CIHM Microfiche Series (IVIonographs) ICMH Collection de microfiches (monographles) Canadian Institute for Historical Microreproductions / Institut Canadian de microreproductions historiques Technical and Bibliographic Notes / Notes techniques et bibliographiques The Institute has attempted to obtain the best original copy available for filming. Features of this copy which may be bibliographically unique, which may alter any of the images in the reproduction, or which may significantly change the usual method of filming are checked below. n n D D D □ Coloured covers / Couverture de couleur Covers damaged / Couverture endommag6e Covers restored and/or laminated / Couverture restaur^e et/ou pellicul^e Cover title missing / Le titre de couverture manque Coloured maps / Cartes g^ographiques en couleur Coloured ink (i.e. other than blue or black) / Encre de couleur (i.e. autre que bleue ou noire) Coloured plates and/or illustrations / Planches et/ou illustrations en couleur Bound with other material / Reli^ avec d'autres documents Only edition available / Seule Edition disponible Tight binding may cause shadows or distortion along interior margin / La reliure serr^e peut causer de I'ombre ou de la distorsion le long de la marge int^rieure. Blank leaves added during restorations may appear within the text. Whenever possible, these have been omitted from filming / II se peut que certaines pages blanches ajout6es lors d'une restauration apparaissent dans le texte, mais, lorsque cela 6tait possible, ces pages n'ont pas 6t6 film^es. Additional comments / Commentaires suppl6mentaires: L'Institut a microfilm^ le meilleur exemplaire qu'il lui a 6t6 possible de se procurer. Les details de cet exem- plaire qui sont peut-§tre uniques du point de vue bibli- ographique, qui peuvent modifier une image reproduite, ou qui peuvent exiger une modification dans la mdtho- de normale de filmage sont indiqu^s ci-dessous. [ I Coloured pages / Pages de couleur I I Pages damaged / Pages endommag6es D Pages restored and/or laminated / Pages restaur^es et/ou pellicul^es Pages discoloured, stained or foxed / Pages d^color^es, tachet^es ou piqu^es Pages detached / Pages d6tach6es I y/j Showthrough / Transparence I I Quality of print varies / n Quality in^gale de I'impression Includes supplementary material / Comprend du materiel suppl^mentaire Pages wholly or partially obscured by errata slips, tissues, etc., have been refilmed to ensure the best possible image / Les pages totalement ou partiellement obscurcies par un feuillet d'errata, une pelure, etc., ont 6t6 film^es k nouveau de fagon k obtenir la meilleure image possible. Opposing pages with varying colouration or discolou rations are filmed twice to ensure the best possible image / Les pages s'opposant ayant des colorations variables ou des decolorations sont film^es deux fois afin d'obtenir la meilleure image possible. This item is filmed at the reduction ratio checked below / Ce document est filmi au taux de rMuction indiquA ci-dessous. lOx 14x 18x 22x 26x 30x J 12x 16x 20x 24x 28x 32x The copy filmed here has been reproduced thanks to the generosity of: Dana Porter Arts Library University of Waterloo The images appearing here are the best quality possible considering the condition and legibility of the original copy and in keeping with the filming contract specifications. Original copies In printed paper covers are filmed beginning with the front cover and ending on the last page with a printed or illustrated impres- sion, or the back cover when appropriate. All other original copies are filmed beginning on the first page with a printed or illustrated impres- sion, and ending on the last page with a printed or illustrated impression. The last recorded frame on each microfiche shall contain the symbol —►(meaning "CON- TINUED"), or the symbol V (meaning "END"), whichever applies. Maps, plates, charts, etc., may be filmed at different reduction ratios. Those too large to be entirely included in one exposure are filmed beginning in the upper left hand corner, left to right and top to bottom, as many frames as required. The following diagrams illustrate the method: 1 2 3 1 2 4 5 i thanks uality libility he 9 filmed 3 on impres* . AIS I on the ires- printed he CON- MD"). at B to be led left to as te the L'exemplaire film6 fut reproduit grdce A la g6n6rosit6 de: Dana Porter Arts Library University of Waterloo Les images suivantes ont 6t6 reprodultes avec le plus grand soin, compte tenu de la condition at de la nettet6 de I'exempiaire film6, et en conformity avec les conditions du contrat de filmage. Les exemplaires originaux dont la couverture en papier est imprim6e sont f ilm6s en commen^ant par le premier plat et en terminant soit par la dernidre page qui comporte une empreinte d'impression ou d'illustration, soit par le second plat, selon le cas. Tous les autres exemplaires originaux sont film6s en commengant par la premlAre page qui comporte une empreinte d'impression ou d'illustration et en terminant par la dernidre page qui comporte une talle empreinte. Un des symboles suivants apparaitra sur la dernidre image de cheque microfiche, selon le cas: le symbole — ► signifie "A SUIVRE", le symbole V signifie "FIN ". Les cartes, planches, tableaux, etc., peuvent dtre film6s A des taux de reduction diff6rents. Lorsque le document est trop grand pour dtre reproduit en un seul ciich6, il est film6 d partir de I'angle sup6rieur gauche, de gauche d droite, et de haut en bas, en prenant le nombre d'images n6cessaire. Les diagrammes suivants illustrent la m6thode. 1 2 3 4 5 6 MICROCOPY RESOLUTION TEST CHART (ANSI and ISO TEST CHART No. 2) ■ 6.3 4.0 2.0 1.8 .-4 :dg.^L JED IN/HG E I^.Tj '653 East Main Street ~^ RochestLr. New York 14609 USA SSS (716) 482 -0300 -Phone SSS (716) 288 - 5989 - Fax inc EJ ELEMENTARY ALGEBRA FOR SCHOOLS WITH ANSWERS HY H. S. HALL, M.A. BORIIKKLY SCHOLAR OF CHUIKT's COLLEHK, CAMJIRIDOK MASTER OF THE MILITARY SIDE, CLIFTON COLLEGE .*'♦ AND -. .. Wtol^, B.A., M.k, 'ot.B _*■-■' ' i UE, CJ^il4li tlljyt ■^G&OLAR OF 'trinity COLUEfi SOME-Mlfi-^ASSISTAN^ ■ilASl'ER AT MAULBoioDGH C».LEOE \ ■ - .s. »'-' A. AND W. MACKINLAY ILonlion MACMILLAN AXD CO., Limited NEW YORK : THE MACMILLAN COMPANY ^.n.-* , J PDA Property of the Library University of Waterloo FROM In setting of eleniei should (1 symbols, stood. 1 thought i succession ilhistrate adopted i the comni upon the of frequeii As rcj determine( to introdi subject ar as easy ei opinion tli resolution operations simple exp acquainted encounteri] at tlie san EXTRACT FROM THE PREFACE TO THE FIRST EDITION. i I In setting before a beginner the real and perplexing difRculties of elementary Algebra, there is some fear lest first lessons should degenerate into a mere mechanical manipulation of symbols, uninteresting and uninstructive, because little under- stood. This well-known danger led us to devote special thought to the question of order ; to consider, in short, what succession of the various parts of the subject would best Illustrate its bearings at an early stage ; and we have finally adopted an arrangement, which-if it varies somewhat from the common use of elementary text-books-is at least ],ased upon the experience of many years, and embodies the result of frequent consultation with our colleagues and other teachers As regards the earlier chapters, our order has been determined mainly by two considerations: first, a desire to introduce as early as possible the practical side of the subject and some of its most interesting applications, such as easy equations and problems; avA secondly, the strong opinion that all reference to compound expressions and their resolution into factors should be postponed until the usual operations of Algebra have been exemplified in the c. . of simple expressions. By this course the beginner soon becomes acquainted with the ordinary algebraical procpsga, v^ithont encountering too many of their diniculties ; and he is learning at the same time something of the more attractive parts of P- r/ PREFACE. the subject. Again, by postponing Resolution into Factors until the student has acquired some freedom and readiness in the use of symbols, we are enabled to treat this important section, and all the processes to which it gives rise, more adequately than is possible where factors are introduced and disposed of in one short early chapter. We had originally intended to arrange the chapters in an order that might be followed without deviation from beginning to end ; but in the course of the work this was found impossible without very extensive subdivision in some parts, and needless repetition in others. We have therefore marked with an asterisk all artides and examples which may conveniently be omitted by a student who is reading the sub- ject for the first time, and we have occasionally added a note suggesting the most suitable place for a section which may have to be postponed. June 1885. The distinctive features of the Seventh Edition are :— (1) The definitions of dimension, degree, homogenemis expres. sion are transferred from Art. 10 to Art. 24. (2) Greater prominence has been given to the fundamental Laws of Algebra (see Arts. 22, 29-32, 46-48). With this object parts of the chapters on Multiplication and Division have been re-written. (3) A short section on the use of Detached Coefficimts has been given on page 37. (4) A fuller treatment of the Remainder Tlieorem and its applications will be found on pages 238, 237. PREFACE. ▼ (5) Five new sets of Mtscellaneo^is Examples have been added at convenient intervals, beginning with one on page 32, which replaces Examples IV. c. With the exception of this change and a combination of Examples XI. b and XI.c, which now appear as one exercise, all the original examples will be found under their . old numbers and with the answers unaltered, even where the examples themselves have been improved. As the book has been entirely reprinted, improvement in many typographical details has been secured ; in particular, without any appearance of crowding, space has been found for additions to some of the more important exercises. At the same time it will be found that the pagination is very little altered, the only difference being that the introduction of miscellaneous examples has placed the beginning of some chapters a few pages further on in the book than in previous editions. It is believed that the few alterations in the text will all be justified by usage, and that they will cause no inconvenience to those who are familiar with the book in its old form. The very favourable reception accorded to previous editions encourages the hope that, in its present more complete form, the work will be found suitable as a first text-book for every class of student, and amply sufficient for all whose study of Algebra does nut extend much beyond the Binomial Theorem. I regret that by the early and lamented death of Dr. Knight I have been deprived of the advice and assistance which would have been invaluable in the final arrangement and revision of this our first joint work. H. S. HALL. Jan. 189?. CHAP I. II. m. IV. V. VI. VII. VIII. IX. X. XI. : XII. ; ] XIII. { XIV. ] XV. 1 XVI. I XVII. I XVIII. I XIX. F XX. T. XXI. A CONTENTS. CHAP I. II. III. IV. V. VI. VII. VIII. IX. X. XI. XII. XIII. XIV. XV. XVI. XVII. XVIII. XIX. XX. XXI. DKnNiTioN.s, Substitutions . • • • » Negative Quantities. Addition of Like Terms Simple Brackets. Addition Subtraction . • • • • Miscellaneous Examples I. Multiplication . • • • • Division " • • . . Removal and Insertion of Brackets Simple Equations • . . . Symbolical Expression Problems leading to Simple Equations Highest Common Factor, Lowe.st Common Mul TiPLE op Simple Expressions . Elementary Fractions Miscellaneous Examples II. Simultaneous Equations Problems leading to Simultaneous Equations Involution Evolution ■ • • • J Resolution into Factors Miscellaneous Examples III. , Highest Common Factor . Fractions Lowest Common Multiple Addition and Subtraction of Fractions PAOR 1 9 18 20 23 26 38 46 52 61 69 74 76 80 83 93 98 102 112 126 128 136 144 148 ir. - m Vlll CONTENTS. CHAP. XXII XXIII XXIV. XXV. XXVI. XXVII. XXVIII. XXIX. XXX. XXXI. XXXII. XXXIII. XXXIV. XXXV. XXXVI. XXXVII. XXXVIII. XXXIX. XL. XLI. XLII. XLIII. i( Ml.SCELLANEOrs FRACTIONS Miscellaneous Examples IV. Harder Equations. • • • Harder Problems .... Quadratic Equations . Simultaneous Quadratic Equatfons Problems leadimj to Quadratic Equation Harder Fractions . * • • • Miscellaneous Theorems and Examples Tjie Theory of Indices Elementary Surds Ratio, Proportion, and Variation Arithmetical Prooression . Geometrical Progression Harmonical Pro«ression Miscellaneous Examples V. . Theory of Quadratic Equations Permutations and Combinations Binomial Theorem. Logarithms • • • Scales of Notation Exponential and Logarithmic Series Miscellaneous Equations Interest and Annuities Miscellaneous Examples VI. Answers . page 160 172 178 186 192 202 209 21.^ 222 240 253 270 287 294 301 307 31] 319 329 340 349 355 361 367 371 393 ALGEBEA. CHAPTER I. Definitions. Substitutions. grLtefg^^it'rlor'lhTJh^^^^^ Vl'- ^"t'^^^^^c, but with processef are dZted bT iVm "'"^ in arithmetical value, algebraical quan?fti{frr?de^ d ^v^S';"^^^'^f^"^^« have any value we choose to assign to them^ ^ "^' ""^''^ "^"^ alpTLrantthrih^^^^^^^^^^ those of our own values a symbol mav re«respnf "o restriction as to the numerical piece of work itSs the sam. vll "^If^^ood that in the same we say « let a = 1 " w?do nn?r.. if i^^-oughout. Thus, when alwavs, but only in the mrh^r *^'^* "" T'^ ^^^« "»«^alue 1 Moreover, we mVy^^ta't^ ^^ t Vi^^^^^^^ them any particular numerical vaW Wi In -^ assigning to such operations that Algeb"rlr% 11^4^^^^^^ '' ^ -«^ symbolsT - 'V'l'^^"^"?f!\°^ ^'ff ^-. premising that the Arithmetic. Also ' for fvT,^ .^ *^'^ ^^'^^ meanings as in »ot..^ When „„ .ig„ p„„,,,, , ,^^„ y,^ ^.^__ ^ .^ ^__^_^_^^ * « ALGEBRA. [chap. i 3. Expressions are either simple or compound. A simple expression consht^ of one term, as 5a. A compound expression consists of two or more terms. Compound expressions may be further distinguished. Thus an expression of tivo terms, as ^a-26, IS called a binomial expression ; one of three terms, as ^a~Ab+c, a tlinonual; one of more than three terms a multi- nomial. Simple expressions are also spoken of as monomials. 4 When two or more quantities are multiplied together the result is called the product. One important difference between the notation of Arithmetic and Algebra should be here remarked In Arithmetic the product of 2 and 3 is written 2x3, whereas m Algebra the product of a and h may be written in any of the forms axb, a.b, or ab. The form ab is the most usual, llius, If a=2, 6 = 3, the product ab = axb = 2x3 = 6: but in Arithmetic 23 means "twenty-three," or 2 x 10 + 3. 5. Each of the quantities multiplied together to form a pro- duct IS cal ed a factor of the product. Thus 5, «, b, are the factors of the product 5ab. 6. When one of the factors of an expression is a numerical quantity, it is called the coefficient of the remaining factors. Ihus, m the expression 5ab, 5 is the coefficient. But the word coeflicient is also used in a wider sense, and it is sometimes convenieiit to consider any factor, or factors, of a product as the coefficient of the remaining factors. Thus, in the product 6abc, 6a may be appropriately called the coefficient of 6c. A coethcient which is not merely numerical is sometimes called a literal coefficient. Note. When the coefficient is unity it is usually omitted. Thus we do not write la, but simply a. j *. lua 7. If a q uantity be multiplied by itself any number of times, the product IS called a power of that quantity, and is expressed by writing the number of factors to the right of the quantity and above it. Thus ^ ^^j ax a is called the second power of a, and is written a^ ; ctxcixa third power oi a, ^3 ! and so on. ' The number which expresses the power of any quantity is called Its index or exponent. Thus 2, 5, 7 are respectively the indices of a^, a\ a'. ^ ^ a squared"; a^ is read "«, cubed"; Note, a' is usually read " z* is read "a to the fourth" and so on. I 8. Th coefficient Exampl By 3a Bya^ qua: Thus, Example Here whe Example Here Note. T 9. In factors of 3x4 mean ill each cas In a sim and it is product of In like r of the two have there nch, bac, bt product of what ordei however, t( Fraction; kept in the Example. When the index is unity it is omitted. Thus we do not a\ but simply a. Thus a, la, a^, la» all have the write same meaninir- I.] DEFINITIONS. SUBSTITUTIONS. 3 a The beginner must be careful to diatinguisli between coejfficient and index. Example 1. What is t' -■ difference in meaning between 3a and a3' By 3a we mean t;e . .duct of the quantities 3 and a. By a? we mean the • . rd power of a j that is, tlie product of the quantities a, a, a. . i vi/ x mc Thus, if a = 4, 3a = 3xa=3x4 = 12; a3=:axaxa = 4x 4x4=64. Example 2. If 6 = 5, distinguish between 4?>2 and 26*. Here 462=4x6x6 = 4x5x5 = 100; whereas 26''=2x6x 6x 6x 6 = 2x5x5 x5x5 = 1250. Example 3. If a = 4, a; = 1, find the value of 5««. Here 5a:;»=5xa:xa;xa;xa:=5x 1 x 1 x 1 x 1=5. Note. The beginner should observe that every power of lis l. 9. In arithmetical multiplication the order in which the factors of a product are written is immaterial. For instance 3x4 means 4 sets of 3 units, and 4 x 3 means 3 sets of 4 units • ni each case we have 12 units in all. Thus In a similar way, 3x4 = 4x3. 3x4x5=4x3x5=4x5x3; and it is easy to see that the same principle holds for the product of any number of arithmetical quantities. In like manner in Algebra ab and ha each denote the product ot the two quantities represented by the letters a and h, and liave therefore the same value. Again, the expressions «6^, acb bac, bca, cab, cba have the same value, each denotiijtr the product of the three quantities a, b, c. It is immaterial in what order the factors of a product are written ; it is usual However, to arrange them in alphabetical order. ' Icpn^lf^i^vl'^i coefficients which are greater than unity are usually Kept in the form of improper fractions. Il Example. If a = 6, a; = 7, s = 5, find the value of ~axz Here 13 13 10^ ^axz= ,-7; X 6 X 7 X 5=273. il li If f 10 10 19 ■ I 4 ALGEBRA. EXAMPLES I. a. If «=7, 5=2, 0=1, .r=5,^=3, find the value of [chap. 1. 14a:, 6. ¥. 11. 7c«. 2. ur'. 7. 3ft2, 12. 9M. 3. 3ax. 8. 2a:a, 13. 86cy. a« 4. 9. 6c4. 14. 7y"'. If a=8, 5=5, 0=4, ^=1, ^=3, find the value of 5. 56y. 10. V. 15. 8*2. 16. 9xy. 21. c^. 26. a*. 17. 863, 22. hK 27. 6^. 18. 23. 28. .Sari 19. 24. 29. If a=5, J = l, c=6, ,«;=4, find the value of 31. #*^. 32. ^ax. 33. 3- 34. 2<'. 36. 7-. 37. ^', ijcicx. 38. iftca;. 39. 2.3 1) • 20. 7y*. 25. y». 30. ^bxy, 35. 8». A«5 40 — . *"• 64 a '^l^t^nZ^:To^TjtT^^^ r% ™^^i^P"^^ *^^ether is written a^6W3. And convlrselv ?<^3^^?^ ' J^""' ^<'^bbbcddd ^^xaxaxaxcxd ^J Y''"''^^'^'^' ^a^c^a has the same meaning £^mmple 1. If x=5, y=.3, find the value of 4xy. 4a;y = 4x52x3'' =4x25x27 =2700. ^'xample 2. If a=4, 6 = 9, a.=6, find the value of ^-^\ 86£2^8j<9><62_8x9x36 ^^"^^ 27a» 27x43 ■""27^6T =1=1^ must b? :r-rtXsi"t *r' v'^ "^^^^ p-^-* A factor ol usually' canecU^ro g^^^^^^^ ^'^''''' "^«^ ^--• ThIreVr'^et^:>J'o;ht.'-^(;^S^ T^^ ^ ^^ ^-^"r. Again, if c/o, tl j'o3^iS7S.eT&5^XtuT^^^^^^^ ^' ^' f"'^" a and /> ni-iy have. "^iciuit ao c -o, uiiatever values Note. Every power of Is 0, 1.] DEFINITIONS, SUBSTITUTIONS. 6 EXAMPLES I. b. If a=7, ( -.2, c=0, x=5, y=3, find the value of 5. i6-.r. 6. #6V. 7. t.,^. 8. a«o. 9. a^cy. 10. 8:^:3^. H. ^r^^j:,^, jg. ^xV- If « = 2, 6=3, c= 1,^=0, y=4, r=6, find the vahie of 13. 3aV 86 17. 3a26", 2a-p 21. 25. 7r 64r«' 14. 8a6'^ 9ga 15. 6a3o 62 ' 16. 4cr2 9a3 18. |6a''. 19. 86« da'- 20. Sa'c*-. 22. 3»26. 23. 2''a\ 24. c*6». 26. 27a» 32' 27. 64 28. 6'- r** given expression. Thus the square root of 81 is 9, because 92=81 The square root of a is denoted by Va, or more simply ^a is IS1;^.S; ttteS'ffl If th'ef ^' ^"^ '^^^^^-" the given expression ' ' ^^^^'' ^*^^'' P°^^^ ^« «q»al to The roots are denoted by the symbols V, V, l^, etc. Examples. ^27 = 3 ; because 33=27. ;^32 = 2 ; because 2' =32. The symboVis sometimes called the radical sign. Example 1. Find the value of 5V(6a36H when a=3, 6 = 1 c=8 5V(6a36M = 5x^/(6x33xPx8) = 5xV(6x27x8) = 5x^1296=5x36=180, £«.0e 2. Fu,a the value of ^/(g), „,„„ ,^,, j^3_ ^^^ m ^^^//9x9x9\ 9 "Vv 1000 ;-io' 6 ALGEBRA. EXAMPLES I. c. 1. 4. 7. 10. 13. 16. 19. If .=8 0=0, ^.=9, .=4, .,= 1, find ihe'value of v/(2a). 4/(8a;y), ak )' V(is)- V( 2. 6. 8. 11. 14. 17. 20. N/(l-a:). V(f)- 3. 6. 9. 12 15. 18. 21. 22. 25. 28. 31. 34. Ifa=4 A=l ^_9 7 - ^''^ 6 7/4^3, ''-\f=''^=^'^ = 8,fi„dthevalueof 6v/(463) 1 '3£3\ 4'y)' v?/" 23. 26. 29. 32. 35. 1s/(5dx). 24. 27. 30. 33. 36. x/(c5y). 1 ^(8a62c)- sjd\ [chap. \/(2aa?). 2arV(2tty). l(kax^\ sions which contairmore tlS ? n. T'"^""'^' ^^^"« «f expres- can be dealt with sfng J t he n^:?\ ^V^'^' '^'^ ^ combining the terms the numerirJ li f ?/ ^^^""' ^"^ bv won IS obtained. ""werical vahie of the whole expres- c -5«=5x5x5x5 = 625- 4c = 4x5 = 20; ' 2c3=2x53=2x5x5x5 = 250- 3c2=3x 52=3x5x5 = 75 Hence the vahie of the expression =625-20 + 250-75 = 780. !•] DEFINITIONS. SUBSTITUTIONS. vJ*' ^/ ^'\^ ^ any term whicli coutaius a zero factor is itself zero, and njay be called a zero term. Example 1. If a = 2, 6 = 0, x=5, y = 'A, find the value of The expression = (5 x 23) - + (2 x 5^ x 3) + = 40+150=190. Note. Tlie two zero terms do not affect the result. Example 2. Find the value of ^x^ - a:hj + lahx - fjr^, when a=5, 6 = 0, a;=7,y=l. ^a;2 - a2y + 7a6a; - 12/3=4 . 72 - 52 . 1 + - :^ . P = 29|-25-2l Example 3. When p=% r=C, ^-=4, find the value of 1 » /27 , + 1)- 2x36 9x4" =3x|+8-2 = 6i. to 'the foUoTvi:ig?l„r"'''^^ "-^ ''"^-" *°'"'' Wattentfon .t„u ''"i' "'"* ''^P<>''""'<:o cannot be attached to neatness of noMo en,pl„y the s.gn of equahty in any 'vague anytoextt i„ «,. ^t""'"' fe ""^Pf^sio'is are very short the signs of eaualitv n> the steps o the work should be pfaced one nndlXXr ^ frolu ttl°rl?rWor''?M?'',' °"' ''T "^^ '"^f '°"°-» advisable fr-.jj 1 . ' u . purpose it wili sometimes be its be seen l^t'' '"•''" ^^^P'""""""' i 'h' importance of 8 Ifa=2, 6=3.^= 1. 0a + 5b~8c + 9a. 3. 5a + 3c-2b + 6d. 5. Sab-3cd+2da-5cb + 2db. 7. 3abc-2bcd + 2cda~4dab. 9. 3bcd + 5cda~7dab + abc. 11. 2a2 + 363_4c4_ ALGEBIU. EXAMPLES I. d. 3, c=l,,;.-.0, find the numerical [CiUl'. value of 2. 3o-46 + 6c + 5rf. 4. a6 + ftc + ca-rfa. 6. a^c + bcd + cda + dab. 8. ^bc + 3cd~ida + 5ab. 10. a2 + i3 + c2 + d2. 12. 04 + 64_c4. 15. 3a6c-62,_6^5 ^4. I^c^ - a^ _ ^3 _ |„j;,^^ 16. 2a^+2.3^2c.^,2.._2.c-o,,_,,„_2a. 17. c^+iad*-3a^+b^d 18. «H26^+2o^^.^.+2„,^ 19. 2o^ + 2„^ + 2.^_4c. + 6a.c.. " 21. 6a6-4„,._^^,^,_^^ 22. «*-c«+ft^-^+2a6-2c^. 23. 2a6-f63 + 3ac-2c-rf + _4^„rf. 24. 12o6'«c-9rfs + 3„j,2^ '''-'' '=''^=^'^^=9,^=4, find the value of 25. |a- 163 + |y2^ 26. ^M-Sf-i y^ cxy 30. kj(bxy)~lb^+^. 32. ^^^- 12a2a; ; Nega 16. Ii Itemed t( (signs + i [as l| + 7 which th< jsign - is which no The same ision 7a + j represent , 17. In : greater tl were the In Algtbi terms exc i stand alou Hence a guantitiei I expressed irrespectiv This ide trations. , (i) Sup total gain loses £100 The corr and the ne debt, that is CHAPTER II. Negative Quantities. AdditioxN or Like Terms. 16 In his arithmetical work the student has been accus- tomed to deal with numerical quantities connected by the :riU7f 'alle^ifl'"'"^ '\' ^f "^^^ -^ expr:ssiorf such 1 fTi ''"• s"^ .~"^* "® understands that the Quantities tn whicTi the sign + is prefixed are additive, and those ^wfch the s.^n - ,s prefixed are subtractive, while the first quant tvlV to vvjich no sign is prefixed is counted among the aSdUh'^'teLs It T^V. rSr^'^ in Algebra ; thus in using the expres sion 7a + 36 -4c -20? we understand the symbols 7a iLnd^htn represent additive quantities, while 4c and 2c; are subtractive I crJLr^L^'lll'''"^'' *'r.f "™ ?^ ^^' ^^^^^^^« terms is always greater than the sum of the subtractive terms; if the reverC 'inllSvrf '^' r««»It ^ould have no arithmeUcal me^n ig ' ternt^excTed Z7of' T ""a^^'^^'^'' ^""^ «^ '^' subtractiv^e Z^A 1 5*^ ^i t^'® additive, but a subtractive term mav stand alone, and yet have a meaning quite intelligible ^ I auStiHfia" aJfbraical quantities may be divided into positive ! quantities and negative quantities, according as they "Te expressed with the sign + or the si^n - ; ani this s quUe irrespective of any actual process of addition and suUraction traSns! ^^ "'^^ ^^ "'^'^' '^'^'''' ^^ *^"^ ^" *^« «^"^Pl« i""«- (i) Suppose a man were to gain £1D0 and then lose P'jn T,i« oteliSrli^^?^ fV' ^.^ «^«^ gains V7rafd\'n Joses ±,100 the result of his trading is a loss of £30 The corresponding algebraical statements would be £100 -£70 =+£30, £70 -£100= _ £30, and the negative quantity in the second (Lse is interpreted as a debt, that IS, a sum of money opposite in character to tfie positfve > .1 10 ALGEBRA. [CHAI-. quantity, or gain, in tlie first case • in f.,.f ;f . i possess a subtractive qualitv m\ m '^ "i*>' ''« ^^"i to on ^r t^nsactiona, r^l;:;';l;^tLKS;?.tn[ a4 a'^S^S foo1"afdrC;;f r" ^7"^. ^^^ *" -^^• backwards, his distance from M ^^""^^''^'^ ^'^^ then 70 yards yards. But if he first walk? ^n «tart.ng.point would be 30 yards backwards lu'dsSceflrH?.' "T"''^' ^"^^ *''«» ^^^ 30 yards, but on tke o.^jT^ft' SfeTeVr '^^ '^' 100 yards - 70 yards = + 30 yards, 70 yards- 100 yards= -30 yards! st:Xg ,i'in?irtre"sU'e^. but'^t^^ 1?'"^^^ ^^^ ^''^> negative signs into account, 'we see^L?^'5^ .the positive and the starting point eavnl 7-^ IX u 5 , "30 is a distance from to the distfnc'e'repSnt S by'^S''^ SusThf'^ ""' ^'^'^'V^^^ "Manroth\?'rrT'^^^^^"'^^ ^''" pStt'r:\i':sre!;j£t'^.at^^^^^ ^"^ '^ -"» >- IS always opposite in chararfpr fn ^V^-f.^^^tractive quantity absolute vile. Ii oth^ word« Itf^'^r^ ^."^"tity ^f equal addition. '^'"'^^^ subtraction is the reverse of Addition of Like Terms. otherwise they are mUed unlik* ?/ ' *o'^.^'^ '^^^'^ ^^^ 3a'62,-4a362aremirsTl,-ki f ^^'"? ^""^ ^^ '* ^a^fi, 2^26 pairs' of unliK S In th rdnT\'^ ^«'' ^^'^ ^^^ cono^der the addition of like term? '^'"P*'" ^" ^^^" «»>.V The Eules for adding like terms are Rule II. If all tke terms are positive, add the coefficients. Example. Find the value of 8a + 5a o. ., , 8a + 5a=13a. Similarly, 8a + 5a+a+2a + 6a=22a. Rule numeric Exam^ Her q' or sa Thii! Rule] gether se coefficient residts, p of tke «?«j Examp 17 Examp Thee tivi We lie terms ins convenier This pr 19. W the result Thus 1 of 11a, -! Exampli The SI Note. 1 opposite si II.] ADDITION OF LIKE TEUMS. n Rule III. If all the terms are negative, add the coeificienta numericaUy and prejix the mi'mis sign to the sum. * Example. To find the sum of - 3a;, - 5a;, - 7a;, - x. Here th, word ,s«»i indicutes the aggregate of 4 suhtructivo qi.antitics of hke character. In other words, we have to tale away successively 3, 5, 7, 1 like things, and the result is the same as taking away 3 + 5 + 7 + 1 such things in the aggregate. Thus the sum of -3a;, -5ar, -7a;, -x is -16a;. Rule IV. If the terms are not all of the same sian, add to- gether separateh/ the eoejieients of all the positive te^ms and the coejicients of a I the negative terms; the difference of these two results, preceded hv the sign of the greater, will give the coefficient oj the sum required. •* Example 1 The sum of 17a; and - 8a; is 9r, for the difference of i 7 and 8 is 9, and the greater is positive. Example 2. To find the sum of 8a, -9a, -a, 3a, 4«, -11a, a. The sum of the coefficients of the positive terms is 16. negative 21. The difference of these is 5, and the sign of the greater is ne«a. tive ; hence the required sum is - 5a. We need not however, adhere strictly to this rule, for the terms may be added or subtracted in the order we find most convenient. This process is called collecting terms. 19. When quantities are connected by the'siLnis + and -. the resulting expression is called their algebraical sum. ofT?r ^^?~?Q'' + ^^''=r^" '*^^^' ^^^* the algebraical sum or Ha, -27a, 13a is equal to -3a. Example. Find the algebraical sum of fa, 3a, - ^a, - 2a. The sum =3fa-2|a = lia =2«- Note. The sum of tw'o (quantities numericallv eoual but wifh opposite signs is ze^-o. Thus the sum of 5« and -5a is 0. 12 Find the sum of h 5a, 7a, lla, a, 23a. 8. '7b,i'M,,]lh,9l,oh, ALGEBRA. EXAMPLES II. [ciur. li. B. - 3x. I -3y, -7y, - «>-, -ilr, -7a:. 9. -116, -%, ~ y, -% -4y. a 36, -b. 11. 2Cy -11^, -15y,y, _3y^2y. 12. 5/, -9y; -^, 2J/, - 13. &, -3«. 2. 4x-, a:, 3x, 7a:, 9a:. 4. 8c, 8c, 2c, 15c, 19c, 100, , 6. -56, -66, -116, _i86' -c, -2c, -50c, -]3c. 5a:, -a?, -3a;, 2a;, -a;. 10. «, -«. - 30/ 5s. 14. 7y, -lly, 16y, -3y, -2y. 16. 5a:, -7a;, -2a;, 7a;, 2.;, -5a:. 16. 7a6, -3a6, -5ab,2ab,ab. Find tlie value of 17. -9a:2+liaJ2 + 3a.2_4^ 18. 3a^a;-18a2a; + a2i 'X. a^ 19. 3a3-7a3-8a3 + 2a3-ll 20. 4x^~5a^-8x^~ 7ar^. 21. 4a262-a262_7„a63^g^2jo_^2^3^ 22. -9a:«-4a:*-12a;4+i3^4_7^4 23. 7a6crf - 1 labcd - 4labcd + 2abcd. 24, 25. 26. ._ 1 ^a;-;^a; + a: + |a;. 5b + lb~p + 2b-^b + lb. |a.-2-2^-|a:2 + a~' + lx2+ii-a;2. 27. - 28. -a6-ia6_ 29. ._ 3 ^a6-»a6-^a6-in6 + „6 + •c, c. ^a;-fa; + |a;-2a; + -V-a;-4 TH a6. 30. -|ar»-3a:2_ F a:^-ia;2-a;2 ^a; + a;. CHAPTER III. Simple Brackets. Addition. 20, When a number of arithmetical quantities are connected together by the signs + and -, the value of the result is the same m w))atever order the terms are taken. This also holds in tlip oase of algebraical quantities. Thus a-6 + c is equivalent to a +c-h, for in the first of the two expressions b is taken from a, and c added to the result • in the second c is added to a, and b taken from the result. Similar reasonuig applies to all algebraical expressions. Hence we may write the terms of an expression in any order we please. Thus it appears that the expression a-b may be written in the equivalent form -b+a. To illustrate this we may suppose, as in Art. 17, that a repre- sents a gain of a pounds, and -b a loss of b pounds : it is clearly immaterial whether the gain precedes the loss or the loss pre- cedes the gain. * 21. Brackets ( ) are used to indicate that the terms enclosed within them are to be consid-red as one quantity. The full use of brackets will be considered in Chap. vii. ; here we shall deal only with the simpler cases. 8 + (13-f 5) means that 13 and 5 are to be added and their sum added to 8. It is clear that 13 and 5 may be added sepamtely or together without altering the result. Thus 8 + (13 + 5) = 8 + 13 + 5 = 26. Siinijarly «+(6-f-c) means that the sum of b and c ia to be I I I a i > H . it If' r 'i'hus -{h\-c)-^a-\-h-\rC. • i ir\ 1^ ALGEBRA. ^^^^^ therefore take 5 from the ^sult "'"'^'' ^'"^ ""'^^ Thus 8+(13-5)=8-|. 13-5 = 16. Similarly aHb-c) means that to a we are to add b, diminished Thus a+(b-c)==a + b~c (.. Ill hke manner, ^^z' Conversel,r'"^^^'-^-^=^^*-^^^— /• ('^^ Agam. a-b+c=a+c-b, i^t^'Sl = the sum of a and c- 6, L Art. 20.] ,, , =thesuraof aand -i+c, FArt 901 therefore a-b+c=a + (-b + c) 1.., •• fj By considerinof the results (]\ f9\ /"}\ VTx •...•..(4). following rule : ^ ^' ^ ^' ^'^■^^ ^^^ ^® ^^® ^^d to the ^■^'^l\X^'''j''!' '^■^^™«'^ ^«*V/«V» 5mc/{^e<* «;? preceded bv th^ imo^e^, remamml nnalflrld ^ ^ ''"''-^ ^''"'^ '''*^^*'^ ^^« thJ^o^X?" "-'+^-^+^ -^^ be written in any of «+(-J+c-c^+e), «-6+(c-fl?+e), «-^> + c + (-c;+e). tak?the''s'umTr:ii\"-^L^lX"^ « -^ -^ *" ^ and . are subtracteS^ipaSj:!;-;! ^olll J.m^ S ^^'^^^'^"' ex^eK'L"vt7^ m;r£/ier"^ ^? ^^ f"^^-^^ ^•- doing we shall have taken aw.T. '""f ^^^ f " ^ ^ but by so add c to a-/. Thus ^ '''' '''"'^'' ^"^* '""«t therefore In like manner, <^~{b-c)=a-b + c. a-b-{c~d-e) = a-b~c + d+e Accordn^gly the following rule may . e enunciated : III.] ADDITION. 15 Rule. When an expression within hracJcetn is preceded hii the 'T'l"' i r * "*"•'' ** ^^^o^^<^ ^ftJ^e »m of every term xoithin Conversely- Ant/ part of an expression may be e7iclosed within brackets and the sign - prefixed, provided the sign of every term within the brackets be changed. Thus the expression a-b + c+d-e may be written in any of the following ways, "^ a-( + b-c-d+e), a-b-(-c-d+e), ci-b+c-(-d+e). We have now established the following results : I. Additions a7id subtractions may be made in any order. thii8a + b-c+d-e-f=a-c+b + d-f-e = a-c-f+d+b-e. Subtract •'^"^^^'^ ^^ *^^ Commutative Law for Addition and II. The terms of an expression may be grouped in any manner. Thus a + b-c + d-e-f={a + b)-c+{d-e)-f ^a+ib-c) + (d-e)-f=a + b-(c-d)~(e+f). Subtrl(Jtion!°'^" ^^ *"^ Associative Law for Addition and Addition of Unlike Terms. 23. When two or more like terms are to be added together we have seen that they may be collected and the result expressed as a single like term. If, however, the terms are unlike they can- not be collected ; thus in finding the sum of two unlike quantities rt aiul b, all that can be done is to connect them by the sitrn of additionandleavetheresult in the forma + 6. " . Also by the rules for removing brackets, a + (-b) = a-b: that IS, the algebraic sum of a and -6 is written in the form a-b It will be observed that in Algebra the word stm is used' in ^rktUT'^ f^^ in Arithmetic. Thus, in the language of .ntnmctic, a-b signifies that b is to be subtracted from a" and ears that mean, ng only ; but in Algebra it is also taken to i^ean rie sum ot the two quantities a and -b without any regard to the relative magnitudes of a and b. ^ I III ■i" 4 ' ^ < j» f'il ■;l ^\ -31 ■ ; » ; jf r * r> r " t ■ r fa I 16 ALGEBRA. ^^^Plel. Findtheoumof3a-5..2.;2a + 3.-,. .,,^,, -3a-56 + 2c + 2a + 36-rf-4« + 26) =3a + 2a-4a-56 + 36 + 26 + 2c-rf = a + 2c-rf, by collecting like terms. foIWinll^'r "■ '■''"^''^'•' '»"« convenient,^ effected by the MV m« (te o« the le/l • "' "'''' """* "'»" %!«. in the third and f™, Jk ","«'" 'o"™ I brought down „Ttho,!?"ha„gr"""'" " -5a6 + 6ftc-7ac 8a6 +3ac-2a(^ -2a6 +4ac + 5a(^ z3aM^Jc____+4af^ -2ab+7bc +7 ad 3a ~ 5b + 2c 2a + 3b -a j:^+26 » +2c-rf Find the sum of EXAMPLES HI. a. 1. a + 26-3c; -3a + 6 + 2cj 2. 3a + 26-c,- -a + 36 + 2ci 3. -Sx + 2i/ + z', x~3i/ + 2z'' 4. -^ + 2y + 3.;3:.-y + 2^;.a: + ay_, 8 ifl " , ' ^^«-106 + 4(!; a + 206-c. 2a-.36 + c. 2a-ft + 3c. 2«+y-33. 2a: + 3y-g. 10. 20/;f^_i aa; + 26y-, P~20q + r; p + q- P + q-20r. ~3ax + 2bi/ + 3cz. ADDITION. Add together the following expressions • 11. -5ab + 6bc~7ca;8ab-4bc + 3ca; ~2ab~%c + 4ca I5ab-27bc-eca; Uab-mc + lOca; 45bc - 3ca - 49ab, 5ab + bc-3ca; ab-bc + ca; ~ab + 2ca + bc pg + gr-rp; -pg + gr + rpipg-qr + rp. x+y + z', 2x + 3y-2z; 3x~4y + z. 2a-36 + c; I5a-2\b~8c; 24b + 7c + 3a 7aft-I35c + 8ca,. -5ab + 9bc~7ca ; -7bc-ca + 2ab. 47:.-63y+.; - 25a: + 15y - 3z ; -22cc + 15z + 4Sy -m-2c + 23a; -9a-M564-7c; -I3a + 36-4c. Dimension and Degree. Ascending and Descending Powers. the degree of th^ term S th^ ^l^^'^f T^^^^'^ ^'^ ^^""d : t/»'ee dimensions, or of the i,W iv. ^ "^"^^ ''^^ '^ ^^id to be of > ;?.. dimensions, or ofthefif^tgrT' ' ""^ "^ ''« ^^^ ^« ^e % jare^aToSL^^f ettl^ "^' ^^""*^^- ^^ ««^*« and ^^^^ 17 '12. [la ill [15. il6. 1 17. jl8. [19. l2a I* I " I « r f >-«.^,cocty/t u/ me mmh, dearee a»ul ,»2^ >7;2 i • ■ "«--u is an [the fifth degree. But it is soml-^fl" ^* f /' "'' ^-^J^^^^^^ow o/ dimensions of an expressL wifK "^'^"^ *^ ^P^^k of thi letters it involves X Snrti/'^'^ ° '°^°« «"« «f the is said to be of th^ee Sj^t^^'t '"P'''^^'" «^-*^+c.-rf itstrmTaro'f l^'S^Zt^S^ '' be homogeneous when all thM:u5t"^of^S^^^^^^^^ - -Hl^e terms; [pressed by a single term5>nf,^n.uTbe 1-''-^ ?^ '^""^-* >" «^- i Similar.!,, Ai ' 1 , ' '"'- ^^ ^^^i- in tne rorm ^.t-^ + .lrZ 5«?r-SL*^f fe^^-^-^ --^ ?f 5a^5^ -3«i3, and -i't cannot be abridged ' '^P''''^°" ^^ ^" ^^« ^^^P^^^t form and E.A. g r pi 18 ALGEBRA. In adding together several algebraical expressions containing terms with different powers of the same letter, it will be found nnllT^? fwT.r^^ ^" ?^P»:ef ions in descendmg or ascending S les ' ^^^^ ''^^^'" ^^ *^'^ following 4a;-2a:3 + 3a;2; 3ar»-9a;-^ 3ar»-5a:2 + 6a; + 7 2a:2_9a;_8 -2«3 + 3a;2 + 4a; ~3a:3-2a;2-7ar + 3 a;-a;2-a:3 + 4. In writing down the firat expression . we put in the first term the highest I power of X, in the second term the next highest power, and so on till the last term, in which x does not appear. The other expressions are arranged in the same way, so that in each column we have likt powers of the same letter. Example 2. Add together Sab'- 8a9 + 563; 9a%-2a^ + ab\ 263 + a3; 5a^b - ab"' - 3a» i 1 -263 + 3a62 + „3 - ab^+ 5a^-3a^ 6ft' +8a3 a¥+ 9a'-b-2a^ 3b^ + 3ab^+lia%i-ia^ Here each expression contains powers of two letters, and is arranged according to descend- ing powers of b, and ascending powers of a. 4. 5. 6. 7. 8. 9. 10. 11. EXAMPLES III. b. Find the sum of 1. 2ab + 3ca + 6abc; -5ab + 2bc-5abc; 3ab-2bc-3ca. 2. 2x'-2xy + 3y^; iy^^ + 5xy ~ 2x^ ; x''-2xy-Qy\ 3. 3a2 - lab - 462 ; - 5^2 + 9^5 _ 3^2 . 4^2 + „ j ^ 5^2. X-^ + xy-y^; -z^ + yz + y^; -x^ + xz + Z^. -x^-3xy + 3y^; 3x^ + 4xy-5y^; x^ + xy + y\ xr'-x^ + x-l; 2a:2_2a; + 2; -3a:3 + 5a;+l. 2a;s-a;2-a;; 4ar' + Sx^ + 7a; ; -6aP-6x^ + x. 9x2-7a: + 5; - 14a;2+15a:-6 ; 20a;2 - 40a; - 17. 10a:3 + 5a; + 8; 3a:3_4a,2_6. 2a:3-2a:-.S. a^-ab + hc; ab + ¥-ca; ca-bc + c\ 5a3 - 3c3 + rf3 . ^3 _ 2a3 + 3^3 . 4^3 _ 2^3 _ 3^3. ADDinON. Find the sum of 12. Q^-2x + l;2xS+x+6;x^-7x3+2x-4. 13. a3 - a3 + 3a ; 8a^ + 4a^ + Sa ; 5a^ - 6a» - Ha. 14. ^+V'-2xy}2z^-3y^-4yz;2x^-2z^-3:^. 15. x->-2y^ + x; y^-2a? + y; x'^ + 2y^-x+f. 16. x^ + 3x^y + Zxy'^; - 3x^y - Gxy^ - a^ ; 3x^-y + 4xy^ 17. a» + 5a62+63. b^ - lOab^. _ ^3 . 5ab^-2b^ + 2r-^. 18. aH'-4aHy-5a.V; 3a;V + 2;*;3y3 - 6ary* ; 3x^y^ + 6xy*-y^ 19. a3-4a26 + 6aftc; a^ft- iQafec + c^; i^ + g^sj^^^^ 20. ^^-4x^y + 6xy^; 2x^y-3xy^+2y^ ; j^ + 3xhj + 4xy^ Add together the following expressions : J21. ^a-^b; -a + f6; |a_j. 122. -^a-l6; -§a + |6; -2a-6 123. -2a + |cj -ia-2b;^b-3c. j24. -'#a-^c;2a-36; i^6-c. |25. |a:2 + lary-iy2. -cc^-2^+2f.. |;,= _<^_ 6^2 126. 3a2-|a6-|62; -fa^ + gaft- 252 . -^a^.^b + b^ (27. |a:^-^:«y + TV; - f *^ + i|:.y - y. ; la:«-«.y+ 1^2. 128. -|a:3+5„^o_|^2^. ^_3^7^^2+i„2^. _i^^ + 42^, 129. ^x-^-ixy-7y^'; t^y + ^%=^; -|:^2 + 4y2. [30. Ia3-2a=6-fi3. .la^b-^abH2b^; -^a^+ab^+if^. 19 r ^ i^fM'-r. il CHAPTER IV. Zx'-'Jxy + lyi Subtraction. 26. The simplest cases of Subtraction have already come under the head of addition of like terms, of which some are negative. [Art. 18.] Thus 5a-3a= 2a, 3a - 7a = - 4a, - 3a - 6a = - 9a. Also, by the rule for removing brackets [Art. 22], 3a-(-8a)=3a + 8a = lla, and -3a-(-8a)= -3a+8a ■ar'-3a:" + 2a; + Subtraction of Unlike Terms. 27. The method is shewn in the following example. Example. Subtract 3a - 26 - c from 4a - 36 + 5c. The difference =4a-36 + 5c-(3a-26-c) = 4a - 3& + 5c -.*]« + 26 + c -4a-3a-36 + 26 + 5c + c = a-6 + 6c. The expression to be subtracted is first enclosed in biackets with a minus sign prefixed, then on removal of the brackets the like terms are combined by the rules already ex- plained in Art. 18. It is, however, more convenient to arrange the work as follows, the signs of all the terms in the lower line being changed. by addition. 4a -36 + 5c -3a4-26+ c a- 6 + 6c The like terms are written in the same vertical column, and each column is treated separately. Rule, Change the sign of every term in the expression to he subtracted, and' add to the other expression. Note. It is not necessary that in the expression to be subtracted the signs should be actually changed ; the operation of changing signs ought to be performed mentally. CHAP. IV.] Example 1. 5a;- + xy 2x^- + 8xy-7y" ^x^-'lxy + lyi SUBTRACTION. 21 From hx'^ + xy take 2x--ir%xy-'Jy-. In the first column we combine mentally bx"^ and - 2a;2, the algebraic sum of which is {ix\ In the last column the sign of the term - ly" has to be changed before it is put down in the result. Example 2. Subtract Sx"^ - 2x from 1 - x^. Terms containing different powers of the same letter being tinlike must stand m different columns. -x" 3a;2-2a; + 1 -ar*-3a;- + 2a: + l In the first and last columns, as there is nothing to be subtracted, the terms are put down without change of sign. In tlie second and third columns each sign has to be changed. The re-arrangement of terms in the first line is not necessary, but V^ ""ZllToix '* ^'^^^ *^^ ''^^"^'^ ""^ subtraction in descend- ! I 2. 3. 4. 5. 6. 7. 8. 9. 1 10. Subtract EXAMPLES IV. a. 1. 4a - 36 + c from 2a- 36 -c. a-36 + 5c from4a-86 + c. 2x-8y + z from 15a: + lOy ~ I82. 15a - 276 + 8c from 10a + 36 + 4c. - 10a? -14y + 15z from x-y-z. -llab + Qcd from - 106c + a6-4crf. 4a - 36 + 15o' from 25a - 166 - 18c. - 16a: - 18y - 15z from - 5x + 8y + 7z. a6 + cc?-ac-6(Zfrom a6-f-crf + ac + 6rf. - a6 + cc? - oc + 6fZ from a6 - cc? + oc - 6d, From 11. 3a6 + 5cc? - 4ac - 6bd take 3a6 + 6cd - Sac - 5bd. yz-zx + xy take -xy + yz-zx. - 2ar' - a:2 - 3a: + 2 take a:3 ~ a: + 1. - Sx'^y + l^xy- + lOxyz take 4a;2y - 6xy" - Bxyz. |a-6 + |ctakela + A6-ic. |a: + y-stakela;-|y-i~. - a - 36 take #a + i6 - ir 12. 13. 14. 15. 16. 17. 118. 19. -I 20. •i^-ry + xV take -|a: + 4y- !« - |y - 5z take |a: - fy - -\fz'. TU^* ■^a: '-ty-itake|a;-|y-|-. r t 22 11 ALGEBRA. [OHAP. 2. 3. 4. 5. 6. 7. 8. 9. 10. From EXAMPLES IV. b. « 1. 3x1/ - 5yz + 8zx take - ixy + 2yz-l Ozx. -SxY+lBx-'^y+lSxy^ take 4xY + 7x^i/~8xy^. ~8 + 6ab + a-b^ take 4~3ab-5a-b\ a-bc + b-ca + c-ab take Sarbc - 5Pca - 4c^ab. -7a^b + Sab^ + cd take 5a"-b - 7a¥ + 6cd. -8x'^y + 5xy"-xY take 8xhj - 5xy^ + xhf. 4a;2_3a; + 2 take -5a:2 + 6a;-7. ar» + lla:2 + 4 take Sa;'-^ - 5a; - 3. ~8a^x^ + 5afi + 15 take 9aV-8a:2-5. Subtract 11. a:^-a:2+a; + l froma:»H-a;2-a; + l. 12. 3a:y2 _ 3^2^ + 3,3 _ ^3 from a:3 + 3^.2^ ^ g^j^^o ^ ^ , 13. 63 + c3 _ 2abc from a^ 4- 6^ - 3abc. 14. 7a?y2 _ y3 _ 3^2y ^ 5^, jj,^j^ g^ ^ ^_^2j, _ ^_^^, _ ^^ 15. a^ + 5 + a;-3a;3from5a:<-8a:3-2a;2 + 7. 16. a3 + 63 + c3 - 3abc from 7a6c - Sa^ + Sjs - c^. 17. l-a: + aH'-a:*-ar'froma^-l+a;-a;2. 18. 7a* - 8a2 + 3a^ + a from a^ - 5a3 _ 7 + y^s, 19. lOa^ + 8ab^ - 8a'P - ¥ from 5a?b - 6ab^'- 7a^bs 20. a^-b^ + 8ab^-7a^ from -Sab^ + lSa'b + l^. From 21. ^a;9 - ^xy - ^^ take - ^x^ + xy- y\ 22. |a2-|a-i take -|a2 + „_i^ 23. -1x2 - |a; + ^ take |a; - 1 + ^x^ 24. |«'-' - %ax take ^ - |a;2 _ ^^^ 25. |a:3 _ 1 3,2,2 _ y2 take iar^ - |y2 _ 1 ^^3. 26. |a3 - 2aa;2 - 1 a^a; take ^a^ar + ^a^ - ^aa^. IV.] MISCELLANEOUS EXAMPLES. I. 23 MISCELLANEOUS EXAMPLES I. 1. Simplify (1) 4x-2x^- (2x - 3x"-) ; (2) 3a - 46 - (36 + a) - (5a - 86). a\-?76tdr66.' '""-''-'^ ^"^^ 26-a-^7c add the sum of 3. When a; =3, y= 2, s=0, find the value of 4. Define mrfea:, coefficient. In the expressions 4a;2 + 3a.- 2a.-3 + a;3 X- + 1X, find (1) the sum of the indices, (2)^the sum of the coefficient' 5. From Sar" + 3a: - 1 take the sum of 2a; - 5 + 7a;2 and 3a;2 + 4 - 23^5 + a;. 6. Subtract 3a - Ta^ + Sa^ from the sum of 2 + 8a3-a3 and 2a3 - Sa^ + a - 2. 7. Distinguish between like and unlilce terms. Pick out the like terms m the expression ^ "''*' a3 - 3a6 + 6^ - 2a^ - a2 + 363 + 5^^, ^. 7^2 to|;ther t yfald i" ""' ""^"^ "^^^^ *' P°''^^^' *^^' '■^'"^'^ °^ ^^^^^S 9. Subtract 5a;2 + 3a; - 1 from 2a:3, and add the result to 3a;2 + 3a;-l. Xlll '-UZ^\ "^ P'""^' ^ '''''''' '' represented by +a, 2a^b^y thT^^rn'oTs" Indif""'' ^^"'^^^ ^"^^ ^'^^"^^ ^^ ^--^^-S -f^J^:^i^o\^-ti:tT2j%r^^^ ^' ^'^ """^ °^ '^' . M !-f l! r 1.; If ? f r r i t I; r^& Add the sum of 2y-3y2 and l-5y3 to the remainder left when l-2y^ + yi3 subtracted from 5y^. 14. Explain clearly why x-{y-z)=x-i/ + z. 15. If a; =4, y=S, 2 = 2, a =0, find the value of 3a;2 - 2ys - aa; + 5aa; Y 16. Simplify 2a - 6 - (3a - 26) + (2a - 36) - (a - 26). I 24 ALGEBRA. [CHAP. IV. 17. Find the algebraical sum of the like terms in the expression 5a' - 4a% + b^ + 6a'6 + 7a62 - 3aV} + 4a6» + Sa^fc. 18. A boy works x + i/ sums, of which only y-2z are richt • how many are wrong ? e v , .ivfw 19. In the expression 3a^ - 7a^b + h*, point out the highest power, the lowest power, the positive terms, and the coefficient of aK -„??*«f J*^® f "o^V'T^^^^^-^y"' ^'"^ ^^"^ <^he remainder to the Bum of 4x1/ - a;2 - 3y2 ajid 2x^ + 6i/. 21. If x=l, y = 3, 2 = 5, 1^=0, find the value of \/(3a?y) + ^(5xz) + ^{Syiv). ■rJ?y, ^^*^ -^ the degree oi a term in an algebraical expression? term? "'^P''''''^" ^^'-3x^a^ + a\ what is the degree of the negative refS'tfroJ!fc-4V""' of 6a- 76 + c and 36 -9a, and subtract the 24. If x=3, y=4, p = 8, g = 10, find the value of «yp + — ^+2a. 25. If a; represents the date 10 A. d. what will - 3x stand for ? result bt st'^+f *^'' ^"^'-7* + ^ ^"^ 2a:3 + 5^_3^ ^„^ ^i^j^j^,^ ^j^^ 27. In the expression Fstergr^e ettt"?xp"tS?'"' """* '" "-"K^-- ^hat anf 6 ofe?rdta£iaf,'' '"'"'""' "■' ''=«''= "^ "■« »«"■ "' « 29. A man walks 2a - 6 miles due North from a fixed point SuorjSe;^^^?,^^^' ■""- ^- «-*^ -''^' - "'""-i Tal^'l?^^^** expression must be added to Sa:^. 73,^2 to produce p CHAPTER V. Multiplication. 28. Multiplication in its primary sense signifies repeated addition. Thus 3x4 = 3 taken 4 times = 3 + 3 + 3 + 3. Here the multiplier contains four miits, and the number of times we take 3 is the same as the number of units in 4. Again axh — a taken h times =a+a + a+..., the number of terms being h. Also 3x4=4x3; and so long as a and h denote positive whole numbers, it is easy to show that axb = bxa. 29. When the quantities to be multiplied together are not positive whole numbers, we may define multiplication as an operation performed on one quantity which when performed on unity produces the other. For example, to multiply 4 by f, we perform on f that operation which when performed on unity gives if ; that is, we must divide f into seven equal parts and take three of them. Now each part will be equal to -^, and the 5x7' 4x3 result of taking three of such parts is expressed by ■= — =. 5x7 Hence Also, by the last article, 4 34x3 5^7~5x7' 4x3 5x7 3x4 3 4 '7x5~7^5* 4 33 4 5^7~7^5* I It I •J- 4 ) " I'll I > r i V f r c r c I 26 ALGEBRA. [chap. or fractional ^'° ^"^ P^'^'^'^^ quantities, integral In the same way it easily follows that ahc=axbxc ={axb)xc={hxa)xc = hao = bx(axc)= bxcxa =bca; Example. 2ax3bxc = 2x3xaxbxc = 6abc. 4 w A^at' '^' ^^''''' '^ " ^''^^"^^ '««^ ^^ ^'•''"/'^^ in any Thus abcd=axbxcxd = {ab)x (cd) = ax(bc)xd=ax (bed). This is the Associative Law for Multiplication. 31. Since, by definition, a^=aaa, and a^> = aaaaa, .'. i /1\ Also (a - b)m-= m + m + vt + ... taken t/ - i times =(m + m + w + ... taken « times), diminished by (m+m+7n+... taken b times) =am, — bni _ /■2). Similarly ((« - 6 + c)??i =am- bm + cm. Thus it ajjpears that the product of a compound expression by a single factor is the algebraic sum of the partial products of each term of the compound expression by that factor. This is known as the Distributive Law for Multiplication. Note. It should be observed that for the present a, h, c denote positive whole numbers, and that a is supposed greater than h. Examples. 3(2a + 36 - 4c) = 6a + 96 - I2c. (4a;2 - 7y - Sz^) xZxy'^=\ 2jB»y= - 21a;2r' - 2ixy^;^. EXAMPLES V. a. Find the value of 1. 5x' X 7a-*5. 2. 4. 6a;j/2x5a:3. 5. 7. 2a363 X 2aW. 8. 10. 5a^6«xa;V. il. 13. Za*lPa? X 5a%x. 14. 16. 5ar»y3x6a3a:8. 17. 4a^ X Sa". Sa^ft X h\ 5a% X 2a. arV' X %a^X*. 4:a^bx X 76'^a;* 2x^ X ar^y'. 3. 7a6 X Sa'ftl 6. 2a6c X 3a(^. 9. 4a263x7a». 12. «6c X xyz. 15. 5a-a; x 8cx. 18. 3a3a^7xaVy9. 14 -«.: 4 »4 1 " i •'J r I i hi f * -n \ > ■ , « ! ; : 9 r' •? f > * 1 \ f » , i 1 t - 28 ALGEBRA. Multiply together : 19. ab + bc and a^b. on «; . -. 21. 5a; + 3y and 2:.2. S* ^f "/^^ and «:^. 23. bc + ca-ab and abc f/ "" + *'-^= ^nd a3ft. 27. 6a3ic-7a6=c^andaV. * ^^"^^^+7^^^ and 8:.V. 34 j,^''";*^ ^^^**°° Of Compound Expressions. Xc+d)=a(c+d)-b(c + d) = (c+d)a~(c+d)b =ac + ad-(bc+bd) S-iiarl,,!^ writing. -^r:i:;-^-^^ (4> (<^ + bXc-d)=.a(c~d) + b(c-d) = {c-d)a + {c-d)b Also, from (2) =<^c~ad^bc~bd , /gv ^^-^)i<^-d)^a{c-d)-b{c~d) =={c-d)a-{c~d)b =ac-ad-(bc-bd) ( + «')x(+c)=+ac. i~h)x{~d)=+bd i~h)x{+c)=.-bc. Of Si^^S riXtio:.^^ ''^'' -^^' - ^"own as the Rule Kule of Siffjis TLp 7 v.] MULTIPLICATION. 29 3f: .The rule of signs, and especially the use of the negative multiplier, wil probably present some difficulty to the beginner Perhaps the following numerical instances may be useful ifi illusl rating the interpretation that may be given to multiplication by a negative quantitv. ^ ^'^aviuu To multiply 3 by -4 we must do to 3 what is done to unity to obtain -4 Now -4 means that unity is taken 4 times and he result made negative; therefore 3x(-4) implies that 3 S to be taken 4 times and the product made ne<^ative But 3 taken 4 times gives +12 ; .-. 3x(-4)=-12. Similarly .-3x -4 indicates that -3 is to be taken 4 times second V/r ''"^ ' '^ 'P''"^^"^^ ^'''' -12, and Te Tl'"s (-3)x(-4)=+12. Hence, multiplication by a negative quantity indicates that we are Note on Arithmetical and Symbolical Algebra. 36. Arithmetical Algebra is that part of the science which ^Z: t T^' 'r'^'^' ''^1'^ ^I^^^^^^^"« arithmetica hM, tel ligib e Starting from purely arithmetical definitions we are enabled to prove certain fundamental laws. ' Symbolical Algebra assumes these laws to be true in every case and thence finds what meaning must be attadial to ymbols and operations which under unrestricted conditions no sTlml ?I ^" ^"t»'«^«<^i?^l "^^aning. Thus the results of A rs! 33 and 34 were proved from arithmetical definitions which S'^and :i7^t \'. ^' ^■''''T '^^'^'^ mmibers, such a a>b and c>d. By the principles of symbolical Algebra we a sume these results to be universally tn/e when all reftricLZ thereby'' ' """^ "'''P' ^^'' interpretation to which we are kd •mJthrff nr^-"'" ^vf ^Z ^PP^^ ^'^« ^^^^' «f Distribution 2sl''"[s';fArr3rK^^^^^^^ "^^' '''''''''''' '^ *« ^he symbols i,,V' To ^'^'"iliarize the beginner with the i)rinciples we havo some of the symbols denote negative quantities. M* -Mm' ■ 1 W 1 ■ 1»' g\u m ' " !•;:: rii ri» s <« vi ■ *• < > •a - ' ■** ■ ia « '« • : a l-i ^^ V ? «* : r ' r ' c r m t p» P. 30 Example I. Here ALGEBRA. [chap. Ifa=-4, findtjjevalueofas a3=(-4)3=(-4)x(-4)x(-4)=_64 shf Taf^T^^^^^^^^^ signs it' .ay easily be Here -S^^.:r;l'^p:-':^^^^ W^;!"^' ! (-2)"= -8. If -_ EXAMPLES V. b. a~ -2, 6=3, c= - 1, .^.= _ 5^ ^^^^ g^^^ ^^^^ ^^^^^^^ ^^ 1. 5. 9. 13. 17. - 7a36c. -5a2ft2c2 2. 6. 10. 14. 18. 8ahc\ -2a*bx. - lc*xy. - lah^ 3. 7. 11. 15. 19. -5c\ -b'^c?. ~ Saar*. 8c*x\ 4. 8. 12. 16. 20. 4c^x''', Ifa=-4 j=,_o _ ^- ''*^- 21. 3a2 + 6^_4 ' ' ^- -^' /=0, ^=4, ^ = 1, find the value of /a^-263-J: g- 2ai2-36c2 + 2/l.. 2a3- 363 + 7,^4. g- V(ac) - 3^(xy) + ^/(i2c4)_ 28 * 23. 25. 27. 29. 30. 3a22/3_5j23._2c.t. 362y»-462/_6c%. 'V(«ca:)-2^/{i23^)_6V(c2y). signf^nd'L W^fnSs"^^^' '"^'^^^^" ^""'^^^^t^ the rule of Example \, Multiply 4a by - 3ft By theruleof signs the produet is negative; also 4ax3.= 12... Example 2. Multiply -Sail^ by -a63r "a^'SstS^^trp?Xt-f^^^^^^^^^^^^ '•« ^«^^«^^ and by the ^-...^e 3. Find the continued prodLt :/3:2,, .2,,,. _,,. 3a26 X ( - 2a3ft2) ^ _ ^^^^a . | This result, however ' mav b. ( - 6a«ft3) , ( _ „ j,^ ^ ^ g^/^^ j written down ^t onTe Ifor ^ ' ,-. J^6?- ^'*^ complete product I ^'''^ ^ ^aSfts x «&«= 3a«//, ^ ^'' i ^''^^ 5^ ^J'« rule of signs the re I quired product is positive! |H aa; a ^B a26 ) H 5. -a6( H 3a:2/- B -X- 1 ^^' -db H '^^' 5x^y H 15. -5x1 H 17. -13a ■ 19. dbc- H Find til I ^^' 2a-{ ■ ^' §«-- I ^' - 5„2 1 ^' -|a:3 '% 39. Tl 1 ^°^^' of th M or more te hi I' (i.' v.] MULTIPLICATION. 31 Example 4. Multiiply 6a?-^^a% ^ab^- hy -^ab\ The product is the algebraical sum of the partial products formed according to the rule enunciated in Art. 37 • thus (6a3 ■ t«'* - !«*') X ( - |a6-) = - ^a*b^' + |a-^63 + 3^2j4. 3. 5. 7. 9. 11. 13. 15. 17. 19. EXAMPLES V. Multiply together : 1, ax and -3ax. a^b and -ab"^. -abed and -^a^bh^d/^. 3xy + iyz and -12xyz. -x-y-z and -3a:. -db + bc-ca and -abc 5x''y-6xy^- + 8xY and 3xy. C. 2. 4. 6. 8. 10. 12. 14. -2a6a; and -7abx. 6x-y and - ]Oxy. xyz and -5a;2y%. a&-6c and a%c\ a"-b'^ + c^ and aic. -2a26-4a62 and -Ta^fta. - 7a:3y _ 5a.y3 and - Sa^^yS. - 5xfz+3xyz^ - 8xY. and xyz. 16. 4^:^22 - 8a:yz and - 12;^i;23, abc - a%c - ab'^c and -ak\ 36 + 4c and - |a. Find the product of 21. 2a 23. 25. 27. |a-i6-c -fa;V and and -faa;. and -fa2 + aa:- -ia:2 + 2y=. a:- 18. 20. 22. 24. 26. 28. 9>xyz - lOar^yz ' and - xyz. -a^c + b^ca-c^ab and ~ab. 3x-2y-4: and ~^x. ja'^x-~^ax^ and -^a^x. -Ixy and -3a:2 + |a:y. -ix^.V'' and T:x^-iy\ Wlf'nf ^il"^ ''^'"^*' -^ '^'^- f ™*>' ^^ extended to the case where orifetn^nrinsttc': ''''''''''''' '''''''^ ^^^^^ ^- (a-b+ c)m = am - hm + cm ; replacing m by .r-y, we have {(^-h + c){x-y) = a{x-y)-b{x-y) + c{x-y) = (a.t' - ay) - {hx - by) + {ex - cy) = ax-ay-hx+hy-J(-cx-cy. nnvVJ''^^ ""'"^ ^^^^^- ^^'^ S^"'^^^ ^"^^ f«^ multiplying toc^ether any two compound expressions. f ^ ^ i-UoCLuei of the second. Mhen the terms multiplied together hnL HU o-JJ? PWx to the prodicct the sign +, when unUke prekx'^l-J^e product. This process is called Distributing the Product. ; a M f I V J r r I ( 32 ALGEBRA. [chap. product beinVdetmL^b^fe^-fforiln;.'''^ "^ °' -^" ^a:am2>^e 1. Multiply a: + 8 by a; + 7. The product =(^^8)(a: + 7) = x^ + l5x + 5G. J he operation is more conveniently arranged as follows : X + 8 a; + 7 x^+ 8x + 7a; + 56 by addition, a;Hl5a; + 56 ^a:ami>/e 2. Multiply 2x - 3y by 4x - 1y. 2x - 3y 4a; - 7y 8^^M2^y 8a;2-26a;y + 21y2. We begin on the left and work to the right, placing the second result one place to the right, so that like terms may stand in tlie same vertical column. by addition, Find the 1. x + 5 3. x-l 5. a; + 7 7. x + Q 9. a;-12 11. a?- 15 13. -a;-2 15. -a; + 5 17. a; -17 19. -rr-16 21. 2a; -3 EXAMPLES V. d. product of and a; +10. and a; -10. and a; -10. and x-Q. and a;- 1. and a; + 15. and -a: -3. and -a; -5. and a; + 18. and -a- + 16. and a; + 8. 2. a; + 5 and a;-5. 4. a; -7 and a; + 10. 6. a; + 7 and a; + 10. 8. a; + 8 and a:-4. 10. a; + 12 and a;-l. 12. a;- 15 and -a; + 3. 14. -a; + 7 and a;-7. 16. a;- 13 and a; + I4. 18. a; +19 and a; -20. 20. -u; + 2l and a; -21. 22. 2a;+3 and a;-8. 25. 3x-L 27. 5x-(i 29. 3a;-5 31. a -2b 33. 3a -6 35. x + a 37. x-2a 39. xy~ai 4 . We Example 2x'~ 2x - 5 ()X>- 4a;^-J - 15a;2+] 6ar'- 19a;2 I v.] MULTIPLICATrON. Find the product of 1 23. x-5 and 2a; -I. 125. 3a; -5 and 2a; + 7. |27. 5a; -6 and 2a; + 3. 129. 3a; -5y and 3a; + 5y. J31. a -26 and a + 36. 133. 3a -66 and a -86. I35. x + a and a;- 6. 137. a; -2a and a; + 36, |39. xi/~ab and xij + ab. 33 24. 2a;-5 anda;-!. 26. 3a; + 5 and 2a; -7. 28. 5a; + 6 and 2a; -3. 30. 3a; -5y and 3a; -5y. 32. a -76 and a + Sb. 34. a -96 and a + 56. 36. x~a and a;+6. 38. ax -by and ax + by. 40. 2pq-Zr and 2py + 3r. 4.. We shall „„. give a few examples of ..eater diflieuUy. 13.' T- 5' ^"'"*-° •"■:""«' °' 8^-^-5 and 2.-. hx - 5 lOa.^- 4a;^-]0a; ^J5a;2+]0a; + 25 Bar' -19:^2 :j:^ terms are nlarpr] ,•„ f k" ^ i>y - o ; like the resufts added ^' '^'"' ''''"'""« ^"^ i^^a;a/«^>/6 2. Multiply a _ 5 + 3, by „ + oj. a - 6 + 3c a_+__26 a^- ab + 3ac ___2a6___^^262+66c a"+ «6 + 3ac- 262 + 66e prol'ss ^'MuS^Src^^^^^^^^^^^^ - "- «- ordinary b' the rules of Arithmetic. ^ ''^ fractional coeflicieati ^'-ctmple. Multiply ^a2-|a6+=6= ,,y ^„^,^,. ia:"- Ub +|62 ^a + ^6 K.A, i :> : ■» ■« 1 :^ •.*.!' '» a ^ I Ifl « f ■ ''« » ! * 3 ^ r 1 V f (" r •J r 1 c r p g t;i« fx f^' J ' 31 ALGEBRA. [chap. 43. If the expressions are not arrancred accordin'r to nowpr, ascending or descending, of some common letter, | Jea?raT' ment will be found convenient. ' ^^^^^^" ge- Example 1. Find the product of 2a=' + 462-3a6 and 3a6-5a= + 46= 2a-- 3ah +462 - 5a'+ 3ab +4b^ ~10a* + 15a^-20aVi^ + 6a%~ 9a-b^ + 12aP 8a^-h'^-12a¥+lGj,i ■10a* + 2la'^b-2la"b' + 1G6-* The rearrangement is not i necessary, but convenient, because it makes the collec- tion of like terms more easy. Example 2. Multiply 2xz -z^ + 2x-^ - 3yz + xyhyx-y + 2z 2a;^+ xy+2xz-3yz --a _a; - y +2z 2aN^ xV + 2xh - Zxijz -~^ -2x^-y -2xyz -xyH3y'^z+ yz^ _^^z + 2xyz + 4xz^ ~Gyz-2-2z^ 2s?- a:V + 6a;2z - Zxyz + Zxz' - xy^^ 3fz - 5^/^Z^3 Multiply together EXAMPLES V. e. 1. 3. 5. 7. 9. 11. 13. 15. 17. 19. 20. 21. 22. 23. a + b + c, a-jb-c. a^-ab + b\ a"'-i-ab + b\ a?-2x'- + 8, x + 2. x^ + xy+f', x-y. 16a2 + i2a6 + 9j2^ 4a -36. a^Ha;-2, x'^ + x-^S. -a^-Va%-a?b\ -a-b. 2. a-26 + c, a + 26-c. 4. x"' + 3y% x + 'iij. 6. x'-zY- + y\ x"- + yl 8. a^-2ax + ix% a- + 2aa;-f-4£ 10. a^x-ax'^ + xi-a^ x + a 12. 2x"''-3a:2 + 2a;, 2x'' + 3x + o X- - 3xy ~ y\ -x^~ + xy + y\ i o ys „2i/^ / ' ,. ^ ' ,,7- ab + cd + ac + bd, ah + cd ~ ac - Id ~^ay- + 4ab^+Voa% 5a^bH ah'-'' ~ 3b\ 27x^-36ax^ + 48a"-x-64a% 3a.' + 4a 2i. a?- oab-b\ a^ + 5ab + b^ In each of t !• The proi . .2. The first I binomial expre 3- The thir ! two binomial e , 4. The mid Jumierioal quai second terms oi [chap. ng to powers, a rearrange- igement is not t convenient, ios the collec- terms more v.] -y + 23 -2f -2=3 ~c. + 2aa; + 4a:- , x + a. H 3x4-2. K + 3. f - 3y. a^i-^ + Js. MULTIPLICATION. Multiply together 25. «=+6Hc^-ic-.„-„,,„,.j^^; 26. -.^-V + ^/^ + a:V + a.•■'-:ty, a;+y 28. 3a- + 2a + 2aH-l+a^ a'-'-2a + I 29. -a^-=' + 3aay-9ay. -ax-3«2/=. 30. -2..V + .^^-3.y + .^-2.^^ .H2.y + y, 31. |«^ + ^a+i, ^a_i. -f^y- 33. -|a;2 + a;w-i-5»/2 1, 35 32. la;^-2a; + #, 1^?+^ « - 2 ^e.rt o?,m,'}?*f,,^^^^^^^^ Although the ^ explained, it is of the r^t 1st Ln.o . ^'K f% ^"^*^''«d« ^I^-eady «oo„ learn to write down \t pZ /cTl^V^^^^^^ iJns IS done by observinn- i rapidly b?/ inspection. I lie terms in theVoXet "rise an^, T^ ^^^^ coelHcients of i .en the combination of the nmnerllT "'f >'"'^^ *'>^>^ »'««"lt bu)ommls which are multijl-edJiS^;;"^^^^^^^^^ iu the two ='^■--15.^•+6G. I» each of those results we notice that : 2 .pf ' f ";^"^t ^«"«i«fc« of three terms, f bin;miarexXsSs." "" ^""'"^^ «^ ^^^ «rst terms of the two ! t.^ b^Kp^SLii "'^ P^^'"^^ ^' ^'- --n^ terms of tho jmiterSl q^u^Sies'Ttaken wfth^r^^'^^"^ ^'^'^ ^"'^^ of tl>e second terms of the ^^^o^o^^t.^Z^^^^^^^ ''^'''^ "^ «- ■'a ■'I -«4V ••31 ; » 36 ALGEBRA. The intermediate step in the work may be omitted, and the products M^ntten down at once, as in the following examples : Cv + 2)(.v + 3) = x- + 5x + G. (^-3)(.f+4)=^2+A--12. (A-+6)(^-9)=A'2-3.r-54. By an easy extension of these principles we may write down the product of ani/ two binomials. ^ Thus (2^+%)C^-y) =2x^^ + 3x1/ -2a.y-3f' =2^H 0.^-3^2. {3x - 4y) (2x +^)=Gx^-8si/+ 3xy - Af = Qx^-6xy-A^j\ • (^ + 4)(.j? - 4) = a.-2 + 4*' - 4a^ - 1 6 =a;2-16. (2.r + 5ij){2x - 57/) = 4^2 + iq _ -^q _ ^q 2 =4i-2-25^'2 Write down tlie 1. (x+8){x-5). 4. (a;-l)(a; + 5). 7. {a;-4){a;+ll). 10. (a-l)(a + l). 13. (a -8) (a + 4), 16. {a + 3) {a + 3). 19. (a? -3a) (a; + 2a) 22. {x + 4i/){x~2i/) 25. (a + 36) (a +36). 28. (2a; -5) (a; +2). 31. (3a;-l)(a;+l). 34. (4a; -3) (2a: + 3). EXAMPLES V. f. values of tli, and the others follow m descending order. 6:i-''-4ar»-20a;3-7a:3_20ie + 10. Example. Multiply 3a^ + 2a^b + ia¥ + 2b* by 2a^- - h\ 342+0+4+2 iJjfO-^1 O + 4 + 0T8 + 4 „___^i:i2-0-4 -2 6 + 4-3 + 6 + 4-4-"2 We write a zero coefRcient to represent ?u'tho"fi T"^^^"i"g «-'''^ which is^absei"t 111 the first expression. Similarly, the term containing ah is represented by ! /.ero coelhcient in the second expression 6a« + Aa% - ^^aHfi + fia-V." + Art-h* - Aa¥ - 2b\ of multiplication. ^ '"^^^ practised in the ordinary fnll process CHAPTER VI Division. produces «. This opemt.on of division is denotej by «-J ". s^fcsittfeor"^^ °^ ^^p---™ « « called ;„t Div,si„„ is thus tl,e inverse of m„Iti„lica«„„, and This statement n,ay also'be erprrss"ed verbally as follows ■ qnotientxdivisor=dividend Ja-rh have been estSS f^ Ml^ieSl^St 47. T'A, Rule of Signs holds fo, dMon. Thus I'H a x(-S) « ^~b. -a _^ -=-6. — a _ ^ —v. Hence in division as well as ninltiplLtion nice sicins j)roducc -u unlike signs produce ~. We remove from the divisor and dividend the factors com- mon to both, ji c -a in arith- metic. CHAP. VI.] DIVISION. 39 Division of Simple Expressions. 48. The method is sliewn in the following cxaniplos : Example 1 Since the pro.luct of 4 and x is 4:r, it follows that ivlien Ax IS divided by x the quotient is 4, or otherwise, 4a;-ra; = 4. Example 9.. Divide 27a» by 9a'. The Ciuotient = ^"''=27aaaaa =:3aa =3a2. Therefore 2W -r Oa' = So'. Example 3. Divide 35a^b'^c^ by '7ab''-c'"'. The qnotient = ?^^^^^-_^-5„„ c-'JaV I We see, in each case, that the index of any letter in the quotient I ^sthe chfferenee of the indices of that letter in the dividend and ■ dmsor. Tins is called the Index Law for Division! The rule may now be stated : Rule. The index of each letter in the quotient is obtained h, To the result so obtained prefix with its proper siqn the quotient of the coefiieient of the divictend by that of the divisor. ^ Example 4. Divide 45a«6 V ^y _ ^a%x\ The quotient = (- 5) X ftS - »62 - 1^4 - 2 = -5a%x\ Example 5. - ^la^b^ -f ( - Ta^ftS) = 35 Thus, by the rule a' 4- «' = a' - 3 = ^o . but also aHa-^ = % =1. ludicei explained m the chapter on the Theory of i 1 40 ALOEnUA. [chap. DivWon of a Con>^„„d Expression by a Siinplo Expression. tpT,:"" "^ '"^ ^"«''/;;s,sr fife ^^^ '-^-^ * iiiis follows at once frcm Art. 33. Example.' *y + jx y j .- _ .^x= - 4x + lOy - 3xi/. 3. 5. 7. 9. 11. 13. 15. 17. 19. 21. 23. 25. 27. 29. 31. 33. 1. 3x^ by ar" -SSa;" by 7arl ^Y hy xY 4a"-b"'c^ by ahhl -rt'c" by -^o^ -lOarVby -4^.^2. .35a" by 7a7. 7a26c by -7a8ic. lC6=^a;2 by ~2xi/. x-~2xy by a;. -24a.-''-.32a,'» by -S-r). cC'-ah-ac l)y -«,. Sr'-9a;V-12.ry2 j.^ -3a2 + ,|]a^,_6„(. \^y _ EXAMPLES VI. a. ^x. ^Y-3xYhy -^a^^a -2aV + ^a'«^i.4 K,. 7_3., The reas( into as ma ((uotient is ThuH.f-'+]] namely .1" + thus we obt Example 2 V..] DIVISION. 41 Division of Compound Expressions. 50. To divide one compound exprosmon by another. t/.w;«7;.o,'^J^r"f J''"''"" ^^^ ,^^^^idend in ascending or uimxnainrj jwuos oj some common letter as\„XZl::ri '"*"' *" ^™» "'' ''""''-"i - -""'H 'erm. i^r&fr "^"""'^ "" "" "" '™' '■'•« "■" *■-•« are Example \. Pivide a:2+ila; + 30 by a: + 6. Arrange the work thus : put the produo't :.= + ». „„aer the a&lld. 'Iv^ f^T^l'i ^^ """ a; + 6)a~!+Ila; + 30(a? by subtraction 5ar + 30 The entire operation is more compactly written as follows • a: + 6)a:2 + l 1:^ + 30 (a; + 5 a.--+ ()g 5a? + 30 5^ + 30 nnnti^nf If Parts as lua ^ be convenient, and tlie com' '-to na inPlv Sip . ^ is^'liVHlod by the above process into two parts n !L w •• • T^ ''•'*+^^' ^"^' ^acJi "f theV is divided bv xli'. thus we obtain the complete quotient a;+ 5. ' ^^ ^ ' Example2. Divide 24ar"-65ary + 21y= by 8:e-3y. 8^ - 3y ) 24x3 - 65.rv + 2 1 v^ f 3r - 7,y 24a;2- 9a:.y " " -5Ga:yf2l7/2 -56a:y + 21y-' < •! r e I 42 Divide ALGEBRA. EXAMPLES VI. b. [CHAr. 1. xH^x + 2 by a?+L 3. a^'-lla + so'by a~5 5. 3«3+10a; + 3 by a; + 3. 7. 5a:2+lia; + 2 by a; + 2* 9. 5a;2+16a; + 3 by a; + 3* 11. 4:t^ + 23a. + 15by4aM-3. 13. 3a;- + a:-14 by a; -2 15. 6a:2-3ia. + 3g i,„ g.^'- 7 2. a:=-7.r + i2 by a; -.3. 4. a2-49a + 600 by a-2o 6. 2x^+\ix+5 by 2a:+l in !t-'i^^ + 21by2a;-^3. 10. 3.3 + 34.+ ,! by 3a:-f-l. 12. 6.2_7^_3^,y2^_g 14. 3a?2-a;-l4 by x + 2. 16. 4.2 + .- 14 by x + 2. 17. I2a2-7a:^_i2a,2bv'J« a,. Jo* ^'''' + *^- ^^ by . + 2. ^. ^^-4.y-,5y^ ,; ,^,!j^ 20. 9aHCac-35c3 by 3a + 7c. 26. 16«3_46a3 + 39a-9by8a 3 ^' fjf ^^t"'"-'^ ^^ ^=^-2. 28. 16-96. + 2l6.3-216.3;8,f; b^' 2 3a: ^ ^" '"' ''^ ^-^- th'Lii:;\CSof'i;:-tL'L^^^^^ *^ -- - -inch ^^7t ^^-^««---^ + 4.^-5.3-._,5by2.= . + 3 2.4-5a^'- 5a;2 -43^5- 8.2- a? -10.2 + 5.-15 52. Sometimes it will bo f^., 1 ^'™"'* Divide 2»'+I0-,6,,-39..H.,5„, hv « 7 4«-14a2^- Ja-* l^-:_8a2^-J0a3 - «a'-^+l2a'+15«4 _1_^5^1+12a''+15ffl4 . ) .i vij DIVISION. 43 53. We add a few harder cases worked out in full. Example 1 . Divide a: « + ^a* by x- + 2xn + 2a"' x^ + 2xa + 2a^):^ + 4a* ( x'-^ - 2xa + 2a= x^ + 2x>a + 2x^a^ -23T^a-2x'a- Z_^^a-ix^aPj^4xa^ 2x^a^ + 4xa^ + 4a* 23^a^jjxa^ + 4ai Example 2. Divide a^ + ¥ + c^- 3db^hy^b+T~ a + b + c)a^-3abc+ b^ + cHa^-ab-ac + b^-bc + c^ a^+ a% + a^c ~ a^b- - a^b- ~a^c - -ab^- -a^c + -a^c 3abc abc ab^- 2abc abc- -ac" ab^~ ab^ abc + ac- + P + 63 + &2C abc + ac^-b^c abc - IPc - bc"^ ac- + bc^+ c3 .|;ts|"V=S!^>^^^^ — ive re. The result of this important divifio^n will\e referred to later ^.ay^;ti7bX%ef "^"*^ are fractional the ordinary process Example. Divide ^a^ + ^\xf' + ^y^ by ^x + ^y. h^ + h)i=^+r\^f+j\y^-lx^''-l^y+iy. In tlie examples i;ivou hitlierto the dr f-rr br- h .t tuutaiiied in the div dpi.J wi Vi j- • ^ ^'" '^'^'''" ^^aet y York should be ca ed on m M I]'" ^•"'«'?"/'' "?t exact th^ dimensions [Ait. 24TtL the divisor. ''"""^"' " ^^ ^°"- c < M i S i u Divide EXAMPLES VI c. foHAP. ^^-12 by .= .3.,,. 2. 2//3- 3. 6»i3-^2_ 5y-l. 4. 6a'-i3a4^4^ 6. ** + ar' + 7r2-6a; + 8 + 3a2 by 3a3_2a2- a'-o«-8a2 + i2a_ by a:--f-2a; + 8. a. i:-t'!'??=t' ^-'^ -y 9 .by a2+2a-3, 10. »-^-4a:4^3^_^g^2_ 11. 30x* + iix3_ 12. 30y+9-7iy3 14. 15 + 2?«4-3i aH3a + 2. by a;^-.r+3. (-2 by a;2-3a; + 2 3« + 2 by a;2-a,_2. 4 + 3a;3 + 6. + 1. 82«2-5a: + 3by 2a:- t!lt~^T''}y'y'-^^y 1k + 2 by 3P-;t ic „, ^ r^^ + ^2-7a:3 by a:2 + Q_o^ 16. x^~2x*~4arf 17. 192 -a:4 + 19*2 by ar-i-7 a?'' + 2-3.r. m^ ••c + 5. ^ .^^:^:-:^!^ 19. a^ 20. ■ar^y f. ' + a:4y- 78r^V + 45a:y3+i4y4by2.r2+ r/:-^:+^^-s^ by «.-./_ «"-&" by o3_^3 21. 23. 24. 25. rS- 27. 28. I-a3_ ^V + :^-2a:y2 + y3 ^y y- 5a?y+7y^ oc' ■{- xy ~ if; 22. a;»-7/9 y' by ar» + a:V + ^y: 2a?. 29. «'H2a«M+6i2 by „.;2a26^ I-a«-Sar'-6aa:byl-«_ Find ti.e quotient of by a-^h~\ '+y'- 26. ai2-fti2 by a^-fi! 30. ¥T«'-TV«''+-Aa- jci - 3a?. ry2^3 31. 32. Aa^-i'slT ^«-(rV by '^''' + ^-oO^ by |„2^ 1 rtC. TB 33. 36a;3+i,,2, I 34. Xwd ^f;+T~4a:^j-6xi-yi;y6^_r 5 _ 2 4 3 siiiax* by §a~^x. SV ','' VI.] DIVISION. 49 55. The following examples in division may be easily verified • they are of great importance, and should be carefully noticed. ' 'Jt- ■y X- ofi — y^ x+y, x'^+xij+y'^, x-y and so on; the divisor being x-y, the terms in the quotient all positive, and the index in the dividend eWier odd or even. II. x+y .7 ' ./ » -~^=x^-x^y+xY-xy^+y\ x+y x'+y'' -^y -^x'^-^y+xY-xhf + xhf-xf+y^ and so on ; the divisor being x+y, the terms in the quotient alternately positive and negative, and the index in the dividend always odd. — ~-=X-' x+y -yy ■y' x+y —3fi-x'^y+ xy"^ - y\ x+y ^~=x^-x^+X^y^-xy+xy*-7/, and so on ; the divisor being x+y, the terms in the quotient alternately positive and negative, and the index in the dividend always even. ,x. • J ^^' '^^'^ expressions x^+y% x^+y\ x^+y^, ... (where tne index is even, and the terms both positive) are never divisible l>y x+y or x-y. AH these different cases may be more concisely stated as follows : "^ (1) a;"-?/" is divisible by x-y if n be any whole number. (2) A-" + ?/" is divisible Iw x+ii if « l»o nu^f V'^ wi.^i« ^, — 1,__ )X *'„ ~^ ?^ divisible by x+y if n be any even whole number. (4) X +y» 18 never divisible by x+y or x-y, when n is an even whole number. ^ u^ 1 4 lit J *' 4 1 w ', ■ m < It . ») m ' M I ( '» ■ ■ 11 •1 c ;> ; - » ' ■ a •• . f ' ^ K H ; tt r 1 : I ; v 1 -tr ^M r ■ 1 r '' 1 c. I r I p) w^^ c H p* ^H p ' 1 j 1 1 ' jH 1 ' CHAP. VI] if r! If CHAPTER VII sr'°vir-'' '-^^- ' S htti^ '"F'^ *™"''" V ^' ihll Sometimes a liup p.iL "^ . ^^^ '" common use a.i the symbols to be co^Zl ^ ^ "^^-^""^ ''^ drawn «; -e meanin, as .-(hS^;!.; ll!;- ::|i^i:^-l;^tb the Removal of Brackets. |"sid; paiv/aTrhfcS'titj/L^I"^'^^ ^^^^ ^<^ l^egin witli tbe tl.e rules already given "^ Zt sT'^?'" "^ ^"^^^««'«" ^^Xph ^--rnplel. S-pUfy, by removing brackets the "^•"-^"^^---oneby::^^^^^^^^^^^^ ^-«-i>^e2. Simplify t,Heoxpre.io„ T]ieexpression=-r-2a;-/q„ o o ^^^ + -^-'- L -a;- % + 2a;-3y-3a; + 2v +arl CHAP. VII.] REMOVAL AND INSERTION OF BRACKETS. EXAMPLES VII. a. Simplify by removing brackets 1. a-{b-c)ra + {b-c) + b-{c-ia). 2. a-[b + {a-(b + a)}l 3. a-[2a-{36-(4c-2«)}]. 4. {a-{b-c)} + {b-{c-n)}-{c-{a-b)}. 5. 2a-{5b + l3c-a])-{5a-[b + c]). 6. -{-[-(a-6r7)]|. 7. -[a~{b-{c-a)}]-[b-.{c-{a-b)}l 8. -(-(-(-a^)))-(-(-y)). 9. -[-{-(6+c-a)}] + [-{-(c + a-&)}]. 10. -5x-[3i/-{2x-{2y-x)}]. 11. -{-(-a))-(-(-{-a;))). 12. 3<'i-[a + b-{a + b + c-{a + b + c + d)}l 13. -2a-[3a: + {.3c-(4y + 3a; + 2a)}]. 14. 3a;-[.5y-{62-(4a;-7y)}]. 15. -[5x-(Uy-3x)]-[5y-{3x-6i/)l - [15x - { Uy -{\5z + ]2y) - (lOx - 157,)}]. 8x-{lGy-[3x-{l2y-x)-8y] + x}. -[x-{zi-{x-z)-{z-x)-z}-xl -[a + {a-(a-x)-(a+x)-a}-al -[a-{a + {x-a)-(x-a)-a}-2al m i7 16. 17. 18. 19. 20. 58. A coefficient placed before any bracket indicates that Note. The line between the numerator and denominator of a fraction is a kind of vinculum. Thus —^ is equivalent to ' (a; + 5). thf W'„r ^T''''°" "^ ^^'^^^™' ^(^ + -^) i« °ft«° written v^T],. Joot of fS ''^^"g/^Sarded as a vinculum indicating the squafo root ot the compound expression x + y taken a.-i a whole. 'J^'lius v'25Tl44 = v/Ie9=13, whereas \/25 + v'Ii4 = 5 + 12 = 17. i: M.I 1 ■ I »t I ^ • I m ■ < ; 1 m ; ^ 1 r ii 1 ■ > I ^ 1 i^ » 1 ■ •< *! ■ B « j 1 ■ < ' 1 e ; > 'i " • » ; : ^ • • 11 • ' M . ? ■ ' ,«♦ i i r 1 V 5' f r r £ { !' r #■ £ m ' ■ P. 1 I ■ t ■ i 1 is -'If i I : il IP ALGEBRA. fOHiO>, thf L.?"^^^-- ^'^ - advisable to si.plif, f„ the cou.e of i^xample. Find the value of ■the expression = 84- 7r-lii>. 4/ i-r -84 - „ -* -^7a; + 3(8-9+5:,)|j = 84-7l-1U.-4{-17:. + J5x-3}] = 84-7[-lla:-4{-2a;-3n = 84-7[-lla: + 8a: + 12] = 84-7[-3a; + I2] =S4 + 21a;-84 =2Ia:. «te^'i^fe Sd^^S^ff P-^- the „u.W Of EXAMPLES VII. b. Simplify by removing brackets 1. «-[2H(3c-3a-(a-,5))+2a-(64-3c-)]. 6. 36-{5a-(;6a+2(10a-&)j|. ^^^■'^• 7. «-(ft-c)-[a-6-c-2(6 + o}l 8. 3a2-[6a2-(8fi3_(9,2_2a2)}] 15. 2(3fi-5a)-7[a-6{2-5(o-Mn 18. -10{a-6[a-(6- la c)]} + 60(6-(c + a)| -2(a + a:)]}. vn.) KmoVAl AND INSERTION OF BRACKlrn,. 49 19. -3!-2[-4(-<.)]) + 5j-2[-2(^„)n 20. -2!-[-(:t-yW + {-2r-(a:-y)]j. a l(l"<-')-»(»-o)}-(^f-ir'.}..(,_„_|,„_^,j Insertion of Brackets. Art.21,22; for conv^t^ ^^p^irX^" — ated iu ^''(ickets remaining unaltered ^ V ^er^/ term within the Inltf7n/Te stn "- JSri/"^^ .^^ ^«^^«^^^ ^^ithin term within the brackets he cSlged. ^ "^ '^'' ^'^'^ 'f '''<^y Emmples. a-l + c-a-e = a~h + (c -d~e). l^jBt The tern, of an expression can be bracketed in various ^^-auple. The expression a.-6. + ,,_„y^, may be written (ax h-r\Mf . J-^^V qf (""^ -»!/)- {bx~ by) + (ex ~cy), .^rt^^^^^^^ \o every tern, within a [ofth^>^expressiin witS tl'e bmckef '''^ '"''"'^ '' ^ """l^^P^'er D < 4S f'jf 4} 50 ALGEBRA. [chap. Example 1. In the expression aa^-cx + 7-dx^ + bx -c- da^ + bx^-2x The expression = (oa^* - rfar') + (6a;2 - dx^) + {bx - ex - 2x) + (7 - c) = x^{a-d) + x-(b-d) + x{b-c-2) + (T-c) = {a-d)x^ + (b-d)x^ + (/j~c-2)x + 7-c. In this last result the compound expressions a-d,b-d b-r o are regarded as the coefficients of x^, x\ and x respectively.' ' I ExampU 2. In the expression - a?x - 7a + a^ + 3 - 2a: - aft bracket I together the powers of a so as to have the sign -before each bracket The expression = - {p?x - a?y) ~(7a + ab}- {2x - 3) = -a'^(x-y)-a(7 + b)-(2x-3) --{x-ij)a^-(7 + b)a-(2x-3). n ill '.I! EXAMPLES Vil. c. In the following expressions bracket the powers of x so that the signs before all the brackets shall be positive : ' 1. ax* + bx^ + 5 + 2bx-5x^ + 2x*-3x. 2. Sbx^-7-2x + ab + 5ax^ + cx-4x'-bx^. 3. 2-7x^ + 5ax--2cx + 9ax^ + 7x~3x\ 4. 2cx'^-3abx + 4dx-3bx*-a^x^ + x*. In the following expressions bracket the powers of x so tliatl the signs before all the brackets shall be negative : ' 5. ax^ + 5x!^-a^x*-2bx^-3x^-bx*. 6. 7x^-'Sc^x-aba^ + 5ax + 7a^-abcx^. 7. ax^ + a^x^-bx'^-5x^-cx\ 8. 36V -bx-ax*-cx*- Qc'x - 7x*. Simplify the following expressions, and in each result re-jjroup the terms according to powers of .r : i 9. axr^ - 2cx - [6a;2 - {cz - dx ~ (bx' + Sex'') } - (ea;^ - brj) ]. 10. 5ax^ -7(bx- ex"') - { Qbx"- - (3ax^ + 2ax) - 4car»}. 11. ax^-3{- ax' + 3bx - 4[|car' - |(aa; - 6a;2)] } . 12. x^ - 4hx* - 1 [l2ax - 4[ibx* - 9 (f - b:^) - ^ax^)']^. IS. oc{x~b~x(a-bx)} + ax-x{x-x{ax-b)}. |vn.] REMOVAL AND INSERTION OF BRACKETS, 51 more conveniently written by LZ^f' A \^:;'^^^ T^ ^'^ [powers of some common letter '° ^' according to Exam2>k 1. Add together ax^ - 2bx^ + 3, bx- cx^ - a;'-' and o^-ax- + cx. Thesum = ax-3-26,x-2 + 3 + 6a;-car'-a:2 + a;''-aa:» + rx = ct^''-cx^ + x^-ax"'-2bx"--x"' + bx + cx + 3 = (o-c + l)ar»-(a + 26 + ]).rH(i + c)ar + 3. Example 2. Multiply aa;^ - 2bx + 3c by ^;a; - g. The product = (aa;2 - 26a,- + 3c) (/;a; - 5) = «i>x-^ - 2bpx' + 3cpx - aqx^- + 2bqx - Sc^ = apx^ - (2bp + aq)x^ + (3cp + 2bq)x - 3eq. EXAMPLES VII. d. 1. aa^-2ca;, bx^-ca^, cx^-x. 2. a:2-a;-i, ax^-bx\ bx + x^. 3. aV-5a:, 2aa;2-5ax3, 2x=^-bx^-ax. 4. aa:2 + 6a:-c, g-ar-r-^jaj^, a:2 + 2a; + 3. 5. px'-qx, qx^-px, q~ar\ px^ + q^s^ 6. ax" + bx+l and c:e + 2. 7 ...2 9^ . , „ , Q , , /. ca. -Ja? + 3 and aa;-6. a. ax'-bx-c and px + g. q 0,^2 o^ i , , liA , „, ^ y* -^ -3a:- 1 and 6ar + c. 110. «a;2-26a; + 3c and a;-l 11 2 « 1? .3a. « . "• ^* -2a;-g and ax-3. 12. a:'' + aa:- - 6a; - c and ct-^ - ax'- ~ bx + c. 13. aa«'-ar2+3a;-6andaa,-3 + a:-:> + 3a; + 6. V t til -1" ■ I u i, ■ t (I ' ■ ' M ■ f r 11 \\» i» -I c:> r i r r IK I i m CHAPTRR VIII. If! Simple Equations. 64. An equation asserts that two expressions are Pnnal !.„♦ we do no usually employ the word equaLnTZ wTdeltnt"' Tlie parts of an equation in the nVht unA l«.f*- r.f +i,^ • and s called an equation of condition, or more usnallv".. of the present chapter is to explain how to treat an eqult^oi o the snnplest kind in order to discover the value which SS it fi,®®* J^^^ ^^"®^ ^^°^® ^'^^"6 it is required to find is o^]U] to It, form an extremely inii sorts of mathematical prob mination of some quantity ' , — „ "l-'v^Kiiuiuiis onus iportant part of Mathematics. All Mems consist in the indirect deter- more equations, and secondly to solve these ennnf ;nn« V CHAP, VI U.] SIMPLE KQVATIONS. ti .k®®fl .^!1 equation which involves the nnknown qnantitv in the fir<^ degree is called a simple equation. It is us a[ S denote the unknown quantity by the letter x. 1. If to equals we add equals the sums are equal. 2. If from equals we take equals the remainders are equal 3. If equals are multiplied by equals the products are equal. 4. If equals are divided by equals the quotients are equal. 70. Consider the equation 7.^'= 14 Dividing both sides by 7 we get ^=2. [Axiom 4.] Similarly, if 2= -6, multiplying both sides by 2, we get A • . ., . ^=-12. [Axioms.] Again, in the equation 7.f-2.r-.r=23 + 15-10. by collectino- terms, we have 4.r=28. i-ouecting .-. a: =7. 71. To solve 3.r - 8 = .*• + 1 2. Tl.is case differs from the preceding in that the unknown £nt^lr''''T both sides of the equation. We ca^,, 1 .Tevr transpose any term from one side to the other by sTmnlv changing Its sign. This we proceed to si. ew. ^ ^^ Subtract X from both sides of the equation, and we get ,,,. „, , , 3a;-.^'-8=12. [Axiom 2.] Adding 8 to both sides, we have r,, , 3.r-.r=12-f8. [Axiom 1.1 oKl r """ ^''^ "^'^^'■5 ^"'J -« ''»« bee., -enioved from one side and appears as +8 on the other. T J*f ^;^i'!^.* *^^<^ «!™'lar «teps may be employed in all cases. — I.. ,. ^e raa^- enunciate the ioliowing lule : tv/S; A ^fr u'^'l ^^ . *''<'''<^Posed from one side of the ''1^^<^iion to the other hj changing its sign. ^ I :* <: M.I m I •! ■ I ■t t ' « , If » 1 fi* 1 K k' :■ i »" Ir -^c ?^ r «e i KM . le^ 1 i • ■ •# 64 AI/4EBRA. [CITAP. It appears from this tlmt we may change the siqn of even, tmmn an equaUon ; for this i.s equivalent to traiflolf/a clmng^pSia^ ''"" ""'"^' '^" ''^'''- '^"'^ l^-ft-hand'nrel:.' Example. Take the equation -3x-l2^x- 24 Transposing, -a; + 24 = 3a:+ 12, which 18 the original equation with the sign of every term changed 72. To solve "n-3: '4"^5' Thus, multiplexing by 20, . . 10,v-G0 = 5,r + 4.t': transposing, lo.^ _ r,.t: - 4.,; == 60 ; «nw- ^^.i^'^ "o\v give a general rule for solving any simi.le equation with one unknown quantity. ^ ^ ^ r,7?'Jt}f) ^"'*'' '^."f^^^^^'-y. clear of fractions ; then transime ^xamp^e 1. Solve 5(x - .S) - 7(6 - a?) + 3 = 24 - 3(8 - x) Removing brackets, 5x - 15 - 42 + 7a; + 3 = 24 ~ 24 + 3a,' • transposing, 5x + 7a; - .Sa; = 24 - 24 + 1 5 + 40 _ 3 . ' .-. 9a; = 54; x = (i. Example 2. Solve 5x- (4a;-7)(3a;-5)=6-3(4a;-9)(a:-l). Simplifying, we have ' 5a;-(12a;2-41a; + 35) = 6-3(4a:2_i3a, + 9,. And by removing brackets 5x--12a;2 + 41a;-35 = 6-12a:3 + 39a;-27. Erase the term - 12a;= on each side and transpose ; " oaTiix ,3ya;_u-i;7-f-a5; .'. 7a;=14; x = 2. When a; =2 VIII.] SIMPLE EQUATIONS. 55 Note. Since the - sign Ixsforo a bracket affects every \ vm within it, in the first line of work of Kx. 2, we do not removo tiic brackets until we have formed the products. Examples. Solve 7a;-5[a:-{7- 6(a;-3)}] = 3a; + l. Removing brackets, wo have 7a:-5[a:-{7-6a;+18}] = 3a: + l, 7a:-5[a:-25 + 6a;] = 3.r+l, 7a;-5a;+12r>-30a: = 3a:+l ; transposing, lx-5x- SOx - 3a: = 1 - 125 ; .-. -31a:=-124; a;=4. 74. It is extremely useful for the beginner to acquire the habit of verifying, that is, proving the truth of liis results. The habit of applying such teats tends to make the student self-reliant and confident in his own accuracy. Ih the case of simple equation^ ^,ve have only to shew tliat when we substitute the valu^ ol .< in the two .sides of the equation we obtain the same result. Example. To shew that « = £ "U'^^fiei) ti; equation 5a;-(4a;-7)(3a.--5) = '"- 3/4r~9)(a:-l). When x=2, the left side 5x - (ix - 7) (3a; - 5) = 10-(8-7)(6-r>) = 10-1=9. The right side 6 - 3(4a; - 9)(a; - 1) = 6-3(8-9)(2-l) = 6-3(-l) = 9. Thus, since these two results are the same, a; =2 .satisfies the equation. «a EXAMPLES VIII. a. Solve the following equations : 1. 3a;+15=a; + 25. 2. 2a;-3 = 3a;-7. 3. 3a; + 4 = 5(a?-2). 4. 2a: + 3 = 16 -(2a; -3). 5. 8(a;-l) + 17(a;-3)=4(4a:-9) + 4. 6. 15(.r-l) + 4(a; + 3) = 2(7 + a:). I--' 56 ALGEBRA. Solve the following equations: 7. 5x-6{x-5) = 2{z + 5) + 5{x-4). 8. 8{x-3)-{6-2x) = 2{x + 2)- 9. 7(25 - a;) -2x= 2{3a? - 25). 10. S{m-x)~{78+x)='29x. 11. 5a?-17 + 3a;-5 12. 7ar-39-10a; + ._ 13. 118-65a;- 123 = 1 [ciiAr. >(5~u-). 6a;-7-8ar+115. /ar-39-10a; + 15 = 100-33a: 14. 157-21 + 26. 5a; + 35 -120a?. {a: + 3) = 163-15(2a;-5). 15. 170-lS(a:-10) = 1.58-3{a:-17) 16. 97-5(a; + 20) = lll-8(a; 17. a?-[3 + {ar-(3 + a:)}] = 5 8(a; + 3) 18. 5a; -(3a? -7) -{4 19. i4x-{5x-9)~{4-3x~, 20. 25a;-19-[3-|4x-5n = 2a; -(6a? -3)} = 10. 21. (a; + l)(2a+lj = (. (2a; -3)} = 30. }] = 3a;-(6«;-5). 22. (a? + l)2_(a;2_i)^ :a; + 3)(2a; + .3)-14, 23. 2(a; + I)(a; + 3) + 8 = (a;2 - 1 ) =a;(2a; + 1 ) - 2(a; + 2) (a; + 1 ) (2x+l)(a; + 5). + 20. ^. C(a?=-.Sa; + 2)-2(a?2-l)=4f 25. 2(a;-4)-(a;2 + 26. (.r + 15)(a;~3)-(a?2-_-^, 27. 2a;-5{3a?-7(4a;-9)} = 66 >+l)(a; + 2) -24. a:-20) = 4a;3-(5a? + 3)(a?-4)- 6a? + 9) = 30-15(a?-l). 64. 28. 20(2-a;) + 3(a;-7)-2[; 29. l-t^-J^~8~2{8-S{5-x)~x}]=: > + 9 - 3(9-4(2- a?) }]=22. 30. 3(5-6a;)-6[a;-5{I-3(a;- 31. (a:+l)(2a; + 3)=2(a; + l) 0. ' + 8. 15. 5)}] = 23: 32. 3(a;-l)2-3(a:2_2)^^,_^^ 33. (3a;+l)(2a?-7) = 6(a;-3)»+7 34. ^"^-8a; + 25=a;(a-4)-25(a;-5). 35. a;(* + l) + (a.+ j)(^^2) 36. 2(a; + 2)(a;-4) = a?(2a;+] 37. (a:+l)2 + 2(a; + .3)3=3a.( 38. 4(a; + 5)2-(2a;-f-l)^ = 3( 16. (a; + 2)(x + 3) + a;(a? + 4)-9. 39. 84 + (a;4 4)(a;. 40. (a;+l)(a? + 2)( )-2I 3a?(a; + 2)+3,'] 5) + 180. 3)(a? + 5) = (a;4-l)(a; + 2)(a; + 3) a; + 6)=a;3 + 9a;2 + 4(7a;-l). ^'^io^oUl7&sri!::.,i^T-^^^<' most useful vni.] Example Multiply ] thus removing bri transposing, collecting te: Note. He line between as a bracket. 76. In c multiply th clear of fraci Example 2 Multiplyin transposing, Now clear ( thus 77. To 80 expresij the d but it is oftei Examjtle 1. Expressing clearing of frai transposing, m Vlll.] Example 1. Solve 4- SIMPLE EQUATIONS. a;-9 X 1 «T x^ Multiply by 88, the least common multiple of tlic denominators; t'lus 352-ll(a;-9) = 4A--44; removing brackets, 352 - 1 1 a; + 99 = 4a; - 44 J transposing, - Ha;- 4a; = -44-352-99; collecting terms and changing signs, 15a: = 495; •• a;=33. 37 — 1 Note. Here --g^ is equivalent to -|(a;-9), the vincvlum or "Z bSeT. ^^ArtTsT' "' ^^^"""^•"■•^*^^ ^--g *he same effect «,7u' 1 ^" ^®^^^" ^Y^'f ^*^ 7^'^ ^"^ ^«""^ more convenient not to multiply throughout by the l.c.m. of the denominator, but to clear ot fractions in two or more steps. Example 2. Solve ^ + ^^^ _ 5a: - 32 a: + 9 3 ' 35 ~ 9 Multiplying throughout by 9, we have 18a: - 27 28 3a:- 12 + 35 18a:- 27 . 9a: + 81 6a:-32-^+81 28 ' transposing, — -^-^ + ^^±J1^ ^ 2a: - 20. Now clear of fractions by multiplying by 5 x 7 x 4 or 140 ; ^^"^ 72a: - 1 08 + 45a: + 405 = 280a: - 2800 ; /. 2800 - 108 + 405 = 280a; - 72a: -45a: ; ••• 3097 = lG3a:; a:=19. ovn,l*«. 1^ ^'']^^- ^^r*'0"» f J'ose coefficients are decimals, we may bXit is of .rff "''''^" "' ''"^^^^ ^^^"«"''' ^"^^ P'-o^^^^J as'before^ but It IS often found more simple to work entirely in decimals. Example 1. Solve •^x+-2^-\x=\-^ -'15x -~. Expressing the decimals as vulgar fractions, we have , . ,^ t^ + i-^a^=l^-fa:-^; Clearing of fractions, 24a: + 9-4a: = 68-27^-12. transposing, .- ^, 12-9. 47a: =47; . a:=l. 41 «•.! 4 at. I i .« I M « jij m < '1 '5 ir <;» r^ ALGfifillA. fcHAP. viii. of the other eide. ^ '* "^^ '« tramf erred to the nuvieraZ Jhe ready application o£ these principles will be found ver. Example 1. then Example 2. then If If 3« 14' 9 '35' 9x14 x=}r;;-^=\i. 35x3 5* 5= -a:; :..-r. anftt^LtiSi?4roSte^^^^ EXAMPLES VIII. c. p- I 1 2 3 I -. i 4 -?-«> 7 3 1 '• 2x~-%- ''' 1=-^ - -t ^ 13- H 16. ^^=15 ? • 8 2a?- ^^' l6=2^- eq^SiL:'"" '''^ ^'^'"'^•^ '' ^ -^»«h «atisf3. the following 2. 5. 8. 11. 14. 17. 3_ X 7 "14' ^_29 17~5T' _4«_ 1 3 --2* 13 050: 2T~84- .-A.- 4 21a:- "f 18-42* 3. P^= Oft 56 8a: ^' 15 ="3- 6. 9. 12. 15. 18. 21. 3^6 5 a?* — = -?. 15 45' i.-25 3a;~27" -7_ 1 3G^^ 35 5a;* 3a: "27* 19a: 57 7 ~49" CHAPTER IX. Symbolical Expression. 79. In solving algebraical problems the chief difficulty of the beginner is to express the conditions of the question by means of symbols. A question proposed in algebraical symbols will frequently be found puzzling, when a similar arithmetical question would present no difficulty. Thus, the answer to the question •' find a number greater than x by a " may not be self- evident to the beginner, who would of course readily answer an analogous arithmetical question, " find a number greater than 50 by 6." The process of addition which gives the answer in the second case supplies the necessary hint ; and, just as the number which is greater than 60 by 6 is 50 + 6, so the number which is greater than xhy aia x+a. 80. The following examples will perhaps be the best intro- duction to the subject of this chapter. After the first we leave to the student the choice of arithmetical instances, should he find them necessary. Example 1. By how much does x exceed 17 ? Take a numerical instance; "by how much does 27 exceed 17?" The answer obviously is 10, which is equal to 27 - 17. Hence the excess of x over 17 is a; - 17. Similarly the defect of x from 17 is 17 - «. Example 2. If x is one part of 45 the other jMirt is 45 - x. Example 3. If x is one /actor of 45 the other factor is — • X Example 4. How far can a man walk in a hours at the rate of 4 miles an hour ? In 1 hour he walks 4 miles, In a hours he walks a blines as far, that is, 4a miles. Example 5. If £20 is divided equally among y persons, the share of each is the total sum divided by the number of persons, or £ — . 41 -v*.' « 111 J « I tl I M ■ «4 m i ^ m mA »«5 € > r: 62 ff ALGEBRA, fOHAP. lOHAP. ren^afnderS. '^ ^ '^'^'^^^ ^Y tl,e quotient is 2, and ti.e that ia, 17 g So if .y be di icied by D b.J ,u remainder 7,-, we have ' ^ *^^ 'i"°*'«'^t be (? and tiie or AT-n Thas, if the diviaor is a. til t-^' the dividend i/^il ''"°*''"' '^' ^"'' ^'^^ ^^^maind^r .. What i?]ius won Jhas lost, ^mtsgr + 20xshiilings. 1 2. 3. 4. a 6. EXAMPLES IX. a-. WI.«tm,„tbe„Hedt„a,.t„„,ake„. the S«,e:.'^ ^= "■"*" '"'o two part, a,K, „„e ,«rt <« . ..h„, , -. Ja/L';i,V«fiTe»T °' ''^o "-"''- •>« ". and if the sn.aUer he i. tfe ofL*; ™"- »' '« ■"'"■'- te c ana „„, „, „,,„ ,, ^ ,^,^^^ 13. What is the exees, of 90 over:.. if "^.'""'""""hdoes.rexoe^.lsn. iH|i ^:V-V fOHAP. ■ IX.] 2, and tJie 1 "' ■ 18. ? and the ■ 19. ■ take? ■ 20. ■ 21. SYMBOLICAL EXPRESSION. 6S "' il ' with £p m shillings m In X yeurs a man will be 36 years old, what is his present age? How old will a man be in a years if his present age is x years? If X men take 5 days to reap a field, how long will one man « What value of x will make 5x equal to 20 ? What is the price in shillings of 120 apples, when the cost of a score is x pence ? 22. How many hours will it take to walk x miles at 4 miles an hour ? 23. How far can I walk in x hours at the rate of y miles an hour? 24. In X days a man walks y miles, what is his rate per day ? 25. How many minutes will it take to walk x miles at a miles an hour ? 26. A train goes x miles an hour, how long does it take to go from Bristol to London, a distance of 120 miles ? 27. How many miles is it between two places, if a train travelling p miles an hour takes 5 hours to perform tlie journey ? 28. What is the velocity in feet per second of a train which travels 30 miles in x hours ? 29. A man has a crowns and h florins, how many shillings has he ? 30. If I spend X shillings out of a sum of £20, how many shillings liave I left ? 31. Out of a purse containing £a and h shillings a man spends c pence ; express in pence the siim left. 32. By how much does 2a3 - 5 exceed a; + 1 ? 33. What number must be taken from « - 26 to leave a - 36 ? 34. If a bill is shared equally amongst x persons and each pays 3«. 4rf. , how many pence does the bill amount to ? 35. If I give away c shillings out of a purse containing a sovereigns and h florins, how many shillings have I left ? 36. In how many weeks will x horses eat 100 bushels of oats if one horse eats y bushels a week ? 37. If I spend X shillings a week, how many pounds do I save out of a yearly income of % ? 38. A bookshelf contains x Latin, y Greek, and z English books: if there are 100 books, how many are there in other languages ? 39. I have X pounds in my purse, y shillings in one pocket, and V pence in another ; if I give away half-a-crown, how many pence have I left ? 40. In a class of x boys, y work at Classics, z at Mathematics, and the rest are idle : what is the excess of workers over idlers ? V « Ml • J W 4l 1 i '^ ■1 1 «) '• 1 : , ^ V , "■ _ •« I Ml C^5» ■ * M ■•I ■"••*« '€:> - " ^« Mi , i 4 ' m ; *i '^ ■r , . i '■ » '«■ b - ' f f p^ r la :■ lU « ' :" c "■ \ M r ■ i-h i' • i % . € r wo € ' \ : m» 64 \ i ALGEBRA. [CHAP. xn a? years the son's ace will be w-i- ^ .rJ u ^ J-™^. 2. rind the Simple inte^, „„ ,, ,„ „ ^^^ ^^ ^ ^^ Interest on £100 for I year is £/, f . £1 £k ^loo' ioo* '. Interest on £i- for ji years is £^' r, 100' findTor^'ny a1rr;:4^:;tl;?t'^i?n^^^^^^ -^ « ^eet high; and how many s^are Jards ol pap^r for' "he Vaui"'''^^ ''' '^^ ^«-' (1) The area of the floor is 3xr/ square feet ; .-. the number of square yards of carpet required is 3^=.^. (2) The perimeter of the room is 2{3a: + y) feet • ^ ^ .-. the area of the walls is 2a(3x+y) square feet ; •■• ""'"^^'" °^ ^^^^'-e yards of paper required is 2a(3^_+y| ai'^crwhat'isThe nliSli"? " ""'"^"^ '^^•""^"g ^o™ the' left are The number is therefore'eq^lt^^hSSl^^tt^^^^^^^ If th ^- • =I00a + 106 + c. 100c + 106 + (,. Example 5. What is n Wh Becutive numbers of which the leasUs i? "'" ^'"^""^ °^ ^'^^^^ ''^n- The numbers consecutive to n are « + l, „ + 2; •■■ *^«°«ai^«i(« + !) + (« + 2) 3n + 3. And the product=w(« + i)(„4.2^ DC.] SYMBOLICAL EXPRESSION. 65 We may remark here that any even number may be denoted by 2«, where n is aio/ positive whole number ; for this expres- sion is exactly divisible by 2. Similarly, any odd number may be denoted by 2/i + l ; for this expression when divided by 2 'leaves remainder 1. Example 6. How many days will a men take to mow b acres if c boys can mow a acres in 6 days, and each man's work equals that of 11 boys ? Since c boys can mow a acres in 6 days ; •■• 1 boy he days, :. n boys, or 1 man, — days, 7v he ••• amen i^ days, an •' ' he .'. amen lacre... -|r.daysj therefore a men can mow b acres in -^ davs a'n ■' EXAMPLES IX. b. 1. Write down four consecutive numbers of which x is the least. 2. Write down three consecutive numbers of which y is the greatest. 3. Write down five consecutive numbers of which x is the middle one. 4. What is the next even number after 2ra ? 5. What is the odd number next before 2a; + 1 ? 6. Find the sum of three consecutive odd numbers of which the middle one is 2n + l. 7. A man makes a journey of x miles. He travels a miles by coach, b by train, and finishes the journey by boat. How far does the boat carry him ? 8. A horse eats a bushels and a donkey b bushels of corn in a week : how many bushels will they together consume in n weeks ? 9. If a man was x years old 5 years ago, how old will he be w years hence ? 10. A boy is x years old, and 5 years hence his age will be half tluit of his father. How old is the father now ? 11. What is the age of a man who y years aco was m tim.es as old as a child then aged x years ? 12. A 'a age is double B'b, B'b is three times (7s, and C is x years old: lind^'s age. 13. What is the interest on £1000 in b years at c per cent. ? E.A R S 5 .1' ■ « |i C ;> .1 , M r « ! i ii ■» I 66 AI/IEBRA. [chap. m 14. What is the interest on £x in a years at 6 per cent. ? 15. What is the interest on £50a in a years at a per cent ? anium ?^^* '' *^^ '"*^'^'^ °" ^^'*'^^ '" "^ '"^"*^« ^<^ ^ P^'' ««»*• Per 17. A room is x yards in length, and y feet in breadth • how many square feet are there in the area of the floor? ' 18. A 8(,uare room measures x feet each way : how manv sonarp yards of cir-, , • i ; be re.iuired to cover it? ^ """^ ™"'">^ «'i"'^'^« of rar ^ ^^"^ ^ ^T long and X yards in width : how many yards of car :. .s a , -e -, ide wdl be required for the iioor ? 6 Bt wll'f^ '^^u^^ '''"J '" P°""'^^ °^ carpeting a room a yards Ion. 6 feet broad with carpet costing c shillings a square yard ? ^ « ^^' *,^T ""^r^ y*^'^^ °^ ^^^Pet a: inches wide will be reouired to cover the floor of a room rj feet long and z feet broad ? ^ 22. , A room is a yards \, », yards broad ; in the middle there is a carpet r feet .-,.^.,.re : how inan> .quare yards of oi^dotl will be required to cover the rest of the floor? a iSies hT'J Ws'?""'^"' ''^^ ^ P"'''''' "^""^^ ^ *^ """"*«« *^ ^^ ^'"^Iks 20^milef rn'c'K;:?" '' '"'^' ' P'"^" *° "'^^'^ ^ ""'^ ^^ '^^ --'ks mo'v'; thr'ouglit onT to^nc?"" '" ' '^"""' "^^ """^ ^^^ '^^ '^ 26. A train is running with a velocity of x feet nor second • how many miles will it travel in y hours ? ^ ' ma^-n^ricTe^ lly ?""" '"'^ ^'^ "^«" ^ ^^^ ^^ --' ^^ -', me^'dot'rhoTrsT" ""' ^' "^"""^ '' '^' '" ^ ' '^"^^ ^'-^ , waS:^^:i^^?s^crs r;L:f ^^ ^"^ p^°^"«« ^^ -*--^ ;«?^' /'!^^ '" ^'^^ '"'"^"y y^'"'"'' '^ principal of £a will product £d interest at r per cent, per annum. piotiuct t^ *82. In Example 6, Art 80, we prov. >d Lprssii7'^^'e'rf ''''' /^ '^ f°'' Bt;tement a general relation expiessin, '^.e co .lection tween a niniber, its divisor ai-i the resulting quotient and remainder. "'visor, ai i tne This is an ovari-'nl" "f - • • i . _ _ 8t-itAniPnfoT -''"^^'P- --^^ « --^^J ':;:purxanc class of algebraical statements V ,> -ti ^^ forrmil ,, and it m. be worth while briefly to foreshadow fheir use and ; vpHcation, not only in Alffebni but also in other branches of Mathe^ .atics ^nd ele Jentar/p 'Sics whence 7)=1 IX.] SYMBOLICAL EXPRESSION. 67 Dkfinition. a formula is a rela, i cbtabliahed by reason- mg among certain quantities, any on. f which may in turn be regarded as the unknown. Thus in the formula above mentiojieu, if e velocity, and time; and all these can be solved without ditticnlty by common sense reasoning. At the same time we ZIX'^'^.^^-''^ *J'l^'S °"'>^ particular cases of the genJa formula s^vt, in which a denotes tie space described by a body which moves with uniform velocity t> for a time t. ^ In this formula, if t denotes the number of seconds the bo , f 'i £ I ( and 5x = 180, .'. a: =36, the greater part, 60 -a; = 24, the less. 70 ALGEBRA. Ti lll^ tOHAP. Example 3. Divide £47 betwppn A n n „ t-i. i. , £10 more than B, and i £8 n^e than a ^' "^ '^^* ^ "^^^ ^*^« Hence a? + (a: + 8) + (a; + 8+10)=47j a:+a;+8+a;+8 + 10=47, dx = 2\ ; so that G has £7, 5 £15, A £25.' "*"" ^ ' if eth't'sf -cosl 7T:;d S duck t '" '."^i".^. «^^«° ,-'^ ^-J^^ ' birds bo^ht was J& :rov^'t4":,^f etrfd hety^^*^^ ""'"'^ «^ q«intre:tpre1se^Vl'^^^^^ "««"^n^. ^^P^*-- *« W all stance it whA^^^^^^^^^^^^^ - ^he^p-ent in- du?ksf '"^^ ""'"'^'•^ "^ «^««^' *^'^° 108 -. is the number of SiBce each goose costs 7 shillings, x geese cost 1x shillings. shilHnls!''^ ^"'^ '"''' ^ ''^"""Ss. 108 - X ducks cost 3(lo8 - .) Therefore the amount spent is 7a: + 3 (108 -a;) shillings; ?sl64'aiings" ''• *" ''^' *^^ ^'"'^•^"* - -^- £28. 4., that Hence 7ar + 3(108-a;) = 564; 7a: + 324 -3a; =564, 4a; =240, -* • .o^-.':«;tt„"rtMsrcE. Let £'» age be j. years, then ^'« age is 2x years. y»rs'^r.hr;'erve''*rS(;x;"'^'\-''' ='■'■' '"-''' 2a:-I0 = 4a;-40, 2a;=.30; •*■ a:=15 so that 5 is 15 years old, A 3o'years. a Se/o"ft^i'X""Pl!;.*.'^^ unknown quantity .represents careful to avoid hcainnina^^i}::!;"' "^"-V: j ^"'^ ^he student must ])o vague and inexact *''*' *^"''''' ' *"" '^"^ «t'^^— "^ -' or any statement so 11 , i„; X.] PROBLEMS LEADING TO SIMPLE EQUATIONS. 71 EXAMPLES X. a. 1. One number exceeds another by 5, and their sum is 29 ; find them. 2. The difference between two numbers is 8 ; if 2 be added to the greater the result will be three times the smaller : find the numbers. 3. Find a number such that its excess over 50 may be greater by 11 than its defect from 89. 4. A man walks 10 miles, then travels a certain distance by train, and then twice as far by coach. If the whole journey is 70 miles, how far does he travei by train ? 5. What two numbers are those whose sum is 58, and difference 28? 6. If 288 be added to a certain number, the result will bo equal to three times the excess of the number over 12 : find the number. 7. Twenty-three times a certain number is as much above 14 as 10 Ih above seven times the number : find it. 8. Divide 105 into two parts, one of which diminished by 20 Bhetll be equal to the other diminished by 15. 9. Find three consecutive numbers whose sum shall equal 84. 10. The sum of two numbers is 8, and one of them with 22 added to it is five times the other : find the numbers, 11. Find two numbers differing by 10 whose sum is equal to twice their difference. 12. A and B begin to play each with £60. If they play till ^'s money is double B's, what does A win ? 13. Find a number such that if 5, 15, and 35 arc added to it, the prodyct of the first and third re'sults may be eijual to the square of the second. 14. The difference between the squares of two consecutive num- bers is 121 : find the numbers. 15. The difference of two numbers is 3, and the difference of their squares is 27 : find the numbers. 16. Divide £380 between A, B, and C, so that B may have £30 more than A, and C may have £20 more than B. Yl. A sum of £8. Ms. is made up of 124 coins which are either florins or shillings : how many are there of each ? 18. If silk costs six times as much as linen, and I spend £9. 8s. in buying 23 yards of silk and 50 yards of linen : find the cost of each per yard. 19. A father is four times as old as his son ; in 24 years ho will *Mi!y be twice as old : find their ages. 20. A is 25 years older than B, and .^'i's age is as much above 20 iis B'& is below 85 : find their agea. 4 •«*>; i *' t u E '^^ «i 2 it m * »t !..^f ^ IK 9J \ ill r n ALGEBRA. [Clliil'. th| tips rs :lv;i^iLraU'*"" """ '^"^°' ^ ^"^ "^^ shmLgs^ '"Thelumbef ofTh^r^^^^^ crowns, half-crowns, and number of crowns and twice the S "'^."^ 7".^? ^o"^ times the were there of each ? ^^ ""'"'^^^ «^ shillings : how many he??e /wyjr^tt'e'Urafoli S^i.^ SulV^^' ^^ «- years 24. In a cricket mnt.l. .k , ' "'"'"' Present ages, the remainder™ tlTc'rc wi ii;::.rbvt"'"%°J *" -"-• «-' 1 A S^lTn^t^L^rb^-St'i'th'lT'^uV^'. '^'' ' « "■» hafUn iSc'reStVfee? llZtJ*' ^Tl''"- l-^ « '-'! « -ch «0 square fect : find 4e Stin'JSro^^'^Tc tr"'^^'''' ''^ f.^«o,MU„et?:;n°r '"'°'''™' "'''■='' '"»'' *» ovations witi, Let ^ be the sn»I,er number, then :.+4 is the greater. One-half of the greater is represented bv '(.-xii , of the less by ia. «'"'='"'y2(''+"), «nd one sixth Hence multiplying by 6, 2(*' + 4)-ia;=8; 3a;+12-a:=48; .-. 2a;=36; and ' JaZ.}^' ^•'*' '®'''' number, _'Tffil''8f,rX,-«''"''' ^ >-"- JSO-. shilling. Hence 180-a;=|(84 + a:); 10.«0-6a:=420 + 5ar. r,„ - •■• a- =60. Iherefore B wins 60 shillings, or £3. wijf. , aiKi X.] PROBLEMS LEADING TO SIMPLE EQUATIONS. 73 EXAMPLES X. b. 1 Find a number such that the sum of its sixth and ninth parts may be equal to 15. 2. What is the number who' e eighth, sixth, and fourth parts together make up 13? '■ X. o' P^^^^ ^^ "^ number whoso fifth part is less than its fourth part by 3: rina it, '■ au^' /i"? a "«™^er such that six-sevenths of it shall exceed four- fifths of it by 2. 5. The fifth, fifteenth, and twenty-fifth parts of a number together make up 23 : find the number. 6- , Two consecutivo numbers are such that one-fourth of the less exceeds one fifth of the greater by 1 : find the numbers. .1 '^' '^ali ""'"^^^ differ by 28, and one is eight-ninths of the other: find them. * 8. There are two consecutive numbers such that one-fifth of the greater exceeds one-seventh of the less by 3 : find them. I ?^ 7}^^ ^}^T consecutive numbers such that if they be divided by 10, 17, and 26 respectively, the sum of the quotients will be 10. 10. A and 5 begin to play with equal sums, and when B has lost five-elevenths of what he had to begin with, A has gained £6 more than half of what B haj left : what bad they at first? 11. From a certain number 3 is taken, and the remainder is (livuled by 4 ; the quotient is then increased by 4 and divided bv 5 and the result is 2 : find the number. 12. In a cellar one-fifth of the wine is port and one-third claret • besides this it contains 15 dozen of sherry and 30 bottles of hock- how much port and claret does it contam ? 13. Two-fifths of ^'8 money is equal to B's, and seven-nintbs of Bsis equal to C's ; in all they have £770 : what have they eacli ? U.^A, 5, and (7 have £1285 between them : A'a share is greater than five-sixths of ^'s by £25, and (Ts is four-fifteenths of B't- find the share of each. 15. A man sold a horse for £35 and half as much as he cave for It, und gained thereby ten guineas : what did he pay fcT the horse ? A?i ,'^^®,^J *< e s ^ M * ** m H ■J ' -^^ € :> m ^% m : H m 1 ^ Qt » - » a ' ^ r 'i lU V' IP' '•i r ^ € \ C r 'Ma c m CHAPTER XL Highest Common Factor, Lowest Common Multiple OF Simple Expressions. Highest Common Factor liighMt power ol i thT,vai drwdol'^^ ' -vKle »■> a, a'; b> is the factor. "^""^ ' ' ' ' V ; and c is not a rom»,o,i coemcient to the a^bSa? 1^^^ t =S So^ ^ ^^ ^ ^ is 1a^^/X i^:o^tr'oiT:^J^,^' ''^'^y' ^^«'A. 28.3.^ I2I tt.' r?*'f '^'"'"''" "^^^""'^ «f the numerical coefficients • '"'^tit^lSr^. -^ ^«"- -h^«h divides ever;r of EXAMPLES XT. a. Find the highest common factor of 5. 5a»6^ I5a6c'-'. 6. ^xY-z\ Uxifz. 8. 7a264^5^ Uab-^c^ 4. a/^c, 2a62c 7. 4a%-k'\ 6a%\^ 9. 15«yc2^ ]2a;2ys2 IL idax^, 63ai/\ 56az^ 13. a^xY, Pxy^- c-Vy. 10. ^i^-^a-z, IVOX^i/z, 125X1/. 17. 16a«i.\-7, 60aWc8, SyaVr'^c-'. 10. 12. 16. 18. Sa^a:, 6ahxy, lOaftary. 17a62c, 34a2ic, Slaftc^, 2ia%'<(^, Gia^Pc% 48«.W(r». 35aVft, 42a3<.ft2^ go^c^fc-'. 'chap. XI.] LOWEST COMMON MULTIPLE. 75 Lowest Common Multiple. 91. Definition. The lowest common multiple of two or more algebraical expressions is the expression of lowest dimen- sions which is divisible by each of them without remainder. The abbreviation L.C.M. is sometimes used instead of the words lowest common multiple. 92. In the case of simple expressions the lowest common multiple can be written down by inspection. Example 1. The lowest common multiple of a\ a\ a", a« is a\ Example 2. The lowest common multiple of a^b\ aV^, a^W is aW; for a^ is the lowest power of a that is divisible by each of the quantities a?, a, a^ ; and V is the lowest power of h that is divisible by each of the quantities h*, h^, V. 93. If the expressions have numerical coefficients, find by Arithmetic their least common multiple, and prefix it as a coefficient to the algebraical lowest common multiple. Example. The lowest common r:ultiple of 2\a*x^y, 35aVy, 28a%y* is 420a*ary ; for it consists of the product of (1) the least common midtiple of the numerical coefficients ; (2) the lowest power of each letter which is divisible by every power of that letter occurring in the given expressions. 1. 4. 7. 10. 13. 15. 17. EXAMPLES XI. b. Find the lowest common multiple of abc, 2a\ 2. x^y"^, xyz. 5a%c\ AaJfic. 5. ac, he, ah. 8. 2a:, 3y, 4z. H. a^hc, bha, c^ab. 2a;V, Sxy, A«?y*. ^*h-c\ 5aWc^ a\ hc\ ch\ 3X2, 4y2^ 3^2, 14. 16. 3. 6. 9. 12. 5a\ 6c?>2, 36c3. 1os*y, 8a?i/^, 2a:V ^xh/z, 4x^y^. 12ah, 8xy. 2ab, 3hr., ica. la\ 2ab, 3b\ 35a^c\ 42a^cb% SOac^b^ 18. Q6a*bh\ 44a%*c\ 24a%h*. Find both the highest common factor and the lowest common multiple of 19. 2abc,Sca,4bca. 20. 2xy,4yz,Qzcjy. 21. 9abc,3b"-c.cab. 22. i:ia:'bc, 39a='6c2. 23. 17xyz\ 51a;2y. 24. l^x^yh, 25xih\ 25. Sab, 2bc, 5cab. 26. ITm^V, 51mV' 27. aPy\ yh*, z^xf. 28. \5prY, 20mY 56y^z^p ^°* P "56 a« -"6^ 4 «4.l i »• > «) E < ^ » « «» M ;ii ^ei € :> IB « ^ « -#• ■ •( » . rt BSI y 1' I BU f tr> r €■ € c I 78 f 1: ij ALGEBRA. [CIIAP. Reduction to a Common Denominator ^^mple. Express with lowese common denominator the fraction, O' c M„lr;?^ngThrn\Sll^^ denominators is 6.,:. which'^is^eVired tSX deno'rJin^'r ^^ '^'^ ^'^^tor equivalent fractions denominator Garyz, we have tlie EXAMPLES XII. c. Expresa », equivalent f,ucM„,« „it,. common denominator- 1. 5. 9. 13. 2a" 2^ a 2a b b' 3'e' a h he* ca 2x % 3y' 2x 2. 6. la 14. 4x y 4«' 5re' 6 nab "*• 26' ? n It P '' 2a:' 3i- 4. 8. a 4a 3a 56' lo?- 11. -, a; 3 y a c Q b' rf' ^- 3a:' 6a;" 5b ^^- 76' 2k" 12. ^, y 3a:. y X 2 b a a 3' 9" 16. — I* Addition and Subtraction of Fractions. ^araw^j/e 1 . Simplify ~ + ^x- 1-^. mi - 3 4 ({ ■ The least common denominator is 12, The expression = ^^' ' ^-^ " ^^.r 15a; 12 5a: 12 ~ 4 xn.J Example 2. The expn Example 3. The expr< Note. The erasing eaua cancelling e reducing fract tions he must numerator an( Thus in 6as "3 and not the because it on fraction is the When no i be understocM Example 4. If a fractic before combii Example 5. Simplify tl 1. a; a; 2 3' 5. ^,y 2"^ 5' 9. X y 7 21" 13. 2a: 4a: ELEMENTARY FRACTIONS. 79 ExamvU2. Simplify ^-f^-^^.^, rt^, . 6ab-5ah-ah » Tiie expre88ion= — j-^^ = j— =0. Example 3. Simplify 2? 3ca»* The expression = 3 „ ^, and admits of no further simplification. Note. The beginner must be careful to distinguish between erasing equal terms with different signs, as in Example 2, and cancelling equal factors in the course of multiplication, or in reducing fractions to lowest terms. Moreover, in simplifying frac- tions he must remember that a factor can only be removed from numerator and denominator when it divides eacli take^i as a whole. Thus in 0^3 3 , c cannot be cancelled because it only divides cy and not the whole numerator. Similarly a cannot be cancelled because it only divides 6ax and not the whole numerator. The fraction is therefore in its simplest form. When no denominator is expressed the denoninator be understood. 1 may Example 4. 3 g'^Sa; a'^^ \2xy-a'^ 4y~ 1 4y~ 4y If a fraction is not in its lowest terms it should be simplified before combining it with other fractious. Example 5. ax xhf ax Zxy 2 X 3' .3aa; - 2x EXAMPLES Xn. d. Simplify the following expressions : 1. 5. 9. 13. X X 2 + 3' ? + y 2"^ 5* X y^ 7 21" ^ 4x 3 "'■■5' ^' 4 5" - a 6 ^' 4-6- in — A „ a a ^- 3" 4 4. 2»_5 3 X 7. 11. 771 P n l2" Q 2m n "• 15 ~5' 16 48 14. 5x 4x IK ^_7? ^^' 6 12" 12. 16. Hm n 12" 36* 2a_46 5 15* 2 1 « t aJ E il » H 4 tl C :> a ■ *f at kc «« *► , « im '^ r 1 ■u ? ■. r« r if €' »:. ^ ^^' ,^ ,»» lg (BK 1?^ 1 1 « 80 ALGKBRA. Simplify the following expressions : 18. *• 17 a a a 23. ?-|^. a 6 _ _ :r X 4 8 + l2" 21. —-£.4.* 2*- ^.. ^rf• 25. a+^. 27. 3a2-a' aa;- 36 28. aH-. a 29. ^^-??. da; x^ 22. ^4.£. « 8 +12~4' 26. a;-^-'. 30. i>3-~. MISCELLANEOUS EXAMPLES IL {Chiefly on Cliapters I.-VIIL) 4a-+7a:^6T '"P"'""" '""^^ ^« ^^^'^^^ to 4a- -3.^2 to produce the value of I +'^4^^^^.^''' ^=a: + y + 2, and C=.10ar + y- find 3. If a: = 3, y=4, 2 = 1, find the value of 4. Simplify by removing brackets 5. Multiply ar» + a:2 + 3a: + 5 by a.-" - a; - 2. 6. 8 jjve the equations : ^'' (2) 5a:- 15 = 17a; + 21. 7. Divide ar»-l0a;2 + 9 by a:2-2a;-3, 8. Simplify 7a-46-{5a-3[ft-2(a-'6)]}. hasLic'e\rmrn^tt^^^i"^"-^^--^^«. ^ ^as 2a:-3.. ... ^ 10 Pin^.v, " 7''^"^'*"^' "^^^k^^'^ve^.^, and C together? lU. J?md the sum of \-2xA.y& ^^ o,^ e o .. 6 '^' • the result in descending poweS of i. ' ^^'-7^-2, arranging 24. Solve the xir ] msCETXANEOUS EXAMPLES, ti. 11. Write down th- folio wing products : (1) (a: + 17)(a;-3); (2) (3a; -8) (8a: +3). 12, Solv ' tho equationB : (1) 7ar-3-{7-6a;)=r3-3a;-i6ar + 8). (2) (5x+i ,^x-2)~'iT-3){3x-l) = lO-ax■\- 81 13. Prom the sum of 3ab, -6a6. 2ab. lab -ftnh «nKf..o«f +u-. 14. When a-:4, 6=3, c=2, find the numerical value of 2a + 6 (2c a) 15. From what expression must Ua•^-5ab-^hc be subtracted so as to give for remainder 76(a ■\-c) + 5a^t auocracwa so 16. Multiply arHeara + Saj-S by a:2-2a:+4. 17. Simplify 12a-[6a-2{3a-4(6-a)}-(9a+8 j. 18. Solve the equations : (1) 3(2ar-l) + 2(3a;-2) + 3=4(ar-6); ^2) ^(a:+l) + |.(a: + 3) = |(a; + 4) + I6. Verify the s. ition in each case. 19. Divide 3p' + 16p* - 33^)3 + 14^2 by p^+1p. 20. Add together a-\-2b-(2c+d), 3a-(6-2c)+2d, and 2a-[6-(2c-3rf)]. mlk 9l?.fel\^'"''«^°" "^"«^ 7a:3-6ar'-oa: be added so as to 1 ^:u ^?u* ^*^^® ^'^ ? ^^^ "^*^« *^e product of a: + 1 and 2a: +1 less than the product of a: + 3 and 2ar + 3 by 14 ? •'^ "^ * ana ja: + 1 23. When a=2, 6 = 3, c = l, gr=4, r=6, find the value of SoV - 3«2* + 2»-a8 - c***. 24. Solve the equation : _ aj-13 6a; + l 2 Shew also that x=3 does not satisfy the equati «l 1 ti 41 , »i ' t« r ' n ■i :i« f 11 M 1 ' *i m 1 • 41 € > lu r i' c r CIS K , « s f MICROCOPY RESOLUTION TEST CHART (ANSI and ISO TEST CHART No. 2) 1.0 I.I 1.25 y^ III 2.8 |2.5 11111=== ■ 63 i-^^- BIUu |2.0 1.8 1.4 1.6 ^ /APPLIED IN/MGE inc 1653 East Main Street Rochester, New York 14609 USA (716) 482 - 0300 - Phone (716) 288- 5989 -Fox 82 ALGEBRA. [chap. XII. 25. A horse can eat 9m + 2n bushels of corn in a week- how many weeks will he be in eating 12m^ - Imn - lOn^ bushels ? ' 26. Subtract the sum of 2a:3-3a; + 4 and -Sa^ + 2x-T from 43:3-3x2+0; -6- [2a^-(a:-6)]. 27. Find the value of a^ + ¥+c^- 3abc, when a=l,6=4, c=-5. 28. Solve the equations : 2x x-6_S(x A, ^ ' 15"^ 12 ~l6\2~^J' Divide 3^6-37^4+35^3+7^2+2 by y{y-l)(y + 4)-2. 29. iu^' A ^^^'^^^ ^ll'-^O between A and B so that for every half-crown that A receives B may receive a shilling. 31. Find the value of (a-l-6)2 + (6 + c)2 + (c + a)2 when 0= -1, 6= -2, c=3. 32. Multiply (2nv' + 8)(m + 2) by 3m -6. 33. Divide the product of x-2, x + 3, and 2a;-7 by the sum of 3 {x^ - 2ar - 2) and 5a; - a;2 _ jg 34. A man walks at the rate of a miles an hour for p hours • he then rules for q hours at the rate of b miles an hour. How far'has he travelled, and how long would it have taken to ride the same distance at c miles an hour ? Also work out the result supposing p = 7, g = 3, a = 4, b = 9, c=U. 35. Solve the equations : (1) f-^=2te-|(2:r+10fV)i 36. An egg-dealer bought a certain number of eggs at Is. 4d. per score, and five times the number at 6.-!. 34. per hundred. Se sold the whole at lOd. per dozen, gaining £1. 7s. by the transaction. now many eggs did he buy ? "^ CHAPTER XIII. Simultaneous Equations, 100. Consider the equation 2a? +5^ =23, which contains two unknown quantities. From this we get 5y = 23 — 2.r, that is, 2/ = ^^^ (1). From this it appears that for every value we choose to give to .V there will be one corresponding value of y. Thus we shall be able to find as many pairs of values as we please whicli satisfy the given equation. For instance, if ^=1, then from (1) y=— . 27 Again, if .r= -2, then ^ = -5- ; and so on. But if also we have a second equation of the same kind, such as 3.r + 4y = 24, we have from this y- 24 - 3;V .(2X If now we seek values of .r and 1/ which satisfy both equations, tae values of y in (1) and (2) must be identical. Therefore 23-2£_2j4-.3^ 5 4 ' Multiplying up, 92-8:p=120-15.r ; .-. 7^=28 ; Substituting this value in the first equation, we have 8 + 5^ = 23; .-. 5y = 15; ••• 1/ = ^ and A- =4./ Thus, if both equations are to be satisfied by the same values of .V and ?/, there is only one solution possible. I r... 'il * Hi m .< ■** m 1 i:» Ik ' «l » ■. « e» ■ ^ g. S f r €' t c r KB c IfM im ■5 if I > 84 ALGEBRA. [OHAP. 101, Definition. When two or more equations are satisfied «L«u ^""^ values of the unknown quantities they are called Simultaneous equations. We proceed to explain the diflPerent methods for so)^'"'>'- aimrl attention to the simpler cases in which the unknown quantities are involved m the first degree. ^ »"w".8 ml?hn^ ^? the example already worked we have used the method of solution wAich best illustrates the meaning of the term mnultaneous equation; but in practice it will be found that this 18 rarely the readiest mode of solution. It must be borne m mind that since the two equations are simultaneous^ true any equation formed by combining them will be satiSl by the values of ;r and y which satisfy the oriSal eqSLns 2;^ W^'1 Tk" ^^T^^' ^' *^ ""^^^^ an equation^whicSvdves one only of the unknown quantitiea "ivoives 103. The process by which we get rid of either of the un known quantities is called elimination, and it must be effected propS ^^^" "''"'^"'^ *" '^' ""'"^« °^ *^^ «q""'?ons Example \. Solve 3a?-f 7y=27 5a? + 2y=16 (1). (2). 15a;+35y=135, 16a;+ 6y=48; subtracting, 29y=87; .-. y=3. equrtions^ "-'' '"^''""*' '^'' ^^^"^^ °^ i' ^^ "'^^^'^ of the given Thus from (1) 3a; + 21 =27; and p"e preoiinu cAampiu, if we subatitute 3 for " '"' 5a; + 6=16 y m (2), we have ir=2, as before. Substitute i Xlll.] SiMlJLTANEOtJS EQUATIONS. 85 Example 2. Solve 7a;+2y=47 (1), 5a;-4y=l (2). Here it 'vill be more convenient to eliminate y. Multiplying (1) by 2, 14a;+4y=94, and from (2) 5a; - 4y = 1 j addtTig, 19a; =95; .'. a; =6. Substitute this value in (1), .-. 35 + 2y=47; and a;=5./ Note. Add when the coefficients of one unknown are equal and mh.:z m sign ; subtract when the coefficients are equal and like in 8'gn. E^'ample 3. Solve 2a; = 5y+l 24-7a;=3y (1), (2). from Sl^^ThL^^'"''"**^ "" ^^ substituting in (2; ita value obtained 24-^(5y+l)=3y; .'. 48-35y-7=6y; /. 41=41yj und from (1) a;=3.J 104. Any one of the methods given above will be found suthcient ; but there are certain arithmetical artifices which will frequently shorten the w rk. Example 1. Solve 171a; -213y= 642 114a;-326y=244 (I), (2). n£'it^Jtm ■'^\ *"? "* *''*/***^" ** ^^'^^^^^ ^^«t«r 57, we shall make the coefficients of x m the two equations equal to the least i.onmy,i mxdtiple of 171 and 114 if we multiply d) by 2 and (2) byT rru..- - . _ Thus subtracting, that is, and therefore from (1) 342a; -426y= 1284, 552y=552; y=i, a;=5. r C r <) Si), I i * ■ i) [ i Ml € > I ,! 1:' i 1; \i^ \ 86 ALGEBRA. [chap. ExampkQ. Solve 127a;+ 59y=1928 (j) 59a: + 127y=1792 !..!.'.......!..(2).' By addition, 186a; + 1 86y = 3720 ; •■• « + y = 20 (3) Subtracting (2) from (1), 68a; -68?/= 136 ; .'. X-l/ = 2 , n\ Thus, by an easy combination of (1) and (2), the problem is reduced, to the solution of the eauations (3) and (4). From ei we obtain by addition 2a: = 22, an^ by subtraction 2y = 18 Therefore a;=ll, and y=9. EXAMPLES Xin. a. Solve the equations : ^' ^L'Vt-'g' 2' ?;:?^=!!' 3. 4a;+7y=29, ^q?''~7riS' 5- 5^ + 6y = 17. 6. 2a; + y=10, 3a;-7y=19. 6a; + 5y=16. 7a;+8y=53. 7. 8a;-y=34, 8. 15a:+7y=29, 9. 14ar-3v=39 ^ + 8y=53. 9a:+15y=39; 6a;.-17y=35: 10. 28a;-23y= 33, 11. 35a; + 17y=86, 12. 15a;+77t/-q2 63a;-25y=101. 56a;- 13|;=17: '^ 55^-33^=22 ^^' ?^:lr72' ^*- ^1^"-^= ^' 15. 39a;-8y=99, 7«+5y=74. 28a;--27y=199. 52a; -15^=80. 16. 5a;=7y-21, 17. 6y-5a;=18, 18. 8a;=5y, 21a;-9y=76. 12a:-9y= 0. 13a;=8y + l. ■^^•li«-S'i ^- 19=« + 17y= 0, a. 93a;+15y=123. 12y=5a;-l. 2a;-y=53. 15a; + 93y=201. u ^° n u ^® ^^*^ ^ ^''^ "^^^^ ^" '^^^<^^' before proceeding to solve, It wiU be necessary to simplify the equations. Example!. Solve 5(a; + 2y)-(3a; + lly) = 14 (i), 7a;-92/-3(a;-4y)=i38 ..!!.... !..(2).' From(l) 5a; + %-3a;-lly=14; ••• 2a;-y=14 (3), From (2) 7a; - 9y - 3a; + 12y = 38 ; . . ^.o -ray = 00 ^4)1 ^••omO) ^ 6a;-3y=42. By addition 10a;=:80 ; whence a;=8. From (3) we obtain y=2. Xiii.] SIMULTANEOUS EQUATIONS. 87 Example^. Solve 3a;-^ = l^ (1), %ti_l(2a:-5)=y (2). Clear of fractions. Thus from(l) 42a?-2y + 10=28a;-21; .-. 14a;-2y=-31 (3). From (2) 9y + 12- 10ic + 25=15>/; .-. 10a?+6y=37 (4). Eliminating y from (3) and (4), we find that , 14 =*^=-T3- Eliminating x from (3) aiid (4), we find that 207 Note. Sometimes, as in the present instance, the value of the second unknown is more easily found by elimination than by sub- stituting the value of the unknown already found. EXAMPLES Xni. b. Solve the equations : 1. ~+y=i6, 2. 1+1-5, 3. ^-y=3, a:+|=14. a:-y=4. a;-^=8. 4. a;-y=.=5, 5. §+f=10, 6. x=Zy, 1-1=2. |+y=60. |+y=34. 2 1 11 ^* C*"l2^=^^' ^' 2*~5^=*' 9. 2a; + y=0, 4a?-y=20. ^a;+^y=3. gy-3a;=8. 10. f+|'=lf, U. 3a?-7y=0. 12. |-|=0, a:+|=4f. |a?+gy=7. 3a:+^y=I7. 'I 'I M -41 -a*.! ii.i E- «a Ml M « ■** )l w ■. '«« m \ isa i? N ■ ^ |, 1 K^ 1' r-' r c £ I ; ! i- ALGEBRA. . [obap. Solve the equations : 13. l+f=3a:-7y-37=0. 14 a?+l .^;/-5 ar-y To" 2~^~a' 15. ^3^8-y^3(^+^) ,fi a: « ^ 5 4 8- 16. j3-| = 6a:-10y-8=0. ^^ot^^^Z'^^^^^^^ -»>;oh contain tlJree equations "'""" ""^"°^" quantities we must have he solved by the rul^alreadv2Z^ %t "^ • ?'^^' ^^^'''^ ^«^ Example I. Solve 6a? + 2y- 52=13 3a;+3y-22=13 .".'.'.".'.'.".'.'." o)' «, 7a;+5y-3z=26 ,' Choose y as the unknown to be eliminated. ^^' Multiply (1) by 3 and (2\ by 2, 18a; + 6y- 152=39, , ,^ ,. 6a? + 6y- 4z=26: subtracting, 12a:- 112=13 Again, multiply (1) by 5 and (3) by 2, ^^^• 30ar+10y- 252=65, ,^ ,. 14a: + 10y- 62 = 52; Bubtractmg, 16a:-192=13 ,,, Multiply (4) by 4 and (5) by 3, ^^^' 48a;-44z=52, v^ ,. 48a?- 572=39: subtracting, jg^^jg' and from (4) ^Ig'l .^'^Xnif^rJS^^^^^^ proposed equations. Thus in tli^ nrnol.^f "• ? ^--JJubination of tlie ■n (I) and (2) we h.ve two ewy equations in y and ? ®"'«'>"°"°S XiiT.] SIMtJLTANfiOUS EQUATIONS. g9 Some modification of the foregoing rule may often be used with advantage. Example 2. Solve k-1=^+1=-+2 2 6 7 ' 3+2-lA From the equation ^ _ i - ?? 4. i 2 6 * we have 3x-y=12 (1). Also, from the equation ? _ 1 = ^ ^ 2 2 7 we have 7aj-2z=42 (2). And, from the equation - + -=13 we have 2y + 3z=78 (3). Eliminating 2 from (2) and (3), we have 21a: + 4y=282; and from (1) 12a; - 4y =48 ; whence a;=10, y=18. Also by substitution in (2) we obtain z= 14. Example 3. Consider the equations 5x-Sy- 3=6 (1), 13a:-7y+33 = 14 (2), 7a:-4y=8 (3). Multiplying (1) by 3 and adding to (2), we have 28a;-16y = 32, or 7a?- 4y=8. Thiis the combination of equations (1) and (2) leads us *- an equation which is identical with (3), and so to find x and y Wt .. e but a smgle equation 7a;-4y=8, the solution of which is indet r- mmate. [Art. 100.] In this and similar cases the anomalv arises from the fact that ihe equations are not independent ; in other words, one equation is deducible from the others, and therefore contains no new relation I)etween t e unknown quantities which is not already implied in tho other equations. 1: i > < » * * i E ' ^ C Ml m « <** m i SI i» ' •» » > M K» • '4 r » n-d P C' * r r • ••^ m.\ .1 t.mk:M I m 1 . ■i 1 fm 1^ . ■ M M 1 1 1 1 fmm 1 1 90 I I' I: V h ALGEBRA. EXAMPLES Xm. c. Solve tlie equations : 1. 3. 5. 7. 9. 10. x + 2y + 2z=U, 2 2x+ y+ z=z 7, 3a: + 4y+ s = 14. a? + 4y + 3z = 17, 4 3a: + 3y-,' z=16, 2a: + 2y+ z=li. 2a?+ y+ 2=16, R x + 2y+ 2= 9, «+ y + 2z= 3. 3a; + 2y- 2=20, Q 2a; + 3y + 62=70, ic- y + 62=41. 3a?-4v=63-16, 4a;-y-2=5, a;=3y + 2(3- 1). 5a: + 2y=14, y-6z=-15, a; + 2y + z=0. a; + 3y + 4a=14, ar + 2y+ z= 7, 2a:+ y + 2z= 2. 3a:-2y+ 2 = 2, 2a; + 3y- 2=6, x+ y+ 2=6. «-2y + 3z=2, 2a?-3y+ s = l, 3a;- y + 2z=9. 2a: + 3y + 4z=20, 3a; + 4y + 52 = 26, 3^ + 5y + 6z=31. 11. x-U=6, y_|=8, z-|=10, 12. 13. y + z_2 + a; x + y ~i~~~S~~~2^' a? + 2/ + 3=27. y-z y-x ~3~ 2~~°^~**» y+s=2a:+l. 14. 2a?+3y=5, 2z-y=l, 7a;-9z=3. 15. 2^''^+'2-5)=y-z = 2a;-ll = 9-(a:+22). [chap. 16. a:+20=|?+10 =2z+5 = 110- (y + z). to un^ty £ch7s rid fo ^1 *^!,P^°^««* ^^ ^^o quantities be equal if aT-1 TanA^i t ^^ *^^ reciprocal of the other. Thus itab-l, a and h are reciprocals. They are so called because a-y and 6=-; and consequently a is related to 6 exactly as h is related to a. The reciprocals of ^ and y are 1 and 1 respectively, and in solving the following equations we consider 1 and I as the unknown quantities. ^ ^ xni.] SIMULTANEOUS EQUATIONS. 91 Exampkl. Solve ?-?=! (1), X y ^^' •* Multiply (1) by 2 and (2) by 3 ; thus X y X y * 46 adding, "^=^'^5 multiplying up, 46 = 23a;, x = 2', and by substituting in ( 1 ) , y = 3. Examph2. Solve ^ + ^-3-^=5 (1). 5"% ^^^' i-^+r^A- (3). clearing of fractional coefficients, we obtain from(l) hl-l = ^ W, from(2) ?-l =0 (5), a; y from (3) 15_3^60^32 ^gj^ a; y z Multiply (4) by 15 and add the result to (6) ; we have 105 42 -_ — +—=77; X y ' ^f\ (\ dividingby?, — + -=11 (7); X y from (5) ^--=0} X y 33 „ •• X ' .'. x=3a from (5) y=l. [ from (4) 2=2. J m '« «. "^ 4 *i Mi 1 *< Itt ■i * »' m i *' E • t ii< t '^.l* m « Al m ;« ■•,•» e :> m •• n ^ ' t m i i!l m ' M » ; « •a ' §? r « ■u en f r tf c t •a. IC 1 i i ■ ^ 1 f 1 H: ^2 ALOEfeftA. ^EXAMPLES XUI. d. (chap, xiii Solve the equations 5. 6 X + 7=3, X y 16 1 X -.- = 44. a; V-'' v« y) 10. 3 5__8 a? y~15' t-ia i-?+4= a; y 1 1 .--r + I=0. y 3 z X *• 2- l-?=a a y 2. 14 « y = 3. a? y ^' y a? 42' 8. 25^24^ « y ''» X y~^' 6. ? + .^=30. a; y a: - ^* a; ^7='- y 9. i4-27. 20 11. V 15. ?-?«5-3^7 15_ « y a a; v^^z' m- X :42. 14 15 X =1. -^ + 1-2 12. 2y-x=4a;y, y 2a; 4 3 y~x' 14. - + ^+1=36 « y^a ^» 1,3 1 ;:+,-. -r=28. a? y 2 Proble 108. In { seen that it : unknown qu of problems always be m contain as n: to be (leterra Example 1. fifth of whose Let X be thi Tlien By additioii The numbei Example 2. £3. 6s. 6rf., ai £4. 6s. 2d. ; fi Suppose a p and .... Then from \ Multiplying Subtracting And from (1 whence .*. the c( and the c( CHAPTER XIV. Problems leading to Simttltaneoits Equations. 108. In the Examples discussed in the last chnpter we have seen that it is essential to have as many equations as there are unknown quantities to determine. Consequently in the solution of problems which give rise to simultaneous equations, it will always be necessary that the statement of the question Bhould contain as many independent conditions as there are quantities to be determined. Example 1. Find two numbers whose diflference is 11, and one- fifth of whose sum is 9. Let X be the greater number, y the less ; Tlien a;-y=ll (1), Also ^±y=9 or a;+y=45 (2). By addition 2y=56 ; and by subtraction 2y = 34. The numbers are therefore 28 and 17. Example 2. If 15 lbs. of tea and 17 lbs. of coffee together cost £3. 5s. 6d, and 25 lbs. of tea and 13 lbs. of coffee together cost £4. 6«. 2d. ; find the jrice of each per pound. Suppose a pound of tea to cost x shillings, and coffee y Then from the question, we have 15x + 11y=65l (1), 25a; + 13y=86^ (2). Multiplying (1) by 5 and (2) by 3, we have 75a; + 85y=327|, 75a:+39y=258^. Subtracting, 46y=69, And from (1) 15a; + 25^ = 65^, whence 15^=40; .*. the cost of a pound of tea is 2f shillings, or 2s. 8d., and the coat of a pound of coffee is l4 shillings, or la. 6d, 2 « iil I »i i ^ > M ; m •» •a 1* M » 9 i 1 pi f r tf €' ! C r va in. ir' \ii f Ih I-' '§ i \z m 94 ALGEBRA. [chap. Let X be the number of oranges, and y the number of apples. 2x X oranges cost y pence, 5y y apples cost j| pence 3- + l|=182. .(1). Again, 5a; oranges cost 5a; x|, or -|^ pence, and I apples cost f x ^. or | pence, 10a; 5w ^„^ Multiply (1) by 5 and subtract (2) from the result • •• ll2-48>=380; .(2). 95y 48 = 380; ,^ •■• y=192, and from (1) a;=153. Thus there were 153 oranges and 192 apples. Example 4. If the numerator of a fraction is increased bv 2 an,! the denommator by 1, it becomes equal to |; and, if the numeral TA ttTactt: '-' ''-' ''"''^^''' 'y ^' ^^ ^— eqiTri; Let X be the numerator of the fraction, y the denominator; then the fraction is -. y From the first supposition. from the second, « + 2_5 y+i~8' x-l 1 y-i~2' These equations give a;=8, y = lo. Thu3 the fraction is 4- lo .(1), .(2). [chap. ■ j^j^ ■] PROBLEMS LEADING TO SIMULTANEOUS EQUATIONS. 95 .(1). .(2). Example 5. The middle digit of a number between 100 and 1000 is zero, and the sum of the other digits is 1 1. If the digits be reversed, the number so formed exceeds the original number by 495 ; find it. Let X be the digit in the units' place, y hundreds' place ; then, since the digit in the tens' place is 0, the numbe- will be represented by 10(^ + a:. [Art. 81, Ex. 4.] And if the digits ci'e reversed the number so formed will be represented by lOOa; + y. .: 100a; + y-(100y+x)=495, or 100a; + y-100y-a; = 495; .-. 99a;-99y=495, that is, x-y=5 (1), Again, since the sum of the digits is 11, and the middle one is 0, we have x + y=ll (2). From (1) and (2) we find x=8, y=3. Hence the number is 308. ^M .(1), .(2). EXAMPLES XIV. 1. Find two numbers whose sur 's 34, and whose difiference is 10. 2. The sum of two numbers ii» , 3, and their difference is 37 ; find the numbers. 3. One third of the sum of two numbers is 14, and one half of their difference is 4 ; find the numbers. 4. One nineteenth of the sum of two numbers is 4, and their difference is 30 ; find the numbers. 5. Half the sum of two numbers is 20, and three times their difference is 18 ; find the numbers. 6. Six pounds of tea and eleven pounds of sugar cost £1. 3s. 8d., and eleven pounds of tea and six pounds of sugar cost £1. 18s. 8d. Find the cost of tea and sugar per pound. 7. Six horses and seven cows can be bought for £250, and thirteen cows and eleven horses can be bought for £461. What is the value of each animal ? 8. A, B, G, D have £290 between them ; A has twice as much as 0, and JB has three times as much as D ; also C and D together have £50 less than A. Find how much each has. 9. A, B, O, JJ have £270 between them ; A has three times as mnch as C, and B five times as nmch as D ; also A and B together have £50 less than eight times what G has. Find how much each has. » r !i; 96 ALGEBRA. [chap. i thirteen years ago : flndtheifagef '^'"^ *" """» ^'» "«« w ^ doe, m „ „e ho„«. How .nanrnfcS rh"^ ''pSTou'f r W. How many mZ do:"t^wSk'';'rTo^'' "^ ''°'' "• -" by^teSn7the1igi^^^^ find the numbers. ^ ' ^"*^ *^^ diflference of the digits is 6: . find the numbers. ™^** ""y reversing the digits is 27: 20. A certain number of two diai+a .a *i, digits, and if 45 be added to 'r^he d Sts wH?f *'""' ^^% «""» ^^ '^« number. ^ "'8»8 wiU be reversed : find the s«m^fi'dtl?s%"n7SVblTubt ''.T} ^^ ^« ^^g'^* «-- the reversed: find the^umbet '"^^"**^<*** fr««> it the digits will be tha^if l^p^^iX^erTtt^edTto^^^^^^^^ B^nHings, and he observes pounds he would gZ £5 "^ . but U^t^' ^°? *^" ^^^"^"g^ '"*« half-sovereigns and the shillings inf. ^if'"'''^' ^^^« *"™ed into \«1. 13.. 6d.^ What sum ht he f ^*"-«ro^« he would lose 23. In a bag containinfr Woofe on^ .t-i--^ i of white is equal to a thfra at'tL^^.l c^^^^^'J""^^ ^^e number whole numbe? of balls eSds thrP. fT ''.u^ ^^*'^ ' ^^^ t^i'^^ *''« by four. How many bXS fh^^TonLYnr"''^^ °^ '^"' '^^^"^ [C"AP. ■ XIVJPROBLEMS LEADING TO SIMULTANEOUS EQUATIONS. 97 ars, and one iheir ages, fs than one 's age waj I B does in I more than per hour ? s in twehe es in seven )minator it or. I from the es to ^ on lator. uces to}; luired the bction the m its de- er formed gits is 6 : lifference its is 27 : urn of its find the imes the 8 will be observes ngs into tied into jld lose number vice the ck balls zero. 24. A number consists of three digits, the right-hand one beinc 0. « the left-hand and middle digits be interchanged^^^^^ isdimuushed by 180; -f the left-hanS digit be halved Ind the mk die S "KtSe ^S ' ^"^^^^^^^^^ ^« --her is diminSJ^ ^ 25. The wages of 10 men and 8 boys amount to £1. 17s -if 4 S mffanS Toy r ''' "°^^ *^^" ' '^^^«' ^^^^ ^ '^^ -i- of 26. A grocer wishes to mix spice at 8.9. a pound with another sort at 5. a pound to make 60 pounds to be sold at 6.. a poSnd • whit quantity of each must he take ? l^uuim . wnao 27. A traveller walks a certain distance; had he cone half a nnle an hour faster, he would have walked it in fonr fiff ho rtl^ tune : had he gone half a mile an hourl We'r h" wS W Lt^ 2| hours longer en the road. Find the distance. 28. A man walks 35 miles partlv at the ratp nf d. m;iA» „« i. and partly at 5; if he had walk^ed a?6 mHes Tn^Lurth n L waS at 4, and vice versa he would have covered two miles more iStho same time. Find the time he was walking. ^° fn??;. T70 persons 27 miles apart, setting out at the same time are together in 9 hours if they walk in the sanie direction, buUn sTows If they walk m opposite directions : find their rates of walking ZSi::l^T^^ '^ °"^-*^^^^- ^^' ^^- -t'piJtet^^ ocr?ainLVe?cilX?oarby1ia'fuTi5T5o r! 'fr^* ^^^ ^ ■v.\ ■ U"^ K.A. .m\ r i .1 «■ < >^ ' II ^i i 6 :> Mr ■ II •1 4) m i i^n •» ' *« i» s M • '4 i' M m-i ■'.t (T-s r i^ • C r |.1B m Its, l«" a If CHAPTER XV. Involution. 100. Definition. Involution is the general name for multi- plying an expression by itself so as to find its second, third fourth, or any other power. ' Involution may always be effected by actual multiplication Here, however, we shall give some rules for writing down at once ^ (1) any power of a simple expression ; (2) the square and cube of any binomial ; (3) the square of any multinomial. 110. It is evident from the Rule of Signs that (1) no even power of a7ii/ quantity can be negative ; (2) Bxiy odd power of a quantity will have the same siqn as the quantity itself. ' Note. It is especially worthy of notice that the square of every expression, whether positive or negative, is positive. 111. From definition we have, by the rules of multiplication, (a2)3 = a2. a2. a2 = a2+2+2=„6^ (-3a3)* = (-3)V)* = 81ai2 Hence we obtain a rule for raising a simple expression to any proposed power. ' Rule. (1) Raise the coefficient to the required power hi Arith- metic, andprej?\« 3^V* 31. Ta-'y^ Write down the value of each of the following expressions : 33. (3a263)4. 34. (_„2^)6. 35^ (_2^yj„_ gg n y. ^•©- 38. (-y. 39. (-^y. 40." '12. By multiplication we have (a + by=(a + b)(a + b) = a^ + 2ab + b^ (1) (a-bf=(a-b)(a-b) =a2-2a6 + 62 ^g). These results are embodied in tlie following rules : sum^fL^'r f'" ^^""•^'' "-^ ^'^l i"^''^ ^-^ '""* quantities is equal to the suni of their squares increased by twice their product. toth^sum oTtL^i'^''' ^•^i'^-' ^^rr/ «/ i^'^ quantities is equal tne sum of their squares diminished by twice their product. ^ : :•: Mi "* 1 (> ■ : m ■ r 'ii 5 • ^< % 1) M 6 *♦ -> m , .:.t Be *» it» : M i» ■ f i' 1 m^ V er- w ii € t r 100 ALGEBRA. [CUAP. h • Example 1. {x + 2i/)'^=x^ + 2.x .2y + (2yf = a^ + 'ixy + 4y-. Example 2. (2a3 - 362)= = (2a')2 - 2 . 2a» . 362 ^ (3^2)2 =4a«-12a%2 + 96^ 113. These rules may sometimes be conveniently applied to find the squares of numerical quantities. ^ «ipp"eu 10 Example 1. The square of 1012=(1000 + 12)= = (1000)2 + 2. 1000. 12 + (12)2 = 1000000 + 24000+144 = 1024144. Example 2. The square of 98 = ( 100 - 2)2 = (100)2-2. 100. 2 + {2)2 = 10000-400 + 4 =9604. two^steT^^ " considerably shortened by the omission of the first 114. We may now extend the rules of Art 112 thus : (a + 6+c)2={(a + &)+c}2 =(a + 6)2+2(a+ft)c+c2 [Art. 112. Rule 1.1 = a2 + 62 + c2 4- 2a6 + 2ac + 2hc. In the same way we may prove (a-6 + c)2=a2+62+c2_2a& + 2ac-2&c {a + b + c+df=a^ + h^+c'^ + cP + 2ah + 2ac + 2ad+2hc + 2hd+2ccl. In each of these instances we observe that the square con- sists of ^ (1) the sum of the squares of the several terms of the given expression; given (2) twice the sum of the products two and two of the several terms, taken with their proper signs; that is, in each prchict the sign IS -f or - according as the quantities composing it nave like c^ unlike signs. ' *' Note. The square terms are always positive. The same laws hold whatever be the number of terms in the expression to be squared. Rule. To find the square of any multinominal : to the sum of the squares of the several term^ add twice the product (with tL proper sign) of each term into each of the terms that follow it lii I' XV.] INVOLUTION. 101 Ex. 1. (a;-2y-3z)2 = a~2 + 4y2 + 922_2.a..2y-2.a;.3s + 2.2y.33 = x2 + 4y2 + 922 _ 4^y _ 6ar= + I2yz. ^05.2. (l+2a;-3a;2)2=n.4a:2 + 9a:4 + 2.1.2x-2.1. 3x2-2. 2a:.3ar» = 1 + 4x3 + 9a4 ^ 4a, _ g3.2 _ 2 2a^ = l+4x-2a;2-12ar' + 9x*, by collecting like terms and rearranging. EXAMPLES XV. b. Write down the square of each of the following expressions : 2. a -36. 3. a;-5y. 4. 2ar + 3y. 6. 3a: + 5y. 7. 9a; -2y. 8. 5a6-c. 10. x-abc. 11. ax + 2hy. 12. x^-l. 14. a + 6-c. 15. a+26 + c. 17. a:2-y2-s2. jg^ ary + yz + zar. 20. x^-x + l. 21. 2a:2+3a;-l. 23. 2a; + 3y + a-26. 24. m-n-7>-»y. 1. a + 36, 5. 3a; -y. 9. vq-r. 13. a-6-c. 16. 2a -36 + 4c. 19. 3/j-25 + 4r 22. x-y+a-h 1 25. S«-26+j. ^ 4 26. |-36-|. 27. 1^2 - a; +^. Also 115. By actual multiplication we have (a + by =(a + b)(a+ b)(a + b) =a^+Sa^b+3ab^+b^ (a - 6)3 = a3 - 3a^b + 3ab^ - b\ r.llu °^'^''^'"g ^1»« }^^ of formation of the terms in these results we can write down the cube of any binomial. Example 1. (2a; + y)3=(2a;)3 + 3(2a;)2y + 3{2a;)y2 + y3 = 8ar' + 12ar2y + 6a;y2 + yi. Example 2. {3a; - 2a'^f={3xf ~ 3(3a:)2(2a2) + 3(3a;) (2a^f - {2a?f = 27ar« - 54a;V + 36a;a* - Sa". EXAMPLES XV. c. Write down the cube of each of the following expressions : 1. ^ + a 2. x-a. 3. a;-2y. 4. 2a' + y. 13. a-S. 10. 4a;2-5y2 14. ^ + 2. 11. 2a3-.362. 12. Sar' - 4.11* 15. x" 4y* -3a;. a 16. g + ar. 11 IB I - f r it 'I' I.TO It*' I I ^^P f CHAPTER XVr. Evolution. 117. By the Rule of Signs we see that (2) wo ng^a^m quantity can have an even root • qutet'e^f "'' °' ^ ^"^"*^*^ ^- ^^« --« -gn as the Example. s/9aF^= ±3ax^. the posrtiPv;'rot.'^'P*''' ^'^'"'^' "« «^*" «=-«"« °»r attention to Examples. J^=a%\ because {a%'^f=a%*. si -3?z=-a?, because {-x^f= -aP. y/^=c*, because {c*)^=c^. 4/81a;»2=3arJ, because (Sx^)*=8W^. v„ii?" "^?"^ .*^® foregoing examples we may deduce a ffenenl rule for extracting any proposed rJot of a sim^e expressS : Examples. U -Mi?--\x\ V 25c* -5?a* CHAP. XVI.] EVOLUTION. 103 Write down 1. 4a%*. 5. 81o«68. 9. 64a^". 3245" ^^- 169y«" Write down 17. 27a%h^ ^' "125' Write down 25. 4/(a8a;W). 28. ^/(729a"6«). 128 EXAMPLES ZVI. a. the square root of the following expressions : 2. 9aV. 6. 100a:8. 10 ^ 14. 81a" 366"' a 25zy. „ a"68 11. ig * 256xY 15. 8. a%h^\ 28V ^25"* 400a«6«' 12. ^ 289/)" 16. Slx^v'"' the cube root of the following expressions : 18. -8ai269. 19. 64a«2/V2. 20. -343a"6". 22. 8a;» 729y«' 23. 125a«^>« 216a;y 24. - 27£27 642/ i63" 31 •• A^a the value of each of the following expressions : 26. ^(«'yi). 27. ^(32a;V<'). 29. 4/{256a8a;M). g^^ 4/(-a;'Y»). 10 /aso^ 32 33 119. To find the square root of a compoimd expression. Since the square of a + i is a^+^ah + V^, we have to discover a process by which a and h, the terms of the root, can be found when o?-\-'iah + h^ is given. The first t^m, a, is the square root of aK Arrange the terms according to powers of one letter a. The first term is a\ and its square root is a. Set this down as the first term of the required root. Subtract a? from the given expression and the remainder is 2a6 + 62 or (2a + 6) x h. Thus, 6, the second term of the root, will be the quotient when the remainder is divided by 2a + h. This divisor consists of two terms : 1. The double of a, the term of the root already found. 2. 6, the new term itself. The work may be arranged as follows : a' + '2ah + h'^{a+h a^ 2a+6 2a6+62 2a6+6a s - c ftSfm I * «i IS :> on ' 1-1 m #1 m • y.i Be " «« » 5 *! M • SI! i' .J; »■< en ' r w III I 104 ALGEBRA. [cnxp. Example 1. Find the square root of Ox' - 42a;y +49ya. 9a;»-42a:y+4V{3r-7y toSfSit. ""^ ''"'"■° ™' °' '^ " ^'^ ""» "■" fa the to, term in the root and in the dTvTs'or Wo novf r. u'^' " m *''' ^^'^ '''^^ found, i!t^^r.is.rj:i:iz:ZtS'^;^°°' "^ •'«" Example 2. Find the square root of 25x^a^ - I2a?a3 + 16x* + 4a*- Qix^a. Rearrange in descending powers of a-. 16x'J-24ar'a+26a;V-12a:aS+4a<(4a;2-'Sa;a+2aS 8x^-3xa 8x'^-6xa + 2a^ -24x^a + 25x^a^ -24xr^a+ 9x-a'^ lQx'a^--12xa^+4a* 16a;2a2-12A.-a' + 4a^. 'iZLt «"""■""-=■•., by 8^», the first term „ fthe *fi 't. and the foot is found. '"Mraot. There is no renminder, .,i^l XVI.] EVOLUTION. 105 EXAMPLES XVI. b. Find the square root of each of the following expressions : 9a2+12a6 + 4&a. 253^- 30x1/ + f)y\ l-2a3 + a«. 4ar* - 12a:3 + 29«2 _ 30a; + 25, x^-4x^ + 6x*-4x+l. l-10;i;+27ar»-10«»+a;«. 1. x'' + 4a.' + 4y«. 2. 3. a^-10xy-i25y\ 4. 6. Sl3^+18xy+y"-. fi. 7. x*-2Th/^ + y*. 8. 9. o*-2a3 + 3a'»-2a+l. 10. 11. 9ar*-12ar»-2ar» + 4a; + l. 12. 13. 4a« + 4a3-7a2-4o + 4. I4. 15. iaP + 9y^ + 25z' + l2xy-30i/z-20xz. 16. 16a;« + 16x'- 4x8 -4aJ» + «!<». 17. a,-6 - 22x4 + 34«3+ 121x2 -374.r + 289. 18. 25»*-30ax8+49a2a:2-24a3a;+16a*. 19. 4x* + 4xV''-12a;V+y*-62/V + 92*. 20. Qab^c - Aa?hc + a^h^ + 4a2c2 + 9fc V - 1 2ahc\ 21. - 66V + 9c* + 6* -12cV + 4a* + 4a262. 22. 4a;* + 9y*+13xV'-6a;y3-4xV- 23. 67x2 + 49 + 9x*-70x- 30x3. 24. 1 -4x+ 10x2- 20x3 + 25x*-24x» + 16x«. 25. 6acx» + 462a^ + a2xi« + 9c2 - 126cx2 - iaha?. takWrChap^'xS?:f '^'' °^ ''"' '^'P*'' "^^y ^' J^^^'P^"^^ '^"^ *120. When the expression whose root is required contains tractiona.1 terms, we may proceed as before, the fractional part of the work being performed by the rules explained in Chap. xii. *I2I. There is one important point to be observed when an expression contains powers of a certain letter and also powers of Its reciprocal. Thus in the expression the order of descending powers is ^'+7^+2^+4+- + ! + X or 8 and the numerical quantity 4 stands between a: and -. The reason for this arrangement will appear in Chap. xxx. m m it ( m 1 , Mi ' « . «• ft ^ w li m I ', M ;il t "*! C :> m < f, 1 « • 1 1 "> . ..1 ■* -•« ' » ■i *» tap • s? r ^ ■u 15' cm r r c: r ».* K ftX If' 1 1 « : im^ 106 ALGEBRA. Example. Find the square root of 24 + l^y!-^-*' 4.*' ^V ■^ V y^'lc" Arrange the expression in descending powers of y •^ * y y* \ X y [cnAp. ^-4 aa y .^. x+24 -f^ + 16 8 ?f-i-^ Here the second te-'m in the root. -4, arises from division of ~ by -, and the third term, f, arises from division of 8 by ^:- thu8 8-?y=8x^=?. X 8y y ^EXAMPLES XVI. c. Find the square root of each of the following expressions : 1. ^-aeH9. 3. i^'-^^y^- 5. 7. 26 ar» 2x - — + 4. 4y^ y 64a;2 32a? . 9. M+g-^+l- 2. 4-^+% y y A «' , 10a; „„ « 9? J2ax a? 8. ^-24-25 25 ^ + 9^- 1 11. 13. 15. -3a3+|+a4-5a + ga^ a * a3 a3 a: X a' 4 ^3^9^ X* 10. a:* + 2a;3-a; + i. 4 12. ^-2x + l+^:^^Q^, 14. a;4_2.^^_^ 1 2 2 !ft V XVI.] EVOLUTION. 107 ^^ g^'bx"^ 25 ~16a"*"9a»* 16 4 * 17. 16m*+^m>« + 8m»+sn»+ , s + 1. 18. 4ar* + 32x^ + 96 + ^ + ^. ♦122. Tofirid the cube ro< ■' of a compuund expression. Since the cube of a + b is a^ + :ia^ + 3ab^ + b\ we have to discover a process by which a and h, Mie terms of the root, can be found when a^ + Sa^b + Sab^ + P is given. The first term a is the cube root of a\ Arrange the terms acoording to powers of one letter a ; then the first term is a», and its cube root a. Set this down as (he first term of the required root. Subtract a^ from the given expression and the remainder is 3a^b + Sab"- + P or (.3a2 + 3ab + P) x b. Thus b, the second term of the root, will be the quotient when the remainder is divided by Sa^ + Sab + P. This divisor consists of three terms : (1) Tliree times the square of a, the term of the root al ready found. (2) Three times the product of this first term a, and the new term 6. (3) The square of b. The work may be arranged as follows : a^+3a^b+3ab'^ + bHa + b a' 3(a)2 =3a2 3xax6= 4-3rij (6)2 = +62 3a^+3ab + b- Sa^+3aP+¥ 3a^b + 3aP+P Example 1. Find the cube root of 8x^ - 36«2y + 54xy^ - 27y^ 8a;3 - 36x^ + 54xy^ - 272/' ( 2a; - 3y Bar' 3(2x)2 -,12.^2 ox2a:x(-3j^)= ~18xy ( - oy)- = + V 12a;2-18xy + 9y3 -36x^ + 54xy-^-^y3 I * tf i«* s ■1 f c ^> "" ^ w « < •* 4 sjt» •h * •* m s M ta ' W S' ' 1 IM ,. f 108 ALGEBRA. [chap. n I If "Ml Jj to 0) "S S « S 3 '=' H I * S o >>«0 « cr"T3 «8 2 S « § ^°fj^ o a O "O^ eS 2 <- I o- S 2 S I 2 5.2 2.S J2 w + + MM X X M e<3 .1 I' 05 ^ S1O4. £ •3 2 :g <„ M ec:3 °'« ^'3 a< 0,73 a I Find the 1. a-i + Sa^ 3. aV-3 5. Ma? -I 6. l + 3a; + 7. 1-6* + 8. a3 + 6a2/ 9. 8a«-3ft 10. y6-3y» 11. 8a« + 12 12. 27x«-5 13. 27a:«-2 14. 24x^3/2 + 15. 216 + 34 # 123. W terms occur Example. Arrange tli XVI.J EVOLUTION. 109 •EXAMPLES XVI. d. Find the cube root of each of the following expressions : 1. a^ + 3a^ + 3a+l. 2. 3. a^3r^-3a''xY + 3axi/*-yO. 4. 5. 64a3 - lUa% + lOSab"^ - 2763. 6. l + 3x+6x^ + 7x^ + ex* + 3cf^ + afi. 7. l-6x + 2lx^-Ua^ + 63x*-5ia^ + 27a^. 8. o» + Qa% - 3a\ + I2ab'^ - 12abc + 3ac^ + Sb^ - \2b'^c + 66c2 - c\ 9. SaS-SGa' + eea^-eSa^ + SSa^-Qa + l. 10. /-3y» + 6y<-7y3 + 6y2-3y + l. 11. 8a8+12aJ>-30x*-35x3 + 45a^ + 27a;-27. 12. 27«« - 54x«a + 1 1 7x4a2 - 1 16ar»a3 + 1 1 Ix'^a* - 54xa<^ + 27a«. 13. 2'Ja^-27a^~18x* + na^ + Qx^-3x-l. 14. 24a:^y2 + 96a;2y4 _ g^^ ^ ^.e _ gg^s + 54^6 _ 5Q;j^y^ 15. 216 + 342a;2 + 171a;4 + 27a,-8-27.T«-109ar'- 108:c. *1 23. We add some examples of cube root where fractional terms occur in the given expressions. Example. Find the cube root of 54 - 27x^ + 4 - ^• Arrange the expression in ascendinr] powers of x. 8 36^_. 0-7 3/^2 ^-^ + 54-27ar^(^-,- 8 3x -er 3x i x{-3x)= - ( - Zxf a;' > 12 18 x + 9ar2 -p + 54-27ar' 12 18^Q 2 -f + 54-27ar« .3 1 . . m ■ A 1 •n M £ ' il «i ru f Ml ■» * •! IK -■11 ■i 1*1 c ;> OK ^' 11 im *' UN i i.i «* »a R> > ;l.» £» r • '4 e^ r:-^' r .. t' » c £ Ji V . 110 ALGEBRA. [chap. •EXAMPLES XVI. *».a^aurusa XVI. e. Rnd the cube root of each of the following expre^ion, ; x^ 2aP ^42* 2. 27 + -3- ■*■**+ ^* 6. arS-9^ + 27_27^ 27 64y3 "V y + ~-8. X a^' x^ 7. g+^V^-4-^+6jf_3^ 6. ^-6«*+12a%S-8y6. 8. x^ y x'^ x^ '^' X' 27 3 + 2ar-7 + ^-27,27 "' a' a X a;2"t-^ 112+i2?2_48a2 8a3 10. ^-M^%240a_,,,_^6o/l2j a;3' 11. 66 , 6a X 160 + ' a «■' - + a + ?^-7 + £!_3a2_3^2 j3 a«' fe3~ 62 a- a 12. ^^^-SO^.^O^^S^i,^ lOSar y V ~T^~'^^'^ y ■27+ 48a^ and cube roots in ArithmeHp n^7i ®i ^'''' extracting square methods we have e.pll!l;e"d1;.%?,rpt;„1 rp"tef '"'*™'«'' Example I. Find the square root of 5329. («mhf ^'^ ''"' hetween 4900 and 6400 .(.,♦ ■ i . (80)', Its «,uare root consists of two Ss °Vf- ^""f^" <™''''»^«e factors. resolution of the expression into its the restttrofexp^^^^^^^^^^^ Sf.P-cipal rules by which be effected. P^essions into their component factors may by dividing each term se3ktpltlTl''°i' ""^^ ^^ simplified the quotient within bmcJete^^jfc /J!"'^ ^f ^°'*' ^"^ ^"^losing outside as a coefficient ' ''°"''"°'' ^^^<^°r being placed ^J^^e 1. The ^rms of the expression 3a»-6a. haveacom.on A 3aa-6a6=3a(a-26) EXAMPLES XVIL a. Resolve into factors : 1. a^-ax. 4. a^-ab^ 7. 5ax~5a^x\ 10. ic3-a^. 13. 16a;+64xV. 16. 10a:S-25a;V- 19. a^'-JcV+ary^ 21 7a-7a»+14a<. 5. 7p2+^. 8. 3x^+3^. 11. 6a; -25a;V 14. I5o3-225al 17. 3s^-s^ + x. 20. 3a<-3a364-6a262^ 3. 2a-2al 6. 8a;-2ar'. 9. ic2+ary. 12. l5 + 25x^ 15. 64 -81a;. 18. 6a;3+2a^ + 4a:». 21. 2aV-6x2«/2+23;2/3. 23. 5x^-10a^a^^i5as^^ 25. 38aV+67a4ar'. II, 5^. ^ CHAP. XVII.] Resolution into factors 113 « 127. An expression may be resolved into factors if the terms can he arranged m groups which have a compound factor common. Example \. Resolve into factors x^- ax + &a;- aft. Noticing that the first two terms contain a common factor .r and the last two terms a common factor i, we enclose the first two terns in one bracket, and the last two in another. Thus x" -ax + bx -ab = (x"^ -ax) + (bx- ab) = x{x-a) + h{x-a) = {x-a) taken x times plm (x - a) taken b times = (a; -a) taken (a; + 6) times = {x-a){x+b). y^ /.' ' ■ ' ' Example 2. Resolve into factors 6^2 - Oax + Abx - 6a6. 6x2 _ ^^ ^ 4j^ - 6a& = (6x2 - g„^) ^ ^^^^^ _ ^^^ = 3x{2x-3a) + 26(2x-3a) = (2x-3a)(3x + 2?>). Example 3. Resolve into factors I2a2 - 4a6 - 3ax2 + bx\ 12a2 - 4a6 - 3ax2 + 6x2= {I2a2- 4a6) - (3ax2 - 6x2)' = 4a(3a-6)-x2(3a-fc) = (3a-6)(4a-x2). Note. In the first line of work it is sufficient to see that each nair contams some common factor. Thus, in the last example bv a different arrangement, we have ^Aanipie, oy a 12a2 - 4a6 - 3ax2 + &x2= (12a2 - Sax^) - (4a6 - 6x2) = 3a(4a-x2)-6(4a-x2) = {4a-x2)(3a-6), EXAMPLES XVU. b. Resolve into factors :• 1. a^+ab+ac + bc. 2. 3. a'^c^ + acd + abc + bd. 4* 6. 2x + cx + 2c + c2 g' 7. 5a + a6 + 56 + 62. 3 9. ax~bx-az + bz. 11. 13. mx- my-nx + ny. a^-ac + ab- be. a^ + Sa + ac + Sc. x2-axf 6x-6a. ab-by-ay+y\ qs. 'Mx + ay + 2bx + by. r- A. 10. pr + qr-pa- 12. mx - ma + nx - na. 14. Sax-bx-Say + by. tS$ 1 : Mfl 1 * ' na ■V a mi ' li $ Mi " «i m tm M| ^. c :> m ^ If m *( m ;;) •ft *ii *»■ » J-f tm - SJ r « 114 ALGEBRA. [eiu,.. Resolve into factors : 15. ex'+3xy-2ax-ay. 16. mx-2mi/-nx + 2ni/. 17. ax^-3bxi/-axy + 3by\ 18. x^ + mxy - 4xy - 4my^ 19. ax^ + bx^ + 2a + 2b. 20. a;'' - 3x - :ry + Sy. 21. 2x*-a^ + 4x-2. 22. 3a^ + 5x2 + 3x+'6. 23. ar4+a^ + 2a; + 2. 24. y^-y^+y-l. 25. axy + bcxy-az-bcz. 26. /^x^ + g^x^ - ^^ - af^ 27. 2aa;2 + 3axy_26:,y_36y2. gS. ama^^ + fc^^cy _„„a;y _ j^^^2 29. ax-bx + by + cy-cx-ay. 30. a2a; + a6x + ac + a6y + 62y + 5c Trinomial Expressions. 1 28. ^Before proceeding to the next case of resolution into tactors the student is advised to refer to Chap. v. Art 44 Attention has there been drawn to the way in which, in forming the product of two binomials, the coefficients of the different terms combine so as to give a trinomial result. Thus, by Art. 44 (x+5)(a;+S)=x'+8x+15 (i)| (x~5)(x-3)=a;^-8x-\-15 (2)' (^+6)(a;-3)=^ + 2.r-15 1.(3),' {x-5)(x+3)=x^-2x-15 (4) We now propose to consider the converse problem : namely the resolution of a trinomial expression, similar to those which occur on the right-hand side of the above identities, into its component binomial factors. By examining the above results, we notice that : 1. The first term of both the factors i,« n-, 2. The product of the second terms of the two factors is equal to the third term of the trinomial ; e.g. in (2) above we see that 15 IS the product of -5 and -3; while in (3) -15 is the product of + 5 and - 3. \ / o 3. The algebraic sumo{ the second terms of the two factors is equal to the coejicient of x in the trinomial ; e.g. in (4) the sum of -5 and +3 gives -2, the coefficient of ^t- in the trinomial. In applj^ing these laws we will first consider a case where the tmra term of the trinomial is positive. Example 1 . Resol ve into factors a;" + 1 1^; + 24. The second terms of the factors must be such that their product and +3 ^""' "^^^' ^^'^^^ '"'^^^'^ *^** *^^y must be +8 .-. a;2 + lla;-|-24=(ar + 8)(a;-|-3). i;'i XVII.] RESOLUTION INTO FACTORS. 116 factors is ve '.ve see 15 is the Example 2. Resolve into factors x> - 10a? + 24. The second terms of the factors must be such that their product IS +24, and their sum - 10. Hence they must both be neaatfve and It 18 easy to see that they must be - 6 and - 4. negative, and .-. a;2- 10a? +24= (a; -6) (a: -4). Examples. x^-18x+81 = {x-9){x^9) = (a?-9)2. Example i. a?4+10x2+25=(a;2+g)(3.9+5) = (a:»+6)2. Example 5. Resolve into factors ic^-llax+ lOa". The second terms of the factors must be such that their product 18 + i0a2, and their sum - 1 la. Hence they must be - 10a £d - a. .-. a^-llaa? + 10a2=(a;-10a)(a?-a). hi^^^'u "^1 ^'i^'^&^ of this kind the student should always verify EXAMPLES inrii. c. Resolve into factors 1. a^\-3a + 2. 4. a3-7a + 12. 7. a;2-19a? + 90. 10. ?;2_2ia;+i08. 13. x-^-ldx + M. 16. a?2 + 20a? + 96. 19. a;2 + 23a?+102. 22. o2+30a + 225. 25 a'^-14ab + 4:%\ 27. w2-l3mn+40n2. 29. a?2-23a;y+132y3. 31. a?*+8a?a + 7. 33. x'y^-lGxy+Sd. 35. a%2+34a;y + 289. 37. a^-20abz+7^b^xK 39. o2-29a& + 546a 41. 12-7a?+a;2 43. I32-23a; + a;2 45. 130 + 31a;y+;:2y2^ "". 204-29a;2 + a;4. 2. 5. 8. 11. 14. 17. 20. 23. a2 + 2a + l. a?a-lla?+-30. a;2+13a? + 42. a?3-21a?+80. ar'-19a? + 78. a?a-26a? + 165. a2-24a + 95. a3 + 54a + 729. 26. a^ + 28. wi2- 30. a?3- 32. x*+ 3. a2+7a + 12. 6. a;a-15a? + 56. 9. a;2-21a? + 110. 12. a;2 + 21a? + 90. 15. a?2-18a;+-45. 18. a;2-21a?+104. 21. a2- 32a + 256. 24. a2- 38a +361. 34. 36. 38. 40. a?2+ a*b* x* + 42. 20+ Safe + 662, 22»i»+ 1057*2. 26a?y + 169y2. 9a:2y2 + i4y4, 49a?y + 600j/». + 37a262 + 3oo. 43a?y + 390^2. 162a?2+6561. 9a: + a:2, 44. 88 + 19a;+a?2. 46. 143-24a:a + a;2a9 48. 216 + 35a?+a;2. m 1 '■ 1 w§ Hi ' , m E ' i\ 11 5 i> tu •' «i m m •4| •*! ^ : J, ■B i 3-t •K j» 1 m .- ' ft* ^ i v> t ' t» 1 ; r 1 116 ALGEBRA. [chap. i I' 120. Next consider a case where the third term of the tri- normal 18 negative. •' "*^ "' Example!. Resolve into factors a;" + 2a; -35. The second terms of the factors must be such that their product IS - 35, and their algebraical sum + 2. Hence they must have oppo>dle signs and the greater of them must be positive in order to SI Sign to their sum. * w give us The required terms are therefore +7 and - 5. :. a:'' + 2a;-35 = (a; + 7)(a;-5). Example 2. Resolve into factors «" - 3a: - 54. The second terms of the factors must be such that their product IS - 54, and their algebraical mm - 3. Hence they must have opposite signs and the greater of them must be negative in order to che its sign to their sum. ° The required terms are therefore -9 and +6. .-. a;2-3a;-£* = (ar-P^(a; + 6). Remembering that in these cases the numerical quantities must have opposite signs, if preferred, the following method may Example 3. Resolve into factors a;V + 23a:y - 420. Find two numbers whose product is 420, anu *hose difference is 23. Ihese are 35 and 12; hence inserting the signs so that the positive may predominate, we have a%2 + 23a:y _ 420 = (xy + 35) (ary - 12). EXAMPLES XVII. d. Resolve into factora : 1. a?-x-2. 2. a~'+a;-2. 3. 4. a;3+a;-6. 5. a;2-2a;-3. 6. 7. a;2+a;- 56. 8. a;2+3a?-40. 9. 10. a2-a-20. 11. a2-4a-21. 12. 13. a" -4a -117. 14. »2+9a;-36. 15. 16. a;2 + a;- 110. 17. ar2-9a;-90. 18. 19. a8-12o-85. 20. a2- 11a -152. 21. 72. o^ + lxy-myK 23. o^+ax-^>a\ 24. 25. a^-ay-2\0if. 26. a;2+18a7-l]5. 27. 28. a;2 + 16a;-260. 29. a2-lla-26. 30. ar' + 2a;-3. a:2-4a;-12. a2 + a-20. a^ + a?-156. a;2-a;-240. ar - Z2xy - iOoy'. a;2-20a;y-96?/2. aV + 14oy-240. xm] RESOLUTION INTO FACTORS. 117 31. a*-a>b^-56b*. 32. x*-Ux^-5h 33. y*+6xY-27x*. 31 a»63-3a6c-I0c2. 35, a'' + i2abx-28b^x''. 36. a!'-18axy-243xY. 37. x*+13a2a:2-300a*. 38. x*-a^x^-132a*. 39. x* - o'a:" - 462a«. 40. a^^+x'-STO. 41. 2 + x-:>^. 42. 6 + a;-a:». 43. m-x-x\ 44. aSO-a.-a^'. 45. 120-7aa:-a^^'. 46. 66 + 8a:y-a;V. 47. 98-7a:-a:2. 43. 204-5x-sc^. [Examples for revision will be found on page 174.] 130. We proceed now to the resolution into factors of tri- ",% expressions when the coefficient of the highest power is not (3;f + 2)(^ + 4)=3^+14^ + 8 (1) (3.r-2)(a:-4)=3^2_ 14^+8 '(2) (3^ + 2)(^- 4)= 3^2-10.^-8 (3)' (3^-2)(.r+4)=3.r2+10.r-8 (4) we't:7t cTnsTdtlT ^""^^'^ "°^^ '''^'''''' ^'^^^ *'- ^-- winte'tnrflf w^r'T *° ^^^-^ ^ ?""""^^ "^^thod of procedure, it ^ven ablve '''''"""' ^^ ^'^^" ^^'^ ^^ *^e identities Consider the result 3^2 _ 14^ ^ g = (Zx -2){x- 4). TJie first term 3^2 jg the product of Zx and x. llie third term +8 -2 and -4 ■ Again, consider the result Za^-lOx-S={Zx + 2){x-4). The first term 3a;2 is the product of Zx and x. The third term -8 + 2 and -4. The middle term -10.r is the result of adding together the iTe Sr'nf'f." -f ""^ "J^"' ^^d ^*« «^g« is nelatSeLuse tne greater of these two products is negative. J^]- .,^e beginner will frequently find that it lot easv to eSe htCSl'r'r f '^' "rV^^"^'- ^-etice alo^^m «' J * i\ II - Ml ^ !• in ^ 1: as # < Ml i ''' •« * •'•ii » ; SI *m r J ip B.J .1' t:~> 118 ALGEBRA. [chap. (f m t a^ Example. Resolve into factors Tx" - 19a; - 6. Write down (7a: 3) (x 2) for a first trial, noticing that 3 and " must have opposite signs. These factors give 7x^ and -6 for the first and third terms. But since 7 x 2 - 3 x 1 = 1 1 , the combination fails to give the correct coefficient of the middle term. Next try (7a; 2)(a; 3). Since 7x3-2x1 = 19, these factors will bo correct if we insert the signs so that the negative shall predominate. Thus 7a;2-19a;-6=(7a;+2)(a;-3). [Verify by mental multiplication.] 132. In actual work it will not be necessary to put down all these steps at length. The student will soon find that the different cases may be rapidly reviewed, and the unsuitable combinations rejected at once. It is especially important to pay attention to the two followin? hints : ° 1. If the third term of the trinomial is positive, then the^ second terras of its factors have both the same sign, and this sign Vis the same as that of the middle term of the trinomial. ' 2. If the third term cf the trinomial is ysegative, then the second terms of its factors have opposite signs. Example 1. Resolve into factors 14a:2 + 29a;-15 (i), 14ar'-29a;-I5 , (2).' In each case we may write down (7a; 3) (2a; 5) as a first trial, noticing that 3 and 5 must have opposite signs. And since 7 x 5 - 3 x 2 = 29, we have only now to insert the proper signs in each factor. In (1) the positive sign must pr u->minate, in 2 the negative Therefore 14a;2 + 29a; - 15 = (7a; - 3) (2a; + 5). 14a:2 - 29a; - 15 = (7a; + 3) (2a; - 6). L Example 2. Resolve into factors 5a;2 + 17a; + 6 . . 5a;2-17a;+6 !.'."!!!!;!!! In (1) we notice that the factors which give 6 are both positive. "^ (2) negative And therefore for (1) we may write (5a; + )(a;+ ). (2) (6a;- )(a;- ). And, since 5 x 3 + 1 x 2 = 17, we see that 5a;2 + 17a; + 6 = (5a; + 2) (a; + 3). 5a;2 - 1 7a; + 6 = (5a; - 2) (a; - 3). (1), (2). 1. 2a:2 + 3. 4. 3a;2+l( 7. 2a;2 + 7: 10. 5a;--fl 13. 4a;2 + l 16. 2x2 -a; 19. 3a;2+K 22. 3a;2 + 7j 25. 3a;2+lJ 28. 4a;2 + a; 31. 43^" + 2; 31 12a;2.-; 37. 12a:3-I 40. 2Ax^-i 43. 6+6a;- 4& 7 + 10a; 49. 20 -9a; [chap. xvn.] RESOLUTION INTO FACTORS. 119 hat 3 and 2 - 6 for the JOTibination f we insert (1), (2). positive, legative. Note. In each expression the third term 6 also admits tore 6 and 1 ; but this is one of the cases referred to above wiucli the student would reject at once as unsuitable. Example 3. Ox' - 48a:y + Uy"^ = (3a; - 8y) {Zx - 8y) = (3x-8y)'\ Example 4. 6 + 7a; - 6a;2 = (3 + 6x) (2 - x). EXAMPLES XVII. e. Resolve into factors t down all I that the unsuitable 1. 2a;2 + 3x+l. 2. 3a;2 + 5a; + 2. 3. 2x= + 5x + 2. 4. 30^2+ 10a; + 3. 5. 2a;2 + 9a; + 4. 6. 3x2 + 8x + 4. 7. 2a;2 + 7a; + 6. 8. 2a;2 + lla; + 5. 9. 3x2+llx + 6. ) following 10. oa;2^11a; + 2. 11. 2a:2 + 3a;-2. 12. 3x2 + X- 2. vl3. 4a;2 + lla;-3. 14. 3a;3 + 14a;-5. 15. 2x2+15x-8. , then the 16. 2a;2-a;-l. 17. 3x2 + 7a; -6. 18. 2x2 + X- 28. d this sign 19. 3a;2+13a;-30. 20. 6ar2 + 7a;-3. 21. 6x2- 7x-. 3. ± ^ ' 22. 3a;2 + 7a; + 4. 23. 3a;2 + 23a;+14. 24. 2x2-x-15. then the 25. 3a;2 + 19a;-14. 26. 3a;2-19a;-14. 27. 6x2-31x + 35. 28. 4a:2 + a:-14. 29. 3a»2-l3a;+14. 30. 3x2 + 41x + 2G. (1). (2). first trial, 31. 4a!2 + 23a;+15. 32. 2x^-5xy-3yK 33. 8.'c2-38x + 35. 34. 12x2-23a;y+10y' '.35. 15a;2 + 224a;-15 .36. 15x2-77x+10. 37. 12a;2-31a;-15. 38. 24a;2 + 22a;-21. 39. 72x2-145x + 72. 40. 24^-29xy-4y^. 41. 2-3a;-2a;2. 42. 3 + llx-4x2. the proper 43. 6+6x-6aP. 44. 4 -5a; -6x2. 45. 5 + 32x-21x2. 4& 1 + 10x + 3x"-. 47. 18-33x + 5x2. 48. 8 + 6x-5x2. 49. 20-9a;-20a;2. 50. 24 + 37x-72xa. 133. The Difference of Two Squares. By multiplying a + bhy a-bwe obtain the identity {a+b)(a-b) = a^-b% a result which may be verbally expressed as follows : The product of the sum and the difference of any two quantities w equal to the difference of their squares. Conversely, the difference of the squares of any two quantities IS equal to the product of the sum and the difference cf the icco qiiantities. Thus any expression which is the diflference of two squares may at once be resolved into factors. m 1 ■M « Mi » B ' i! ■' ji i i» H * *t M :n ^. € , 'l> . m ^ ir n jt 1 HI .. 1 tec ■>( l» • > fc» r ■ i 'X K.u r-- r -■ €' f C ^i '^ «Bi% 1?" 1 1 1 : 120 ALOEDRA. [chap. Example. Resolve into factors 2i)«-- 16y'. 25x"' - Iby^ = (5x)^ - (iy)\ Therefore the first factor is the sum of 6ar and 4i/, and the second factor is the difference of 5a; and 4y. .-. 26x8 - 16y» = {6t + 4y)(5a: - 4y). The intermediate steps may usually be omitted. Example. l-49c-« = (l + 7c=')(l-7c3). The diflTerence of the squares of two numerical quantities may be found by the formula a^ - b' = (a + b)(a - b). Example. (329)^ - (171)2 = (329 + 171)(329- 171) = 500 X 158 = 79090. i,;; EXAMPLES XVII. Resolve into factors : 1. a:»-4. 4. c3-144. 7. 121 -a;2. 10. y2_25a,2. 13. 3lip'^~4dq\ 16. l-25x=, 19. JoV-36. 22. 9o4-121. 25. a:«-25. 28. 8lx«~25a^. 31. a262-9;c«. 4-x\ x*~16b\ 25-64x2. G4a:2 _ 2528. 16a;i6-9^. 25a;i0-16a8. 34. 37. 40. 43. 46. 49. 2. a2-81. 6. 9-a». 8. 400 -a». 11. 36ar'-2562. 14. 4/fc2-l. 17. a^-4b% 20. a«62-4c»i9. 23. 25a;2-64. 26. l-36a«. 29. a:*a8- 49. 32. x^y^-4. 35. 9-4o2. 38. a:2~25y». 41. 121a8-81a;». 44. 49a;* -16y*. 47. 36aH>«-49a". 60. o26*c«-a;". f. 3. 6. 9. 12. 15. 18. 21. 24. 27. 30. 33. 36. 39. 42. 45. 48. J/'- 100. 49 -c2. 9x^-1. 49-10i)k\ 9a;2 - y\ x*-9. 8la*-49x*. 9ar4-a». a»-64a;«. l-a^b\ 9a* -25b*. 1 - 10062 8lp*z'-25b\ 1 - 100a«6*cl Find by resolving into factors the value 61. (575)2 -(425)2. 52. (121)2- (120)2. 61 (339)2 -(319)'^. jj5^ /753p_j253j2. 57. (1723)2 -(277)2. 68. (1639)2- (739)2. 60. (2731)2 -(269)2. 61. (8133)»-(8131)'. of 63. (750)2 -(250)'. 56. (101)5 -(99)-. 59. (1811)2- (689)-. 62. (10001)2-1. 1. (a + 6)2 4. (a; + 2y 7. (a; + 5c) 10. a2-(6- 13. 9a;2-(2 16. (a + 6)2 19. (a + 6)2 22. (4a + a; 24. l-(7a 26. (a -3a; 28. (0 + 6- Eesolve i 30. (a; + y)2 33. (24a; + 2 35. 9a;2-(3 37. (3a + 1) 39. (2a + 6 41. {x+y- xvn.] RESOLUTION INTO FACTORS. 121 134. "When one or l>oth of the Bquare.s is a compound quan- tity the same nietbod is employed. Example 1. Resolve into fact ^i + 2i)' - 16^;". The sum of a f 2h and 4x is a + 2b + 4a?, and their difference ia a + 2b- Ax. :. {a + 26)2 - 16tt;2 = (o + 26 + ix) (a + 26 - 4x). Examplt 2. Ro.aolve into factors x^ - (26 - 3c)'. The sum of .r .iud 26 - 3c is a; + 26 - 3c, and their difference is a: - (26 - 3c) =x - 26 + 3c, .-. jc^ - (26 - 3c )2 = (x + 26 - 3c) (a; - 26 + 3c). If the factors contain like terms they should be collected so as to give the result in its simplest form Example 3. (3a; + lyf - (2a: - Sy)" = { (3ar + 7y) + (2a; - 3y) } { (3a: + 7y) - (2a; - 3y) } = (3a: + 7y + 2a: - 3y ) (3a; + 7y - 2a: + 3y ) = (5a; + 4y)(a;+10y). EXAMPLES XVII. g. Eesolve into factors : 1. [a + hf-c\ 4. {x + 2yf-a\ 7. (a: + 5c)2-l. 10. a2-(6-c)2. 13. 9a:»-(2a-36)2. 3. (x + yY-Az\ 6. (x + 5a)2-9y2. 9. (2a:-3a)a-9c'. 12. 4aa-(y-2)a. 15. c2-(5a-36)«. 18. (7a;+y)»-l. 2. (a-6)2-c« 5. (a + 36)2-165;2. 8. (a-2a;)8-62 11. a:2-(jr + z)a. 14. \-{a-hf. 16. (a + 6)2-(c + d)2. 17. {a-hf-{x-\-yf 19. {a + hf-(m-nf. 20. {a-nf-(h + mf. 21. (6 - c)^ - (a - a:)^. 22. (4a + a;)2-(6+y)2. 23. (a + 26)2-(3a:+4y)a. 24. l-(7o-36)«. 25. {a-hf-{x-y)\ 26. (a-3a:)2-16y2 27. (2a-5a:)«-l. 28. [a + b~cf-{x-y-^z)\ 29. (3a + 26)a-(c + a;-2y)2. Eesolve into factors and simplify : 30. {x + yf-x\ Si. x'-iy-xf. 32. {x + '&yf-Ay\ 33. (24a: + y)a-(23a;-y)2. 34. (5x + 2yf-{'ix-yf. 35. 9a:2-(3a;-5i/)2 36. (7a: + 3)2 - (,'5a: - 4)2 37. (3a + 1)2 -(2a -1)2, 38. 16a2-(3a + l)2. 39. (2a + 6-c)2-(a-6 + c)2. 40. (a:-7y+2)2-(7y-z)2. 41. (a;+y-8)=-{a;-8)2 42. (2a: + o-3)2-(3-2a;)2 2 • • 2 1 * I II i ji 1 m (T ir» c 1 122 ALGEBRA. [chap. 135. By suitably grouping together the terms, comDoimH expressions can often be expressed as the difference TS squares, and so be resolved into factors. "«it-nce or two Example 1 . Resolve into factors a'^-2ax + si?-^\ a2 - 2ax + a;2 - 46«= {a2 - 2ax + a?)- 46* -{a-xf-{2hf = {a-x+2h){a-x-2h). Example 2. Resolve into factors Oa" - c" + 4ca; - 4a;3. 9a8 - c2 + 4ca; - 4a;2= 9a2 - (cS - 4ca; + 4a;2) = (3a)2-{c-2;c)2 = (3a + c - 2a;) (3a - c + 2a;). Example 3. Resolve into factors 2^c^ - a^ - c2 + fta + rfa ^ 2ac. Here the terms 2hd and 2ac suggest the proner Dreliminnrv arrangement of the expression. Thus ^ ^ prelimmary 26c?-a2-c2 + 62 + cP + 2ac = 62 + 26(^ + rf2_„2+2oc-c2 = 62 + 2W + cP - (a2 _ 2a<. + (.2) = {b + df-{a-cf = (6 + rf + a-c)(6 + d-a+c). Example^. Resolve into factors a;* + a;V + y*. a:* + a;V + y4 ^ (a;4 ^ 2a;V + 2^) - a-V^ = (a;2 + 2/2)2,(^)2 = (a;2 + y2 + ajy)(a;2+y2_^y) = (a:2 + ^y ^ y2) (jp2 _ ^y ^ 2/2). cJpt'xxviif. '''^ ^""P'''""' "'^^ ^^" ^^ ^^^^^^«d to again in EXAMPLES XVII. h. Resolve into factors 1. a;2 + 2a^ + y2_(j2_ 3. a;2-6aa; + 9a2-16fe2 6. ar»+a2+2aa;-j/2 7. ar»-a2-2a6-62. 9. l~x'^-2xy-y\ 11. X- + y^ +2x1/- ixhf\ 13. a;'' + 2a:y+y2_a2-2a6-&2. 2. a2-2a6 + 62_a^. 4. 4a2 + 4a5 + j2_9c3 6. 2ay + a2 + y2_^9, 8. y^-c'^+2cx-x\ 10. c'^-x'^-y'i + 2xy. 12. a2_4„j(>^4j2_9^2g9^ 14. a2_2a& + 62_c2_2crf_rf2. 15. «;«-4aa: + 4a''-62 + 26y-2/2. jg^ f + 2hy + }fl-o?-%ax-9x\ 3£vn.] RESOLUTION INTO FACTORS. 123 17. aP-2x + l-a^-4ab-4b\ 18. da^-ea + l-X'-Sdx-ied^ 19. x'-a^ + y^-b^--2xy + 2ab. 20. a^ + b^-2ab-c'>-a^-2cd. 21. ix^-12ax-c^-k^-2ck + Qa^. 22. a^ + 6bx-9b^x^-10ab-l+25b\ 23. a^-25a;« + 8a2a:2- 9 + 30x3 + 163^. 24. al*-x'-9-2a^x^ + a* + 6x. 25. a* + a262 + fe4, 26. a:^ + 4a;V+16y4- 27. j9* + 9pV + 81g*. 28. c*+3c2rf2 + 4d^ 29. x*+y*-UxY. 30. 4w*-5mV+n*'. The Sum or Difference of Two Cubes. 1 36. If we divide a^ + b^hy a + b the quotient is a^ — ab + b^; and if we divide a^-Phy a-b the quotient is a^ + ab+b^. We have therefore the following identities : a^ + b^=(a+b){a'^-ab + b'^); a?-W= (a - 6)(a2 +ab + b^). These results enable us to resolve into factors any expression which can be written as the sum or the difference of two cubes. Example 1. 8a^ - 27y^=(2xf - (3y)3 = (2x - 3y) (4a;2 + 6xy + 9y% Note. The middle term 6xy is the prodtict of 2x and 3y. Example 2. 64a3 + 1 = (4a)3 + ( 1 )3 = (4a + l)(16a2-4a + l). We may usually omit the intermediate step and write down the factors at once. Examples. M3a^ - 27 x^ =(7a^- 3x) (49a* + 2\a^x + 9x% 8a:9 + 729 = (2x3 + 9) (4x8 - 18a^ + 81). EXAMPLES XVII. k. Resolve into factors : I, ?>^-f.. 2. T> + ir\ 3. x^-l' 4. l + a3 5. 8x3-y3. 6. x^ + 8y\ 7. 27x^ + 1. g. l-8y^. 9. a36'-c9. 10. 8«r' + 27jr'. 11. l-343x». 12. 64 + y3. 13. 125 + 0'. 14. 21G-a'. 15. OS63 + 512. 16. lOOOy^-l. I?:-' & t ii i '■ ,i> « « M -1 •• •*. ^ . y. U» '■ i: •» ^ ' 'HI ' •■ ' KC -• , ia> < 1 u» i' } n.^ ',; cr-/ "i r *r 1 .^ •'ir §;» « «!)« nr J « « ' * 124 ALGEBftA. [chap. V: ■ Resolve into factors : 17. T^ + Urr". 18. 27-100Oa:S. 20. 343-8a:3. 23. 125ar'-l. 26. a?¥<^-\. 29. 8a363+125a:3 32. 64a:«+125y3. 35. a3 + 34363. 38. Ji^q^-21si?. 21. a' + 2763. 24. 216p3-343. 27. 343x3+ lOOOyS. 30. ar'y3-216z'. 33. %x^-z\ 36. ««+ 72963. 39. s3-64y«. 19. a363 + 216c3. 22. 27ar»-642r'. 25. x^jr^+z\ 28. 729a3-6463. 31. a:''-27y3. 34. 216a:«-63. 37. 8a:3_729y8. 40. a:3y3-512. Jfl' ^^^""7, ^"^"«'"di»g this chapter we shall draw attention to a few miscellaneous cases of resolution into factors. ^''®''^"'° Example \. Resolve into factors IGa*- 816*. 16a* - bW={^a? + 962) (4^2 _ g^g) = (4a2 + 962)(4a2-962). Example 2. Resolve into factors x^ - y«. Note. When an expression can be arranged either aa fhp rUf ference of two squares, or as the difference offwo cubes ^ch of he In all cases where an expression to be resolved containq i teS^ouWe riZr/" ™^°? '5? >™^' 'Ms shouM S «. caKen outside a bracket as explained in Art. 126. Example 3. Resolve into factors 28*:^ + 64j:3y - eOa;^^. 28x*y + 64x^y - &)x^y=ix^y{7x^ + iq^ _ jgj = 4a;2y(7a?-5)(a?+3). Example 4. Resolve into factors ar'/)2 - 8y3p2 _ ^^y ^. gg^^^a The expression = p8 (ar* - 8y3) - 4^2 (^J _ gyS) = (a^-8y3)(;)2_4^2j = (a; - 2y ) (a:2 + 2ary + 4y2) (p + 2g) (p - 2q). Example 5. Resolve into factors 43:2 _ 05^,2 ^2x-r5i/ '^^'-25yH2x + 5y={2x + 5y){2x-5y) + 2x + 5y = (2x+5y)(2x-5y + l). 1. a^-y^ i 729/- 6. 2m7i + \ 8. a* + 6*- 10. (a + 6 + 12. ar'-lOi 14. 7?-a^- 16. 21a^' + J 18. &(J?-d 20. 7?-^X 22. aca;2 - J 24. a;* + 4a;2 27. 500a;2y 30. 6- + c2- 32. 14a2a:3- 34. l-(m2- 37. 8x2y + 5 39. 729a76 - 41. ai2_6i2 43. (a + 6)*- 45. (c + d)3- 47. 250 (a - 49. 8(a: + y) 51. a2-63 + 53. a3 + 63 + 55. \{x-y) [Miscellan fnmisk fiirtl XVII. 3 RESOLUTION INTO FACTORS. 125 EXAM! .ES XVn. 1. Resolve into two or more factors : 1. a?-f'-1yz-z\ 2. ^-if^^. 3. 6a2-a?-77. 4. 729y«-64a:8. 5^ a:6-4096. 6. 2mH + 2a:y + w2 + n2-a;2-^3. 7^ 33a~* - 16a: - 65. 8. a*+6*-c*-d^ + 2a262_2c2d2. 9. m^x-^m^y-nH-n^y. 10. (a + 6 + c)2-(a-6-c)2. 12. a:2_i0a;_ii9. 14. ari-aa+2/2_2a;y. 16. 21a^' + 82ar-39. 18. c'^-c^-a^(?d?\a\ 20. ar'-6a;-247. 22. aca;2 - fcca; + adx - bd. 24. x^+ixYz^+iyh*. 27. 500a:2y-2(y_ 30. ?>2 + c2-a2-26c. 32. 14aV-35a3a;2 + 14a4ar. 34. l-(m2 + n2)+2mn. 37.' 8ofiy + 52xy + 6(yy. 39. 729a76-o67. 41. ai2-6i2. 43. (a + &)*-!. 45. (c+d)3-l. 47. 250(a-6)» + 2. 49. 8{x + yf-(2x-yf. 51. tt2 -63 + a_b. 53, aH¥ + a + b. 55. 4{x--y)3-(a;-y), [Miscellaneous Examples IV., p. 174, awe? Chapter XX VIII . will jVTTAsh, jxiTtker pTuciice lu ussolutioji tato Ihciors,'] 11. 4 + 4a: + 2at/ + a:2_ a^-y\ m 1. 13. a^~b^-c^ + (P-2(ad- ■be). 1 ' ^1 15. a2 + a;2-(y2+22)_2(yz -ax). • it 17. l-a^7p-bY + 2abxy. § 19. a'a^-a^y^-b'^mfi+bY' Ml 21. a^x^-<^x»-aY+(^y^- M 23. a?x-b^x + a^y-Wy. 110 ' 11 25. a»63+512. 26. 2a;2+i7a:+35. 28. a» -%a?b\ 29. a?a?- I6a^y\ 31. 5«<-15a:3_9Qa.2^ KM « t 33. a?-\. 1 ' 35. 75x4 -48a*. 33^ 5a*b*-5ab, im 38. 3a;2y2 + 26aa:y + 35a2. IB- ^ *■ 40. a8a:8_64a2i/«. 42. 24a;2y2_30a:y3_36y4. 1 44. a4-{6 + c)* t: 46. l-(a;-y)'. r 48. (c + d)3+{c-d)3. 50. a;2-4y2 + a:-2y. 52. (a + 6)2 + a + &. ir 54. a2-962 + a + 36. « 56. af'y - a;2y3 _ yPyi-S^xy*. « • ■ 126 ALGEBRA. [chap. MISCELLANEOUS EXAMPLES III. ,.J- Subtract 3a3-7a:+l from 2*2 -5a; -3. then 8ubtr«Pf ♦»,» diflFerence from zero, and add this laat result to 2^-^-4 ' 2. Simplify 2{3a-(46-5c)} + 4{4a-(56-2c)}+4{5a-3(6-c)}. 3. Find the product of a3-2a2c + 2ac2-c3 *^^ a3+2a2c+2ac2 + c8. 4. Solve the equations : (2) 9x+5t/=75, 7a?-4y=ll. 5. Find the square root oi 8x*+16x^+l-8x-2x^ + xf' togShef mi: Z"^' "'""'' "-•"• ""'■^'■' »^"'- -"d -gWh part, <^^ 2 + 3 = 4 + 7' 7. If a=4, 6=3, c=2, find the value of 55i:i^%^i:il'+^- «" 6 + c c + a"^a + 6' 8. Divide .. + ?^ + |^^|^^.^ by a:. + ^4 9. Add 5a;2 - 6u; to the excess of 1 over 3a;2 _ Sj.^ j^ 10. Find the factors of (!) a'x^-2ax-15; (2) 4m* - Slp^: 11. Solve the equations : (1) 13a; + lly=i8, (2) 57a;+52y=181, lla; + 13y=30. 76a:-39y = 458. a c^olh^ I??h J!?lf l*f r^' " ?^^' '" ^ ^°"^« »« P ti'^es as fast as ?wr^Ls!it^rn^^^t\re^re^^ ^^« ^^«^- ^-- 13. Find the continued product of Sx"^ - 2a; + 3, 4a; + 5, 7a; - 2. 14. Solve the equations : 15. Write down the square of a;^ + 7a; - 1 1. 16. Res (1 17. Fine 18. Ah sum he the 19. Sim 20. She 21. Exp (1) (2) 22. Soh and shew t 23. Div 24. Wh when 60 a cost as muc 25. Exp in simple : product of 26. If a; XVII.] 16. 17. 18. MISCELLANEOUS EXAMPLES, III. 127 Resolve into factors : (1) a^ + 2ax-bx-2ab; (2) x*+10x^-56y\ Find the H.C.F. and L.C.M. of i9bc^ 2\a^b'^, 56ca^ 63abcK A has £50, and B has £6 ; after B has won from A a certain Bum he then has five-ninths of what A has : how much did B win ? 19. Simplify 56^X4p;^3-64^- 20. Shew that a(a-l)(a-2)(a-3)=(a2-3a+l)2-l. 21. Express by means of symbols : (1) The excess of m over n is greater than a by c ; (2) Three times the square of ab together with the cube of c is equal to p times the sum of m and n. 2i solve 1(3-1) -|(,-f) = .5(i-5), and shew that x=2 does not satisfy the equation. 23. Divide the product of 3x^ - 2xy - y^ and 2x-yhy x-y. 24. What is the price of apples per dozen, and of eggs per score, when 60 apples and 100 eggs together cost 8s. 4d., and 72 apples cost as much as 30 eggs ? 25. Express the product (2x^-13x + 15){x^-4x-5)(2x^-x-3) in simple factors, and thence write down its square root as the product of three binomial factors. 26. If a;=6, ?/ = 7, z=8, find the value of a;-(y-z)-2[x + z-3{-2(2/-l)}] + 4[|-(3-|/)]. 27. Divide 6!ii^ + 51x*y + 128a^y^-mxY-lS0xy*+G3y'^ by 3x^ + 15aPy + 7xy'^-9y^ 28. Solve the equations : 4a; + 2y+z=14, 3a;-y + 2z=3, a; + 7y-2=23. 29. Resolve into two or more factors : (1) a?y-4.xf', (2) 2m*-[-m?v?-3n*. 30. In how many days will a men do - th of a piece of work, the 7/6 whole of which can be done by b men in c days ? If 7/1=4, a =24, 6 = 14, c=18, what is the numerical value of the answer ? .1 * s i a* ii - Si - -11 M •> ■ 11 «£ « f m i» 'n ' «.!► '^ " CBAF. XVII] |)'> CHAPTER XVIII. Highest Common Factor. t38. Definition. The highest common factor of two or more algebraical expressions is the expression ofhinhest dLn sions which divides each of them without remainder «f ^°*t* 7^^ ^^^"^ grecuest common measure is sometimes used instoirl fh^Ohest common factor; but, strictly speaking, tKrm iS Z^TX^"^r' """"^^'^^ ^" «°"fi°«d tS arithTAetiLl JLSes for the highest common factor is not necessarily the greatest common measure in all cases, as will appear later. [Art. 145.] ''°"""°" In Chap xi.^ we have explained how to write down bv inspection the highest common factor of two oi more .Ui finrthTr', ^^ ^"^^"^"r "^^^^^«d ^"1 enable u read l7to find the highest common factor of compound expressions which r:sXd^?o^ir^'^^^^ ^^ ^^^^^^«' ^^ -^-^-- ^^ -% Example 1. Find the highest common factor of ^ca? and 2ca^ + ic-x^, 4ca^=4ca:', 2ca;3 + 4c V = 2ca^2 (a; + 2c) : therefore the H.CF. is 2ca;8. Example 2. Find the highest common factor of 3a2 + 9a&, a^-9ab\ a^ + Ga% + 9ab^ Resolving each expression into its factors, we have 3a^ + 9ab = 3a{a + 3b), a3-Qah^=a{a + 3b){a-3b), ,. , ''' + ^''b + 9ab^=a(a + 3b)(a + 3b)i therefore the fl[.C.F. is a{a+36). '' Therefore factor is a(z Find the 1. a^+ab, 3. 2^2 -2a 5. xr' + x'h/ 7. a^-a'^x 9. a^x + a 11. a2-xa, 13. 20a: -4, 15. x"- + x. 17. ^-2xy 19. x' + ^f. 21. a-2 + 3a;i 23. a:2-18a; 25. 12i-2+a: 27. cV-d5 28. -1"' - xv"^. 29. ah: -an 30. 2x2 + 9a; 31. 3x* + %x- iS.4- CBAF. XVIII.] HIGHEST COMMON FACTOR. 129 139. When there are two or more expressions containing different powers of the same compmind factor, the student should be careful to notice tliat the highest common factor must contain the highest power of the compound factor which is common to all the given expressions. Example 1, The highest common factor of x{a-x)\ a{a-xf, and 2ax{a-xf is {q,-x)% Example 2. Find the highest common factor of aar' + 2a2a? + a3, 2a3^-^a'^z-^a^, 2(ax + a^f. Resolving the expressions into factors, we have ax'^ + 2a?x-\-a^=a{x^ + 2ax-k-a'^) =a(x + af (1), 2ax^ - 4a^x - 6a^ .-= 2a (x^- 2ax - 3a^) ~-2a{x + a)(x-3a) (2), S{ax + a'^)'^=3(^{x + af (3). Therefore from (1), (2), (.3), by inspection, the highest common factor IS a (x + a). EXAMPLES XVIII. a. Find the highest common factor of 1. a'^ + ah, a^-h\ 2. {x + yf, x^-y^. 3. Ix^'-lxy, T^-x^y. 4. ^x'^-^ocy, 4:r2-9.v2 5. x^^xhj, x^ + ys. 6. a?h~a¥, a^b'^-a%t>. 7. a^-a^x, a^-ax"^, a^-ax"^. 8. a^-Aie^, a^ + ^ax. 9. a-bx + ah^x, a%-h\ IQ. 2x^y-Qxy\ x^-O/. 11. a2-x2, a^-ax, a^x-ax?. 12. 4z- + 2xy, 12x^-Sy\ 13. 20ar-4, SOx^ 2. 14. Qbx + iby, 9cx + Qcy. 15. X- + X, {x+iy, arJ+l. 16. xy-y, x*y-xy. 17. x'-2xy + y\ {x-yf. Ig. ofi + a'x, x*-a*. 19. aP + 8}^, 7?^xy~2y\ 20. a^-27a»x, {x-Za)\ 21. .r2 + 3x + 2, x2-4. 22. X2-X-20, x^.g^ + gO. 23. a-2- 18^+45, «2_9, 24. 2x2-7x+3, 3x2-7x-6. 25. I2x2 + x-l, 15x-2 + 8x+l. 26. 2x2-a:-l, 3x2-a;-2. 27. cV-cP, a<;x-2-6ca; + arf«-6d. 29. a^x-a%x-^b'^x, a'6x2-4a62x2 + 363x2, 30. 2x2 + 9x + 4, 2x2+llx + 5, 2x^-.3x-2. j 31. 3x* + 8x3 + 4a.2. ac5 + llx* + 6x3, 3x4- 16x3 -12w2. »• ■a <1 xm il 2 *• •• *t, c ^' t» ^ 11 IS « 1 •« 1 ui ti« >«!» B» '. J.J «:» ■ -•'.,' li;., ; C?^ y r r f c: (••ta ft" 130 ALGEBRA. [chap. 140. The highest common factor should always be deter- mined by inspection when possible, but it will sometimes hap})en that expressions cannot be readily resolved into factors. To find the highest common factor in such cases, we adopt a method analogous to that used in Arithmetic, for finding the greatest common measure of two or moie numbers. 141. We shall now work out examples illustrative of the algebraical process of finding the highest common factor, post- poniug for the present the complete'proof of the rules we use. But we may conveniently enunciate two principles, which the student should bear in mind in reading the examples wliicli follow. I. If an expression contain a certain factor, any mxiltiph of the expression is divisible hy that factor. II. If two expressions have a common factor, it will divide their sum and their difference; and also the sum and the difference of any multiples of them. Example. Find the highest common factor of 4ar' - 3a'2 - 24a; - 9 and Sx^ - 2x^ - 5Zx - m. X 2x 3 4ar'-.Sa;2-24a; 4.'r'-5a;^-21.r 2.rS - 2^2- 3.T-9 Ox 3x-9 Sx-9 8ar'-2a;2-53a;-39 8x^-6x'^-48x-\8 43^2- 6x-21 4x^-- 6a; -18 X- 3 Therefore the H.C.F. is a; -3. Explaiiation. First arrange the given expressions according to descending or ascending powers of x. The expressions so arranged havmg their first terms of the same order, we take for divisor that whose highest power has the smaller coefficient. Arrange the work in parallel columns as above. When the first remainder 4a;2 - 53; - 21 is made the divisor we put the quotient x to the left of the dividend. Again, when the second remainder 2a;2-3a;-9 is in turn made the divisor, the quotient 2 is placed to the ri' in factor x. Removing the simple factors 2a; and 3a;, and reserving their common factor x, we continue as in Art, 141. X 2x ear' -2a;2- 13a; -6 12a;»- a;2-30a;-16 6ar'-8a;2- 8a; 6a;2- 5a;-6 12ar''-4a;2-26a--12 2 3ar^- 4a;- 4 6a;2- 8a;-8 3x2+ 2a; 3a; + 2 - 6a;- 4 - 6a;- 4 Tlierefore the H.C.F. is a; (3a; + 2). • * 41 C 1. 143. So far the process of Arithmetic has been found exactly applicable to the algebraical expressions we have considered. But in many cases certain modifications of the arithmetical method will be found necessary. These will be more clearly understood if it is remembered that, at every stage of the work the remainder must contain as a factor of itself the bicrhest common factor we are seeking. [See Art. 141, I. & II.]. ° Example 1. Find the highest common factor of 3ar» - 1 3a;2 + 23a; - 21 and %x^ + x^-4Ax + 2\. 3ar'-13a;2 + 23a;-21 6ar'+ a;2-44a; + 21 6a;^-26a;^ + 46a;-42 27*2 -90a; + 63 Here on making 27a;2 - 90:t; + 63 a divisor, we find that it is not contained in 3a;3_ 13^.2 + 23a; -21 with an integral quotient. But^noticmg that 27a;2-90a; + 63 may be written in the form tliP ~ o f V"'" ^ bearing in mind that every remainder in w n^p"^"" *"'' "'°^^ contains the H.C.F., we conclude tliat the ii.t.l<. we are seeking is contained in 9(3a;2- 10a; + 7). But the iwo origmal expressions have no simple factors, therefore their a.\u.b. can have none. We may therefore reject the factor 9 and go on with divisor 3.V-- 10a; + 7. •a » 1 m v» J t *.* ' - r \i \ .13.4 133 ALGEBRA. [chap. Resuming the work, we have X 3x^-Ux^ + 23x-2\ 3ar»-10a;3+ 1x -1 - 3x2+ 16a: -21 - Sar^+lOx- 7 2) 6a: -14 3x2-10u: + 7 3a:-- 7jr - 3a; + 7 - 3a; + 7 X Zx- 7 Therefore the H.C.F. is 3a:- 7. The factor 2 has been removed on the same grounds as the factor 9 above. Example 2. Find the highest common factor of and 231?+ x^-x-2 (1), 3a:'-2a:2 + a:-2 (2), As the expressions stand we cannot begin to divide one by the other without using a fractional quotient. The difficulty may bo obviated by introducing a suitable factor, just as in the last case we found it useful to remove a factor when we could no longer proceed with the division in the ordinary way. The given expressions liave no common simple factor, hence their H.C.F. cannot be affected if we multiply either of them by any simple factor. Multiply (2) by 2, and use (1) as a divisor : -Q: 2ar»+ a;2- .-r- 7 - 2 14 ~U -14 14 6x^- 4x^+ 2a;- 4 6ar»+ Sx^- 3x- 6 14ar»+ 7x^- Ix- Ux^-lOx^- 4x - 7a;'-+ 5x+ 2 17 I7a:2- 3a;- 17a;2-17a: nx- 14a;- -119a;2 + 85a; + 34 -119a;2 + 21a: + 98 64)64x-64 X- 1 '.ix llx 14 Therefore the H.C.F. is a: - 1. After the first division the factor 7 is introduced because the first remainder -7xr^ + 5x + 2 will not divide 2ar' + a;^ - a; - 2. At the next stage the factor 17 is introduced for a similar reason, and finally the factor 64 is removed as explained in Example 1. Note. Here the highest common factor might have been more easily obtained by arranging the expressions in ascending powers of x. in this case it will be found that there is no need to introduce a numerical factor in the course of the work. Detached coefficients, as explained in Art. 45, may also be used with advantage here, and will often effect a considerable saving of labour. XVIII, ] ttlGHKST COMMON l-'AOtOll. 133 144. From the last two examples it appears that we may multiply or divide either of the given expressions, or any of the remainders whicli occur in the course of the work, by any factor which does not divide lx)th of the given expressions. 145. Let the two expressions in Example 2, Art. 143, be written in the form 2^+.r2-^_2=(^-l)(2.r2+3.r+2), Then their highest common factor is .r-1, and therefore 2v- + 3x-\-2 and 3j?2-|.^ + 2 have no algeHaical common divisor. If, however, we put .r=6, then 2.r»+a^2_^._2=460, and a.r' - 2^-2 + a; - 2 = 580 ; and the greatest common measure of 460 and 580 is 20 ; whereas 5 is the numerical value of x- 1, the algebraical highest common factor. Thus the numerical values of the algebraical highest common factor and of the arithmetical greatest common measure do iiot in this case agree. The reason may be explained as follows : when x=Q, the expressions 2^2 + 3^+2 and 3.^2+^+2 become equal to 92 and 116 respectively, and have a common arithmetical factor 4 ; whereas the expressions have no algebraical common factor. It will thus often happen that the highest common factor of two expressions, and their numerical greatest common measure, when the letters have particular values, are not the same ; for this reason the term greatest common measure is inappropriate when applied to algebraical quantities. EXAMPLES XVin. b. Find the highest common factor of the following expressions : 1. ar» + 2.c2-l Bar +10, si^-\-x^-\Qx + %. 2. ^3- 5x2 -99a: + 40, a^s _ 6a;2 - 86a; + 35. 3. »3 + 2a;2-8a;-16, x^ + 3a;2 - 8a; - 24. 4. ar' + 4a;2-5a;-20, x^ + Qx^-^x-^Q. 5. ar'-a;2-5a;-3, ar'-4a;2- lla;-6. 6. 3-'' + 3a-2-8.r-24, ar? + ,S.r2 - ,3.r 9. 7. a^-5a2a; + 7aar'-3ar\ a^ - 3aa;2 + 2ar'». 8. ar«-2ar«-4a;-7, a^ + a? -'ix'^-x-\-2. 9. 2ar»-5a~«+lla; + 7, 4ar»-llar' + 25a; + 7. E , »t c u» MB • 't\ «» J» 1 «M • ;; 1 rjj ■*ii lc» ■•', J:» *» Is! K.J f^' r r \ ■c r !»•-» C '3W m;"* If- { 134 ALGEBRA. [CHAV. Find the highest common factor of the following expreswioiLs; 10. 2ar'« + 4a;»-7«-14, 6ari- 10a?2_ 21^ + 35. 11. Sx*-3xr*-2x'»-x-l, 9r*-Sjp-x-l. 12. 2x*-2x^ + x^ + Sx-G, 4x*-2x'* + :ix-9. 13. 3x3 _ 3„^2 + 2a^x - 2a\ 3.^•3 + I2ax^ + Oa^^ f 8a\ 14. 2ar'-9aa:2 + 9a2a:-7a^ 4ar'»-20aa:2 + 20aV- 16a"'. 15. 10ar' + 25aa;2-5ff', 4ar' + 9aa;2 - 20^^: - a^ 16. 6a3+13a2^-9aa;2-10ar', &a«+ 12a2a;- llaa;^- lOar'. 17. 24x*y + 72.ry - 6xY - 90xy*, 6a:V + l^^Y - 4,xy - IS-ry'. 18. 4a:«a2 + lOar^a^ _ eOar'a* + 54a;2a5, 24x«a» + 30^V - l2Qx"-a\ 19. 4«» + 14x4 + 20*3 + 70«2, 8x7 + 28x8 -8x«- 12x4 + 56ar'. 20. 72x3 _ i2aa;2 + 72a2x - 420a3, 18x3 + 42ax^ _ 282a2^ + 270o3. 21. 9x4 + 2xV + 2/*, 3.1-4-8x^ + 5x^-2x^3. 22. x«-x3-x + l, X^ + ixfi + X*-l. 23. 1+X + X3-X«, 1-X*-X« + X'. 24. 6-8o-32a2-18a3, 20 - 35a - 95a2 - 40a3. 25. 9x8 -15x3- 45x4 -12x», 42.f _ 49x2 - 203x3 - 84x4. 26. 3x«- 5x3+2, 2x«- 5x2 + 3. 27. 4x»-6x3-28x, 6x-4+10x3-17x2-35x-14. *146. The statements of Art 141 may be proved as follows. I. If F divides A it will also divide mA. For suppose A =aF, then mA=maF. Thus F is a factor of mA. II. If F divides A and B, then it will divide niA ± nB. For suppose A = aF, B= bF, then mA ± nB=maF± nhF =F{ma±nh). Thus F divides mA ± nB. *\^-^V ^^ ™*^ "°^ enunciate and prove the rule for finding,' the highest common factor of anv two compound algebraical expressions. We suppose that any simple factors are first removed. fSee Example, Art. 142.] ■■ Let A and B be the two expressions after the simplo fnftnra have been removed. Let them be arranged in descending or ascending powers of same common letter ; also let the highest power of that letter in B be not less than the highest power in A. xvm.] HIGHEST COMMON FACTOR. 135 Divide B hy A ; let p be the quotient, and C the remainder. Supnose C to have a simple factor m. Remove this factor, and 80 obtain a new divisor D. Further, suppose that in order to make .-1 divisible by D it is necessary to nuiltinly A by a simple factor n. Let q be the next quotient and A' the reniain(ler Finally, divide I) hy E ; let r be the quotient, and aui)poH.> that there is no remainder. Then E will be the H.C.F. required. The work will stand thus : A)B(p m)C_ D)nA{q qJl E)D{r rE First, to shew that E\^a common factor of A and B. By examining the steps of the work, it is clear that i&' divides B therefore also qD ; therefore qD^E, therefore nA : therefore Ay since n is a simple factor. Attain, ^' divides D, therefore mD, that is, C. And since E divii es A and C, it also divides pA + 1\ that is, B. Hence E divides both A and B. Secondly, to show that E is the highest common factor. If not, let there be a factor X of higher di.nensions than E. Then X divides A and B, therefore /?-; . that is, C ; there- fore 1) (since m is a simple factor) ; therefc -^ J - qD, that is, E. Thus .V divides E ; which is impossible since by hypothefiis. A IS of higher dimensions than E. -^ ^t > Therefore E is the highest common factor. expressions *\^^' The highest common factor of three A, B, C may 1 o obtained as follows. First determine F the highest r immon factor of A and B • next hnd 6' the highest common factor of /'and C: then G will be the reriuired highest common factor of ^, i?, C. For F contains every factor which isconimon to A and B, ?.nr. ,. =. tuo liignest comniuii factor of i'' and C. Therefore G IS t lie highest common factor of A, B, C. •I ft r e ii' f Ml • l| "%. c H' ■■ )' m» * i Ml , 1 tw > i la 1 1 •» 1 r »^ «sm '■; r c \ c I!"' CHAPTER XIX. fjj"!' 'II Fractions. [On first reading the subject, the student may omit the general proofs of the rules given in this chapter.] 149. In Chapter xii. we discussed the simpler kinds of fractions, using the ordinary arithmetical rules. We here propose to give proofs of those rules, and shew that they are applicable to algebraical fractions. Definition. If a quantity x be divided into h equal parts and a of these parts be taken, the result is called the fraction ^ of X. If X be the unit, the fraction ^ oi x m called simi)ly " the fraction ^ " ; so that the fraction ^ represents a equal parts, b of which make up the unit. Note. This definition requires that a and h should be positive Mhole numbers. In Art. 155 we shall adopt a definition which vill enable us to remove this restriction. 150. To prove that ^=^, where a, b, m are positive integers. ay ^ we mean a equal parts, b of which make up the unit ... (I); mb ^2). , ma But .ma that is, Conversely, b parts in (l)=m6 parts in (2); 1 part = m a parts = ma a_ma b mb' ma _a mb ~b' U K CHAP. XIX.] FRACTIONS. 137 Hence, the value of a fraction is not altered if %ve multiply or divide the numerator and denominator by the same quantity. Reduction to Lowest Terms. 151. An algebraical fraction may be changed into an equivalent fraction by dividing numerator and denominator by any common factor ; if this factor be the highest common factor the resulting fraction is said to be reduced to its lowest terms. Example 1. Reduce to lowest terms The expression: 24a\^3c^ 4ac' '6a^x'(3a-2x)~3a-2x' Example 2. Reduce to lowest terms 7: 17#„. 9xy - 12y^ 2x{3x-4y) 2x '3y(3x-4y)-Sy Note. The beginner should be careful not to l)egin cancelling until he has expressed both numerator and denominator in the most convenient form, by resolution into factors where necessary. The expression : 1. 4. 7. 10. Reduce to lowest EXAMPLES XIZ. a. terms : ahx+bx* 2a26-4a62' I 5a%h I00(a»-a%)' .r(2a2-3aa;) a{'ki^x-9a^)' x^-bx x^~Ax-5' i« 3?y-\-2i"y + ix y 15. 18. x*-Ux^-S\ X ^-'2x'-l5" 16a2 oV- ax' + 9ax + 2l0a' 2. 5. 8. 11. 16. 19. acx + cx^' 4a;'^ + Qxy' a^~2xy^ X* - 4a^y- + 4y*' J^±6x ^ + ix + 4' 3. 6. 9. 12. ax a^x^ - ax bx^ + bxy-vby^' {xy-3y'^f 5a; ^ » lOgg^g 3a'^b-^ + (iab^' 14. 3a* + da% + 6a%'' a* + a»6-2a-62 x'^ + xy-2y'' 3x'^ + 23x+l4 3x'-{4lx + 2(i' 17. 20. r-c 2^2+n^2i 3a;3 + 26a; + 35' 27a + a< 18a-6a2 + 2o»* mm I *• e , il i i\ t ^ l» " '<■■ m M *r, s ■■' Tl. Ktt ■■ 1: »« s * m - ' ti* ■t.i R» v 1 ! M» • ''i ^1' K.I J S f" 6 1. t._ r If'' t r 138 ALGEBRA. [chap. ); 162, When the factors of the numerator and denomimfnr cannot be determined by inspection, the fraction ly beTeS to Its lowest terms by dividing both numerator and denonSo by the highest common factor, which may be found by tlie rule given in Chap. xvm. "^ ^ "^ ^"'^^ Example. Reduce to lowest terms 3a;^-13a:a+23ar-21 The H.C.F. of numerator and denominator is First Method. 3a; -7. Dividing numerator and denominator by 3a; -7 we ohtiin .. respective quotients a;^ - 2a; + 3 and 5x^ - a; -3. "^ Thus ?^!^JM±23^z21^(3a;-7)(a;2-2a; + 3) a;2-2a; + 3 15a;8-38x2-2a; + 21 (3a;-7)(5ar^-a;-3)-5^T^r3- Tliis is the simplest solution for the beginner • but in ihW and similar cases we may often effect the reduction w Ion the fraction = ^ :^(3a; - 7) - 2a; (3a;-7) + 3 {3a; - 7) ^^W^)-x{Zx - 7) - 3 (35^7) _( 3a;-7)(a;g- 2a; + 3) (3a;-7)(5a~2-a;-3) _a;|-2a; + 3 5a;a-a;-3' res^lL inf.^'Jl'T ""aerator or denominator can readily be resolved into factors we may use the following method Example. Reduce to lowest terms _ ^+3a;g-4a; rru . 7a;3-18a;2 + 6a; + 5' lhemimerator=a;(a~2 + 3a;-4) = a;(a; + 4)(a;-l) Of these factors the only one which can be a common divisor is « - 1. Hence, arrangmg the denominator, ^""^^o" o»visor is the fraction = oc{x + 4^){x-^Vi^ _ x{x + q{x-lL)_ _ a;(a; + 4) (* - I ) (7a;3 - 1 la; - 6) - T^-rn^Ts- 1. a^-c a» + 3« 3. a3 + 2o a^ + o 5. 4a3 + 6a»+] 7. a;2-S 3a;3 + ^ 9. 4ar»-J x^ + as 11. 16a;*- 4ar'-j 13. 5a;3 + 7ar'-4 15. 3ar»-2 2a;3+ s 154, R Wt' Vv^h- // - 5a;' + 7a; -3 a:3-3a;+2 2a:3 + 5a;2y - 30a;y'' + 27^/^ 4a;3 + 6a:y2-21ya ' l+2a;'' + a;3 4-2a;* l + 3a:2 + 2a;3 + 3x''' 3a3-3a'6^aft'^-6'» 4a2-5a6 + fe2 * 4a;3-10a;'' + 4a; + 2 3a;*-2a:3_3a; + 2- 6a:3 + a;a-5a;-2 (5ar' + 5a;2-3a;-2' 4a;*+llar' + 25 4a;<-9:r2 + 30a;-25* aa;» - 5a V - OOa^y + 40a* «* - 6aar« - Sea'a:^ + 35o»a;' Multiplication and Division of Fractions. 154., Rule I. To multiply a fraction by an integer: «»;'<: ^tV the numerator hy that integer ; or, if the denominator be « ' .« >r'i! Jy the integer, divide the denominator hy it. The rule may be proved as follows : (1) T represents a equal parts, 6 of which make up the unit ; -J- represents ac equal parts, h of which make up the unit ; and the number of parts taken in the second fraction is c times tlie number taken in the first ; that is (2) a _aG fjXo?=T->5 by the preceding case, a b' m 1 * ' 1 * E ii • ^ ii "■ Mi ■n e "■■'* 'Mot •a. * 1 ,f-« . . • 'ISC "* i K» .! «:» r ? k- r-' n i' vi €' \ ■e: r «aa ^ IRk, I**" [Art. 151.] 140 I? » ALGEBRA. [chap. 155. By the preceding article a , ab that is, the fraction | is that which must be multiplied by b in Z^Z.u- f^'u \ . ^"*i ^^ ^^*- 46' *^^ quantity which must from thiP];"^ ^^ ^ J" T^'r *^^ ^^^^^'^ « ^« *he quotient resul ng from the division of a by 6 ; we may therefore define a fraction the fraction ^ w the quotient of a y the preceding case. ft.in«:»«o^^® 7?w To multiply together two or more fractions: mtdtiply together all the numerators to form a new numerator, and all the denominators to form a new denominator. To find the value of Let a c b^d' _a c '"~b''d' XIX.] FRACTTIONS. Ul Multiplying each side hy bxd, we have .t' X 6 X rf=T X -. X 6 X 0? a a , = tXOX c d"" d [Art. 29.] =axc :. xxbd=ac. ;Art. 154.] Dividing each side by bd, we have ac ^~hd' . a c ac '• h^d~hd' ., , ace ace rmlarly l^'dTbd/' and 80 for any number of fractions. 158. Bule rv. To divide one fraction by another : invert the divisor^ and proceed as in multiplication. Since division is the inverse of multiplication, we may define the quotient x, when r is divided by -i, to be such that c _a d Multiplying by - we have .r x -, x - =- x - • c d c b c' iV = ad Hence a . c _ad_a d b ' d~ bc~b c' [Art 157,-] which proves the rule. Example 1. S^^Pl^ty -^^ x ^^^g. 2a2 + 3a 4a2- 6a _ a(2a + 3) 2a{2a-S) 4a3 ^12aH-18~ 4a^ ^' 6(2a + 3) 2a -3 ' 12o * by cancelling those factors which are common to both numerator and denominator. hat " Ml an M ■.*>'* ma ■I* HP,,.- ■r: "*.» <« 142 ALGEBRA. Examph 2. Simplify [chap. 6x^-ax-2a^ x-a . 2x + a ax - a- ^ da^ - 4a^ ~ 3aJ+2a^" m. . 6x^-ax-2a'^ x-a Sax + 20^ Ihe expression = 5 — x „ „ -^ x — - ■ _(3a;-2a)(2.r + a) x-a a{x-a) ^ a(3x + '2a ) {Sx + 2a)(3x-2a)''~2^^T^ = 1. since all the factors cancel each other. EXAMPLES XIX. c. 1. 3. 5. 7. 9. 11. 13. 15. 16. 17. 18. Simplify Ux^-7x 12ar»+24a~2'a;2 + 2a;" x^ - 4a2 2a ax + 2a^ x-2a 16x2 - 9a2 x-2 X x^-4: 4x-3a' x'^ + 5x + 6 a^25j-3 a;2-i "^ a;a_9 • 2a^ + 5x + 2 a^Ax a?»-4 ^2a;2+9i^- g°-14a;-15. a;''-12a;-45 ar'-4x-45 " a;2-6x-27* 262 + 56^262-116 + 15- 64j02g2- .c-2)2 2. 4. 6. 8. 10. 12. 14. a262 + 3a6 . a6 + 3 4a2-l -204- 1* a 2-121 . tt + 11 a2-4 • + 2' 25a2- 62 x{3a + 2) 9a^a^-4x'^ 5a + b ' a:2 + 3a; + 2 ir2 + 7a; + 12 a;2 + 9a; + 20 a;2 + 5a;+6' 2a?2 + 13a;+15 . 2a;2+ 11a; 4.5 4a;2_9 4*2 -1 2a ;2-a;-l 4a;2 + a;-14 2a;2 + 5a; + 2^ 16ar2-49 * a?-6x^ + S6x , a;< + 216a; a;2-49 •a;2-a;-42" X .2_, a;2-4 Spq + z^'(x + 2f xl^-x-20 x^-x-2 . x + l a;2-25 ^ x'^ + 2x-^^W+hx a;2-l8a; + 80 ar'-Ga;-? a; + 5 a;2 - 5a; - 6U ^ a;^ - 15i^+56 "" ^^* a;2-8a;-9 0^^-25 . a;« + 4a;-5 19. Qxy- 20. x2 + a; a;2-a;- 21. 4a;2-l 2a;2 + 22. a;4-g 3-^-43; 23. (a + hf a^ + ab 24. a2+2a a2-62- 25. a;2- a;2 + 24i 26. (o2 + aa a-'-x 27. m' + 4« 3m2n- 28. l + SarJ (2-a;)2 29. a^(a;- (x + 4)2 30. {p + g? {p+q-i 31. a*- «2-2aa 32. aH8a «'»-17« + 72 a;2-r •a;2-9a; + 8" XIX.] 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. FRACTIONS. ix^+x-U _4a^ aT-2 2a^ + 4a? Qxy - Uy ^ x^ - 4 ^ 4x - 7 " Sxa:^^;—^- x'^ + x-2 x^-x-2Q 4a;2-16ar+15 a?-x • Va^-2a;-I5^ a;2 j' x'^-Qx-1 4a;2-l 2a;2 + 3a;+l 2a;»-17a; + 2l 4a;2-20a; + 25' a:*-8a; x^ + 'ix+l . x' + 2a; + 4 x''-4x-5 a;*- a;'* -2a;' a;-5 {0 + 6)2-c3, a ,(a-6)2-c2 a2 + 2a6 + 6^c2 g a _ 2ac + c" - 6^ ^ a;2-64 a;2 + 12a;-64 a;5'-16a; + 64 =:;; X a-2 + 24a; + 128 a:3-64 ' a;2 + 4a; + 16' {a ^ + axf (g - a;)" a''-aa; + a:g 3m% - 5m?i2 - 2n3 ^ Gw^-Swji + w^ "^ 27 ws + j?- 1+853 4a;-a:3. (l-2a;)g+2a; (2 - xf ^ i - 4a;2 • 2 - 5a; +2a;2 ' _^-4)2_ 64-^^(a;2-4a;)3 (X + 4)2'- 4a; 16 - a,-2 * {x + 4)2 ( p + g)2-r 2 ^ pi+pq^pr . p^-pq+pr (p + q + rf (p-rf-q^-{p-qf-rs: a ^x + ac^ a^ + a'^x'^ + x^ \ ;3-ar* a^x-ax^ + x^J' «2-2aa; + a;2 ' \ a a ^ + 8a26 + 1 5a?>2 16a*-17a262 + b* ^ a^+ 2ab - 362 {64a3 - 68) (a3 + 6=* j ^ 4a2 + 21a6 + 562 • oS -^26" +^^2- US mm 1::' 1 ■M ■n ' it am HP j! vm mm fl $ M :ii ■*.. <^ ■ "^^j em ■ ai mm * i m ' *». •» i K» « 1 «J» !• ; »» CHAPTER XX. Lowest Common Multiple. 150. Definition. The lowest common multiple of two or more algebraical expressions is the expression of lowest dimen- sions, which is divisible by each of them without remainder. In Chapter xi, we have explained how to write down by inspection the lowest common multiple of two or more simple expressions ; the lowest common multiple of compound expres- sions which are given as the product of factors, or which can be easily resolved into factors, can be readily found by a similar method. L'xample 1. The lowest common multiple of 6x^{a - x)\ 8a^(a - x)'^ and 12ax{a - «)» is 24a^x^a - x)\ For it consists of the product of (1) the L.C.M. of the numerical coefficients ; (2) the lowest power of each factor which is divisible by every power of that factor occurring in the given expressions. Example 2. Find the lowest common multiple of 3a^ + 9ab, 2a^-18ab^, a3-f6a26 + 9ai2. 3a^ + 9ab = 3a(a + Sb), 2a3 - 18a62=2o(a + 3ft) (a - 36), a^ + 6a% + 9ab'^=a{a + Sb)(a + 3h) = a(a + 36)2. Therefore the L. C. M. is 6a (a -f 3b f {a -3b). EXAMPLES ZX. a. Find the lowest common multiple of 1. X, x^ + x. 2. ar^, a^-3x. 3. 3x^, 4«2-f8.r. 4. 21a^, 7x^{x+l). 5. ar^-l, ^ + x. 6. a^ + ab, ab + bl 7. 4a;2y-y, 2ar2 + a:. g. 6*2 -2a;, 9x'-3x. 9. x^ + 2x, x^ + 3x + 2. 10. x'^-3x + 2, a?-l. U. ar»+4a;+4, ar2+5a;+6. 12. «8-5ar+4, sc2-6a;-|-8. CHAP. XX.] LOWEST COMMON MULTIPLE. 145 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. -a?-e, ic^+x-2, a:2-4a: + 3. x'^-\-x-'2Q, a:a-10a; + 24, x^-x-m. a:Ha;-42, a;^ - 1 1 a; + .30, x^ + 2x-'^5. 2a;2 + 3a; + l, 2x'^-\-6x + 2, a;2 + 3a; + 2. 3a;2+lla: + C, ^x^' + Sx + i, x'^ + ^x + Q. 5a:2 + lla; + 2, 5a;2 + 16a; + 3, a:2 + 5a; + 6. 2a:2 + 3a:-2, 2a;2+15a:-8, a;2+10a; + 16. 3a:2-a:-14, 3ar2- 13a:+14, x^-A. 12a:2+33._42, 12a:» + 30a;2+12cr, 32ar» - 40a; - 28. 3a:* + 26a.3 + 35a;2, 6a;2 + 38a;-28, 27ar' + 27a;2 - 30ar. mx* + 5ot^-5x^, 60a;2y + 32a:y + 4y, 40a;3y - 2a;V - 2a:y. 8a;2-38xy + 35y2, Ax"^ - xy - ^y"^, 2x^ - 5xy - 7 y"^. I2x^-23xy+l0y^, 4x>-9xy + 5y^, 3x^-5xy + 2y\ %a3? + la?x^ - 3a»a;, 3a2a;2 + Ua?x - Za\ 6a;2 + 39aa: + AhdK ^az^y'^-V\\axy^-^y\ Sx^y^ + lxY-Gxy^ 2iax'^-22ax + 4a. {3x-5x^f, 6-7a;-5a;2, 4a; + 4a;2 + a;3. 14a*(o3-63), 2la%^(a-b)\ 6a^b{a-b)(a^-b'^). m* + mhi^ + n*, m?n + n*, (m^-mnf. (2c3-3cd)2, (4c-6rf)3, 8c3-27#. 160. When the given expressions are such that their factors cannot be determined by inspection, they must be resolved by hndiiig the highest common factor. Example. Find the lowest common multiple of 23;* + a;3-20a-2-7a; + 24 and 2x-^ + 3ar'- 13a;2-7a;+I5. The highest common factor is x"^ + 2a; - 3. By division, we obtain 2a;^ + a;3 _ 20a;2 _ 7a, + 24 = (a;2 + 2a; - 3) (2a;2 - 3a: - 8). 2a;* + 3ar' - l.iar' - 7a; + 15 = (a:2 + 2a; - 3) (2a;2 - a; - 5). Therefore the L.C.M. is (a;2 + 2a;-3)(2a~i-3a;-8)(2a~i-a;-5). *161. We may now give the proof of the rule for finding the lowest common multiple of two compound algebraical ex- pressions. Let A and B be the two expressions, and F their highest cunmion lactor. Also suppose that a and b are the respective quotients when A and B are divided by F; then A =aF, B=bF. iiieretore, since a and b have no common factor, the lowest common multiple of A and B is abF, by inspection/ ill s MM 1 * MH *m E Ul § in ; ti "*• c aiB ■' r rlM * ' :'>^ ■ '«« "% , l» " ' ft» . ' r I %■' 1; e; r WIS m ■■gift: It 146 ALGEBRA. [CII.VP. 1^ 11 *162. There is an important relation between the highest comnion factor and the lowest common multiple of two ex- pressions which it is desirable to notice. Let F he the highest common factor, and X the lowest couuuou multiple of A and B. Then, as in the preceding article, A=aF, B=bF, and X=ahF. Therefore the product AB=aF.hF =F. ahF = FX (1). Kenee the product of two expressions is equal to the product of their highest common factor and lowest common multiple. Again, from (1) X^ ABA B . '' F ~ ji^^^-/^^^ > hence the loicent common multiple of two expressions maif be found hy dividing their product hif their highest common factor ; or hy dividing either of them hy their highest common factor, and multiplying the quotient hy the other. *ie3. The lowest common multiple of three expressions A, B, C may be obtained as follows. First, find X the L.C.M. of A and B. Next find Y the L.C.M. of X and C; then Y will be the required L.C.M. of A, B, a For Yia the e::pression of lowest dimensions which is divisible by X and C, and X is the expression of lowest dimensions divisible by A and B. Therefore Y is the expression of lowest dimensions divisible by all three. I » EXAMPLES XX. b. 1. Find the highest common factor and the lowest common multiple of x^-5x + 6, x* - 4, a^- 3x - 2. 2. Find the lowest common multiple of ttb{x'+}) + x(a^ + b^) and ab{x^-l) + x(a^-b'^). 3. Find the lowest common multiple of »y~bx, xy-ay, y^-3by + 2h-, xy-2bx-ay + 2ab, ney-hx-ay + ab. \h K XX.] LOWEST COMMON MULTIPLE. 147 4. Find the highest common factor and the lowest common multiple of a?-\-2x^- 'ix, 2x» + 5x- - 3x. 5. Find the lowest common multiple of 1-x, (l-x-f, (1+a.f. 6. Find the lowest common mtiltiple of a~»-10a; + 24, a;2-8a; + 12, x^'-Gx + fi. 7. Find the highest common factor and tho lowest common multiple ol6s(^ + a^-5x-2, 6ar* + 5x^ - 3a; - 2. 8. Find the lowest common multiple of (6c2-a6c)2, b^ac'-a% oV + 2ac» + c«. 9. Find the lowest common multiple of x»-y^, x^y-y*, y'^(x-y)\ x"- + xy + y\ Also find the highest common factor of the first three expressions. 10. Find the highest common factor of 6a;2-13a; + 6, 2a;2 + 5a;-12, 6x''-x~]2. Also shew that the lowest common multiple is the product of the three quantities divided by the square of the highest common factor. 11. Find the lowest common multiple of x* + ax^ + a^x + a*, x* + a^x^ + a*. ^u- ,^"i'l*]*®„^!.S^®^* common factor and the lowest common multiple of 3x3 _ 7^,2^ ^ 53,^2 _ ^^ ^2^ ^ ^^^2 _ 3^3 _ ^;,^ Sx^ + SaPy + xy^-yl 13. Find the highest common factor of 4x3- 10x2 + 4a; + 2, Sx*-2x^-3x^2. 14. Find the lowest common multiple of a^-b^ a3-63, a^ - a% - ah"- - 2¥. m 15. Find the highest common factor and the lowest common ultiple of (2x2 - 3a^)y + (2a^ - 3y^)x, (2a^ + 3y'')x + {2x' + Sa^)y. ¥;• ,^'"r*^ }^^^ highest common factor and the lowest common multiple of x3 - 9x2 + 26x - 24, x* - 12x2 4 < 7x - 60. 17. Find the highest common factor c x' - 15ax2 + 48a2x + 64^3, x^ - lOax + lQa\ 18. Find the lowest common multiple of 21x(x^-y2)2^ 35(xV-x2/), 15jr(x2 + xy)2. I J * CHAPTER XXI. Addition and Subtraction of Fractions. 164, Havino explained the rules for finding the lowest common multiple of any given expressions, we now proceed to shew how the addition and subtraction of fractions may be effected. 165. To prove a^cad+bc b^d bd We have a V ad , c be 'bd^'''''^d^rd- [Art. 150]. Thus in each case we divide the unit into bd equal parts, and we take first ad of these parts, and then be of them ; that is, we take ad+bc of the bd parts of the unit ; and this is expressed by the fraction «^+*"' bd Similarly, a c_ad+be b'^d bd'' a_c_ad—bc b~d bd ' 166. Here the fractions have been both expressed with a common denominator bd. But if b and d have a common factor, the product bd is not the lowest common denominator, and the fraction —^j- will not be in its lowest terms. To avoid work- ing with fractions which are not in their lowest terms, some rnonifir»oflrv>i nf tVi^ oV./MT.^ 1"!'" ^^ "-1 a T— ^; - -•'- -!! be found advisable to take the lotoest common denominator, which is the lowest common multiple of the denominators of the given fractions. since the tern ni.] ADDITION AND SUBTRACTION OF FKACTIONS. 140 Rule. I. To reduce fractions to thi .r lowest common denominator: Jind the L.C.M. of the given denominaton, and taken for the cominon denominator; divide it hy the denomiruttor of the first fraction, and multivlv the mmerator of this fraction hy the qtiotxent so obtained; and do the same with all the other given fractions. ^ Example. Express with lowest common denominator 5a; , 4a The lowest common denominator is Qax{x - o)(.r + a). We must therefore multiply the numerators by Zx(x + a) and 2o respectively. j \ i -« Hence the equivalent fractions are 15a;3(x + a) Sa" ^ax(x-a){x-\- a) "" Qax {x -a)[x + a)' 167. We may now enunciate the rule for the addition or subtraction of fraction'- Rule II. To add or subivict fractions: reduce them to the (owest common deno ni.iaior ; hid the algebraical sum of the 7iumerators, and retai:' tU corrm on denominator. Example 1. Find the ;U6 0f?^ + ^-i?. 3a Qa The lowest common denominator is 9a. Therefore the ^v prrnBJnn - 3 (2a; + a) + 5x - 4a ■(:, 9a _ 6£ + 3a + 5^j-4a _ 1 1 a; 9a 2a Example 2. Find the value of ^:i^ + fc-'*-?£z2a ^ ay ax ' The lowest common denominator is axy. Thus the nvp..c,.,i^n _ a{x-<2y) + x (3.v - g) - v^^x-2a\ axy _ ctx-2ay-\-^xy-ax-'ixy + 2av axy = 0, since the terms in the numerator destroy each other. i - » 1 * ii ii J» tffPi Jg ;i i ' I'l 'IH ' ^: '•» ■t ii ''i» ' £i' i: ■ 1 tr: t t^ * t .mti '»'aK-.! 150 I « ■« ALGEBRA. EXAMPLES XXI. a. [CHAP. Find the value of 1. 3. 5. 7. 9. 11. 13. 15. x-\ , a; + 3 , .r + 7 5a;- 1 8 o*%; "~ ^ X "^ \ a; _ 7 a;-9 15 "^ 25 x+i a-b b-c c-a +-^ — J-- ab b + c c + a 2a "*■ 46 ap + 2_«--5 17a; 34a; "* a;-3 a~2-9 ca a-b a;+2 "61a;' 8-x^ 5x 10«2 I5a;3 2x-.Sy , 3a;-2z . 5 1 f- — • xy xz X 2. 4. 6. 8. 10. 12. 14. 16. 2a; - 1 a;-5 a; - 4 2a; -3 a; + 2 5a; + 8 9 " 6 "•" 12 ' 2a; + 5 a; + 3 27 8a~»* a + 76 X 2x a -2b a -5b 2a 4a ■ + - Sa g-a; a + x a^-ofi X a 2ax 2a2-68 6'-cg a" 63 c2-a2 2^ a;y a" Sy'^-a;^ , xy + y^ xy^ be ac 3t^y^ 62 a6-c2 be ac ab 2a; - 3a 2a; - a Example 3. Simplify The lowest common denominator is (x - 2a) {x - a). Hence, multiplying the numerators by a; - a and a; - 2a respectively, we have the expression = <^ - ^""^ -^f- - "> 'f''-^^ <^ " ^) (a; - 2a) (a; - a) _ 2x^ - Sax + Sa" - (2a;'' - 5aa; + 2a') (a; - 2a) (x-a) 2ar' - 5aa; + 3a2 - 2a~» + 6a« - 20" (a; - 2a) (x - a) a' {x - 2a) {x - a)' Note. In finding the value of such an expression as -(2a; -a) (a; -2a), the beginner should first express the product in brackets, and then remove the l)racket8, as we have done. After a little practice he will be able to take both steps together. The work .v^ill sometimes be shortened by first reducing the fractions to their lowest terms. Find th( 1. a; + 2"^ 4. 3 a;-6 7. .r + 3 a; + 4 10. x-4 a;-2 13. 1 2a; -3i 15. ia-^ + h 4a2-6 17. x-x^ 19. xy 25x^- 21. ar»-4f a;2-2a 23. 1 a -2a; 25. 3 x«-4" XXI.] ADDITION AND SUBTRACTION OF FRACTIONS, 151 Example L Simplify ^l^^-^'-^^^-. The expression _ x" + ^xy-4 y'^ y ~ x^-lQy^ x + iy _ V? + 5a;y - 4y '' -y(x- 4y) x'^-T6y^ _ot^ + 5xy ~ 4y^ - xy + iy^ ~ 3^16y^ _ x'^ + 4xy _ X x^- lQy^~x-iy' EXAMPLES XXI. b. Find the value of 1 . 1 x + 2'^x + 3: 3 1 10. a:-6 x + 2 .r + 3 _ a; + l x + i x + 2' x-i x-7 x-2 x-5' 11. x + 3 x + 4' a h x + a x + b' a + x a-x a-x a + x a a2 x-5 x-4t a + ■ x-a x^-a .2 _ «2- 12. x-a x-b x+2 x-2 x-2 x + 2' 2x x-'i^x^-9' 1 13. --^-^' x + y 15. 2x-Zy 4a;2-9ya 4a? + h'^ 2a-h 4«2-6a 2a + h' 17. - 9? X X-X^ 1+; 14. 16. 18. x + a x^ + 2a^ X - 2a x^ - 4a^' 2ofi 2a;2 ►•2 _ tfl -y.2 x^-y' x^ + xy 19. 21. 23. xy » + 2ofiy 25x2-yaT-iQ^2y + 2a;y2* ^Z.^ _ x'^ + 2ax-%a'^ x^-2ax x^-4a!^ ' _J (a + 2x^ a~2x~ a^-Sar*' «ix-y) y{x+y) ''"• x{x^-yYy{x^+y^)' 22. 3 a->~l 1 24, WM- xy + y^ x^-xi/ + y x + y a^ + l^ X- y mSJ* O " i" "t~ . /^Tn» X* - 4 ' (u; - 2)9 26. ' a'^-ab + b^ d^ + ab+b'^' I 1 aW-a^) x{x + af • 1,1? 1 «. 2 s ill t •Ml K ■'«* m fee i* It •<*• ., |. Si US' ■• i H ;ff % 152 ALGEBRA. [chap. 168. Some modification of the foregoing general methods may sometimes be used with advantage. The most useful artifices are explained in the examples which follow, but no general rules can be given which will apply to all cases. a + 3 Example 1. Simplify ^ „ - —. — ^. Taking the first two fractions together, we have a + 4 a-3' 8 the expression: a2-16 8 (a-4)(a-3) . 7 ■(a-4)(a-3) {a + 4)(a-4) - 7(a + 4)-8(a-3) ■(a + 4)(a-4)(a-3) 52 -g '(a + 4)(a-4)(o-3)' 1 Example 2. Simplify ^ ^*C "T* iC 1 1 + 1 The expressions 3a;2 + 4x+i* 1 ; + , {2x-l)(x + iy{Sx+'lj\x+l) 3a;+l+2a;-l ■(2a;-l)(a; + l)(3a;+l) 5x '' W-l)(x+l){3x + l)' Example 3. Simplify 1 1 2x 4«3 a ■X a + x a^ + x"^ a*-\-a^' Here it should be evident that the first two denominators give L.C.M. a^-a;*, which readily combines with a^ + x"^ to give L.C.M. o*-x*, which again combines with a* + ar* to give L.C.M. a^-n?. Hence it will be convenient to proceed as follows ; The expreB8ion=^-li^L _ 2x 2x a, ■^ - x^ aF+^ 4a^ 4ar» "a* -3!* a*-\-x* Find th 1. 1 x+y 3. 5 l+2x 5. 10 9-a2 7. 1 2(a-l 9. ^2"^ XXI.] ADDITION AND SUBTRACTION OF FRACTIONS. 153 EXAMPLES XXI. c. 1. 3. 5. 7. Find the value of 1 1 ^_2x x + y x-y s^-y^' 5 3ig 4 - 13a; I + 2a: l-2a; l-4a;2* 10 2_ _1_ 9-a2 3 + o~3-a 1 1 h_ 2(a-b) 2(a + 6) a^-fta 9. ±-.J-^^ ^- 11. 13. : 15. x-2^Zx + %'^x^-^' a;2-9a; + 20 ar^-llx + SO 1 1 2x2-a-l 2x='+a;-3* 4 3 4-7a-2aa 3-a-10a2' 8. 10. • 12. 14. 16. o _1_._J 3a; '- 2x + y'*'2x-y \x^~^' 4 2« 36 m^ *• 2af36'^2a-36~4a2-9P' A -^ ^ ■ 1 '*• 6(ar»-l) 2(^^ + 3(^:ri]* 5 4(3a + 2) 2a 2a -3 6a + 9 ti^^^S a;3 + y3 ar*-g/3 a:«-y« * 1 1__ a;''-7a; + r2 ar'-5a; + 6' 1 3_^ 2a;3-a;-l 6a~8-a;-2' 5 2 5 + a;-18a;3 2 + 6a; + 2a:2' 17 _1 J , 1 *" x + 1 (a;+i;(ar + 2)'^(a;+l)(a; + 2)(x + 3)' 18. 19. 20. 21. 5a; 15(a;-l) 9(a;+3) 2(a:+l)(a;-3)~16(a;-3){a;-2)~16(a;+l)(a;-2)* a + 36 : + ,- a+26 a + 6 4(a + 6){a+26) (a +6) (a + 36) 4{a+26)(a+36)' L_ 4.__2 1 a:2-3a; + 2'*'a;2-a;-2~^"^ X 15 12 aj'^+Sar + e a:" + 9a; + 14 a^»+10a; + 2r 22 -J I 4 , 4« + 2 ar'-l"^2a; + r 2«»+''3x+l* oo _ 5(2a;-3) . 7« 12{3a;+l) ii^Oar + x-i; 6a;- + V*-3 ll(4a;"- + 8x + 25. 24 ^3 a--2 1 • «+2 a;+3"'"a;-r l(4a;- + 8x + 3)* a;-3 a; + 4 a;-4 a;+3"a;2-16' s UK* J .* « «, « *. ' ::? • li * li <*• *9C **> 1 *J» £ f •is 'a 1,1 I'! •'if W t I a. 154 ALGEBRA. 26. 28. 30. 32. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. Find the value of l+2a l-2o 8a l-2a l+2a (l-2af J. 1^ 2x 3-x 3 + x 9 + x^' 1.11 4(l+a;)"^4(l-a:)'^2(l+a;2)- a» 1 1 4 + a:2 1 + .- 2-x 2+x 27. 29. 31. 33. 24« [CIIAP, 3 + 2a;^3-2a; 9-12a; + 4a;2 3-2a;"^3 + 2« __1 1 4ct 2a + 3'^2a-3 ia^+d' 1 a-x 8{a-xy8{a + x) 4{a^ + x^)' 5 5_ X 3-6x 3 + 6x~2 + 8a:2- 1 2o-8a; 3a2 + 48xa"^2a + 8a;' 1 ,+« 1 a 6a2 + 54"^3a-9 3aa-27 _1 1 , a; 8-8« 1 - + X 8 + 8a;^4 + 4a;2 1 1 < + 2 + 2ar«" 18 6a- 18 6a + 18~aF+9 a*TSl' x+l x-l 1 2«8 - 4a;a ■*■ ac* + 4a~* ~ a~* - 4' 1 1 3x8 _ 4a:y + 2^2 + ^2 _ 4^,^ ^ 3^2 " 3a;'« - lOxy + Sy^' 1 2 3a;-2 1^_ x-l rc+l «»-! (x + lf 108 -52a ; _4 12 fl + xV x(B-xf~3-x x'^[3-xj' {a + bf a + 26 + a? {a-hb)x + {x-a)(x + a + b) 2(a;-a) 3(x'^ + a;-2) 3(a;'»-a;-2) 8a;_ x^-x-2 x'^ + x-2 x^-4: x'^ + bx-a'-ab^^ 169. We have thus far assumed both nuraerator and deno- minator to be positive integers, and have shewn in Art. 155 that a fraction itself is the quotient resulting from the division of the numerator by denominator. But in algebra division is a process not restricted to positive integers, and we shall extend this definition as follows : The algebraic fraction ^ is the quotient remlting from the division of ^ by b, where a a„d b may have any values whatever. XXI.] ADDITION AND SUBTRACTION OF FRACTIONS. 156 170. By the preceding article -- , is the quotieht resulting from the division oi -ahy -b; and this is obtained by dividing a by b, and, by the rule of signs, prefixing +. — a ■A^g^-i"? -y- is the quotient resulting from the division of - a by b ; and this is obtained by dividing a by 6, and, by the rule of signs, prefixing -. -a a Therefore •(2). a Likewise — r is the quotient resulting from the division of a by -b; and this is obtained by dividing a by 6, and, by the nile of signs, prefixing - . Therefore a -b' a b (3). These results may be enunciated as follows : 1. If the signs of both numerator and denominator of a fraction be changed, the sign of the whole fraction will be un- changed. 2. If the sign of either numerator or denominator alone be changed, the sign of the whole fraction will be changed. The principles here involved are so useful in certain cases of reduction of fractions that we quote them in another form, which will sometimes be found more easy of application. 1. We may change the sign of every term in the numerator and denominator of a fraction without altering its value. 2. We may change the sign of a fraction by simply changing the sign of every ter7n in either the numerator or denominator. Examj)le\. ^~"' -&+« «-ft Example 2. Example 3. The intermediate step may usually be omitted. y-x -y+x X- y x-x^ 2y - -x + x" 2y - x^-x 2y • Sx 3x 3a? 4-a;2~ -4 + a;2- a;2-4' Kaa lip „. r I i^m if- ',' lit ■•if ft. If f.tfir 166 ALOfeBRA. [chap. Example 4. Simplify _ilL + ^ . «('^-«) firJJ* t '\?^»<*«.nt that the lowest common denominator of the first two fractions is :f -a», therefore it will be convenient to alter the sign of the denominator in the third fraction. Thus the expression = -^ + -^ - °(^^-") x+a x-a x^-a^ _ a(x-a) + 2x{x +0 ; - a{3^- a) a*'"a2 _ aa; -a' + 2a;« + 2 j.t ~ ::^a; + a''^ a;" - a"*' 2g« Example 5. Simplify '- i '^^"^ , ' The expression 5 3*1 3(a;-l) z^- l''2"fFrr) _ 10(£+1_) -fc{;ia;-l} j. g(a;-l) 6(x*-l) _10a;+10-18a; + 6 + 3a;-3 6(a;2-l) Simplify 13 -5a; "6(a»-l)' EXAMPLES XZI. d. 1. 1 1 -5Tr7-,^ + : 1 4a; - 4 ~ 5ar + 5 i - a;8' 3 -^^z2^ + 2(^^4a£)__3a_ a; + a a^-a;^ a;-o" 5. 1 r + ;^ 1 4a; 2a;+l 2a;-l"^l-4a;2" 7 2-5a; 3 + a; 2a;ff^- 1 1 ) a; + .3 S-a?"^ 2r2_9 ' + ; 11 26 + 2 46-4'*'6-66«' 2. 4. 6. 8. 10. _S 2___6ai 1+a 1-a a2-r £j^ aH3a^ £+« a; + a a2-a;2 "*"a;-a' 3a; 2 2 1 - a;2 a; - 1 ~ 5Tl* 3 -2a; 2a; + 3 12 ^x-ro o 2x 4a;* - 9 -^+-i- -1- 6a + 6 6-6a Sa^'-S* 13. 15. 1 2a + 5' 17. a7?+h 2a; -1 19. a- (a -6) 20. 2a {x-a) 21. {a^-b'' 22. x + a 23. 3 a; + a 24. 1 4a3(aH 25. a; x'-y" 26. 6 o{a2- 27. a2-2o "2(a2- 28. 2 + 6 29. 3 8(1 -a;' 30. a; X- 31. a'^ + oc a^c-cs 32. 4a + 66 . .' » ~ V. XXI.] ADDITION AND SUBTRACflON OF FRACTIONS. 157 11. 13. 15. 17. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. aP + yi + a; + ■ y jpi-yi x + y y-a; + 3a +, 1 2a + 5b^25b^-4a^ 2a-iyb ax^+b 2(bx+aai^) ax'^-h 2x^ ■*" l-4a~» ~ 2^T" 12. 14. 16. 18. te'-y' xy-y^ xy xy-x' a:" + 2a; + 4 x^-2x + i x + 2 2-x 2b -a 3x{a-b) b-2a x-b b'-x' '^T+x' a + c b + c (a-b)(x-a) ' (b-a){x-b)' a-c b-c (a-b){x-a) (b-a)(b-x) 2a + y ; + a + b + y x+y-a {x-a){a-b) (x-b)(b-a) (x-a){x—b) 0-V + 1 (a2-62)(a;« + 6'') (ft" - a") (a;^ + a*) (ar^ + ««) (x^ + fe*)' 1 + 4a 1 2a x + a y?-o? a-x ^-{^ct^ 3 1 + + ■ 1 x + a a; + 3a a-a: a;-3a 1 1 + , 1 a« 4a3(a + x) 4a3(a;-a)^2aa(a2 + a~2) a^-^ X y ^v? + 't^ + i xy a;3-ya x^ + ^^ y*-3i^^\x-\-y){7^+y'^S ,+!-, a : + a4 + 6* a" (a2 - 63) ^ 6 (aa + 6') ^ aft (6* - a^) 6^ _ ^s" x'^-a^ a2 -2aa;+x° 2 (o2 - a~») (a-a;)(a2 + 2ax + a:2) ~ 2(x-a)J^ 2ox(a + x) 1 36 ab a + b a-b V^-a^^a? + ¥' 29. ^ L-i + 1-X 8(l-x)^8{l+x) 4(l+x2) 4{x8-l)* 30. - + X +■ X x-l x+l^l-a~« x(a?»-f) 31. 32. a' + oc a^-c^ 2c d^c-Mi S» 158 ALGEBRA. *171. Consider the expression 1 1 [chap. .+. and the expression (a-b)(a-cy(b-c)(b-a)'^{c-a)(c-bj' Here in finding the L.C.M. of the denominators it must be observed that there are not «ir different compound factors to be considered ; for three of them differ from the other three onlv m sign. ■" Thus (a-c)=-(c~a), (b-a)=-(a-b), (c-b)=-(b-c). Hence, replacing the second factor in each denominator by its equivalent, we may write the expression in the form 1 1 1 (a-bXc-a) {b-c){a-b) {c-a){p-c) W' Now the L.C.M. is (6 - c){c - a){a - b) ; _ -{b~c)-{c-a)-{a-b) (b-c){c-a){a-b) _ -b+c-e+a-a+ b {b - c){c - a)(a - 6)~ =0. *172. There is a peculiarity in the arrangement of this ex- ample which it is desirable to notice. In the expression (1) the letters occur in what is known as Cyclic Order ; that is, b follows a, a follows c, c follows b. Thus if a, b, c are arranged round the circumference of a circle, as in the annexed diagram, if we start from any letter and move round in the direction of the arrows, the other letters follow in cyclic order, namely, abc, bca, cab. The observance of this principle is espe- cially important in a large class of examples in which the differences of three letters are involved. Thus we are observing cyclic c:-der when we write b-c, c-a, a-b ; whereas we are violating cyclic order by the use of arrangements such as b-c, a-c, a-b,or a-c, b-a, b-c. It will always be found that the work is rendered shorter and easier by following cvclic order from the begnining, and adhering to it throughout the question. In the present chapter we shall confine our attention to a few of the simpler cases, resuming the subject in Chapter xxix. Find the 1. a (a-b) 2. h {a-b) 3. 2 {x-y) 4. y-i {x-y) 5. b- {a-b) 6. x^^ {x-y) 7. 1-t (a -ft) 8. P- (p-q) 9. p+(, (p-q) 10. (a2-ft2 11. X-] (P-Q) 19 9 + liii.l XXI.] ADDITION AND SUBTRACTION OF FRACTIONS. 159 "^ EXAMPLES XXI. e. Find the value of a .+ {a-b){a-c) {h-c){Jb-ay{c-a){c-h) a (a-6)(a-c) (6-c)(6-a) (c-o)(c-6) : + + , y (x-y)(x-z) {y-z){y-x) (z-x){z-i/) y + z z + x x + y {x-y)(x-z] {y-z)[y-x) (z-x)(z-y) b-e c-a a-h {a-b)(a-c) {b -c)(b -a) (c-a){c-b) xh/z yh X z^xy {x-y)(x-z) {y-z)(y-x) (z-x){z-y) 1 + a i + \ + b \ + e (a-6)(a-c) (6-c)(6-a)^(c-a)(c-6) p-a q-a + : r-a (p-q)(p-r) (q-r){q-p) (r-p)(r-q) p + q-r q + r-p + , — r+p-q (p-q)(p-r) (q-r)(q-p) {r-p){r-q) 10. 11. 12. a» 62 (a2 - 62) (a2 - c2) ^ (62 - c^) (62 - a^) ^ (c^ - a^) (c2 - 62) x + y :+ x + y x + y {p-q){p-r) (q-r){q-py(r-p){r-q)' q + r r+p + : p+g (x-y){x-z) (y-z)(y-xy(z'-x)(z-yi J *. ! "St i ""M " - If s^ I,;. 1' r I I i 15 r CHAPTER XXII. I ■ ■ f J Miscellaneous Fractions. t'^3 • vow propose to considor some miscellaneous ones- tiotw luvo'vuig fractions of a more complicated kind than those alr-^Ht./ d-dcussed. In the previous chapters on Fractions, the nume' \tor and denominator have been reijarded as integers; but cases fre- quently occur in which the numerator or denominator of a fraction is itself fractional. 174. DefLu. .^Jti. A fracti -a of which the numerator or denominator is itself a fraction is called a Complex Fraction. a a Thus abb o X c - are Complex Fractions, c d In the last of these types, the outside quantities, a and d, are sometimes referred to as the extremes, while the two middle quantities, b and c, are called the means. 175. Instead of using the horizontal line to separate numerator and .'enominator, it is sometimes conveniei\fc to write comph X fractions in the forms /b a I ale ""Ic^ ir^ bid- 176. By definition (Art. 119) - is the quotient resultii a c ^ , from th'> division f ^ by ^ , and this by Art. 158 is ^ ; a I c d a O -2(1 c be' 4 OHAP. XXII.] MISCELLANEOUS FRACJTIONS, 161 Blmpliflcatlo- of Complex Fractions. 177. From the precec. g article we deduce an easy method of writing down tlie Birapl tied form of a complex fraction. Multiply the extremes for a new numerator, and the means for a new denominator. Example. a'rx b _a h(a + x) a ^^ ~ ft la" - a;») ~ S"^' ab by cancelling common factors in numerator and denominator. '78. The student should especially notice the following cases, and should be able to write down the results readily. a a a a 1 I 1 a 1^1_1 h_h a a \ a - » i: .1 *• : OV % .MM:. inn 1- I' ' V- 179. The following examples illustrate the simplification of complex fractions. K.A. a c '^^'- H<^'Mti) h d _ ad+hc . ad-jg w •~hcr, ad+hc _od+6c ad -he r f# ■ 162 ALGEBRA. [chap. Im i: II Hi I I ■I I I* I BSR* tel' X »» Exanvjplt 3. Simplify a; a:r*-a'* x" - a*' a 13" 6^2 a Here the reduction may be simply eflfected by multi fractions above and below by 6a, which i« the L.O. denominators. Thiis the expression = 18 + 2a»-12a a« + 3a-18 _ 2(o'-6a + 9 )_ 2(a-3) (a+6)(a-3)~ a + 6 * Example 4. Simplify The numerator a + T A' a-b a+b {a? + b^){a^-b') '(a^ + b>){a'^~h^)' Similarly the denominator = Hence the fraction 4ab {a + b)(a-b)' 4a262 4ab (a2 + 62) (aQ _ 52) • (a + b}(a-b) ,{a + h){a-b) 4a26a ■(a8 + 62)(aa-62) ah 4a6 "«2 + /ja- g the of the Note. To ensure accuracy and neatness, when the numerator and denominator are somewhat complicated, tlie beginner is advised to simplify each separately as in the above example. xxn.] MISCELLANEOUS FRACTIONS. 163 180. In tlu' ca»<> of fractions like tlie following, called Continued Inactions, we begin from the lowest fraction and simplify step by step. Example. Simplify 9a-' - 64 x-l- 1 The expression = 93^-M X 4 + x 9a:3-64 i+x-x 4 i + JC 9a:>-64 9ar3-64 4a;-4-(4 + ar) 3a; -8 a 5.. EXAMPLES xxn. a. Find the value of 1 n m a b m n 2. H X y 3. , 6 x-y' d 4. 5.!:|. 6. 7. l-t x2 8. ""l- 9. -2-. "-2 10. a? 11. n p 12. 1 a;-- a; '4 x+d+- 1 14. ^ 2 3 9 15. 2*2- J 4 t;-6 1 + ^8 1 m !si i 111 If i t i ■f ii I* 164 ALGEBRA. [OHAP. Find the value of ;8-6» a8+&8\ , 4o6 lA -2_./' 1 M 17 ftz^-^±i^\ "*- l~x»'\T^ T+^J' ^'' \a-b a + bj a'^-W 18. 19. ofl-aas+x^ a*+ax+a^ a-x a+x ( ("-M"-^) ». (ifi+¥)-(rf.-^1- 21. 23. a+6 o~6 a-h a+h (a+6)" 3a?-2 3a: + 2 9-^ 25. 27. 29. a~- tt'' a + a-1 a a?+ m n y+r x-2 x~2- X x-\ X-- x-'Z a-b b-i SI. 1- l + o6 l + 6c {a-b)(b-c) (l + ab){l + bc) 22. a , X a^'oa;''' ^ 24. 1+- a; ! + !» + 1-a? 4x+- 4x 1+4 2(a?+y) 6-0?, 28. 30. 1- l+a; ' 1 X X- a;+- X x+- X — X 32. a+6 ^~^?+P xxn.] 33. - 35. a? -a 4.C-2 181. S< as a group Example 182, Si by the den valent fori Example Example In some Example By (livisi Thus the Therefon xxn.] MISCELLANEOUS FRACTIONS. 165 33. 35. a-x a^-ax- (a - xf 1- a X 2-4a; 4.«-2- 4F 34. 36. 2a;'^ + 2a; 3(a:-l) X X a: _. 2 .^ "!' a;-2 a;-4 a:-* . + . a;2-2 1 + 1 2.C-1 1- 1 + 1 4a; -i it" + 1 X a; 1- 1 X 1 X X 181. Sometimes it is convenient to express a single fraction as a group of fractions. ^ 10a;V lOxV I0«V 10a:V =1-1 + ^ 2y X 2x^' 1 82, Since a fraction represents the quotient of the numerator by the denominator, we may often express a fraction in an equi- valent form, partly integral and partly fractional. ^ a; + 2 a; + 2 a; + 2 Example^. 3^, 3(a; + 5)- 15-2 _ 3(a: + 5)- 17 17 x + 5 a; + 5 .-r + S " a + 5* In some cases actual division may be advisable. Example Z. Shew that ^^^7 '^^ ~ ^ =2a; - 1 - * a;-3 x-Z By division, a;-3)2a;2-7a;-l (2a;-l 23?^^ - 6a? - x-\ - a; + 3 -4 Thus tl>e quotient is 2a;- 1, and the remamder -4. Therefore ^.7_Z?Jl1=2u;- 1 - '^ a;-3 a;-3" Q r * ..I, 1 «.'j w -: 1.} >i|||j « > ■51* II I I*. I m m m m I ■ I = i 166 ALGEBRA. [chap. 183. If the numerator be of lower dimensions than the denominator, we may still perform the division, and express the result in a form whict is partly integral and partly fractional. Example. Prove that By division j^=2.-6:r.+ 18^-^. 1+3*2)2^; (2x-Q^+lS!xfi 18a^ 183^ + Mx'^ whence the result follows. Here the division may he carried on to any number of terms in the quotient, and we can stop at any term we please by taking for our remainder the fraction whose numerator is the remainder last found, and whose denominator is the divisor. Thubj if we carried on the quotient to four terms, we should have 2x 1+3.^2 =2a; -6x3 + 18ic»- 54x7 + 162a:» l+3r»* The terms in the quotient may be fractional ; tlius if .v^ is divided by ,v*-a% the first four terms of the quotient are ^+^+j7+^' and the remainder is ^„, 184. Miscellaneous examples in multiplication and division occur which can be dealt with by the preceding rules for tlie reduction of fractions. Example. Multiply x + 2a- ^, ° .. by 2a; - a - -^. 2a; + 3a '' x+a The product=(.+2a-2j^)x(2.-a-^) 2x^ + 1ax + 6a^-a'^ ^^ + ax-a^-'M 2x + 3o x+a 2a;2 + 7aa; + 5a2 2x^ + ax-da^ 2x\Sa x + a _{2 x + oa){x + a) {2x + Sa){x-a) 2.v + '3a x + a = {2x + 5a)(x-a). xxil] MISCELLANEOUS FRACTIONS. 167 EXAMPLES XXIL b. Express each of the following fractions as a group of simple fractions in lowest ^erms : 3. 5. 2ab bc + ca+ab abc ^' 12ax a + b + c abc a%c-Sa¥c + 2abc Qabc 4. 6. Perform the following divisions, giving the remainder after four terms in the quotient : 7, x-r(\ + x). 8. a-r(a-6). 10. l^(l-x+x% n. a;H(a;+3). 13. Shew that p^-^^=a+2b+ """ 14. Shew that x'^-xy + y^ 9. (l+a;)4-(l-a;). 12.. l-r(l-«)«. a-b' 2y^ a? -y^ x-ry x+y 15. Shew that 5^ + to-2 =^^-^5 + ^+2- 16. Shew that 1 + d^ + h^-c^ (a + j, + c)(a + 6-c) 2a6 2ab 17. Divide m Multiply a«-ax. + 4,^-i^by3-^g^. , 16a;-27 , , , 13 a5"-16 -^ a;+4 19. Divide 20. Divide ia + 36_2- J^ by 36 + 6---^. o , OJ2 , 656* , , „, , 136» r I il m 1- a. Mumply ^.......ll^lByl-,,,^^, 9% I m m , « m I ■ I I I m I I 1 I I 168 ALGEBRA. [CHAP. il®?'-ii'^® following exercise contains miscellaneous examples wnich Illustrate most of the processes connected with fractions. EXAMPLES XXU. c. Simplify the following fractions : 1 ^(<»''-^) . r«^-aa; a^+2ax + ■D 36 (c8 - x') ' Ibc +bx'' ■c«^2^^T^3 2 x{x+a)(x + 2a) x(x + a){2x+a) 0- 3a 3 V_i l\ ' h\a-b a+2bj~ 6a aa+a6-26«' 8. 10. 11. 12. 13. 1+X3 fc 6» '^~¥(b + a^ r _2_j._2_ 4a; ^- a;-l'*'a;+l"^-ar+l* 7 1 1 2_ '• a; (a;+l)3 a;+l + l + a?+a;3' Q 2x3-9a;a+27 3»«-81a;+ie2' X / a;*-a-» . ar»+aa; 'i x°-aV /a; a\ la:'-2aar + aa • a;-a J ^ x» + a^~~\a~x) a^ + ax + xr^ ' a^-afl ac»-10ar» + 15a;-8* .IVa 7 g > 15. i+A^- ^ 4 a a a+Ka+2~ 1 a In a; +3 1 1 1 2i!^+9x+9^2 ' 2x-3 9' X--r- 4a; XXII.] 17. a;3 + »=' 18. {1+a.t 19. arJ-a a:3- 21. 2a (x-2a 22. l/a2 + 2W- 2a »/ 1 24. 1 . x+y ■ 25. /'a.r-s 26. \x a XZII.] MISCELLANEOUS FRACTIONS. 169 17. 18. 19. 21. a^+3i^+x+l~ x^-x'+x-i' l-a" (l+ax)^-{a + x)' ar»-3a; + 2 ' 2a x-a ■'2\i-x^TT^)' 20. x^~6x+S 'U^-21ay^+15x + 20' {x-2a)^ x^-5ax + Qa^'^x-3a ^' 2w-xv 2''^r^~\^rrx}' 23. ffj L_^ ^ 2\x-y x + yj 2i, y x + y) a^y + xy^' x+y x+y \_2\x + y x-y) a;2y + «yO' 25. (3-5-^)(3..5-?).(.-|). 26. j?-4- + -L.U/?L!:£_5L^\ \x a+x a-x\'\a-x a + xj' 2x-~ 27. 29. 30. 32. iit- '>r_ \x-l 4x^-1 28. (6+,-^)C6-^)fj;:i^:\ tf a-b T+ab a- a~b 1^5 (a-6)6~ a(a-b) 1 + ab 1- 1-ab -(I-^> y x"^ -y^ 1_1 ^a^ + y^' 31. 5to+I '*"2' y X ~ ' l + ah x + y 1- ay j_a(6-a) , yjx+y)' 1+ab ^"^ l-«y a + ?> . a~b a + b a -I' m m '*«^ r " -mi t I' f>'y i 9 I « (: i 'ti f ■ 170 ALGEBRA. Simplify the following fractions : [chap. 34. (l-x'){l-x^) x{l+x)(l-xf 1 1 ar x^ »• {^-^3(.-i)}.(.-i). 1+- 36. 38. 39. 40. 41. 42. 7>l m m 1 m m''+— WI-1 + m 1^' 37. -+?-l 1+ y g E X 1+t x' X' , X + -+1 r y x-y •«^_^' y~ X ( ' Sx + a^ y 1 + 3*7 1 ^- 9 .33-a;a a^-3x + 1 a;'' 3a;3+l 3 2(a;« + 3)' (a;3_a;)2 *■' 1 r+- a2-2 a2-l^a2+l aa + 2* 1 - + 1 6m-2» 3m + 2n 6m + 2»* s+ + x-1 i(l-xy'^d{l-x)^8(l+x)'^^l+'Wi «.+: 1 1 9(a:-2")'^9(a:+l) 3(.r+l)2 _ l* a; + 2 + - X U^Ay'^-W x^+xy'^x^^' ^ .-i^Mj^::^]: 2/M2s^a;)2 4f-j£-j^2 *"• {2z + ar)2 - y2 -^ (X + y)2 -4z^ + {y + 2z)3 - a;«' 46 (^ - y) (?/ - g) + (y - s) (2 - a;) + (2 - x) (X - 1/) a;{s-«)+y(x-y) + z(y-2) " ' XXII.] 47. a- {a-b) 48. c-\ (a-b) 49. a;2-(2 (3z + x 50. 9y2-( o a"- 52. i^- «) 53. f Is a- -a J xxn.] MISCELLANEOUS FRACTIONS. 47. a -h-c b~c-a + , c-a-h (a-b){a-cy(b-c)(b-ay{c-a){c-h) 171 48. 49. 50. 51. 52. 53. 54. c+a : + a + b b+c (a-b){a-cy(b-c){b-ay(c-a){c-b)' 5 + + ■ (3s + x)2 - 4y3 ^ (a; + 2y)a - 9z^ "^ (2!/ + 3z)2 - x 9y2_(4g_2a;)2 IGg^ - (2.>; - 3y)2 , 4x"' - (3y - 42)2 i+- S- + , (2* + 3y)2 - 16s2 ^ (3y + 4z)2 - 4a~» "^ (42 + 2a;)2 - 9y ^ _ ''^+'^ 1 x-a X x^+a^ X x^-\(jfi 1 a+x 1 a- •» a a^-^a? a a^ + x"^ {x-:a){x + b)-{y + a){y+b) {x-a)(y-b)-{x-b){y-a) x-y a-b U- a + x a-x ax + x^ c^ + ax + x .T-1 . x'-l a; + 3 x + 3 - + x-2 a; + 4 a; + 2 . x + 2, x-2 x-2 55. X-2 + 6 £+3_aH-3 a; + 3 4 a; + l - + a;-3 + iC-l a;~4 + 12 x-3 x-S x + 3 + a;-4 «>• <-^'-l^^{'-.^^}- 57. (Ua)2-fjl + a 1 -a + a l+a + a^ :)- • {aa:»+(6 + e)a;-/p-{<»^'--t-(6-e)a;-/P' I- m m m m W Hi .]> I ^ I' m If <1 ,' •■* ♦ tit If Mi ih ■it if ' 172 ALGEBRA. [chap. 1 1 1 'i I 1 :; tl 1 ' i M 1 i . tf 1 , m n m 1 i\ m \ m i • ii' • ^'t . 4 I I? 1 MISCELLANEOUS EXAMPLES IV. [The following Examples for revision are arranged in groups tinder different headings; each group illustrates one or more of the principal rules and processes already discussed, and for the most part the Examples ^'esent more variety and difficulty than those of the saine type which have appeared in previous exercises,] Substitutions and Brackets. 1. Find the value of J^2b{a^^-b'') ^^®° « = - 4, 6 = - 3. 2. When a = l, 6=-l, c = 2 evaluate the expression \^3a»(6 - c) + 36»(c - a) + 3cr>(a - b). 3. Simplify a(6-c)3-a(6-c)(268-5c + 2c2) + (a6+ac)(68-c2); and find its value when a = l, 6 = 2, c=3. 4. Find the value of ^{5(62 - c2) - a2} + ^3{a(a^-c^)-l} when a=4, 5 = 6, c = 3. 5. Find the value of s/{x^ + y^ + z){x-y-3z) 4- v'^^ when x= -1, y= -3, z = l, 6. When a=0, ft=2, a;=l,y=-3, 2=5, find the numerical value of (1) (a;-y)3-3a(a;-y)2+36{ar'-jr»); (2) {X- a)3 - 62(a; - y + 2) + V(fear2 - axy + y2 + g2)_ 7. If a; = 6, y = 7, s = 8, find the value of ^-' -\7 3\4'7J 7(^--- • --• ^'^^ -^' 3J' (2)4(.-i)}-|{.-3(|-xz>)}-^-,(..^,)}. xxn.] MISCELLANEOUS EXAMPLES 8. Evaluate ^o-ftjc-^c)] r^T§~ when 0=2, 6=-l, c=:l. Find the value of a-[b-{a- (b-c)}-'sf2 a^ + IP+c^ when a=-], 6=6, c=3. 10. When a =4, &= -2, c=^, rf= -1, find the value of (1) a3-63-(a-6)3-ll(36 + 2c)^2c2-^]; (2) ^4c2 -a(a- 26 -rf) - -^6*c + 116W 11. Find the value of when /=|, wi= -^. 12. When 0=1, 6=-l, c=0, evaluate a-[6-c-{2a-26-|(3c-6 )}] 6 178 a- a 13. Simplify and find its value when x=^, y= - ». 14. If x=6, y='J, z = 8 find the value of ,,. 2f /^ a:M vr^'' -.<"- 223) a,-, (2, %{|5-(f-.)}4[3-.{.0-|„-„} 'A ""> [chap. e 1 III m m m t m i IB » It 6- t k m. 174 ALGEBRA. Resolution into Factors. (On Arts. 128-132.) Eesol into two or more factors : m a:»T21a; i 108. 16. a«+6a-91. 17. aP-20xy+%y\ 18. a«62-i4a6-5i. 19. c3+c«-lf3c. 20. mhi-Qmn'>+%\ 21. p*-/)V-'''^«9*- 22. d^-4cPc'--45c^ 23. a^y-xY-i2xy\ 24. m2+28m + 195. 25. 210-a-o2. 26. 57 + IBp? - ?iY. 28. aH7a'-98. 29. 0"+ 54c + 729. 31. «* + 9o2«2+14a^. 32. p'^-3pg-mql 34. K*-2a:3-63a:2. 35. ?A2 + 56c-84. 37. a2-22ac + 57c 38. J^s + 6^/23 -91)/:. 40. 2a%'>+ab-l5. 27. a:^ + 27«"+176. 30. 12 + xy-a^y^. 33. 2a« + 2a3-264. 36. z'' + 34z + 289. 39. 2 + x^-3afi. 42. 35 + 12mn + mhi"^. 43. 11 9 - 10c - c^. 46. 6m2+7w-3. 46. 4a2-8a6-5W 48. 20a:a-9a:2;-20s2. 49. 8a:* + 2x2- 15. 51. 12(a26a-l) + 7afe. 53. 21a:2+2y(5a;-8y). 41. 9p«-24^ + 16. 44. Ga^-5x*+x\ 47. 6^2-13^^ + 252. 50. 12.v2-3r^ + 12. 52. 2{a*b'^ + 5)-da%. 54. 3(6m2-5?t2)+17w7i. (On Arts. 133-137.) Resolve into two or more factors : 55. c>-a>-b^+2ah. 56. a2+26c-63-c2. 57. 125a:3+272r'. 58. 0=^63 + 343. 59^ 51263 -««. 60. aa-4(a;-y)2. 61. 2mn + m'^-l + n^. 62. 8c*-2c''(d+c)^. 63. {a%'--iy>-x'>+2xy-y\ 64. l-64m«. 65. p^+lOOOpY. 66. 6561 -a<. 67. x*-2si^-y^-z^ + 2yz + l. 68. a2-16(&-c)2. 69. c-d-4{c-rf)3 71. 2 + 128 (a + 6)3. 70. p^-16q^-hp-4q. 72. x+3y+x^+27!r\ xxn] MISCELLANEOUS ^XAJfPLES. 176 {Miscellaneom Factors.) Resolve into two or more factors : 73. .r3'a:i^ + .ry«+y3. 74. ,,,,.2 ndx-hd. 75. M - 6a - a'^. 76. 98.r< - T^V -y. TJ. 5' - 14a - a\ 78. ' (w» +p^)~ 'Imp. 79. aft (a;- , 1 ) - a; (a' + V^). 80. 9/^2 - 6k - 18 + C''. 81. a,-3c3 - c3 + a;» - 1. 82 3z2-2a6-a;(6-6a). 83. m3-n»-(a:2-ni?i)(m-n). 84. a(ft« + ra-aa) + 6(a2 + c2-fca). 85. x'-.r' + Sa;* 8. 86. Tvxpress in factors the square root of (a;« + 8a: + 7) {2a;» - a; - 3) (2xH 1 Ix - 21 ). 87. Find the expression whose squ; (2a;2 -xy- 15y2) (4a;2 - 25^/ - lla-y + 15y»). Highest Common Factor and Lowest Common Multiple. 8fl. Find the lowest common multiple of 13afe2(a:3 _ 3(^2^ ^ 2a3), ma?h (x^ + ax- 2a\ 2oh\x^ - a'^f. 89. Find the highest common factor of 2(ar* + 9)-5a;2(a;+l), 2ar'(2^-9) + 81(x- 1). 90. Find the expression of lowest dimensions which is divisible by each of the following expressions : (2x* + ix^) (x2 + 2x -8), (2a:3 - 4x^) (x''-2x- 8), (a;2-4.r)(a:2 + 2a:-8). 91. Find the H.C.F. and the L.C.M. of the three expressions a(a + c)-b{b + c), b{h + a)-c(c + a), c(c + h)-a(a + b). 92. Find the divisor of highest dimensions of the expressions (a + 6)(a-6) + r(c-2a), (a + c)(a-c) + b(b + 2a). 93. Find the expression of lowest dimensions such that the L.C.M of It and 2a- - 3ab + ' is 2a*~Sa%-aVj^ + 3ab^-b*. ■ ft' c ■ 11 W 1 IS| W,W^^'^ MICROCOPY RESOLUTION TEST CHART (ANSI and ISO TEST CHART No. 2) 1.0 I.I 1.25 15.0 "'"== 1" f" 13-6 ■^ 140 III 2-0 lUUU 1.4 1.8 ^ APPLIED IIVMGE inc 1653 Eost Main Street Rochester. New York 14609 USA (716) 482 - OJOO - Phone (716) 288- 5989 -Fox If! If m m m IB IE i Mr, IT I I 176 ALGEBRA. [chap. 94. Shew that x'^ - iy^ is the H.C.F. of the expressions x*-3xY-4y*, «6-6V, and ocfi + S2y^-8x^y^ + 2x*y + l6xi/-lQ3^. 95. Find the lowest common multiple of (a-c)2-(6-c)2, a3+62-2ac-26c + 2a6, a*-b*. 96. Shew that the lowest common multiple of a(a-6)2-ac2, a%-b(b-cf, (a + cfc-b'^c is abc (a* + b* + c*- 26 V - 2c^a^ - 2a%% 97. Prove that ot^-\ba? + 15x^ - 145a; + 84 and a;*-17a,-3 + 101«2_247« + 210 have the same H.C.F and L.C.M. as x*-\Zo(^ + 53x2 _ 83^ + 42 and a^-19jr'+131r»;2- 389^+420. Simplification of Fractions. Simplify 100. — ^+ ^ 99. 2a; -7 2 (a? +2) (a;- 3)2 a;2-9' 1 1 ini {«+1)M^il1)' (a;3-2a;)2-(a;2-2)2 ^"^- (l-a;)2 + l-a;2 + (l+.r)2- 103- (a;- l)(a; + l)(a;2_2)2" 104. 1 1 1 -— a;.2 4 2 1 6a;-2 ■^PIJ---- X 106. -^ L + _l_. i_i ?!::^rf-y' x^~y^ x-y x + y X y ccv(x--y)' xxir.] 108. 109. a* + (ac-f 110 111. 112. 114. '■m. (52 + 14 a; 1-a; i; 1- X — 115. 1 + 116. 1- ab / « + i\ (a + 6) 117. — : B.A. [CJtAP. XXII.] MISCELLANEOUS EXAMPLES. IV, 177 108. ^* + ^'* + "^ <"^ + ^^) a* + h*- ab (g^ + h'') Ua^b'^ 109. (a + bf (a-bf {ac + bd)^-(ad + bcl _ (ac + bdf + {ad -,- bc)^ {a-b){c-d) (a + b)ic + d) 111 a ^-(a:-l)\ a;2-(a;2-l)2 a;2(^_l)2_i 113 112. ^-^ 1+^' l-a^ 1+^ m2 l-a;2"''i + a;* l + a^a 114. 115. 1 X X- 2 ^+2 X 1 tr x + l 1+X + X^ 1 + a; • l+3x + 3x^ + 2a^' 1-X + X l+x "«• ^i^-'0'd¥'"-'^>-H^J 117. b-a ~ ^ _^^(a + 6)3+(6-a)» _J 1_ 6-a a+6 (a + 6)2-(a-/>)2' 118. (r— ^::^L___£z^L_.__A:-a_l (.(a-6)(a-o) (ft-c)(&-a)'^(c-a)(c-6)/ (a-6)(6-c)(c-a) E.A- * « m e « Jl s ! ■r !< CHAPTER XXIII, m • m m ta m 11 r Harder Equations. 186. In this chaptei- we propose to give a miscellaneous col- lection of equations. Some of these wilfs'^rve as a useful exercise for revision of the methods already explained in previous chap- ters ; but we also add others presenting more difficulty, the solution of which will often be facilitated by some special artifice. The following examples worked in full will sufficiently illus- trate the most useful methods. x + 5' Example 1. Solve ., „ ■ Multiplying up, we have (Qx - 3) {x + 5) = (3x - 2) (207 + 7), 6a:2 + 27.C - 15 = 6.X-2 + 17a; - U ; .-. 10ar=l; _ 1 •'• ^-10" Note. By a simple reduction raai.y equations can be brouglit to the form in which the above equation is given. When this Is the case, the necessary simplification ia readily completed by multiplying up, or " multiplying across," as it is sometimes called. Example 2. Solve 8a; + 23 5.r + 2 2aj + 3 20 3.T + 4' Multiply by 20, and we have 5 -1. 8x+23-^^^^^--- ..c+12-20. By transposition, Multiplying across, 3X + 4: 31 = 20(5a;+2) 3a;+4 93.r + 124=20(5a; + 2), 84= 7a;; .-. a: = 12. Nl'i ,neo'.!3 col- ul exercihe ious chap- icnlty, the ial artifice. ;ntly illus- brought to thia is the multiplyiug CHAP. XXIII.] HARDER EQUATIONS. 179 When two or more fractions have the same denominator tney shouid be taken together and simplified. 3. Solve liz_2^+?3^8^-!Lil x + 3 ^ 4x + 5 - x-\3 Example By transposition, we have + 4. 1 **--4=- * 4x + b 7x- x + 3 35 4z + 5 Multiplying across, we have 3^ ■a,' + 3' 1^-~ + 2lx-35=l2x + 7x''+15+^. 137a: 12 a?= - 50; 600 137* Example 4. Solve ^'^ + ^11* = ?Z^ + ilT 7 a:-10^a;-6 x~7 x-£' Tliis equation might be solved by clearing of fractions, but the Transposing, ilZ^ _ ^z^ ^x-'J _x-4: .r-10 x-7 ;• 9 x-a' Simplifying each aide separately, we have (a; -8) (a; -7) -(a; -5) (a; -10) ^ (^:^7)(^j^M^-4)(a?- 9) (a;-10)(a;-7) (ar-9)(^^^6) "' . a.'^-15a; + 56-(a;2-15x + 50) _ a~»-13a; + 42-( a;2-13^36> (a;-10)(«-7) {x-9)(,o^^) ' . 6 6 Ix-I0)(x~7) {x-^)(x-%)' eq^a^l"°^' ^'""'^ *''^ numerators are equal, the denominators must be that is, {x-\Q){x-'!) = {x-^){x-% a2 - 17a; + 70=a:2 - 15x + 54 J .-. 16 = 2a:; t I. 5 « ii I if fl ; » I ^. !«i • 1 >|( .: « « ■»,4 M ■e ■ ^.' 180 ALGEBRA. [chap. I* i li i I i rfffl ■■* m The above equation may alao be solved very neatly by the follow- mg artifice. The equation may be written in the form (X-1QH2 (x-6)J^_(x^)J^ {^^ a;- 10 ''" x-d ~ x-1 "*" a:-9 ' whence we have '+5rVo+l+j!6='+j?7 + '+A' which gives Transposing, 1 1 • v' a;-10"^a;-6 x-1 x-% 1 1 1 1 x-\Q x-1 a;-9 a:-6' 3 3 •■• (a; - 10) (z- 7) "(a; -9) (a; -6)' and the solution may be completed as before. 5a;-64 a»-ll 4a;-55 tc-G Example 5. Solve a;- 13 a;-6 a;- 14 x-1 Wehave 5 + --\3-(2 + ^)=4 + --^-(n-^); 1 •• a?-13 a;-6 a;-14 x-1 The solution may now be completed as before, and we obtain a;=10. EXAMPLES XXin. a. 1. a; + 4 _ a; + 5 3a;-8~3a?-7' 2. 3. - 5. 7. 7 -5a; 11 -15a; 9. l+a; l+3a; 6a;+ 13 3a; + 5 _2a; 15 5a:-25~5" 3a;-l 4a;-2^1 2a;- 1 3x-l~6' X 4 , a;+2 a;+6" 3a; + l _ a?-2 3(a;-2)~a;-l" 3 (7 + 6a;) ^ 35 + 4a; *• 2 + 9a; " 9 + 2a;" a 6a; + 8 2a; + 38 "• 2x+l~a;+12 - a; + 25 _ 2a; + 75 a; -5 ~2a;- 15* in 6a; + 7 _ 1 5a;-5 ^"' 9a; + 6~12"*'l2a; + 8' 31. 33. [chap. he follow- we obtain XXIIl.] 2a- -5 11. 12. 13. 15. 16. 17. 18. 20. 22. 23. 25. 26. 27. 29. 30. 31. + x-S HARDER EQUATIONS. 4x-3 ]81 2x-15 iO - i 1 4(£+3)_8a; + 37 7a; -29 9 ~ 18 ~5a;-12' (2a?-l)(3.r + 8) 6a?(a; + 4) ~^-"- 14. 2:g + 5_2a;+l 5a;+3 5a;+2 =0. 5 n x + S x + 1 2x + 6 2x + 2* 7 _60 10^ 8 x-4~5x-^b~3x-12~F^' -1 30 3 5 4-2a;'^8(l-«)~2-a;"^2-2S' "^ 3 1 6a; + 4| a;+l '^~3a;+2 — r; 5 + 23 x+i' X x+1 x-2 x^^ x-7 x-9 _x-8 x-9 x-B x-7' 19. 21. 30 + 6a; 6Q + 8a; 48 •r+l ■*" a; + 3 -^* + ^+l- x + 5 x + 4' a;-6_^ .r-4 a;-1 5 'x-7 a;-5~a;-16* _ a;-13 a;-15 a;-9~a;-ll a;-15 a;-17' i£+3_^6_ a; + 2 x + 5 x + 3 a; + 9~a; + 5~a; + 8* 4«:J7 10^-13 _ 8a; -30 5a; -4 24. a; + 2 X ■ + X 7 an- 3 x-5 x-\-\' X 'x- -6 -+ +• a;-4 2a;-3 2a;-7 a;-l 5a^-8 6a--44 lOr-j a;-8 a;-2 x~7 x-\ "J^e' 2a- -3 -43; --6 28. a;-2 •3a;--4~^06^r^7' •083 (a;- '625)= -09 (a;- -59375). (2a- + 1 •5)(3a; - 2'25) = (2a; - M25)(3a;+ 1 -25) a--4 •0625 = 56. •3a;- 1 •6+l-2a? •5x--4 2a;- -1 32. •2 + a; -l-Sx' Qo {•3a?-2)('3a;-l) 1 f I ft li it l» « I ! ..; I «.» ,11 i€ 4 m m m m m m 9* B er r €■ c J 182 ALGEBRA. Literal Equations. [CHAP. 187. In the equations we have discussed hitherto the co- efficients have been numerical qua^ tities, but equations often involve literal coefficients. [Art. 6.] These are supposed to be known, and will appear in the solution. Examplel. Solve {x + a){x + b)-c{a + c) = {x-c){x + c) + ab. Multiplying out, we have x^ + ax + bx + ab - ac - c^^x"^ - C' + ab ; whence ax + bx=ac, {a + b)x=ac; etc x= Example 2. Solve a x-a a + b h _ a-b x-b~ x-c Simplifying the left side, we have a[x-b)-b{x- a) a-b ■c {x~a)(x-b) {a-b)x X- a-h Multiplying across, (x-a){x-b) x-c' X ^ I (x-a){x-b) x-c x'^-cx=x^-ax-bx + ab, ax-{-bx-cx=ab, {a + b-c)x=ab; ab x= a + b-c EXAMPLES XZni b. Solve the equations : 1, ax-2b = 5bx-Sa. 3. x^+a'^=(b-xf. 5. a{x-2) + 2x = 6 + a. 7. {a + :v){b + x) = x{x-c). 2x + 3a _ 2(Sx + 2a) x + a ~ 3a; + o 9. 2. a%x -a) + b^(x-b)= abx. 4. {x-a){x + b) = (x-a+bf. 6. vi^im -x)- mux = v?(n +x). 8. (a-b){x-a)=(a-c)(x-b). 2{x-b) _ 2x + b 3x-c ~3{x-c)' 10. XXIII.] 11. ^ 23. 24. [chap. xxni.] LITERAL EQUATIONS. 183 11. 1 i_l_i ' X b a X 13. -=c{a-h) + -. X X 15. x-a _x - h h-x~ a - x' -. 9« 3x 4/> 2x 14. -, r- • h a a ic x-a _ {x-b)''^ 17. j.(.-«)-(-i?;=|(.-|). 18. (a + h)x^-a(bx + a-) = bx {x - a) + ax{x- b). 19. b(a + x)-(a + x)(b-x) = x'^+—. 20. h{a-x)-^(b + xr' + abff^ + lJ=0. 21. x"' + a{2a-x)-^-^'=(^x-^^J + a^ 22. (2a; - a) ^x + ^) = 4.i- /^| - a; V 1 (a - 4a;) (2a + 3.r). -_ a;-a + 6,a;-?> 23. + a* a; - a ■ + ■ x-a x-2b x-b x-a-b' 25. ?> (a; + a) 2x + Sb-a _ 2(.c^ + ?w - h"-) Example 3, Solve ax + by=c .(1), a'x-\-b'y = c' (2), Tlie notation here first used '\^ one that the student will frequently meet with in the course of Iris reading. In the first equation we choose certain letters as the coefficients of x and y, and we choose eomspondi7tg letters u-ith accents to denote corresponding quantities in the second equation. There is no necessary cor -ntion between the values of a and a', and they are as different as u id b; but it is often convenient to use the same letter thus slightly v,,ried to mark some common meaning of such letters, and thereby assist the memory. Thus re, a' have a common property as being coefficients ot a; ; b,b' as being coefficients of y. Sometimes instead of accents letters are used with a suffix, such as «i. ^i, Oa ; &i. K K etc. Il .1 <> il II IS' I' » 4 2 i m m € 184 ALGEBRA. [chap. To return to the equations ax + hy = c (l), a'x + b'y=:c' (2). Multiply ( 1) by 6' and (2) by b. Thus ab'x + bh'!/ = b'c, a'bx + bb'y=:bc' ; by subtraction, {ab' -a'b)x= b'c - he' ; A 0:=^^ (3). ab -ab Aa previously explained in Art. 104, we might obtain y by sulisti- tuting this value of x in either of the equations (1) or (2) ; but y is more conveniently found by eliminating x, as follows : Multiplying (1) by a' and (2) by a, we have aa'x + a'by = a'c, aa'x + ab'y = ad ; by subtraction, (a'6 -ah')y= a'c - ac' ; _ a'c - ac' •' ^~dW^b'' or, changing signs in the terms of the denominator so as to have the same denominator as in (3), _ ac' - a'c , _ ^'c - be' y-ab'-a'b' ^""^ "^-ab'-a'b' Example^. Solve ^:i^ + ?L:^=i m c-a c-b ^ " x+a , y-a _a .^. c a-6 c From (1) by clearing of fractions, we have xic -b)-a(c-b) + y(c-a)-bic-a) = (c-a){c- h), x(c-b) + i/(c-a) = ac-ab + bc-ab + c'^-ac-bc + ab, x(c-b) + y{c-a) = c^-ab (3). Again, from (2), we have x(a-b) + a{a-b) + cy-ca=a{a-b) x{a~b) + cy = ac (4). Multiply (3) by c and (4) by c - a and subtract, x{c{c-b)-{c-a){a-b)}=c^-abc-ac{c-a), x(c^ -ac + a^ -ab)=c{c^-ab-ac-i-a'^) ; :. x—c; and therefore from (4) y=b. XXIII.] X 15. [chap. .(2). (3). ' by sulisti- ) ; but y is have ihe (1), (2). ), W (4). XXIII.] LITERAL EQUATIONS. EXAMPLES XXin. c. Solve the equations : 185 1. axJrhy=l, hx + ay = m. 4, ax + hy=a\ bx + ay=b-, '• ab~ab* X y_ \ a'~h'~a'b'' 10. qx-rb=p(a-y), 13. (a-b)x=(a + b)y, x+y=c. 2. lx + iny=ii, px + qy = r. 6. x + ay = a', ax+a'i/=l. ^' a ft-"' bx + ay^iab 11- S+i' = l' ^ 2/_i m m 3. ax=hy, bx + ay=c. 6. px-qy = r, rx-jiy=q. 9 ?^ + ?2^=3 9a; 6y_„ r — «*' a 12. 7)x + 7y=0, 14. (tt-t)x' + (a + 6)j/ = 2a2-262. (a + 6)a; -(a-6).v = 4a&. 15. a b 3a Qb 3* 16- -M=^. 17. M=i. a b a b' (* + y) (m2 + P) = 2 (m'» + P) + ml (x + y). 19. hx->rcy = a-\-b, ( \ 1 \ , (J. 1 \__2a "•^V^^'STft/ ^^\6-« b + a)~a^-b' 20. («-6)a; + (a + 6)y=2(a2-&2), ax'-62/=a2 + &'^. 21' K«-^+^; K"^'-^)' "■*-^='"'- I I t ,, ^ I)' I.* 4.: CHAPTER XXIV. Harder Problems. I; •9 € ka US lei r !'■ 188. In previous chapters we have given collections cf problems which lead to simple equations. ^We add here a few examples of somewhat greater difficulty. Example 1. A grocer buys 15 lbs. of figs and 28 lbs, of currant.s tor i,\. \s. %d. ; by selling the figs at a loss of 10 per cent., and the currants at a gam of .SO per cent., he clears 2s. M. on his outlav how much per pound did he pay for each ? Let X, y denote the number of pence in the price of a pound of tigs and currants respectively ; then the outlay is 15a; + 28y pence. Therefore l5x + 2Sy = 2m (i). The loss upon the figs is — x Ux pence, and the gain upon the currants is jy x 28y pence ; therefore the total gain is 42y .3a; -5- - 2" p®°<^<' ; . 42y ^x •■ -5~~2" = '^^ (2). From (1) and (2) we find that a;=8, and 2/ = 5 ; that is the fi.'s cost 8d. a pound, and the currants cost 5d. a pound. Example 2. At what time between 4 and 5 o'clock will the minute-hand of a watch be 13 minutes in advance of the hour-hand? Let X denote the required number of minutes after 4 o'clock ; then, as the minute-hand travels twelve times as fast as the hour- hand, the hour-hand will move over ~ minute divisions in x minutes. At 4 o'clock the minute-liand is 20 divisinna behind the liour-hand and finally the minute-hand is 13 divisions in advance ; therefore the minute hand moves over 20+13, or 33 divisions more than the hour-hand. CHAP. XXIV.] HARDER PROBLKMS. 187 ictioiis fif ere a few d currants ., and the is outlay ; i pound of (1). upon tlif s the tigs will the )ur-hand? r o'clock ; the hour- ' minutes. )ur-luuKl, therefore than the Hence Y^a;=33; '.-. 3. = 36. Thus the time is 36 minutes past 4. If the question be asked as follows : ' • At what timeH between 4 and 5 o'clock will there bo 13 minutes between the two hands?" we nuist also take into consideration the case when the minute-hand is 13 divisions behind the hour-hand. In this case the minute-hand gains 20-13, or 7 divisions. Hence which gives .=7^. r Therefore the times are 7tt pas* 4, and 36' past 4. Example 3. Two persons A and B start simultaneously from two places, c miles apart, and walk in the same direction. A travels at the rate of p miles an hour, and B at the rate of q miles ; how far will A have walked before he overtakes B ? Suppose A has walked x miles, then B has walked x-c miles. A walking at the rate of p miles an hour will travel x miles in - x-c - ^* X hours ; and B will travel x-c miles in being equal, we have x_x-c hours : these two times whence p q qx=px-pc; _ pc X- p-q Therefore A has travelled — — miles. p-q Example 4. A train travelled a certain distance at a uniform rate. Had the speed been 6 miles an hour more, the journey would have occupied 4 hours less ; and had the speed been 6 miles an hour less, the journey would have occupied 6 hours more. Find the distance. Let^ the speed of the train be x miles per hour, and let the time occupied be y hours ; then the distance traversed will be represented by xy miles. I m ii 188 m W i € w.. r ALGEBRA. [cnAP. ^ On the first supposition the speed per hour is a; + 6 miles, an« 204 ALGEBRA. [chap. Solve the following equations : 10. xy=-2\m, 11. a;-y=-18, 12. a;y = -]9l4. a;+y=-8. a:y =1363. a;+y=-C5, 13. a;2+y2=89, 14. x^+y'=\10, 15. a:2+2/2=65, «y =40. a;y =13. a:y =28. 16. a:=+2/2=l78, 17. a; +y =15, 18. a; -y =4, a;+y=16. a;2 + y2 = 125. x« + y2=i06. 19. ar2 + y2=i80, 20. a:'' + y2 = 185, 21. « +?/ =13, X -y =6. a; -y =3. x'^ + y'^=^'l. 22. a? +y =9, 23. x -y =3, 24. aP-xy +w2=76. a;2 + a;y + y2=61. a:2-3a;y+2/2= _i9. a; +J =14. 25. l(a:-,) = l. 26. i + l=2. 27. l + i=f,, a:2-4a?y + y2=52. a; + y=2, a:y=12, 28. aa; + 6y=2, 29. x^+pxy-^f'=p-{-2, abxy = l. qxi^ + xy + qy^=2q+l. 205. Any pair of equations of the form x^±pxi/+i/^=a^ (1)^ x±!/ = b (2), where p is any jiumerical quantity, can be reduced to one of the cases already considered ; for by squaring (2) and combining with (1), an equation to find .ry is obtained ; tha solution can then be completed by the aid of equation (2). Example I. Solve x!^-y^-9Q9 (])^ «-y = 3 (2), By division, x- + xy + y^=333 (3); from (2) x^-2xy + y'= 9 ; by subtraction, 3ary = 324, a:y^l08 (4). From (2) and (4) ^=^^' ^^ ~ ^'\ ' y= 9, or -12./ Example2. Solve x!^ + x''y''' + y*=2%\2 (1), x"-\-xy +y2_ gy (2), Dividing (1) by (2) x'i-xy+y'^= 39 (3). From (2) and (3) by addition, x'^+y'^= 53 : by subtraction, a?y= 14 • -^-- ll^if,'] [Art. 204, Ex. 1.] 11. X' 14. * l^it' [chap. xy = -19U, = -G5. xy = 65, = 28. Vy^ =4, = 106. = 13, = 97. xy X + 2/==76, +y =14, y 7 ~12' ay =12, 0), (2), d to Olio of 1 combining solution can (1), (2), (3); (4). (1), (2). (3). 204, Ex. 1.] xm.] SIMULTANEOUS QUADRATIC EQUATIONS. Example 3. Solve x^^y'^ 9 205 ..(1), ..(2). From ( 1 ) by squaring, -,, + -o = \; X' xy y^ 9 2 4 by subtraction, — =-; •^ xy ^ 12 1 adding to (2), + _ + = i . X- xy 2/2 .-. i+i=±i. X y 1 2 1 Combining with (1), ^=3' °" -3' 1 1 2 r3'°'-3' a:=3, or -3, and y=3, or -5. EXAMPLES XXVI. b. Solve the equations ; 1. ar' + j/5=407, X +y =11. 2. a; +2/ =13. 3. a; +2/ =23, a;3 + 2/3 = 3473, 4. ar'-2/»=218, X -y =2. 5. X -y =4, a:3_2/3 = 988. 6. a^'-2/»=2197, a; -y =13. 7. a;2 + a;2/ + y2_ = 2128 :76. > 8. a:2-a;2/ + j^=37. 9. x* + x^y^ + y*= x^~xy + y^ = = 9211 = 61. » 10. a;*+ar'2/^ + 2/4=7371, x^-xy + y'^ = 63. 11. 11 481 a:- 2/- ~ 576' 11 29 x^y~2^' 12. 11 61 a;2'*"2/2""900' a;y = 30. 13. ^ + ^=21, 2/ a; ^' a; + 2/=6. 11 a;-y=4. 15. .34 15 x2.i-J/2-^' x + y=S. 16. x^-y'=5e, x- + xy + y- = 28. 17. 4(^:2 + 2/2) = 17a:j/, x-y^G. 18. ar' + 2/'=126, x^-xy + y^=2l. '* 1. ■ ■''' ' 1 ^ il 1 H 1 1 1 i ii I m m » k r r r^ 206 ALGEBRA. [chap. Solve the equations : "• 3+^=1t*3. 20. ^-i,=91, aj y '^ a; y ^* 206. The following method of solution may always be used when the equations are of the same degree and homogeneous. [See Art. 24.] Example. Solve oi^ + xi/ + 2y'^=7i m 2x2 +2a:y+ 2/2=73 (2).' Put y=mx, and substitute in both equations. Thus x^(l+m+2m^) = 14: (3). and a;2(2+2m + w2)=73 (4)* By division, l+m+2m\ 74. 2+2w+m2~73' .-. 73 + 73»i + 146m2=148 + 148m+74w2; .-. 72m2- 75m -75=0, or 24m2-25wt-25=0; ^ .'. (8m + 5)(3m-5)=0; 5 5 •• "^="8'°' 3- (i) Take ni~-~, and substitute in either (3) or (4). From (3) oc^^i.^+^^ = U; . „_64x74 „. .-. x=±8; 5 .. i/=mx= -■^x=t5. o (ii) Take wt=|; then from (3) 74 ^' .-. x=±3; .-. y=wa;=sx=±5. XXVI.] whence whence r ' [chap. ^s be used neons. e Art. 24.] (1). (2). (3). (4). I XXVI.] SIMULTANEOUS QUADRATIC EQUATIONS. 207 207. When one of the equations is of the first degree and the other of a higher degree, we may from the simple equation find the value of one of the unknowns in terms of the other, and substitute in the second equation. Example. Solve From (1) we have 3x-4y=6 3x2-xy-3»^2^2I .(1). .(2). X _5+4y. and substituting in (2), M+M'_y(5+i^_3 2^21 ; .-. 75 + 120t/ + 48?/2 - I5y - 12^2 _ 27y2 = 189 ; 9y2 + i05y- 114 = 0, 3!/2 + 35y-38 = 0; ••• (y-l)(3y + 38)=0; , 38 .-. y=l, or -yj 137. and by substituting in (1), x=3, or - 9 208. The examples we have given will be sufficient as a general explanation of the methods to be employed ; but in some cases special artifices are necessary. Examplel. Solve x'+4xy + 3x=i0-6j/-4y^ (1)^ 2xy-x^=3 (2)' From ( 1 ) we ha^•e x^ + 4xy + 4j/2 + 3* + 6y = 40 ; that is, (a; + 2y)2 + 3(a; + 2y)-40=0, or (x + 2yhS)(xT2y-5)=0; whence a; + 2y=-8, or 6. (i) Combining a; + 2y = 5 with (2) we obtain 2a;2-5a; + 3 = 0; whence and by substituting in a? + 2y=5, (ii) Combining x + 2y=-R with (2) we obtain x=l, or 5; y = 2, or 1. 2a;2 + 8.r + 3 = 0: whence x = _-4±Vl0 and y=z — -\2tJ10 i 1^ I r € Pi- 1 r ! ^■•■■ k" 'C 'C r 1 If,-' [ 208 Example 2. Solve ALGEBRA. [chap. XXVI. From (1) from (2) by subtraction, 3xij + y=2(9 + x) , 9a;y-6x + 3y = 54; a;V-9a;y + 20 = 0, (a;t/-5)(a;y-4)-0; .•. xy = 5, or 4. (i) Substituting xy=5 in (2) gives y-2a; = 3. From these equations we obtain x--=] y- (ii) Substituting a;y = 4 in (2) gives y-2x = Q. From these equations we obtain x-= ' J^ - — ■d), .(2). • = 1. or -g.l ^=5, or -2.; and 17 1 y=3±V17 EXAMPLES XXVI. c. Solve the equations 1. 4. 7. 6a;-y=17, ay =12. 2. 5. 8. 11. x'^ + xy- y'^-Vxy-- 3a: -y: 3a;2~y2; 2xy-y'^-. .3.C2-2/2: 2A'--ajy= 3a; + 2y=16, a?y = 10. ar + 2y=9, 3y2-5a:2=43. 10. 3a;2-6i/2=28, 3a:!/-4y2=8, 13. x''-Zxy + y'' + \=(i, 3a;2-a;y + 32/2=i3. 15. a;2- 2x^ = 21, 16. x'^-\-Zxy = xy + y'^— 18. xy + 4y^= 18. x'-y^=:m, 19. a:3_y3^ a^V - ^y^ = 42. try (a; -y) = 21. a;2 + 4y2 + 80=15a: + 30y, 22. a:'y--6. :15, :10. 11, :47. ^5, 23 12. 54, 115. 208, 48. 3. 6. 9. 12. 14. 17. 20. x-y~ x^-2xy-3y--- X-^iXI- 5a; + y 1x''--Zxy-y- :10, :84. =1, :17. =3, a?2 + ajy + y"^ 2x^-3xy + 2y'^ 1X7/ - Hx"^ 8y^-9.vy x^ + y^ x-y + xy- x-y + oxy x + y 9a;2 + y2_G3a;-21y + 128 =n. = 10, = 18. = ]o2, = 120. = 84, = 8, =0, -4. \\hence or « I' OHAP. XXVI, .(1), .(2). a;-y=10, r-3!/ = l, f + 92^2 =17. 5x + ]/ = ^, :y + 1/ = ^. r- 8^2 = 10, -9xy=lS. ^5+2/3 = 152, ' + xy-=m-. + 5.17/ = 84, x + y=S, + 128 = 0, X- -4. CHAPTER XXVII. Problems leading to Quadratic Equations. 209. We shall now discuss some problems which give rise to quadratic equations. Example 1. A train travels 300 miles at a uniform rate ; if the rate had been 5 miles an hour more, the journey would have taken two hours less : find the rate of the train. Suppose the train travels at the rate of x miles per hour, then the time occupied is — hours. X 300 On the other supposition the time is — — - hours • .r + 5 ' wlience or 300^300 x + 5~ X .(1); a;2 + 5a; -750=0, (a; + 30) (a;- 25) =0, .'. a; = 25, or 30. Hence the train travels 25 miles per hour, the negative value benig niadmissible. It will frequently happen that the algebraical statement of the question leads to a result which does not apply to the actual problem we are discussmg. But sucli results can sometimes be explained bv a suitable modification of the conditions of the question. In the present case we may explain the negative solution as follows. Since the values a: = 25 and -30 satisfy the equation (1), if we write - X for x the resulting equation, 300 (2), r=^-2 - a; + 5 - a; will be satisfied by the values a;= -25 and 30 300 300 Now, l)y changing X — a r ""~~ r *^ X signs throughout, equation (2) becomes and this is the algebraical statement of thelfoUowing question . A train travels 300 miles at a uniform rate ; if the rate had been miles an hour less, the journey would have taken two hours more-. tiiKl the rate of the train. The rate is 30 miles an hour. E.A. Q « » :» 1 .; I i Is: , ^ s r i ^m € m ^H m m fer ^^^^B ' » ^^^^B fJt ^^H ■ i ^^^^^m li^ ^^B^B . cr? ^^H r ^^H r ^H^ ^ ■c ^B. ^ ir ^^^B ' ^^H ^^^^^H 1 ir!> ^^^^H I -", ^^^^H 1 ^^^^^H 1 ^^^^B e ^^^1 '" i |« 210 ALGEBRA. [chap. Example 2. A person selling a horse for £72 finds that, his loss Eer cent, is one-eighth of the number of pounds that he paid for tlie orse : what was the cost price ? Suppose that the cost price of the horse is x pounds ; then the .X loss on £100 is £g a? Hence the loss on £x is a: x ^, or ^ pounds ; X'' . the selling price is x - ^ pounds Hence «'-^ = 72, or that is, ar'- 800a; + 57600 = 0; (a;-80)(a;-720)=0; .-. a;=80, or720; and each of these values will be found to satisfy the conditions of the problem. Thus the cost is either £80, or £720. Example 3. A cistern can be filled by two pipes in 33^ niiiuites ; if the larger pipe takes 15 minutes less than the smaller to fill the cistern, find in what time it will be filled by each pipe singly. Suppose that the two pipes running singly would fill tlie cistern in X and a; -15 minutes. When running together they will fill [14.— ^ of the cistern in one minute. But they fill —r, or ~ V«^a;-15/ 33i 100 of the cistern in one minute. Hence X X 3 'lOO' -15 100(2a;-15) = 3.T(.r-15), 3.t2- 245a; +1500=0, (a; -75) (3a;- 20) = 0; .-. a; =75, or 6|. Thus the smaller pipe takes 75 minutes, the larger 60 minutes. The other solution 6f is inadmissible. Example 4. The small wheel of a bicycle (1885) makes 135 revolutions more than the large wheel in a distance of 2G0 yards ; ii the circumference of each were one foot more, the small wheel would make 27 revolutions more than the large wheel in a distance of 70 yards : find the circumference of each wheel. or or [chap. that hia loss paid for tlie (Is ; then the conditions of 33^ minutes ; [er to fill tlie singly. 11 the cistern they will fill „ 1 ._ 3 331' or 100 minutes. 5) makes lo') 2G0 yards ; if 11 wheel would distance of 70 XXVII,] PROBLEMS LEADING TO QUADRATIC EQUATIONS. 211 Suppose the small wheel to be x feet, and the large wheel y feefc in circumference. In a distance of 260 yards the two wheels make — and ^ revolutions respectively. Hence 780 780 = 135, or X y X y 52 .(1). Similarly from the second condition, we obtain 210 210 or From (1) whence x+\ y+l 1 1 = 27, 9 x+l y + 1 70 .(2). X: ■ 52y . ■52 + 9y' a.4.1_61y + 52 "^^^-9^^+52' Substituting in (2), ^^±^ -~-L * ^ '' 61y + 52 y+i-W 70x92/2=.9(61y + 52)(y + l), 92/2- ii3y_ 52 = 0, (y-13){9y + 4)=0; .-. y=U, or -| Putting 2/= 13 we find that a;=4. The other value of y ia inad- rcumf'eiencr ^'^"^^ '' * ^'''' '^" ^""'^^ ^^'^^ ^^ feet in Example 5. On a river there are two towns 24 miles apart. Bv rowmg one half of the distance and walking the other half, a man hofr' w f .r'^t^ ^^«^n «t^ea"^ in 5 hours, and up stream in 7 nours. Had there been no current, each journey would have taken the stream *^^ ''**'' ""^ ^^^ walking, and rowing, and the rate of i^uppo.^c that the man walks x miles \ . hour, rows v miles npr hour, an,l that the .trcam flows at the rate of = mi41i^i?hour * 3>) II mm' 1 » 11 il J» ■ m I 11 « 1 „ -1 II >c e ^^^^■. m ^^^H : te ^^^Hm' t% *'* ^H I r ■BE: t"-t, 1 ^'^ ^^B ? ■ ^H > i t ^^^B '. f 1 '« 212 ALGEBRA. [chap. Heuce we have the following equations : 12+J2-=5 „,, X y + z 12+J2.=, ,2,, X y-z 12 + 12^g2 (3) a; y -^ From (1) and (3) by subtraction, --ZTX~i8 ^^^" y if Similarly, from (2) and 3) _I--l = g (o). From (4) 18z=y(y + s) (6); and from (5) 9s;=y(y-2) (7). From (6) and (7) by division, 2=|^ ; whence y-3z; .-. from (4) z = \}v', and hence 2/ = 4|, 1=4. Thus the rates of walking and rowing are 4 miles and 4^ miles per hour respectively ; and the streain flows at the rate of 1 1 miles per hour. EXAMPLES XXVII. 1. Find a number whose square diminished by 119 is cfjiial to ten times the excess of the number over 8. 2. A man is five tunes as old as his son, and the sum of the squares of their ages is equal to 2106 : find their ages. 3. The sum of the reciprocals of two consecutive numbers is 1| : find them. OD 4. Find a number which when increased by 17 is equal to 00 times the reciprocal of the number. 5. Find two numbers whose sum is 9 times their difference, and the difference of whose squares is 81. 6. The sum of a numbesr and its square is nine times tlic next highest number : find it. 7. If a train travelled 5 miles an hour faster it would take one hour less to travel 210 miles : what time does it take ? ^'!l [chap. (1), (2), W. W (5). (6); (7). ind 4i miles e of 1 }, miles [) is equal to 2 sum of the 3 numbers is 5 equal to w> ifferencc, and mes the next 3uUl take one XXVII.] PROBLEMS LEADING TO QUADRATIC KijUATlONS. Jl3 8. Find two numbers the sum of whose sqi \h 74, and lose sum is 12. 9. The perimeter of a rectangular field is 500 yards, and its area is 14400 square yards : find the length of the sides. 10. The perimeter of one square exceeds that of another by 100 feet ; and the area of the larger square exceeds three times the area of the smaller by 325 square feet : find the length of their sides. 11. A cistern can be filled by two pipes running together in 22^ minutes ; the larger pipe would fill the cistern in 24 minutes less than the smaller one : find the time taken by each. 12. A man travels 108 miles, and finds that he could have made the jouiTiey in 4^ hours less had he travelled 2 miles an hour faster : at what rate did he travel ? 13. I buy a number of cricket Italia for £5 ; had they cost a shilling apiece less, I should have had five more for the money: find the cost of each, 14. A boy was sent out for a shilling's worth of eggs. He broke :i on his way home, and his master therefore had to pay at the rate of a penny more than the market price for 6. How many did the master get for a shilling ? 15. What are eggs a dozen when two more in a shilling's worth lowers the price a penny per dozen ? 16. A lawn 50 feet long and 34 feet broad has a path of uniform width round it ; if the area of the path is 540 square feet, find its width. 17. A hall can be paved with 200 square tiles of a certain size ; if each tile were one inch longer each way it would take 128 tiles : niul the length of each tile. 18. In the centre of a square garden is a square lawn ; outside this IS a gravel walk 4 feet wide, and then a flower border 6 feet wide. If the flower border and lawn together contain 721 square feet, find the area of the lawn. 19. By lowering the price of apples and selling them one penny \ ^^1" cheaper, an applcwoman finds that she can sell 60 more than she used to do for 5s. At what price per dozen did she sell them at first ? ti^'^'tJ^^*' rectangles contain the same area, 480 square yards. Ihe ditrerence of their lengths is 10 yards, and of their breadths 4 yards : find their sides. „ ^1- . There is a number between 10 and 100 ; when multiplied by the dicit on the left the product is 280 ; if the sum of the digits be niultiphed by the same digit the product is 66 : required the ml ii ii SI 4 I! ^ I 214 ALGEBRA. fcnAP. XXVII. 22. ^ A farmer having sold at I'm. a head, a flock of sheep which cost him X shillinga a head, finds that he has realised x per cent. profit on his outlay : find x. 23. A tradesman bought a number of yards of cloth for £5 ; he kept 6 yards and Hold the rest at 28. per yard more than he gave, and got £1 more than ho originally spent : how many yards did he buy? 24. If a carriage wheel H^ ft. in circumference takes one second more to revolve, the rate of the carriage per hour will be 2^ miles less : how fast is the carriage travelling ? 25. A broker bought as many railway shares as cost him £187') ; he reserved 16, and sold the remainder for £1740, gaining £4 a share on their cost price. How many shares did he buy ? 26. A and B are two stations 300 miles apart. Two trains start simultaneously from A and B, each to the" opposite station. The train from A reaches B nine hours, the train from B reaches A four hours after they meet : find the rate at which each train travels, 27. A train A starts to go from P to Q, two stations 240 miles apart, and travels uniformly. An hour later another train B starts from P, and after travelling for 2 hours, comes to a point that A had pa8se(^45 minutes previously. The pace of B is now increased by 5 milfes an hour, and it overtakes A just on entering Q. Find the rates at which they started. 28. A cask P is filled with 50 gallons of water, and a cask Q with 40 gallons of brandy ; x gallons are drawn from each cask, mixed and replaced ; and the same operation is repeated. Find x wlien there are 8|- gallons of brandy in P after the second replacement. 29. Two farmers A and B have 30 cows between them ; they sell at different prices, but each receives the same sum. If A had sold his at B'a price, he would have received £320 ; and if B had sold his at A' a price, he would have received £245. How many had each ? 30. A man arrives at the railway station nearest to his house ij hours before the time at which he had ordered his carriage to meet him. He sets out at once to walk at the rate of 4 miles an hour, and, meeting his carriage when it had travelled 8 miles, reaches home exactly 1 hour earlier than he had originally expected. How far is his house from the station, and at what rate was his carriage driven ? HAP. XXVIl. heep which X per cent. for £5 ; he in he giive, ircU dill he one second 36 2§ miles tiim £1H7.'»; ; £4 a yiiare trains start ition. The shea A four travels. 3 240 miles in B starts >int that A IV increased g Q. Find sask Q with lask, mixed nd X wiien acement. i; they sell (4 had sold lad sold his had each ? his house carriage to 4 miles an id 8 miles, y expected, bte was his CHArTER XXVIII. Haudek Factors. 210. Iv Chapter xvii. we have explained several rules for resolving algebraieal exjjre.ssioiirt into faetorH ; in the present chdpter we shall continue the subject by discussing cases of greater difficulty. 211. By a slight modification some exj)ressions admit of being written in the form of the difference of two s(juares, and may then be resolved into factors by the method of Art. 133. Example!. Resolve into factors a;* + x-^- + y*. x* + xY + y* = {x* + 2xY + y*)- xhf = {x'^+y^r-(xyf = {x'^ + y^ + xy) {x"^ + y'^- xy) = (x^+xy + y^)(x'^-xy + y'^). Example 2. Resolve into factors ar*- ISar-y^ + O,!/*. ar* - \oxY- + 9y* = (x* - 6xY + ^y*) - ^^Y = (a:2-3yT-(3ary)2 = (a;2 - 3y2 + 3.ry) (a;2 - 3y2 - 3xy). 212. Expressions which can be put into the form o.'^i-g may be separated into factors by the rules for resolving the sum or the difference of two cubes. [Art. 136.] Example 1. ^ - 276« =(-Y- (36*)' 8a2 %^ a 8 )• Example 2. Resolve a?x^ -~-s(? + ^ into four factors. = (a'i-\\fx3- I' = (a.l)(a-l)(.J)(.H|V^). 216 ALGEBRA. [ClIAK r c we m fer e^ : r- Examiile 3. Resolve a" - 64a3 - ai + 64 into six factors. The expression = a^ia'^ - 64) - (a" - 64) = (a«-64)(a3-l) = (a3 + 8)(a3_8)(a3-l) = (a + 2)(a2_2a + 4)(a-2)(a2 + 2a + 4)(a-l)(a2 + « + l). Example 4. a(a-l)a;2-(a-&-l)xy-6(6 + l).v2. .{aa;-(6 + l)y}{(a-i)a; + ^,y). , . *^°*^' , /" examples of this kind the coefficients of x and y in thu binomial factors can usually be guessed at once, and it only remains to verify the coefficient of the middle term. 213. From Example 2, Art. 53, we see that the quotient of a^+b^+(^-3abc by a + b + c is a^+b^+c'^-bc-ca-ah. Thus a^+b^+c^-3abc=(a + b + c)(a^+b'^ + c'^-bc-ca-ah)...(l). This result is important and should be airefully remembered We may note that the expression on the left consists of the sunl ot the cubes of three quantities a, b, c, diminished by 3 times the product abc. Whenever an expression admits of a similar arrangement, the above formula will enable us to resolve it into factors. Example 1 . Resolve into factors a^-b^ + c^ + 3abc. a^-b^ + c^ + 3abc=a^ + {-b)3 + c^-3a{-b)c = (a-b + c){a^ + b'^ + c'^ + bc-ca + ab), - b taking the place of b in formula (1). Example 2. iB3-8y3-27-18a;y = ar' + (-22/)3 + (-3)3-3.c(-2y)(-3) = {x-2i/-3){x'^ + 4if + 9-6y + 3x + 2xy). IL EXAMPLES XXVIII, a. Eesolve into factors : 1. .r4+ 16^:2 + 256. 2. SlaUda^-^ + b*. 3. «* + y*-7a;V- 4. m* + n*-lSm-7i". 5. X*-ex^^ + y*. 6. ix* + 9y*~93xY 7. 4»r+9/i^-24mV'. 8. 9x* + 4y* + lhvY- 9. x*-19xY+25y*. 10. 16a* + b*-28a^'\ 43. 'A 47. [chap. rs. I){fi2+a + l). i-l)x + bij}. ' and ij in the only remains quotient of - ca - ah. a-ah)...{\). eniembered. 3 of the sum by 3 tinie.s of a similar D resolve it ca-Vab), l(-3) 'ixVlxy). XXVIII.] 27 11 -—-\ ^** 729 ~ HARDER FACTORS. 217 12. 216a3--|'. 15. J25 + 1000- 13. — + 1^. Resolve into two or more factors : 17. .cV + Sajy^-SarJ-yS. js. 4»j?t2 - 20n» + 45n»i2 - 9m3. 19, «6(a:2+l)-f-a:(a2 + 62), gO. y232(,^_ 1)^^,2(2^ .^ij. 21. a-' + (a + 6)aa; + 6a;2. 22. 2m{m'^+l)-7n{p^ + n% 23. 6&x(a2+l)-a(4a;2-f-962). 24. (2a2 + 3/)a; + (2a;2 + 3a2)y. 25. (2.r2-3a2)y + (2a2_3y2)_^.^ 26. a(a-l)a;2 + (2a2_l)a: + a(a + l). 27, ^x--(4a + 2b)x + a^ + 2ah. 28, 2a2x-2-2(36-4c)(6-c)i/2 + a&a;y. 29. (a2-3a + 2):c2 + (2a2-4a + l)a; + o(a-l). 30, a(a + l)x- + {a + b)x7/-h{b-l)y'''. P + c^-l + 3bc. 32. a^ + 8c3+l-6ac. a^ + b^ + 8c^-6abc. 34. a^ - 27?'=' + c» + 9a?)c, «'-^'-c^-3a6c. 36. 8a'' + 2W + c^-18abc. Resolve a;8 + 81a;^ + 6561 into three factors. Resolve (a" - 2^262 - 6^)2 - ia^b^ into four factors. Resolve i(ab + cdf - [a? + W-c^- d^f into four factors. 31. 33. 35. 37. 38. 39, 40, Resolve 3? - ^ into four factors. 41. Resolve x^^ - y^^ into five factors. 42. Resolve x^^ - y^^ into six factors. Resolve ijito four factors : 43, 45. 47 w -^,-8x-a^ + 8jfi. x^ + yf' + Ux^ + M. xy'' x^y^ 1 V'J 32 9^2+4- 44. 46. 48. a,-9 + x3y«-8^y3-8/, 4a- 4/v'i 0^:1 62 Resolve into five factors 49. ^^ + ar»-16x3-16. Si if 11 s ! I' ' 50. 16.c7-81.r'-16ar' + 81. il 218 ALGEBRA. [chap. ^^^^M im ^^H r ^^^^^H Kt' ^^^^^H EW ^^H[ ■ r ^^H c ^H ■c ^^^B ' ^■^^^^^H ' r ^^^^^^^B ' WE 4^ ^^^^H '\ I ' - ^^^m : ^^^H' - ^^^^^^B ^^1^1 1 !■ 214. The actual processes of multiplication and division can often be partially or wholly avoided by a skilful use of factors. It should be observed that the formulae which the student has seen exemplified in the preceding pages are just as useful in their converse as in their direct application. Thus the formula for resolving into factors the difference ef two squares is equally useful as enabling us to write down at once the product of the sum and the difference of two quantities. Example 1. Multiply 2a + 3b-c hy 2a-3b + c. These expressions may be arranged thus : 2a + (3b-c) and 2a-(3b-c). Hence the product ={ 2a + (36 - c)}{ 2a - (36 - c)} = (2a)'-'-(36-c)2 [Art. 133.] = 4a2-(962-66c + c2) = 4a2-962 + 66c-c2. Example2. Multiply (a^ + a + l)a;-a- 1 by {a-\)x-a?-Va-\. Theproduct = {(a2 + rt + l)a:-(a + l)}{(a-l)a;-(a2-a+l)} = (a3-l)a:2-{(a4 + a2.fl) + (a2-l)}a; + (a3+l) =={a?-\)x^-{a* + 2a'^)x + a^ + \ = {a?-l)x'^-a\a^ + 2)x + a^+\. Note. The product of a- + a + \ and a^-a + l is a^ + a^+l and should be written down without actual multiplication. Examples. Multiply {3 + x-2x^)'^-{3~x + 2x'^f (1), by (3 + x + 2x^f-{3-x-2xY (2). The expression (1) = (3 + x-2z^ + 3-x + 2x'^)(3 + x-2x'^-3rX-2x'^) = 6{2x-4x'^) = 12a;(l-2a;). The expression (2) = {3+x+2x^ + 3-x-2x'^)(3 + x + 2x'^-S+x + 2x'^) = 6 (2a; + 4x2) = 12a;(l + 2a;). Therefore the product^ 12x(l - 2x) x 12a;(l +2a;) = 144x8(1 -4«2). [chap. XXVIII.] HARDER FACTORS. 219 iivisiou tan of factors. bhe student as useful in ;he formula ?a is equal In- duct of the [Art. 133.] -a- + a-l. + 1)} + 1) + a2+l and (1), (2). x^) Example 4. Divide the product of 2.x2 + a;-6, and dx'^-dx+l by 3x2 + 5a;-2. Denoting the division by means of a fraction, the required quotient _ (2x' + x-6)(6x^-5x+l) 3a;2 + 5a;-2 ^ (2x -S){x + 2)(Sx - \)(2x -I) (3x-l)(x + 2) = {2x-3){2x-l). Example 5. Shew that (2x + 3i/-zf + {3x + 7y + zf is divisible by 5(x + 2y). The given expression is of the form A^ + B^, and therefore has a divisor of the form A + B. Therefore is divisible by that is, by or by {2x + 3y-zf + (3x + 7y + z)^ (2x + 3y-z) + (Sx+7y + z), 5x+l0y, 5{x + 2y). Example 6. Find the quotient when a=^ + 8 - 56 (256^ - 6a) is divided by a - 56 + 2. The expression = a^ + 8 - 1 256^ + 30a6 =a3 + (-56)3 + (2)3-3.a(-66){2) = (a-56 + 2)(a2 + 2562 + 4 + 106-2a + 5a6). .'. the quotient is a^ + 256^ + 4 + 106 - 2a + 5a6. Example 7. If a; + y = a, and x-y = b shew that 4 (o^ - 6xY + y*) = 6a262 -a*- b\ = (a;2- 2/2)2 _l(4^y)2 [Art. 213.] =^{{x+y)(x-y)f--A{x + yf-(x-yff = (a6)2-l(a2-6T; A 4(.'C*-6xV + 2/<) = 4a262-(a2-62)2 =6a262-a*-6*. ii I! ■.; I ,» ! r / m Iflf 1)'* 1^ f tin I c SB if' } 220 ALGEBRA. EXAMPLES XXVIII. b. Find the product of 1. 2a;-7y + 3a and 2x + 7i/-3z. 2. 3.r2 - ^xy + 72/2 and Sx"^ + 4xy + 7y\ 3. 5x' + 5xy-9if and 5x'^-5xy-9y'^. 4. 7a;2 - 8.ry + 3y2 and 7x2 + g^.^ _ 3^2. 5. ar' + 2a;V + 2a;«/2 + y3 ^nd x"'-2x^y + 2xy^-y^. 6. (a; + 2/)2 + 2(a; + 2/) + 4 and {x + y)"--2(x + y) + 4. 7. (l+a; + 2a;2)2-(l-a;-2a;2)2 and (1 +a;- 2x^)2 _ (j _a,^Oc2)2. 8. (a2 + 3a-l)2-(a2-3a-l)2 and (a2 + a + l)2-(a2-a + l)2. ar'-4a;2 + 8x'-8 and a^ + 4x- + 8x + 8. x^-6ax-+l8a'^x-2'Ja^ and .r' + 6a;c2+i8a2_^ + 27a3. [CJIAP. x2 a2 , a;2 a2 x~a and x + a-i . ax ax 9. 10. 11. 12. (2a;2 + 3x + 1 )2 - (2a;2 _ 3^; _ 1 )2 ^nd (x^ + 6x- 2f - (x^ - 6a; + 2)= Find the continued product of 13. x^ + ar + a"^, x'-ax-\-a\ a^-a?x''- + a\ 14. I-X + X"^, \-\-X + X', l-X^ + x\ \-X^ + 3(». 15. {a-xf, {a + xf, (a2 + x-2f. 16. {\-xf, (l+a;)2, (l + a:2)2, (l + x^- 17. a:2 + 4a; + 3, a;2 + a;-2, x2-5a; + 6. 18. a;2 + 2a;-3, x^-5x + %, a:2 + ,3x + 2. 19. x + 2, a;2 + 2ic + 4, x-2, ic2-2;c + 4. 20. Multiply the square of a + Zl) by a"-^h + Oft^. 21. Multiply ^ (a -?>)2 + i(fc-f)2 + 1 (c-a)2 by a + 6 + c. 22. Divide (4a; + 3y - 22)2 - (3a; - 2y + 32)2 by x + oy- 5s. 23. Di vide a;8 + 1 ^*x^ + 256a8 by a;2 + 2aa; + 4a2. 24. Divide (3a; + 4y - 2z)2 - (2a; + 3y - 42)2 by .r + y + 2z. 25. Divide the product of a;2 + 73;+10 and a; + 3 by x^ + 5x + (S. 26. Divide 2a; (a;2 _ 1 ) (a; + 2) by a;2 + a; - 2. 27. Divide 5» (a - 1 1 ) (x^ - « - 156) by ar* + a;^ - 132a;. [CJIAP. XXVIII.] HARDER FACTORS. 221 28, 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. Divide a,-«+19ar»-216 by (x!^-3x + 9){x-2). Divide (5x^-3x-6f-(2x'^-lx + 9)'^ by the product of 3.1--5 and x + 3. Divide a^-h^ by the product of a' + ah + b"^ and a^' + aW + lA Divide (x^-3xh/T--(3xy''-y'f by (x-y)^ Divide (x^ - yzf + Sy^r' by x^ + yz. Divide \Sxy + l+ 27ar' - 8y^ by 1 + .3a: - 2i/. Divide {2x'^ + 3x-l)^-{x^ + 'ix + 5)'^ by the product of 3a: + 4 and x + 2. Divide the product of 6a2-23a + 20 and 22a2-81a + 14 by 33a2-50a + 8. Divide the product of x^ + (a-b)x-ab and x^-(a-b)x-ah by x'^ + (a + b)x + ab. Divide a^-8y^-9x(3x^ + 2ay) by a-3a:-2y. Divide 27 - Sx^ - Qiy^ - 12xy by 3 - 2 (a: -» 2y). Shew that (2a:-3y+ 1)3-(1 -3a; + 2?/)3 jg divisible by 5(a:-y). Shew that the square of a: + l exactly divides {a? + x'^ + ^f-(a?-2x + 3f. Shew that 2b + 2(1 is a factor of the expression {a + b + c-vdf-{a-b^-c-(lf. Shew that (3a;2-7a; + 2)'-(ar»-8a; + 8f is divisible by 20^-3 and by x + 2. Shew that {7a;2 + 3x--3)3 + (5.r2-4a:-3)3 is divisible by 4a- -3 and by 3j: + 2. Shew that the sum of the cubes of 2.r2 - 5a; - 9 and a;^ + Ga- - 5 is divisible by the product of 3a: + 7 and a: - 2. If x + y—m and x-y = n, express a? + y^ in terms of m and n. Iix + y = 7n and x-y = n, shew that 16 {x* - Ixh/ + y*) = (5to2 - n^) (5?i2 - m% Find the value of a;* + a:V + 2/* when a: + y=2a, x-y = 2b. lix-\-y = 2a and x-y = 2b prove that X* - 23xhf + y*=(7a^- 3b''') [W- - 3a?). Find the value of x^-Alxhfvy* in terms of ;> and q v.hen il ■II Ifc « ^1 •*:!t x + y=p and x-y = q. 50. Find the value of a"* - 2a-''i/ + 2a,y - ?/* when x=a + /» and y = a-b. ' . r i M € « ■h «» » » g ■U CTf ■ r r 'C r «te ■M. iti. t «."■■• ii CHAPTER XXIX. Miscellaneous Theorems and Examples. 215. Examples upon the simple rules, e.g. Division, Hif^hest Common Factor, Evolution, etc., frequently occur whicli cannot be neatly and concisely worked without a ready use of factors and compound expressions. These we have hitherto excluded as unsuitable for the student until he has gained confidence and power by practice. We propose in the present chapter to briii + {a^ + ab + b-)x'^ + {a^ + b^)x + a^b" by x- + ax + b-. 8. 2Px!^-2(3m-'in){m-n)y- + lmxy by lx + 2{m-7i)7/. 9. (a^ + a-2)x^-(2a + l)xy-{a^ + a)y^ hy (a-l)x-ay. 10. x> - (a-b -2)x- - (ab + 2a-2b)x -2ab by {x-a)(x + 2). 11. (x + l)^ + i(x + l)^ + 6(x + \)* + 4{x+lf+l by a;2 + 2a; + 2. 12. (m+l)(bx + an)b^x^-{n + l)[mbx + a)a^ by bx-a. Find the H.C.F. of 13. (m^-3m + 2)x^ + {2m^-im + l)x + m{m-l) and m(m-l)x- + {2m'-l)x + m(m + l). 14. mpx^ + (mq-np)x^-{mr + nq)x + nr and maa^ - {mc + na) x^ - {mb - 7ic) x + nb. 15. 2ap^ + {3a - 2b) p^q + {a- Zb)pq^ - bq^ and 3ap^ - (a + 3b)p'^q + (2a + b)pq^ - 2bq\ 16. acx'^ + {bc + ad)x--\-(bd-\-ac)xJrbc and 2aca;3 + (26c -ad)x'^- (3ac + bd)x- 3bc. ■»i.i, ii J.) \» J* < i f (■• / h 11 224 ALGEBRA. Hi. f m ■ r I:. ^^^^^B ^H'' ^ :' ^1 ^ Bi m ^^^^a rF^ iS [CHAr. Find the H.C.F. of 17. 2a^r^-(4b + 3)ax^' + 2(3h-ac)x + Sc(n\d 2a2^ + (2/> - 3) aa;2 - (4ao + 3i) a: + 6c. 18. 2ax^ + (4a2 - 1 ) hx'^ - (-lah^ + 3<')x - 6ahr. and ax3 - (3 - 2a2) hx"^ + (2c - Qab^) x + 4ahc. Find the L.C.M. of 19. ai^-px^ + {q-l)x^+px-q and X*-qx' + {p-l)x'^ + qx-p. 20. ^{io+l)a;2 + a:-/>(;>-l) and jo(^ + 2)a;2 + 2a;-j92 + i. 21. {a^-5a + 6)x'^ + 2{a-l)x-a(a + \) and a(a-3)x^+12x-{a + l)(a + 4). 217. We add some miscellaneous questions in Evolution. The fourth root of an expression is obtained by extracting tlu- sqnare root of the square root of the expression. Similarly by successive applications of the rule for findiiiir the square root, we may find the eighth, sixteenth ...root TIr' sivth root of an expression is found by taking the cube root of the square root, or the square root of the cube root. Similarly by combining the two processes for extraction of cube and square roots, certain other liigher roots may be obtained. Example 1. Find the fourth root of 8-ar» - 216.r3y + 2l6xY- - ^xy^ + I6i/. Extracting the square root by the rule we obtain 9x'^-12xy + 4i/"; and by inspection, the square root of this is 3x - 2y, which is the required fourth root. Example 2. Find the sixth root of (.4y_a(.-i)(..4,).„(..-iy. By inspection, the square root of this is (-4)-K-^)' which may be written a^-3x + 5; a-' x^ 1 X--, X and the cube root of this is which is the required sixth root. 218. In Chap. vi. we have given examples of inexact division. In a similar manner when an expression is not an exact square [chap, qx-p. jlutioii. [•acting tlit- 'oi' finding root. The be root of :racti()n of s may be 12xy + 4y"', t division. ,ct square xxrx.] MISCELLANEOUS THEOREMS AND EXAMPLES. 225 or cube, we may perform tlie process of evolution, and obtain aa many terms of the root as we please. Example. To find four terms of the square ror t of 1 + 2a; - 2x'^. l+2x-2x2(l+x-|a;2+|;c3 2+x 2x-a»2 2x+ a;2 2+2x-lx^ -3x2 -3x2- ■3ar» + jx* 4 2 + ac-3a;2 + |ar» 4 3.v^ + 3x*-^c^ + ^:^ -^x* + ^x^-^i figures at most ; also a is a number of 2n + \ figures (the last n of which are ciphers) and thus 2a contains 2/i + l figures at least; ami therefore ^ is a proper fraction. From the above investigation, by putting 7i=l, we see that two at least of the figures of a square root must have been ob- tained in order that the method of division, which is employed to obtain the next figure of the square root, may give that figure correctly. Example. Find the square root of 290 to five places of decimals. ^96(17-02 1 27 3402 190 189 10000 6804 3196 Here we have obtained four figures in the square root hy the ordinary method. Three more may be obtained by division only, using 2 X 1703, that is 3404, for divisor, and 3196 as remainder. Tlius 3404)31960(938 30636 13240 10212 30280 27232 3048 And therefore to five places of decimals ^290 = 17 "02938. When the divisor consists of several digits, the method of con- tracted division may be employed with advantage. Again, it may be noticed that in obtaining the second figure of the root, the division of 190 by 20 gives 9 for the next figure ; this i.s too great, and the figure 7 has to be obtained tentatively. Tliis is one of the modifications of the algebraical rule to which we referred in Art. 124. X' ,}■' 1. 2. 3. 4. 5. m^ [chap. clividod bv isetl by '-— r neglecti:!;^ , the rest of i 2« figures b n of wliich least ; and we see that 7e been ob- Huploywlto that figure of decimals. root by the ivision only, iiuler. Thus XXIX.] MISCELLANEOUS THEOREMS AND EXAMPLES. 227 *22l. If the cube root of a number consists of 2n + 2 fimires n'hrn the lust ii + 2 of time have been obtained L the onUnaru method, the remaining n may be obtained h) division. Lot # denote the given number ; a the* part of the cube root already found, that is the first ;i + 2 figures found by the coninu,n rule, with n ciphers annexed ; x the remaining part of the root. Then ^N=a->rx\ N-a^ 3a2 •=A' + X'' a r3 0) Now N-a^ is the remainder after m + 2 figures of the root represents by a have been found ; and 3a-' is the divisor at the same stage of the work. We see from (1) that .V-a^>lividei by 3a^ gives .r, the rest of the quotient required, increase,! by - + ~ We shall shew tliat th is expression is a p'oper fraction, 80 that by neglecting the remainder arising from the division we obtain .r, the rest of the root. division, By supposition, x is < 10", and a h > io-"+' ; rt i«<10^ai; thatis, <-; and hence 3fr -, is < 10^ 3x10"+*' a^ 1 a'^3a-'"^^10"^3"xl0"+i' + and i.s therefore a proper fraction. . n lA" > I I3S. thod of con- )nd figure of giire ; this is jly. This is I we referred EXAMPLES XXIX. b. Find the fourth roots of the following expressions : 1. •'•^- 28.^3 + 294.1-2- 1372x + 2401. 2. lG-^ + 2:|-A + J_. m m^ m'^ m* 3. a* + Sa^v + 1 6x* + S2ax^ + 2ia'^x'^. 4. l+4x' + 2a:2_8ar»-5a.'^ + 8.r' + 2.t-8-4.r7 + a;S. 5. 1 + 8x + 20*2 + 8.r' - 26x4 - 8a?= + 20;c0 - Sx^ + x\ 228 ALGEBRA. [chap 'S w i M M Find the sixth roots of the following expressions : 7. afi-l 2ax» + 2400*0^2 - 1 92a''x + 60a V - 1 60aV + 64a8. 8. a« - ISa'^x + \35a*x' - 5i0a^x* + 1215aV - 1468ax-» + 729a:*. Find the eighth roots of the following expressions : 9. a* - Sx'^y + 2Safiy- - 56x-y + lOxy - 56x-y + 28xV _ Sxy^ + y>i 10. {x* + 2{p-l)xr^ + (p'^-2p-l)x^-2{p-\)x+l)*. Find to four terms the square root of 11. l+x. 12. 1-ar. 13. 4 + 2a:. 14. 15. a^^-x. 16. ar^+a^. 17. a*-3x\ 18. Find to three terms the cube root of 1 1 - .r - X'. 9a2+12aa-. 19. ap-a\ 22. l-6« + 21x2. 20. 8 + a?. 21. 23. 27a:«-27x*-18x<. 24, :3+9a:. 64-48j; + 9x» a' 1- If., Identities and Transformations. *222. Definition. An identity is an algebraical statement which is true for all values of the letters involved in it. Examples. a^ + b^={a + h){a^-ab + b^). 3i^+y^+z^-3xyz=(x + y+z)(x'^+y^+z^-yz-zx-xi/). *223. An identity asserts that two expressions are alwavs equal ; and the proof of this equality is called " provint,' tlie identity." The method of procedure is to choose one of the expressions given, and to shew by successive transformatiuiis that it can be made to assume the form of the other. Example 1. To prove that he (b - c) + ca{c - a) + aJ) (a - h)= -(b-c){c-a)(a-b). The first side = bc(b-c) + c^a - ca^ + a% - ab"^ = 6c(6-c) + a2(&-c)-a(&2-c2) = (b--c){hc + a'^-a(b + c)} z={b' c) {Uc + a'^-ab- ac} z={b-c){a(a-b)-c(a-b)} = {b' c){a-b){a-c) = - (6 - c) (c -a)(a- b), changing the signs of tlie factor a-c, so as to preserve cyclic order. [Compare Art. 229, Example 3.] or [chap XXIX.] IDENTITIES AND TRANSFORMATIONS. ,8_ 729a:«. I : 9a2+12ax-. + 9ar. -48x + 9a;''. al statement lit. \ are always proving the one of the isformatioiis ,-&). The expression on the left-hand sitlc can be readily put in the following forms : Hence we have the following resuUa : 6c(6-c) + ca(c-a) + a6(a-6)=-(/i-c)(c-a)(tt-6)j a2(6-r) + i2{c-a) + c'(a-fc)=-(6-c)(c-a)(a-ft); a [}? - cy + 6 (c2 - a2) + c (a3 _ b^) = (/, .. c) (c -a){a- b). These identities are of such frequent occurrence that they should be carefully noticed and remembered. Example 2. If 2s=a + 6 + c prove that 1,1,1 1^ ahc 8-a s-h a-c~s ii(.'i-a)(n-b)(>i-c)' The first 8ide= (~ h- -i.^ + M _ 1\ \>i-a s-bj \8-c a J _ s-b + a-a , «-» + c ~{3-a){s-b) 8(s^ — •^~ f* - '' c ~(«-o)(s-6) 8{a~-c) = ? + _J1_ {«-a)(s-6)^«(s-c) _ f a(8-o) + (a-CT) (a- />)■» \ s(s-a){a-b)(8-cY) _ c{./-s)U(z-uf=0. t K 1 Jl ^. t-n- 230 ALGEBRA. [CHAP. en- r r r i Now since the square of any quantity is always positive, each of the expressions {x - yf, {y - zf, (z - u)"^ is positive. Hence their sum cannot be zero unless each of them be separately equal to zero. .'. x-y=0, y-z=0, z-u=0; or x=y=z=u. Note. The student should be careful to notice the difference between the conclusions to be drawn from the two statements (a;-a)a+(y-6)2=0 (1), and {x-a){y-h)=0 (2) From (1) we infer that both x-a=0 and y-b=0 simtdtaneously, while from (2) we infer that either x-a=0 or y-b=0. "EXAMPLES XXIX. c. Prove the following identities : 1. b{x^+a^) + ax{x^-a^) + a\x+a)=(a + b){x+a){x^-ax-^a'^). 2. {ax+byf + {ay- bxf + c^x^ + c V = (a-^ + y^) (a^ + &2 + c^). 3. [x+yf+Z{x+yfz-{-Z{x+y)z'^+z^ = (a; + 2)3 + 3 (a; + z)^ + 3 (a; + 2)2/2 + ys. 4. (a+6 + c)(a6 + 6c + ca)-a6c = (a + &)(6 + c)(c+a). 5. (a+6+c)2-a(6 + c-a)-6(a + c-&)-c(a + 6-c)=2(a2 + 62+c2). 6. {x-yf+{x + yf + 'i{x-yf[x-{-y)^Z{x+ynx-y)=^x\ 7. x^y-z)-vy\z-x)+z\x-y) + {y-z){z-x)[x-y)=(i. 8. a3(6-c) + 63(c-a) + c3{a-&)=-(6-c)(c-a)(a-6)(a + 6 + c) 9. If a? + y+z = 0, prove that y? + y^ + z^=Zxyz. 10. Prove that (&-c)3 + (c-a)3 + (a-6)=»=3{6-c)(c-a){a-6). If 2»=a+6+c, shew that 11. (s-a)''+(s-t)= + (s-c)2 + ,<(2=o2 + 62+ca. 12. {a-af-\-[s-bf+{s-cf + ^bc=s^. 13. 16s(«-a)(a-6)(8-c)=26V+2c2a2+2a''6a-o«-64-c«. 14. 2(.9-a)(8-6){s-c)+a(s-6)(s-c) + 6(s-c)(«-o) + c(s-a)(s-6)=a6c. If a+6+c=0, shew that 15. (2a-6)3 + (26-c)3 + (2c-a)''=3(2a-&)(2&-c)(2c-a). Ifi. a' 62 = 1. 2tt- + &o ■ 262 + ca 2o- + a6' 17. Prove that {a;+y+2)3+{a;+y~s)3+(a;-2/ + a)3 + (a;-y-2)3=4u;(x-2+3/+3;2). [CHAP. I'lve, each of 36 their sum o zero. e difference ments (1), (2) itUtaneovMy, x+z)y'^ + i^. ■ + b + c): a-b). ibc. ). 2+ 3/- + 3^2). XXIX.] IDENTITIES AND TRANSFORMATIONS. 281 18. If a + & + c = s, prove that (5 - 3a)3 + {s- 'Sbf + (s - 3c)» - 3 (s - 3a) (s - 36) (a - 3c) = 0. 19. If X = b + c-2a, Y = c + a-2b, Z=a + b-2c, find the value of X'+Y-' + Z^-SXYZ. 20. Find the value of a {o? + bc) + b{b'^ + ac) - c(c^ - ab) when a = -7, 6= -08, c=-78. 21. Prove that {a-bf + (b-cf + {c-af = 2(c-6)(c-a) + 2(6-a)(6-c) + 2(a-?;)(a-c). 22. Prove that «2(ft3 _ ^s) + 52(^3 _ ^3) + c2(„3 _ ^3) = {a-b){b-c){c-a){ab + hc + ca) = a?(b- cf-hb\c - af + c?{a - bf = -[a%'^{a-b) + b'^c^{h-c) + c-a'^{c-a)l 23. If (a + 6)2+(6 + c)2 + (c + d)2=4(a& + 6c + cd), prove that a = b=:c=d. 24. If x-a + d, y = b + d, z=c + d, prove that x^ + y^ + z^-yz-zx-xi/ = a^ + b^ + c"-hc-ca~ab. 25. li a + b + c=0, prove that 1 + - 1 ,=0. ^I 62 + c2 - a2 "*" o2 + a2 _ 62 -^ a2T62^a 26. It a + 6 + c=:0, simplify ^(62 + c2-a2) + ^(c2 + a2-62) + ^^a2 + 62_c2). ca a6 27. Prove that the equation {x-af + (y-bf + {a^ + b'i-l)(x'^ + y^-l)=0, is equivalent to the equation {ax + by - \f + (hx - ayf=0 ; hence shew that the only possible values of x and y are a h 2? , If 2 (a;2 + a2 - ax) {,f + b"-- by) = x^ + orb^ shew that {x - afiy - bf + (bx - ayf=0, and therefore that x=a, y=:b are the only possible solutions. *224. We shall now give some further examples of fractions to illustrate the advantage of arranging expressions with regard to cyclic order. [Art. 172.] '•II « ^1 it il r * 1 r I 232 ALGEBRA. [chap. to r KB- r I < 1 4 Example. Find the value of o. 6 i + ; (o-6)(a-c)(x-a) {b - c){b - a){x -b) {c - a)(c -b){x - c)' Changing the sign of one factor in each denominator, so as to preserve cyclic order, we get for the lowest common denominator, (a -b){b- c){c -a)(x- a){x -b){x- c). The whole expression has for its numerator -[a{b-c){x-b){x-c)+ + ] or -[a{b-c){x'^-(b + c)x + bc}+ + ]. Arrange it according to powers of x ; thus coefficient of x^= - {a{b - c) + b(c - a) + c{a -b)} =0; coefficient of a; = (a (6^ - c^) + 6 (c^ -a^) + c (a^ - 62)} = (b-c){c-a){a-b); [Art. 223.] terms which do not contain x = -{abc(b-c) + dbc{c-a) + abc{a-b)} = -a^c{b-c+c-a + a~b} =0. Hence the expression = {h-c){c-a){a-b)x (b-c)(c- a) [a -b){x- a) (a? -b){x- c) X {x~a)(x-b)(x-c)' Note. In examples of this kind the work will be much facilitated if the student accustoms himself to readily writing down the follow- ing equivalents : (b-c) + (c-a) + {a-b) = 0. a{b-c) + b(c-a) + c{a-b)=0. o2(6 - c) + 62(c - a) + c2(a- 6)= -(a-b){b-c){c-a). be (6 - c) + ca (c-a) + ab{a-b)= -{a- b) (b -c){c-a). a{b^-c'^) + b{c^-a^) + c{a^-b'^) = (a-b){b-c){c-a). Some of the identities in xxix. c. may also be remembered with advantage. ♦EXAMPLES XXIX. d. 1. 2. a {„.-^h}{a-c)'^{b-c)(b-a)'^(c-a){c-b)' be . ca . ah + ; (a-b){a-cy{b-c){b-ay(c-a){c-b)' 4. 9. 10. n. 12. 13. 14. 15. 16. [chap. b)(x-c)' .tor, 80 as to mominator, ] >)(x-c) ch facilitated n the follow- I !-a). :-a). a). mbered with XXIX.] CYCLIC ORDER. 233 A (a-6)(a-c)^(6-c)(6-a)"*'(c-a)(c-6)* 4. 5. 6. 7. a^ : + + , (a-6)(a-c)'^(6-c)(6-a)'^(c-a)(c-6)' a{b + c) h(a + c) cja + b) {a-b)(c~ayla-b)(b-c)'^(c-a)(b-c)' 1 1 a{a-b){a-c) b(b-c){b-ay c(c-a){c-b)' &c , eg ah Q («-^)(a:-c) (a;-c)(a;-a ) (a;-a)(a:- 6) «>• (a-6)(a-c)"^(6-c)(6-ar (c-a)(c-6")* 9. 10. 11. hc{a + d) ca{b + d) , ab{c + d) : + rr rvi r + : (a-b)(a-c)'^{b-c)lb-a)'^(c-a)(c-b)' 1 1 : + 7f (a-6)(a-c)(»-a) {h-c){b-a){x-hy(c-a){c-b)(x-c)' a' 6' (o-6)(a-c)(a; + a) (6-c)(6-a)(a; + 6)'^(c-rt){c-6)(a;+c)* ^^* (a - 6) (a - cr (6 - c) (6 - a) + ' (T^Hc - 6)' 13. 14. 15. a^{ h-c) + ¥{c-a) + c^a - b) {b-cf + {c-af + (a-bf ' aHb -c) + b^c -a) + c'^ja -h) + 2 (a - h)(b-c){c-a) (6 - c)3 + (c - a)3 + (a - b^ aHb -c) + b^c -a) + c^a - b) a^b-c) + bHc-a) + c^{a-b)' ,g a^7»~c)3 + &2(c-a)3 + c2(a-6)3 la-b)(b-c){c-a) 17. 18. -(&-c) + i(c-a) + l(a-&) bc\c bj^ca\a cj'^ab\h a) t:i U 1,1 > « * •ill '^ .; ( ■» ! tllK It; m ^^^■l:-' « li ^^^^^^■'^ « 1 ^^^H'- to 1 ^^^H" n 1. ^^^H U "' ^^B ' r ^^^^H ■ K< ^^^^H . CW ^H' ' r .:, ^^B' ^ V ^ f 4 ^H':. c ^■/: r ^ ^^^^^H -: »ac ^^^H j M> ^H 1 "' ^^^^^^B ' r 1 H^^^^B • ^^^H, ( rt ^^H '« ^1 •• ll« ft.1 234 ALGEBRA. [chap. *225. Tojmdwhen x' + px^+qx + r... is divisible hy x^+ax + b Divide (1) by (2) in the ordinary way ; thus a^+ax+h\afi+px^-hqx+r •(1), •(2)- {ja-a\x'^+ (q-b) x + r Cp-a)x^+a' x+(p-a) (j>-a) x+b{p-a) This is {(q-b)-a(p-a)}x+r-b(p-a) .... Now if the remainder is zero the division is exact the case when {(.q-b)-a(p-a)}x+r-b(p-a)=0, or y_ Hp-a)-r q-b-a(p-a) Hence when x has this vahie, (l).i8 divisible by (2). But if in (3), q-b-a{p-a)=0, and also r-b(p-a)=0, the remainder is equal to zero whatever value x ina^/ have. Thus x^+pf+gx+b is divisible by x^ + ax+b for all values of *-, provided that q-b-a(p-a)=0, and r-b(p-a)=0. *226. To find ike condition that x^+px + q may be a perfect square. Using the ordinary rule for square root, we have x^+px + qix+^ X'' 2^+f px->fq px^-^ <1- P' If therefore x^+px+q be a perfect square, the remainder, P q~^, must be zero. Hence q~~=^0, or p' = iq, is the condition required. [chap. (1), ' (2). p-a) (-3) ct. Thia is lave. Thus allies of X, be a perfect remainder, d. XXIX,] misceixanb:ous theorems. 235 *227. To prove that x^+px^ + qx'^+rx + s is a perfect square i/(q-^) =4s and r2=p2s. The square root must clearly be a trinoniial expression of the form x''\-lx-\-m ; if therefore we put o(^ +pa^ + gx^ + rx+s= (a-^ + Ix + m)\ we have, on expanding the right-hand side x^ +px^ + qx^ ■\-rx-\-s=x^-ir'^lx^-\-x^ {l^ + im) + 'ilmx + m\ Since this is to be true for all values of x, we may assume that the coefficients of the like powers of x are the same ; hence 'il=p, Hm=r, P+2m = q, m?=i From these equations, by eliminating the unknown quantities I and m, we shall obtain the necessary relations between jt), q^ r, and A Thus we have <3=a« The object of the present chapter is twofold : first, to give general proofs which shall establish the laws of combination in the case of all positive integral indices ; secondly, to explain how, in strict accordance with these laws, intelligible nieaninsfs may be given to symbols whose indices are fractional, zero, or negative. We shall begin by proving, directly from the definition of a positive integral index, three important propositions. 232. Definition. When m is a positive integer, a"* stands for the product of m factors each equal to a. 233. Prop. I. To prove that a'"xa''=a"'+'', 7chen m and n are positive integers. By definition, a^=a.a .a... to m factors ; a" :« . a . a ... to n factors ; a" xa"=(a .a. a = a .a .a Cor. to in factors) x (a . a . a . . , to n factors) to m + 7i factors =a'"+", by definition. If JO is also a positive integer, then «"* X a" X a" = a^^"** ; laws. and so for any number of factors. CHAP. XXX.] TIIK THEORY OF INDICES. 241 finition of a m m and n 234. Prop. IL Improve that a"'-^a■•=a»-^ when m and n are positive integers, and iii > n. ./»».,,»_« (t.a.a u . a — - = . . . to ni factors a" a. a. a to jt factors —0"a.a to m-n factors To prove that (a™)" = a"", when m and x\ are 235. Prop. III. positive integers. («"•)" = «'". «'". a"* to w factors =(a. «. a ... to m factors) (a. a. a ... to m factors) ... the bracket being repeated n times, = «•«.« to nin factors • ?^®" ''I'T ^^® *''^ fundamental laws of combination of HjdiccH and they are proved directly from a definition wliich is and ''S m/"" °" supposition that the indices are positive liiit it is found convenient to use fractional and negative indices such as a* «-?, cr, more generally, «f, «-« • and these have at present no intelligible meaning. For it is plain that the dehmtion of «-, [Art. 232], upon which we based the three pri! or '!^' a)/i" ^'"^ '^ "° '''"°^'' ""PP^^^^^^^ ^^'^^«n '>^ i^ fractional, Now it is important that all indices, whether positive or hi. w'^Jf^^'"" ''^ fractional, should be governed by the same Jcuvs. We therefore determine m-anings for symbols such as mII?" V" *h«,^?"o^'i«?r way : we assume that they conform to «. fimdamental law, «-x «"=«"•-', and accept the^meaning to ^^lnch this assumption leads us. It will l,o found that the m pXI'ii.' aXif '^'" ^^'"^ ""^'^ *^'' "^"'"^ ^''''^^ enunciated inf^ln. ^^ ^"^ "* '^^'''^"' ■^'"' ^'' P ''"'^ q ^--^^".^ i^o*^"'^'^« Since a- X a" = «♦»+« is to ue true for all values of «? and ??, by replacing each of the indices m and n by ^ we have 5-' a« xai —at »=a». E.A. Q IS Sj ■A * .■.,«! , 1 i i 'ii 242 ALGEBRA. [chap. It »' l» f ■tc> r c c r 1 B i- 2 P P ?P £ ^a£ !* Similarly, a^ xw xa9=d9 xa9=a'' 9=a9. Proceeding in tliis way for 4, 6, q factors, we havp P £ P 12 a^y.aiy.a^ to g* factors =a» ; p that is, {a9)i=a». Therefore, by taking the g"" root, p or, in words, a» is equal to " the j"" root of a^" Examples. (1) x^ = l/aT^, (2) a*=iya. (3) 4^=V4''=V64 = 8. (4) a^xa^ = a^'^^=a^. (5) Fxili=F+*=:it^^. (6) 3a^6* .^4aV = 12a*■*'*6*■*■^=l2aM. 238. T'o find a meaning for a**. Since a'"xa"=a*^'^'' is to be true for all values of m and n, by replacing the index m by 0, we have aOxa"=aO+" = a"; a" = 1. Hence any qiutnttty with zero index is equivalent to 1 Example. x^-''xaf-^=x^-<^+'-'>=o(P=\. 239. To find a meaning for a~°. Since a*" x a"= a'"^" is to be true for a?i! values of m and ??, by replacing the index vi by — /i, we iiave a-«xa"=a-"+" = a*'. But a"=l [chap. a hnvf of m and n, ol f wi and n, by XXX.] hence and THE THEORY OF INDICES. 243 «"= a" From this it follows that Kx\y factor may be transferrprl f.nm he numerator to the deno.ninatoi of an expressio^ or vl^^^^^^^^ by merely changing the sign of the index. ' Examples. (1) a:-»=Jr,. (2) ~=y^=^y. y * (3) 27~*= — = ?- -i _1 1 27)^ 4^(27)2-4/3S-32=9* 240. T'o/^roi;. ^Aa/ a"'-|-a»=a-Vor a/^ r..?«.. 0/ m and n. a''-ri" = a'"x~ a" ■f-a^xa-^ a"*-", by the fundamental law. Examples, (i) a''-ra'=o3-B=a-2=-l (2) c--c-^=c^+*=c'^. (3) a;«-*-f«»-<'=a;«-6-(a-c)-_a;c-j_ 241. The method of finding a nipnnnio. f.^« o i, 1 meanings, and then prove the r Ip« f.!^ /i • ^ inboLs give them the process is reversed Hi^rrvK ^"^ t^^eir combniation. Here whii?. they areToTonfor^^^^^^^^^^^^^^ -d.^he W to symbols are determined. ' ^^^ nieanmgs of the pr^dples'^w^hav^SS^^^ ""^ ^"^^-^^ ^'- ^'^-nt Examples. (1) ^*~^ ~ ^^ (2) 2a^xa^xGa~^ 9a~'i = 4 M-i+F^_4_, 4 xa' =3""=^- ill' li Ji 11 r •m * I ?.s 244 ALGEBRA. [CTUP. pi! I'll i c £ ''' 1 » 1 * « i«> i 1 tst 1 l» ^ i mj tr- i . ^ r c (4) 2Va + A: + «^=2a* + 3a* + a* /.~2 a =5a*+a^=a^(5+a2). EXAMPLES XXX. a. Express with positive indices : 1. 2a;"* 2. 3a"^. 3. 4a;-%3. 5. 4a-2 6. 7. 4. 3-fa-2. 9. 2a;*x3x-^ 10. l-f2a~* U. a;y2xa;-». 12. a-^oj-USx. la 14. 15. 16. *U~i ^x 17. a-^x-^^a-\ 18. ^/a-H4/a. 19. ^a-^-^^cf. Express with radical signs and positive indices : 2(J. 24. X 2a^ 21. a"*. 2 25. 3- 6"* 22. 5x"*. 26. V- 23. 2a~». 1 27. 1' a;"« 28. a-^x2a-* 29. a;"'-f2a-l 30. laT^ x3a-\ 31. 2a-' a ' 32. a rh 3a' 33. 4x-^ _x' a; i* 34. ^.p. 35. -V^v'"'. 36. >-'-f>-''- 37. '^a:x^^2. 3S. ^a;-r'';ya:». 39. >H^a2. 40. ^a" x >" t '^'a'". l>i\ [CTUP. a-2a:-H3x. _i 2a X. 1 XXX.] THE THEORY OF INDICES. 245 Find the value of 41. 16*. 42. 4-i 43. 125*. 44. 8" 45. 36" 4a ^.. 47. mK 4S. {±y\ 49. (51)1 50. (S)-l 243. To prove that (a"')-=a- w nnivermlhj true for all values Case I. Let n be a. positive integer. Now, whatever he the value of m («"*)" = «'". «"'.«'" to n factors = a'''+''' + '''+ to 1 tonus Case II. Let m be unrestricted as before, and let n be a positive fraction. Eeplacing n by ^, wliere p and q are positive integers, we have (a'")»=(a'")9. Now the J*"" power of (a'»)7= ' <'»)7}7 Hence by taking the (7"" root of tJiese equals, p [Case T.] [Case I.] [Art. 237.] Case III. Let m be unrestricted as before, and let n be nrtu ^^auve quantity. Replacing n by -r, wh«;e r is ^^^.t, V^ 1 a mr' [Art. 239.] [Case II.] =a~""'=a""'. u."Slft°,?ie:"-''^''- '''• «"=«""'- •-«» ahewn to bo ii IT' -if "Hi* I' 246 ALGEBRA. [chap. Examples. (1) (bh^=h^''^=bt (2) {{x-2)3}-4 = (x-«)-»=ar". (3) {x'^-y =x^-'> =a^+<'. 244. To prove that (ab)''=a''b'', whatever he the value of n; a and b being any quantities whatever. Case I. Let w be a positive integer. "Now (ab)"=ab.ab.ab to w factors '=(a.a.a... to nia.ctora)(b.b.b... to 7i factors) Case II. Let w be a positive fraction. Replacing, n by ^, where p and q are positive integers, we have (ab)"=(ab)'i. p p Now the 5"" power of («&)» ={(«&)»}» =(a6)P [Art. 243.] ' =afb» p p =(as6«)». [Case I.] p p p Taking the 5"" root, {ab)i=^aib9. Case III. Let n have any negative value. Replacing n by — r, where r is positive, =a-'-6-* Hence the proposition is proved universally. The result we have just proved may be expressed in a verbal form by saying that the index of a product may be distributed over its factors. ^VVC. .^^U lIlUCA 13 ll\!U VllDtliwUl.lVC ^-VCl. XjllXZ IZ.T ttlO t-fl till ItJ'tr-.. sion. Thus («* + &^)'' is not equal to a + 6. Again (a" + 6")' is equal to 'slai^ + &", acd cannot be further simpiifittd. 15. [chap. XXX.] THE THEORY OF INDICES. value of n ; . factors) ing.M by I, [ahyi. [Art. 243.] [Case I.] )lacing n by 1 in a verbal »e distributed fft^jT is equal 247 Emmplea. (1) {yzY-^zxY{xy)-'=y^-<'z<*-ez<:3fx-<=y-o — Old- 2c ^n — if '' • (2) {(a-6)*}-^x{(a + 6)-*}' = (a-fc)-«x{a+6)-« = {(a-6)(a+6)}-t' = (a--62)-« 245. It should be observer! that in the proof of Art 244 the quantities a and 6 are toholli/ unrestricted, and may themselves involve indices. Examples. (1) {x^y~^^{x^y-^)~^=x'^y-^^x'^ys 4 (2) (^-^.!^^'\^(oh-^ , ^lab^\ -2\6 = (aV^-fo6~^)8 EXAMPLES XXX. b. Simplify and express with positive indices ; 1. (n/^3)6. 2. (SI^FY)-^ 3. (:>f'y-»)3x{:r»y2)-. 7. {^(7P)'}-* 8. tim^-\ 9. (4a-^varT* a;x^a; "jl^. 12. (^a:»^;ya;pa. 14. 'v'aFV^x(a-i6-2c-4)-i 10. (.r-fXya;)". 15. ^^c a;'*x(a^a!-i)-« 4»;}.6 16. ^so-i^y^^slyHJx. « 1 \ '^ « 1^ -1 ' .I' -t. -T4«aiM m ' m e iXt » r If 248 ALGEBRA. [chap. Simplify and express with positive indices : 17. (fr%a?)-8xVa;-Va-^ 18. \/a^+*62^^-r(a«6"")*. 19. ^f{a + brx{a + b)-^. 20. {(a;-t/)-3}n^{(a;+2,)«}3. 21 r»:-!^V'-/'i^V 22 /^" /fe^V.^H" 23. (a"'*^ yJax-^*/x^ ) . 24. 4/{^+6)8 x (a^ - ftS)"*. 25, « n/„2n 29. (a«»-ir+i+'^^. 2g \^(a363 + ct6) 30. {.Arv^'. 35. K=xr-7=rT-x 32. (a;*2/ a: a:n*-62\a+» «• (£)"*<^"/' 37. 2n+i >< 2«-i 4-'** 3.2"-4.2''- '' 2" - 2"-i ~* 36. 38. '-x'z- 2"+i 4n+l (2n)'»-l • (2"-l)'»»+l* 3"+*- 6 . 3«+i 3^2x7 246. Since the index-laws are universally true, all the ordinary operations of multiplication, division, involution nricl evolution are applicable to expressions which contain fractional and negative indices. 247. In Art. 121, we pointed out that the descending powers of X are »3 r2 r 1 ^ ^ ^ X X X" A reason for tliis may be seen if we write these terms in the form Ov.J. ^, x^, xK x^, x-\ x-\ x-\ i\m [CIUP. 1 _1 -(a"6 ")t. r{{x+y)"}K •=-62)"'-^. VP3 ya:2" a; 1* +1 )' )n+r lie, all the aluti(»ii niid u fractional ling powers in the form XXX.] THE THEORY OF INDICES. Example 1. Multiply ^x~^ + x + 2x^ by x^ - 2. Arrange in descending powers of x. 249 X ^^x'-^ ■{■Zx'^ x^-2 x'^-\-2x +3 -2a; 4a;»-6«~'^ x^-^x^+Z -6a;"i Example 2. Divide 16a-3- 6a-2+5a-i + 6 by l+2a~K 1 6a-3+ 8a-2 -14a-2+ 5a-i -14a-2- 7a- 1 Example 3. Find the square root of --+-3^-2x + -| + .r3-V(a:^2/-i). 2x*-2ary~* 2^?-4a;y-*+^ ^-4rV* + 4a;V + a;V-2a; + |(x^-2a:y-^ + ^ -4x^2/ ^ + 4*%-i -4a;^y~^ + 4x-jy-J x^y^-2x+-i 4 4 of netaWe^ndtereiVi'S^^^ he observed that the introduction « ive maices enables us to avoid the use of algebraical fractions. ill' ^1' ua ■ HM I 1 ^M ■1 ! r; 1 ■ m-' ' u 1 m\ '^ H ^H « f 1 ^■i Wm P^ i »•., 1^ 9''' ? '■■ ■ m'' i 1 ^M ^L ^ i -.;. 1 || ' 260 ALGEBRA. EXAMPLES XXX. c. 1. Multiply 3a;' -5 + 8a;" ^ by 4a;* + 3a;"*. 2. Multiply 3a*-4a*-a~^ by 3a* + a"*-6a~^. 3. Find the product of c'+2c-'-7 and 5-3c-*+2c*. 4. Find the product of 5 + 2x''-; Sa;-"^' and 4sf'~3a;-«. 5. Divide 21a; + a;^ + a;^ + l by Sj.-* + 1. 6. Divide 15a - 3a* 2a~ ^ - >: «■ i by 5(v' (- 4. 7. Divide 16«-»+6a-3 -^5a-l-6: by 2a-i-l. 8. J); vide 5b^ o6* _ 46"* _ 4b" i - 5 by 6* - 'J}~ K 9. Divide 21a'^ -^ 20 27a' - iGa^* by ,3a^ - 5. 10. Divide 8c-» - 8c"-' 5c'' - 3c-3" by £:« - 3c"". Find the square root of 11. 9a;-12a;*+10-4a;~* + a;-i. 12. 25a* + 16 - 30a - 24a^ + idJ. 13. 4a:» + 9a;-" + 28-24a;~^ IGx^. 14. 12a«+4-6a3«+a**+5a2*. 15. Multiply a*-8a~^ + 4a~^~2a^" by 4a~^+a^+4a"l 16. Multiply 1 - 2 ^a; - 2a;^ by 1 - 4/a;. 17. Multiply 2^-J-^ by 2a -3^^ -a"*. 18. Divide ^+ 2a;* - 16a;"^ - — by a;* + 4a;~* + 4- 19. Divide l-^a--^ + 2a^ hy 1 - a*. a- [CHAP. 23. 20. Divide 4sy^- 8x* - 5 + ^ + 3a;" ^ by 2x^ - ]^x - ~. six ^ ^ 4/x |iiMp->*i-,;a [chat. XXX.] THE THEORY OF INDICES. Find the square root of 21. 9.r-''-18a:-\/y + ^-6^(g)+y2 22. ^sJ^-l2*J{j^y) + 25^y - 24:^K) + IGx" V 23. 8l(^+l)436^(xV^-l)-158!^'. 21 ^'ll+i+J.^ + LzMVayy 251 248. The following examples will illustrate the formulje of earlier chapters when applied to expressions involving fractional and negative indices. * £ _* _p *_* * p * _5 PS. Example I. (a* -5*) (a * + t »)=o* *-a~*6« + a*6 ^-fca"* = l-a~*6«+a*6~«-l * _£ _* £ =a*6 "-a *6». Example 2. Multiply 2a;'2i'-a;^ + 3 by 'Hx'^p+xP-Z. The product ^ { 2x^p -(xp-Z)}{ 2x^p + (a;P - 3) } = (2«2p)a_(a;P-3)2 = 4ar*^-a:2i' + 6a;P-9. Example 3. The square of 3x^-2-a;~^ =9a;+4 + x-i - 2 . 3a;* . 2 -2 . 3x* . a;"* + 2 . 2 . a;~* = 9a; + 4 + a; -1 - 12a;* - 6 + 4a;~* = 9a; - 12a;* - 2 + 4a;~ * + a; -1, by collecting like terms and rearranging. Example 4. Divide a^ + a~^' by a^ + oT^. The quotient =(a''-" + a~^)-j-(a^ + a~?) sa^-l+a-". ft'-l •m • 11,.. B.' E I f? |Q» I m r r c r I I 252 ALGEBRA. EXAMPLES XXZ. d. FTnVe down the value of 1. {x^-7)(x^+3). 3. (7«-9y-i){7x+9y-i). 5. (a'^-2a-^)2. (a I \2 9. (l»*-a-i)'. 11. (a'-^-a-*y. [chap. XXX. 13. {(a + ft)*+(a-6)T}2. 2. (4a;-5a;-i)(4a;+3a;-»). 4. (^"'-2/'»)(»-"'+y-''). 6. {a'+a')^ 10. (3a^y-6 + 5a;- V) (3^:"^* - 5x-"y->') 12. (a;«-a;~«+a;)3. 14. {(a+6)*-(a-6r*}2. TFnVe ofoiwi the quotient of 15. a; -9a by a?*+3a*. 16. a;^-27 by a;* -3. 17. a'^- 16 by a* -4. 19. c^-c-'hy c*-c~^. 21. a*'-afi by a^^ + a^. 23. a;^-l by a;*-l. Find the value of 25. (a;+a;^-4)(a; + a;* + 4). 27. (2-a;* + a;)(2t-a;^+x). 29. — -— » 31 a?^-4^a;-« 4'a?+4+4«~^ 18. ar^+8 by a^+2. 20. l-8a-3 by l-2a-\ 22. x-*~l by a;-i + l. 24. a:^+32 by a:"+2. 26. (2a;^' + 4 + 3.c"*) {2a;"^ + 4 - 3.r" '), 28, (a'+7 + 3a-»)(a*-7-3a-«). 30 =^-7a:^" .(., 2\-^ ^- x-5Vt:-i4"v+;/s; • n a^ + ah Ja 32. ab-¥ ^a-b' [CHAP. XXX. f''-5x-"y->') •1 hi-3x~h J-Sa-'). 2W CHAPTER XXXI. Elementary Surds. 249. Definition. If the root of a quantity cannot be exactly obtained the root is called a surd. Thus V2, 4^5, 4/a3, Vo^+F are surds. By reference to the preceding chapter it will be seen that these are only cases of fractional indices ; for the above quanti- ties i^ight be written 2* 5* a*, (a2 + &2)* Since surds may always be expressed as quantities with frac- tional indices they are subject to the same laws of combination as other algebraical symbols. 250. A quantity may be expressed in a surd form without really being a surd. Thus iLv^ or x^', though apparently a surd, can be expressed in the equivalent form a;\ 251. A surd is sometimes called an irrational quantity: and quantities which are not surds are, for the sfi,ke of distinction, termed rational quantities. 252. In the case of numerical surds such as ^2, ^5, ..., although the exact value can never be found, it can be deter- minedto any degree of accuracy by carrying the process of evolution far enough. Thus x/5 = 2-23G068 ; that is ^/5 lies between 2-23606 and 2-23607 ; and therefore the error in using either of these quantities instead of ^/5 is less than •00001. By taking the root to a greater number of decimal places we can approximate still nearer to the true value. _ It thus appears that it will never be absolutely necessary to mtroduce surds into numerical work, which can always be carried on to a certain degree of accuracy ; but we shall in the present chapter prove laws for combination of surd quantities which will enable us to work with symbols such as ^2, ^/^, l^a, ... with absolute accuracy so long as the symbols are kont ifi their surd form. Moreover it will be found that even wlure lipproximate numerical results are required, the work is considerably simpli- I; 11 264 i\ ALGEBRA. [CHAP. p ! .. E •J Ik m a I* r V, t .l fled and shortened hy operating' with surd svmbols, and after- wards suLstituting numerical values, if necessary. 253. The order of a surd is indicated by the root symbol or surd index. Thus ^x, ^a are respectively surds of the th'ird and w"* orders. The surds of the most frequent occurrence are those of the second order; they are sometimes called quadratic SUrds. Thus V") V«, iJx-\-y are quadratin surds. J54. It will frequently be found convenient to express a rational quantity in a surd form. A rational quantity may be expressed in tlu iorm of a surd of any reqxared order In raising it to the power whose root the surd expresses, and prefixing the radical sign. Thus '■•^ = V25 = 4/1 25 = 4/625 = ^5" ; 255. A surd of any order may be transformeJ into a surd of a different order. Examples. (1) ^2 = 2^=2T'^=i^2*. (2) Va=a'' = aP«=';:^a». 256. Surds of different orders may be transformed Into surds of the same order. This order may be any common multiple of each of the given orders, but it is usually most convenient to choose the least common multiple. or^T'""^''' ■^''^''^'' ^'*'' ^^'' ''"' ''' '""^' °^ ^''' '*'"*^ ^"^^'<^«^ The least common multiple of 4, 3, 6 is 12; and e) • • ^ssing Ihe given surds as .surds of the twelf * li order they become 12/a", '^lb\\la}\ 257. Surdn of different orders mav be arranged accoi ■'iig to magnitude by transforming them into surds of the .sam urder. E.cample. Arrange ^a, ^^6, 4/IO according to magnitude. The least commci multiple of 2, 3, 4 is 12; and, expressing tht; given surd as surds of the twelfth order, we have v/3 =i4/3« =14/729, 4^6 =^6< =i^l29( •Vio = I'^/l 03 = V-/! nfio Hence ar. nged in aso nding order of magnitude the surds are V3, 410, ;^/6. 7. U. [CHAP. Is, and after. >t symbol, or of the third those of the lurds. Tluw to express a 1 of a surd of root the surd itoasurd of 'd into surds multij)le of mvenit'iit to same lowest j».'ff8sing the iccor''ng to am -rder. iude. pressing the surds are XXXI.] Express as si 1. xi ELEMENTARY SURDS. EXAMPLES XXXI. a. <.f the twelfth order with positive indices : 255 4. a ,-f 2. a-^i-a 5. ' -i 3. >/aa^ X sIa-'^x-\ Express as surds of the w"- order with positive indices : 7. ^^^ 8. X". 9. a*. 10. V«^. -4 12. I .-1' 13. V- 14. a- 11 a/ ^ Express as surds of the same lowest order : 15. v/a. >". 16. ^a\ x/a. 17. ^/^r'', ^a^e^ M/a:5. 18. i^a:*, '^x^'. 19. '^^, ;/^. 20. V^, ^^^. 21. V5, 4^11, 4/13. 22. 4/8, v/3, >. 23. 4/2, 4/8, ^4. 258. T' root of any expression is equal to the product of the roots oi the separate factors of the expression. For Vah={ahf =«"&", [Art. 244.] Similarly, Vahc = ;;/« . ^Ih . J^fc ; and so for any number of factors. Examples. (1) 4/15 =4/3 .4/5. (2) 4/a«& = 4/a« . ^b = a^^j. (3) ^50 =x/25.^/2=5^2. Hence it appears that a surd may sometimes be expressed as the product of a rational quantity and a surd ; when so reduced ilie surd is said to be in its simplest form. ThuR iho simplest form of ^12? '- ", 'e. CouYiMly, the coefficient of a ad v,iv' be brought under the radical sign by first reducing it to ,:..: form of a surd, and tlien muln;,iying the surds together. ■•.ft « 1.1 »'« » • li J) l> '■m I «' li I 256 ALGEBRA. [cnAP. i< ft' I m I r f Examptei. (1) 7^/5=^,/49. >/5=v'245. (2) a^h=^a?.^b=^^b. When 80 reduced a surd is said to be an entire surd. 259. When surds have, or can be reduced to have, the 8,'ime irrational factor, tliev are said to be like ; otherwise, they are said to be unlike. Thus 6^3, 2^3, ^^3 are like surda. But 3^2 and 2^3 are unlike surds. Again, 3^20, 4^/5, A/f are liko surda ; 3^/20 = 3^4 .^5 = 3. 2^5 = 6^5 ; for and 260. In finding tlie sum of a number of like jurds we reduce them to Uieir simplest form, and prefix to their common irrational part the sum of the coefficients. Example 1. The sum of 3;^20, 4^75, -rp \'o Example 2, Thesumofa; i/Sa^a + y^ - y^a - z \/zFa =x.2x^a + y(~y)i^a-z.z^a = {2x'^-f--z^)^a. 261. Unlike surds cannot be collected. Thus the sum of 5^2, -2^3 and ^6 is 5^/2-2V3+,^'n, and cannot be further simplified. EXAMPLES XXXI b. Express in the simplest form : ^H^H|^# ^^^^^^BfarSB^BK t ^H^^^^py sn ^H' ^ r f Hi' -^ ^■^ [' r ^^^^^^^H t*: ■ )£ ^^■.. 1 263. If the surds are not in their simplest form, it will save labour to reduce them to this form before multiplication. Example. The product of 5^32, ^'48, 2v'54 =6 . 4 V2 X 4^3 X 2 . 3^6 = 480 . ^/2 . ^/S . ^/6 = 480 x 6=2880. 264. To multiply surds wliich are not of the same order: reduce them to eqiiivalent surds of the same order, and proceed os before. Example. Multiply 5;^ by 2,^5. The product=5 SJ'H^ x 2^53= 10^^x5^= 10^500. 265. Suppose it is required to find the numerical value of the quotient when ^5 is divided by ^7. At first sight it would seem that we must find the square root of 5, which is 2*236 ..., and then the square root of 7, which is 2-645 ..., and finally divide 2-236 ... by 2-645 ... ; three trouble- some operations. But we may avoid much of this labour by nuiltiplyiuif both numerator and denominator by »/?, so as to make tlie denonii- nator a rational quantity. Thus '^-'d^ v^._ ^>^5x7 V35 n/7 J7^^7~ 7 ~1" Now V35 = 5-916... . x/S 5-916... s/r •845 .. 266. The great utility of this artifice in calculating the numerical value of surd fractions suggests its convenience in the case of all surd fractions, even where numerical values are not required. Thus it is usual to simplify ^^ as follows : ajh _ ajh x sJ c_a\/bo The process by which surds are removed from the denomi- nator of any fraction is known as rationalising the denominator. It is effected by multiplying both inimerator and denoniiiiator by any factor which renders the denominator rational. AVe «hall return to this point in Art. 270. [chap. n, it will save cation. ► x 6=2880. same order: nd proceed an rical value of d the square 3t of 7, which ;hree trouble. tiplying both J tlie denoiui- culating the nieiice in the lines ai-e not ws : the denonii- enominator. denonuMatdr ll. Wt' shall XXXI.] ELEMENTARY SURDS. 259 267. The quotient of one surd by another may be found dtomS^^ ''^ "^"^^ ^ ^ ''^^^ -^^ -^--"-g the Example 1. Divide 4v^75 by 25v'56. Thequotient = l;4^=-^^^^^ - ^^^^ 25^56 25x2^14" 5^14 _ 2^3x^/14 _ 2^ ^/42 5^14x^14 5x14" 35"' Example^. ^^^bx^c_m j^ EXAMPLES XXXI. c. Find the value of 1. 2va4x^/21. 4. 2^15x3^5. 7. 21V384-r8V98. 10. ^168x^147. 13. Ov/68x6Va 2. 3^8x^6. 5. 8^/12x3^/24. 8. 5V27-r3V24. 3. 5^a X 2^3. 6. \/xT2 X i^^^^. 9. -13Vl2S-f5Ve5. 11. 5^128x2^432. 12. 6Vl4-f2V21. 14. Mi-f;^. IK 3v/48 . 6v^4 ic 3^/S2 4 /a^' _ 2^98 •7^/22- 15. 5;/iT2-;/W 17. J-^JI^^JJ^ a-bya-b V(a-( (a-6)B '7^-?fi4r^7l^ fi^\f 'V^^=,^'^'^2^^' x/5 = 2-23607, ^/6 = 2-44949, v^7-2 64575 : find to four places of decimals the numerical value of 18. 22. 14 V2* 60 1 19. 25 23. 144^^6. ^' vm 27. V243" 20. 24. 28. 10 v/24-v/3. 25 V252' 21. 25. 29. 48 1 2v'3' /256 snr^V ■,"'!" ", havcj:onfined our attention to simple surds, such as 4/5, •/«., v/^. An expression involving two or 7al vr ' '"'''' '' 'f •''^ "^ compound surd ; thus V?- 3 J6 • Va+ ^6 are compound surds. ^ '^ * H il Jit 3:» ill ifll i 260 ALGEBRA. !» 1 Pr m ! [chap. 269. The multiplication of compound surds is performed like the multiplication of compound algebraical expressions. Example 1. Multiply 2Jx-5 by 3,^a;. The product = 3 sjx{2^x ~ 5) =Qx-\5sJx. Example 2. Multiply 2^l^ + 'isjx by fj^-tjx. The product= (2 V5 + ^i>Jx) (v/5 - ^x) =2^/5 . Vo + 3n/5 . x/a; - 2^5 . s/« - 3 ^a: . sjx = 10-3a; + »y5z. Example 3. iMnd the square of 2^x + sJT-4x. {,2^x-{-sJT^Axf={2sJxf+{>J'r^f + 4^x . s/f^^ =ix + l -'ix+i'Jfx-ix^ =T + 4\^fx-ixK kt » ■» ! ^ 1 Kj s?v . r : : ' c i 'to« ' .'S 1 li 1 Is EXAMPLES ZXXI. d. Find the value of 1. (3>Jx-5)x2s/x. 3. is/a +^)x '■Jab. 5. (9^3 + 3^/2)2. 7. (3V5-4v/2)(2V5 + 3V2). . 9. (x/a; + s/x-l) x >Jx-l. 11. (\/^+^-2Va)^. 13. (sla + x - s/a - xf. 15. (v/2 + v/3-V5)(V2 + v/3 + ^/5). 16. (V5 + 3 v/2 + V7) (v/5 + 3 ^2 - ^7). Write down the square of 17. \f2x+a-sf2x'^. 18. sfx^-2y^+s/xHW"' 19. Vi 2. {sjx-^a}x2^x. 4. (Va:+y-l)x\/a; + y, 6. (v/7 + 5^3) (2v7- 4^/3). 8. (Sy/a - 2Jx) (2 ^a + 3^/a:). 10. (vi + a - six -a) X \^a; +o. 12. {2sla-^!r+idf. 14. (V'a+a; - 2) (^/o+'k - 1 ). m + >i + \lm - 11. 21. 3a;V2-3\T^a:3. n^. va' 20. 3Va2 + ft2-2 22. \'4SMn[-v'4^-l. 6". [chap. XXXI.] ELEMENTAIiY SURDS. 261 is performed ^ressions. . sjx Ix a+x-l). 270. One case of the multiplication of compound surds deserves careful attention. For if we multiply together the sum and the difference of any two quadratic surds we obtain a rational l)roduct. Examples. ( 1 ) {^a + ^) (^'a -^lh) = {>Jaf - {^hf=a - h. (2) (3^/5 + 4V3)(3x/5-4V3) = (3Vc'^)2-(4V3)2 = 45-48= -3. Similarly, (4 - sja + h) (4 + V^+i) = (4)2 - {»J^+bf= 16 - a - ft. 271. Definition. When two binomial quadratic surds differ only m the sign which connects their terms they are said to be conjugate. Thus 3V7 + 5v/ll is conjugate to 3^7-5^11. Similarly, a - >Ja^-x^ is conjugate to a+s,la?-x\ The product of two conjugate surds is rational. [Art. 270.] Example. (3 V" + Vrc - ^a) {SJa- sjx-9a) = (3 v/a)^ - {\fx~-^W= 9a -{x- 9a) = lSa-x. 272. The only case of the division of compound surds which we shall liere consider is that in which the divisor is a binomial quadratic surd. If we express the division by means of a frac- tion, we can always i-ationalise the denominator by multiplying Dumerator and denominator by the surd which is conjugate to the divisor. Example 1 . Divide 4 + 3 v'S by 5 - 3 ^2. ^ 20 + 18 + 12^2+15^2 _ .38 + 27s/2 2.5-18 ~ 7 Example 2. Rationalise th« denominator of 62 li^+a The expression= -,=iL - x'^E^'' (a2 + 62)_aa sVa'+fe'-a. " I 'Pi 1:1 'S I it 262 ALGEBRA. [CHAP. Example 3. Divide ^/3i:_>^ by Z|i>|. v/3 + V2^ V3-V 2 (V3)2-(V2)2 14- 12+8^3 -7v/3 1 _87_ 7-2^5* 2 + ^3 =2-,y3, on rationalising. Example 4. Given ^5 = 2-236068, find the value of Rationalising the denominator, 7^=^^£3|^^ = 3(7 + 2V5)=34-416408. It will be seen that by rationalising the denominator we have avoided the use of a divisor consisting of 7 figures. EXAMPLES XXXI. e. Find the value of 1. (9^2 -7) (9^2 +7). 3. (5^8 -2^/7) (5^8 + 2^7). 5. Wa + 2^){^a-2y/b). 7. [s/a+x - i^a) (s/a + x + sja), 8. {'J^T3q-2^q){y/^T3q + 2^q). 9. ('Ja + x + s/a^'x){sja-i-x-s/a-x). 10. {5>Jx^-3y^+7a) (5 V^^3y2 - 7a). 2. (3 + 5V7)(3-5V7). 4. {2vll+5V2)(2Vll--5v'2), 6. (3c-2yfx)(3c + 2^x). 11. 29-r(ll + 3V7). 12. 13. (3vA2-l)-^ (3^2 + 1). 14. 15. (2x-sf^)^{2sf^j-y). 16. sJa . sJa-\-sJx >Jx 17. nationalise the deuoniinator of 18. 17-^(3^+2^3), (2^/3 + 7v/2)-r (5^3 -4^2). (3+v/5)(V5-2)-f(5-v5). 2v/15 + 8 . 8V3-6V5 5+;^15 •5;v/3-3v/5" -Q 25.v/3-4V2 20. 10v/6-2V7 3^6 + 2^7' 21. s/7+v/2 9 + 2^14* 27. 33. [CHAf. 87 7-2V5' 08. lator we have /7). Vll-5v'2). ). V3-4V2). ^(5-v'5). V2 7iT XXXI.] 22. ELEMENTARY SURDS. 263 25. 27. 2v^3 + 3V2 6+2^6 * \/9+^-3 ^9+^ + 3 23. ?/" 24. a^ x + sjx^-y^ "' s/x'^ + a^ + a 2'Ja + h + ZsJa-b 26. 2sla + h->Ja-h OQ 3 + v/6 ^- 5^3-2^12-^32 + ^/50' Givpu v/2 = l-41421, V3 = l-'73205, ^5=2-23607: find to four places of decimals the value of 29. 33. 2 + V3' 30. 3 + V5 x/5-2' 7v/5 + 15 V5-2 v/5-1 ^8 + ^5' Qi i/^jVS «o s^5-2 '*^* 4 + V15' "^^^ ir^5* 34. (2-^3)(7-4V3)v(3V3-5). 273. The square root of a rational quantity cannot be partly rational and partly a quadratic surd. If possible ^et then by squaring, sln=a+sj'>n ; n=a^ + m+2av^m J n — a^ — m isjm=- 2a that is a surd is equal to a rational quantity ; which is im- possible. 274. If x+^y=B. + ^), then will x = a a7id y = b. For if X is not equal to a, let x—a + 7n; then a+7n+>Jy = a+>Jbi tJiiitis, slb = m+s!y', which is impossible. ' [Art. 273.] Therefore and consequently, If therefore we must also have x=a. y=b. x+sfy—a + s/b, x — ^y = a — ^b. 275. It appears from the preceding article that in any equation of the form X+JY=A^^in (1), we may equate the rational parts (.-i eacli side, and also the irrational parts ; so that the equatior» (1) is really equivalent to two independent equations, X=Ai and'F=J3. .J''li i.;, If • m * m ir ^^1^ ! ; I) .J I fit M E Hf C ■ \' ■• an tt 11. i : It i 1 , » ts r b tw r f e: ^ IffLS 264 ALGEBRA. f,,,,,. 276. // v/a+7b = Vx + ^y then will ^a^ b = V* - sfy. For by squaring, we obtain .-, a=x+'i/, ^Ih^'ijxy. [Art. 275.] Hence a-jh=x-2sjxy+y^ and sfa-sjb=^x-s/^j. 277, To find the square root o/a+^b. Suppose sf^iT^h=^x+^y ; then as in the last article, ^+y=<^ (1), ^s[xy=Jb (2). =a ^~b, from (1) and (2), ,-, x-7/ = ^a^-b. Combining this with (1) we find x='^±#H», and y=t:^ V^c J^°^- ^^f ""i^J"*^^ J"^^ ^°""^ ^«^ ^ -'^nd y, it appears tliat each of them is itself a compound surd unless a^-b is a perfect square. Hence the method of Art. 277 for finding the square routof a+ V6isof nopractical utility except when a^ - 6 is a perfectsquare. Example. Find the square root of 16 + 2^/55. Assume sA6 + 2V55=v/-^- + ^y. ^hen ie + 2^55:=x + 2^f^y + y. x + y = ie (1), 2^ = 2^55 (2). .-. {x-yf=(x + y)-i-ixi/ = 162-4x56, by (1) and (2). = 4x9. ••• a:-y=±6 (3). From (1) and (.^) we obtain x=ll, or 5, and y=5, or 11. That is, the required square root is ^11+^/5. [OUAI- [Art. 275.] XXXI.] ELEMENTARY SURDS. 263 (1). (^). u (1) ami (2). -b) appears tliafc ') is a perfect square root of Jrfectsquare. (1), (2). yr (1) and (2). (3). In the same way we may shew that Note. Since every quantity has two square roots equal in n.acni- tude but opposite m sign, strictly speaking we should have the square root of 16 + 2^55= ±(^11 +^5), 16-2^55= ±(^/ll -^5). However it is usually sufficient to tak e the positive value of the square root, so that in assiuning sja-^lb=^x-^y it is understood that X IS greater than y Witli this proviso it will he unnecessary in any numerical example to use the double sign at the stage of work corresponding to equation (3) of the last exaniple. 279, When the binomial whose square root we are seokiiiff consists of two quadratic surds, we proceed as explained in the following example. Example. Find the square root of ^^175 - ^/147. Since >/175-Vl47-V7(x/25--^/21) = ^/7 (5-^21). •■• v/n/176-V147 = 4/7 . x/S^V^l. And, proceeding as in the last article, V5^V2r=^^-^|; ••■ Vv/I75-v/147=t/7(^5- ^1). 280. The square root of a binomial surd may often be found by inspection. Example 1, Find the square root of 11 + 2^/30. Wo have only to find two quantities whose sum is 11, and whose i)ro(luct 18 30 ; thus 11+2^/30=6 + 5 + 2^/6x5 __ = (v/6 + V5)2. .-. v/n + 2"^'30=v/6 + V5. Example 2. Find the square root of 53 - 12^/10. First write the biuomial sn that the surd part has a coefTicicnt 2 ; *'"'^^ 53 - 12^10=53 - 2^360. nro)ln ""T^^-^ !u ^"'^ ^^'"^ O»'"^titios whose sum is 53 and whose product 18 360 ; these are 45 and 8 ; •ini Wt'l |i: ; • E ■I < I B ( t 1 '« ■* |:^< ai 4 1 m 4k 1 ' /3-2V2. 23. x/4 + SV3. 25. y 19 + 873. 26. 78 + 2^15. 28. 7ll +4^.76. 29. 715-4^14 18. 2^5 + 3i. 21. 248 + .32^60. 24. 76-^75. 27. ^9-2^14. 30. J'2d+1Q'22. Equations involving Surds. 281. Sometimes equations are proposed in which the un- known quantity appears under the radical sign. Such equations are very varied in character and often require special artificos for their solution. Here we shall only consider a few of the simpler cases, which can generally be solved by the following nietliod Bring to one side of the equation a single radical ternrby itself: on squaring both sides this radical will disappear. By repoatiug this process any remaining radicals can in turn be removed. or [CHAI-. ing biuoiiiiiil 2V7. -4^33. ■5V7. ) + V40. surds : j+3i. + 32^60. ^J5. -2^14. hich the un- jch equations il artifices for )f the simpler ^in"' Tuefcl'.odi jrni by itself: By repeating removed. XXXI.] EQUATIONS INVOLVING SURDS. 267 Example 1. Solv^ Transposing Square both sides j then Squaring, (1). 2v'a;-V4jrii = i. 4a; -4^/0. + 1=4.6- 11, 4Va: = 12, .: x=9. Example 2. Solve 2 + \^x^ = 13. Transposing '^^=^=11. Here we must cube both sides ; thus a; - 5= 1331 ; whence 05=1336.' Examples. Solve s/x + 5 + >/3x + i=sfl2xTl. Squaring both sides, a; + 5 + 3a: + 4 + 2\/(u; + 5)(3a; + 4) = 12a;+l. Transposing and dividiiig by 2, \'{x + 5)(3x + 4)==4:X-4: {X + 5) (3a; + 4) = 16x-2 - 32x- + 16, 13a;2-51a;-4 = 0, (a;-4)(13x- + l) = 0; •■• ^=^'''^-iT If we proceed to verify the solution by substituting these values in the origmal equation, it will be found that it is satisfied by J= 4, bit not by x= -j^. But this latter value will be found on trial to satisfy the given equatjon^if we alter t he sign of the second radical ; '^x + 5~\/3x + i=s/T2xT'l. On squaring this and reducing, we obtain -V(a; + 5)^;c + 4y=4.r-4 (2); wollf //T^*"'''" J*^ [^-^ ^"'^ i' i- m It . il 1 H w t ifciK r I 1 ^1. 1: 268 ALGEI'.IIA. [chap In order that all the s allies found l>y the solution of tl' equation may be applicable it w ill be uucesBar^ to take into acct unt botli signs of the radicals in luu given equation. 3. 6. 7. 9. 11. 13. 15. 17. 19. 21. 23. 25. EXAMPLES XXXI. g. Solve the equations : 1. \/«^=3. \'5x-l = 2'JxT3. 2^5a;-35 = 5\/2x^T. V2a: + ll=^5. s/x + 3 + ^x=5. 2. -5/4^^=6. 2\/r-^- 3N/8a^T2=0. 'Jd3^-in^=3x-2. sf4x^-1x+i=2x-l^. \'8^33-3=2>/2a;. 6. 8. 10. 12. 14. iO-\f25T9x=3^x. \/x-^ + 3=\fz + U. slix + b- ^x = 'Jx + 3. sJ^x + Vf - sj2x = \/2x + d. Vr2x^W3a;-l = \/27a:-2. Va:-(-2+\/4x+l- \/9a;+7 = 0. i^x + >j4a + x = 2 s/b + x. 27. 5 \/70x + 29= 9 ^/I4x^l5. 29. 16. 18. 20. 22. 24. 26. 28. >s/Qx^=3>/x + i-2. s/?5x-29 - sfix^U = 3 ^/x. 's/Sx-n + sjl^x = \/l2a:-2.S. ^/x + 3 + \^ +8 -\/4a;+2i = 0. \/a^ x- + \ '' + a; = v^a + v/6. Vx8-3a;2 + 7a;-fi=a;- 1. v/8^3+l2^+r2^rrr=2a:+l. 30. v^T+«+'v/r^=^2, _ 282, When radicals appear in a fractional form in an equa- tion, we must clear of fractions in the ordinary way, combining the irrational factors by the rules already explained in this chapter. Example 1. Solve 6Va;-ll^Vg+l 3^Jx >Jx + 6 ' Multiplying across, we have Qx + 25 >Jx - 66 = 6a; + 3 sjx, that is, 25 v/a; - 3 Jx =66, 22v'a:=66, v/r=3, «=9. 1. a 5. 7. oi XXX .) EQUATK .VS INVOLVING SURDS. 5 2G9 Example 2. Solve ^9 + 23; - \'2x = , Cloaring of fractions, 9 + 2.r - s/l ">T2«) = 5. 4 + 2a:- „ t-2i). Squaring, 16 + 16a; + 4a;2 == i ^ f. ix\ 16 = 2.1, a;=8. II EXAMPLES XXXI. h. Solve the equations : 1. 6^/ar-21 ^ 8Va;-l l 3v/.r-14 4Va:~13" - 2^/a;-l Jx-2 5. -=A _. 2./. + I ,/--! 7 12Va?-ll_ 6v/a; + 5 o Va;-23_6 v/a;-1 7 ^* 'S^/x-8~2^/x-6' 4 o \/a;-f3_v/a: + 9 *• x/a:-t2~Va; + 7' ft gy/g-T ^_ 7v/a;-2 6 8. \TT^ + v/a:=-fi l+K 2 9. \^^+Va;=-|r 10. V^-V^^=-7i= 11. N'a; + 5 + V.r=-^. Va; 12. 2^/x-s/4x-3=-rJ=. \'4x~3 13. 3v':r = -^=L= + V9a;-.32. 14. v/a;-7 = -r^. 15. (n/»+11)U/x-11) + 110=0. 16. 2^/x='^4i^' 17. 3v'x-l = 2^0: -3 3^3;+ 7 + 6. 18- ^=34-' 2 19. l..+ -^ + ^i_,=0. 20. 2=.^^+^ + ^2-ar _2ac-3_ \'a;-2+l \^2 + a;-\/2-a7 =2sa;-2-i. 22. .r-G + v/a; v/a;-2 V« + 3" [P'urther practice in surd equations will be found objj 3o3.J 07i pages i\ 'li» r^ !•■ MICROCOPY RESOLUTION TEST CHART (ANSI and ISO TEST CHART No. 2) 1.0 1^ llll-^-^ l£ 11111=^ ^ IK in |3.6 u Ijo ■ 4.0 IZ Km IX 2.5 2.2 2.0 1.8 J .APPLIED ilVHGE I 1653 East Main Street Rochester, New York 14609 USA (716) 482 - 0300 - Phone (716) 288- 5989 -Fax nc 11 as M -! ! ■ ' W i 1* \ [ ■ , » » *j i t?5 r c CHAPTER XXXII. Eatio, Proportion, and Variation. Batio. 283. Definition. Ratio is the relation which one quantity bears to another of the same kind, the comparison being made by considering what multiple, part, or parts, one quantity is of the other. "^ The ratio of ^ to i? is usually written A : B. The quantities il^ J ^^® ^^^^^^ *'^® ^^'''"''' ^^ *^^^^ ^^tio. The first term is called the antecedent, tlie second term the consequent. 284. To find what multiple or part A is of B we divide A by B ; hence the ratio A : ^ may be measured by the fraction ^, and W3 shall usually find it convenient to adopt this notation. In order to compare two quantities they must be expressed in terms of the same unit. Thus the ratio of £2 to 15s. is measured by the fraction ?-^ or -. 15 3 Note. Since a ratio expresses the nwnher of times that one quantity contains another, every ratio is an abstract qimitity. 285. By Art. 151, «=^ • mb and thus the ratio a :6 is equal to the ratio ma-.mh : tliat is, the value of a ratio remains xmaltered if the antecedent and the consequent are multiplied or divided hy the same quantity. 286. Two or more ratios may be compared by reducin^^ their equivalent fractions to a common denominator. Thus suppose a : h and x:y are two ratios. Now ^' = p, and •^==— ; l,ence the ratio a : h is greater than, equal to, or less tban 'the ratio X : y according as ay is greater than, equal to, or less than hx. N. one quantity \ being made luantity is of he quantities first term is uent. we divide A the fraction ;his notation. be expressed )2 to 15a. is les that one mtity. mb ; that is, dent and the itity. ducing their lius sujjpose hence ~by ' m the ratio i than b,v. CHAP. XXXII.] RATIO. 271 287. Tlie ratio of two fractions can be expressed as a ratio of a two integers. Thus the ratio | : ^ is measured by the fraction ?, ad c d or ^; and is therefore equivalent to the ratio ad : he. 288. If either or botli, of the terms of a ratio be a surd quantity then no two mtegers can be found which will exa^t measure their ratio. Thus the ratio ^2 : 1 cannot be exSy expressed by any two integers. exdcuy 289. Definition. If the ratio of any two quantities can be expressed exactly by the ratio of two integers the qiantitfes are said to be commensurable ; otherwise, they are slS to be incommensurable. , j- aic sd,iu lo oe Although we cannot find two integers wiiich will exactly measure the ratio of two incommensurable quantities, we can always find two integers whose ratio differs fJom that required by as small a quantity as we please. ^equiiea ^/5 2-236067... X= 4 = -559016... 559017 Thus and therefore 4 V5 559016 4 1000000 and < 1000000' fl~ if ''^'^^''^^ ^l'?* ^^ '^"'^'''"&' *'^« ^\^^m^^\^ further, any degree of approximation may be arrived at. ' ^ fn.???' 5^^1/iTioN. Ratios are compounded bv multinlvin^ together the fractions which denote them ; or bV mu It p y S Stslr'f antecedents for a new antecedent, afid tl e S^e? quents tor a new consequent. Example. Find the ratio compounded of tlie three ratios 2a : 3&, Qah 5c2 a. The required ratio J^x^x~ = ~ 3b 5c^ a 5^;" 291. Definition. When the ratio n • h i« cnmnounc'V TvJfh at1o of VT^''>^ '-f^ '' f If.' ^"^ ^-^ -lied tC dupiica e ratio ofa.b. Similarly a^ : b^ is called the triplicate ratio of a : 6. Also a^ : h- is called the subduplicate ratio of « : 6. m A ! f ' m^ PHi i iC m i 272 ALGEBRA. [CHAr. Examples. (1) The duplicate ratio of 2a : 3?) is ia^ : %\ (2) The siibduplicate ratio of 49 : 25 is 7 : 5, (3) The triplicate ratio of 2j; : 1 is Sa:^ : 1. 292. Definition. A ratio is said to be a ratio of greater inequality, of less inequality, or of equality, according as the antecedent is greater than, less than, or equal to the consequent. 293. If to each term of the ratio 8 : 3 we add 4, a new ratio 12 : 7 is obtained, and we see that it is less than the former because -=- is clearly less than ^• 7 •! This is a particular case of a more general proposition whicli we shall now prove. A ratio of greater ineqiiality is diminished, and a ratio of less ineqvmity is increased^ by adding the same quantity to both its terms. Let - be the ratio, and let '^Jl_ be the new ratio fornifd by adding x to both its terms. i + .^• Now a a ■ __ ax — b.v b" b+.);~b{b+.v) _x{a — b)^ b{b+x) and a-& is positive or negative according aa a h greater or less than b. Hence if a > b. a a + x , 1) b+x* a a+x b b+x^ and if a< b, which proves the proposition. Similarly it can be proved that a raii greater inequality is increased, and a ratio of less inequality is di-^inished, by taMng the same quantity from both its terms. 294. When two or more ratios are equal, many useful pro- positions may be proved by introducing a single svmbol to denote each of the equal ratios. The proof of the following important theorem will illustrate the method of procedure. if [CHAV, ' •. 962. 7:5. :1. tio of greater rding as tlie consequent. I, anew ratio 1 the former asition wliicli I ratio of less ty to both its io formc'i] liy is greater or er ineqnalit}] led, by tnh'ng ' useful pro- e svmbo! to all illustrate xxxn.] RATIO. a_c_e_ b~d~f~" 273 each of these ratios = ( tplS^L+L^l+^Y \pb» + qd'' + rf» + .../' where p, q, r, n are any quantities whatever. Let h~d~f ^' then .■I a=hk, c=dk, e=fk, ... ; whence pa''=pb"k'', qc'' = qd"k'', re" = rf"k", ... ; • y^"+gc"+ye " + ... _ pb"k'' + qd"k'' + r f"k" +... " pb" + qd» + rf'+...-' pb'' + qd" + rf'^+77. . f pa"+qc'' + re'' + ... Y J g ^ . By giving different values to p, q, r, n manv particular cases of this general proposition may be deduced ; o.- thev mav be proved independently by using the same method. For instance, if T=|=J, each of these ratios =~^,t^, : a result which will frequently be found > eful ExampU 1. If -=f find the value of E^^^, 5a; -3y y 4^3 7a? + 2y Ix , - 21 „~29' r ^ ~7~ + 2 y i Example 2. Two numbers are in the ratio of 5 • 8 If 9 be -{(led to eacli they are in the ratio of 8 : 11. Find the numbers. Let the numbers be denoted by 5x and 8x. Then 5x + 9^ 8 . iix + 9 11' .-. .r = 3. Hence the numbers are 1.5 and 24. £.A. S Wl w 1 **,- 1 «'» 274 ALGEBRA. [chap. Example 3. If ^ : iB be in the duplicate ratio of A+x\ B-\-Xt prove that x'^ = AB. By the given condition, { ■ „ ) ~~r* :. B{A+xf=A{B + x)'^, A'^B + 2ABx + Bx'^=AB^+2ABx+Aoi^, x^{A-B) = AB{A-B); .'. x^=AB, since A- B is, by supposition, not zero. EXAMPLES XXXII. a. Find the ratio compounded of 1. The duplicate ratio of 4 : 3, and the ratio 27 : 8. 2. The ratio 32 : 27, and the triplicate ratio of 3 : 4. 3. The subduplicate ratio of 25 : 36, and the ratio 6 : 25. 4. The ratio 169 : 200, and the duplicate ratio of 15 : 26. 5. The triplicate ratio oi x : y, and the ratio 2y^ : Sx^. 6. The ratio 3a : 4b, and the subduplicate ratio of b* : a*. 7. If a; : y = 5 : 7, find the value of x + y : y-x. 8. If -=3|, find the value of ^f- y ^x — oy 9. If 6 : a= 2 : 5, find the value of 2a - 3?) : 3& - a. 10. If r = 7, and - = =, find the value of ' ^ ~- ^ . 6 4 y 1 prove that each of these I'atios is equal to b~d~f '4. 2a2c + 3c3e + 4e2o_ 2b'^d + M\ + 4pd' xxxir.] RATIO AND PROPORTION. 275 :25. .:26. x^. *:a\ 16. Two numbers are in the ratio of 3 : 4, and if 7 be subtracted from each the remainders are in tlie ratio of 2 : 3. Find them. 17. What number must be taken from each term of the ratio 27 : 36 that it may become 2:3? 18. What number must be added to each term of the ratio 37 : 29 that it may become 8:7? 19' W63^=^=^,shewthat2? + s + r=0. 21. ^^ T — ^ — 'fi shew that the square root of a%-2c^e + 3a*(^ e^ . ace V - 2rf«/+ Zh*cdh'' ^^ ®^"^' ^ hUJ' 22. Prove that the ratio fe + mc + 7ie : Ih + md + nf will be equal to each of the ratios a :h, c: d, e :f, ii these be all equal ; and that it will be intermediate in value between the greatest and least of these ratios if they be not all equal. OQ Tf ^■^~*y cx-az z + y ^, ^' ^^ 7^r^"fc^^^r^=^Ti' ^^^" "^'^^ ^^""^ ^^ *^^^«« fractions be equal to -, unless 6+c=0. ni jf2x-^y z-y x + Sz equal to - ; hence shew that either x=y, or z = x + i/. t H ''I 3^1 IJ(I» Ai' U. Proportion. 295. Definition. When two ratios are equal, the four quantities composing them are said to be proportionals. Thus " 3=^ then a, b, c, d are proportionals. This is expressed by saying that a is to 6 as c is to d, and the proportion is written or a:b:',c:d , a:b = c:d. The terms a and d are called tlie extremes, b and c the means. a- > 1 "» I- 1 ! i c -« •I ]'-a «■ ' , ■i m S; 1 i w •■ 1 1 ' 1 - t 1 : ti ■' ; i^ r fa 276 ALGEBRA. [chap. 296. If four qtiantities are in proportion^ the prodtict of the extremes is equal to the product of the means. Let a, 6, c, d be the proportionals. Then by definition 7~^ ' whence ad=hc. Hence if any three terms of a proportion are given, the fourth may be found. Thus if cr, c, d are given, then &= — G)nversely, if there are any four quantities, a, h, c, d, such that ad=hc, then a, 6, c, d are proportionals ; a and d being the extremes, h and c the means ; or vice versS,. 297. Definition. Quantities are said to be in continued proportion when the first is to the second, as the second is to the third, as the third to the fourth ; and so on. Tlius a, b, c, d, are in continued propoi'tion when bed If three quantities a, 6, c are in continued proportion, then a :b=b :c ; .-. ac=b\ [Art. 296.] In this case b is said to be a mean proportional between a and c ; and c is said to be a third proportional to a and 6. 298. If three quantities are proportionals the first is to the third in the duplicate ratio of the first to the second. Let the three quantities be a,b,c; then t = — n n. h n n ri^ Now a _a b _a a_a^ c~b^c~b^b~¥'' that is, a:c=a^ ib^. 299. If a : b=c : d and e : f =g : h then mil ae : bf=cg :dh. . Fnr «-^ and ^-^ •'• ^^M' °^ ae:bf=cg'.dh. m [chap. Todiict of the i given, tlie lien 0= — . c b, c, d, sucli 1 d being the n continued he second is 30 on. ThuH rtion, then XXXII.] PROPORTION. 277 Cor. If a:b=c:d, '"^"d h'.x=d:y, then a\x=c :y. This is the theorem known as ex cequali in Geometry. 300. If four quantities, «, h, c, d form a proportion, many other proportions may be deduced by the properties of fractions^ Tlie results of these operations are very useful, and some of them are often quoted by the annexed names borrowed from ijreometry. (1) li a:h = c:d, then h : a=d : c. [Invertendo.] For 1=^ ; therefore 1-|=1^^; that is, k==i. a c ' or b'.a=d:c. (2) If a : b=c : d, then a : c=b : d. For ad=bc ; therefore ^-^ . cd cd ' [Alternando.] ;Art. 296.] that is, ?-^. c~d' 8tween a and d6. or a.c=b:d. Irst is to (lie (3) If a : b=c : d, then a + b : & = c+(? : d Pnv ?_^ . ^u *„„„ « . , c . , [Co??tjoowe?jc?o.] that is. a+b _c+d b dT' ^^ a + b:b = c+d:d. (4) If a : 6=c : rf, then a-6 : b=c-d : c/. For ^=«, therefore?- 1=5-1. [Z)iV endo.] that is, or 6 * (^ a — Z» (! — t? a-b : b = c-d : d. t i-. ^ II' ;m m '•nil kil:) r^ 278 ALGEBRA. [chap. I I IB 't « 1' • 4 ki » » 1 r Ke tw r %' c r fao ^ ito,: 1 '"'• t - 1 f^^ (5) If a: b=--c:dy then a+& : a-6=c+c/ : c-fi?. a + 6 c + fl?. For by (3) and by (4) .*. by division, b ~ d a — b_c — d , a+b c+d a-b c — d' or a-\-b:a-b — c-\-d:c~d. Several other proportions may be proved in a similar way. 301. The results of the preceding article are the algebraical equivalents of some of the propositions in the fifth book of Euclid, and the student is advised to make himself familiar with them in their verbal form. For example, dividendo may be quoted as follows : When there are four proportionals, the excess of the first above the second is to the second^ as the excess of the third above the fourth is to the fourth. 302. We shall now compare the algebraical definition of proportion with that given in Euclid. In algebraical symbols the definition of Euclid may be stated as follows : Four magnitudes, a, b, c, d are in proportion when pc=.qd according as pa^qb,p and q being any positive integers whatever. I. To deduce the geometrical definition of proportion from the algebraical definition. Since ?=§, by multiplying both sides by ^, we obtain o d q pa_pc . qb qd' hence, from the properties of fractions, pc^qd according as pa ] which proves the proposition. 'qh or [chap. XXXII.] PROPORTION. 279 ilar way. 3 algebraical fth book of amiliar with ndo may be he first above rd above the definition of ay be stated hen pc = qd jers xohatever. portion from »btain II. To deduce the algebraical definition of proportion from the geometrical definition. Given that pc ^ qd according m pa ^ qb, to prove a b c ''d; If ^ is not equal to -„ one of them must be the greater. it c Supi)ose i> -fl then it will be possible to find some fraction 2 which lies between them. Hence and «>2 b p a p •(1), •(2), *>om (1) pa>qb; from (2) pc q h a ^ ah _d c cd ab 'a^ + b^~c'^d'^'(P' Solve the e([uatiori3 : 19. 3x-l:Qx-7 = 1x-lO:9x + lO. 20. a;-12:y-f3 = 2u,-19: r)y-13=5: 14. x^~2x + 3_x^-3x + 5 ^ 2.f-l 21. -. 2x-3 3a; -5 22. ^•f4_ a;2 + 2.c-l x" + a; + 4' 23. If {a + b-3c-3d)(2a-2h-c + d} = (2a + 2b - c -d){a -b -3c + 3d) prove that a, h, c- d are proportionals. 24. If o, h, c, d are in continued proportion, prove that a : d=a^ + b^ + <^ -. fts + c^+rfs. 25. If i is a mean proportional between a and c, shew that 4a - 96' IS to 4^2 - 9c*> in the duplicate ratio of a to 6. 26. If a, 6, c, d are in continued proportion, prove that 6 + c is a mean proportional between a + b and c + d. 27. If a + h:b + c = c-\-d'.d + a, prove that a = c, or a + 6 + c + d=0. Variation, 303. Definition. One quantity A is said to vary directly as aiiotlier B, wlien the two quantities depend upon each other in such a manner that if B is changed, A is changed in the same ratio. The word directly it often omitted, and A is said to vary Note. as B. 304. For instance : if a train moving at a uniform < ,ate travels 40 miles in 60 minutes, it will travel 20 miles in 30 minutes, 80 miles in 120 minutes, and so on j the distance in each ca^being increased or diminished in the same ratio as the time. This is expressed by saying that when the velocity is uniform the distance is proportional to the time, or more briefly, the dutance varies as the time. Hi 1 1 v^ m 282 IS • E i m r 1 u ■i I ALGEBRA. [chap. Again, if we refer to the general formula of Art, 84, we find that- = v is a relation connecting the space described by a body which moves for a time t with uniform velocity v. That is, if «i, 52, *3... be spaces described in times t^, t^^ ^g ... respectively, we have ^=^=!?^ =y. ^1 ^2 ^3 From this it appears that the ratio of any value of s to the corresponding value of t is constant, that is, remains the same whatever numerical values s and t may have. * This is an instance of direct variation, and s is said to van/ as t. 305. The symbol oc is used to denote variation ; so that A cc B is read " A varies as B." 306. If A varies as B, then A is equal to B multiplied hy some constant quantity. For suppose that a^, a^, ag..., 6j, 62, 63... are correspondinrj values of A and B. Then, by definition, # = #;^ = #;- = ^; and «i Oi a^ 62 «3 ^ so on. A = 7- =7-=..., each being equal to -^ 1^2 O3 // Hence that is, any value of A th^^^i^sponding value oFB '^ ^^^^>^' ^'^ ^'^""'^ -D=w, where m is constant. .-. A=mB. 307. Depin rTioN. One quantity A is said to vary inversely as another B when A varies directly as the reciprocal of B. [See Art. 107]. Thus if A varies inversely as B, A = -^, where m is constant. JtS The following is an illustration of inverse variation : If 6 men do a certain work in 8 hours, 12 men would do the same woik in 4 hour.'?, 2 men in 24 hours ; and so on. Thus it appears that when the number of men is increased the time is proportionately decreased j and vice vers&. or ll [chap. ■t. 84, we find >ed by a body -'. That is, if respectively, e of s to tlie ins the same said to vtm/ ion ; so that i'pUed hi/ some orresponding XXXII.] VARIATION. 283 l?U d so i B' 3ame ; on. ry inversely procal of B. 8 constant. on : If 6 men ame work in tppears tlsat portionately 308. Definition. One quantity is said to vary jointly as a number of others when it varies directly as their product. Thus A varies jointly as B and C when A =viBC, where m is constant. For instance, the interest on a sum of money varies jointly as the principal, the time, and the rate per cent. 309. Definition. A is said to vary directly as B and in- versely as C when A varies as ^. 310. If A. varies as B when C is constant, and A varies as C ichen B is constant, then ivill A vary as BO when both B and C vary. The variation of A depends partly on that of B and partly on that of C. Suppose these latter variations to take place sepa- rately, each in its turn producing its own effect on A ; also let a, h, c be certain simultaneous values oi A, B, C. 1. Let G be comtant while B changes to b ; then A must undergo a partial change and will assume some intermediate value a', where a' b ^^•'• 2. Let B be constant, that is, let it retain its value b, while C changes to c ; then A must cc lete its change and pass from its intermediate value a! to its faiitil value a, where a' a' C c A a' a' X -: a B G =vx-; c A: a ~bc' BG. A varies as BG. .(2). From (1) and (2) that is, or 311. The following are illustrations of the theorem proved ui the last article. The amount of work done by a gii;m number of men varies directly as the number of days they work, and the amount of work done in a given time varies directly as the number of men ; tiierefore when the nimiber of days and the number of men are both variable, the amount of work will vary as the product of the number of men and the number of days. DID iiff \U ! ig m : I m I I i f I J » . -I € n B r p! r c r 1 »■. I 1 284 ALGEBRA. [chap. Again, in Geometry the area of a triangle varies directly as its base when the height is constant, and directly as the heirrht when the base is constant ; and when both the height and base are variable, the area vai'ies as the product of the numbers representing the height and the base. Example 1. li A !i» 'In w tat » r f 286 ALGEBRA. [chap. XXXII. 20. Given that the area of a circle varies as the square of its radius, and that the area of a circle is 154 square feet when the radius is 7 feet ; find the area of a circle whose radius is 10 feet 6 inches. 21. The area of a circle varies as the square of its diameter ; prove that the area of a circle whose diameter is 2i inches is equal to the sum of the areas of two circles whose diameters are 1^ and 2 inches respectively. 22. The pressure of wind on a plane surface varies jointly as the area of the surface, and the square of the wind's velocity. The pressure on a square foot is 1 lb. when the wind is moving at the rate of 15 miles per hour ; find the velocity of the wind when the pressure on a square yard is 16 lbs. 23. The value of a silver coin varies directly as the square of its diameter, while its thickness remains the same ; it also varies directly as its thickness while its diameter remains the same. Two silver coins have their diameters in the ratio of 4 : .3. Find the ratio of their thicknesses if the value of the lirst be four times that of the second. 24. The volume of a circular cylinder varies as the square of tlie radius of the base when the height is the same, and as the height when the base is the same. The volume is 88 cubic feet when the height is 7 feet, and the radius of the base is 2 feet ; what will be the height of a cylinder on a base of radius 9 feet, when the volume is 396 cubic feet ? 25. The altitude of a triangle varies directly as its area and inversely as its base. A triangle, 2 square yards in area, standing on a base of 13|- feet, has an altitude of 2| feet : find the altitude of a triangle whose base is 1 foot 4 inclies, and whose area is 2 square feet 96 inches. 26. The expenses of a school are partly constant and partly vary as the number of boys. The expenses were £1000 for 150 boys and £840 for 120 boys ; what will the expenses be wlien there are 330 boys. [chap, xxxii. le square of its [uare feet when whose radius is iiameter; prove inches is equal ameters are 1|, i jointly as the vind's velocity. in the wind is the velocity of s 16 lbs. le square of its ! ; it also varies lains the same. ratio of 4 : 3. of the lirst be 5 square of the ne, and as the me is 88 cubic I of the base is r on a base of t? ! its area and yards in area, de of 2| feet : foot 4 inches, partly vary as )0 for 150 boys uses be when CHAPTER XXXIII. Arithmetical Progression. 312. Definition. Quantities are said to be in Arithmetical Progression when they increase or decrease by a cominon differ- ence. ■" Thus each of the following series forms an Arithmetical Progression : 3, 7, 11, 15, 8, 2, -4, -10, a, a+c?, a-VM, a + 3c^, The common difference is found by subtracting any term of tlie series from that vMg\i folloxcs it. In the first of the above examples the common difference is 4 ; in the second it is -6 • in the third it is d. ' 313. If we examine the series a, a-Vd, a+2d, a + 3d, we notice that in any term the coefficient ofd is always less by one than the number of the term in the series. Thus the 3"^ term is a + 2(^ ; 6"" term is a + 5c^; 20"' term laa + lQd; and, generally, the p-^ term is a + (p - \)d. If n be the number of terms, and if I denote the last, or ^i'" term, we have l=a + {n--[)d. 314. To find the s^m of a number of terms in Arithmetical J rogression. Let a denote the first term, d the common difference, and n the number of terms. Also let I denote the last term, and s the required sum ; then ' s^a + (a + d) + (a+2d) + +(l-2d) + (l-d)+l ; and, by writing the series in the reverse order, s=l+{l-d) + (l-2d)-{-.......^r(a + 2d)+(a+d)+a. i» c i ii: 1 m f K ' i m « '|i ^ ,i 1 Its y m m fer r em r 288 ALGEBRA. [chap. Adding together these two series, 2s=(a + l) + (a + l) + {a + I)+...... ton terms .-. s=l(a + l) (1); and l=a+(n-l)d (2), /. s=^{2a+{n-l)d} (3). 315. In the last article we have three useful formulfe (1), (2), (3) ; in each of these any one of the letters may denote the unknown quantity when the three others are known. [See Art. 82, Chap, ix.] For instance, in (1) if we substitute given values for «, n, I, we obtain an equation for finding a ; and simi- larly in the other formulae. But it is necessary to guard against a too mechanical use of these general formulae, and it will often be found better to solve simple questions by a mental rather than by an actual reference to the requisite formula. Example 1. Find the 20"" and SS"" terms of the series ' 38, 36,34, Here the common difference is 36 - 38, or - 2, .-. the 20"> tenn = 38 + 19 ( - 2) = 0; and the SS"" term = 38 + 34 ( - 2) = -30. Example 2. Find the sum of the series 5| , 6f, 8, ...... to 17 term3. Here the common difference is l^ ; hence from (3) The sum =~|2x^+16xl|.| =^(11+20) _ 17x31 ~ 2 =263^. 7. [CHAP. terms (1); (2), (3). formulic (1), may denote mown. [See stitute given a ; and simi- ;uard against . it Avill often lental ratlier BS , to 17 terms. XXXin.] ARITHMETICAL PROGRESSION. 289 Example 3 The first term of a series is 5, the last 45. and the sum 400: find the number of terms, and the common difference If n be the number of terms, then from (I) 400=1 (5 + 45); v.Jicnce n = \Q. If d be the common difference, 45 = the IS"* term wlience d=2^. EXAMPLES XXXIII. a. 1. Find the 27"' and 41" terms in the series 5, 11 17 2. Find thb IS"- and 109'" terms in the series 71, 70 69 " * 3. Find the 17"- and 54'" terms in the series 10 llV n' "" ' 4. Find the 20'" and 13"" terms in the series -3 -2 -l"' 5. Find the 90'" and 16'" terms in the series - 4' 2-5 ' 9 6. Find the 37'" and 89«" terms in the series - 2-8, 0,' 2'8,"'..*. . Find the last term in the following series : q nl' q' di*' ^ T,T ^' ^' ^' - *' - ^° ^^ terms. n. 27,34, 4 1, ...to 11 terms. 12. a:. 2^, 3.r, ... to25 terms. 16. n-d,a + d,a + Sd,... to 30 terms. 14. 2a - b, 4^. - 36, 6a - 56, . . . to 40 terms. Find the last term and sum of the following series : 15. 14,64, 114, ...to 20 terms. 16. 1, 1-2, 1-4, ... to 12 terms. 17. 9, 5, 1. ... to 100 terms. 18. i. -1 -| ... t^ 21 terms. 19. 3i, 1, - 1 J, ... to 19 terms. 20. 64, 96, 128, ... to 16 terms. Find the sum of the following series : 21. 5, 9 13, to 19 terms. 22. 12, 9, 6, ... to 23 terms. 23. 4 5| 6|,„.to37t.ermH. 24. lO^ 9, 7I, ... to 94 terms. ^. -3, 1. 5, ... to 17 terms. 26. 10, 9f, 9j, ... to 21 terms. ^ P, 3p, 5p, ... top terms. 28. 3a, a, -a, ... to a terms. k1 ""' ■•• *° " *^''™'' ^' -^^' -^' «' ■ • top terms. 290 ALGEBRA. m. 11* 1 1; It r erf r ^^^^H '■ ; . ■ 1 iri. ^^1 ; I r-> ^^^^H - • ' I ^^^^^M -■ i :; 1 ^^^H';;: i-« 1 '1 ^^^^^^^■'^ ^^^^^^K^ ' ^^^^K> -' t ^. i ; 6!i r ' I ^" -16 = the 100"" term = 17 + 99(i; The 30'" term = 17 + 29 =7- (-1) ^- I li i 4 I: liCKI. 1 ^ Ml 11 .II =1' If Vf 292 ALGEBRA. [OITAP. Example 2. The sum of three numbers in A. P. is 33, and their product is 792 ; find them. Let a be the middle number, d the common diflference ; then the three numbers are a-d, a, a + d. Hence a-d + a + a + d=3S; whence a =11, and the three numbers are 11- d, 11, 11 + rZ, .-. 11(11 +rf)(ll-d) = 792, 121-fi'=72, d=±7; and the numbers are 4, 11, 18. Example 3. How many terms ol' the series 24, 20, 16, must be taken that the sum may be 72 ? Let the number of terms be n ; then, since the common difference is 20 - 24, or - 4, we have from (3), Art. 314, 72=|{2x24'i7.-l)(-4)} =24»-2»(n-l); whence ii^ - 13« + 36 = 0, or (»-4)(n-9) = 0; .•. n = 4 or 9. Both these values satisfy the conditions of the question ; for if we write down the first 9 terms, we get 24, 20, 16, 12, 8, 4, 0, - 4, - 8 ; and, as the last five terms destroy each other, the sum of 9 teinis is the same as that of 4 terms. Example 4. An A. P. consists of 21 terms ; the sum of the three terms in the middle is 129, and of the last three is 237 ; find the series. Let a be the first term, and d the common difference. Then 237 = the sum of the last three terms =a+20d+a + 19d+a + 18d = 3a + 57d; whence a + 19rf=79 (1). Again, tlie three middle terms are the 10'^, 11"', 12"' ; hence 129= the sum of the three middle terms =a + 9d+a + 10dta + lld = 3a + 30c?; whence a + 10rf=43 , (2). From (1) and (2), we obtain d = 4, a=3. Hence th« series is 3, 7, 11 S3. 15. 16. 17. 18. 19. 20. 21. LI [OHAP. is 33, and tlieir rence ; then the U+d. 50, 16 must nmon difference istion ; for if we ?, 4, 0, -4, -8; im of 9 terras is iim of the three ; find the series. ice. Then ns .(1). erms ii)>* How many terms must be tsiken of 15. The series 42, 39, 36, to make 315? 16. The series -16,-15,-14, to make - 100 ? 17. The series 15|, 15 J, 15, to make 129? 18. The series 20, ISf, 17^, to make 1621? 19. The series - 10^, - 9, "- 7^, to make - 42 ? 20. The scries - 6i, - 6f , - 6," to make - 52*. ^^' '^^"'find them """"^^""^ ^" ^■^- '^ ^^' ^^'^ ^heir product is ^" '^sonir,?!fift''fl"?lf "'' '" ^•^- ^« 1-' '-^"^^ th« S"'" oi their squares is 66 ; find them. m !JX I # -1 1 \ ^ : " Z i -tt^ . wb fi 4ft «!■ r f* • C .** : ' I mi ^^^^^1. «l ^^^H •t ^^^H » ^^^H<' » ^^m f ^^^^^H ■u ^^^^H t" ^^m r ^^B f f ^R c H r ^^^^^^B IK ^^H; ^ ^^^Hv ■ . 1 tv ^^■:> ! t ^^^B^'"' -» ?» « I*:) 1 CHAPTER XXXIV. Geometrical Progression. 321. DEFixiTrof* Quantities are said to be iu Geometrical Progression when they increase or decrease by a "onstant factor. Tlins each of the following series forms a Geometrical Pro- gression : 3, 6, 12, 24, 1-11-1 ' 3' 9' 27' a, ar, ar^, ar^, The constant factor is also called the common ratio, and it h found by dividing any terra by that which immediately precede, it. In the first of the above examples the common ratio i^ 2 ; in the second it is - ^ ; in the third it is r. o 322, If we examine the series a, nr, ar\ a)^, a;**, we notice that in any term the index of r is always less hy one than the member of the term in the series. Thus the 3'^ terra is ar^ ; the 6"" term is a?-*^ ; • the 20"' term is ar^o ; and, generally, the p"" term is ar'^~'^. If n be the number of terms, and if I denote tho last, or n^ term, we have I = ar"~\ 11 3 Example. Find the 8* term of the series -s, 5, -t, The common ratio iss-rf -5), or -^; .'. theS«'term=-lx(-|y 1 2187 ~ " 3 ^ " 128 "138" 11 Geometrical "onstant factor. oroetiical Pro- ratio, and it i) iiately prcccdi , nion ratio i;^ 2 ; ays less hy one tho last, or n^ 3 "4' ' ClIAP. XXXIV.] GEeMETUICAL PROGRESSION. 295 323. D-^FiNiTiox. Whon three quantities are in Geometrical ProgrcsMion the middle one is called the geometric mean be- tween the other two. Tojind tho geometric h in between two giv-,! 'quantities. Let n and b be the two quantities; O the geometric mean. Then mnce a, O, b are in (i.P., each being equal to the common ratio ; whence 0=^s'^. 324. To insert a given mmber of geometric means Nttveen (ii'o given qtiantities. Let a and b be the given quantities, n the number of means. Ill all there will be n + 2 terms ; so that v/e have to find a scries of w + 2 terms in G.P., of which a is the first and b the last. Let ;• be the common ratio ; tlien 6 = the0i + 2)"'term = „;.«+l. ,...+i_^. / — ) a ■■ -0 bY^ ' (n Hence the required means are ar, ar\ af\ where r Ins the value found in (1). Example. Insert ^ geometric means between 160 and 5 Wo have to find 6 terms in G.P. of which 160 is the first, and 5 the sixth. Let r be the common ratio ; then 5 = the sixth term = lG0r5; • ^ L • whence, by trial, and the means are 80, 40, 20, 10 1' — — • 296 LI,- II I m m i> B 19 r b P» r '3. r ( '■' ALGEBRA. [CIIAP. 325. 7b /^c? ^Ae «mw &/ a number of tenns in Oeometrical rrogression. Let a be the firat term, »• the common ratio, n the number of terms, and 5 tlie sum required. Then s=a+ar-\-ar^+ + ar"--+ar''-* ; multiplying every term by »•, we have rs =ar-}rar^+ + ar»-2 + ^^^-i + a,r\ Hence by subtraction, rs-s=ar'^ -a ; .'. {r-\)s=aif-l); -^(^J-0 s=- r-1 •(I). Changing the signs in numerator and denominator [Art. 170.] 1 -7" (2> Note. It will be found convenient to remember both forms in\m ih^ll ^' "^'"^ ^^^ ^" "^^^ '^^^^^ ^"^^^^^ ^^^° *' ^^ /wz^tye and greater Since ar"-i=;, the formula (1) may be written rl-a a form which is sometimes useful. Example 1. Sum the series 81, 54, 36 to 9 terms. The common ratio =^=|, which is less than 1 ; hence the sum j±M. -i =-{•-(!)"} =243- 512 =236|j. [chap. in Oeometrical the number of XXXIV. ] (1). r [Art. 170.] (2). )th forms given tive and greater nils. GEOMETRICAL PROGRESSION. 3 297 ExamiHtl, Sum the series g, -1, | to 7 terms. 3 The common ratio =x -^ • hence by formula (2) the sum ^iL-A-^Zj. U-l |[iH-?lHl 128 J 5 2 _2 2315 2 3^ 128 ^5 _463 EXAMPLES XXXIV. a. t Find the S"" and 8"- terms of the series 3, 6, 12, ... . 2. Find the 10'" and 16'" terms of the series 256, 128, 64, ... , 3. Find the 7"" and 11«> terms of the aeries 64, - 32, 16 4. Find the 8'" and 12'" terms of the series 81, - 27, 9, ... . 5. Find the 14'" and 7'" terms of the series -L .1 ^ 64' 32' 16 6. Find the 4'" and 8"' terms of the series -008, -04, -2, ... . Find the last term in the following series : 7. 2, 4, 8, ... to 9 terms. 8. 2, - 6, 18, ... to 8 terms. 9. 2, 3, 4|, ... to 6 terms. 10. 3, - 3^, 3', ... to 2« terms. 11. X, ar», a*, ... to /) terms. 12. a;, 1 , 1, . . . to 30 terms. X 13. Insert 3 geometric means between 486 and G. 14. Insert 4 geometric means between - and 128. 9 15. Insert 6 geometric means between 56 and - — . 16 16. Insert 5 geometric means between if and 4i. bl * 111 lit 298 P ■ I 1 ■ . tt-J ! P • ^ « m »' i' ( I ..!> 1 » 4 11 1 M ■»'■«! i t ^ 1^^ b B (1* to ltd r r ■^ 1 '■'■ 1 11 1 ALGEBRA. tcHAf. Find the last term and the sum of the following series : 17. 3, 6, 12, ... to 8 terms. 18, 6, - 18, 54, ... to 6 terms. 19. 64, 32, 16, ... to 10 terms. 20. 8-1, 27, '9, ... to 7 terms. 21- 72' 24' 8""' *°^*®^'"^' 22. 4|, U, 2,... to 9 terms. Find the sum of the series 23. 3, - 1, g, ... to 6 terms. 24. J, \, | ... to 7 terms. 2 1 2' 3' 9' 1 1 ^' ~5' 2' -§»••• to 6 terms. 26. 1, -g. j, ... to 12 terms. 27. 9, -6, 4, ... to 7 terms. 28. | -^ .■^, ... to 8 terms. o b ^£4 29. 1, 3, 32, ... topterms. 30. 2, - 4, 8, . . . to 2p terms. 1 "i 31- ^'^'js'-^^^ ^^^^^' 32. v/a, v/«^ V«^ •.. to a terms. 1 fi 33. ;;y2' ~ ^' ;^' •■• *^° ^ ^^""^- ^' ^^' *^^' ^v^^, ... to 12 terms. 326. Consider the series 1, 1 i^, -\, The sum to w terms 2" =), dividing every term by ahc, llll c b~b a which proves the proposition. 331. Harmonical properties are chiefly interesting because of their importance in Geometry and in the Theory of Sound : in Algebra the proposition just proved is the only one of any importance. TIifira-4k~iiQ_gp.naiaJfQnmiln for the sum of an" "'"^ber. of fluantitie8.iiLJI^aimiicaI.Prog.resaion. Questions in H.P. are generally solved by inverting the terms, and making use of the properties of the corresponding A.P. ri I ! c m m t :4 < s ,P » a r b r ■c f ; 1 302 ALGEBRA. [chap. Example. The 12"" term of a H.P. is |, and the 19"' term is — : find the series. Let a be the first term, d the common difference of the correspond- ing A. P. ; then ^ 5= the 12"' term =a + Ud; and y =*^^ 1^"" *®^'" = a + \M; whence cl=- a = -. Hence the Arithmetical Progression is -, -, 2, 2, and the Harmonical Progression is - - z ?. ^ 4' 5' 2' 7' 332. Tojind the harmonic mean hetiveen two given quantities. Let a, 6 be the two quantities, H their harmonic mean j then -, •^j J are in A. P. , 1_1_1_ 1 • H a~b W 1=1+1 rt + 6 333. If A, G, H be the arithmetic, geometric, and harmonic means between a and 6, we have proved ^ = -2- 0)- G'-v/^ (-2). ^^=^ (3> Therefore ^//=^ . ^^ 2 a+6 = 6^2. that is, 6* is tlie geometric mean between A and //. liii [chap, 19'" term is 1: the correspond- ven quantities. monic mean ; and harmonic (I). (2). (3). XXXV.] HARMON ICAL PROGRESSION. 303 334. Miscellaneous questions in the Progressions afford scope for much skill and ingenuity, the solution being often very neatly effected by some special artifice. The student will find the following hints useful : 1. If the same quantity be added to, or subtracted from, all the terms of an A. P., the resulting terms will form an A.P. with the same common difference as before. [Art. 312.] 2. If all the terms of an A.P. be multiplied or divided by the same quantity, the resulting terms form an A.P., but with a new common difference. [Art. 312.] 3. If all the terms of a G.P. be multiplied or divided by the same quantity, the resulting terms form a G.P. with the same common ratio as before. [Art. 322.] 4. li a, h, c,d, be in G.P., they are also in continued •proportion, since, by definition, <^_h _c _ _1 b~c~d~ ^7' Conversely, a series of quantities in continued proportion may be represented by x, xr, xr^, Example 1. Find three quantities in G.P. such that their pro- duct is 343, and their sum 30^. Let -, a, ar be the three quantities ; then we have and From (1) .-. from (2) Whence we obtaii -xaxar=343 r a{^-+\ + r^ = 91 3 (1), .(2). a3=343, a-7; 7(l+r+r2)=^r. r=3, or I 3' and the numbers are f, 7, 21. m 11. Ill .' I,. ! E m t a r m t 133 ■ I! f" I 304 ALGEBRA. [chap. Example 2. If a, h, c be in H.P., prove that -r^. — , -^ calsoinH.P. o + c c+a. a+h are also in H. P. Since -, r. - are in A. P., a c a + b + c a + b + c a + b + c ' » ^ -» — 5 — I are in A.P. ; a c a + h * + -^) l + -j^, 1 + — — are in A.P. ; b + r a a 6 + c' a + c c + a' « + ?> c c a + b are in A. P. ; are in H.P. Example 3. The ji"- term of an A.P. is =+2, find the sum of 49 terms. ^ Let a be the first term, and I the last ; then by putting n=l, and n=49 respectively, we obtain 1 4Q 4Q = 1^x14=343. Example 4. If a, 6, c, d, e be in G.P., prove that b + d k the geometric mean between a + c and c + e. Since a, 6, c, d, e are in continued proportion, s 6~c~rf~e' a+c b+d :. each ratio Whence b+d c+e' {b + df=(a + c)(c+e). [Art. 204,] EXAMPLES XXXV. 1. Find the 6'" term of the series 4, 2, 1^, .... 2. Find the 21" term of the series 2i li|^ lA. - • 3. Find the 8'" term of the series 1^, \\\, 2j\ 4. Find the «'" term of the series 3, 1|, 1 19. 20. 21. 22. 23. 24. 25. 26. [chap. _o_ b c ' + c' c+a' a+l 5 . • • » ■ ■ » > . • » ind the sum of ttmgji=l, and at b + d is the [Art. 204,] XXXV.] HARMONICAL PROGRESSION. 305 Find the series in which 5. The IS'" term is ~=, and the 23^ term is 1. ^ 41 6. The 2»^ term is 2, and the 31" term is — . 81 7. The 39'Herm is 1 and the 54"' term is -i. Find tlie liarmonic mean between 8. 2 and 4. 9. 1 and 13. in 1 and 1 11. - and T. 12 anA ._L_ 10 « * • x+1/^^^x^- 13. a^ + i/vmlx-ij. 14. Insert two harmonic means between 4 and 12. 15. Insert tJiree harmonic means between 2f and 12. 16. Insert four harmonic means between 1 and 6. ^'^' ^llSi^?! geometric mean between two quantities A and H shew that the ratio of the arithmetic and haJnonfc means of ieZt oi G IZT ''^ ^^^^^ '' ^^^ -^^-"« and CZdJ 18. To each of three consecutive termq nf n n t* +1, i r the three i, added. Shew l^'Ztti ^^^ ^^ Sum the following;; series : 19. l + l| + 33^+ toGterms. 20. l + l|+2^+ toGterms. 21. (2a + x) + 3a + {ia-x)+ to^Uerms. 22. ^l-li+g- toSterms. 23. 1| + 1|. + |4. to 12 terms. ''• " a™s„!;-^iii«"e„';i''.«;S-/-r *"' ^'^-«' '» '"" 26. If a^ ^=, c2 be in A.P. F,A, prove that b + c, c + a, a + b are in H.P, I ill ^ c 1* : « ■ 1 ' kr 1 D a i lu O) 1 f • 1 : 1 I f ff ■«* i r «K ^; ' ' 1 1 t'^. ; ! !•'. t i -- . ( - '« ki iCj&l^^ >ll ■ I Sti 306 ALGEBRA. [chap. 27. If a> &> c be in A. P., and o, p, y in H.P., show tlmt a+c_ a+y bp 07 28. If « be the arithmetic mean between b and c, and b the geo- metric mean between a and c, prove that c will be the harmonic mean between a and 6. 29. If —Q—) by —Q~ ^ ^" H.P., then a, b, c are in G.P. 30. If a, b, c, d, e be in f).P., prove that c(o + 2c + e)=(6 + d)*. 31. If o, 6, c, d ... be a series of quantities in G.P., shew that the reciprocals of a? -b"^, W-c^, c^-d^, ... are also in G.P. ; and find the sum of n terms of this latter series in terms of a and b, 32. If a, b, c be in A. P., and 6, c, d in H.P., then a, ^, c are in ad d' H.P., and b, -j-, d are also in H.P. 33. If 9 be the geometric and a the arithi^etic mean between m and n, and if i^ be the arithmetic mean between vi- and 71^, prove that a^ ia the arithmetic mean between (f and /-•* 34. If a, b, c, d be in G . P. , prove that {b-c)^=ac + bd - 2c.d. 35. If ci, b, c, d be in G.P., prove that (a + rf)(a-6)2 : a{a-c){a-d) = a-b + c : a + 6 + c 36. If a, b, c be in H.P., prove that 1 1 1 111. a b-^c b c + a c a + b are also in H. P. [chap. that XXXV.] MISCELLANEOUS EXAMPLES. V. 307 t f i It ', and b the geo- t c will be the G.P. c)=(6 + d)2. '. , shew that the so in G.P. ; and ries in terms of !no, T, c are in I mean between jan between ??;- lean between y" i-bd-2ad. + 6 + C MISCELLANEOUS EXAMPLES V. and c=432. (Chiejl^ on Chapters XA\YII.—A\YA'V.) 1. Simplify <"'^'^ \'l' , and find its value when a=2, 6=3 yfL eirteim! *'' "'" ''-' ' ""^ '^ '"^'-^^ ^^ -'^^-ting 3. If y = j, shew that w a (1) ^ 2a + 36 2o + 3rf 3a - 76 3c - 7rf (2) «^-^^^(« + 2c)(a + 3r) ^^'-w'^ (6 + 2rf)(6 + :irfy *• " 63-c = c-:^=^r:^. prove that (1) x + y + z = 0; (2) (h + c)x + {c + a)y + {a + h)z = 0. 5- -^^1 = 2 = 1' P'"^^^ *hat \f5x^T»^^T7z'^=5y. divellv^iL*'flIi'''«r'"i''- ^'''^ ""•*^^«'-«. of whicli tlie first varies (inectly and the second inversely as a-, and if v = 7 when x-o and »/= -1, when x=l, shew that ' "' ' " ^ '' '"''^en a -2, ^ 6 7. Simplify ^45 + ^8 -V80+^/I8+Vr=V^. 8. If 3^-+ 10 has to 9a: + 4 the duplicate ratio of 5 to 7, find x. ^' ^^ b^d""/' P''°^'® ^^^^^ ^^^^^ ^'^^^° is equal to (1) '/4ac-;^-3ce3 + 2a^ . ."/WVe -«<««/•+ 7^=~ i^ 308 ALGEBRA. [CHAl'. M I '5 ' It » I' tin r c r K ,** » 11. If 3a + 6fi: 3a-56 = 3c + M: 3c-6rf, prove that a : fe = c : rf. 12. Reduce to their simplest forms: (1) >s ^ ^ (a -6)* 13, When a:= -^, find the value of X' + ax + a'^ x^ - ax + a? ar'-a'- 14. Simplify (1) 52 m^^' ofi + a^ (2) 2"+* - 2 X 2» 15. Find the ratio compounded of the ratios ^+6 • a» - i« a2 - 6» " (a 4- &)a* 16. If a, h, c be three proportionals, prove that (1) a{a + b) : b(h-a) = b(h + c) : c(c-6); (2) (a + 6 + c)(6«-ftc + c2)=c(a2+62+c2). 17. If a : 6 : c = xi/ : x^ : yz, prove that x : y : z = ab : a^ : be. 18. If jo : 7 be the duplicate ratio oi p-r: q-r, prove that r is a mean proportional between ^> and 5. 19. li a I b = c : d, prove that (1) « + <•: a + 6 + r + rf=a : a + 6; (a-b)(b-d) (2) (a -/>)-(c -(/) = ' 20. Shev/ that any ratio is made more nearly equal to unity by adding the same quaiitity to each of its terms. 21. If a; varies as y + ::, and s varies as a;; and if a: = 2 when y = 4, fiud the value of y when a:=l. f^F. [ClIAl*. XXXV.] _2x2" «x4 ■ &); ■,=ab : a" : hr, -r, prove that ?• equal to unity by nd if x=2 when MISCELLANEOUS EXAMPLES. V. 309 22.. If a» + 3y : •2x-3y=2ft'' + 36a : 2a»-3/>», then jc has to y the diiphcate ratio of ct to 6. ^ ait^renoo"3. ''" ^'^* ''^ '' ' '®""'' '*'^'*'° "'"" '^ '"^ ^"'^ common 24. The sum of 10 terms of an A. P. is Hf), and the sum of its fourth and nmth terms is five times the third term; determine the scries. ' '"''^"'""«' 25. Find the value of v/l9T 4^21 + V7 - s/1'2 - v'->932"728. 26. Sum to 10 terms each of the series (1) 5+10+15H20+ ; (2) 5-10 + 20-40+ 27. If P- = ^^ = . , shew that bz-ci/~cx + az~ayTbx^ ap + bq-cr^O, and xp- i/(j + zr=0. 28. The sum of five numbers in arithmetical progression is 10 and tlie sum of their squares is GO ; find the numbers ' 29. Find the sum of n terms of the progression 3 + 2U2iV+ 30. Find the ninth term of the harmonic series whose first ard thud terms are 3 and 2 respectively. 31. Simplify 32. Sum to 71 terms 3 9 15 21 2 + 2+2"+2""^' and find five consecutive terms of this progression whose sum is 187 J. iqS '^^^ f *^"" of an Arithmetical Progression is double the 13 term ; sh ,w that the 2-" term is double the 10"> term. :.i fl :l f ' r m H C r r re , ■•■<. I I : II •« '%,i 310 ALGEBRA. [chap. XXXV. 34. Sum the following series : (1) (a-2a;) + 2(rt + a;) + 3(a + 2a;) + to 18 tema. (2) 3|^-5|f-l-8f- toTterms. 35. Shew that the sum of 2n terms of the series 3 9 "^27 81 ~ 243 729 *" 2' ST"*" IS g{l-(-l)»3-2'.}. 36. If J— ^, ^, ^-— are in A. P., then a, h, c are in G.P. 37. The last term of an A. P. is ten times the first, and the last but one is equal to the sum of the 4"' and 5'\ Find the number of the terms, and shew that the conmion difference is equal to the first term. 38. Sum to 2n terms each of the series (1) 1-3 + 9-27+ ; (2) 1-3 + 5-7+ , and write down the last term of each series. 39. Find two numbers whose arithmetic mean exceeds their geometric mean by 2, and whose harmonic mean is one-fifth of the larger number. 40. Find an infinite geometrical progression, whose first term is 1, and in which each term is twice the sum of all the terms that follow it. 41. The arithmetic mean between two numbers is to the geometric mean as 5 to 4, and the difference of their geometric and harmonic means is -*- : find the numbers. 42. If X, y, 2 be in G.P,, prove that «2y V(.^-3 + y-3 + 2-3) = j,3 + 2^3 + j3. CHAPTER XXXVI. The Theory of Quadratic Equations. 335. In Chapter xxv. it was shewn that after suitable re auction every quadratic equation may be written in the form ax'^ + bx + C=r:0 /JN and that the solution of the equation is ^, ~b± sJW^^^c ^ Ta (2). wiU^^^fp''iL"f°''' ^7''" I?'^^ important propositions connected tTe type^ coeftcients of all equations of which (1) is 336. ^ A quadratic equation cannot have more than two roots. For, if possible, let the equation ax'^ + hx + c = have three iTtC': r^' ^- .^'"" "'"^^ ^^'^ -' these vires mus satisfy the equation, v/e have aa^ + ha + c = n\ a&"' + hf3+c=^0 (2)' ay^ + by + c=^0 (3) From (1) and (2), by subtraction, a(a^-/3'~) + b(a-f3)==0; divide out by a-/? which, by hypothesis, is not zero ; then Similarly from (2) and (3) .■. by subtraction a(a-y)=0; nolttur""'^jP ''"?v: ^^ J^yP^thesi's, a is not zero, and a is not equal to y. Hence there cannot be three different roots. ^ Kr m. » u i r is: I v.. 1 l*' I iHi 312 ALGEBRA. [chap, 337. The terms 'unreal', 'imaginary', and 'impossible' are all used in the same sense : namely, to denote expressions which involve the square root of a negative quantity, such as V^, \/^, sf^. It is important that the student should clearly distinguish between the terms real and rational, imaginary and irrational. Thus ^25 or 5, 3^, - f are rational and real ; ^7 is irrational but real ; while \^ - 7 is irrational and also imaginary. 338. In Art. 335 let the two roots in (2) be denoted by a and /?, so that h + s'b^-4ac o_-^- '^l^^^^ac . a=- 2a ' ^ 2a then we have the following results : (1) If b'^-4ac, the quantity under the radical, is positive, a and /8 are real and unequal. (2) If b^-4ao is zero, a and /J are real and equal, eadi reducing in this case to -~. (3) If b^ - 4ac is negative, a and (3 are imaginary and unequal. (4) If 62 _ 4Qg jj, a perfect square, a and /? are rational and unequal. By applying these tests the nature of the roots of any quadratic may be determined without solving the equation. Example 1. Shew that the equation 2a;2-6a; + 7 = cannot be satisfied by any real values of x. Here a = 2, b=-G, c-7; so that 62-4otc = (-6)2-4.2.7=-20. Therefore the roots are imaginary. Example 2. For what value of k will the equation 3x^ - Qx + k-.Q have equal roots ? The condition for equal roots gives {-6)^-4:. 3. k=0, whence k=3. Example 3. Shew that the roots of the equation x^-2px+p^-q^ + 2qr-r^=0 are rational. The roots will be rational provided ( - 2p)- - 4(p- - q- + 2qr - r-) is ft perfect square. But this expression reduces to 4{q^-2qr + r'^), or i{q- r)\ Hence the roots are rational. XXXVI. j THE THEORY OF QUADtlATIC EQUATIONS. 31 3 on3x"-6x + k-.0 (q^-2qr + r% or 339. Since a^ -f>+ '¥^4ac rf_-b-Jb^-4ac 2a P- we have by addition 2a a + iS= -b+Jb'^-Aac-b-Jb'^-4ao 2a 2a a .(1) and by multiplication we have a« =( - ^ +\/6=^ - 4ac) (-b-Jb^- 4ac) 4a2 _4ac_c ~4^2-- (2). By writing the equation in the form ^•2+-^ + -=0, a a ' these results may also be expressed as follows : In a quadratic equation cohere the coefficient of the first term is U7lit9/, (i) the sum of the roots is equal to the coefficient of a: with Its sign changed ; (ii) the product of the roots is equal to the thiixi term. Note. In any equation the term wliich does not contain the UHknown quantity is frequently called the absolute term. 340. Since -^ = a + f3, and - = aR a ' a ^ tiie equation x'^ + -x + - may be written a a '' x^~(a + fiyv+a/3=0 (1). Hence any quadratic may also be expressed in the form .r2-(simi of roots) .2? -[-product of roots =0 (2). Again, from (1) we have (.t-a)(.r-/3)=0 (3). We may now easily form an equation Avith given roots. r - n sum of roots = 4, product of roots = 1 ; We have the equation is X :2-4«+l=0, by using formula (2) of the present article. n.f iJ" ^^1? ""^'l*^^. °^ ^''^- ^^^ ^^6 ™"st important, and they are generally sufficient to solve problems connected with the va&0)'.;H'A''(2M'fr "° "" o'^'-P-+?=0, find th. We have a + p=p, ap=q. :. a2+i32=(a + /3)2-2a/3 =p^-2q. ■^g^'"' a^ + /3^=(a + |8)(a2 + |82-o/3) =2?{(a + ^)a-3a^} =P(^'-35). Example 2. If «, /3 are the roots of the eciuation Ix"^ + mx + h =0, find the equation whose roots are -, ^. j3 a We have sum of roots = - + ^=?l±^, ^ a a/3 ' product of roots = ^ • - = 1 » fi + ^=0, find the XXXVI.] THE THEORY OF QUADRATIC EQUATIONS. .*, by Art. 340 the required equation is or apx'^-{a'^+^)x + ap=0. As in the last example 0^+0^ = — W~> ^^^ *P=7* .'. the equation la j x^ - ■ — j^ — a; + y = 0, 315 or nix"- - ( w= - 2jiI) x + 7d=0. Example 3. Find the condition that the roots of the equation ax'^ + bx + c=0 should be (1) equal in magnitude and opposite in sign, (2) reciprocals. The roots will be equal in magnitude and opposite in sign if their sum is zero ; therefore -- = 0, or 6 = 0. a Again, the roots will bo reciprocals when their product is unity ; therefore - = 1 , or c = a. a Example 4. Find the relation which must subsist between the coefficients of the equation px'^ + qx + r=0, when one root is three times the other. We have p p but since a = 3/3, we obtain by substitution 4/3=-^, 3S2=-. P P From the first of these equations j8-=t|^, and from the second '■ 16p2-3y OP 3q^ = Wpr, wiiich is the required condition. 342. The following example illustrates a useful application of the results proved in Art. 338. 31 6 ALGEBRA. tCHAP, I : ■;3 P rf i C ^H ^^^^^B ' a < ■ r ^^^^H ' lu ^^^^H ,' tw ^^■'^ r ^^■'i f ^H' c ^H r ^^^^^■^ «35 ^^■^ ^ ^H' 1 t;,; ^^H ^H 1 ots of the cor- )n investigated the equation tveen them. le greater. [Art. 339.] factors x-a^ therefore tlie ' has the same 'ession bo that of a. XXXVI.] THE THEORY OF QUADRATIC EQUATIONS. 317 Case II. If a and yS are equal, then ax^+hx+c—aix-af, and {x - a)" is positive for all real values of x ; hence ax^+bx+c has the same sign as a. Case III. Suppose that the equation ax^+hv+c=0 has imaginary roots ; then ax'^+hx+c=a\x^+-x+-\ y. a a) but since h'-Aac is negative [Art. 338], the expression is positive for all real values of x ; therefore ax^+hx+c has the same sign as a. EXAMPLES XXXVI. Find (without actual solution) the nature of the root« of the following equations : 3. ^a;2=14-.3a«. 6. (a; + 2)2= 4a; +15. 10 1. x^ + x -810=0. 2. 8 + 6a?=5a:2 4. x''-\-7=ix. 5. 2a;=a;2 + 5. Form the equations whose roots are 7. 5, -3. 8. -9, -11. 3 5 2' 6' 11 2 4 o 5 9. a + b, a-b. 12. 0, ' 13. If the equation x^ + 2{l + l)x + k-=0 lias equal roots, what is the value of k ? 14. Prove that the equation Smx"^ - (2m + 3n)x + 2n = has rational roots. 15. Without solving the equation .3.c2-4a:- 1=0, find the sum, the diflference, and the sum of the squares of the roots. 16. Shew that the roots of a(x^ ~l) = {h-c)x are alwn,ys real. JiJS J^ li I c « ^ I I ( .P ' • 3 i ' 5,1 li i>, t » 318 ALGEBRA. [chap. XXXVI. Form the equations whose roots are 17. 3 + ^5, 3-V5. 18. -2+V3, -2-^3. 19. a b "5' 6- 20. 5(4 ±V7). 21. a+6 a-6 a -6' a + b' "^^ 26' 2^- If o, ^ are the roots of the equation px^+qx + r=0, find the VcliU6S 01 23. o2 + ^2. 26. o-'+zS^. 24. {o-)8)8. 27. a^^ + a"-ft\ 25. o2/3 + a^-'. 28. 5+^-. p a ^' ".2''' ?>,T^*^n T*f P^ ^\-P^ + 'l=0. and a", ^ the roots of x^-l^x + Q^O, find P and Q m terms of ^^ and q. 30. If a, ^ are the roots of x'- - ox + 6 = 0, find the equation Mhose roots are ^, P 31. Find the condition that one root of the equation ax'^-\-bx-\-c=Q may be double the other. 32. Form an ef.uation whose roots shall be the cubes of the roots of the equation 2a; (a;- a) =:a2. 33. Prove tha; the roots of the equation (a + o)a;2-(a + 6 + c)a; + |=0 are always real. 34. Shew that (a + 6 + c)a;2_2(a + 6)a; + (a + i-r)=o has rational roots. 35. Form an equation whose roots shall be the arithmetic ami harmonic means between the roots of x--px-{-q = Q. 36. In the equation px"^ -\- qx -V r =Q the roots are in the ratio of / to m, prove that {P + m'^)pr + Im [^2pr - q") = 0. 37. Shew that if a: is real the expression ^;:^ cannot lie between 3 and 5. 2a;-8 38. If X is real, prove that - ^ ^ot^ f can have all values except such as lie between 2 and a;2-2a;-l 3 Z! + r=0, find the equation M'hose bes of the roots CHAPTER XXXVIT. Permutations and Combinations. 344. Each of the arrangements which can be made by taking some or all of a number of things is called a permutation. Each of the groups or selections which can be made by taking some or all of a number of things is called a combination. Thus the permutations which can be made by taking the letters a b, c, d two at a time are twelve in number : namely, ab, ac, ad, be, bd, cd, ba, ca, da, cb, db, dc ; each of these presenting a different arrangement of two letters. The combinations which can be made by taking the letters cf, b, c, d two at a time are six in number : namely, ab, ac, ads be, bd, cd ; each of these presenting a different selection of two letters. From this it appears that in forming combinations we are only concerned with the number of things each selection contains ; whereas in forming permutations we have also to consider the order of the things which make up each arrangement; for mstance, if from four letters a, b, c, d we make a selection of three, such as abc, this single combination admits of being arranged in the following ways : abc, acb, bca, bac, cab, cba, and so gives rise to six different permutations. 345. Before discussing the general propositions of this chapter the following important principle should be carefully If one operation can be performed in m wai/s, and {when it has been performed in any one of these vmys) a second operation can ttien be performed in n ways ; the number of ways of performing the two operations will te m x n. if J f J if 320 ALOEHRA. [OIIAP, ^^H I 'Si ^^^^^B a m ; ^^^^B 1 ^^^1 1 <■ ^^^H f E ^^^^H m ^^B i ^^^^H ■ . ^^H ' • . H| M 12 in ^H » ^^^^B w ^^^H to ^^^B t» ^^^H a ^^1 ^ : ^ i ^^^^H ' b ^^^^^H tn ^^^^^B r ■^■/ rC ■■ . , Ji'C ^B r ^^^^^H »* ^^■' % ^^^B ■it^ ^H !'"^ ^^H 1 1 ^^^H 1 ^^^^B '• ■ , 1 : ^1 If the first operation be performed in any one way, we can asaociate with this any of the n ways of performing the secoml operation : and thns we shall liave n ways of performiiifr the two operations without considerinjr more than one way of ptu-forniif v the first ; and so, corresponding to each of the m ways of ijtr- forming the first operation, we shall have n ways of perfori'iiiKr the two ; hence altogether the number of ways in which the two operations can be performed is represented bv the proiluct mxn. ' '■ Example. There are 10 steamers plying between Liverpool and Dublin; in how many ways can a ma-; go from Liverpool to Uul.lin and return by a different steamer ? There are »g8 taken r at w(7i-l)(7i-2) (w-r + 1). a Umeifl'^''^ """'^^'' ""^ P^^™"<^^'0"« ^^ '* tilings taken all at n(n-lXn-2) to w factors, or n(n-lXn-2) 3.2.1. It is usual to denote this product by the symbol |«, which is read " factorial n^' Also the symbol n ! is sometimes used for In, 347. We shall in future denot t?- number of permutations of n thmgs taken r at a time by the .ymbol »/^„ so that "Pr=n(n-lX7i-2) (n-r+\); also "P. = [w. In working numerical examples it is useful to notice that the Mrson 111 J ; tlie thud m 4 ; and the fourth fn 3: and since each nf usg-stfoft^ninTdTgit'sTrr^ "T^-^ """ "^ 'o™"' "-^ .'. the required result = 9p„ =9x8x7x6x5x4 = 60480. fetr^' ^T^'^c S^ »'eq"'r«l number of combinations. inen each of these combinations consists of a ffroun of r Hence "(7, x(^ is equal to the number of arrangemmts of n thmgs taken r at a time ; that is, • "C? =-!'-" l)^!.* ~ 2). . .(w - r + 1) E.A. X "-- II §1 323 ALGEBRA. m 4 t I » t ■'mt « isr » » l^r. t lu . r» r t ':' € r we 1 fci [OHAP. Cor. This formula for "CV may also be written in a different form ; for if we multiply the numerator and the denominator by |n-r we obtain n(n-l)(n~2) (%-r+l)x[wj-r r \n — r or l» \r \n-r Bince«(n-l)(n-2) (n-r + l)x( w-r -»[w. Example. From 12 books in how many ways can a selection of 5 be made, (1) when one specified book is always included, (2) when one specified book is always excluded ? (1) Since the specified book is to be included in every selection, we have only to cnoose 4 out of the remaining 11. Hence the number of ways = »€. = ^]^}P^„^^,^ = 330. •' * 1x2x3x4 (2) Since the Epecified book is always to be excluded, we have to select the 5 books out of the remaining 11. Hence the number of ways ="C.=?l^il^^4:?4^= 462. "1 x2x3x4x5 349. The number of combincUions of n things r at a time is eqtcal to the number of combinations ofn things n-v at a time. In making all the possible combinations of n things, to each group of r things we select, there is left a corresponding group of n-r things; that is, the number of combinations of n things r at a time is the same as the number of combinations of n things tt — r at a time ; This result is frequently useful in enabling us to abridge arithmetical work. Example. Out of 14 men in how many ways can an eleven be chosen ? The required number ="Cji - ^8- 1x2x3"-^^*- If we had made use of the formula "CT,!, we should have had to reduce an expression whoso numerator and denominator each con- tained 11 factors. 350. In the examples which follow it is important to notice that the formula for permutations should not be used until the suitable selections required by tJie question have been made. XXXVIT.] PERMUTATIONS AND CO^rBINATIONS. 323 in every selection, :ions of n thiiiirs Exam})le 1. From 7 Englishmen and 4 Americans a committee of 6 13 to be fonned : in how many ways can this be done, (1) when the committee contains exactly 2 Americans, (2) at least 2 Americans' Ju ^!n '^!!trT^^'" ''^i "^^^2 •" "^^'."^ ^''° Americans can Le chosen iLSnT/r V"!"7*u* r^' •" ^'"^''^ ^^« Englishmen can be oUrsecInS ; h^ent ""' ''^ '™' ^^^^^ ^" ^^ ^^^^^'-^^^ -■'»' -^ the required number of ways = *C2 x ^C. = Li |7 _ [7 (2) We shall exhaust all the suitable combinations by forming all the groups containing 2 Americans and 4 Englishmen ; then EnglTshnl"' ' ^ EngTishmen ; and lastly 4 ALricanS and 2 The Hum of the three results will give the answe- reriuired number of ways = ^C^ x 'C4 + *Cs x 'Cg + *C^ x ^O [2l2'^l4[3 Hence the [4 tl ,_^ = 210 + 140 + 21=371. In this example we have only to make use of the suitable formulae ZTnlZT'' ^r ""^ri '^^^ ^^"^^••"ed with the possible IrrZe menta of the members of the committee among themdelves. Example 2. Out of 7 consonants and 4 vowels, how many words can be made each containing 3 consonants and 2 vowels? Tlio number of ways of choosing the three consonants is ^a and the number of ways of choosing tlie 2 vo« s is *C.; and since'each of the first groups can be associated wit ich of the second ?he r'^is^'c^gT^^Jr ^""'''' '"^ ""''"- "^ ' consonants rnd'2 Further, each of thr o ^^^ contains 5 Liters, whic) iiiav be arranged among'' sinlSways. Hence ^ the required number of words = -i=- x ^ x 1 5 =5 "[7 =25200. EXAMPLES XXXVII. a. 1. Find the value of "P^, 7Pg, 8q^^ ^q^^^ "v| (2?alIoTtK *S:rf rXrTSS^ ^ "■"•' "^ '»""« <" 3. If"Cs:"-i(7,=8:6,flnd«. u 1 |-.i ■ m J: V : i| : 1 '':m \ ■j m t m « » B r r : Ra I I i ■I '^ 324 ALGEBRA. [chap. 4. How many different selections of four coins can be made from a bag containing a sovereign, a half-sovereign, a half-crown, a florin a shilling, a franc, a sixpence, a penny, and a farthing ? ' 5. How many numbers between 3000 and 4000 can be made with the digits 9, 3, 4, 6? 6. In how many ways can the letters of the word volume be arranged if the vowels can only occupy the even places ? 7. If the number of permutations of n things four at a time is fourteen times the number of permutations of n - 2 things three at a time, find n. 8. From 5 masters and 10 boys how many committees can be selected containing 3 masters and 6 boys ? 9. If «'C;=«'OU,o, find -Gj„ '«C,. 10. Out of the twenty-six letters of the alphabet in how many ways can a word be made consisting of five different letters, tw o of which must be a and c ? 11. How many words can be formed by taking 3 consonants and 2 vowels from an alphabet containing 21 consonants and 5 vowels ? ^ 12. A railway carriage will accommodate 5 passengers on each side: in how many ways can 10 persons take their seats wlien two of them decline to face the engine, and a third cannot travel backwards ? 351. Hitherto, in the forraulse we have proved, the things have been regarded as unlU'e. Before considering cases in which some one or more sets of things may be like, it is necessary to point out exactly in what sense the words like and imlih' are used. When we speak of things being dissimilar^ different, nn- like, we imply that the things are visibly unlike, so as to be easily distinguishable from each other. On the other hand we shall always use the term like things to denote such as are alike to the eye and cannot be distinguished from each other. For instance, in Ex. 2, Art. 350, the consonants and the vowels may be said each to consist of a group of things united by a coinmon characteristic, and thus in a certain sense to be of the same kind ; but they cannot be regarded as like things, because there is an individuality existing among the things of each group which makes them easily diatinfuishable from each other. Hpiice, in the final stage of the example we considered each group to consist of five dissimilar things and therefore capable of 15 arrangements among themselves. [Art. 346. Cor.] [chap. i can be made from lalf-crown, a florin, thing ? ) can be made with le word i}olume be places? 8 four at a time is - 2 things three at committees can be al)et in how many rent letters, two of g 3 consonants and its and 5 vowels ? mssengers on each their seats wlien tiird cannot travel roved, the things ing cases in which it is necessary to ':e and unlike are i7rtr, different, nn- like, so as to be 16 other hand we such as are alike each other. For I the vowels may ted by a common be of the same gs, because there each group which ither. Hpju'p, in i each group to *e capable of [5 XXXVll.] PERMUTATIONS AND COMBINATIONS. 325 352. To find the number of ways in which n things may he arranged among themselves, taking thetn all at a time, when p of the things are exactly alike of one kind, a of them exactly alike of another kind, r of them exactly alike of a third kind, and the rest all different. Let there be n letters ; suppose p of them to be a, q of them to be 6, r of them to be c, and the rest to be unlike. Let ^ be the required number of permtitations ; then if the p letters a were replaced by p unlike letters diflferent from any of the rest, from any one of the x permutations, without altering the position of any of the remaining letters, we could form \p new permutations. Hence if this change were made in each of the x permutations, we should obtain x->(.\p^ per- mutations. Similarly, if the q letters h were replaced by q unlike letters, the number of permutations would h& xx\px \q. In like manner, by replacing the r letters c by r unlike letters, we should finally obtain a; x [£ x [^ x [r permutations. But the things are now all different, and therefore admit of In permutations among themselves. Hence a;x|£x[£x[rj=[w; .,-hL. which is the required number of permutations. Any case in which the things are not all different may be treated similarly. Example \. How many different permutations ca:? fie made out ot the letters of the word assassination taken all together ? We ha\ J here 13 letters of which 4 are s, 3 are a, 2 are », and 2 are n. Hence the number of permutations - _B^ = 13.11.10.9.8.7 .3.5 =1001 X 10800=10810800. that is. X = \ w i i I H I 1; I. E I P 4 *f < ; > ** e K^ HI { I :' lu ^ r 326 ALGEBRA. [CHAP. Example 2. How many numbers can be formed with the digits *» 2, 3, 4, 3, 2, 1, so that the odd digits always occupy the odd places ? The odd digits 1, 3, 3, 1 can be arranged in their four places in \± 12 [2 -— ways .(1). The even digits 2, 4, 2 can be arranged in their three places in [3 ^ways (2). ^ Each of the ways in (1) can be associated with each of the ways m (2). •' 14 13 Hence the required number = J=j^ x ==6 x 3= 18. 363. To Jind the number of permutations of n things r at a time, when each thing may be repeated once, twice, up to r times in any arrangement. Here we have to consider the number of ways in which r places can be filled up when we have n different things at our disposal, each of the n things being used as often as we please in any arrangement. The first place may be filled up in n ways, and, when it has been filled up in any one way, the second place may also be filled up in n ways, since we are not precluded from using the same thing again. Therefore the number of ways in which the first two places can be filled up is w x w or n\ The third place can also be filled up in n ways, and therefore the first three places in n^ ways. Proceeding in this manner, and noticing that at any stage tl.v index of n is always the same as the number of places filled up, we shall have the number of ways in which the r places can be filled up equal to »*". Example. In how many ways can 6 prizes be given away to 4 boys, when each boy is eligible for all the prizes ? Any one of the prizes can be given in 4 ways ; and then any one of the remaining prizes can also be given in 4 ways, since it mo v Iw obtained bv the boy who has already received a prize. Thus two prizes can be given away in 4? ways, three prizes in 4^ ways, and so on. Hence the 5 prizes can be given away in 4", or 1024 ways. :ie ill [chap. led with the digits apy the odd places? }ir four places in (1). ' three places in (2). h each of the ways = 18. ?/ n things r at a tmce, up to Y ways in which r •ent things at our en as we please in and, when it has may also be filled 1 using the same in which the first lys, and therefore ; at any stage the tf places fiiUed up, e r places can be be given away to and then any one rn, since it niov be prize. Thus two in 4^ ways, and so )r 1024 ways. XXXVli.] PERMUTATIONS AND COMBINATIONS. 327 354, To find for what value of r the number of combinations ofn things r at a time is greatest. _. „^ n( n-l)(n-2) (n-r+2){n-r+l) S^"C« ^-= 1.2.3 (r-l)r .r.A nr n(n-l){n-2) (n-r+2) . and Cr-i- 172.3 (r-1) '•C;="Cr-iX n-r+l The multiplying factor ^— -^^ — may be written — 1, which shews that it decreases as r increases. Hence as r receives the values 1, 2, 3, in succession, "CV is continually increased, until 1 becomes equal to 1 or less than 1. Now ^^ - 1 > 1, so long as ^>2 ; that is, ^>r. We have to choose the greatest value of r consistent with this inequality. (1) Let n be even, and equal to 2m ; then n+1 2m+l 1. -^=-^- = '^+2' t.'id for all values of r up to m inclusive this is greater than r. Hence by putting r=m=|, we find that the greatest number of combinations is "Cn. 3 (2) Let n be odd, and equal to 2m+l ; then and for all values of r up to w inclusive this is greater than r ; but when r=m + l the multiplying factor becomes equal to 1, and "C„+x=''(?^ ; that is, "^+1 ="<^«r3 '* and therefore the number of combinations is greatest when the things are taken ^^^, or ^^ at a time ; the result being the same in the two cases. 'Hfii' 1 328 ALGEBRA. [chap, xxxvtt. EXAMPLES XXXVU. b. all\he^it™:fTe"tUs' ^ "^'^'^ ''"^ "^ -^^e from (1) irresistible, (2) phenomenon, (3) tittle-tattle. d,4s ?74T? f f r w""""^"' '^^ ^ ^""«^ >^y »«i"« the seven aigits J, d, 4, 3, 3, 1, 2 ? How many with the digits 2, 3, 4, 3, 3, 0, 2 ? 3. How many words can be formed from the letters of thPuL-i S^nu^rn, so that vowels and consonants occur iwtely i^eaJh iJ: How many different arrangements can be made out of thp letters of the expression a%h^ when written at full len|th ? 7. There are four copies each of 3 different volumes- find t}i« number of ways m which they can be arranged on oneTelf C„?;w" J""^ ""^"^ '"I^y? ''^^ ^ P^^sons form a ring? Find the number of ways in which 4 gentlemen and 4 ladies can sit af. round table so that no two gentlemen sit together fK?*i i" ''''''' I"*"^ '""Y^ ^*" * ^'«^d of 4 letters be made out of the letters a, b, e c, d, o, when there is no restrictiZas to the number of times a letter is repeated in each word ? fV,J^' ^''m "?*^y arrangements can be made out of the letters of ^erzi^^X:^r ''''' '"^^ ^'^'^^^'^^^^^ --p^ thesr^f^r;^^; row on1,o^rSp'Tl'°"'^'^'i^^ ^^8^*^ r°' ^'^ ^l^^*" °«« «an only Snth: c^; h^lrT.i^r, ^"^^ °" «'™^^ "^« = - how many way^ 12. Shew that "+^C,-="(7, + ''0;_i. 4 ^n bowP^'I'n wf^" ^''' *° ^ ^^T" ^''^'"^ 13 men of whom only fncTul'rW 2'bTwre?sT "'^' ''" '''^ *^^" '^^^ '"^^^ "P ^ ^ ^' «3fi« i" ^'^""^ many ways can 7i men be arranged in a row if two specified men are neither of them to be at either extremity of ihl [chap, xxxvtt. 11 be made from (^ using the seven 2,3,4,3,3,0,2? ters of the word Jrnately in each :stinct positions, ■r of signals that n persons, when ;h may receive ? lade out of the length ? lumes; find the tie shelf. ing? Find the es can sit at a Je made out of ction as to the f the letters of le first, fourth, 1 one can only low many ways I of whom only le up so as to I a row if two tremity of the CHAPTER XXXVIII. Binomial Theorem. 3B5. It may be shewn by actual multiplication that =^+(a+b+c+d)a,^+(ab+ac+ad+bc+bd+cd)jc^ +(abc+abd+acd+bcd)x+abcd m. We may, however, write down this result by inspection ; for the complete product consists of the sum of a number of partial products each of which is formed by multiplying together four letters, one being taken from each of the four factoiB. If we examine the way in which the various partial products are formed, we see that (1) the terra .r* is formed by taking the letter x out of each of the factors. (2) the terms involving x^ are formed by taking the letter x out of am three factors, in every way possible, and one of the letters a, b, e, d out of the remaining factor. (3) the terms involving x^ are formed by taking the letter x out of anu two factors, in every way possible, and two of the letters a, 6, e, d out of the remaining factors. (4) the terms involving x are formed by taking the letter x out of any one factor, and three of the letters a, b, x d out of the remaining factoi's. (5) the terra independent of x is the product of all the letters a, Of c,d. Example. Find the value of (a; - 2) (x + 3) (a; - 5) (a; + 9). The product =ar*+{-2 + 3-5 + 9);e3 + ^-6 + 10- 18- 15 + 27- 45)i»;2 ^ .. , „ +{30 -54 + 90 -135) a; +270 =.T* + 5*3- 47x2 -69j; + 270. oo,u;-r^iv 330 ALGEBRA. [OHAP. i ! i 1* > m: ' 356. If in equation (1) of the preceding article we suppoae b=c=d=a^ we obtain We shall now emjiloy the same method to prove a formula known as the Binomial Theorem, by which any binomial of the form x + a can be raised to any assigned positive integral power. _ 357. To find the expansion of (x+a)" token n is a positive integer. Consider the expression (x+a) (x+b) (x+e) (x+k), the number of factors being n. The expansion of this expression is the continued product of the n factors, x+a,x+b,x+c, x+k, and every term in the expansion is of n dimensions, being a product formed by multi- plying together n letters, one taken from each of these n factors. The highest power of x is x", and is formed by taking the letter x from each of the n factors. The terms involving ^-^ are formed by taking the letter x from awy n-1 oi the factors, and one of the letters a, 6, c, ... k from the remaining factor ; thus the coefficient of a:"-^ in tlie final product is the sum of the letters a.b.c, k: denote it by^x. The terms involving af*-^ are formed by taking the letter x from ani/ n-2 oi the factors, and two of the letters a,b,c,...k from the two remaining factors ; thus the coefficient of x"-"^ in the final product is the sum of the products of the letters a,b,c,...k taken two at a time ; denote it by S2. And, generally, the terms involving ^r"-*" are formed by taking the letter x from any n-r of the factors, and r of the letters a, 6, c, ... ^ from the r remaining factors ; thus the coefficient of af-"" in the final product is the sum of the products of the letters a,b,c,...k taken r at a time ; denote it by Sr. The las* m in the product isabc ...k; denote it by S„. Hence (x+a) (x+b) (x+c) (x+k) =x^+Six"-^+S^t^-^+... +,V~'"+ ...+Sn-lX+S„. In Si the number of terms is w ; in *S'2 the number of terms is the same as the number of combinations of n things 2 at a time ; that is, "C« , in S3 the number of terms is "C3 ; and so on. Now suppose 6, c, ... k, each equal to a ; then ^S*! becomes "CiO ; ^2 becomes "C^^^; S3 becomes "Csa^ ; and so on ; thus [OHAP, bicle we suppose irove a fornmla ' binomial of the integral power, ■ n is a positive nued product of ery term in the )rmed by multi- these n factors. I by taking the ;ng the letter x bters a, b, c,...k : of a^-^ in the ,,..k; denote it ing the letter x ters a, h, c, ,,.k jfficient of x^-^ s of the letters rmed by taking r of the letters he coefficient of its of the letters be it by S^. nher of terms is igs 2 at a time ; 1 so on. becomes "Cia ; thus ■...+ CnCf J n{n~\){n-2) , „ , XXXVm.] BINOMIAL THEOREM. 331 substituting for ""C^, "Cg, ... we obtain {x + af = A'" + woo;'-! + ^^(^-L)^ V"" 1 72. 3 " "" I • . . -r w J the series containing n+\ terms. • T'l^- ^^ • j\^ Binomial Theorem, and the expression on the right IS said to be the expansion of {x+a)\ 358. The coefficients in the expansion of (x+aY are very conveniently expressed by the symbols "d, "Cg, "C3, . "C We shall, however, sometimes further abbreviate them by omitting n, and writing (7„ C^, C,, ... (7„. With this notation we have If we write - a in the place of a, we obtain (x - af =^x"+ Ci( - ay -» + C^- afxf'-^ + C,{-afx--^+... + Cn{-aT = J?" - Ciax^-' + C^a^af'-'' - C^a\v^-^ ■{.... +( - l)"C„a». Thus the terms in the expansion of (^+a)" and {x~aY are numei'icam the ^me, but in (x - a)" they are alternately positive and negative, and the last term is positive or negative according as n IS even or odd. ° Example 1. Find the expansion of {x + y)\ By the formula, the expansion = a:" + ^C^T^y + ^G^Y + 8(7^x3^3 + «C'4X V + ^C,xy ^ + sCgy' =x^ + QxPy + 15a;V + 20ary + 15a:V + 6a;y» + ys, on calculating the values of «Ci, ^Cg, ^Cg, Example 2. Fiad the expansion of (a - 2a:)7. (a - 2a:)'^ = a^ - 7C,a»(2a:) + •'G^a^{2xY - W^a*{2x)^ + to 8 tei-ms. Now remembering that "a="C^„-r, after calculating the co- efficients up to 7C3, the rest may be written down at once ; for W- ^; "5 = ^02; and soon. Hence (a - 2xr = a' - 7a«(2ar) + f^|a»(2a:)2 - '^^^i^a*(2x)^ + . . . = a' - 7a«(2a;) + 21a''(2a:)2 - fi5aH2x)^ + 35amz)* -2la^2xf + 7a(2x)<^~(2xy = o' - Ua^x + 84a«a:2 - 280a*a;N SGOa^a^ - 672a'-'a;' + U8ax^ - 1 28a;'. m ^^^^^^■'' ^ i 1^ en ^p|r f ^^^^^bL ' ' -1* ^■^: , i::'C r las t ^^^H"' :ltx : ; !" ^^m'' 1 t '■ ■ 1 HH^B^ ■ ' -I ■■^■.'^ 1 ^^H;; ^ ■ 332 AIXi^EBRA. [CHAf. 369. In the expansion of (x + a)", the coefficient of the second term is "Ci ; of the third term is "d; of the fourth term is "C3; and so on ; the suffix in each term beina one leas than the number of the term to which it applies ; hence "C> is the co- efficient of the (r+1)"' term. This is called the general term, because by giving to r different numerical values any of the coefficients may be found from "Or ; and by giving to x and a their appropriate indices any assigned term may be obtained. Thus the (r+l)"" terra may be written -Cr^-'-a'-, or ^(^-1)0^-2) (^-r+l) ^_,^^ [r In applying this formula to any particular case, it should be observed that the index of a is the same as the suffix of C, and that the sum of the indices ofx aud a is n. Example 1. Find the fifth term of {c« + 2a:«)". The required term =^''C^a^\2a?)* ^ 17.16.15.14 1.2.3.4 ^ ^ =38080a"x". Example 2. Find the fourteenth term of (3 - a)". The required term = "<7i3{3)'»( - ap ="C2x(-9a»3) [Art. 349.] = -945a". 360. The simplest form of the binomial theorem is the ex- pansion of (\+x)\ This is obtained from the general formula of Art. 357, by writing 1 in the place of x, and x in the place of a. Thus (1 -f a;)"= 1 +"Ci^+«(72a;2+ ... +»C^+ ... +"0.^ , , , n(n-l) „ . = l + nx+ \ 'a ^-\-. .+^; V. the general term being w(n- l)(n-2) {n-r +\) 361. The expansion of a binomial may always be made to depend upon the case in which the first term is unity ; thus W-t)' / =«"(H-2)'', where z=^. JO Vi I i: [CttAP. b of the second h term is "C^ ; less than the "Cr is the co- j^eneral term, ;s any of the \g to .V and a r be obtained. I, it should be of C, and that XXXVIII.] BINOMIAL THEOREM. 888 [Art. 349.] em is the ex- leral formula ; in the place I be made to ity ; thus Example. Find the coefficient of x" in the expansion of (a* - 2x)". We have (x2-2x)"»=a;»'/'l -^V"; and, since ic^ multiplies every term in the expansion we have in this expansion to seek the coefficient of the terin which -HT' JUbttiua —y Hence the required coefficient="C4(- 2)* 10.9 8. 7 1.2 3. 4 =3360. xl6 EXAMPLES XXXVm. a. Expand the following binomials : I. (a: +2)*. 2. (x + 3)\ 8. (« + «)'• 4. (a-a:)» 5. (l-2y)». 6. (^+1)*- ■'■ (-1)' 8- (-1)' <^ («-l)' Write down and simplify : 10. The4"'termof(l+3!)W II. The6"'termof(2-y)8. 12. The 5* term of (a - 5ft)'. 13. The i&^ term of (2x - 1)". 14. The 7"- term of A- iY°. 15. The e"* term of ( 3a? + 1 Y. (2 3 \« sa-g-J • 17. The 23-» term of (^+^, 18. The 10"" term of (x" - xf. 'M I € : > pj L flirFli^-f*^ 331 ALGEBRA. [OHAP. la 19. Find the value of {x - ^3)* +(x+ ^3)*. 20. Expand (\/r^+ 1)» - (VTT^ - 1)». 21. Find the coeflBcient of a:" in (ar»+2a?p. 22. Find the coefficient of a; in (x^ - ^ Y*. 23. Find the term independent of a; in ( 2x^ - - ) 24. Find the coefficient of a;-» in (^ - ^Y^ 362. In the expansion of (1+ ')" the coefficients of terms equi- distant from the "beginning and end are equal. The coefficient of the (/•+ 1)* term from the beginning is "(7^ The (r+iy term from the end baa 9j + l-(r+l), or n-r terms before if, ; therefore counting from tb© beginning it is the in-r-vXf term, and its coefficient is "(7„_„ which has b^eii shewn to be equal to "C;. [Art. 349.1 Hence the proposition foUowa 363. To find the greatest coefficient in the expansion of (1+x)". •' The coefficient of the general term of (1 +xf is "C, ; and we have only to find for what value of r this is greatest. By Art. 354, when n is even, the greatest coefficient is "C,, ; and when n is odd, it is "CU, or "C^h^; these two coefficients being equaL 364. To find the greatest term in the expansion o/(x+a)». We have (pi!+aY=af{\ + -X -, therefore, since a?" multiplies every term in [l + -)" it will be sufficient to find the greatest term in this latter expansion. Let the J-'" and (r+l)'" be any two consecutive terms. Lhe v''+i)* term is obtained by multiplying the r"* term by [Art. 359.] w— r+l a |;thati,by(»-±i-l)|. [chap. of term» eqid- inning is "C*^ + 1), or n-r Jgiuning it is lich has been le proposition expansion oj I "C^ ; and we st. icient is **C„ ; sro ccefl&cients )/(x+a)» ■ j , it will be pansion. !utive terms, r"* term by [Art. 359.] XXXVIII.] BINOMIAL THEOREM. 886 n+l The factor — 1 decreases as r increases; hence the (r+l)* term is not always greater than the r"" term, but only until f — \\~ becomes equal to 1, or less than 1. Now (^-l)?>I,solonga,^±i-l>?i that is — — >- + l, or '^ — -^~>r. r a x+a 'in ^^ ^^ .i" ^® *" integer, denote it by » ; then if r=« the multiolying factor becomes 1, and the (p + l)"' term is equal to the jo* ; and these are greater than any other term. If — r~ ^ not an integer, denote its integral part by q ; then the greatest value of r consistent with (1) is g : hence the (q + 1)"" term is the greatest. Since we are only concerned with the mimefically greatest term, the investigation will be the same for {x-af ;' therefore in any numerical example it i-D unnecessary to consider the sign of the second term of the binomial. Also it will be found best to work each example independently of the general formula. Example. Find the greatest term in the expansion of (l+4a;)8, when X has the value -. Denote the r* and (r+ 1)'* terms by TV and TV+i respectively; then hence that is, TV+i>7V, so long as ^x^>I,. The greatest term ia Uie aixth, and its value I value 36-4r>3r or 36>7r. ol r consisteri with \ =nx(iy='C3.(|y= is 5 ; hence the crreatest .'i7344 M ^9R 1 243 ); ! R. ill i E * 1 1 1 m 5 « *» ; , • , *» 'I -«• ^b - P> ^- r s, * I ' %' '; V, r 336 ALGEBRA. COHAP. 365. To find the mm of the coefficients in the expansion o/(l+x)'. In the identity (1 +xy^\^-C^x+C^'»-{-C^-^..,^C,^^ put :r=l ; thus = sum of the coefficients. Cor. C, + C2+C3+... + C„=2"-1 ; that is, the total number of combinations of n things tahhq tome or all of them at a time is 2" - 1. 366. To prove that in the expansion of (1 +x)% the mm of the coefficients of the odd terms is equal to the mm of the coefficients of the even terms. In the identity (1 + ^)* = 1 + (?,.r + C^r^ + C^a^s + . . . + C„a^", put ^= - 1 ; thus 0=l-(7, + C2-C3 + C4-(76+ ; /. I + C2 + C4 + = r', + C3+(75+ 367. The Binomial Theorem may also l)e applied to expand expressions which contain more than two terms. Example. Find the expansion of (a' + 2a; - 1 f. Regarding 2a: - 1 as a single term, the expansion = (a;')" -V 3 (ar»)8(2« - 1 ) + 3ara(2a: - 1 )2 + (2a- - 1 )3 =a;« + ftrs + 9^ _ 43JJ _ 9^.2+ 6j; _ 1, on reduction. 368. For a full discussion of the Binomial Theorem when the index is not restricted to positive integral values the student is referred to the Higher Alpebra, Chap. xiv. It is there shewn that when x is less than unity, the formula is true for any value of n. When n is negative or fractional the number of terms in the expansion is uuliiuited, but in any particular case we may write down as many terms as we pieaae, or we may find the coefficient of any assigned term. - [OHAP. the expanaion things tahing , the sum of the he coejicienis of ■... + C„afy lied U> expand 1)3 ction. rheorem when les the student is there shewn a^ + .. terms in the we may write the coefficient xxxvtll.] BINOMIAL THEOREM. 337 Example 1. Expand (l + x)~' to four urina. 1.2 1.2.3 '^+- = l-3a; + 6ar«-l0a:8+.... Example 2. Expand (4 + 3a;)^ to four terms. (4 + ar)*=4«(l + ^)^=8(u^)^ ^S + Ox + f-^a-'-^la-f ... 369. In findii.^ the general term we must now use the fornmla n(n- l)(n~2) (n-r+l) . xmueninfull ; for the symbol "C, cannot be employed when n 18 tractional or negative. Example 1. Find the genera) term in the expansior of (1 +x)^ The(r+1)»' _ l(-l)(-3)(-5) ( -2r-f3) , The number of factors in the numerator is r, and r-1 of these are negative ; therefore, by taking - 1 out of each of these negativa factors, we may -rite the aK-ve expression ^4- (-l)r-iL^.-_5,^.12r-3) ^r |i) ii^ i ^i I * «■» 111 ,h » n F 19 r c c } i I* ^ n m I? p i' 338 ALGEBRA. [OHAP. Example 2. Find the general term in the expansion of (1 - x)-\ The(r+l)>ner«=:<-^><-^)<-°> <-^-''+V (-xr 1 • <2 • o r _(r+lHr+2) . 1.2 ^» by removing like factors from the numerator and denominator. 370. The following example illustrates a useful application of the Binomial Theorem. Example. Find the cube root of 128 to five places of decimals. (126)*=(58+l)*=5(l+i)* -!i(^A.^ 111,5 ^^■h-.) -.1 1_1 1 1 1 _, 1 2«_1 2»^1 2? ~ "^3*1^ 9'l06"^8l'i07~- =:5+!2i •00032 , -0000128 ■^ 3 ^ 9 "'■ 81 ■• =6+ -013333 ...- -000035 ... + ... =6 -01329, to five places of decimals. EXAMPLES XXXVIII. b. In the following expansions find which is the greatest term : 1. («+y)" when x=4, y = 3. 2. (a?-v)* when x=9, y~A. 3. (l + «)* when x=%. V -i: [chap. lion of (l-x)-». aif inominator. )f ul application 1 of decimals. Is. reatest terra : XXXVITI.] BINOMIAL THEOREM. 339 1 (a-46)"^ when a = 12, 6=2. 5. (7a: + 2y)^ when x^8, y=li. 6. (2a; + 3)" when a;=|, n = 15. 7. In the expansion of (l+a;)* the coefficients of the (2r+l)"' and (r + S)*"" terms are equal ; find r. 8. Find n when the coefficients of the 18"' and 26"' terms of ( 1 + ar)" are equal. 9. Find the relation between r and n in order that the coefficients of the (r + 3)*'" and (2r - 3)"" terms of (!+»)»» may be equal. 10. Find the coefficient of af" in the expansion of ( a:" + i ) . 11. Find the middle term of (l+x)*» in its simplest form. 12. Find the sum of the coefficients of (x + yY^. 13. Find the sum of the coefficients of \Sx + i/)^. 14. Find the r"" term from the beginning and the r"" term from the end of (a + 2a;)''. 15. Expand (a* + 2a + 1 )3 and (a;" -ix + 2)*. Expand to 4 terms the following expressions : 16. {l + x)\ 17. (i+x)* 18. (l+a;)*. 19. (l+3.r)-2. 20. (l-^)-\ 21. (l+3a;)-*. 22. {2+x)~». 23. (l+2a;)-*. 24. {a-2x)-i Write down and simplify : 25. The 5"- term and the lO"- term of (1 +«)"* 26. The 3-* term and the II*'' term of (1 +2x)^. 27. The 4* term and the {r+1 )"" term of ( 1 -r x)-". 28. The 7"' term and the (r + 1)*"" term of (1 - x)^. 29. The (r + 1 )"■ term of [a - bx)-^, and of ( 1 - na;)»- Find to four places of decimals the value of 3Q. S/F^. 31. ,^620. 32. m. 33. i^.^. «« 3 *'i <~ 'y, > Xi ft CHAPTER XXXIX. Logarithms. 371. Definition. The logarithm of any number to a given base IS the mdex of the power to which the base must be raised in order to equal the given number. Thus if a'^iV, a? is called the logarithm of iV to the base a. Examples. (!) Since 3<=81, the logarithm of 81 to base 3 is 4. (2) Since 101 = 10, 102=100, 10^=1000 \^ ''f'^''^ numbers 12, 3, ... are respectively the logarithms of 10, lUO, 1000, ... to base 10. . ' 372. The logarithm of N to base a is usually written log^iT, 80 that the same meaning is expressed by the two equations Example. Find the logarithm of 32^4 to base 2,^/2. Let X be the required logarithm ; then by definition, (2^2f=32;^4; (2.2*)'= 2».2 .*. 2*'=2'+*. hence, by equating the indices, L=? J 2 5 a-=^=3-6. o 1 ^^^V_^^®" ^^ '^ understood that a y irticular system of Io«arK.i.ii3 ig in use, the suffix denoting Uie base is omitted. Ihus m arithmetical calculations in which 10 is the base, we usually write log 2, log 3, instead of log,o2, logj^B CHAP. XXXIX.] LOGARITHMS. 341 nber to a given must be raised =-A^, a? is called » base 3 is 4. » >garithms of 10, written hgaN, > equations (2. at system of ie is omitted. the base, we Logarithms to the base 10 are known as Common Loga- rithms; this system was first introduced, in 1615 bv Briggs a contempoi-ary of Napier the inventor of Logarithms'. Before discussing the properties of common logarithms we shall prove some general propositions which are true for all logarithms mdeptndently of any particular base. 374. The logarithm of 1 is 0. For a9=l for all values of a; therefore log 1=0, whatever the base may be. 375. The logarithm of the base itself is 1. For a^=ai therefore log„a=l. 376. To find the logarithm of a product. Let MN be the product ; let a be the base of the system, and suppose x=\ogJ[, y=\oga^^; 80 that a'=M, a'^K Thus the product MiV- a*xa>'=a'*» ; whence, by definition, logaMN=x+i/ = \ogaM+logaK Similarly, log„ir.VP= log„ J/+ \og,,N+ hg^P ; and so on for any number of factors. Example, log 42 = log {2 x 3 x 7) = log 2 + log 3 + log 7. 377. To find the logarithm of a fraction. Let 2^ be the fraction, and suppose x=\ogaMy i/ = \ogaN; so that a'=M, a»=N. Thus the fraction whence, by definition. M a* A-y lOge tr ■N -X — 19 =logai/"-log«i\r. I Mi: i-!l; lie t *■ i : E I s^ c ' >' ■ r 1 , i 1 k 1 i ■*: f-'- k f iM BMi» Ji' 342 ALGEBRA. tOHAP. 15 Example, log (2|) = log y = log 15 - log 7 =log(3 X 5) - log7=log 3 + log5 - log7. 378. To Jind the logarithm of a number raised to any power integral or fractional. ' Let logaiAf) be required, and suppose X = logaMy SO that a'-M; then M»={a'Y=a^; whence, by definition, \oga{M^)=px\ that is, loga(3/^)=;?Iog„J/: Similarly, Iog„(JfO=^log,i/: Example. Express the logarithm of ^ in terms of log a, log 6, and log c. log^a=log^=loga^ - log(cW) 3 q ■=:gloga~(logc''+log6a)=gloga-51ogc-21og6. 370. From the equation 10*=^^, it is evident that common logarithms will not in general be integral, and that they will not always be positive. For instance, Again, 3154 > 103 and <10*; /. log 3164 = 3 + a f raccion. •06 > 10-2 and <10-»; .-. log -06 = - 2 + a fraction. 380. Definition. The integral part of a logarithir is called the cnaracteristic, and the decimal part is called the mantissa. The characteristic of the logarithm of any number to the base 10 can be written down by inspection, as we shall now shew. KXXDC.] LOOARITfiMS. 343 rmsof log a, logb, - 5 log c - 2 log b. 381. To detenntne the characteristic of the logarithm of any number greater than unity. It is clear that a number with two digits in its integral part lies between 10* and lO'* ; a number with three digits ni its in- tegral part lies between 10^ and 10* ; and so on. Hence a number with n digits in its integral part lies between 10""* and 10". Let iV be a number whose integral part contains n digits ; then .*. log N={n - l)+a fraction. Hence the characteristic is w - 1 ; that is, the characteristic of the logarithm, of a number greater than unity is less by one than the number of digits in its integral part, and is positive. 382. To determine the characteristic of the logarithm of a decimal fraction. A decimal with one cipher immediately after the decimal point, such as -0324, being greater than '01 and less than •], lies between lO"'^ and 10~* ; a number with two ciphers after the decimal point lies between 10~' and 10"^ ; and so on. Hence a decimal fraction with n ciphers immediately after the decimal point lies between 10-'"+*> and 10'" Let Dhf a, decimal beginning with n ciphers ; then D~ 10-('»+l'+» fraction . .*. log 7)= -(w+l)+a fraction. Hence the characteristic is-(w+l); that is, the characteristic of the logarithm of a decimal fraction is greater by unity than the number of ciphers immediately after the decimal point and is negative. 383. The logarithms to base 10 of all integers from 1 to 200000 have been found and tabulated; in most Tables they are given to seven places of decimals. This is the system in practical use, and it has two great advantages : (1) From the results already proved it is tvident that the characteristics can be written dowii by inspection^ so that only the niantissse have to be registered in the Tables. (2) The mantisscB are the same for the logarithms of all B^ ff^ ii 1 ^ ■ ff ^ if k ' ' M'^- ■i { i ^ r 'IS! : J. HI 1 !' ' « * If 1» fr» %« » ,*■' B 17 i 1 lU (■ P» r f ( 344 ALGEBRA. [OHAf. S^Z ^^'t ^7^i^^ ^™? significant digite; so that it is sufficient to tabulate the mantissae of the logarithms of integers. Tixm proposition we proceed to prove. 3m. Let iV be any number, then since multiply ins or dividing by a power of 10 merely alters the position of the tSH" J^Ii;. '''?'';l?^ changing the sequence of figures, it follows that^ X 10^, and JVjl(y, where p and rr are any integers, are numbei-s whose significant digits are the sam« m those of JV. Now (1). T 18 portion log (iTx 10*)=log iT ! ;j lot; :( =log^+M Again, log(A^4-10«)=Iogir-^/loglo =logiV-g' ^2). .„J?i\^ an integer is added to log if, and in (2) an intege subtracter from log . -. ■ that is, the mantissa orVe imal pS-t ot the logtuithm revaains unaltered. In this and the th^ee pxectdiiig articles the mantissse have ^sen supposed positive. lu orapr to n^mv^^ the advantages of liuggs system, we arrange or.v work so aa aiwam to keep the mantusa positive, so that wiien the mu.fcissa of any logarithm ha,« beer, taken from the Tables the characteristic is prefixed 'vitli It i appropriate sign, according to the rules already given. w,?/fl ^^ i*^ T^ ""^ *. ".eg'^^^ye logarithm the minus sign is fij t T {^^.<'f'<\^^H'''^^c, ar.d not before it, to indicate that the characteristic alone is negative, and not the whole expression. 1 hus 4-30103, the logarithm of -(K^ % is equivalent to - 4 + -30103 and must be distinguished frou -4-30103, an exprelsion in which both the integer and the decimal are negative.^ In work- ing with negative logarithms an arithmetical Irtifice will some- times be necessary m order to make the mantissa positive. For instance, a result such as -3-69897, in which the whole expres- sion IS negative, may be transformed by subtracting 1 froni the characteristic and adding 1 to the mantissa. Thus -3-69897= -4+(l - -69897) =4-30103. Example 1. Required the logarithm of 0002432. In the Tables we find that 3859636 is the mantissa of 1ob2432 bvl^t'S tCoh"' well as the characteristic being omittd^f ^S. ^g- _^. - is-.tio or .,„e iv-^aiiLi;iii oi tne given number .*. log 0002432=4-3859636. XXXIX.] LOGARITHMS. 34<{ Example 2. Find the value of %/ -000001 65, given log 165 = 22174839, log 697424 =5 -8434968. Let X denote the value required ; then logx=log(-00000166)*=i log( 00000165) =^(6-2174839) ; the vmnHssa of log ;00000165 being the same as that of log 165. and the characteristic being prefixed by the rule. Now g (6 -2174839) = i{ 10 + 4 -2174839) = 2 8434968 and -8434968 is the maHtissa of log 697424; hence x is a number consiBtmg of these same digits but with one cipher after the decimal pomt. [Art. 382. J Thus a;= -0697424. 386. Suppose that the logarithms of all numbers to base a ^e known and tabulated, it is required to find the logarithms to Let N be any number whose logarithm to base h is required. Lety=logftiV; so that 6*=^; .-. loga(6')=log,i^; that is, 3'loga6=log„iV; ^=i^^^°g-^' or *^^*^=BP^^°g<»^ (1> Now since iV and 6 are given, log„^and log,6 are known from the Tables, and thus logs iV may be found. Hence it appears that to transform logarithms from base a to K ■ as \( to r 1«V ;i r I' '!«!* y^. lOHAP. Cor. If in equation (1) we put a for iV, we obtain logja = 5 T X logatt = , f ; .-. logftaxloga6=l. 387. The following examples illustrate the utility of loga- rithms in facilitating arithmetical calculation. Example 1. Given log 3= -4771213, find log{(2-7)'x (•81)*-f(90)*}. The required value =31og|^+| log ^-| log 90 =3(log33-l) + |{log3*-2)-|(log3»+l) = (^-T-i)^og3-(3.|.|) =4 -6280766 -5 -86 =2-7780766. The student should notice that the logarithm of 6 and its powers can always be obtained from log 2 ; thus log5=logY=loglO-log2=l-Iog2. Example 2. Find the number of digits in 875^^ given log 2 =-3010300, log 7 =-8450980. log{875i6)=161og(7xl26) = 16(Iog7+31og5) = 16(log7 + 3-31og2) = 16x2-9420080 =47-072128 ; hence the number of digits is 48. [Art. 381.] Example 3. Given log 2 and log 3, find to two places of decimals the value of x from the equation 6'-»*.4*'*-*=8. Taking logarithms of both sides, we have (3 - 4a;)log 6 + (a; + 5)log4=log8 ; .-. (3-4a;)(log2+log3) + (a; + 5)21og2=31og2) XXXIX.] LOGARITHMS. 347 .'. x(-41og2-41og3 + 21og2) = 31og2-31og2-31og3-101og2; X _ 101og2 + 31og 3 21og2 + 41og3 4-4416639 2-5105452 = 1-77.... places of decimals EXAMPLES XXXIX. 1 Find the logarithms of ^32 and -03125 to base 3/2, and 100 and -00001 to base -01. 2. Find the value of log4512, log, 0016, loga^, log«343. 3. Write down the numbers whose logarithms to bases 25, 3, -02, 1, -4, 17, 1000 *** g» -2, -3, 6, -I, 2, - s re^'pectively. Simplify the expressions 4. log Va-Vc«' 5. Hii^y.i^)'}. 6. Find by inspection the characteristics of the locarit' iv i of 3174, 625-7, 3-502, -4, -374, -000135, 2322065. 7. The mantissa of log 37203 is -5705780: write down the logarithms of 37 '203, -000037203, 372030000. 8. The logarithm of 7623 is 3-8821259 : write down the numbers whose logarithms are -8821259, 6-8821259, 7-8821259. tb Given log 2 =-3010300, log 3= '4771213, log 7 = •8450980, find e value of 9. log 729. 12. log 5-832. 10. log 8400. 13. log 1. log "-^m. 14. log -3046. 4s I 348 f : l! ;. « ■' |;j ■ r^ » c t IB i, % ■ i K 'i l» i Ir ■^ r K< ,. 1 r -: .f 1' r-f i ■ I .! M ii£3*it.^ ^'^ i^ ALGBBRA. [OHAP. XXXIX. 16. Shew that logll + log^-2log^ = log2. 16. Find to six decimal places the value of .225 ^, 20 , 512 Iog224-21ogj^ + log— . 81 /588 X 768 '686 X 972* r lie. ^'""^^^^ lo«{(10-8)* x (•24)^(90)-'}, and find its numerical lu. Find the value of log ( 4^126 . >/m -r ^l0O8. ^/m). 19. Find the value of log \/- 20. Fie yoie uoiiiOfcx of d'gits in 42*". 21. Shew that (^V*^ is greator than 100000 22. How many ciphers are there between the decimal pomt and the first significant digit in /^y****? 23. Find the value of V-OHJOS, having given log 398742=6-6006921. 21 Find the seventh root of •00792, having given log il = 10413927 and log 600«^77= 2-6998179. 25. Find the value of 2Iogg + Iog^?_3j^g45 Find the numerical value of x in the following equations : 26. 3*+'= 405. 27. 10''-*' =2'-*'. 28. 5*-^ =8. 29. 12a'-M8'-a'=1458. m^ [OHAP. XXXIX. J find ita numerical CHAPTER XL. Scales of Notation. decimal point and vexx )98179. t5 18" ig equations * • S7-«'=1468. 388. The ordinary numbers with which we are acquainted in Arithmetic are expreHsed by means of multiples of powers of 10 ; for instance 25 = 2xlO + r); 4705-4x103 + 7x102+0x10+5. This metliod of representing numbers is called the common or denary scale of notation, and ten is said to be i\\v radix of the scale. The symbols employed in this system of notation are the nine digits nid zero. In like manner any number other than ten may be takf^n aa the radix of a scale of notation ; thus if 7 is the radix, a number expressed by 2453 represents 2x7H4x72+5x7+3 ; and in this scale no digit higher than 6 can occur. 389, The names Binary, Ternary, Quaternary, Quinary, Senary, Septenary, Octenary, Nonary, Denary, Undenary, and' Duodenary are used to denote the scales corresponding to the values hoo, three, ... twelve of the radix. In the undenary, duodenary, ... scales we shall require symbols to represent the digits which are greater than nine. It is unusual to consider any scale higlier than that with radix twelve ; when necessary we shall employ the symbols ', c, ^as fligitH to denote ' ten,* 'eleven ' and ' twelve.* 8ymlj( e8i)ecially worthy of notice that in every scale 10 is the not for « ten', but for the radix itself. ^ 390. The ordinary operations of Arithmetic may be per- i. uiea in any rle ; but, bearing in mi ' rafc the successive ptnvers of the liwii x are no longer powern ten ■ determining xXiQ carrying figures we must not divide by tm. • by the radix nfthe scale in qu tion. Y 1 IP 360 ALGEBRA. [CHAP. I ^ H jij j f. -^ Example 1. In the scale of eight subtract 371632 from 630225, and multiply the difference by 7. 630225 136473 371532 7 136473 1226235 Explanation. After the first figure of the 8"btraction, since we cannot take 3 from 2 wo add 8 ; thus we have to take 3 from ten, which leaves 7 ; then 6 from ten, which leaves 4 ; then 2 from eight which leaves 6 ; and so on. Again, in multiplying by 7, we have 3 X 7 = twenty-one =2 X 8 + 5 ; we therefore put down 5 and carry 2. Next 7x7 + 2 = fifty-one = 6 8 + 3; put down 3 and carry 6 ; and so on. Example 2. Divide 15e/20 by 9 in the scale of twelve. 9 ) 15e<20 lee96...6. Explanation. Since 15 = 1 x 7^+5 = seven teen- x9 + 8, we put down 1 and carry 8. Also 8x !7'+e = one hundred and seven=«x9 + 8; we therefore put down e and carry 8 ; and ao on. 391. To express a given integral number in any proposed scale. Let iV be the given number, and r the radix of the proposed scale. Let cfo, a,, a^, ... a„ be the required digits by which iV is to be pressed, beginning with that in the unit's place ; then iV= a„r" + a„_,r"- > + . . . + aj^ + a^r + ao. We have now to find the values of a^, ai, Oa, ... a„. Divide Nhy r, then the remainder is a,,, and the quotient is a„r"-* + a„_ir"-2 + . . . + ojr + aj. If tliis quotient is divided by r, the remainder is aj ; if the next quotient a-i ; and so on, until there is no further quotient. Thus all the required digits a„, ui, 0,2, ...an are determined hy successive divisions by the radix of the proposed scale. ex XL.] HCALES OF NOTATION. 351 Example 1. Ikpreas the denary number 5213 in the scale of 7)6213 7)744.. .6 7)106. ..2 7]J5...1 2 ..1 Thus e2l3=2x7*+lx73+lx7«+2x7+5, and the number required is 21125. Example 2. Transform 21125 from scale seven to scale eleven. e )21125 e )1244 ...< e)61...0 3...< .'. the required number is StOt. Explanation. In the first line of work 21=2x7 + l = fifteen = lxc + 4; therefore on dividing by e we put down 1 and carry 4. Next 4 X 7 + 1 = twenty-nine = 2 X e + 7 ; therefore we put down 2 and carry 7 ; and so on. 392. Hitherto we have onlv discussed whole numbers ; but tractions may also be expressed in any scale of notation ; thus "26 in scale ten denotes :r7r+:r^ ; 10 lO'' "25 in scale six denotes |+^ ; 6 6* •25 in scale r denotes - + -„ ; Fractions thus expressed in a form analogous to that of S^Z oT'SH/'"^'*''^^ ^""^ f ""^ radix-fractions, and the point IS called the radix-point. The general type of such fractions m scale r is ^ *i , ^2 . 6a . ^Ty'ii '"^1- ; where 6„ b.^ 6g, ... are integers, all less than r, of which any one or flfjore may be z^ro. > ^ . ;> u^ I' I 362 ALGEBRA. [OHAP. »>'' t : > :r i ;i 303. To express a given radix-fraction in any proposed scale. Let F be the given fraction, and r the radix of the scale. Let 5i, 6-2, 63, ... be the required digits beginning from the left ; then ^-'^^^+?+' • We have now to find the values of 6,, 62, 63, Multiply both sides of the equation by r ; then r/^=6. + ^2+^^+ r r' Hence 61 is equal to the integral part of rF; and, if we denote the fractional part by /^i, we have ^.=^+^+ Multiply again by r ; then 6^ is the integral part of rF^. Similarly by successive multiplications by r, each of the digits may be found, and the fraction expressed in the proposed scale. Example 1. Express ^ as a radix-fraction in scale six. o gX6— J— 5 + j, tx6 = ^-^ = 1+h^ 2 1 2 x6=3. gig .*. the required fraction = ^ + xj + p = "Sl.^. Example 2. Transform 1606 '7 from scale eight to scale five. Treating the integral and the fractional parts separately, we have 5 ) 1606 -7 5 4*8 1-7 part recur ; hence the After this required number is 1^102'4). 5)264.. 5)44.. .2 .0 6)7,. .1 1 .. .2 e dicits in the fractional rF; and, if we XL.] SCALES OF NOTATION. 353 sefSy 735? ^"^ ""^^^ ^^^ ^ ^^^ septenary number 2403 repre- Let r be the radix of the scale required ; then 7r2 + 3r + 5 = 2x7» + 4x7H3 = 885; that is, 7r2+3r-880=0; whence r=ll or -— . 7 Thus the scale is the undenary. ./^?*"^-? ""7 ''''^' 0/ wo^a^WTi of which the radix is r, the sum of the digits of any whole number divided by r-1 will leave the same remainder as the xohole number divided by r-1. .ittfJ^.f^'lf f? number, «o, «i, a„ a„ the digits be- digirf then '" P ^'^' ^"^ '^ ^^'^ '""^ ^^ t^« iV=ao+«ir+a.r2+ + a„_ir«-i + a„r"; 'S'=ao + ai+a2 + +a„-i+a„. .-. ^'S^ai(r~l) + a,(r'-l) + + a„_i(r'-i-l) + a„(r"~l> Now every terra on the right-hand side is divisible by r- 1 ; '• ^:^M-=an integer=/suppose; that is, --_ =/+^_- j-; which proves the proposition. ecur ; hence the EXAMPLES XL. 1. Add together 352, 21435, 3505, 35 in the scale of six. 2. From 35260013 take 7471235 in the scale of eight. 3. Multiply 31044 by 4302 in the quinary scale, 4. Find the product of the undenary numbers 9^83 and 3<7 E.A. 2 ' ALGEBRA. [CHAP. XL, I. i '1 I . t I Divide 31664435 by 6541 in the scale of seven. Find the s ]uare of 3024 in the quinary scale. Express 75013 in the nonary, and 5210 in the quaternary Transform 987504 to the scale of twelve. Express the octenary number 76543 in the denary scale. Transform 54321 from scale six to scale seven. Express the duodenary nuniber te in the binary scale. Express a thousand and one in powers of two, and one 364 6. 6. 7. scale. 8. 9. 10. 11. 12. hundred thousand in powers of eleven. 13. Express the sum of the septenary numbers 532, 2106, 3261, 63 in the uudenary scale j also express the diflFerence of the ternary numbers 2021 121 and 1221212 in the same scale, and find the product of the two results. 14. Find the difference between 53774 in the scale of 8 and 32875 iu the scale of 9, expressing the result in the denary scale. 15. Express 131-890625 in scale eight. 16. Transform 1001*12211 from the ternary to the nonary scale. 17. 18. 19. Express the octenary fraction "2037 in the scale of 4. Express ks *^d =^ as radix fractions in scale 6. 587 Reduce the undenary fraction =j^ to its lowest terms. 20. In what scale is a hundred denoted by 400 ? 21. In what scale is 647 the square of 25 ? 22. In what scale are the numbers denoted by 432, 565, 708 in arithmetical progression ? 23. In what scale are the numbers denoted by 22, 2 "6, 34 in geometrical progression ? 24. Find the square root of 443001 in the scale of 5 ; 2434524 in the scale of 7 ; and <986679 in the scale of eleven. CHAPTER XLI. Exponential and Logarithmic Series. 395. TriE advantages of common logarithms have been ex- (lained m Art. 383, and in practice no other system is used 3ut m the first place these logarithms are calculated to another base and then transformed to the base 10. In the present chapter we shall prove certain formulae known as the Exponential and Logarithmic Series, and give a brief explanation of the way in which they are used in constructing a table of logarithms. 396. To expajid a' in ascending powers ofn. By the Binomial Theorem, if n>\, » [2 n'^ ' [3 — ^+' = 1+^4 ^(^-^)/(-^)(^-|) 12 (1 —+. .(1> By putting a;= 1, we obtain . ,.. 1-1 (i-i)(i-?) (•+s) =1+1+-^'+ ^ "g "U » ('-^r={('-i)T= hence the series (1) is the x'^ power of the series (2) ; that is, \1 \1 1 + 1- n,\ n)\ rJ H IF }■ II If . 1 • II ii H: K • 1t ^HBI'' ;■ » ^^■i ^^^^^B^'' 1* ^^^B/' ' t l' ^^^^Ht: tjl ^B- ( ' f - ^^^^^B^f t ^Hi: 1 .f it. Hi ' ^^^■'. ^H- 1' ' ' mHk'' ^H; 1 if I If i 366 ALGEBRA. [chap. and this is true however great n may be. If therefore n be indefinitely increased we have /p2 /p3 •=(>+i+i+E+ )'• The series ^"^^"^ 12 "^13 ''"IT"'' is usually denoted by e ; hence ^ ^ ^ Write ex for x, then \l \1 Now let e"=a, so that c=log,a ; by substituting for c we obtain xHloSeaf . a^ (logeaf 11 L3 t* • ••• »< a'=l+x\ogea + This is the Expo7iential Theorem. 397. The series »+»+i+i+H+ ■ which we have denoted by e, is very important as it is the base to which logarithms are first calculated. Logarithms to this base are known as the Napierian system, so named after Napier theix inventor. They are also called natural logarithms from the fact that they are the first logarithms which naturally come into consideration in algebraical investigations. When logarithms are used in theoretical work it is to be remembered tbat the base e is always understood, just as in arithmetic}*': '^ork the base 10 is invariably employed. From the series the approximate value of e can be determined to any required degree of accuracy ; to 10 places of decimals it is found to be 2-7182818284. [chap. If therefore n be )■• jstituting for c we mt as it is the base Logarithms to this named after Napier ul logarithms from hich naturally come >ns. ,1 work it is to 1)0 ierstood, just as in employed. e can be determined jlaces of decimals it XLI.1 EXPONENTIAL AND LOGARITHMIC SERIES. 357 Example 1. Find the sum of the infinite series We have e=l + l+^+^+^+ and by putting a;= - 1 in the series for e', we obtain '-'='-'4-i^[E- hence the sum of the series is ~{e + e-^). Example 2. Find the coefficient of x^ in the expansion of -■^. e* -^={a -hx)e-* =(.-m{i-.+|-|.....(^V...}. The coefficient required = \ ^ . « _ ^~^y~^ 5 \r_ \r-\ 398. To expand log,, (1+ x) in ascending powers of x. From Art. 396, In this series write 1 +x for a ; thus (1 +xy = l+.yiog.(l+.r)+|'{logXl+^)}2+|'{log.(H-^)}3 + (1). Also by the Binomial Theorem, when ly- (1 +^)'= 1 +y.rf ^^-^>ar2+^l^llfc2) 12 \± ^-\- (2). HP If ! it i - •.V, « 35§ ALGfi^tlA. tciiAf. Now in (2) the coefficient of y is ^+t:2 ^ +~rT^'^+ 1.2.3.4 ^+ ' X^ X^ 3^ that is, 0? — -5-+-5- — -^+...>... . Equate this to the coefficient of y in (1) ; thus we have x^ a? 3^ log«(l+^)=.r--2+-3— -^+ This is known as the Logarithmic Series. 399. Except when x is very small the series for logXl+^) is of little use for numerical calculations. We can, however, deduce from it other series by the aid of which Tables of Logar- ithms may be constructed. 400. In Art. 398 we have proved that ■ log,(l+^)=:F-~-!-^--... ; replacing :i; by -x, we have lOg,(l-,r)= -a;— 2 --g- .... By subtraction, Put 1+^ n+1 ■X n , so that •^=g— XY 5 ^® *'^"^ obtain l0g.(H+ 1) - l0ge^ = 2 {2^ + g^^:^3 + 5^2^^^^^^^^ From this formula by putting n=\ we can obtain log,2. Again, by putting w = 2 we obtain )og„3-logg2; whence log,3 is found, and therefore also log^O is known. Now by putting n=9 we obtain logglO-log^D ; thus the value of logJO is found to be 2-30258509... . To convert Napierian logarithms into logarithms to base 10 we multiply by r— ~i7^» which is the modu/m [Art. 386] of the tciiAf. i>x.+ . lus we have iries for logXl+^) We can, however, h Tables of Logar- s obtain can obtain lo(^,2. g,2 ; whence log,3 -log„9 ; thus the irithms to base 10 3 [Art. 386] of the XLI.] EXPONENTIAL AND LOGARITHMIC SERIES. 359 common system, and its vahie is , or -43429448... ; we shall denote this modulus by //,. By multiplying the last series throughout by fi we obtain a formula adapted to the calculation of common logarithms. Thus that is, log,.(..+ l)-log„n = 2(g^+3-^^+^L_+..| From this result we see that if the logarithm of one of two consecutive numbers be known, the logarithm of the other may be found, and thus a table of logarithms can be constructed. EXAMPLES XLI. L Shew that <^> ^ =^-rs-r <2) '-^=i+4+[|+[i:H- 2. Expand log vl+a; in ascending powers of x, a Prove that log«2 = ^ + l + ^ + ^+ 4. Shew that 5. Prove that 1+3* OS logY^~='Lc+^x^ + ~z^ + 20x*+ 6. Shew that if a; > 1, log./^l=log.-^,-±,-^- ■ 1 C ; ^^* 1 m *■ ,m »* ! 1 1*1 '■ ■ H : if ■^ I r ' t ; f . :lti|i 360 ALGEBRA. [chap. xli. . 6" 63 6* 8. If express 6 in ascending powers of a. 9. Calculate the value of ^e to 4 places of decimals. 10. Prove that log.(l + a;-2a:2)=ar-^'+-^|?-l^+ and find the general term of the series. U. Prove that c-i=2^i +|. + -| + ^ 12. Prove that log.3=l+3i2-2+5^+7^+ ' 13. Shew that log.(l-3a; + 2x2)-i = 3x + ^ + 3r' + -^+ and find the general term of the series. 14» Prove that the expansion of log, ( 1 - x + a;^) is ~ "^ 2 ■*■ 3 "^T 5 "T" 15. If aj > 1, prove that i.J_4.J_. __!_ 1 . 1 » "^23:2 ■^3x3'^ ~a;-l~2(a;-l)2"*"3(x-l)4"-' f '■ i I; CHAPTER XLII. Miscellaneous Equations. 401. Many kinds of miscellaneous equations may be solved by the ordinary rules for quadratic equations as explained in Art. 202 ; but others require some special artifice for their solution. These will be illustrated in the present chapter. Example 1. Solve a;2-6 Write y for ; thus whence that is, Thus X y + -=6, or y2-6y+5=0; y=5, or 1. .. =0, or =1 • X X ' x^-5x-6=0, or x'^-x-G=0. x=6, -1; or x=S, -2. Example 2. Solve S^+^ - 65 = 28 (3^ - 2). This equation may be written 3^ . S^* - 28 . 3* + 1 =0. By writing y for 3', we obtain 272/2 -28y + 1=0; that is, (27y- l)(y- 1)=0 whence Thus •and therefore y = ^, or 1. 3^=^=3-3, or 3'= 1=30. x= -3, or 0. 362 ALGEBRA. [chap. I I i P Example 3. Solve 2a:» - 3 \^2a;''' - 7^ + 7 -^ 7« - 3. On transposition, {2x^ - Ir) - 3 \''2x^^fx + l = - 3. By putting \f2x'^ - 7a? + 7 = y, so that 2ar'-7a: + 7=y^ we obtain {y«-7) 3y=-3, or y^-3y-i=0; whence y=4. "r -1. Thus '»j23c^^^Tx+f=4, or ^'•2x'^-7x + 7= - 1 ; thatia, Sa;"- 7a? -9=0, or 2a;2_7a; i 6=0. 9 From the first of these quadratics we obtain a; — 5, or - 1, and from 3 the second a; =2, or 5. It should be noticed that in this solution we have tacitly assumed y to be the positive value of the exprf^ssion \''2a;2 - 7x + 7, so tliat the roots obtained from the solution of >j2x'^-Tx + 7= -1 will only satisfy the original equation in the modified form obtained by changing the sign of the radical. Thus x=5, or - 1 satisfies 2a;»-3v^2a;2-7a; + 7 = 7a;-3. and a; = 2, or 5 satisfies 2x'^ + 3 v'2a;'^ - 7a; + 7 = 7a; 3. EXAMPLES XLII. a. Solve the following equations : 1. x'+aj+l^-o'f-. x^ + x 2. ^-l + "a; -^«' 3. HM^^D- 4. 8a;8 + 65ar» + 8=0. 5. x + 8 5 3x + 14 a; + 12 x + 4~ 3x- + 8* 6. 4' + 8 = 9.2'. 7. 3a;-6 ll-2a; ,1 5-x ' 10-4a;~'*2- 8. 3Va;-3a;"*=8. 9. (.-2)%.-|=. 10. 27a?*-l=26a;* 11. 7\/^^-V21ar+12=2v/3. 12. 42^+^16=65.4*. 13. a?+2=\^+a;\/8'- X. 14. 3M3~^=2. tttl.) MISCELT .MEOtrS KQUATIONS. 86d or - ] , and from 15. 2xa-2aj + 2v/2*"-7^ + 6=5a;-6. 16. a^ + 6\^x'-'-2a; + 6 = ll+2x. 17. 2 A-2 ~0x + 2 + 4a; + 1 = x« - 2x. 18. \/4 T2J+7 = 12x2 + 6^ _ 1 j 19. 3a?(3-a:) ll-4v^^3j+5. 20. a;3-x + 3N'2a;»-3a; + 2=| + 7. 23. (a-6)a;N (^-r)a; + c-a -0. 24. a{b-'C)x'^ + b{c-a)x + c{a-h)=0. 25. Va-ic + s''6-a; = v^a + 6 -"2.f. 26. -1_.J_=JL a-a; 6-.r a-c h 27. \^^ + Vi^=-^ -~L=. xjx-i^ "Jx-p 28. V(x-2)(x-3) + 5-y|I|=Va;2 + 6.i- + 8: 29. Va;2 + 4a; - 4 + \!x^ + 4a; - 10 = 6. 30. ^lx^-^!J^)=^b^. 402. No general methotls can be given for the solution of simultaneous equations containing two or more unknowns. The simpler cases have been considered in Chai)ter xxvi. ; the follownig examples illustrate useful artifices to be omployed in special cases. Example 1. Solve a; + y = 4 (n (a;2 + y2Kar' + 2/3) = 280 (2). We have x'^ + if={x + yf-2xy = n~2xy; and !^ + y^ = [x + yf -^xy{x + y)^U-\2xy. By substituting in (2), we obtain ( 16 - 2xy) (64 12a;y) = 280 ; that is, Zxh/' - 'iOxy + 93 = } ii i-* whence a?y=3, or 31 fr- MICROCOPY RESOLUTION TEST CHART (ANSI and ISO TEST CHART No. 2) 1.0 1.1 USA Li \.i.6 ^ BiUu [ 2.5 2.2 2£ 1.8 ^ APPLIED Ifvt^GE inc 1653 East Main Street Rochester. New York 14609 USA (716) 482 - 0300 - Phone (716) 288-5989 -Fax i: IE m m I it ^ P i-- • ■« I i tit I Hi H b a 1C^ liS:!j.U, It Ml! If ^ ^64 ALGEBRA. [CHAt. Thus or „ J- whence we obtain * J. x!/=3;j y=l,orS'J 4, 1 x=2±^f^; 31 }- whence - Example 2. Solve a;2//22 = 225, a:2/V = 75, a;V'=45. By multiplying the three equations together, we have x^j^z" = 225 X 75 X 45 = 5" X 35 ; whence a;^2=5x3=15. By squaring this equation and dividing by each of the given equations in succession, we obtain 2 = 1, x=3, y=5. Example 3. Solve the equations x' + xy + xz = 48, xy + y»+yz = 12, xz + yz+z'^=8i. These equations may be written x{x+y+z)^48, y{x+y+z) = l2, 2(a;+y+2)=84. By addition, (a; + y + 2)(a;+y + 2) = 144; whence x + y+z=±\2. On dividing each of the given equations in turn by this last equation, we obtain x=±4, y=±l, 2=db7. It is clear that the roots must be taken cither all positively or all negatively. "^ Example 4. Solve ar+jr-2=14 n\ y^+z^-x^=46 (2), yz=9 (3), From (2) and (3), {y-z)^-x^= 28. Put M for y - 2 ; then this equation becomes v?'- 0^=28. Also from (1), M + a;=14; by division, u-x=2', when( 07=6, and u—8. Thusy-2=8, and 1/2 = 9; whence « = 9, or -1 ; 2 = 1 the solution is a; =6, y=9, 2 = 1 ora;=6, y=-l, 2= -9. or - 9 : and foHAJ. or 1; or 3 ;} i have ach of the given +22=84. +2) = 84. urn by this last 1 positively or all (1), (2), (3). 2=1, or -9; and = -9. XLIl.] MISCELLANEOUS EQUATIONS. 365 1. 3a; -2y = 11, 9a;2_4y2_209. EXAMPLES XLU. b. 2. a^ + 2/8=91, a:2y + a;y2=84. 3. sk' - y3=336, x^y-xi/^= 70. 4. x^+ xy + y^:=84, X +s/xy + y =14. °' a^ xi/~y^~9' 8 24 5"^y~3" 8. ar»y+y'a;=20, i +i =5 a; y 4 10. iifi-xy + x=Z5, . xy-y^ + y=15. 12. {x-yf=Z-2x-2y, y{x-y + l)=x[y-x + l). 14. Find the rational roots of (1) x + y= 5\ {aP+y^){x^+y^)=io5)' 6. X-+ a;y + y2=189, X -sixy + y = 9. 7 2 A_2_ 9. a;2-7a!y + 43/2=:34, 2x + y ar-3y ^ 2 a;-3y 2a;+y ^* 11. (a: + y)2 + 3(a;-y)=30, a;y + 3(x-y) = ll. 13. a:3+i^81(yHy) v?-{-x= 9(y3+i). (2) (a;2 + y2)^a;3-y3) = 260 «-y= 2) 17. a;V,=72, a;y22=48, a:i/22=96 1^' yj^y^y^^y- I5' ic+y , a;-y 152 ; -f. . six-y sIxTy 15 18. a:y2=30, a;yM=120, a,'2M=20, y2M=24. 19. y2+2a;=13, za;+ a:y=25, xy+yz=:20. i >i t 1 • ' il 1 i^ i .!■» * *J ' ;fl i %, 111 ■ It i*. I- 11, 366 ALGEURA. [CHAP. xui. 20. y{x+z) = U2, z{x+y) = 132, x(y+z)=90. 21. (x+a){y-b)=2, (y-o)(2 t-c)-3, (z *-c)(a;+a)=6. 22. (a;+y)(a: + z)=63. {f ^z){ij+x)sz42, (2+a;)(2+y)»54 23. a;^-a:y-2;z=14, xj/-y^-yz=6, xz-yz-z^'=i. 24. a;(33-2y)=42, y(a:-2z)=4, 2(a: + 52/)T34=0. 25. a;y + a; + y=ll, y2+2/+z=3, za;+z + x=2. 26. {x+yf-z^=Q5, x^-{y + zf=l3, x+y-z=5. 27. z + x=i9xyz, x+y=5ocyz, y+z=8xyz. 28. a;2 + i/2 + z2=84, a;+y+2=14, xy=z^. 29. a+y-z=l, a;'2-y"'«+ 22=15, ccz=12. 30. y+z-x=9, a:2-2/2_z2_i5^ ^2 = 3, 31. a;2+y2+22=i33, j, + 2_a;=7^ yz=a:>. 32. 3'=9J'-», l63-*=8J'-2. as. 2»-i = 16'-, 3' =9% /5^^=^5'F2. 34. x'-{y'zy>=a^ y^-{z-xf=h\ z-'-(x-y)^=c\ [OHAP. XLII. CHAPTER XLIII. Interest and Annuities. 403. Questions involving Simple Interest are easily solved by the ordinary rules of Arithmetic ; but in Compound Interest the calculations are often extremely laborious. We shall now shew how these arithmetical calculations may be simplified by the aid of logarithms. Instead of taking as the rate of in- terest the interest on £100 for one year, it will be found more convenient to take the interest on £l for one year. If this be denoted by £r,and the amount of £l for 1 year by £^, we have R=l+r. ^4C4.^ To jmd the interest and amount of a given sum in a given time a* compound interest. Let P d8no':e the principal, R the amount of £l in one year, n the number of years, / the interest, and AI the amoun^;. The amount of P at the end of the first year is PR ; and, since this is the principal for the second year, the amount at the end of the second year is PR x R or PR-. Similarly the amount at the end of the third year is PR^^ and so on ; hence the amount in n years is PR/* ; that is, M=PR"; and therefore I=P{R"-l). Example. Find the amount of £100 in a liunrlred years, allowing compound interest at the rate of 5 per cent., payable quarterly; having given log 2 =-3010300, log 3 =-4771213, log 14-3906 = 1-15808. The amount of £1 in a quarter of a year is £ (l + \. — ) or £~. \ 4 100/ 80 The number of payments is 400. If M be the amount, we have /Cl\-iOO E ill i f * »* lit! I" I I 1 II 368 ALGEBRA. [CHAP. .-. Iogitf'=logl00+400(log81-log80) = 2 + 4U0{nog3-l-31og2) =2+400(-0053952)=4-15808 ; whence Jlf= 143900. Thus the amount is £14390. 12s. Note. At simple interest the amount is £600. 405. To find the present value and discount of a given sum due in a given time, allowing compound interest. Let P be the given sura, V the present vahie, D the discount, R the amouat of £1 for one year, n the number of years. Since V is the sura which, put out to interest at the present time, will in n years amount to P, we have P=VJi"; .-. V=PR-^, and 2)=P-r=P(l-/2-"). Annuities. 406. An annuity is a fixed sura paid periodically under certain stated conditions ; the payment may be made either once a year or at more frequent intervals. Unless it is otherwise stated we shall suppose the payments annual. 407. To find the amount of an annuity left unpaid for a given number of years allowing compound interest. Let A be the annuity, R the amount of £l for one year, n the number of years, M the amount. At the end of the first year A is due, and the amount of this sura in the remaining n — \ years is AR^~^ ; at the end of the second year another A is due, and the amount of this sum in the remaining n — 2 years is JjR""^ ; and so on. .-. i/'=^/2"-^ + ^/i:"-2+ +AP?-\-AR+A =^(1 + 72+722+ to ?i terms) ,72"-l 408. To find the present value of an annuity to continue for a given number of years allowing compound interest. Let A be the annuity, R the amount of £1 in one year, fi the number of years, V the required present value. [chap. 80) I) )8; t of a given mm , D the discount, * of years. it at the present riodically under made either once 3 it is otherwise ft unpaid for a . for one year, n e amount of this b the end of the f this sum in the AR+A y to continue for est. in one year, ?4 ue. XLIII.] ANNUITIES. 369 The present vahie of A due in 1 year is AR'^ ; the present vahie of A due in 2 years is AR-^ ; the present value of A due in 3 years is AR-^ ; and so on. [Art. 405.] Now F is the sum of the present vahies of the different pay- meats ; ' "^ V=^AR-^ + AR-^-\-AR-^ + to n terms ■ =AR-^ ^A R-\ ' Note. This result may also be obtained by diviclinc the value of M, given m Art. 407, by 1{\ [Art. 404.] Cor. If we make n infinite we obtain for the present value of a perpetual annuity R-l~ r • 409. If niA is the present value of an annuity A, the annuity IS said to be worth ;;? years' purchase. In the case of a perpetual annuity 7nA--- ; hence 1 100 m =- = —-- • r rate per cent. that is, the number of years' purchase of a perpetual annuity is obtained by dividing 100 by the rate per cent. A good test of the credit of a Government is furnished by the number of years' purchase of its Stocks ; thus the 23 p c Consols at 96^ are worth 35 years' purchase ; Russian 4 p.c. btock at 96 IS worth 24 years' purchase ; while Austrian 5 p.c. btock at 80 is only worth 16 years' purchase. ^ 410. A freehold estate is an estate which yields a perpetual annuity called the re7it ; and thus the value of the estate is equal to the present value of a perpetual annuity equal to the rent. It follows from Art. 409 that if we know tlic nunib(3r of years purchase that a tenant pays in order to buy his farm, we obtain the rate per cent, at which interest is reckoned by dividing 100 by the number of years' purchase. ' E.A. g^ ' !| '•in t t I i i€ ■m « m ' m .fa ir € r It 1 » . 1 1 • I! i 3 ■1 370 ALGEBRA. [CHAP, XUli. EXAMPLES XLIII. 1. If i.i the year 1600 a. sum of £1000 had been left to accumu- late for 300 years, find its amount in the year 1900, reckoning compound interest at 4 per cent, per annum. Given log 104 = 2 01 70333 and log 12885-5 = 410999. 2. Find in how many years a sum of money will amount to one hundred times its value at 5^ per cent, per annum compouud interest. Given log 1055 = 3 023. " 3. Find the present value of £6000 due in 20 years, allowing compound interest at 8 per cent, per annum. Given log 2= -30103, log 3= -47712, and log 12876 = 4-10975. 4. Find at what rate per cent, per annum £1200 will amount to £20000 in 15 years at compound interest. Given log2 = -30103, log3=-47712, and log 12063 = 4 08145. 5. Find the amount of an annuity of £100 in 15 years, allowing compound interest at 4 per cent, per annum. Given log 1-04= -01703, and log 180075 = 5-25545. 6. A freehold estate worth £280 a year is sold for £7000 ; find the rate of interest. 7. If a perpetual annuity is worth 40 years' purchase, find what an annuity of £300 will amount to in 10 years at the same rate of interest. Given log 10-25 = 1-01072, and log 1280 = 3-1072. 8. Find the present value of an annuity of £900 to continue for 20 years at 4| per cent, compound interest. Given log 1-045 =-01912, and log 41458 =4-6176. 9. A man borrows £20000 at 5 per cent, compound interest. If the principal and interest are to be paid by 20 equal annual instal- ments, find the amount of each of these ; having given log 105 = 2 021-2, and log 3767 = 3-576. 10. A man has a capital of £100000, for which he receives interest at 3| per cent. ; if he spends £7000 a year, find in ^yhat time he will be ruined. Given log 2 = -301 , log 3 = -477, and log 2=1 -362. [CHAP. XUII. been left to accumu- jar 1900, reckoning iiven 410999. >ney will amount to ir annum compound 1 20 years, allowing iriven J75 = 4-10975. . £1200 will amount iven 63 = 408145. in 15 years, allowing iiven 5-25545. sold for £7000 ; find ' purchase, find what 3 at the same rate of 10 = 3-1072. £900 to continue for iven 4-6176. mpound interest. If > equal annual instal- g given i-576. p which he receives X year, find in what = 1-362. MISCELLANEOUS EXAMPLES VL 1. Simplify h-{b-{a + b)-[b-(b-^rn,}] + 2a}. 2. Find the sum of a + b-2(c + d), b + c~S(d + a), and c + d-4{a + b). 3. Multiply 2^-\ly by z-^^y. 4. If a: = 6, y = 4, 2 = 3, find the value of ^2^^%+!. 5. Find the square of 2 - 3a; + x\ 6. Solve j + -==2. 7. Find the H. C. F. of a^ - 2a - 4 and a^-a^- 4. 8. Simplify :2^+ 26 a= + 6. 9. Solve a + b a~b a^-b^' 1 y ^X g- 3 ^°' '^radJf??n''ffi''' '^7" ^ "r^u'^' '^^''Se places when 18 is added to the number, and the sum of the two numbers thus formed is 44: find the digits. numuers 11. If a=l, b=-2, c = 3, d=-i, find the value of a%' + b'^c + d(a -b) 10a-{c + bf ' 12. Subtract -x'^' + if-z^ from the sum of l=>^ + ly'> ly"- + lA and 13. Write down the cube of a; + 8y. 14. Simplify 3 4 • xU^Xy X* yi y x^-^y"^ xy 15. Solve |(2a.-7)-?(a:-8) = l£+J+4. IG. Find the H.C.F. and L.C.M. of a^ + r' + 2a;-4 and a;3+3a;'-4. Ih) f m i M \ , I* ! •i ' If : S72 ALGEBRA. 17. Find the square root of 4a* + 9 ( 1 - 2a) + Sa^ (7 - 4a). 18. Solve y= x+a , h' x= 2 y + h ■"-3 a + 3 19. Simplify (-^--^Ui±^ ^ "^ \a; + a x-aj x' + a: i9 ax' 20. When 1 is added to the numerator and denominator of a 3 certain fraction the result is equal to ^ ; and when 1 ig subtracted from its numerator and denominator the result is equal to 2 : find the fraction. 21. Shew that the sum of 12a + 6^>-c, -7a-b + r, -^nd a + b + Qr, is six times the sum of 25a + 136 - 8c, - 13a - 136 - c, and -lla + 6 + lOc. 3 1 22. Divide x^-ocy + Y^y"^ by x-^y. 23. Add together 18 l^-g^l' + z)}, 24. 25. 26. 27. Find the factors of (1) 10x2 + 79a; -8. (2) 729a^-2/«. 4a;-118 „ , 2a;- 1 , 5a; + 3 „ Solve — = — + ,- =3 6 17 11 Find the value of (5a-36)(a-6)-6{3a-c(4a-6)-62(a + c)}, when a=0, 6=-l, c=g. Find the H.C.F. of 7x3 -10x2 -7a; +10 and 2ar»-x2-2x+L x^ - Ixy + 12y2 ^ x^ - bxy + 4t/2 28. Simplify 29. Solve x2+5a;y + 6j/^ * x^ + ocy-^y"^ 3a6x+ y= 961 4a6x + Zy 30. Find the two times between of a watch are separated by = 96"| = 176/* 7 and 8 o'clock when the hands 15 minutes. 7 -4a). denominator of a \ ; and when I ig minator the res\ilt • + <•, i^nd a + h + 6r, -13a- 136 -r, and ^-y)}' -r a+c)}, ir+l. ck when the hands MISCELLANKOUS EXAMPLES. VI. 373 « 31. If a=l, 6= -2, c = 3, d= -4, find the value of \^ - 46 + a* - \/c3 + i^ Hh^Trf. 32. Multiply the product of | .r' - 1 ary + y" and ^ a? + y by a:3 - Sy*. 33. Simplify by removing brackets a*-{4a3-(6a2-4a + l)} - [ - 2- {a^ - ( - 4a'' - ^i^^4a)}-(^a - 1)]. 34. Find the remainder when 5a;* - 7ar' + Sx^ - x + 8 is divided by a; -4. 35. Simplify |,-^ 36. Solve xl-^y'^ ^xy-y -tfl xy X*-y* y x-n 3 +y = 18 4 37. Find the square root of 4a:« - 1 2x* + 28ar' + 9a;2 _ 42a; + 49. 38. Solve -OCGa; - -491 + -7233:= - -005. 39. Find the L. C. M. of x^ + y^, Sa:^ + 2xy- y\ and ar» - a;2y + a:ya. 40. A bill of 25 guineas is paid with crowns and half-guineas and twice the number of half-guineas exceeds three times that of the crowns by 17 : how many of each are used? 41. Simplify (a + 6 + c)2-(a-6-i-c)2-f-(o-f-6-c)2-(-a-f6-j-c)» 42. Find the remainder when a* - 30^6 + 2a262 - 6* is divided by tJ* — ab-\- 26^. 43. If a=0, 6 = 1, c= -2, f/=3, find the value of (3a6c - 26cc?) ^d-^bc-c^bd + Z. 44. Find an expression which will divide both 4x2 -|- 3a; -10 ^^^ 4a;3 -f 7x2 _ 3a; _ 15 without remainder. 45. Simplify a + ri2- a-b a2 fta 2a262 ^ TT "a2 + 62 « 6 46. Find the cube root of Sar* - 2arv 4- ^ - -^, 47. Solv( 9a;-i-8y=43a;y 8a:-t-9y=42a;y }• 6 216 IH' ^■f': - ■ ,«: H^" : 1' ■ m 1 1 ^ J ':■ m f c 4 '*■ t » i » i P i! 'm* b r r f -: ii 14 374 48. 49. 50. 61. ALGEBRA. o- 1*3 2 ar-7 '^ ^ x-i x-5 (a?-2)(a?-3) Find the L.C.M, of Sr* + 38a;2 + 59x + 30 and 6ar'-13a;2-13x + 30. A boy spent half of his money in one shop, one-third of the remainder in a second, and one-fifth of what he had left in a third. He had one shilling at laat : how much had he at first? Find the remainder when x' -10x^ + 8x'^-^a^ + 3x-ll is divided by x'^-5x + 4. 62. Simplify 4|a-|^fe-*?^j-|l(2a-6) + 2(6-c)}. 53. Ifa = ^, 6 = 1, c = T, prove that (a -i^b){^a + h)^a-b = Sc* 54. Find the L. C. M. of a;^ _ 7^ + 12, 3x^ -Qx-9, and 2«2 - 6a; - S. 55. Find the sum of the squares of ax + hy, bx-ay, ay + hx, by-ax\ and express the result in factors. cfl 01 X y 3x - 5z z 7y ^ 56. Solve g + |=-4-=g+j|=l. 57. Simplify ^^^^,-^5-^,. If a-b __1_\ ■2\a2 + 62 a-bf' 1. Solve «'-(3^--ro-)=^(2a; + 67) + |(l+|\ 58 59. Add together the following fractions : -4x X' -a;2 x^ + xy + y^' a^-y^' y^(x-yf' x^y-y*' 60. A man agreed to work for 30 days, on condition that for every day's work he should receive .3s. 4d., and that for every day's absence from work he should forfeit 1«. 6d. ; at the end of the time he received £3. lis. : how many days did he work ? 61. 62. 3aH5 43a;2 770,-3 33a; ^2 -ix*+-^ — 4-i>y 2+3-a;v Divide -^- + 27- , 4 4 Find the value of 1 10 f {7x -%)}] when x= -^ and «=2, sp, one third of the lat he had left in a uch had heat first? c»-7ar' + 3«-ll 18 -c)}. and2a;2-63:-8. , hx-ay, ay + hx, \ 7 '•^y-y*' ition that for every hat for ev'ery day's ; at the end of the 1 did he work ? r + 3-a;v -42r')]] MISCELLANEOUS EXAMPLES. VL „ Q. vf lOx-11 lOx-1 , x9-2x + 5 63. Simphfy 3^_^,- r)- 3(xHx-fl) ^-^-f)(,.,iy 375 n,.3 «•■(;' 64. Find tlie cube root of ,~ af^-' , 'x^ + '' — x*--t,3^. h c 7' „ „, 4a;-17 , lOx-13 8.r-30 r^x-i 65. S"lve--^-+-2^3-=^^-_^+^-j. 66. Find the factors of (1) x^ + 5x^ + x + 5. (2) x'-2xy-32^^ 67. Solve 1 68. Simplify? ^(x+y) + 2z=2l 3a;-^(y + 2) = 66 x + i^{x + y-z) = S8 x + 2y 3a,-2 f 63^7/ + 70?/2 r^-y 2x' + 3xy-35if 69. Find the square root of - (.36 - 2c - 2a)»{ 2(rt + c ) - 36 }. 70. The united ages of a man and his wife are six times the united ages of their children. Two years ago tiieir united ages were ten times the united agets of tiieir children, and six years hence their united ages will be three times the united ages of the children. How many children have they ? 71. Find the sum of . , x-^'-Sxy-lf', 2y2-?2/3 + .2^ xy-lyHy^, and 2xy-^7/^ 72. From { (a + 6) (a - a?) - (a - h) (b - x) } subtract (a + bf - 2bx. 73. If a = 5, 6 = 4, c = 3, find the value of ^6abc + (6 + c)'* + (c + a)» + (a + bf-(a + b + c)\ 74. Find the factors of (1) 3x^ + Qx'^-l89x. (2) aH2ci6 + 62 + a + &. 75. Solve px = Qy\ {p + g)x-{q-p)y=zr j' /y J-?L 76. Simplify ST''- y 2x2 + !Ky + r<^' i f ; «: ' m ■j 4 4 • . k i SJ : a • ^! ''I j: 1 r ,ir :|' Qrw ^S 1 1 376 77. Solve ^+ 1 ALGEBRA. 2a;- 15 a; + 7 2(a; + 7) 2a;-6' I70 T> J iK^ — a;' — 2a? + 2 78. Reduce -^-g— ---— to its lowest terms. 79. Add together the fractions : 1 1 2a;2-4a; + 2' 2a;2 + 4a; + 2' and 1-x^ 80. A number consists of three digits, the right-hand one beinc. zero. If the left-hand and middle digits be interchanged the number is diminished by 180; if the left-hand digit be halved and the middle and right-hand digit be interchanged, the number is diminished by 336 : find the number. a Divide l-S=,+ f^-||.,-^^. by l-x-^^. 82. If iJ = l, g=2, find the value of (pg + g") - (p - q)JW±2na+^ 'ip + q-{p-(q-p)) ' ' 83. Multiply ^-5a:H|-t-9 by l-ar-f 3. 84. Find the L. CM. of (o26-2a62)2, 2a2-3a6-262, and 2(2aHahT-. as. Solve?^=^ + 3^. x+l 4a; + 4 3x+T 86. Reduce ^^^^+J^^ to its lowest terms. 87. Find the square root of 4a*-}- 9 ( a2 + ^) -f 12a(a2 -I- 1) + 18. 88. Solve 3^ 2w „„ I 89. Multiply X 1/ a 3x + 4y + ll^ by 10a:-3y-li^V. ^ ,, X X- ^ [+y • « 131 MISCELLANEOUS teXAMtLES. Vt. 377 -X' it-hand one beiiif } interchanged the id digit be halved, interchanged, the iber. 14 I5^ ~^x\ C'+ahf. . 90. A bag contained five pounds in shillings and half-crowns; after 17 shillings and 6 half-crowns were taken out, thrice as many half-crowns as shillings were left : find the number of each coin. 91. Find the value of 5(a-h) 2{3a-{a-|-&)}-|-7{(a-26)-(oa-26)}, when a= -^h. 92. Divide 3a:* - 5ar' -F 7x-2 - 11a; - 13 by 3a; - 2. 93. Find the L.C.M. of 94. Resolve into factors : (2) -a:2-f2a;-l+a:*. (1) a?-m\ x + a a; + 3a 95. Solve ,- =-. x+b x+a+h 96. Simplify (1) 35aW-49ft3^3 65a'*6c-91a»6V' 97. Solve 7a;-9y-f-4z=16 ar-f .v _ x + y + z 3^2 2a:-3y-f42-5 = 2y 2t/a-2y-60 * 98. Simplify y y''-6y+5 o 2v ^ y+1 y-2\ x^^ — s)' y + 2j 99. Find the square root of 4ag - 12a?) - 6hc + 4ac + %'^ + c^ 4a2 + 9c2-12ac 100. The express leaves Bristol at 3 p.m. and reaches London at 6 ; the ordinary train leaves London at 1.30 p.m. and arrives at Bristol at 6. If both trains travel uniformly, find the time when they will meet. 101. Solve ( 1 ) •^x+ -753; - -1 6 = a; - •,583a; -f 5. (2) 37 ,+ i % a;2-5a;-^6"^a; -2~3-a;" m m lEi ' i !! M IS Ik i ■* I I! 5' i3 I *| 37d 102. ALGEBRA. Simplify (1) « + a^ , + -,-^-'^ + ^^ (2) (l+a;)2-r/l+ 2 \. 1 ^-^'+rTlT^^J 103. 104. 105. 106. 107. 108. 109. 110. 111. 112. Find the square root of also the cube root of the result. Divide 1 - 2a; by 1 + 3a; to 4 terms. I bought a horse and carriage for £75 ; I sold the horse at a gain of 5 per cent., and the carriage at a gain of 20 per cent., making on the whole a gain of 16 per cent. Find the original cost of the horse. Find the divisor when (ia^ + lab + bb^)^ is the dividend, 8 (a + 26)2 the quotient, and b'^(9a + Ubf the remainder. Solve (1) 5a;(a;-3)=2(a;-7). (2) : + 6 = -^^ + a;-2^a;-r ah {x-l)(x-2) If ^=a + & + i^'. and y = ^ + —,. 4{a + by * 4 a + b prove that (a; - a)2 - (3^ - 6)2 = b^ Find the square root of 2o Solve a + x -, + ■ a-x 3a Subtract -„- a2 + aa; + a;2 a^-ax + x^ x(a* + a^x^+x*)' x + S , a; + 4 a;2Hf-a!-l2 from a;2-a;~i2' and divide the difference by 1 + J ^^l ^ ^^-. •^ a;2 + 7a; + l2 Find the H.G.F. and L.C.M. of 2a;2 + (6a-106)a;-30a6 and 3x^-(9a + 15b)x + i5ab. 113. Sulvc (1) 2cx--abx + '2ahil — icdx. (2) X 2(a;+3) 9 J — X' 8a;- 1 x^-9 4(a;-3j' MISCELLANEOUS EXAMPLES. VI. 379 f 20; old the horse at a ain of 20 per cent., . Find the original is the dividend, the remainder. 2" b)x+4:5ab, 114. 115. 116. 117. 118. 119. 120. 121. If a=l, 6=2, c=3, d=4, find the value of 6a + c* + d'' + (a + 6)(6 + c) I rode one-third of a journey at 10 miles an hour, one-third more at 9, and the rest at 8 miles an hour ; if I had ridden half the journey at 10, and the other half at 8 miles per hour, I should have been half a minute longer on the "way • what distance did I ride ? The product of two factors is {3x + 2i/f-(2x + 3yf, and one of the factors isx-y; find the other factor. If a -I- 6 = 1, prove that (a^ - 62)2 _ ^^3 + ^,3 _ „j^ Resolve into factors : (1) a^+y^+3xy{x + y). (2) m?-n^-m{m?-n^-) + n{m-n)^. Solve (1) a;3-y3=28j (2) x-'-Qxy+Uy^^Q^ x'^ + xy + y^=7 y Find the square root of (a-b)*-2{a^ + h'i)(a-b)^ + 2{a* + ¥). Simplify the fractions : '-f lly2^9| x-3y = iy (1) 1 a^-- a-' a + 1 (2) (■4)-(-iy I X X a + 1 122. FindtheH.C.F. of a% + b^Cr-abc-ab^ and ax^ + ab-a^-bx^. 123. A constituency had two-thirds of its number Conservatives : in an election 25 refused to vote, and 60 went over to the Liberals ; the voters were now equal. How many voters were there altogether ? 124. Solve 125. Simplify 126. Divide (1) (2) (1) (2) x^ , ,„ , > 2ax — -T -f (a - 6) = =-. a+6 ' a+h U? X y V" 2yi {x+lf-ix 1 + yl+fz^^] ^(i_ ^_+ y^::z^\ (x + l)*-(x-l) 1)3 i4' x* + {a-l)a^~{2a + l)x'^+(a'^+ia-5)x+3a + Q by x^-3x + a + 2. i I i« IB ^1 I r! it 41 ! I I 1 1-U ! E 380 127. 128. 129. 130. 131. 132. ALGEBRA. Resolve into factors : (1) a^+5xy-24y^ + x-3p. (2) n<--. X Find tlie square root of p^ - 3q to three terms. Solve (1) £z^_£z^ = i£^ EZr x-6 x-7 x-2 x-'6 (2) ax + l = by + l = ar/ + bx. Find the H. C.F. of 3x^. + (4a -2b)x- 2ab + a« and a^+{2a- &)«9 - (2a6 - ««) ar - a'b. Simplify (2) x^ '^'&4 133. 134. 135. 136. 137. 1.^ At a cricket match the contractor provided dinner for 24 persons, and fixed the price so as to gain 12A per cent upon his outlay Three of the cricketers being absent, the remaining 21 paid the fixed price for their dinner, and the contractor lost Is. : what was the charge for the dinner? Prove that a;(y + 2) + J + ?^ is equal to a, if y X x^-^ and y=?^. y + l ^ 2 Find the cube root of jk3 - 12a;2 + 54a; - 112 + 1^ - ^ + -^. X ar^ ar Find the H.C.F. and L.C.M. of ar' + 2aa;2 + a2a; + 2a3 and a^-2ax'^ + a?x-2a\ Simplify (1) 42{^^-?^^|_56|'^ril^-?£z%\, Eesoh e 4a2{a^ + ISaft^) - (32a« + 96V) into four factors. l-m Solve (!) ^J^^\=^'jz^_2^ <') :iy-''' x^.=''' ^ - 140 MISCELLANEOUS EXAMPLES. VL 381 ,_4 ae [ns. i' and led dinner for 24 ain I2i per cent. i_ being absent, the ir dinner, and the "or the dinner ? ;-2a3. 2x - %) 3ur factors. 0. 139. Shew that the diflference between x-a x-b x-c ^ x-a x-b^x"^ is the same whatever value x may have. 140. Multiply a;^ + 2y^+32^ by a;*-2y*-3z* 141. Walking 4^ miles an hour, I start 1^ hours after a friend whose pace is 3 miles an hour : how long shall I be in over- taking him ? 142. Express in the simplest form (1) (8* + 4*)xl0"*. 143. Find the square root of (2) 3="'x9 144. Simplify y X yy yx /i) /_iE \_\ ot^jzl {x-lfjx + lf + x^ ^ ' \x-\ jc + iy *a;6 + l • 2) -f— ^ 2ay^y'^' a-y ]"" x'^ + x^+\ ^ a^-a?y'^ . a*-2a?y + aY -\ \ a? + y^ "^ a^-ay + y"^' ]' 145. Find the value of (1) \^ + \/50-Vl8 + v/i8. (2) \/35 + 14^6. x-h x-a 2{a-b) 146. Solve 147. Shew that (1) .--— J-. x-a x-b x-{a + h) (2) 2^ + 3y =11 4a;2 + 9a;y + 92^2 =>n. = 11 J (a + 6)3-c» [b + cf-a^ {c + af -¥ (a + b)-c b + c-a c + a-'b ' is equal to 2(a + b + c)^ + a^ + b'^ + c'^. 148. Divide a-x + ia^x^ -'ia^x^ by a^ + 2a^x^^-x^. 149, Find the square root of (a-l)4+2{a*+l)-2(a2 + l)(a-l)'5. i I t 382 ALGEBRA. i 4 ■*■ ( ■ l> ■ 4 » • M »' <» 1 .n, «i "I r ■9 8" 'IX t I- ; 150. How much are pears a gross when 120 more for a sovereign lowers the price 2d. a score ? 151. Shew that if a number of two digits is six times the sum of its digits, the number formed by interchanging the digits is five times their sum. 152. F) nd the value of 1 1 (a-fc)(6-c) (h-c){a-c)~{c-a){Jb-a)^ 153. Multiply „,^ 12 + 41a; + 36a;2, _ . , 26a? - Sa:^ - 14 3 + o.c 7— s by 5 - 2x + — -„—. 4 + 7a; 154. If X- - = \, prove that x'^+-^=i, and oi?---^^^, 155. Solve (1) ?| + -^ = l(a: + 5). 3-4a; 1 (2) 2x2 -3/ = 23 '-3w2, 2 = 231 2ary-3y2= 3 J" im. Simplify (1) l|V20-3V5-^g. N^/4^V^. (2)^(^Y. y "Va;''/ X* 157. Findt}ieU.C.F.oi{p'i-l)x^ + (3p-l)x-p(p-l)vind p{p + l)x^-{p^-2p-l}x-{p-l). 158. Reduce to its simplest form 1 ^ ', ^ , y 2/2 3V i' / 159. Find the square root of (1) l-22n+i + 42". (2) 9«-2.C" + 4". 160. A clock gains 4 minutes a day. What time should it indicate at 6 o'clock in the morning, in order that it may be right at 7.15 p.m. on the same day ? 161. If .r=2 + ^2, find the value of x"'-^^. X e for a sovereign times the sum of ging the digits is -a) Ax ,=4. r7 p-\) and f4«. should it indicate t may be right at MISCELLANEOUS EX/MPLES. VL 162. Solve 383 (2) '^l + a; + N/l -a;^g six-b ]Jx ' '-' ^/i + x-s'l^' 163. Simplify (/> - a) (c - a) "^ (c - 6) (a • 6) "'' (a - c){6 - c)' 164. Find the product of ^^5, i^2, ^/80, i^5, and divide 8-V5. 3^/5-7 v/5+1 ^ 5 + V7' 165. Resolve 9x V - 576y2 _ ^^j? ^ 256a;2 into six factors. 166. Simplify 1- (1) a'' {x + af x{x + 2a) (2) (x + a)(x-a) ' (x'-a^)(x + af /2 r.3(7n.-«)ic r4(r-s) , r^-s^ \~\ n "^ L~7F+"s) ~ ~ I "21^ ~ 4(??i2-n2) J J' 7{r + s) jp 167. Simplify (1) {a'^-pY^\^^^-. (2) ^14^132. 168. Find the H.C.F. and L.C.M. of 20a^ + a;2-l, 25x^ + 5x-'»-ar-l, 25x^-10x''+l. 169. Solve (1) a + x + sBaxTx'^^h. (2) a; + 9f + ^ = 8. 7 + -8- 170. The price of pho<^ographs is raised 3."?. per dozen, and cus- tomers consequently receive seven less than before for a guinea : what were the prices charged 1 171. If (a + l)^3. prove that o^ + - ^ = 0. 172. Find the value of x + 2a X ^a iah 2b-x'^2b + x'^x'^-4t)'^' when X-- ah a + b' m 384 ALGEBRA. 173. Reduce to fractions in their lowest t' i" (1) /I 1|1\./ cc + y + z ^~\ + l ^ ' \x y z) ' yx^ + y^ + z^-xy-yz-zx x + y + zj ' ,2) A__56_ + -i2_\/ _56__J2^\ ; + 4 x + 3 174. Express as a whole number (27)^ +(16)^- ^ 175. Simplify ^2_ (8)-* ' (4)-'^ • + (1) n ; + n 1 - ic" 1 - a;-» (2) ^97-56x/3. 176. Solve 177. Find the square root of 178. Simplify .. X - 4a scj-Sa _x + Qa X + 5a a;-3a x-4a~x-ia x-3a' (2) 3a;2 + a^ + 32/2=8i | 8xa-8a)y + 8y2=l7|j" a'^'* + 2a'".r» + a;-» (1) — pxVacx^-x r-. (2) 1 3v/3-" J 179. A boat's crew can row 8 miles an hour in still water ; what is the speed of a river's current if it take them 2 hours and 40 minutes to row 8 miles up and 8 miles down ? 180. lia=x^-yz, b=y^-zx, c = z^-xy, prove that a^~bc = x{ax + by + cz). 181. Find a quantity such that when it is subtracted from eacli of the quantities a, b, c, the remainders are in continued proportion. 182. Simplify 1 \ a^-f (1) Ix + y- X -„- X- (2) a 2(7a;-4) ,+ xp I x^-y a; -10 2{4a;-l) 6x2-7a: + 2 6a;2-a;-2 ix'^-\ MISCELLANEOUS EXAMPLES. VL Z8p x+y + zj^*' 3^/3. 3\/3-« J [1 water ; what is in 2 hours and 40 a? hat racted from each are in continued 183. Find the sixth root of 729 - 2916a;8 + 4860«* - 4320^8 h 2160.<;« - 576x1" + 64x«. 184. Simplify (1) 1 1 x + sfi^^l X->s/^^' (2) ^16 + 4/8I - ^ -512 + iyi92 - 7^9. 5 185. Solve (1) 6-- 6- 186. Simplify H- X (2) a;V + 192=28u,y'i x- + y = 8 J* h-c c-a 5L~A._ a2 - (6 - c)- 62-(c-a)2 "*" c-^ - (S - ?;)2' 5 (1) a;-15-2-+— ^.j-6. * a; - 15f 187. Solve (2) 2{a; + y-i) = 3(x-i-y) = 4. 188. If a;y=a6{a + 6) and x^-xy + y^=a^ + ¥ prove that (M)(f-I)=«- 189. Find the H.C.F. of (2tt2 - 3a - 2) a;2 + (a2 + 7a + 2)a; - «2 _ 2(t and (4/2x + s/2{2x-1)--ILz, \2x by -i=. + \/2(2a;-l)-\/2x. 191. Divide a'^¥ + b*c'^ + c*a^-a%*-hh*-c^a* by a^fe + ft^c + c^-aft^-fec^-ca'. 192. Simplify (1) 10a; -1 2(;f+l) 6(a;-l) 'S(x"+x+l) K.A. .;,, / \/a; + a _ \' x - a \ (\^a; - a Va- + a j 2b s/[x + af-ax h '■ II; i ; I 386 ALGEBRA. ' 1 i V « •». 193. If p be the (liffercnce between any quantity and its reciprocal, 7 the (litfercnce between the square of the aanie quantity and the square of its reciprocal, shew that 194. A man started for a walk when the hands of his watoh wore coincident between three and four o'clock. When ho finished, the hands were again coincident lietwecn five and six o'clock. What was the time when he started, and how long did he walk ? 195. If n be an integer, shew that 72"+i + l is always divisible by 8. 196. Simplify 197. Find the value of ^^' 7-3^/5 "^7 + 3^5' 198. If a + h-\c + d = 28, prove that 4:(ah + cdf-{a^ + b'^-c'^-cP)'^=l6{s-a){.i-b){s-c){H-d). 199. A man buys a number of articles for £1, and sells for £1. Isv all but two at 2d. apiece more than they cost : how many did he buy? 200. Find the square root of 2(Slx* + y^)-2(9x^ + y^)(3x-y)'^ + (^x-ij)\ 201. If a; : a : : y : 6 : : 2 : c, prove that (be + ca + abf (a-^ + y" + z") = (fts; + ca; + ay)" {a? + b'^ + c^). 202. If a man save £10 more than he did the previous year, and if he saved £20 the first year, in how many years will Iiis savings amount to £1700? 203. Given that 4 is a root of the quadratic x^-'5x + q=0, find the value of q and the other root. and its reciprocal, Aic same quantity t of his watoh were clock. Wlicn ho ; Iwtwecn five and 3 started, and how B always divisible b)(H-C){.i-d). lid sells for £1. Is'. f cost : how many -y)*. previous year, and any years will liis x^"5x + q=0, tind MISCELLANEOUS EXAMPLES. VL 887 204. A person having 7 miles to walk increases his speed one mile an hour after the first mile, and finds that he is half an hour less on the road than he would iiavo been had he not altered his rate. How long did he take ? 205. liia + b + c)x={-a + b+c)y = {a-b + c)2={a+b-c)w, shew that y z w X 206. Find a Geometrical IVogression of which the sum of the first two terms is 2l|, and the sum to infinity 4j. 207. Simplify (-ir('-ir 208. A man has a stable containing 10 stalls ; in how many ways could he stable 5 horses ? 209. In boring a well 400 feet deep the cost is 2,i. 3d. for the first foot and an • additional penny for each subsequent foot: what is the cost of boring the last foot, and also of boring the entire well ? 210. If a, p are the roots of x^+px + q=0, shew that p, q are tlie roots of the equation a;2 + (o + /3 - aj3) a; - a/3(a + /3) = 0. 211. Multiply together the duodenary numbers tele and ete. 212. If x + z = -= ; determine the ratios x : y : z. y X z-y' " 213. If a, b, c are in ii.p. ahew that ^3.3 2\ . 9 25 ac \a^b cj\cb aj^b'^ c 214. Fiiid the number of permutations which can be made from all the letters of the words (1) Consequences, (2) Acamania. I ■' i !| ii in w ■' ■ :a « ' t i ! ;• «) ■■5 »' 1 t l> iw • *! 1! 1 %. 1(1 • It ■" »i * •\ "i ■ ^ i\ 1' t' ■• 5 r ' , ; ^ ' ! 1 m ,r- i-Jte: it. . ; ; , .;^^^p^^Jij II J| i r ■l[ ii '. r5l iff ,Aja ta^ 217. 218. «^88 ALGEBRA. 216. Expand by the Binomial Theorem (ia-Xtf', an(! find the mimeiicany greatest term iii the expansion of (I la:)", if a; = r, and n = 7. 216. When x=-^, find the value of l+2a? ^ l-2a; l+Vr+2x i-\^r^* Simplifv '^^ ~ ^^^ a;'-m _ jc»_-a5_ Solve the equations : (1) (x^-5x + 2f=x'^-Bxl-22. Prove that (y-a)3+(a?-y)» + .3(a;-y)(r-3)(y-2) = (a?-2)s. Out of 16 consonants and 5 vowels, how many words can be formed each containing 4 consonants and 2 vowels ? If h - a is a harmonic mean between c-a and d-a, shew that d-c ia & harmonic mean between a~c and h - c. In how many ways may 2 red balls, 3 black, 1 white, 2 blue be selected from 4 red, 6 black, 2 white and 5 blue ; and in how many ways may they be arranged ? The sum of a certain number of terms of an arithmetical series is 36, and the first and last of these terms are 1 and 11 respectively : find the number of terms, and the common difterence of the series. Expand by the Binomial Theorem n) (s-fV; (2) (i-2,yto5term3. !a what scale is the denary number 418 represented by 1534 ? 219. 220. 221. 222. 223. 224. 225 )xf', and find the .nsion of ( 1 i x)", ah (r.-h) lany words can be I vowels ? id d - a, shew that \h-c. k, 1 white, 2 hhie nd 5 blue ; and in >f an arithmetical se terms are 1 and , and the common :erm=. MISCETXANEOU.S KXAMPT.KS. VF, 389 226. 227. 228. 229. mid find t!io value of resented by 1534 ? „. ,., 2s'Vd ir)\/2r r.v^ Simplify - rr-^ X ,— -f , ^- .•5/^7 4v/l5 7^48 g-Tg^, given that V5^2-2.S6, By th<> Binomial Theorem find tlie cube root of 128 to nix places of decimals. There are 9 books of which 4 arc fJreek, .'{ are Latin, and 2 are P'nglish ; in how many ways could a selection ))e made so as to include at leasf one of each laugiwge ? Simplify ... \/45x3-v/80?t-\/5a^ ^^' yx''-x+\ xa + x + lj'l a-'-l "' ^^rr 230. Form the quadratic equation whose roots are 5 ^/(J. If the roots of x^-px + q=.Q are two consecuti ■ iiuegers prove that p^-i" ~\=Q. 231. Subtract 4-72473 fr om 7-C41 in the scale of eigh.t ' find the square root of <08404 in the scale of twelve. 232. Find logjrt 128, log4 \ T28, logjj \ ; and having given Iog2=-.3010:i00 and log 3= -4771213, find the logarithm of 00001728. 233. A and B start from tho same point, B five days afte A ; A travels 1 mile the fiif^' day, 2 miles the second, 3 n ea the third, and so on ; i^ ravels 12 miles a day. When will they be together? Explain the double answer, 234. Solve the equations : (1) 2'=8»+i, 9*=3'-9; (2) z'^y"^, 2^ = 2x4', .r + y + 2 = 16. 235. The sum of the first 10 tr ms of an arithmetical series is to the sum of the first 5 ter s as 13 is to 4. Find the ratio of the first term to the conui m difference. ■*» ,Bi « I; i i 390 ALGEBRA. 236. Find the greatest term in the expansion of (l-a;)"^ when 237. Five gentlemen and one lady wish to enter an omnibus in which there are only three vacant places; in how many ways can these places be occupied (I) when there is no restriction, (2) when one of the places is to be occupied bv the lady ? ^ ^ 238. Given log2= -.301030, log 3= -477121, and log7= '845098, find the logarithms of -005, 6-3, and (^^ ' Find X from the equation 18^-*'= (54 ^2)^-^. 239. If P and Q vary respectively as y- and y^ when s is constant, and as 2* and 2* when y is constant, and ii x = P + Q, find the equation between x, y,z\ it being known that M'hen y=2 = 64, a;=12; and that when y = 43 =16, a- =2. 240. Simplify , 133 _, 13 , 143 , 77 I0g-gg- + 2l0gy-l0g_ + l0gj^. If the number of permutations of n things 4 at a time is to the number of combinations of 2ra things 3 at a time as 22 to 3, find n. ^^a'^'^-W^^^b^c' P^*'^'® ^^^^ ^^ ^^ either the arithmetic mean between 2a and 2c, or the harmonic mean between a and c. If "C, denote the number oi combinations of n things taken r together, prove that Find (1) the characteristic of log 54 to base 3 ; (2) logio (-0125)* ; (3) the number of digits in 3*^. Given logio2= -30103, logio3= -47712. 241. 242. 243. 244. 1 of (1-a?)"* when ntcr an omnibus in ices ; in how many ) when there is no is to be occupied by log 7 =-845098, find when z is constant, id if .r = P+(?, fiiul known that when 16, a-=2. 7_ ri' gs 4 at a time is to igs 3 at a time aa ther the arithmetic )nic mean between 245. 246. 248. MISCELLANEOUS EXAMPLES. VL 391 Write down the (r+l)"" term of (2^^^^ - ar*)^, and express it in its simplest form. At a meeting of a Debating Society there were 9 speakers ; 5 spoke for the Government, and 4 for the Opposition. In now many M'ays could the speeches have been made, if a member of the Government always speaks first, and the speeches are alternately for the Government and the Opposition ? 247. Form the quadratic equation whose roots are a + 6 + >/a^ + 6^ and 2a6 a + 6 + \/a2 + 62' A point moves with a speed which is diflerent in different miles, but invariable in tlie same mile, and its speed in any mile varies inversely as the number of miles travelled before it commences this mile. If the second mile be described in 2 hours, find the time taken to describe the n"' mile. 249. Solve the equations : (1) x2(^,_c) + aa:(c-a) + a2(a-6) = 0, (2) (a;2 - im ^-j?) {qx -\-pq +y ) = gur* + jj^q- +p\ 250. Prove by the Binomial Theorem that 3 3.5 3.5. 7 '*'4'''4.8"'"4.8.12 + ad i»/.=^/S. of n things taken r ). 3; ligits in 3*^. m ■I I %t ;• I J! : !■ ■ *; • ^ ;: It i» ii ■« . II : 1? f 1 1. ANSWERS. ii I a. Paqb "' '1' t 'IS ; "i;*'^ ■ i 1 i ■ ■ K 394 ALGEBRA. I. d. Page 8. 1, 19. 2. 0. 8. 7. 4. 11. 6. 21. 6. 6. 7. 18. 8. 36. 9. 6. 10 14. 11. 85. 12. 96. 13. 36. 14. 0. 16. 0. 16. 12. 17. 24. 18. 43. 19. 4. 20. 8. 21. 12. 22. 0. 23. 1. 24. 6000. 26. 3|. 26. 10. 27. 5h 28. 29. 37. 30. 6. 31. 4- 32. 5h II. Page 12. 1. 47a. 2. 2ix. 3. 39&. 4. 151c. 6. -26ar. 6. -401). 7. - ny- 8. -66c. 9. -20&. 10. 2x. 11. 0. 12. -16/. 13. -s. 14. 7y. 16, 0. 16. 2ah. 17. a;2. 18. - 14a^.r. 9. -21a3. 20. -16x3. 21. 0. 22. -19^" 23. - 'iSabcd 24. 11 -g-a;. 26. 8 26. -3&. 27. -a:2. 28. 5 1 29. h 30. -5r^ III. a. Page 16. 1. 0. 2. 4ffl + 46 + 4c. 3. 0. 4. 4:X + 4'!/ + 4z. 6. 3a + 56-2c. 6. b-c. 7. 39a-5i + 4c. 8. 5c. 9. 3aa;-3fey + 3c2 10. 2'2p-18q-20r. 11. ah. 12. -20a& + ca. 13. 5ab + bc. 14. pq + qr + rp. 16. 6*, 16. 20a. 17. 2a?i/ + 223;. 18. 14a6--116c. 19. 13z. 20. a + & + c. III. b. Page 18. 1. ahc. 2. x'^-vxy-'ry'^. 3. a? + Zab-2b\ 4. yz + zx + xy. 6. 3.'c2 + 2ary-2/2. 6. -2ar' + a;2 + 4a; + 2 7. x^- Ix. 8. 15a'2-32x--18. 9. 15ar^-4a:2 + 3a:-l 10. a? + b^ + (?. 11. a? + ¥ + c^ + d^. 12. ofi + x^ + x + Z, 13. 9a3 - 3a2. 14. Sx'-^ - 22/2 - 2xy - 4yz - Zxz. 16. -ar3 + a;2 + 2//2 + y. 16. ^x"y + xy'^ 17. 2a?h. 18, x^ - x*y - y^ 19. a? + b^-¥. 3 4, 15 24. 5a-=o-^rC. o O 4 26. |a' + s«&-l'''- 000 28. -^ar' + ^aa;'- + ^a-a;. 30. -a3-5a2& + ;[a62 + 63, ii 4 rV. a. Page 21. 2. 3a - 56 - 4o. 6, lla; + 13y-16z. 8. lla; + 26i/ + 22s. 1. -2a -2c. 4. -5a + .306 -4c. 7. 21a - 1.36 - 33c. 8. lla; + 26i/ + 22s. 9 10, 2a6-2crf + 2ac-26cZ. n. 12. 2xy, 13, -3^''-x2-2a;+l. 14. -\2x^y + 2lxy'^+\5xyz. 15. 1 3 2 4^+2^-3 16. -x^--y-:.z 17. 5 10, 1 3. 13a:+lSy-19z. 6. 12a6-106c-10crf. 2ac + 26rf. - cd -ac- bd. 1 3, ,5 ga-gi + gc. 18, x-y + -=z. 10 4 4 19. -g.;-^2. 20. 5 , 13 -6=*^+6^- IV. b. Page 22. 6. 7, 9, 11. 1. 7a:y-7ys+18a;2. 3. -12 + 9a6 + 6a2i2 -12a26 + 15a62-5crf, 20a262+16a2?,. ar' + 3a;2 + 5a; + 7. 2. 4. 6. 8. 10. 2a;2-2x. 12. Qx'^y + 2y\ 14. Sx^+lOx^y-lOxy\ 16. 16. - 4a3 + 463 _ 2c3 + I0a6c. 17. 18. 4a«-7a*-5a3 + 9a"-a-7. 19. 20, - a3 + 22o26 - 16a62 + 263. 21. -l2xY- + 8x^y + 2lxi/. -2a^bc + 6h^-ca + 5c^ah. ~l6x^y + l0xy^-2xy. 9a:2_9a; + 9. -17a2a;2 + 13a:2 + 20. 13. a? -c^-ahc. 4x*-5a^-2x--x + 2. -x^ + 2x* + x^-x^ + 2x- 4 1 2a;2-gary--y2 22. 2 7 1 7/— -rt, 23. 26. '^ar* 4^-2*^^-6^ _1 2_? 7 26. itt' 24. 3 - -„.a 5a:- + 7;aa; 8 6 a'a; S- iaa;' F"i I 19' '•■ l Hi In jk |» '» .t ^\ M I" « •*, ,ir ■IS. ' r k It, I- ■■ 1 J % 39( ALGEBRA. Miscellaneous Examples I. Page 23 1. (1) 2j; + a;2; (2) -3a + ?;. 2. 2a+2c. 3. (1)21; (2) 108. 4. (1) 11 ; (2) 18. 5 7ar' - 10a;3. 6. 8ct'* - 2a 9. 2x3 -2x\ 11. 2a -(36 + 5c). 12. 47; 12. 13. -7/2.fy, 16. 36. 16. 0. 17. a-h. 18. a; + 2s. 20. 7x7/. 21. 8. 23. 4a. 24. 118 25. 30 B.C. 26. 2ar' -2x. 28. a + b-(c-d). 29. a + b miles south of 0. 30. 2a:2 + 7a;-3. V. a. Page 2?. 1. 35a:'. 2. 20a". 3. 56a<63 4. mx*y^ 6. 8am. 6. Ga%c*. 7. Aa%\ 8. 10a%. 9. 2Sa''b^ 10. 5a*V^x^y\ 11. Ga^x-'^/^. 12, dbcxyz. 13. Wa'^Wx* 14. 28a='63a^. 16. 40a2ca;2. le. 30a»ar«i/3 17. 2a:y. 18. 3a«a;Y8. 19. aV;3 + a^'^c. 20. 20a362^ -28a262a!4. 21. 10a;3 4 6a;V 22. a5ft + a3j3_„3jf,c2. 23. ab''c'^ + a%c^-a%\. 24. 20a46c3 + I2a%^c^ - Sa^bc^ 25. I5x^y + 3a;^2_2iarBy« 26. 48a^i/3_ 40aV + 56a;3y''. 27. ea^^c -7a»64c2. 1. 6. 8. 11. 13. V. b. Page 3(;. 1. 36. 6. -12. 11. - 16. 16. 500. 21. 40. 26. 3. 2. -48, 7. -9, 12. 375. 3. 5. 8, -24. 13. 500. 4. -24. 9. - 168. 14. 140. 6. - 16. 10. 480. 15. - 2000, 17. -180 18. -56. 22. -63. 23. 118. 19. -1000. 20. •224. 24, -130. 26. -54. 27. 1. 2. 28. 0. 29. 29. 30. - 13. -Sa^x^. 3a36*c5rfe, a^ftV-a^ftZfl*, 15a;3y2 - 18a;22/3 + 24a;3^, 16. -fixhf^z^-\-^xhfh^-%v^y'^z n, 91a;V-{ 105a:i'y*. 19. -aafeV + aSftV + a^iV. V. c. Page 31. lAaPV^xK 3. -a%\ 6, -5oi?y*z\ 7. 9. ^x^+^xy + Zxz. 10. 12. 14. 16. j2ai2 4. - moc^y"^. -36.rV'^-48.T?/V a%c - ah^c + abc^, 14a*6» + 28a3i4. 56icV + 40»V' -48a:«2/%5 + 96ar*i/V, -8a;Va^ + 10a;*«/V. 20. a^^-a^bh + a^b^c^. i« 'AGE 23 2a-\-2c. 2a -(36 + 5c). 0. 17. a-h. 4a. 24. 118. a + b-(c-d). 2a;2 + 7a;-3. 4. 30a;2. 36. a;2 + aa? - 6a: - a6, - 6a6. 38 a'x^ 49 3. 6. 9. 12. 16. 18. 21. 24. 27. 30. 33. 36. a;2-17.i; + 70. a:2+l7a; + 70. .r2-13a: + 12. -ar2+18a:-45. a:2-25. a;2-a:-380. 2a:2+l3a:-24. 2a;2-7a: + 5. 10a:2 + 3.r-18. 9.r2-.30a:7/+ 252/2, 3a2-30a6 + 4862 a;2 - aa; + 6a: - ab. 1. a2 + 2a6 + 6a_c» 3. a'» + a262+6*. 6. a:*-4a;2+8.r+16. 6. 8. a* + 4a2a;2+16a:^. 9. 11, xf^ + 2x^-7x'^-8x + l2. 13. a6+a363. 16. a» + 4a6*. 17. -x* + 4x^y-x'^y'^-4xy^-'y*. 19. x*-2x'^y'^ + y\ 21. 75a»63 1 28a«6'' + 1 3«'-^6« - 12«/; 23. a^-25a262-10a6'i-6^ 26. a3 + 63 + c3-3a6c. 40. 4p^q^ - 9r2. V. e. Page 34. 2. a2- 462 + 46c 4. ar' + 4:e2y + Sxy"^ + 12y^ ar*- V.4. r xfi + y^. 7. 64a8~276*. 10. a;* -a* 12. 4a;" - ar' + 4a:. 14. xfi-2af^-4a^+\9x'^-3}x + 15. 16. 8a;3_27y3. 18. a^~a*b* + 2aW + }A 20. a2^a + fS^p _ ^a^a _ jj^p^ '. 22. 81a:«-256a*. 24. x^ + Sxy + y'^-l. 26. ■Jifi + y\ 27. a;" + y'". M I % 1 >. ' I* ' ii li a* 4 «« I :> 398 28. a« - 2a3 + 1 31. 33. 1. 6. 9. 13. 17. 21. 23. 27. 30. 32. 2,3, 86. 4^ 36^ ^I6 1. x2 + 3a;-40. 4. a;2 + 4a;-5. 7. a;2 + 7a;-44. 10. a2-l. 13. a2-4a-32. 16. a'^ + 6a + 9. 19. a:2 - aa; - Ga". 22. x'^ + 2xy-8y'^. 26. a^ + Qab + 9b^ 28. 1x^ -X- 10. 31. 3a;2 + 2a;-l. 34. 8a;'-, 6a; -9. 37. 9a;2-3a;y-22/2. 40. 25a;2-9a2. 3a;. xy\ - 86%. 2. 6. 10. 14. 18. a;*-7a;3-f4a;2. -3a;2 + 5a;. 24. - a + 6 + c. 28. ALGEBRA. 29. rt2a;3 + 27aV 32 30. afi-\-2oi?y^ + 'j/^, 1 , 5 a , 1 1 84. |x4_3„a^ + l„2^2_2 86. -:a*-\-a^, 4 a* V. f. Page 36. 2. a;2 + 5x-6. 6. a:2 - 2a; - 63. 8. ar»+2a;-8. 11. a2 + 4a-45./ 14. a2 - 64. 17. a2-121. 20. x^ + ax-ma?7 23. a;2 - 49^*. 26. a2 + 5 + (3a + 26) a; - 36= apx^ + {aq -bp)x'^-{bq + cp)x- eg. ^bx^-{Sb-2c)x-'-{b + 3c)x-c. ax^-(a + 26) a;2 + (26 + .3c) a; - 3c ANSWl,R8. 401 -2c~2d 6. 21a + ^, -2a + 6b + 2c-2d. -lla-26. ■-227a + 21W>f81. - lOffl. -i^n - 2fi. u. apx ^-(2a + 3 (6-ag) X \q. 12. sfi- (a2 + 26) x^ -t iL'ac + 62u,'. - '•-. 13. aV + (6a -l)x< + (9-2A)a;3 -62. 14. .tS- (a2+2ft).'r« + (2fflc + />2 + 2 r/),c^-(2W + c»)a;2+(/2. VIII. a . Page 55. 1. 6. 2. 4. 3. 7. 4. 4. 6. 3. 6. 1. 7. 5. 8. 3. 9. 15. 10. 1.3. 11. 13. 12. 5. 13. 1. 14, 16. 16. 10. 16. 30. 17. 5. 18. 1. 19. 2. 20. 1. 21. 1. 22. 2. 23. 3. 24. 1. 20. 4. 26. 3. 27. 3. 28. 3. 29. 1. 30. 4. 31, 7. 32. 3. 33. 4. 34. 4, 36. 1, 36. 1. 37. 2. 38. 2. 39. 1. VIII. b. 40. 2. Page 58. 1. 20. 2. 15. 3. 8. 4, 16. 6. 25. 6. 17, 7. 13. 8. 10, 9. 7. 10, 4. 11 1 7" 12. 1 f 13, 5. 14. 7. 15. 8. 16. 10. 17. 6. 18. 8. 19, 7. 20. 25. 21. 3f 22. 8. 23. 12. 24. 5. 26, 5. 26. 12. ..\ 28 -h\. 29. 8, 30. 66§. 31. 7. 32. 7. 33. 2. VIII. C. 34. 12. 36. Page 60. 27. 36 5. 1, -a* 2. 6. 3. 10. 4. -6. 6. 9f. 6, U^ 7. -12. 8. 3 8* 9. 14. 10, -| 11. 4 5* 12. 2 21 13, 1^- 14. 2 3' 15. l-J. 16. 12. 17. 3f 18. % 19, 5 7* 20. 18 IX. a. Page 62. 1. y-'C. 2. a 3' 3. 56. 4. M-\ Ic. 5. 2& 6. 100-^. a. 20-c. I I IS' "■I I il •I' 1 il 1 > 1 1 c i " ' i« k ■ i Rj r -f :€ J. t" 1 i i t p.. 1 1 t , 402 3 ALGEBRA. 9. >■ 10. 600 X 11. x+11. 12. c-20. 13. 90 -.r. 14. x-SO. 16. 20. 16. 2a;. 17. 36 -a;. 18. x+a. 19. 5a; (lays. 20. 4. 21. X 2* 22. X 4' 23. xy miles. 24. ^ milea. X 25. a 26. 120 hour X i. 27. 5p. 38. 44 a; 29. 5a + 26. 30. 400 -X. 31. 240fi t- 126 - c. 32. x~6. 33. b. 34. 40x. 36. 20a + 26-c. 36. 100 X1/' 37. y- 13x 6 • 8C 1. 100-a;-y-2. 39. 240a; + 12y + z- 30. 40. 2y + 2z- X. 16. 18. IX. b. Paob 65. 1. X, x + 1, x + 2, x + 3. 2. y- -2, y- -1, y- 3. x-% x-\y X, a; + l, a; + 2. 4. 2?t + 2. 6. 2a; -1. 6. 6m + 3. 7. x-a-h miles. 8. n{a + b). 9. ar + y 1-5 10. 2a; 4- 3. 11. mx + y. 12. Ox. 13. £106c. 14. ax „a3 ^20" T 16. £a;V ^'SO' 17. 3a,y. 18. f • "■ f • 20. abc 60" 21. 4v2 a; 22. '•^4- - I- 24. ^ hours. 26. 22a 156" 26. !g_^. ^. ^days 28. yz. 29. y lOr 30. lOOp ar X. a. Page 71. 1. 17, 12. 2. 13, 5. 6. 15, 43. 6. 162. 9. 27, 28, 29. 10. 3, 5. 13. 5. 14. 60, 61. A £100, ^£130, C£150, Silk Gs., Linen l.s. 21. 60, 10. 23. 25, 5. 25. 15 ft.. 12 ft. 22. 20 h 24. 123 3. 75. 4. 20 miles 7. 1. 8. 50, 55. 11. 15, 5. 12. £20. . 15. 6, 3. 17. 53 flor ins, 71 shillings. 19. 48, 12 20. 65, 40. uilf-crowns, 5 ( jrowns, 10 aliillings. runs, 10 byes, 5 wides. 26. 18 ft., 10 ft. ANSWKHS. 403 12. c-20. X. b. Page 73. 16. 2x. 1. 54. 2. 24. 8. 60. 4. 35. 20. 4. 0. 75. 6. 24, 25. 7. 224, 252. 8. 49, 50. 24r. - inilea. 9. 50, 61, 52. 10 . £33. 11. 27. X 12. 90 Port, 150 Claret. 13 . A £450, fl £180 , C £140. 28. 44 1*. A £525, /i£600 , C £160. 16. £4i I. v' 16. 12 ft. 18 ft. 17 . £12000. 18. 44. 24()a r m - C. 35. 20a + 2b-c. XI. a. Page 74. tS. \00-x-y-z. 1. 2ah. 2. xV- 8. 2xijh. 4. abc. + 2s-a 6. Sab. 6. 3xy23. 7. 2a%'^c^. 8. 7a62c3. 9. ^xhfz\ 10. 2ax. 11. 7a. 12. 17a6c. 13. xy. 14. 8a%'^c". 16. 25a;v/. 16. bx. 17. SaWc\ 18. abc. -2, y -hy. 6. 2a;- 1, XI. b. Page 75. 9. x + y\-5. £106c. In examples 19-29 the H.C.F. stands first ; the L.C.M. second. 13. 1. 2a-bc. 2. x^y^. 3. \23rYz. 4. 20a%'^c\ 17. 3xy. 6. \^a*b^c\ 6. 24abxy. 7. abc. 8. a%'^cX 4yz 9. \2abc. 10. \2xyz. 11. 12a:V-- 12. A2anfi. 21. 13. oT-y^c^ 14. ii(ia-b"'c\ 16. 12jcY. 16. 56.rV". X 22a \5b' 100?) ar 17. 2\Qa%\\ 18. 2Ma*h*c\ 19. ac, \2abc. 20. 2.y, 12OT/2. 25. 21 be, 9ab'^c. 22. 13a-/io , 39a^bc^. 23. 17a;y, 5lx-ijz^. 24. 5xyh, ns^yW 26. 6, 30a6c. 26. \1vi-p-, 5\m*n*r>*. 30. 27. y^, x^y^T?. 28. 5/72, mm^p^q^. 29. 36F m2»t^, 216Fw3»» \ XII. a. P\OE 76. 4. 20 miles. 1. 1 26' 2. a 5x *• -A- 8. 12. 50, 55. £20. 6. 22 xy 6. 3a be 8 '^''^ 15. 6, 3. 1 IS, 71 shillings. | 9. 4?t 10. 5m2/,2 6?f» ' 11 '^ 12. '^*^ 5y^ 20. 65, 40. 13. yz^ 14 a^c^ IK "7 i« 2»;)» •owns, [0 shiiliiij^s. 2x JLr^* W mp-'' 'Am ' Avides. Oft. 17. 5ay*' 18. 3 4abc' ,9 2;»2„i2 20. =yt ■7 i i f *l R <; V in ] |i ' f : 404 ALGEBRA. XII. b. Page 77. 6. 14. 2crf2 36 ■ 2x ' nx*' 6x^yz 5a 2. 6. a' be 9mnp 3. 7. 10. 3. 16. 9^f 11. Hl. ia 16. 8. 6c ■ X 12 17. 4, 8. vWy 10a;2' 1463 ISc^y* 400a: 44V' , 7a " 86( 18. y\ i» 7acy 1 *?i_y 2a ■ ^ ad, he, 2b(l hd 7 3/;, 2p 6a; a;?/ 1^ 8ac, 3a6 106c ■ *• "6- XII. c. Page 78. Zx'^y ' f, Qac, 62 5. 9. 13. 17. 21. 26. 5a; + 2y ~I0 Zx-jf 21 ■ 22^ 15' 11a W 31a; , 36"' ac + 6 36c 3. 6. 8. 2m, n Qx ' abc 12. -^^ y^ 3x--;/ xy 9ac, 562 16. 2. ^ 20 216c XII. d. Page 79, 3 13. 16. ac, 262 26c • 5m, 4/) ~"20nr" 10. 4a;2, 9v2 6a;y 18, 3a6, a2 9a aa;, 6 a;2 6. 10. 14. 18. 22. 26. 29. 3a -26 12"' 3a + 6 "39""' 9^ 20" 5x 24' 17a; 24' ocz-y ar* - 2^" 2a^^ ' 7. 11. ja_ 12' 3ot - 2?t 24 ■ 3/5 -g 48 16. -. 4 19. 23. 27. 30. 7a; T8' bx-ay ab a2-.362. 3a 4. 8. 12. 16. 20. 24. 28. 2a;2-15 3a; ■ 2m - 3n 15 ■ 1 5?n - n ~36 6a -46 ~T5~* 5a; T* 96a; + 2ai/ ~3a6 a^ + 63 • a Wc' 13 1463 400a: lacy Sbdx 18. y2 ac, 21}"^ 2bc ' 5m, 4/> ~'20n~' 10. ax, I) x^ %xy 18. 3ct&. g'^ 9a 4. 8. 12. 16. 20. 24. 28. 2j;'^-15 3a; * 2m - 3n 15 ■ 1 o??i - ?i ~36~' Sa - 46 5x T* 96a; + 2ay 3a6 ■ a? + b^ a 29. 31. ANSWERS. Miscellaneous Examples II. Page 80. 40o 1. 4. 7. 10. 12. (1) 3a;2 + 7a;-8. a2 + 62 + c2. a:2 + 2a;-3. 4a;2-6a;-l. 1 1 2. 13.:;. 3. 20. 5. ar^-Ua;- 10. 6. (1) ^; (2) -3. 8. ~4a + 56, 9. 5a;. 11. ( 1 ) a;2 + 1 4a; - 5 1 ; (2) 24.^2 - 55a,- - 24. (2) 1. 13. - ab. 14, 16. 16a2 + 2«6. 17. 29a. 20. 6a + 2c-2rf. 23. 1935. 26. 3a; -9. • 8 5' ar5 + 4a;'* + 48a; -32. (1) -2; (2)41. 19 2afl -x^-x. 22 16 18 21 24. 4. 27. 0. 3y3 _ 9^2 + 2y - 1. 30. A £800, B £320, 14. 32. 6w*-96. 33 3;j3-5y,2 + 2/>. 1. 25. 4m - 5n. 28. (l)-]5; (2)4. a; -2. 34. ap + bq miles ; "~-^ hours. Numerically, 55 miles ; 5 hours. 36. (l)p; (2)7tV36. 4320. XIII. a. Page 86. 1. a; = 2, y=i. 2. x= 3, y = 5. 3. a;=2, y= 3. 4. a; = 4, 2/=-l. 6. x= 1, y=2. 6. a; = 3, i/= 4. 7. a;=5. y=6. 8. a:= 1, y = 2. 9. ^=3, y= 1. 10. a;=2. y=\. 11. x= 1, 2/ = 3. 12. a;=l, y= 1. 13. a;=7, y = 5. 14. a;=10, y = 3. 16. a; = 5, y = 12. 16. x = l, y=8. 17. «= 6, y=8. 18. a:=5, y= 8. 19. x=- -7, y=-3. 20. a;=17, y=- 19. 21. a; = l, y= 2. 1. a;=12, y= 8. 4. a; = 20, y=15. 7. a; = 20, y = 60. 4 A .. r» ■■ iu. x= ,j, y— o. 13. a;= 3, y= -4 XIII. b. Page 87. 2. a;=10, y= 6. 6. a; = 45, y = 35. 8. a;=14, y=15. 11. y= 3. 14. a;=19, y= 3. 16. a;=13, y= 7. 3. a;=18, y=\2. 6. .r=51, 2/ = 17. 9. a;= -2, y= 4. 12. a;= 5, y= 4. 16. a;=12, y=-4. ^H; i .e ^m ■ [ 1 4 5' • 1 !■ ^^^^H' y * ! ', ^H ^^^^^^K ! ^r ', i\ ^^H ''^ :m» ^H 1 ^ *' 1 ' ^^ ^B/ C|;> ^^^^^^^^H[ -.' I'n • *: ^^^^B ;: ' 11 M ^^^^^^^B' «( ^^^^■o' » « ^^H'' : -^^ ^^^■' ' - 1 if ^^^^^H' «* r» ^^^^^^^H: CO ! -■ ^^h' l. , . ^^^H ' ' 1': r ^^^^^^^H ^^■^^^E ^^^^^^M 1- ^^B. !■ ' ^H'" ^^^^^^1 1 '^' ^^■^ ^ fe^ ■ 1 i ^^H' i s ^^^H ^^^H' 1 ^^^^^^H. w^ 406 1. X 3. X 6. X 7. X 9. X 11. X IS. X 16. X ALGEBRA Xni. c. Page 90. = 1, y= 2, 2= 3. = 2, yr= 3, z= 1. :9, y= 2, 2= -4. :5, y= 6, 2= 7. :2, y=-2, z= 5. = 8, y=10, 2 = 14. = 6, y= 8, 2= 5. :6, y= 2, 2= 1. 2. a;= -2, y= 4, z= 1. 4. «= 1, y= 2, 2= 3. 6. x= 3, y= 2, 2= 1. 8. x= 1, y= 2, 2= 3, 10. x= 4, 2/= -3; z= 2. 12. ic= 3, 2/= 9, 2 = 15. ,. _ 3 2 5 14. cc- 2. y- 3, Z- g. 16. a: = 35, y = 30, 2= 25. 1. a:: 4. X- 7. a; 10. X 13. a; 15. a: :5, i/ = 1 3. 1 ^3' y= ? = 2. y=-3. = 9, y = 25. _1 1 ~2' ^~3' ^'' = 3, y=-2, 2=1. XIII. d. Page 92. 2. a:=2, y = 7. 6. x=7, y=6. 8. x= -0, y=4. 3. a;=3, y=2. 1 1 3' ^ = 5- 2 3 ^=3. y=5- 11. a;=j, ^=.r 1 14. X- 12. 1 1 1 ^=5' 2^=6' 8' ^~12' "~ 16" XIVo Page 95. 1. 22, 12. 2. 55, 18. 3. 25, 17. 4. 63, 23. 6. 23.17. 6. Tea 3s. 4c?., Sugar 4rf. 7. Horse £23, Cow £16 8. A £140, i?£60, G£K, D£20. 9. A £99, 5115, C£33, D£23. 10. A 36 years, B 14 years. 12. A 5 miles, B 4 miles. 14. i^. 25 ^"^ 26" 16. 2. 15' 11. A 55 years, 5 21 years. 13. G 3| miles, D 4| nriles. 17. ^. 18, 28, 82, 21. 72. 22. £5. lis. 19. 85, 58. 20. 27. 23. 8 white, 12 black. 24. 860. 26. Man2.s. 6rf., Boy Is. 6d 26. 201bs.,40 1ba, 27, 15 miles= 28. 8 hours. 29. 6 miles, 3 miles an hour. 31. £500. 30. 10s., Is. ed. 32. 3 miles, i^ miles an hour. = 4. z= 1. = 2, z= 3. = 2, 2= 1. = 2, z= 3, = -3. 2= 2. = 9, 2=15. 2 ~ 3' 5 = 30, 2= 25. 3. X- = 3, y=2. 6. X-- 1 1 = 3' 2^ = 5* 9. x- 2 3 = 3' 2/=4- 2. X-- 1 1 ^5' 2^=6' >' ^= 16' 4. 53, 23. )rse £23, Cow £16 115, C? £33, Z) £23, rs, B 21 years. 38, D 4| miles. 18, 28, 82. 22. £5. lis. trf., Boy Is. M. iQurs. 3d. ty miles an hour. ANSWERS XV. a. Page 99. 407 1. 6. 9. 13. 17. 21. 26. 29. 33. 37. 9a2&8. IGarV-". 49a2fe2 9 ' - I25a%^ 1 27i/«" 81a«6i% 24.3a;2o 2. a6(;2. 6. 25^4^10, 10. ^a*b^ 14 9a'»ft« 10fi»a* 18. leOciOa;^. 322/ i¥' 22. 26. 30. 34. 38. 27:c9. 27a-i5 12r)a»' 256a;2^ 6561^8' 3. 49a-b*. 7. 4a2ft2c'«. 11. 4a;* 9y 16. to 19. 1 16a8* 23. 64a;i2. 27. -216ai8. 4. 3. 12. 121fi'»c«. 16 9x*y'^' 16. 4x'Y- 20. 9^. 25a;e 24. - 27a''«'>3 31. 34.3xY2. 36. - 32ic'>\ 39. - 2187' 28. 32. 36. 40. -— ai« 27 ■ 1 128a"* 64.r'Q 729a!''»' 1. 4. 7. 10. 12. 14. 16. 17. 18. 19. 20. 22. 23. 24. x^-lOx7j + 2rnj-. 9x^ + S{)xy + 25y\ p^(f - 2pqr + r\ XV. b. Page 101. a? + Cmh + 9h\ 2. a"-Qah + m\ 3 4a'2 + ]2.r2/ + 9t/2. 5. 9x'^-%xy + y\ 6 8ia;2--.36a-y + 42/'-i. 8. 25a%''' - \Oahc + c^ 9 a;2 - 2a/;<-x + a-b^c'^. 11. a'-^x- + Aahxy + 4/>-V-. a;4-2a;2 + l. . 13. a^ + lfi + c'^-2ah -2ac + 2bc. a2 + 62 + c2 + 2a6-2ac-26c. 16. a^ + 46^ + c^ + 4c-,t + 2ac + 46c 4a.2 + 9/>2 + 16c2 - 12aft 4- 1 6ac - 246r. a;'* + ?/* + 2* - 2a;2y2 _ 2x'^z^ + 2y222. xV + y^2 + 22a;2 + 20-2/% + 2a;2|/2 + 2:ry22. 9;,2 + 4^2 + 16^ _ i2;,5 + 24;)r - 1 ^r. o(^-2x^^-Zx^-2x+\. 21. 4ar*+12x3 + g_^2_6^^j_ a;2 + 1/2 + a2 + ^y2 _ 2^.^ + 2ax - 2hx - 2ay + 2hy - 2ab. 4a;2 + 9f + a^ + 46^ +I2xy + Aax - 8bx + 6ay-\ 2by - 4ab. m2 + 11^ +p^ + 52 _ 2mn - 2mp - 2mq + 2np + 2nq + 2pq. -.0 n ac 25. ^+462 + --_2a6+-_6e. 26. ^ + 96- + H-2a6-a + 96 16 9 4.T* 4a;* 27. -1 ----^-3a^2-3a; + J Iffl I ■ i ,J i: Z S! it 408 ALGRBRA XV. c. Page 101 « 1. 3. 5. 7- 9. 10. 11. 12. x^ + 3ax^ + Sa'x + a\ 2. x^ - Sax'- + Sa^a; - a». x^ - 6a;2v + 1 2xif - %y^ 4. Sar* + \2x'y + Qxy^ + y\ 27.r'-135.r2y + 225a?.v2-125jr^ 6. a^ft'' + Sa^ft^g + Sa^^^a .,. ^3, . 8a363 - 36a2/>2r + 5^bc? - 27c3. 8. 125a3 - *J5a?hc + ISoftV -?>»c''. x^ + 1 2a; V + 48.T V + 64?/«. 64a;« - 240a;V + 300a;V - l^V- 8a» - 36a662 + 54^3^4 _ 27//. 125a;« - SOOajiy + 240xV - 64yi2. 13. a3-2a2i + |a?,2_^;,3, 14. ^a3 + 2^2 + 4« + 8. 16. ^a;«-«5 + 9a;''-27a;3. 16. 2Y6a' + g«-^ + 2aa;2 + 8a;3. i! ^ i t*. 1. 6. 11, 16. 2L lOa:*. a%4 XVI. a. Paqk 103. 3. Zx^y. 3. 5a;y, 4. 4a26c3. 6. 9a'fe*. 7. aioftSc^ 8. a*hc\ 9. 8ary. 10. 6 a 18 12. 5 13. 18^ 1.V 14. 9a9 16^^ 66«' I7/>' -uct u_^ ^^ 3^2^^^ jg_ _ 2^4^3^ ^g^ 4»y ,^ 20. - 7aW 9a^y ■ 5 " 26, xhj^ 27. 2a.'y2. 31 2 23. 28. 32. 6a;y 3a3&. 24. - 25 33, 30-" 4y2i- 2aa,-8. 26. a^or^. 30, ~xY'- a' b^c*' XVI. b. Page 105. 1. a;+2y, 60 9a; +y. 9. a^-a+l. 12. a;2-2a; + l. 16. 2x + Sy-5z. 2. 3cn-2i!)„ 3. x-5y. 6. .5a; -3y. 7. x^-y^. 10. 2a;2-3a; + 5. 11. 13. 2a2 + a-2. 14. 4. 2a; -.3y, 8. l-a\ 3a;2-2a;-l. 1 -5x + x^. 16. 4ar' + 2a;4-arl 17. .r'-lla; + 17. 18. •6a:2-3aa; + 4a2, 19. 2a;''^ + j^-^ - 32 20. ab-2ac + ith( 21. 2a2 + 62_3c2. 22. 2a-2-ary + 3y-. 23. 3a;2-5a; + 7. 24. l-2a: + 3a;2-4ar'. 26. ax^-2bx'^+iic. ^b'^c + Sabc' + c^ ■ ^a^ + ia + S. o i%c\ 6. 9a%\ XT^y^. 'fi' 10. 6 a 18 10. --— ^- 17/>' cV^"., 20. -7ci''Z'". la-"^. r4' 30. O 1» 4. 2a;-.%, 8. l-a3. 3a;2-2aT-l. 1 - 5a; + x^. .53-lla;+17. a6 - 2ar + .S/;^-. 3a;2-5.« + 7. 2fta;2 + 3c. ANSWERS. 409 XVI. c. Page 106. 1. 6. 10. 13. 16. 1-3. 2. X a y 6 a^ a a? I — — , X a 3a 1 2a; X ~6'^3a 2- a; y* 3. l + y. 4. ?^+2 32/ + 2. 8. 1. a+1. 4. 2m -1. 7. l-2a;i-3a;2. 10. y^-y + l. 13. 3a;2-a;-i. 1. 4. a? --1 2 .3» 4y -2. 10. X ^ V y X X a 3ic_^ T) 3a; . a'-g^ + g- 14. a;2-a;+v 4 17. 4m2 + f« + l. XVI. d. Page 109. 2. a; + 2. 6. 4a - 3i. 8. a + 26-c. 11. 2a;2 + a;-3. 14. a;2-2a;y + 42/2. XVI. e. Page 110. 2. 1^-2. 6. a; - -. X 8. 1-1+5. o a; .. a . h 11. T--1+-. o a y 9. 12. 16. 18. 3. 6. 9. 12. 16. 2y 8^2 *• a:2-3a; + l. 2-2a:+3. 2a:2 + 8 + ^,. x^ ax - y'. l+a; + a;2. 2a2-3a + l. 3a;2-2a;a + 3o2 3a;2-a; + 6. 8. 2a; 6. --2ir'. y " 9. 12. ?-4 + ?« a a; 1. a(a2-a;). 6- ^^(7^3+1). 8. a;2(3 + aj8j^ 11. 5a;(l-5a?y). 14. 1.5a2(l - 16a2). 17. a;(3a^!-a;+l). 20. 'ia^d'- ah ■\-2h% 22. 3a;(2a;2-3a;y + 4y2), 24. 7a(l-a2+2a3). E.A, XVII. a. Page 112. x^(x-\). 3. 2a(l-o). 6. 2a;(4-a;). 9, a;(a; + y). 12. 5(3-t-5a;2). 18. 27 (2 -3a;). 18. 2ar'(3 + a; + 2ar'). 21. 2a;y2(a;y-3a; + y). 23. 5a;'(a~'-2a2~3a8). 19a3a;2{2ar' + 3a). 4. a{a-h% 7. 5aa;(l-a2a;). 10. si?{x-y). 13. 16a;(l+4a;y). 16. 5a?(2-b3ey). 19, x{aP — x^' .'-."2), 2d I 1 i M'. B ■ ii 4 «» ■I «■' )« •I m ■ . 1^1 ( I I III 410 ALGEBRA. XVII. b. Page 113. 1. {a + h)(a + c). 2. {a-c)(a + b). 3. {ac + d) (nr 1- h). 4. (a + 3)(rt + c). 6. (2 + c)(x + c). 6. (x-a)(x + 5). 7. (5 + b)(a + b). 8. (a-y)(h-y). 9. {a-b){x-z). 10. (p + q)(r-.'i). 11. (x-y)(m-n). 12. (x-a)(m + n). 13. (2x + i/)(a + h). 14. {Sa-b)(x-y). 15. (2x + y)(3x-a). 16. (x-2i/)(7n-n). 17. (ax-3by)(x-y). 18. (x + my)(x-4y). 19. (a + b){x'^ + 2). 20. (x-S)(x-y). 21. {2x-\)(x' + 2). 22. (3a; + 5)(a;2+l). 23. {x+l)(a^ + 2). 24. (y-l){y' + l). 25. (a + hc){xy-z). 26. (/2 + r,2)(a:2-a). 27. {2x + 3y){ax-by). 28. {ax+by)(mx-ny). 29. {a-b-c)(x-y). 30. (a + b)(ax-iby + c). XVII. c. Page 115. 1. (a+l)(« + 2). 2. (a+l)(a + l). 3. (a + 3) (a + 4). 4. (a-4)(rt-3). 5. (x-5)(x-6). 6. (x-l)(x-8). 7. {x-9){x-\0). 8. (x + 6){x + 7). 9. (a:- 10) (a; -11). 10. (x-9)(x-\2). 11. {a;-5)(a;-16). 12. (a; + 6)(x + 15). 13. (x-7){x-\2). 14. (a;-6)(a;-13). 15. {.'c-3)(a;-15). 16. (x- + 8)(a:+12). 17. (a:-ll)(a;-15). 18. (a:-13)(x--8). 19. {x+ll){x + G). 20. (a -19) (a -5). 21, (a-16)(ffl-16). 22. (a+l5)(a + 15). 23. (« + 27)(a + 27). 24. (a-19)(a-19). 25. (a-1b){a-7b). 26. (a + 2b){a + 3b). 27. {m~5n)(m-8n). 28. (m-77i)(m- 15«) .29. (x-Uy)(x-\2y) 30. (x-l3y)(x-\3y]. 31. (x'^+l)(a,'2 + 7). 32. {xH2y^){xH7y'') 33. (xy-3){xy-13). 34. (x + 242/)(u; + 25y) 35. (xy + ll)[xy + n). 36. (a262+25)(a262+l2) 37. (a-5te)(a-lo/>a;) 38. (x + \3y)(x + 30y) 39. {a -2b) (a -21b). 40. (a:2 + 81)(x2 + 81). 41. (4-x)(3-x). 42. (5 + a:)(4 + a^). 43. (l'2-a;)ai-a-). 44. {8 + a-)(ll+a;). 46. (2Q + xy)(5 + xy). 46. (l3-ara)(ll-a:a). 47. {ll~x^)(\2-x% 48. (27 + x)(S + x). XVII. d. Page 116. 1. (a;+l)(:r-2). 2. {x + 2)(x-\). 3. {x + 2){x-3). 4. (a; + 3)(a;-2). 5. (x + l)(x-3). 6. (a; + 3)(a:-l). 7. {x + 8){x-7l 8. (.r + 8) {t - FiY 9. {x + 2){x-G). 10. (a + 4) (a -5). 11. (a + 3) (a -7). 12. (a + 5) (a -4). 13. (a + 9)(a-13). 14. (a; + 12) (a; -3). 16. (.■r+13)(a;-12). 16. {a; + ll)(x-10). 17. (a; + 6)(x-15). 18. {.r+15)(a--16). {ac + d) {nr i h). (a;-a)(ft; + 5). {a-h)(,x-z). (x-a)(:m + n). {2x + y){Zx-a). (x + my) {x - 'iy). (2x-l)(ar' + 2). {y-i){y' + i). {2x + iiy){ax-b7/). (a + b){axihy + c). (a + 3) (a + 4). (x-1){x~8). (re- 10) (a; -11). (a; + 6) (a; + 15), (a;-3)(a;-15). (a;-13)(a;-8). (a-16)(a-16). (a-19)(a-19). {m - 5n) (m - 8?i). [x-Viy){x-mj). {xy-Z)(xy-n). (a2fc2+25)(a262+l2) {a-2h)(a-Tih). (5 + a;)(4 + x). {2Q + xy){5 + xy). (27 + .r)(8 + a?). (a; + 2) (a; -3). (a; + 3)(a;-l). (a; + 2) (a; -6). (a + 5) (a -4). (a;+13)(a;-12). (.r + 15)(a'-16), ANSWERS. 411 19. (a + 5)(rt-17). 20. 22. {x+\2y){x~riy). 23. 26. (a+14y)(a--irM/). 26. 28. (a; + 26)(a;-10). 29. 31. (a- + 7//'^)(re2_8/>2). 32. 34. (ai!> + 2c)(aft-5c). 36. (a + Qxy)(a-21xy). 38. (a;2+lla-;)(a;2_]2a2). 40. {a;5 + .30){a;3_29). 41. 43. (11+ a;) (10 -a;). 44. 46. (5 + a;?/) (13 -a;?/). 47. (a-?/ + 3)(a7/-8). (.f + .3y)(a--.3,5//). (,r + 4.y)(.r-24y). (av4-24)(«7/-10). (//- + 0a;2)(y'-i-3a;2). (rt + 8)(a-19). 21. (x + 'ta)(x~^(t). 24. (a; + 23) (a; -5). 27. (a + 2)(a-13). 30. (a;2 + .3)(.r2-i7). 33. 35. (rt+14/^a;)(a-2/>.i-). 37. (,T2 + 25a2)(.r2-l2a2). 39. (.T2 + 21a2)(x2-22a2). (l+a;)(2-a-). 42. (2 + a-)(3-a;). (20 + a;)(19-a-). 46. (ir) + rta')(8-aa;), (14 + a.-)(7-.r). 48. (17 + a-)(12-a:). 1. (a;+l)(2.r + l). 4. (a; + 3)(3a;+l). 7. (a; + 2)(2.T + 3). 10. (a; + 2)(oa;+l), K'. (a; + 3)(4a:-l). 16. (2a:+l)(a;-]). 19. (.17 + 6) (3a; -.')). 22. (.3a; + 4)(a;+l}. 26. (a; + 7) (.3a: -2). 28. (4a; -7) (a; + 2). 31. (a; + 5) (4a; + 3). 34. {Zx-2y){^x-by). 37. (12a; + 5)(a;-3). 40. (8a; + y)(3a;-42/). 43. (2 + 3.r)(3-2a;). 46. (7 + 3a;)(l+a;). 49. (.5 + 4.x-)(4-5a-). XVII. e. Page 119. 2. (;e+l)(3a; + 2). 3. 5. (.r + 4)(2a;+l). 6. 8. (a; + 5)(2x-+l). 9. 11. (a; + 2)(2a;-l). 12. 14. (a; + 5)(3.r-l). 15. 17. (a; + 3)(3.r-2). 18. 20. (2.r + 3)(3a;-I). 21. 23. (a; + 7) (3a; + 2). 24. 26. (a?- 7) (3a: + 2). 27. 29. (a; -2) (3a; -7). 30. 32. {2x + y){x-^y). 33. 36. (15a;-I)(a;+15). 36. 38. (12a;-7)(2.r + 3). 39. 41. (2 + a;)(l-2a;). 42. 44. (4 + 3a;)(l-2a;). 45. 47. (6 -a:) (3 -5a;). 48. 50. (8-9.e)(3 + 8.T). XVII. f. Page 120. 1. (a; + 2)(a;-2). 4, (c + 12)(c-12). 7. (ll + a,')(ll-a;). 2. 6. 8. (a + 9) (a -9). (3 + a) (3 -a). (20 + a)(20-«) 3. 6. 9. (a; + 2)(2a;+l). (re + 2) (3a; + 2). (a; + 3) (.3a- + 2). (a;+l)(.3a;-2). (a; + 8)(2a;-l). (a; + 4) (2a; -7). (3a;+l)(2a;-.3). {2a; + 5) (a; -3). (3a; -5) (2a; -7). (a;+lS)(3.r + 2). (2.c-7)(4a;-5). (15x'-2)(a;-5). (8a; -9) (9a; -8). (3-a;)(H4a;). (1 + 7a;) (5 -3a;). (4 + 6a;) (2 -a;). (y + 10)(7/-10). (7 + c)(7-c). (a; + 3a) (a; -3a). 10. (y + 5x){y-5x). 11. (6a; + 5ft) (6a; - 5i). 12. (.3a;+ l)(.3a'- 1) 13. {Qp + lq){^-l:i) 14. (2^-+l)(2,l--l). 15, 16. (l + 5a;)(l-5a;). 17. (a + 2ft) (a - 2/.). 18. (7 + 10A-)(7-10H (3a; + 2/) (.3a; -y). Si e! : It '1 %l » r r : 1: *ml j I! J i I f ; 1 , y^ fl 412 19. 21. 23. 2S. 28. 30. 33. 36. 38. 41. 43. 45. 47. 49. 61. 53. 56. 67. 69. 61. {pq + 6)(pq~6). 20. (x^ + 3)(x^-3). 22. (5x + 8)(5a;-8). 24. {x^ + 5)(a^-5). 26. (9«3 + 5a)(9ar'-5a). {a + 8x3)(a-8ri). 31. {l+ab)(l-a(}). 34. (3a2 + 5i2)(3o2-5fc2). (a; + 5y)(a;-5y). 39. (lla + 9a:)(lla-9x). (8x + 5z^){8x-5z^). (9?>V + 5&)(9/j2s3-56). (6a:i8 + 7a7)(6a;i8-7a'). {5ar' + 4a*)(5x-'^-4a*). 1000x150 = 150000. 1000 X 500 X 500000. 1006x500 = 503000. 2000x1446 = 2892000. 2500x1122 = 2805000. 16264x2 = 32528. (3a;2 + a)(3a;2-a). (x2a + 7)(a;"a-7). ALGEBRA. {ab + 2cd)(ab-2cd). (3a2+ll)(3a2-ll). (9a2 + 7^2)(9a2-7ai) (l + 6a=')(l-6a-''). 27. 29. ,..„,.„„„ , ,. (ab + 3x^){ah-Sx^). 32. (xV + 2)(x33/»-2). {2 + a:)(2-a;). 36. (3 + 2a) (3 -2a). 37. (x'^ + ib){x^-ib). (1 + 10/>)(1-10?>). 40. (5 + 8a;) (5 -8a:). 42. {p<^ + Sa'^)(pq-8a^), (7x2t-4j/2){7a;2-4y2). (4*8 + 3y3){4a:8- 32/3). (1 + I0a362c)(l-I0a3?>2c). (a6V + a:8)(a6V -a;8j, 241x1=241. 658x20=13160. 200x2=? 400. 2378x900 = 2140200. 3000x2462 = 7386000. 10002 x 10000= 100020000. 44. 46. 48. 60. 52. 54. 66. 68. 60. 62. XVII. g. Page 121. 1. {a + b + c){a + b-c). 3. {x + y + 2z)(x + y-2z). 6. {a + Sb + 'ix){a + 3b-ix). 7. {a; + 5c + l)(a; + 5c-l). 9, (2a;-3a + 3c)(2«-3a-3c). 11. (x + y + z)(x-y-z). 13. (3a; + 2a-36)(3x-2a + 3?)). 15. (c + 5a-36)(c-5a + 3/^). 17. {a-b + x + y){a-b-x-y}. 19. (a + b + m-n)(a + b-7H + n). 21. {b-c + a-x)(b-c-a + x). 23. (a + 26 + 3a; + 4i/)(a + 26-3a;- (l + 7> «■ m. 2. (a-6 + c){ot-6-c). 4. (a: + 2y + a)(a; + 2y-a). 6. (aj + 5a + 32/)(a: + 5a-3y). 8. (a-2ic + 6)(a-2a;-ft). 10. {a + b-c){a-b + c). 12. (2a + y-z)(2a-y + z). 14. (l + a-6)(l-a + 6). 16. (a + b + c + d)(a + b-c-d). 18. (7x + y+l)('Jx + y-l}. 20. (a-« + 6 + m)(a-n-ft-?Jt). 22. (4a + a; + 6 + 2^)(4a + a;-6-2/)- 4y). 25, (fif. -?) + .r- j,'){«-??-a; + i/). 27. (2a-5x+l)(2a-5a;-l). 26. (a-3x + 4y){a-3x-4y). 28. (a + fc-c + a;-3/ + s)(a + 6-c-a; + 2/-a). 39. (3a + 26 + c + a;-2i/)(3a + 26-c-a; + 2?/). 30. y{2x + y). ANSWERS. 413 . {x''a + l){x"-a-n {^y8 + 2)(ar'y3_2). (3 + 2a) (3 -2a). (5 + 8a;) (5 -8a:). '■npq-Sa'). 2)(7a;2-4y2). b'^c)(\-\Oa%^c). ;8)(a6V -«8). 241. = 13160. f400. )0 = 2140200. t62 = 7386000. [0000=100020000. )(a-b-c). a)(x + 2y-a). ■Sy)(x + 5a-3y). b)(a-2x-b). ■)(a-6 + c). z){2a-y + z). >)(l-a + b). ■ + d)(a + b-c-d). ■l)(7x + y-l). ') + m)(a-n-h-7n). ■b + y)(ia + x-b-y). ^-y){a-b-x + y). + l)(2a-.5a;-l). 81. 34. 87. 40. y(2«-y). 82. {x + 5y)(x + y). 33. ilx(x + 2y). 1. 3. 6. 7. 9. 11. 13. IS. 17. 19. 21. 22. 23. 24. 26. 27. 29. 30. {fix + y)i2x + 3y). 35. 5y(6x-5y). 36. (12.f- l)(iir + 7) 5a(a + 2). 33. (7a+l)(a-l). 39. 3a(a + 26-2c). x(x-Hy + 2z. 41. y{2x + y-m). 42. a(4.e + a-6). XVII. h. Page 122. 2. {a-/> + a,')(a-6-a;). 4. (2a + 6 + 3c)(2a + 6-3c). 6. {a + y + x){a + y-x). 8. (y + c-a:)(y-c + a:). 10. {c + x-y)(c-x + y). 12. (a-2/> + 3ar)(a-2/>--3ac). 14. {a-b + c + d){a'-b-c-d). 16. (.y + ft + a + 3.r)(y + 6-a-3.i-). 18. (3(1-1 +a; + 4(/)(3a-l-.i--4(:/). 20. {a-b + c + d){a-b-c-d). {x + y + a){x + y-a). {x-Sa + 4h)(x-3a-'ib). {x + a + y){x + a-y). {x + a + b){x-a-b). {l+x + y){l-x-y). {x + y + 2xy)(x + y-2xy). {x + y + a + b)(x + y-a-b) {x-2a + b-y)(x-2a-b + y) {x-\+a-\-2b)(x-l-a-2b). {x-y + a-b){x~y-a + h). {2x ~3a + c + k){2x - 3a - c - k) {a - 5b + 3bx - 1 ) (a - 56 - 3bx + 1 ). (0.2 + 4a;2 + 5ar5 - 3) (a^ + ix^ -50^^ + 3). {i^-a^ + x-3){x^-a'^-x + 3). ' (a2 + ab + 62) («2 _ „^ + ^2). 26. (a.-^ + 2xy + 4^2) (^2 _ g^y + 4,,2,. {x^ + 3xy - 2/2) {x2 - 3.ry - y% ' (2»i? + 3w7i + ?f2) (2,„2 _ 3„„j ^ ,j2)^ XVII. k. Page 123. 1. 3. 5. 7. 9. 11. 13. 16. 17. 19. (x-y){x^ + xy + y-^). (.x--l)(.r2 + a:+l). (2a:-i/)(4a:2 + 2a;?/ + y2). {3a;+l)(9a:2-3a:+l). {ab-c){a%^ + aic + c^). (l-7a;)(l + 7a; + 49.r2). (5 + a)(25-5a + a2). (a6 + 8)(a262-8a6 + 64). {x + Ay) (.r2 - 4.T7/ + 1 6y2). {ab + 6c) (a262 - 6o7 - 7) (.36?>2 + 42/j + 49). 26. (a/>c-l)(a262c2 + a6c + l). ■ \ ! ct- - i ^"' ' 1 r ■i- . i M i ■'- i-| i 1 ■] < p J ? 1 ul 1 414 27. 28. 29. 30. 32. 34. 36. 38. 40. ALGKBRA. 1. 2. 3. 4. S. 6. 8. 9. 11. 13. 16. 17. 18. 19. 20. 22. 24. 26. 28. 30. 32. 34. 36. 38. 39. {Ix + my) {49a;2 - IQxy + lOOy^). (9a - 4ft) (81a2 + 36aft + \m). (2ah + 5x) (4a%a - lOafta; + 25a;2). (xy~6z) {xhf + Qxyz + .SGs^). 31. (4a;«+5y)(16a.'*-20x-2y4 25y2). 33. (6x2 _ i) (36a;4 + 6^2ft + ^2)_ 35. (a2 + 9/,)(a4_9aaft + 81ft2). 37. (/>(7 - 3.C) (^2^2 + 3^ga; + 9x^). 39. (a;y-8)(xV + 8a?y + 64), (*a-3y)(ar* + 3a;-V + 9y^). (2* -s2)(4x-- + 2.f;i2^, •.-»). (a + 7?>)(a2-7aft + 49/'2). (2« - 92/2) (4a;2 + \%xy^ + 81 y*). (Z - 4^2) (Z2 + 42^2+161/^). XVII. 1. Page 125. (a + y + s)(a-y-3). (x- + 2/2) (a;- - xyz + y25;2) (a; _ yz) (a;2 + a.-y2 + yh^), (3x-ll)(2x- + 7). (3i/ + 2x) (9y2 - exy + 4.f2) (3y - 2a;) (9^/2 + Qxy + 4a^!). (x + 4) (a;- - 4a; + 16) (x - 4) (a;^ + 4^; + 16). {m + n + x-y){m + n-x-\-y). 7. (lla:+13K3a:-5). (a2 + />2 + c2 + rf2) (a2 + /,2 _ c2 _ j^). 4a(6 + c). (a:- 17) (a; + 7). {x-y + a){x-y-a). (?u - 7t) (w"^ + mn + n2) (a; + y). 10. (a; + 2 + a-y)(a: + 2-a + y). 12. {a-d + h-c){a-d-h-\-c). 14. (a + a; + y + z) (a + a; - y - z). (l + aa;-fty)(l -aa; + fty). {cd-\){cH'^ + cd + \)(c+a){c-a). {a + b){a-b){x + y){x'^-xy + y^-)(x-y){x^ + xy + y'^). 16. (7a; -3) (3a; + 13). (a; -19) (a; +13). (ca; + c?)(aa!-6). {ay^ + 2yh'^){x'^ + 2y^% (2a; + 7) (a; + 5). a2(a-2ft)(a3 + 2a6 + 462). (6 — c + a)(ft -c-a). 7a2a;(2a;-a)(a;-2a). (1 +m-n){l -m + 7i). 5ab{ab-l){an)^ + ab + \). 21. (x+y)(x-y){a~c){a^+ac+c^) 23. (a + ft)(a-6)(a; + i/). 26. (ah + 8)(a'^b'^-8ah + 64). 27. 20y(5a; + ?/)(5a;-y). 29. a;3(„^4y)((j_4^)_ 31. 5a;2(a:-6)(a; + 3). 33. (a;4''+l)(x2 + l)(a;+l)(a;-l). 36. 3(5a;2 + 4a2)(5a;2-4a2). 37. 4y(2x + 3)(x + 5). (3xy + 5a) {xy + 7a). ah {3a + h) (9a- - Sab + b-) {3a - ?/) (9(t- + 3alj + b"). 40. a^ax + 2y) (a2a;2 - 2axy + 4?/2) (aa; - 2y) (a^x'^ + 2aa;y + 4/). 41. (a2 + 62)(j,4 _ (42^,2 4. 64) (ti + ft) {ct2 _ aft + 62) (a - 6) (a2 + aft + 62), •ANSWKUS. 415 tx- + 2.1-^2 + , -,4). '(3ar-6). + 7). \{x-y-a). a; + 13). y){a-c){a^+ac+c^) ■b)(x + y). '■b^-8ah + 6'l). /) {5x - y). (a - Ay). (a; + 3). ' + l)(a?+l){a;-l). i2)(5a:2-4a2). l(aJ + 5). 2axy + 4?/2). !)){a2 + a6 + fe2). 42. 44. 45. 47. 49. 61. 03. 60. 1. 4. 6. 9. 11. 13. 10. 16. 17. 21. 22. 24. 20. 28. 29. 30. 1. 6. 9. 13. 17. 72 7. 8. 4 4 %2{4x-f3y)(.f-2i/). 48. {(a + ft)»+ l}(a + />+ l)(a + 6-l). (r + rf-r,{(r + ^Z)a + c + rf+l}. 46. (1 -a; + y){l -Ki- y + (a;- ,/)«}. 2{5(a-6) + l}{2.)(«-^)2-5(a-/>)+U. 48. 2c(c'yid% Oy (4x-2 + 2a,'y + f). flO. (.r - 2y ) ( j- + 2y+\). {a-h){a + b + \). 62. (a + ft)(a + /^+ 1). (a + /.)(a2-a6 + 62^-l). 64. (a + 3/>)(a-3^+ 1). (x-y){2(x-y) + l}{2{x-y)-[}. 66. Jv/(a; + y)(.r-y)(a:- //). ■Miscellaneous Examples III. Page 126. a^-2a:. 2. 42a - 406 + lOf. 3. ««--(••», (I) 12; (2) x = 5, y = 6. 6. x^ + 4x-l. 109 210" 2.C- - X. 10. ( 1 ) (aa; - 5) (ax + 3) ; (2) (2ni? + O^q) (2»i2 _ n/.ry ). (1) a;=-2, y = i; (2) .i-=5, y= -2. 12. 84x2 + 25.t"'+ 101.1-30. ar»+14.r' + 27a;» -154a: +121. (1) (a; + 2a)(.r-/>); (2) {x'+Uy)(x^-Ay). L.C.F.=:7; L.C.M.=3r)28a"/>V'. 18. £14. (1) m-?t = a + c; (2) 3a2/;2 + ("'=2>(m + ?i). 8. 23. 6x-2 - xy - j/2. Apples 4t^. a dozen ; eggs In. M. a score. (a;-5)(2.e-3)(.f + l). 26. 33. 27. 2^2 + Oa-y - 7/A x = 2, y = 3, z=0. (1) xy(x + 2y){x-2y)-, (2) (?» + «)(m -?0(2«i'^ + 3vt2). - days, 24. am "^ ' 8 pb 14. (1) 7; (2) -1 1 19. Sp. a + b. x + y. b(a + b). 2(5a;-l). {x-yr. XVIII. a. Page 129. 2. x + y. 3. x(x-y). 21. x + 2. 26. 3.r + l. 29. x(a-'Sb). 30. 2a; +1. 6. aft (a - ?j). 10. a: - 3y. 14. 3a: + 2y. 18. x'^ + a^. 22. .r - 5. 26. a;-l. 7. a (a -a;). 11. K - X. 15. a;+l. 19. x + 2y. 23. 27. a;- :x + d. 31. a;2(3a: + 2). 4. 2a; - 3y. 8. a + 2a;. 12. 2a; fy. 16. y(x~l). 20. x-3a. 24. a; - 3. 28. x'^ + y. I I. I < B .1 r f . :n5| ( ..1 tiv 1 XVIII. b. Paob 133. 1. x'-Sx+Q. 2. x'^-Ux + T). 8. ^- 8. 4. »«-5. B. x'' + 2x + }. 6. x + B. 7. a»-2aa; + a;'». 8. x + l. 9. «"- 3x + 7. 10. 2a;2 - 7. 11. 3a:2+l . 19. 2x3 -3. 18. 3a;2 + 2a^ 14. x^-a^ + «». 16. ar' + 2aa;-a'. 16, 3a''-aa:-2x«. 17, xy{2x'^ + xy- -3y2). 18. 2aPa^(2x-3a). 19. 2x2(2a; + 7). 20. 6(3x- 5a). 21. 3x^ -2xy+y\ 22. a;* + ar»-l. 23. l+x3- ■X*. 24. 1 + a. 26. a; (3+ 4a;). 26. x'^-2x+l. 27. 2x> -7. XIX. a. Page 137. 1. 3 26* 2. b c' 3. 1 ax -I' 4. 3&»c 5. 2a;-3y 2x ■ 6. 4(a;-y). 7. 1 8. aj 2a + 3a;' T^-^y^ 9. x-3y a;2 + 3a;y + 9y2' 10. X x+l' 11. 3a; x + 2' 12. 5a 36" 13. a;y 14. 3(a + h) 15. x^-\1 16 a?+2y x-2' a-b ■ a;2-5' a:^+a<'y + ?/2 17. 2a; + 3 IB. a{x-i) 19. aJ + 7 20, B+a 3a; + 5 x + 5 a: + 13' 2 " XIX. b. Page 139. The expression in [] is in each case the H.C.F, of the numerator and the denominator. 1. 3. 6. 7. 9. a-2h a + 26 a + 5 a + 4 2a+pb 3a + 5b x-1 x° + a [a^ + ab + b^]. [(a-I)(a-2)]. [(2a + 36) (a -6)]. [a:-l]. [x + a]. 3a:2 + 3a; + 10 ix'^ — ax + a- 11. (2a;-3a)2 [(2x + 3a)2]. 2. 4. 6. 8. 10. 12. x-d x + 2 [(x-in x^ + ^xy-9y^ r2a:-'ii/l l-a; + 2a;2 l-a; + 3a;2 3a'' + 6" 4a-6 2(2a;3-:.g-l ) 3.t^ + ar -t- X - 2 [l + x + ar»]. [a -6]. [x-lj. .Sx2-JK -2 3x^ + x~2 [2x+ 1]. ar + 3. ar'-a^ + T. 2x^-3. afl+2az-a\ 2aPa'(2x-3a). 3ar»-2x2/ + y»> 1+a. 2a;a-7. 4. « 8. 12. 16. 20. 20{tt-'5)* X 5a 3b' a?-f-2y x^+xy + i/ 3+a ~2~^ ANSWKRS. 417 '. of the numerator in. ^J2a; 3y]. [l+a; + a;2]. -6]. -'Ux-n -2 - [2a;+ 1]. 18. 16. 1. B. 9. 13. 17. 21. 26. 29. 1. 6. 8. 11. 13. 15. !'(. 19. 21. 23. 26. 27. 29. 30. 6a;+2 [a!»-3]. 7a?-4 3(a?-:ja Ua?-4 ff) 2(a;+3a)fx + 4a) 14. [x - 2a]. 16. 2x'» + 3x + 5 ^2^a a(a; + 8«) a:(aj + 7a) 3.C + 5]. [x^-l3ax + 5a-]. « XIX. c. Page 142. T_ 12' 4a; + 3CT x + 2 ' X x^ 2. 6. 10. ah 2o^' 5a -6 ft2 + 36 + 9, 14. 18. x-V\ a;-l" 2j;-1 2x - 5' 1 x-8' a; + 4 a; (a; -4)" 22. 26. SO. a;{3a-2) 2x-l 2a;-3' 1 a; + 7' 1. g-x a + a;' 1. 3. 7. 11. 16. 19. 23. 27. 31. 2. «+2 ar-l* a;+l a; + 6' 8pg-22 X y 1 6" m n a + x 4. 8. 12. 16. 20. 24. 28. 32. g-ll a -2" a:-f-l x + 5' x~l 4j; + 7* X. X. a + b-c b-c-a x{2 + x). a2 16a'-' + 4a6 + 62 XX. a. Pauk 144. 2. x\x - 3). 3. 1). 6. 9. x{x + i). x{x + l){x 6x(3x-l). {x + 2f{x + 3). {x-3)(x-l){x + 2). {x + 7){x-6){x 5). (a; + 2)(a; + .3)(.3.r + 2). (a: + 2)(x + 8)(2a;-l). 12a?(a; + 2)(2a;+l)(4a:-7) 20xV f 3.r + 1 ) (5a; + 1 ) (4.r - 1 ). 24. (x - (/) (3a; - 2y) (4x - Siy). 28. 2axy^{x 3) (4a; - 1 ) (3a: - 2). 28. i2a*h^ (a - bf{a + b) (a^ + ab + b% 7nJhi. (m^-n^) [m - nf. 31. r(b[a + b). .r(a; + l)(x + 12. 14. 16. 18. 20. 22. ]2x^x + 2). i. 21ar»(a; + l). 7. xy{2x + ])(2x-\). 2). 10. (a;+l)(.c-l)(a;-2). (a;-l)(.T-2)(a;-4). (a; + 5)(.r-4)(a;-6). (a; + l)(a; + 2)(2a;+l). (a; + 2)(a; + 3)(5x + l). (a; !-2)(.r-2)(3.r-7). 6x-2(a' + 7)(3a; + 5)(3a--2). {x + y)(2x-7y){4x-5y). 3a^3- (3x - a) (.2x + 3a) {x + 5a). a;2(3-5.r)2(2 + a;)2 8c2(2c-3 ■H,i ' ''B i h '■^ 'W' gd|w mi .. I Ji. i:ll: 1. 2. 4. 6. 7. 8. 9. 10. 11. 12. 13. 16. 16. 17. 1. 0. 9. 11. 14. 1. 4. 7. 10. 13. XX. b. Pagk 146. H.C.F. x-2. L.C.M. {x+lf(x + 2){x-2)(x-3). {ax + b){ax-b)(hx + a). 3. xy {x - a) (y - b) {y - 2b), H.C.F. a;(a; + 3). L.CM. a;(a;- l)(a: + 3)(2a;- 1). (l + xf{l-x)^. 6. (a; -2) (a: -4) (a; -6). H.C.F. 2a:+l. L.CaM. (2a; + l)(a;+l)(a;- l)(3a; + 2)(3a;-2). ab'^c'(c + af(c-af. L.C.M. y«(a;-y)2(.f2 + a;?/+j/2). H.C.F. x-y. H. C. F. 2a; - 3. L. C. M. (2x - 3) (3a; - 2) (a; + 4) (3a; + 4). (x + of (a;2 + ax + a?) (a;^ - ax + a^). H.C.F. 3a; -y. L.CM. {'ix-y){x-{-yf[x-yf. x-\. 14. (a + 6)(a-6)(a-26)(tt2+a6 + 62). H.C.F. a^ + xy. L.CM. («2 + a;t/)(2a; + 3y)(2a;-3»/). H.C.F. (a; -3) (a; -4). L.CM. (a; -2) (a? -3) (a; -4) (a; -6). a; - 8a. 18. \Q5xhf (a; + yf {x - yf. 4(x + l) 19a- -201 XXI. a. 13 (a; -2) Page 150. 25a; 6. 12 12a;2 + 28.r-27 3. 61 56 225 ■ '■ 8:c2 662c + 66c2 + 3ac2 + 3a2c - U% f 4a62 7. 0. 4. 8. 5.r + 31 102a; ■ x^ + yi^ 12. 16. 12a6c a^b"^- 10. 17a; w 3(a + 36) 8a ■ a2 + 3a;2 6^c2 + a2c4 3y + 2s 13. 16. 2aa; lLr»-18a-2-27a;-16 30a;3 a? + ¥ + c^ - ,Sa6c obc 2a; + 5 (.c + 2)(a; + .3)" _2(x-\i3)^ (a:-6y(a; + 2)* 2 (a; + 2) (a; + 4)' 6^ (.r-2)(.r--5)' x + 2y XXI. b. 2 Page 151. a; + 5 6. 11. 14. (a; + 3) (a; + 4)' (a-b)x {x + a)(x + b)' AojX 3. 1 a' ■X a' ax W O' Sax ^4a5 12. IS. (a;-4)(.-c-5) {a + b)x^-2ah {^-a){x-by 8x a;2-4" 5.r + 9 a;'* -9' 4a6 4aa-6» x-3). /-fe)(y-26). -1). 4) (a; -6). ) (3a; + 2) (3a; -2). y- t)(3a;+4). '/?■ lx-3y). (a; -4) (a; -5). 61 . 17a; g 3(a + 3fe) 8a ■ 10 "^ + 3a;2 2aa; - 18a;2-27a;-16 30.r3 ' + c^ - 3a/^c a6c 3. 1 12. 16. (a; -4) (a; -5) (a + 6)a;-2a/) {^-a)(x-b)' Sx «2T4" 5a; + 9 x'^-9' • iab 4^2Zp' 16. 19. 22. 26. 1. 6. 9. 12. 14. 16. 18. 21. 24. 26. 28. 31. 34. 37. 40. 2a;y 5a;2 25a;2-y2 2a^ x^ - y"^' 4(a;-l) (a; -2)2 (a; + 2)" 17. 20. 23. 26. ANSWERS. 2aH» 419 _ 2ax x'^ + a^ 18. 21. x^ + y^ xy(x'^-y^)' 4a2 a;(a; + 2a) 24. 26. 2 x+y l+g 9-a2' 2(13a; + 7) 3(a;2-4)' 2. 6. 10. ax {x - a) (x + a)"^ XXI. c. Page 153. 3. X ix'-y^ 4a; -5 6(a;2'-^l) x^ + y'^ x* + x^^~+'y'^' 1 - 6x2 1 - 4a;2" (a; -2) (a; -3) (a; -4) 1 (a;- 1) (2a; +1) (3a; -2)" 23« (l+2a;)(2 + a;)(5-9a;)' 1 .^ 1 a;+l 1 a; + 2' 19. a + b 7. 11. 13. 15. 17. 20. 4. 8. 4aa + &2 4rt2_96a 12a2-4« + 7 3(4a2-9) ' (a;-4)(a;-6) 2 22. 8a;2 + 4a;-3 a;2+ll (a;-l)(a; + 2)(a; + 3) 32a2 (l-2a)2(l+2a)" 4ar» 81-.t-»' a(a2 + 2aa- + 3a;2) 4(a*-ar») ' 2a((t2 4-32a:2) 3(a^-256ar»)" 36a^ a3-656l' (a;-l)(a;+l)(2.r+l) 26 (a;-l)(2a;+l)(2x + 3)' 17a (l-2a)(4 + aH3 + 5a)' x + 2__ (a;+r)(a; + 3)' 3x + 2 (a;-2)(a;-l)(a; + l)" 1 2x+ r 2a; + 13 23. 27. (a; + 3)(a; + 4)(.i;-4) 9Ga;2 29. 32. 36. 38. (3-2a;)2(3 + 2a;) 72a „« 1 16a* -81' 16a; 16-a;* 7a2 + 45 6 (a* -81)" Q .t;2(x'2-4) 30. 33. 36. 39, l-x* a,-(.37 + 172a;2) (a;-l)(a;+l)» 41. 1. 42. 0. 6(1 -16a;*) ' 1 (3x-y){x~Sy) 43 "^^ 43. -^-_-j. 1 I I 'J ! I i I f ' ^ ik f i Ik 420 1. ALGEBRA, XXI. d. Page 156. x-U 9, 13. 17. 20. 24. 28. 20(a;a-l)' 6. 0. 61-216 12(1-62)- 2x_ x + y 2bx 4x2-1' 2a 2. 6. 10. 14. 18. 1 l-a2" Jx_ 2 3(1 -a2)* _2^ x^-4' x + c 3. 7, 11. 16. {x-a){x-b) a; + 3a x + a' 1 33-3* g 4a2_2562" 19 4. 8. 12. 16. 2x-a ' x + a' 12(2a;+l) 4a:2-9 ' X y h{a + h) x^-¥ x-c (x-a){x-b)' x* a^~x^' a? 21. 0. 26. 0. (a-6)(a3+6»)' 81. 0. 22. 26. 29. 32. 4a3 x^-a*' a« aS-68' 2 + x + Zx^ ,2(l-a;4)' 4a6 23. 27. {x-a)(x-h)' 48a3 (a;2-a2)(x2-9a2)' g-a; « + »* 30. 2(^!±1) X{3fi-\) a2-62- XXI. e. Page 159. 1. 0. 4. 7. 10. 1. 6. 9. 13. 2. 6. 0. 0. 0. rn^-nl na - mb' 3 46- ad bif+c x{x + 3) a; + 4 6c + ca + a6-a2-62-c2 ^ o^ + j/^ + z^-yz-zx-xy (a-b)(b~c)(c-a) ' ' '-.. y)(y-z){z-x) ' 2(6c + ca + a6-a2-62-c2) (a-6)(6-c)fc-a) ^- "• 8. 0. 9. ^(Qr + rp+pq-p>-q^-r>) {p-g){q-r)(r-p) 11. 0. 12. P(y-'^) + Qi^-x) + r{x-y) {y-z)(z-x)(x-y) XXII. a. Page 163 2. 6. 10. o + y y-x a c' nx nx - m 14. _ x + l g ad + b dx-y 7 *lrX' a;2+y2- 11, P»(«rf + 6c) 6(/(/>TO + A;»)" 16 ?!(2£+3) a? + 2 ' 4 ^-^^ h-x 8 g oc + 6 12. a?-1 16. ±, a; 4. 8. 12. - 16 2x-a x + a' 12(2a;+I) 4x2-9 ■ X y x-c (x-a){x-h) 48a=^ (x2-a2)(a;2-9a2)' a-x a + x' 30. 2{^!±i) x(x^-\) y)(y-z){z-x) {q~r){r-p) q(z-x)+r{x-y) (z-x)[x-y) x-hc h-x o. J. OC + O il. 12- V - 1 I 16. \ 17. 21. 25. 29. 33. 1. 4. 7. 9. 11. 17. 20. 1. 4. 7. 9. 12 16. a2-62 2{a + b) a-b 2a-l ' x'^-^x+l x'^-^x + l 1 a + oj* 18. 2. 22. a + x. 26. 6+^+2y 8x(y + 6) .30. 2. 34. 4. ANSWERS. 19. 23. 27. 31. 36. 421 a{yz + n) xyz+nx + mz' a-c ] +ac 8x2-1. 20. 24. 28. 32. 2x2-1 1+x 1+X2' 1-x. b a XXII. b. Page 167. 3 9 9x" ^^-^^ be ca ab 2. 6. a2 ax x2 T~"3 ■*"2" 1 1 1 - + T+-. a b c 6. 36. 2x2. a^_3a 36 6^ 26 2 "^ 2 '*'2a" a_2_62 1 6 2 ■'"3' x-x2+x^-x*; Rem, x'^. 8. 1 + 2x + 2x2 + 2x3 ; Rgm. 2x*. 10. , 6 62 68 ^ b* l + -4--Ti + -3; Rem. -=. l+x-x^-x*; Rem. x*. x-3 + --^,; Rem. ^. 12. 1 +2x + 3x2 + 4x3; Rem. 5x4-4x». X x^ "^ x-3 x-4 a2-462 a + 36' 4 (c - x) 3 (a + x)' 8xy(x2 + y2) (x2-y2)2 • 1 X2' 18. 3(a-2x)2. 19. (2x-3)(2x + 7) 62-36-2 6-6 • 21. 6 XXII. c. Page 168. 2. 6. X{X+ 1)2(1 +X + X2) 2r + 3 3(x + 6)" a" (a - x) (a + x)2 2(a"'^ + a+l) a(a + l)(a + 2)' 10. 13. x(x + a) ___ 4x(2-x) (x-l)(x3+l)* 8. 6x + a ax +6' (x + l)5 3. 3x3 + 6x2-x-8 6. 1- X + X2 1+X + X2' 11. 14. 1 a2+a6-262' 1-X8' ax8(x2 + a2) aj-y a^-y 16. 2{3-2x) 17. 1-x* 422 ALGEBRA. ^! hi 18. 21. 26. 29. 34. 38. 41. 44. 48. 62. 66. 1. X {x-2af 9x--. X ab 19 (2a?-l)(a^+l) {x + 2,){x-l) a(a^ + x^) (x-a){a + xf' 27, 20. x-i 22. 26. 30. 36. 39. a 2^2 23. 1 1 X. a + b' 1. x(x + l) X^ + 'iX+l' l+x + x"^ (l + z)(l + ic^)(l-xf 1. 46. 1. a^ + b^ + c^-bc-ca-ab (b-c)(c-a)(a-b) 2i/ + a + b. 7(x-4) 31. (-iT 12 {a*-4){a*-iy 2x{2x-i)i' X. 32. 36. 40. 24. 28. bx. 33. 1. 37. 4x^ - 5.r - 5' 1. 6* 6a + a2" 2 • 1. 3m2 2a: *^" (?;-2)(a; + l)2" 46. 1. 47. 0. (3m + 2?i)(9m2-n2)" 1 x-vy 43. 49. 1. 60. L 61. 0. 28 (a; + 4) 63. (2aM:^;)(^) 5,. . a% 9(a: + 3) 4(a;-l) 66. a; + 3. 67. l+a-a3. 68. --, Miscellaneous Examples IV. abc{})-c) ; -6. -22- 2. 6 3. Page 172, 4. 7. e 5 3' 6. (1)232; (2)-29. T. (1) -19; (2)0. 8. 1. 9. --^ 10. (1) -12; (2) 1. 11. 14. (1) 1 ; (2) 21. 16. 17. {,x-%y){x-\2y). 18. 20, Ji(OT-3?j)(m-3n). 22. (c«2 + 5c2)(rf + 3c)(rf-3c). 24. (7n + 13)(TO + 15). 26. 27- (a;2+16)(a;2 + ll). 28. SO. (9-a:y)(8 + a;y). 32. {p-Viq){p + ^q). 33. 3 10" 98a: -2y; 19^. (a -7) (a + 13). c(c + 13)(c-12). 1. 12. 8|. 13. (a: + 9) (a: +12). 16. (a6-17)(aZ< + 3). 19. 21. (?)2 + 7q-2)(^2_8^2) 23. xy{x^^y)l^x-1y). (14-a)(15 + a). 26. (19-^?) (3 + /?(?). (a2+i4)(a2_7), 39. (c + 27)(c + 27). 31. (a2 + 2a:2)(a2 + 7a;2). 2(a3+i2)(a5_ll). 34. a;2(a;-9)(a: + 7). 36. (6c + 12) (6c -7). 36. (2+ 17)(s+17). 37. (a -3c) (a -19c). 38. ys(y-7)(y+13). 89. (2 + 3a.-3)(l-a:)(l+a; + a;2). 40 (2a6-5)(a6 + .3). ar-2 [3m + 2n)(9m^-n^)' 1 43. x + y ). [^ 61. 0. ;8(a; + 4) 9{x + 3y i-a^ 68. --. e QE 172. 7. .. I '■ »• -^. 98a; -2y; 19^. (a -7) (a + 13). c(c + 13)(c-12). 7j2-8g2). {^-7y). (19-/J'?)(3 + ?J?). (c + 27)(c + 27). %^ + 7x% (z+17)(s+17). (2a6-5)(a6 + 3). 41. 44. 47. BO. 62. 64. 66. 68. 60. 62. 64. 66. 67. 69. 71. 72. 74. 76. 78. 80. 82. 84. 86. 87. 89. 91. 92. 97. 98. 101. 106. ANSWERS. 423 (3p-4)(3p-i). 42. x^{2-x)(3-x). 46. (6p-q){p-2q). 48. 6(2y-l){y-2). (2a-b-5)(iiV>-2). (9m-5n)(2m + 3ii). {a + b-c)(a-b + c). (a64-7)(a262-7a6 + 49). (a + 2x-2y)(n-2x + 2i/) (5-f«i?j)(7 + wiH). 43. (17 + c)(7-r). (2?w + 3)(3m-l). 46. (2a --5b) (2a + h). (5x + 4s) (4.1- - 5s). 49. (2x^ + 3) (4x'^ - 5). 51. (3ab + 4)(4ah--3). 63. (lx + Si/)(3x-2y). 65. (c + a-b)(c-a + b). 67. (ox- + 3y) (25x-2 -I5xy + 9y-). 69. (8b-a^)(64b'^ + 8ba'^ + a*). 61. (7n + n + l)(7n + 7i-l). 2c^(3c + (l)(c-d). 63. (a%^-l+x-y)(a'^b'^-l-x + y). (l + 2m)(l-2?n)(l-27tt + 4m2)(l+2m + 4m2). 2)3(1 + 109)(l-10g + 100g2). 66. (8l+a'^)(9 + a)(9-a). (x'^-l+y-z)(x^-l-y + z). 68. (a + 46-4c)(a-4& + 4c). (c-rf)(l + 2c-2rf)(l-2c + 2rf). 70. (p-4:(/)(p + 4q + l). 2[l+4a + 46][l-4(a + 6) + 16(a + 6)2]. (x + 3y){l^x^'-3xy f 9y^-). 73. (cx-d)(ax + b). 76. (Ux^ + y'')(lx^-y% 77. (l+TO + p)(l -TO-;>). 79. (3fe-c + 4);(36-c-4). 81. (3x-b)(x h2a). 83. (a; + ?/)(a;2-|-i/2), (7 + a)(2-a). (17 + a)(3-«). (bx-a)(ax-b). (c + l)(c2-c + l)(a; + l)(a;-l). (m - n) (m + ?i + a;) (m + n- x). ia + b)(c + a-b)(c-a + b). (x + 2)(x'^-2x + 4)(x'^+l)(x + l)(x-l). 86. (x+l)(x + 7)(2x-3). (2x + 5y)(x-3y)(2x-5y). 88. 325a363(a:2_a2)'2(a; + 2a). 2x2 - 9a; + 9_ go. 2a;=' (a;2 - 4) (a;2 - 16). H.C.F. =a + 6 + c, L.C.M. =(a + b + c)(a -b)(b - c)(c -a). a + b-c. 93. (a-6)2(a + />). 96. (a*-¥)(a + b-2c). lI.C.F.=(x-'J)(x-3), L.C.M.=(x-\)(x-2)(x-3)(x-4)(x-5)(x-7). 1 (1- a;)2 2 a; 2a; -y 99. 102. 9 a;' 110. 1. 114. 107. y-x. 111. 1. (a;2-9)(a;-3) _4 (1-X2)2' 108 100. 6.r+l 103. 1. 104. 0. (2r+l)2(2x-l) X 9" 105. ah. 109. 2(ac + bd)(ad + bc). 112. (a;2i2)(u;''| 1) 113. x+l 115. a;(l + x-a;2). 116. 3af)C a + b' X f-g 117. a + &. 118. 1. I I.V. ' y~ a + b' =26. rb pq-r* qr-p" ANSWERS. 425 11. 12. 14. 16. 18. 20. _ pn ql-pm* y~''mp-lq x=a + b, y = a-b. ab' + dV y ab' + a'h' x=m + l, y=m + l. x=a + b, y=a-b. 13. a;-£i«±^) „_<'(«-&) " • "~~2^° 16. 2a ' y- x = Za, y= -26. 17. x=a, y=0. io a b 19. :.=-^, y=-. 21. a; = ««-/,», y=:„3.^^.3^ 1. 6. 8. 11. 14. 16. 18. 20. 22. 25. 28. 31. 33. XXIV. Page 188. 40. 2. 60. 3. 55. 4. £2. 12«. Silk 9s. Calico 9d. per yard. 6. 54. 7. 42. 48,23. 9. 2I3-V past one. 10. IT^V past three. 32,^' past six. 12. 5}^' past two. 13. 378, 2ia 15 persons ; 5 shillings. 16. 8 yards at 4s. 6d. ; '16 yards at 4s. ^'' ^^- 17. 3 miles per hour. '^4- 19. 2i miles per hour. 21,-y and 54/y past seven. At 5/^-' past. 21. JO P.M.; halfway. 23. 1^ hours. 24. £200. 30 miles. 26. £36000. 27. £200. 4 and 3 gallons. 29. | and f of a pint. 30. J^ miles 111 and 126 miles. 8 12' c - 6 and a - c lbs. ,%, ^, ^ yards. 2a b •' 32. Coffee to chicory as 7 to 2. 36. 60 milea. XXV. a. Page 194. 1. 5. 9. 13, 17. 21. ±5. 3,7. 9, -4. 4, -17. 7,6. 3 1 2'~3* E.A. 2. ±4. 6. ±8. 10. 9, -8. 14. 13, -12. 18. 23, -1. 1 22 4. 3. 3, -25. 7. 3, -6. 11. 31, -11. 15. 11,-17. 19. 6, -|. 23. :l-9. 4. 1, -25. 8. 2, -7. 12. 20, - 11. 16. 8, 15. '"• 3'-5- 2 k lii ;i ! I I G Si! i '*. ij J m I t i v 1 1 -. M 426 \ ALGEBRA L* XXV. b. Page 197. 1. ^.-5. 2. 11 " 11, 3. 3. 3 7 3, g. 4. 15 2 6. I- 6. 2 -i^ 7. ^ -5 8. 2' •^• 9. 13 11 3' 3" 10. 7 2 4' 3" 11. 3 4 4' "5- 12. 8' ^• 18. 5 1 "7* "3- 14. 9 3 lO' "5- 16. 13 2 6' 3' 16. 3 ' 3, --. 17. a a 3' "5- 18. 3a a y "3- 19. 7/fc A; 3' 2' 20. 5k 2k 21. 4c 5c ■3' ~T 22. 3, -3. 23. 5 5 5, -g. 24. ^■l 25. 3, -1. 26. ^■ 27. 4 " *' 2- 28. 7,2. 29. 11, 2. 30. *'t- 31. 13 2 " 13,3. 32. ^'I3- 33. 2 39 34. 3.4 36. 12, -2. 36. 5 23 5, y. 37. 3a, -. 38. -.^- 39. ab a. . ' a -2b 1. |, -3. 6. •4, - 7, -; 1 18. ^,-1. 17. 21. 9^ 5 10' 6" 3 2 XXV. c. Page 201. 2. |, -5. 6. Z^^. 10. 1±^. 7 3 4' 2" 8 3 3' "4- 5 7 14. 18. 3. 1, 7. 1, 11. i. -2. 16. ^. -3. 19. I -14. 23. ?^. -^. 4. 8. 3±v/29 17±./89 10 12. 3, 11 6 16. -g, -4. 20. A, _5. 12' 8 24. 9a 4a 26. 56 lb 76 56 _1R, 26. 4?, -^. 27. 2a, 26 6 6 28. 2a, -8. 4. ¥.-2. 8' 8. ^3. 12. ^ -3 8' ^• 16. 3 ^ 3, --. 20. 5k 2k ~T' ~T 24. 4.1 ' 5 28. 7,2. 32. fi 40 ' 13 36. 5 23 5, y. 4. 3±^/29 „ 17±^/89 10 12. 3, - 11 16. 4 -■ 20. 5 3 12' 8' 24. 9a 4a 4' 3" 28. 2a, -8. ANSWERS. 427 • 29. ft 2a + 6 "' 3 • 30. 0. ''-2 a 31. ±2, ±1. 32. ±2, ±3. 33. 1, -2. 34. 3, -2. 36. ±4, ±1 36. ±a, ±6. 37. 2, -3. 38. ±3, ±4. 39. 3,-2,4,-3. 40. 4a, -2a, a. XXVI. a . Page 203. 1. ar=17, 11; y- = 11, 17. 2. x=37, 14; y=14, 37. 3. a;=53, 21 ; y- =21, 53. 4. x=14, -9; y=9, -14. tf. a;=27, -19 ; y= = 19, -27. 6. a;=43, -25; y = 25, -43. 7. a;=71, 13; y= = 13,71. 8. a;=33, -41; y = 41, -33. 9. a;=52, -74 ; y= = 74. -52. 10. a;=43, -51; y=-51, 43. 11. «=29, -47 J y= =47, -29. 12. a;=22, -87; y=-87, 22. 13. x=±8, ±5 ; y= = ±5, ±8. 14. a^=±13, ±1 ; y=±\, ±13. IS. a;=±4, ±7 ; y= = ±7, ±4. 16. x=13, 3; y=3, 13. 17. a;=10, 5; y= = 5, 10. 18. a; = 9, -5; y=5, -9. 19. a;=12, -6; y= = 6, -12. 20. a: = ll, -8; 2/=8, -11. 21. a; = 9, 4; y= =4, 9. 22. a;=5, 4; y=4, 5. 23. x^l, -4; y= =4, -7. 24. a:=10, 4; y=4, 10. 28. ar=12, -2; y= = 2, -12. 26. x=\ ; y=l. 27. a;=4, 3; y= = 3, 4. 28. 1 x=- ; a' 1 29. X=±l', y= = ±1. XXVI. b. Page 205. 1. x='l, 4; y=4, 7. 2. ar=8, 5; y=5, 8. 3. a:=14, 9; y=9, 14. 4. a;=7, -5; 2/ = 5, -7. 6. a;=ll, -7; y=7, -11. 6. «=13, 0; 2/=0, -13. 7. a:=±6, ±4; y=±4, ±6. 8. a7=±7, ±3; y=±3, ±7. 9. x=±9, ±5; y=±5, ±9. 10. ic=±9, ±3; 2/= ±3, ±9. 11. 6 8 8 6 ^=5' 3' y=3' 5- 12. x=±^ ±5; y=±5, ±6. 13. a;=4, 2; y=2, 4. 14. a:=7, -3; y=3, -7. 16. a;=5. 3; y=3, 5. 16. T =r 4 — O . 4i — O A - •*, - , y~£-y — -r. 17. a; = 8, -2; y=2, -8. 18. ap=5, 1; y=\, 5. 19. tt;=5, 1 ; 2/ = l, 5. 20. x-y -I- 6' 5' -i. -[ ^11 If 111 ( ! 428 1. 3. 0. 7. 9. 11. 13. 10. 16. 17. 19. 20. 21. 22. ALGEBRA. XXVI. c. Paor 208. a?=4, -p y = 3. -20. 2. x=±3', y=±2. 10 x = 12, 8; y=2, -2. 4. x=2, y ; y=5, 3. «=4, 7; y=l, 10. 6. a;=4, -3; y = l, _i 71. _. 112 3' 3 8. x=±2,±-j-^;y=±l,±^^ x=2,^; y=-7, -| 10. a;=±4, ±6; y=±2, ±4. 3 1 13 a;=±3, ±4; y= ±2, ±5. 12. x=±^,±^', y=±^,±^. a;=±2, ±1; y=±l, ±2. 14. a;=±2, ±5; y=±3, ±6. x=±1, ±>JS', y=±2, T3V3. 23 x=±3, ±36; y=±5, T^- x=5, 3; y=3, 5. 18. x=7, -6;- y=6, -7. a; = 6, -2; y = 2, -6. a;=7, 1, 4±v'28; y=l, 7, 4^x^28. 1. 6. 10. 13. 17. 20. 24. 27. 29. 1. 3. 5. x = 4, 3, 6, 2; y = ^, 2, 1, 3. x = 2, I 4, I ; y = 2, 6, 1, 12. XXVII. Page 212. 13. 2. 45, 9. 3. 7, 8. 4. 3. 6. 15, 12. 9. 7. 7 hours. 8. 7, 5. 9. 90 yards, 160 yards. 55 feet, 30 feet. 11. 36', 60'. 12. 6. 5 shillings. 14. 12. 16. Ninepence. 16. 3 feet. 4 inches. 18. 121 square feet. 19. Fourpence. 40, 12 ; 30, 16 yards. 21. 56. 22. 50. 23. 25. 6§ miles. 28. 75. 26. 20, 30 miles an hour. 40 and 45 miles an hour. 28. 10 gallons. .^, 16; ^, 14. 30. Distance, 12 miles ; rate, 8 miles an hour. XXVIII. a. Page 216. (!K2+4a; + 16)(x2-4x + 16). 2. {9a^ + 3ab + b'^)(da"-^ab + h% (x2 + Zxy+ y^) {x^ - 3xy + y^). 4. (m^ + imn - v?) {m"^ - 4mn - n% {x^ + 2xy - 2/2) (x'^-2xy- y% ANSWERS. 429 y=±2. y = 5, 3. ; y=\, -|. 6; y=±2, ±4. 1 _ ,1 ,3 2' y~^2^ 2' 5; y=±3, ±6. y=6, -7. 6. 15, 12. yards, 160 yards. 12. 6. !. 16. 3 feet. 19. Fourpence. 23. 25. 30 miles an hour, gallons, lite, 8 miles an hour. 6. 7. 8. 10. 11. 13. 16. 17. 19. 21. 23. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 46. 46. (2x2 + 9.ry - Sy^) {Ix^ - Oxy - Sy-). (2m2 + 6mu + Sm^) {2m^ - em>i + Sti^). (3x'^ + xy + 2y^)(3x'>-xy + 2y-'). 9. (x-' + Sxy -5y'){x--Sxy-5y'^). (4a2 -Qab + Ifi) (4a2 + 6a6 + h"^). (^lo)(t^2«..10o). 12. 14. 16. (mn .\(7n?n- mil ,\ Vs x)\^'^2'^ x'')' (m - 5n) (2n + 3m) (2m - 3w). (xH'^ + y'^){xy + z){xy-z). (mn-p){pm-n). (2x + Sy)(aUxy). {ax + (a+l)}{(a-l)x+a}. (y-Sx)[x + y){x-y). 18. {ax + b){bx + a). 20. (a^ + bx)(a + x). 22. (3a6-2a;)(2aa;-36). 24. {2x-3y)(a^ + xy). 26. (a:-a){3a;-a-2/>). {ax + 2(b-c)y}{2ax-{3b-Ac)y}. {{a-l)x + a}{{a-2)x + {a-l)}. {{a + l)x-{b-l)y}{ax + by). (6 + c-l)(62 + c2+l-6c + c + 6). (o + 2c + l)(a2 + 4c2 + l-2ac-a-2c). (a + b + 2c){a^ + b^ + 'ic^- - ab-2bc -2ca). (a-Sb + c)(a^ + 962 + (.2 + 3„6 + 35^ - ca). (a-6-c)(a' + 62 + c2 + a6-6c + ca). (2a + 3b + c^ (4a2 + 96^ + c2 _ 6a& - 36c - 2ca). (x*-9x' + 81)(x^ + 3x + 9)(x^-Sx + 9). {a* - 4a%'^ - b*) (a^ + b^){a + b)(a-b}. {a + b + c-d){a + b-c + d){c + d + a-b)(c + d-a + b). (a* + ^8) (x* + y*) (a;2 + y2) (^ + y){x-j). (a ' x3y3 + y»){;xfi - ary + y«)(a:2 + xy + y^)(x'^-xy + t/2)(a: + y){x - y). r . r Ij (- - 1 J (a - 2a-) (a2 + 2aa; + 4a;2). i^a + 2,2V /.^ _ ^2yi ^ yi\ (^ _ oy) i^^ u- Ory -f 4jr-2). (a;2 + 4) (x^ - 4x-2 + 16) (ar + 1 ) (x2 - a: + 1 ). (^5)(^5)(« + ^)(a^-a6 + 62). i i in i * S ' 11 i >f;i I . it, f i 1 ? ! I 430 47. 48. 49. 60. ALGEBRA. 1. 8. 6. 8. 11. 14. 16. 18. 31. 28. 20. 28. 31. 33. 36. 37. 88. 46. 49. {x+\){x^-x+\)(x^ + 4)(x + 2){x-2). (x-l)(x'^ + x + l) (ix^ + 9) (2x- + 3) (2x - 3). XXVIII. b. Page 220. 4x2 _ 49^2 + 42y2 - 9z2. 2. Ox* + 263- V + 49y*. 25a;<-115xV + 81y*. 4. 49ar*-64a;V + 48.'7/3-9y». 3fi-%f^. 6. (x-\-y)* + ^x + yY + m. 7. 16x''*(l -4x2). lU. 48a«(a*-l). 9. a:2 3a2 . 7.2 _ ^ .T« 64. 12. 64x^(9x2-1). 13. ai2-3a8x^ + 3aV-a;»2, a:«-14x* + 49x2-36. . 19. x8 - 64. 20. 22. 7x+y + =. 24. 16. 17. 36. l + x^ + x". «« 14x^ + 49x2 2- 552)0^2 _ 5^2). 60. 27. {7x-3)(x x*-4x'^yz + 7y'h^. (x+l)(x-3). {x-a){x-b). (3a2 + &«)(a2 + 362). 16a63. a-b. XXIX. a. Page 223. 1- x + c. 2. x2-ax + 6. 3. x2 + 26x-ax-2ai*. 4. x2-{;3 + g)x + 25'(;?-g'). 6. x^-(m + n)x + m(m-n). 6. ax + a + L 7. x2 + 6x + a2. 8, 2f.x-(^m-4p'^». 9. (a + 2)x + (a + l)y. 10. x + b. 11. (x + l)6 + 3(x + l)* + 3(x+l)2 + l. 12. (m + l)62a;2+(n + l)(m+l)a6x + (?i + l)a2 y + 48.'7/»-9y\ 16x^(1-4x2). x8 + a*x*+a'. a*-18a"-b'' + 8\b' 5x + 7!/-Gz. 5(a;-13). -1). 30, a-b. 3). b). '+36'). x'^+2bx-cix-2ah. v+m(m~n). 2lx- — f 3w. - 4«.\ n.. ANSWERS. 13. (m-\)x-\- m. 14. mx - n. 16. x-(y)-l)}{(?> + l)a; + ^;}{(/) + 2)xH /> + !}. 21. {{(' -Z)x + a + \}{(a-2)x - a\{ax - {a + \)\, 431 IB. a/) - hq. 18. a; + 2a&. ZXIX. b. Page 227. 1. x-1. 2. M 1 m 3. a + 2x. 4. \-Vx-x\ 6. l + 2x- - x\ 6 1+x. 7. a; -2a. 8. a- -3x. 9. x-y. 10. X2+(P-1)X-1. 11. '-I- X2 .,-' 8+I6- ^^- X2 l-x-g- X3 "2' 13. ^2 16^64 14. X 5x2 5_jj 2 '8 16" 16. X X"^ 01^ " 2a 8aa IGa"' 16. a2 a* '*'"^2x 8xa^ 17. 2 3ar» 9x4 27x« " 2a^ 8a^ 16ai»" 18. „ , ., 2.c2 , 4x3 3" + 2^-3a+9a2- 19. a^ o* ^ :rr2~9x5* 20. 2 X X2 "^■^12 288' 'Jl. 22. l-2x + 3x2. 23. 3x2- x-1. 24. XXIX. C. Page 230. 19. 0. 20. 0. 26. 0. XXIX. d. Page 232. 1. 0. 6. 1. 9. d. 3. 1. a + 6 + c. 6. 10. 12, (a + & + c)2. 13. o6c* (x-a){x-b){x-c)' a-\-b-{-c 7. -1-, 8, 1. abc 11. x- (x+o)(x + ?>)(x + c)' I'liHl 14. s. IB. a + 6 + c. \^ 16, 6c + ca + a&. 17. a^c. 18. (6+c)(c + a)(a+?>). 1 1 ■ 1^ 1: ; '■ 3 (! 1 m ' ^ ■ #! 432 1. 5. 9. ±3a. ALGEBRA. XXIX. e. Page 238. 2. 10. 6. 3. i^. a a2 7. a=c, 6=-x + 2, 10. 12. 16. 17. 19. 21. 23. 24. 25. 26. 27. 30. c=a(6-aa)2, c?=(6-a2)3, whence c'^=a^,- 34. 1 a2 10. 15. 20. 25. 30. 35. a* 2yK 21_ 4. V 16g' 2. 8. 6. 11. &3=27c2 13. 32. ;-4). 7 + 7). :-ll). 2x-3). 29. 6 or |. 36. -37a3. 6. 4* 10. a* 2" 16. 2yl 20. ^ar». 26. 24/63 30. 21 36. ^lo>\ 36. ^a'^. 41. 8. 46. 625- 37. 2«/a^. 38. 42. — . 32 47. 9. ANSWERS. 1 43. 25. 48 3 433 39. 4-. SiJa 44 1 44. -. 40. ^a". «■ 2ra- 49. 27 2- - 8- XXX. b. Page 247. 60. 2187 128' 1. a«6». 2. V y 4. 1 6. 16ac*. 1 11. 16. 21. 26. 1 1 a3 7. ^. 8. a;'. 3ax 13. 1 2 18. a62. 22. 6^. 23. a;^. 9, 14. a^. 19, a + 6 24, '■ 9a2a^* 10. a;"-!. 16. x''+\ 20. a + 6 (a -6)^ 26. c^. 27. -^. 28. a6(?>*'-a«)'^. 29. «"<»-'' + a. a"" 30. a^'f-i' + ic''-^ 34. a;'y «^. 31. a*"(P-9\ 32. a,-*. 36. 2"'. 1 33. t^ 36. -. 37. 4. 38. I. 4 XXX. c. Page 250. 1. 12x* -20a;* + 41 - 15a;" '^ + 24a;"*. 2. 9a^ - 9a* - 25 + 23a~ * + 6a"*. 3. 2c2'-9C-34 + 31c-'-6c-2'. 6. 7a;*-2a;^ + l. I. ott -+ /ci " i I). 9. 7a2' + 3a'-4. 11. 3.T*-2 + a;"*. 4. 8x^ + Ux^ - Sxr" - 9x-^. 6. 3a^-3a"^+2a-J. 8. 5i* + 4&* + 36'"^ + 26"*. 10. c2"-l+c-2». 12. 5tt*-3a^ + 4. 434 ALGEBRA. 13. 2x^-i + 3x~'^. 16. a2 + 2a - 16a-2 - 32a-^ 17. 4a^-8a*-5 + 10a~* + 3a~i 19. l-2a-2ai 21. 3a;-2-3a;-Jy* + y. 23. 9x^1/-^ + 2x^i/~ * - 9. 14. a^-3a'-2. 16. l-a;*-2a;* + 2a;^. 18. x^-2x^ + 4x~^-8x~K 20. 2a;*-3a;~^^-a;"T\ 22. 2x^-3y*-|-4a:~*y^. 24. la;-i + l-3M^. 4 »! I ,' ^ XXX. d. Page 252. 1. a; -4a;* -21. 2. IGx^ - 8 - 15a;-2. 3. 49a;2-81y-2. 6. a2^-4 + 4a-2^. 6. a'^' + 2a'^^'' + a'. 4. a;'"y-'»-a;-'»y»«. a 1 7. «»-aj ' + ja;-2«. 8. 20ar^y'»+13-15x-2»y-2*. 9- gCt^-o + a"* 10. 9a;2<' + 15jr'*-15y-24_25a;- 2 2 11 12. x«+x~'^ + x^-2 + 2x^^»-2x^~a. 11. a*«-ii»=-^ + a-^+a-2«. 13. 2a + 2(a2-62)* 14. a + 6 + (a-6)-i-2(a + 6)V-6ri •a _ Q/,7 4 „ , o.,-J 16. x^-3a^. 18. a;2*-2a:«+4. 21. rt^'-as. 23. x^+x + x^ + x^ + 1. 26. iB2 4.2a:* + a._i6. 17. a" + 4. 20. l + 2a-i+4a-2 16. a; + 3a;^ + 9. 19. c*+c~^. 22. a;-3-cc-'> + a:-i-l. 24. x*«-2a:3n + 4^2n_8ar" + 16. 26. 4a;^ + 16a;*+16-9a;~*. 27. 4-a:* + 4a; + a:2. 28. a2«-49-42a-^-9a-'^. 29. o*(a*-26*). SO. 1. 31. 2 x^-2 x^+2 82. 4. . i I 1 12 /~4 5. '^( a' XXXI. a. Page 255. 2. 6. IS'a' 1 ia/«.4 Jl/^ » .^ 4. '>». 8. ^j/a:'"'. ^t - 3m^ Sy\ 49a;2-81y-2. -2A 25a;-2«. 2 + 2a;^"^«-2a;^"«. -6)V-6r*. a'' + 4. l + 2a-i + 4a-2 + 4«2»-8ar" + 16. 29, a*(a^-26*). 82. a i 4. '>". 8. ^a;»'». 9. ;5;/a «/«9 10. J_. ANSWERS. 435 n. yx'%^. 12. ^a». 13. a,?y2« 14. '\/aV^ 16. i^a", is/a'". 16. i^a», w/a». 17. ?J/a;», ?y.r^«, ?y:r«. 18. ^ar«, ?2/.ei«. 19. '^'^^ =^'^3. 20. ?2/^3^, ^^V. 21. 4/125, 4/121, 4/13. 22. ^64, 8/81, 4/6. 23. 4/2, 4/2, 4^2. XXXL b. Page 256. 1. 12^2. 2. 6. .15^6. 6. 9. 54/5. 10. 13. -Sxy^/Ix. 14. 17. 2{x-y)s[xy. 20. 4/864. 21. 24. ^/9^. 26. 7v/3. 24^5. -94/3. 4/760. VI 3. 44/4. 7. 35^5. 11. Qa,Ja. 16. ccy'^j^x^. 18. ^242. 14 26. \/8^. 22. 28. ^a%^ 29. y/at 30 32, X 36. -15^3. 40. 174/2. 43. 6V7-15V6. 34. x/7. 38. 11 4'3. 33. 14^5. 37. 74/7. 41. 20^/3-13^2. 44. ^^W3 XXXI. c. Page 259. 1. 14^/6. 2. 12^3. 3. lOs/Sa. e. 288^2. 3/ 6. 4'a:2-4. 7. 3^3. 9. VJ3. 13. ofcVoft. 10. 144/9. 14. 33 10' 4. 64/2. 8. 74/3. 12. Sab'^JSab. 16. (a + &);^a. 19. ;^980. 23. v/56. 27. 4^. 31. \/52Tp. 36. -12^11, 39. 0. 42. 3^/6. 4. 30^3. 8. ^. 11. 2404/4. 12. 16. 1^/2. 16. 10^ v/6. 2^/2 • a i h I \ ? •i 41 «*: ■ § r i r IS 1 i I'l \ i ' nil 1,1 i ! t i >■ i 436 17. a- a; ALGEBRA. 18. 9-89947. 19. 1118035. 20. 3-77964. 21. 19-69592. 26. -28867. 29. -40316. 22. 26-83284. 26. -04472. 23. 58-78776. 27. -2566. 24. -81649. 28. 1-57485. XXXI. d. Page 260. 1. 6a; - 10 V^. 4. x + y-^/x + y. 7. 6 + VlO. 10. x + a-s/x^-a^. 12. l + Sa-4\/a + 4a2. 14. a + x + 2-3's/a + x. 17. 4a;-2\/4«2^a2. 19. 2m + 2\fm^-n\ 21. 63-18a;\^4-4»2. 2. 2x-2s/ix. 8. as/h + h^a. 6. 30+12V6. _ 6. 6^/21-46. 8. 6a -6a; + 5\/aar. 9. ic-l+V^^T^, 11. 5a + fl?- 4 Va^ + aa;. 13. 2a-2sJ^~^\ !«• 2^/6. 16. 16 + 6^/10. 18. 2a;2 + 2\/a?4-4y4. 20. 13a2 + 56-> - 12\/^*364; 22. 8x2-.2vAl6ar4-l. 1. 113. 6. a -46. 9. 2a;. 12 3^7-2^/3 3 16. ^. y 19. 5+V6- XXXI. e. Page 262. 2. -166. 3. 172. 6. 9c2-4a?. 7, x. 10. 25(a;2-3y2)_49„2^ 16, ?>!^. 17. >^"^ 5 ' a-x 4. -6. 8. 2^9 - (7. 11 "-3x/7 2 14. 2+V6. 18. 4+;v'l5. 22. 3V2-2V3. i-v'r^ 20. 8-v/42. 23. x-slx^~y\ 26. 27. 21 V7-v/2 5 * 24. sllc^+ai-a, 7a + b + 8»Ja^^^ x'^ 18 + a;2- 6x^9+^2 X' 29. 2-^3 =-26795. ?^. V5-v'3= -50402. 83. ^^^=1 11803. 3a + 56 28. V3. 30. 11+5^5 = 2218035. 32. ^/o + 2 =4-23607. 84. ?v|z5= .09807. 20. 3-77964. 24. -81649. 28. 1-57485. ciijb + b^a. 6^/21-46. J. J6. 16 + 6^/10. 4y*. 4. -6. 8. 2p-q. 1 ll-3v/7 4. 2+^6. 8. 4+^15. 1 v/7-v/2 ■ ~"5 •18035. 07. 07. 1. ^5-sJ2. 4. V3+V2. 7. 4^/2-3. 10. 1^5 + 1. 13. 4/3(^2+1). 16. v/2+1. . 19- ^(n^'^ + D- 22. ^/2-l. 26. 4+^3. 28. 2V2 + x/3. 1. 14. •I 11. 144. 16. 12. 1 21. 26. ANSWERS. XXXI. f. Page 26G. 2. V10+V3. 6. 3^7 + 2^3. 2v/5 + 3va 2-1^3. 4/2(v'3-l). v/5 + 1. 437 8. 11, 14 17, 20. v/3-x/2. 23. ^/3 + l. 26. V5 + V3. 29. 2^2-^/7. 3. 6. 9. 12. 16. 18. 21. 24. 27. 30. v/7-1. v/10-2V2. 2V11-V3. ^5(^/2+1). 1 (v/5 + 1). 4^2(^5 + ^3). v/5-1. v/7-v/2. V11 + 3V2. XXXI. g. Page 268. 2. 33. 7 IZ 12. 2. 17. 1. 22. 1. 3. 20. 8. 9. 13 121 25" 18. 9. 51 0,a-h. 27. 10. 23. 2. 28. i. 4. 44. 9. 7. 25 19. 8. 24. (/>-rt)2. 29. 2. 1. 49. 6. 64. 11. 4. 16^ 3. 21. 6. 2. 4. 7 ^ ^' 9' 12. L XXXI. h. Page 269, 4 1 8. 3. 49. 1 121 If- 17. 6. 22. 361. 13. 4. 18. 25. 'I 14= 19. 50. 1 4* 6. 10. 13. 56 16. 5. 20. 12. 26. 30. 2a- 6' ±1. 6. 16. 10. 9. 15= XL 20. |. 5 1. 6. 438 ALGEBRA. XXXII. a. Page 274. 1. 6:1, 2. 1:2. 3. 1:5. 4. 9:32. 6. 2a; : 3y. 6. 3& : 4a. 7. 6:1. 8. 1 6" 9. 4:1. 10. 17 : 7. 11. 3 : 4. 12. 5:4. 16. 21, 28. IT. 1), XXXII I. 18. 27. Page 280. 1. be. 2 *^^ a 3. 5yl 4. 6. 6. 4x. 6. 12a:y2. 7. x'. 8. ah. 9. 4a:2 10. ea^x. 11. 9a?)2. 19. 8or|. 5 or 0. aa -=17. y= = 1L 21. 2 or 0. 22. 16. 14. 19. 22. 54. 1 5' 25a" =272/2 y=Qx-3x^+afl. 20 irHes per hour. XXXII c. Page 285. S. ^^7. 3. 35. 4. 21. 8. 18. 6. 10. 12. 3a = 56. 13. 5x=7y. 16. a'^=b\ 16. ±6. 17. 20. 20. 346| square feet. 23. 9 : 4. 26. £1960. 24. l^feet. 26. 4 feet. XXXm. a. Page 289. 161, 245. 2. 59, -37. 3. 34, 89i. 4. 16, 9. 574|, 93^. 6. 98, 243-6. 7. 43. ' 8. -49. -40^. 10. 7-2. 11. 9-7. 12. 25a;. a+51d. 14. 80a -796. 16. 964, 9780. 16, 32, 252. -387, -18900. 18, -9f, -99f. 19. -41», -361. 544, 4864. 21. 779. 22. -483. 23. 980|. -6569^. 26. 493. 26. 140. 27. p^ " aa(4-o). 29. "^^^.""^ 30. pn(n-4). ^^ ^n 3 2 -• / -i-J -r- •--? •'• 25,-3. 33. 16,-1. 34. 24, 2*. 36. 14,-1. 20, 4. 37. 7, 2rt. 38. 20, -2x. 4. 9: 32 8. 1 6* 12. 5: 4. 4. 6. 8. ab. 19. 8or|. 5 or 0. 22. S. 10. 56. 13. 5x=7y. 17. 20. set. 26. 4 feet. 4. 16, 9. 8. -49. 12. 25a?. 16. 3-2, 25-2. L9. ~411, -361. 23. 980|. 27. p\ 31= 30. 3. 35. 14, -1. ANSWERS. XXXni. b. Page 293. 439 1. 4, 11, 18, 2. 13, 10, 7, 4- 1, -|. -2, 3. 3, 1, -1, ... 8. 4, 5i 7 6. -11, 4, 19, 7. 43. 8. -95. 9. -6|. 10. 68,65, 26. 11. 91f, 90^, 70^^. 12. -6^, -6y\ -2t\-. 13. 6-4, 5-6, -5-6. 14. 8^, 8^, ...... 2h 16. 14 or 15. 16. 8 or 25. 17. 9 or 86. 18. 13 or 20. 19. 7 or 8. 20. 11 or 24. 21, 12, 13, 14. 22. 1, 4, 7. 23. 7, 11, 15, 19, 23. 24. 2, 5, 8, 11, 14. 25. 131. 1. 48, 384. 4. -J- -J 27' 2187 7. 512. 10. -2!^. 13. 162, 54, 18. "• 27' 9' 19. 1, 127|. 1458 i*"** 1001 26. 28. 1 pioos 1458' 1281 2560" 4369 XXXrV. a. Page 297. 2 1 -L * 2' 128" 6. 128, 1. 8. -4374. 11. x^-\ 14. l, % 8, 32. 17. 384, 765. 23. 2|^. 6. 1, 625. 9 243 16" 16. -28, 14, . 18. -1458, -1092. 7 8* 26. 1365 2048" 8192 31. 40(3 + ^/3) 29. i(3^-l). 32. V"(«°-l) . a-1 34. 364(^6 + ^/2). 21. 30g, 4&g-. tl 24. 1 601 ^'4 ^^H 27. k58 II 30. |(i-2n 585^/2-292 2 • 11 33. II Ii > H r 440 t AmEBRA. XXXIV. b. Page 300. • 1. 27. 2. 24. 3- 1- 4. 1. 6. 1. »S- 7 27 •• A- '• i "• w "• 1- "•1 \ "• 2T 14 5 5 5 2187 8' 4' 2'-- "• 256' 729 243 128' 64' ■"• 25' q 1 -• 1. ... 1 •"• ^ -|'^ "• 75,60,48. ( . Ii ».' 20. 4' 4' 12' 23. 9(3^6 + 2^ 2) 21 y^y^zl) + b7i{n + l). 22. 142±?V2 46 24. 27t2(2n+l) 8V S*-"-/ XXXV. Page 304. 1. f. 14' 3. -4. w 5. -i. -1.1 > *} « 6. 4. 2, 1^, .... 7. 1 -27' 1 26' -^ »• ^|. 1 <' 1 10. 1R 1. n. 2 a + b' tB 1 12. 1 1 1 1 X 13. — -■ ■■» a; 14. 6|. 7i «- »' ' 1024* "• 'X' 21- f{{/' + 3)o-(^-3)a:}. 22. 1 *« 23. -18. 31. «'"-6"'^" ■« ^ - 1 •' 62n-2(„2-_ jap 1. ^ aM Miscellaneous Examples V. Page 307. ; 1- 7. 4V2. 8. 5. 10. 52, 78, 91 yards. 15, 12. (1) a~» + a;-2*. (2) (a + 6)-. 13. - "■^^ 21. 2. 25. 1. 28. -2,0,2,4,6.29. 18[l-(|)"l '' "• (^) S-.<2) I 24. 1, 4, 7. .., 7a ' ' a2 ; '"' 8 23. -5,-2,1,4,7,10,13. 26. (1) 275; (2) -1705. 30. 1. 31. a + &. 2315 o« 3)i2 63 69 75 81 87 ooik ^^- ^- •2'2-Y'T'^- ^- <1> 9(19a + 64a;). (2)2315. 81 18. (1) .9=L:i3^; ,^c.-32«-l. (2) 37. 10, 39. 1 and 9. 40. l + l+^H- (2) s= -2»; f'=l-4w. 41. 8 and 2. 64 65' 12. 25 66* J7 6' 729 128' 243 6C 1, 48. 91?! 140 + 9 V2 8 W"} 5- -~, -1,1,.... I. 2^|. 9. 1«. 14. 6|, 7|. 20. 17|. a2n _ ja„ 307. 52, 78, 91 yards. '" „- ,'^> I- , 7, 10, 13. ) -1705. 1. 81. a + b. 84a;). (2) ^. -2n; ^=l-4w. } and 2. ANSWERS. XXXVI. Page 317. 441 1. Rational. 2. Rational. 3. Equal, but opposite in sign. 4. Imaginary. 6. Imaginary. 6. Equal, but opposite in sign. 7. ar»-2a;-15 = 0. 8. a;^^ 20a; + 99 = 0. 9. x^-2ax + a^-b'^=0. 10. 12a;2 - 28a; + 15 = 0. 11. 15x2 + 2oa;-8a2=0. 12. 8x«-7x=0. 13. 16. Sum -, difference -^' , sum of squares ~ 17. a;^ - 6a; + 4 = 0. " o 9 18. ar»+4a;+I=0. 19. 30a;2 + (6a-56)a;-o6 = 0. 20. 4a;2-16a; + 9 = 0. 21. (oa-62)aa-2(a2+62)a; + a2-?>2=0. 22. 4abx''-2(a^ + b'i)x + ab = 0. 1 2* 23. q^-2pr 24. 27. 2?f(3Pi:z£'). 28. 9(3^nz2l) g'-4pr 2g _qr ^^ g* -ip r q^ + 2ph^ P' P* P" 30. 62a;2-(a3-3a6)a; + & = 0. 32. 8«2-20a3a;-a«=0. ph" 29. P=p(p^~Sq),Q = (i^. 31. 262=:9ar. 36. 2/>a;2 - ( />« + 4^) a; + 2/jg = 0. XXXVII. a. Page 323. 1. 120, 5040, 56, 300. 2. (1) 2:y20. (2) 5040. 3. 8. 4. 126. 6. 6. 6. 36. 7. 7 or 8. 8. 2100. 9. 455, 816; (r=15). 10. 242880. 11. 1596000. 12. 504000. XXXVII. b. Page 328. 1. (1) 9979200. (2) 151200. (3) ?Mfi.S20. 2. 420, 360. 3. 18. 4. 1023. 6. m». f. 168168. 7. 34650. 8. 120, 144. 9. 1296. 10. 180. 11. 11520. 13. 78. 14. (n-2)(n-3) [>t-2. XXXVIII. a. Page 333. 1. a;* + 8ar» + 24arJ + 32a;+16. 2. a^+15x*+ 90a;3 + 270a^ + 405a; + 243. 3. a' + 7a«a; + 21o'x2 + 35aV + 35a3ar» + 21aV + 7aa;'' + x'. 4. a<^- 5a*x + lOa^x^ - lOa^;"^ i 5t>-.r* - a^. 8. 1 - lOy + 40y2 - 8(hf + 80y ^ - Z2t/>. 6. 16a:* + 16ar'y + 6xY + xi/ + ^. E.A 2F ill y. : 1 Tjin r' 'I' ■■■K , * ■: i ■ i ^^B|' » 442 7. 8. 9. 10. 13. 16. 19. 22. ALGEBRA. 64 - 96x + 60a^ - 20a:8 + ^ - -1*+^. y 21a8 I89a» 945a* 2SS5a^ 5103a« 5103a 2187 a9.r» + da'a^ + 36a Vy* + 84a33V + I2&a3?y* 126ar'« 84ar3y8 36a;Y 9a^8 y9 + — -— + --3- +— r5-+-;,.7- +,;«• a" 220««. 5440x8. -20. 11. -448j/». 21 ( a*" 2300622 14. 17. X 16 ic*^ 36x2 + 18. 1001 ~ 256 ■a'. 20. 32-40x2+100*. 23. 7920. a" a' a* 12. 21875ffl»6*. ,- 5103x^a» IB. _j^. 18. -24310x88. 21. 24. 1. 4. 7. 8. 12. 14. IB. 16. U. 20. 22. 24. XXXVm. b. Pagk 338. TheS"". 2. The9'\ 3. The 7"'. B. The 11*\ 6. r= 7 ; excluding the value r= 4, 'vhich makes the terms tlie same. 1 2m n = 40. 9. 3r=,3w + 2. 10. 11520. 1025024 "~81— The 2"^ and JJ-^. The O"" and 7"'. Mil 65536. 13. 262144 n(,n-\) (ft-r + 2) n_^+i -,, ,,vr-i |r-l ^ ' n(n-l)......(n-r + 2) ^,_i (2a;\M-r + 1. [r- 1 a« + 6a» + 15a* + 20a8 + 15a2 + 6a + ■ j;«- 12x» + 54.'r'*- 112x3 + 108x2- 4HX + 8. 11. |2n i — I — '*' • 2 3 , 8 - '''5*'"25^ ^12.^^ ••" l+.3x'2 + 6x*+10x6+.... 1 3 3 .. 5 . 8-r6^+l6^"'32^"'-- '**'• 1 3 I + 7X 4 '28 X'"'- .... 19. l-6x+27x2- 108x8+.... 21. 1-12X + 90x2-540x3+.... 3 2 ;'' 3 1 "" it? "r ^ •*'*'* K*^ "!■•••• 2 l/3x 15x2 35x3 X 315 l(,* + ~:+2a2 2a3 +••■;• ^''- 128^' a^^ 230945 g 65536 ^ ' ANSWERS. >3a 2187 y 2187f)a%'». 5103ar«a» ~16 -24310ar». 11520. 1025024 81 ■ The 2°-» and 3^^. The 6"* and T\ he terms the same, 271 11. |7l[» X". ra;2-i08a:»+.... K).r2- 540x3+.... :: •» -J 4- 230945 65536 x9 443 26. 28. 29. SO. 2^' 256 a;W. 27. -4ar', (-!)'•(/• + !)«;»•. 21 ^ 1.3.6...{2r-,3) , 1024 :«-|, ^a;-- («-l)(2n-l). .J(r-nK-]l a'-H-i^' 1^-^^ -V. 4-96967. 31. 4-98998. 32. 1 987,34. XXXIX P UE .347. ^^ IK 1 5 9 33. -100504. 3 S ^- i' -^' -i. I 8. 6. 7. 8. 9. 12. 16. IS. 5, I, 125000. 1, -1. 2-89, -01. 4. ^loga. -^^^SJ/' 6. 3, 2, 0, -1, -1. -4 1 1-5705780. 5-5705780. 8-5705780. 7-623, -000007623, 76230000. 2-8627278. lo. 39242793. n. T-4082400. •7658178. 13. -8644286. 14. T-4841414! Iog7 + 41og3 = 2 -7535832. 17. 6 log ? ^ Jog 3 -11= 3. 39221 60. g log 2 + g log 3 + 1 log 7 = -4797536. 19. 1 (7 log 2 - 3 log 3 - log 7) = 1 -9661496. 20. 24. 26. 28. 1. 4. 7. 10. 12. 13. Sixty-nine. •500977. 22. 176. 23. -398742. 26. 2 - log 2 - log 3 - log 7 = -3767507. l±21og3-log2^3.^g 27. J:^M?=,.206. log 3 l-log2 :4-29. 29. 2> g2-41og 3 3 -2 log 2' 21og2-41os., 4bg2-iog3 " " ^"^ ^'^'■y ^^^^^y- XL. Page 353. 30213. 36641«. 123807; I a22. 30523. 3. 244332343. 6. 14.320241. 9. 32099. 2. 25566556. 6. 3245. 8. 3e7580. a. 10000011. 29+2« - 2' + 2« + 2s + 23+l ; 6e* + 9e^+eHie + f. 1736; lt5; 328108. U. 667. 16. 203-71 If ! 4 4 y=0. 48, 27. a;2-l. 31. 1. 34, 884. 37. 2a;3-.3a:+7. 41. 8a?). 44. 4a; -6. 2(a;-7)(2a;-7) (a; -2) (a; -3) (a; -4) (a; -5)' (2a; + 3) (4a; + 5) (3a; -5) (a: + 2) (a,- -2). 60. 3s, 9rf. -5606X + 5589. 52. 4a2 -962+ 246c -IGc^. 6(a; + l)(a;-3)(a;-4). 66. 2(a^+b^)(x'^+y'^). x=3, y=2, z=l. 67. 0. 68. a;=-5. -a W^^— T-T,. «0- 24 days. 61. |ar«-5a;2 + j + 9, y'(x-y)^(x^ + xy + y^) •'2 4 E 371. 11" 41 :r=15, y=16. ]2^ 20^+3^' x^-y\ 16. 11. •2 + 2).. a? a-x -35a; + 18y+172, 1. 884. 2«3-3a;+7. 8a?). 4r-5. -7)(2a;-7) -3) (a: -4) (a; -5)' 3s, 9rf. 2(a2 + 62)(a;2 + y2). «= -6. |ar'-5x2 + | + 9, ANSWERS. 447 62. 65. 67. 7a 73. 75. 77. 80. 82. 84. 86. 89. 92. 94. 95. 97. 99. 101. 103. 105. 107. 110. 112. 94. ic = 2i, 63. 6»» T"«^ a;=24,y = 9,s = 5. 68. — „ x + 5y 3. 6. 1 66. (1) (a;2+l)(.c + 5). (2) (x-19i/)(;« + 17y). 2j; _ qr _ VI' 69. (2a-36 + 2c)2 71. x'^ + y'^ + z'^. 72. -2a6. 74. (1) 3x(x + 9)(x-1). (2) (a + & + l)(a + 6). 76. 2^2 a;=8. 640. 7a i^'ti^-2 2.«2 + 2a; + r 79. - 8ar»-y3' 2^ (l-a:'?' 81. i-4^-S^'+y^- 3 1 2' ""'" 4 2a262(a„26)2{2a + 6)2. 5ar»-4e-8 83. '7*"-4ar*+ 77 , 43 ., 33 3 87. 2a^-\ 3a + -. a -.c'--i-a;2_ "_^. + 27. 4 4 85. a; =5. 88. x=2a&, y=3a6. 3x2 + 4x + 24" 3(2a;-y)(5x+4y). 90. 25 shillings, 30 half-crowns. 91. 0. :>^-x^ + lx~. Rem. - -^. 93. 60(j9«-ge). ( 1 ) (a - 2&'>) (a2 + 2a¥ + 46i»). (2) (x^ + x-l){x^-x + l ). a; = a-26. a; = l, y= -1, 2a - 3?> + r^ 96. = 0. 2(y + 5) 98. {y+i)(y-5) 2a -3c (y-i)(y-6)' 100. Twelve minutes past four. (1)5^. (2) -1.102. (1) -^^^^. (2) l+x-xr>. " a' + ax + x^ a^ - 3a H „ : a - -. 3 1 1 a a'' ' a £20. 106. (2a -3?)) (a + 6). 104. l-5a; + 16a;2-45r'. (1) 2 or i (2) 2 or g. a; 109. 7«2-^ + 3, 3_ 2a2 HI. ,3a:2+7a;-l2 (.c2-9)(a;2-l6)' (a;-3)(a;-4j' H.C.F. x-5b. L.C.M. 6 (x + 3a) (x - 3a) (« - 66). 1 ! h t i 4 Uii 1.^ « I i « I -■* i\ in" 1\ i ! , 1 ; ] ; i I ■I' i ' ! ■ I 448 113. IIG. 118. 119. lai. 124. 12S. 127. 128. ALGEBRA. ab (1) g or 2d. (2) 9 or 3. 114, 177. 115. ISmUes. (3a; + 2y )2 + (3a; + 2y) (2x + Sy) + (2x + 3y)2 = 19x2 + 37^^ + jg^a (1) {x + y){x + y)(x + y). (2) mn(m~n). (1) ^=3, n y=-i, -3/ a?-l (2) (1) 1. (2) x' 122. a;=7, -6 y=2, -2 a~b. }• 120. a2^&3 123. 435. (1) x = a±b. (2) a: = 3, y=2. (1) a;(a; + 2/ + z) (2) .3x2+1 126. x^ + {a + 2)x+B, z(x-y + z)' '"' ix (a:2 + j j (1) (x-32/)(ar + 8y + l). (2) a;(x + |Va?-?^ 39 9g2 , J a 1 130. 132. 136. 136. 1 i 138. 1 : 140. 142. 144. 1 146. 148. 152. 155. 166. IBS, 161. 131. (1) x"+f>+''. (2) x^^y 134 M X-4 + -. X x+a. 3 shillings H.C. F. x^ + a8. L. C. M. {x^ + a2) (a;2 - 4a2). (1) -2y. (2) .3. 137. (x-2a)(a;2 + 2ax + 4a2)(2a + 36)(2a-3?>). (1)3. (2) a; =105, y = 210, 3 = 420. 139. The diflFerence is 3. ar'-4y3-9z3-12yil _ 141. 3 hrs. 36 min. ^'^ ^='^b' (2) -=2ior-l^,2.= -li._.^_ (2) -—I-'- 145. (1)4(^/2 + ^3). (2)V21+V14. a^ - 2a^x^ + xt 149. a2 ^ j 0. 163 X2 (1) a;=7 or -11. (7x + 4)(4x-3)' (2) X orl§. 150. Six Shillings. V3 ±5 or ±2v/3, y=±3 or ±^- (1)0. (2)a;T^yl 167. {p + l)x-(p-l). I68. 3a (1) l-2a». (2) 3" -2" 12. 162. '" UVi- <») I 56(x2)/2_i)- 150. onr5.o«,4/^. / U6. 18 miles. 35. '^+(a + 2)x+3. 1} a !)2'^~a2- • ^ 223. 224. 226. 229. 231. 233. 234. 235. 238. 240. 245. 247. 249. 260. Number of terms = 6 ; common difference =2. (1) «o «A An 1 135 „ 405 , 243 . 128 1024 (2) l-^ + g-^ + ^«^ + 2l6^- Senary. 226. 4|; 1-412. 227. 5039684. 228. 315. (1) olbx; (2) X--. 230. x^- 10a; + 19 = 0. 2-71405; tet. 232. 1-75; 1-75; -2; 52375439. B overtakes A at the end of the 8"' day ; then A overtakes B at the end of the 15"' day. (1) .r=21, y=6; (2) a;=4, --, y = S, -"; 2 = 9, ^. a = 2d. 236. The 4"' and S"- terms. 237. 120; 60. ^^ 239. 8x='Jyz + 2y/yz 3-698970; -799340; 1-785248; x = j^. log 2, 241. 14. 244. (1) 3; (2) 1:36564; (3) 22. 246. 2880. 248. 2(»i-l) hours. 5 ,2. 1.4.7-(3r-8) (2„)f-^¥+'-, 3'-[r_ x'^-2(a + h)x + 2ab=0. (1) a, '^ — -1, Tone root is evidently a, and the product of b-c L the roots is '^L^^ f] (2) q, p-q. b-c J (1\~^ 1 -5 ) • f { OLABOOW : PRINTKO AT THE UNIVEBSITY PRESS BY ROBERT MACLEHOSE AND CO. ?,;! 228. 315. 19 = 0. ; -2; 5-2375439. en A overtakes B yi -Q iMi 3 ' ^ ' 9 • 237. 120; 60. r-36564; (3) 22. 246. 2880. lurs. nd the product of Mathematical Works BY • Messrs. HALL and KNIGHT, PUBLISHED BY Macmillan and Co., Limited. IACLEH08E AND CO. SEVENTH EDITION, Revised and Enlarged. Noio Heady. ELEMENTARY ALGEBRA FOR SCHOOLS. By H. S. Hall, M.A., formerly Scholar of Christ's College, Cambridge, Master of the Military Side, Clifton College ; and S. R. Knight, B.A., M.B., Ch.B., formerly Scholar of Trinity College, Cam- bridge, sometime Assistant Master at Marlborough College. Globe 8vo. (bound in maroon-coloured cloth), 3s. 6d. With Answers (bound in green-coloured cloth), 4s. Qd. The distinctive features of the Seventh Edition are :— (1) Greater prominence has been given to the fundamental Laws of Alffehra (see Arts. 22, 29-32, 46-48). With this object parts of the chapters on Multiplication and Division have been re-written. (2) A short section on the use of Detached Coefficients has been given on page 37. (3) A fuller treatment of the Remainder Thorem and its applications will be found on pages 236, 237. (4) Five new sets of Miscellaneous Examples have been added at con- venient intervals, beginnmg with one on page 32, which replaces Examples IV. c. With the exception of this change and a combina- tion of Examples XI. b. and XI. c, which now appear as one exercise, all the origina". examples will be found under their old numbers and with the ans 'ers unaltered, even where the examples themselves have been imps ived. MACMILLAN AND CO., LIMITED, LONDON. 'i Works by H. S. HALL, M.A., and S. R. KNIGHT, B.A. It will be found that the pagination is very little altered, the only difference being that the introduction of miscellaneous examples has placed the beginning of some chapters a few pajs^es further on in the book than in previous editions. It is believed that tiie few alterations in the text will all be justified by usage, and that they will cause no incon- venience to those who are familiar with the book in its old form. ' OPINIONS OF THB PRESS. SCHOOLMASTER—" . . . Has SO many poluts Of excellence as com- ?ared with its predecessors, tliat no apology is needed for its issue, ho plan always adopted by every good teacher, of frequently recapitulating and making additions at every recapitulation, is well carried out." NA TURB—" ... We confidently recommend it to mathematical teachers, who, we feel sure, will And it the best book of its kind for teaching purposes." ACADEMY— "Wo will not say that this is the best Elementary Algebra for school UHS that we have come across, but we can say that we dO not remember to have mQn a better. ... It is the outcome of a long experience of school teaching, and so is a thoroughly practical book." EDUCATIONAL TIMES— " . . . A very good book. The explanations are concise and clear, and the examples both numerous and well chosen," EDUCATIONAL jVcirS— " A book of exceptional value." OPINIONS OF TEACHERS. "I think it decidedly the best of all books on Elementary Algebra yet published. The great merit seems to me to be that, while it is quite 8iiiii>lo and elementary, there are no misleading and inaccurate statements which must afterwards be unlearned. I shall certainly make use of it in my classes, and hope it may come into general use throughout the country."— A. J. Wallis, M. A., Fellow and Lecturer of Coi-pus Christi College, Cambridge. " Wo have examined your Algebra very carefully ; and We a^ee that it is as perfect as a book can be. I win introduce it at St. Pauls as soon as ! can."— C. Pkndlkbury, M.A., Senior Mathematical Master, St. Paul's ScUool. " After employing it with my evening class tliis term, I feel it tO be quite the best Elementary Algebra that has yet appeared.' —li. A. Herman, M.A., Fellow of Trinity College, Cambridge; Lata Frojessor 0/ Mathematics at University College, Liverpool. KEY TO NEW EDITION. Crown 8vo. Hs. Qd. ANSWEBS TO EXAMPLES IN ELEMENTARY ALQE- JBlt-A. Jbcap. 8vo. iSewed. Is. MACMILLAN AND CO., LIMITED, LONDON. KNIGHT, B.A. altered, the only ous examples ha» her on in the book alterations in the 11 cause no incon- old form. zcellence as com- ded for its issue. J recapitulating and to mathematical 3k of its kind for nentary Algebra for do not remember ong experience of k." 'ho explanations are chosen." imentary Algebra lie it is quite siiniilo cements wliich must t in my classc!^, and ■A. J. Walus, M.A., e agree tbat it is , Paul's a* soon as ! Paul's School. eel it to be quite I."— R. A. Hekman, ' of JUathematics at TAEY ALOE- Works by H. S. HALL, M.A., and S. R. KNIGHT, B.A. HIGHER ALGEBRA. A Sequel to Elementary Algebra for Schools. By H. S. Hall, M.A., and S. B. Knight, B.A. Fifth Edition, revised and enlarged. Crown 8vo. 7s. 6d. The Fifth Edition contains a collection of three hundred Miscellaneous Examples, which Mill be found useful for advanced students. These Examples have been selected mainly from recent Scholarship or Senate House Papers. SCHOOL OUARDIAN-" We have no hesitation in saying that, in our opinion, it is one of the best books that have been pumished on the subject. . . . The authors have certainly added to their already high reputa- tion as writers of mathematical text-books by the work now under notice, which Is remarkable for clearness, acciu^cy, and thoroughness. " " It is a splendid sequel to your Elementary Algebra, and I am very pleased to see you have introduced the essential parts of the Theory of Equations in Chap. XXXV., which contains all that is required of the subject for ordinary practical purposes.'— A. G. Greenhill, M.A., Professor of Mathematics, to the Senior Clan oj Artillery Officers, R.A. Institution, Woolwich. ATHENJEUM—"1hQ Elementary Algebra by the same authors, which has already reached a third edition, is a work of such exceptional merit that those acquainted with it will form high expectations of the sequel to it now Issued. Nor will they bo disappointed. Of the authors' Higher Algebra as of their EUmentary Algebra, we unhesitatingly assert that it is by far the best work of Its kind with which we are acquainted. It supplies a want much felt by teachers." ^C^DJfJIfr— "Is as admirably adapted for College students as its predecessor was for schools. Tt is a well-arranged and weil-reasoned-out treatise, and con- tains much that we have not met with before in similar works. For instance, wo note as specially good the articles on Convergency and Divergency of Series, on the treatment of Series generally, and the treatment of Continued Fractions. ■ ■ ■ The book is almost Indispensable, and will be found to improve upon acquaintance." SATURDAY J?^ryj?IF— "They have presented such difficult parts of the sub- ject as Convergency and Divergency of Series, Series generally, and Probability with great clearnci-s and fulness of detail. . . . NO Student preparing for the University should omit to get this work in addition to any other he may have, for he need not fear to find here a mere repetition of the oid story. We have found much matter of intcrL.it and many valuable hints. . . . We would specially note the examples, of which there are enough, and more than enough, to try any student's powers." KEY. Crown 8vo= 10*. M. )NDON. MACMILLAN AND CO., LIMITED, LONDON. I Works by H. S. HALL, M.A., and S. R. KNIGHT, B.A. « .!■ ALaEBEA POR BEQINNEBS. By H. S. H. l, M.A., and S. R. Knight, B.A., M.B., Ch.B. Globe 8vo. 2s. With Answers. 28. 6d. EDUCATIONAL TIMES— "Algebra for Beginneri is dealt with on the same lines as the earlier and somewhat more advanced book. The learner is introduced as soon as possible to the practical and more interesting side of the subject, such as equations and problems, while work which largely consists in the manipulation and simpliacatlon of elaborate expressions is postponed till later on. The exam- ples for practice are copious, and have been newly composed for this particular book ; and, as heretofore, the explanations are clear, concise, and simply expressed. Indeed, without besitation we pronounce thla book the best of Its size which we have seen." UNIVERSITY CORRESPONDENT—" Those masters who have alrcsidy adopted Messrs. Hall and Knight's Elementary Algebra in their schools, will welcome this new work for the use of their junior classes. . . . The numerous exercises for the student are excellent in quality and entirely new. We can unhesitatingly recommend the book to the notice of both teachers and students." SCHOOLMASTER— '"Yo teachers who have had experience of either the Elemen- tary or the Higher Algebra it will only be necessary to say that this book is marked by the same qualities which have brought these works into such deserved repute. To those Who axe still in ignorance of these books, we can say that for clear, simple, and concise explanation, convenient order of subject-matter, and copious and well-graduated exercises, these books have, to say the least, no superiors. Quite early the student is introduced to easy problem work, which can only be looked upon as an advantage. The very numerous exercises are entirely new, so that the book might easily serve as a companion and supplement to the elementary work.' GUARDIAN-" It possesses the systematic arrangement and the lucidity which have gained so much praise for the works previously written by the authors in collaboration." ALOEBBAIOAL EXERCISES AND EXAMINATION PAPERS. With or without Answers. By H. S. Hall, M. A., and S. R. Knight, B.A. Third Edition, revised and enlarged. Globe 8vo. 2s. 6rf. This book has been compiled as a suitable companion to the Elementary Algebra by the same autliors. It consists of one hundred and twenty pro- gressive Miscellaneous Exercises, followed by a comprehensive collection of papers set at recent examinations. SA TURD A Y RE VIE W- ' ' To the exercises, one hundred and twenty in number, are added a large selection of examination papers set at the principal examinations which require a knowledge of algebra. These papers are intended chiefly as an aid to toaehers, who no doubt will find them useful as a criterion of the amount of proficiency to which they must work up their pupils before thoy can send them In to the several examinations with any certainty of success." •jt-jxt-t'i^i-i.-ici x:^ — TTc t.-;ir: 3Lrui2{j:y rccuinumud ihc VOlUmO 10 tOOCaOrS Seek' ing a well-arranged series of tests in algebra." MACMILLAN AND CO., LIMITED, LONDON. [NIGHT, B.A. V /Pks by H. S. HALL, M.A., and S. R. KNIGHT, B.A. L. L, M.A., and 8vo, 2s. With ;h on the same lines >er is introduced as the subject, such as a the manipulation iter on. The exain- 1 for this particular id simply expreased. le best of Its size ave alrcjvdy adopted s, will welcome this nerous exercises for m unhesitatingly Id students." f either the Elemou' y that this book is B into such deserved ooks, we can say venlent order of :ises, these books jdent is introduced Ivantage. The very ht easily serve as a i the lucidity which 1 by the authors in lAMINATION . S. Hall, M.A., ed and enlarged. to the Elementary d and twenty pro- heusive collection I twenty in number, ucipal examinations tended chiefly as an )rion of the amount ifore they can send ccess." mo to teachers soek- ELEMENTARY TRIOONOMETRY. By H. S. Hall, M.A., and S. R. Knight, B.A. Third Edition, containing 300 Additional Miscellaneous Examples. Globe 8vo. 4s. 6rf. EDUCATIONAL }{ByiBtV—"'rho authors have that instinctive knowledge of the needs, both of the pupil tind of the teacher, which only belongs to the practical teacher. ... On the Whole It is the best elementary treatise on Trigonometry we have seen." aUARDIAN— "They are lucid and concise in exposition, their methods are Bimple, and the examples are judiciously selected." LYCEUM-" It is not too much to say of this book, that it is the very best class-book that can be placed in the hands of beginners." SPBAKEJi-"Thcy here present Elementary Trigonometry so far as it can well be treated without infinite series and imaginary quantities. The authors lay iv solid foundation by Insisting on the thorough comprehension of trigonometrical ratios before passing on to other subjects. Logarithms and heights and distances have been treated with special care. . . . The full table of contents is a useful feature of the book." CAMBRIDGE REVIEW—" Messrs Hall and Knight's Algebra has won them a reputation which we think their Trigonometry will sustain." NATURE— "This book can safely be recommended to beginners, and it may, besides imparting to them a sound elementary knowledge of the subject, ingraft an intelligent interest for more advanced study." KEY. Crown 8vo. 8.1. 6d. ARITHMETICAL EXERCISES AND EXAMINATION PAPERS. With an Appendix containing Questions in LOGARITHMS AND MENSURATION. With or without Answers. By H. S. Hall, M.A., and S. R. Knight, B.A. Third Edition, revised and enlarged. Globe Svo. 2s. 6d. " In the Second Edition, the Appendix has been increased by a new series of examples, which are intended to be worked by the aid of Logarithmic Tables. In view of tlio increasing importance of logarithmic calculation in many examina- tions, this last section will be founC especially uaofwl."— From the Preface. CAMBRIDGE REVIEW— " AW the mathematical work these gentlemen have given to the public is of genuine worth, and these exercises are no exception to the rule. Tlie addition of the logaritlim and inpiiKiiratiou questions adds "reatl" to tlie value." " -o-- j )NDON. MACMILLAN AND CO., LIMITED, LONDON. « .. 'V. Ill k i . n. [II By H. S. HALL, M.A.. and F. H. STEVENS, MJV. A TEXT-BOOK OF EUCLID'S ELEMEN" 3, including Alternative ProofH, togetlier with Additional heorem^ nd Exerciaes, claaeified and arranged. J.y H. S. Hall, M. A., uul t H. Stevens, M.A., Masters of the Militar Side, Clifton Cr^kg. BookB I. -VI., XL, and XII., Props. 1 and 3. Olobo 8vo. 4rf. 6d. Also in parts sepav ately as follows : — Book I., . . . 1». Books T. and 11, . . la. 6d. Books I-m., . . 28. 6d. Books I.. IV., . . 33. Books n. and HI., . 2*. Books III. and IV., . 2». Books III. -V:., . . 3». Books IV. -VI.) . , 28. 6d. Books v., VI., XI.. and XII. : Props.! and 3, 2a. 6d. Book Xr la. JOURNAL . «■ BDUCA TJON-" The mosi complete introduction to Plane Geometry baaed on Buolld's Elementa tbat we have yet seen." PRACTICAL J HACBBR-"Ooa Of the most attractive hooka on Ck)ometr7 that has yet fallen In Bo our hands." wwuwwx ^ MJL.TS^'^^B EXERCISES AND EXAMPLES OONTAINE > IN A TEXT BOOK OP EUCLID'S ELEMENTS. Books T -VI. and XI. By H. S. Hall, M. A.T and F. H. Stevenh. M./ , Masters of the Military Side, Clifton College. Crown 8 vo. 8«. 6d. Books I. ^, 6». 6d. Books VI. and XI., 3s. 6d. AN ELEMENTARY COURSE OP MATHEMATICS, com- pnsmg Arithmetic, A ^ebra, and Euclid. Globe 8vo. 2«. 6d. By H. S. HALL and R. J. WOOD. ALGEBRA FOR ELEMENTARY SCHOOLS. Globe 8vo. Parts I and II., 6d. each. Cloth, 8d. each. Answers to Parts 1. and 11. , 4d. each. Sewed. By. F. H. STEVENS, M.A. ELEMENTARY MENSURATION. Globe 8vo. 3*. 6d NATURB—''T\\& largo number of original examples will hn fr.,,,,^ t „-.. assistance by teachera, and the questions, selecTed frmn naners «llv^f?« principal examining bodies, will nrove of sprvi™ nl + Papers set by the capabilities in working out mlnsuSn jJoWems"- *''*" "^ *^" ^^^'^^^^'^ MENSURATION FOR BEGINNERS WITH tttt- urrrkT EDUCATIONAL TIMES--' A considerable »m«u"t "f -r^^— -^ ' j ine wnoie is written with rare Judgment '^d clearaess" "" ' ^"'^ MACMILLAN AND CO., LIMITED, LONDON. I STEVENS, MJL •EMEN"^ 3, inciidin litional heorem^ ni> H. S. Hall, M.A., and Militar Side, Clifton Props. 1 and 3. Globe I.-VI., . . Zs. \yU . . 2s. 6d. , VI., XI., and [:>rop8.1 and 3, 2a. 6d. I • • • 18% te introductiooi to Plane lave yet seen." ctive books on Ooometry A.ND EXAMPLES K OF EUCLID'S By H. S. Hall, M.A., 1 Military Side, Clifton I. '., 6a. 6d. Books .THEMATICS, com- Globe 8vo. 2a. 6d. WOOD. IDOLS. Globe 8vo. ch. Answers to Parts M.A. 3be 8vo. 3*. Gd. OS will be found of great froni papers set by the as tests of the student's mm THE RUDI- lWING. Globe 8vo. "; n "vt ia tuVcruu, aiid Clearness." D, LONDON.