {t-r)-n {t-\-r)' 8. ((?° ~~p^) '' ~ C"*^^ "~ ^*^) i' (^^^^ ~ '^^) Q~i'f~ p^)^^ nq—mp nq — mp LXIII. 1. 2, 1, 3. 2. 8, 4, 6. 3. 2, 1, 3. 4. 9, 11, 13. ., IS. ELEMENTS OF ALOEBRA. m 6.4,0,6. 6.5,-5,5. 7.2,4,0. 8.5,7,9. 9. i (t+c'-rt), &c. 10. ^ {(i+h-\-c) -a, &c. 11. i (A+c), &c. 12. x = y=z = i^^_^]'^'_^~^ liXIV. 1. 12,14. 2. 85. 8. A- 4. 9,7,6, 5. 8, 4. 6. 112J min., 90 min. 7. JE141 Is. 6d. 8. 72, 04, 50, 48. 9. A in 3 clays, B in 6 days, C in 9 days. 10. 24. 11. {p-\-l)n, {pq-l)n, {q-^l)n. 12. Passed 8, rejected 6, sent back 8. 13. 15, 25. 14. 24, 60, 120. 15. 110, 50. 10. 84. 17. A £75, B £85 2s., C £51 6s. At end each has £53 16s. 18. 60 min., 75 min., 90 min. All together in 24|| min. 19. 562i per mille, 9 37 J per mille. 20. A to B 87 miles, B to C 45 -niles, C to A 52 miles. ,1, If i i 272 ELEMENTS OF ALGEBRA. 21. 19 five-frano pieces, 11 two-franc pieces. 22. 10 bought, burnt. LXV. 1. 11, 1 ; 1, 11. 2. 26, 24. 8. 102, 98. 4. 4, 1. 6. 8, 10. 6. 1, 2; -60, 19. 7. 4, 1 ; -i, -8J. 8. 6, 2; -8^, 7H. 9. 86, 88. 10. 25» 21. 11. i, i. 12. 4, 2. 18. 18^, 16^. 14. 2, 1. 16. 3, -6. 16. 7, 8. 17. 2, i; ^V -|. 18. 2 19. Va«+46+a Va'+46-<» 20. iHiV2a«-6a, i6-^V'2a»-6«. SX. iV2a»-6»+i6, iV2a»^fc3-i6. o'- ELEMENTS OF ALGEBRA, 27» 22. 28. pj^ V j5« _ ^abq p — V jb" — iabq 2a ' 26 2a 26 LXVI. 1. 1, 2, 8. 2. 2, 8i, 4i. 8. 3, 4, 5. 4. 1, -2, 4, 6.8,4,5, 6.4,-6,7. 7.-5,8,-2. ■ii' 8. ahe a6+ tic—bc , &c. ii 1 1 9. ^(oH-g-^Xo+t-o)^,; I I. 10. Vbc* yea* w ab' LXVIL 1. ±8, ±1; ±|V6, ±lV6. 2. *2,d=8; ±2, ±1. 8. +10, 4:6. 4. 6,4; 4,5. 6. 9,7; -7, -9. 6. ±4,T8; ±:7|,±7|. s IMAGE EVALUATION TEST TARGET (MT-3) {/ \V4 U.. m 1.0 1.1 |4S US ^ m us Ki u U I. 2.0 L25 i 1.4 6" 1.6 /Q ^/i '^^*^- ^4^ '/ /A Photogra{Jiic Sciences CorporatiGn V 6^ 23 WEST MAIN STREET WEBSTER, N.Y. 14580 (716)872.4503 <^ % ftii itijgasrar ov JMHPiiU 7. ±6,±a; ±aV^,T«V-i. a a, 2; 2,8. 9. i,8; -2, -i. 10. 10, 15; -101, -16i. 11. 12. |; -7, -f. 12. ±10, ±8 ; Tl* V^, ±67 V ^. Lxvm. I. 89. 2, 28, 82. 8. 16, 17. 4. Ih W- «. 88, 65. 6. 19, 10. 7. 10, 9. 8. 848 oubio in., 64 onbio in. 9. i (1+ V6), 1(8+ Vl. 10. i(8+V^)»i(8-'^-^- « II. i (6± V5), ±i VB). 12. 11, 7. 18. 7a 14. 82. 16. 1, -H. 16. g. 17.92. 18.98. 19. 6 and 8. 4W. «600e At 7 p«r cent., iB7000 at 6 per cent. 5 -1.6 T ^1- •*» 0.5" •22. 6 yeifcrt at 4 per eeni, or 8 years at 2i p«r o«nt. 28. First cook in 8 Iidiinh seccma eoek ia 6 mmammor iziiHSJt 8^ T.YTY . 2 2^ 4. The line is bisected. 6. The line is bisected. 11. 2:8:4. 12. 2:8:4. TiXXT. TiXXT.TL 1.4. 8.lf:2. 8. «. «^ 4. ^=:^ (i^-a^. 5. «=»+2a^. 6. 12. 8. 5 seconds. If ab\ LXXIV. 1. 881 2. 8. 8. -86. 4. 8|. r™'?"!'^ iVt a^iatpro>t, y, « from the thtee eqn^tiont. a, 6, —X Beralt. a', ft', -1 0", ft*, -1 =0. 12. atfe-^aflKf^O, ISXSJ 1. -rl6, 8100, -972. 6. -8:4:7:1. 6. 4j 8, 2. 7. 1, 2, 8.