Irving Stringham WENTWORTH'S SERIES OF MATHEMATICS. First Steps in Number. Primary Arithmetic. Grammar School Arithmetic. High School Arithmetic. Exercises in Arithmetic. Shorter Course in Algebra. Elements of Algebra. Complete Algebra. College Algebra. Exercises in Algebra. Plane Geometry. Plane and Solid Geometry. Exercises in Geometry. PI. and Sol. Geometry and PI. Trigonometry. Plane Trigonometry and Tables. Plane and Spherical Trigonometry. Surveying. PI. and Sph. Trigonometry, Surveying, and Tables. Trigonometry, Surveying, and Navigation. Trigonometry Formulas. Logarithmic and Trigonometric Tables (Seven). Log. and Trig. Tables {Complete Edition), Analytic Geometry. Special Terms and Circular on Application, ALGEBIUIC ANALYSIS. ■ solutions and exercises ILLUSTRATING THE FUNDAMENTAL THEOREMS AND THE MOST IMPORTANT PROCESSES OF PURE ALGEBRA. BY G. A. WENTWORTH, A.M., Professor or Mathematics in Phillips Exeter Academy: J. A. McLELLAN, LL.D., Inspector or Normal Schools, and Conductor op Teachers' Institutes, for Ontario, Canada; J. c. glashan, Inspector of Public Schools, Ottawa, Canada, PART I. BOSTON, U.S.A.: PUBLISHED BY GINN & COMPANY, 1889. 9f^3 /" Entered, according to Act of Congress, in the year 1889, by G. A. WENTWORTH, in the Office of the Librarian of Congress, at Washington. All Rights Reserved. Typography by J. S. Cushing & Co., Boston, U.S.A. Presswork by Ginn & Co., Boston, U.S.A. PREFACE. rriHE work of which this volume forms the first or introductory part is intended to supply students of mathematics with a well- filled storehouse of solved examples and unsolved exercises in the application of the fundamental theorems and processes of pure Alge- bra, and to exhibit to them the highest and most important results of modern algebraic analysis. It may be used to follow and sup- plement the ordinary text-books, or it may be employed as a guide-book and work of reference, in a course of instruction under a teacher of mathematics. The following are some of the special features of this volume : It gives a large number of solutions in illustration of the best methods of algebraic resolution and reduction, some of which are not found in any text-book. It gives, classified under proper heads and preceded by type- solutions, a great number of exercises, many of them illustrating methods and principles which are generally ignored in elementary Algebras ; and it presents these solutions and exercises in such a way that the student not only sees how algebraic transformations are effected, but also perceives how to form for himself as many additional examples as he may desire. It shows the student how simple principles with which he is quite familiar, may be applied to the solution of questions which he has thought beyond the reach of these principles ; and gives complete explanations and illustrations of important topics which are omitted or are barely touched upon in the ordinary books, such as the Prin- 800573 IV PREFACE. ciple of Symmetry, Theory of Divisors, and its application to Fac- toring, and Applications of Horner's Division. A few of the exercises are chiefly supplementary to those proposed in the text-books, but the intelligent student will find that even these examples have not been selected in an aimless fashion ; he will recognize that they are really expressions of certain laws ; they are in fact proposed with a view to lead him to investigate these laws for himself as soon as he has sufficiently advanced in his course. Nos. 8, 9, 10, and 11 of Ex. 1 afford instances of such exercises. Others of the questions proposed are preparatory or interpretation exercises. These might well have been omitted were it not that they are generally omitted from the text-books and are too often neglected by teachers. Practice in the interpretation of a new notation, and in expression by means of it, should always precede its use as a sym- bolism itself subject to operations. Nos. 23 to 36 of Ex. 3, and nearly the whole of Ex. 15, may serve for instances. By far the greater number of the exercises is intended for prac- tice in the methods exhibited in the solved examples. As many as possible of these have been selected for their intrinsic value. They have been gathered from the works of the great masters of analysis, and the student who proceeds to the higher branches of mathematics will meet again with these examples and exercises, and will find his progress aided by his familiarity with them, and will not have to interrupt his advanced studies to learn theorems and processes properly belonging to elementary Algebra. In making this selec- tion, it has been found that the most widely useful transformations are, at the same time, those that best exhibit the methods of reduc- tion here explained, so that they have thus a double advantage. The present volume ends with an extensive collection of exercises in Determinants. These present under new forms and from a dif- ferent point of view the greater number of the theorems proposed, and many of the general results obtained, in the earlier chapters, and to these they add many important propositions in other subjects ; as, PREFACE. for example, in the method of least squares, in linear, homograph ic, orthogonal, and homaloid transformations, and in the degeneracy and the tangency of quadrics. The second volume will treat of factorials and the combinatory analysis ; finite differences and derived functions, both direct and inverse, of explicit functions of a single variable ; expansion, sum- mation, reversion, transformation, and interpolation of series ; the arithmetic, harmonic, and geometric series of integral orders, includ- ing the theta-functions ; recurring series ; binomial, logarithmic, and exponential series ; hyperbolic and circular functions ; trigonometric series, direct and inverse ; Legendre's, BessePs, Lame's, and Heine's series and their associated functions ; double series ; infinite prod- ucts ; continued fractions ; indeterminate equations ; theory of num- bers ; inequalities ; maxima and minima ; binomial equations and cyclotomic functions ; transformation of binary forms ; theory of the quintic and of higher equations ; theory of substitutions. The whole will close with a chapter on the fundamental postulates and the general laws of algebra, illustrated by examples and problems in matrices, polar algebras, and ideal arithmetic. In this second part of the work the authors hope to be able to give numerous historical notes and bibliographical references for the use of students who desire to pursue the subject further, or to consult the original memoirs. A companion volume to the present is in course of preparation for the use of private students and of all who have not the advantage of instruction by a specialist in mathematics. The companion will contain proofs of the theorems employed and solutions of the exer- cises proposed in this volume, the whole accompanied by hints on the best method of attacking problems, and on the selection of pro- cesses for their reduction. Notwithstanding that the utmost care has been taken in revis- ing the proof-sheets, there doubtless remain many errors both in the examples and in the exercises. The authors would feel grate- VI PREFACE. ful to teachers and students for notification of all errors which may be discovered, and also for suggestions in relation to the im- provement of the work. Messrs. J. S. Gushing & Co. deserve special mention for their masterly skill in overcoming all the difficulties in the typography of this work, and for their excellent taste and judgment exhibited in the beauty and elegance of these pages. G. A. WENTWORTH. J. A. McLELLAN. J. C. GLASHAN. Note. It is due Mr. Glashan to state that the main part of the work on this Algebra has been done by him. G. A. Wentworth. J. A. McLellan. CONTENTS. CHAPTER I. Substitution, Horner's Division, Etc. p^^^^ Numerical and Literal Substitution 1 Fundamental Formulas and their Applications ... 13 Expansion of Binomials 26 Horner's Methods of Multiplication and Division, and their Applications 27 CHAPTER II. Principle of Symmetry, Etc. The Principle of Symmetry and its Applications . . 39 The Theory of Divisors and its Applications ... 49 Factorimo. chapter III. Direct Application of the Fundamental Formulas . . 77 Extended Application of the Formulas .... 88 Factoring by Parts 100 Application of the Theory of Divisors .... 105 Factoring a Polynome by Trial Divisors .... 114 CHAPTER IV. Measures and Multiples, Etc. Division, Measures, and Multiples 130 Fractions 140 Ratios 155 Complete Squares, Cubes, Etc 164 Vlll CONTENTS. CHAPTER V. Linear Equations of One Unknown Quantity. PAGE Preliminary Equations 171 Fractional Equations 173 Application of Ratios 179 Resolution by Rejection of Constant Factors . . . 183 Higher Equations which are Resolvable into Rational Linear Factors 191 CHAPTER VL Simultaneous Linear Equations. Systems of Equations 202 Application of Symmetry 207 Particular Systems of Linear Equations . . . .211 CHAPTER VII. Quadratic Equations. Pure Quadratics 226 Quadratic Equations and Equations that can be solved as Quadratics 231 Simultaneous Quadratic Equations 238 CHAPTER VIII. Indices and Surds. Indices and Surds 263 Complex Quantities 273 Surd Equations 277 CHAPTER IX. Cubic and Quartic Equations Cubic Equations ........ 297 Quartic Equations 305 CONTENTS. IX CHAPTER X. Determinants. Definitions and Notation . Transformation of Determinants Evolution of Determinants . Multiplication of Determinants . Applications of Determinants PAQE 315 321 332 342 348 ALGEBEA. CHAPTER I. — SUBSTITUTION. Ex. 1. 1. If a=l, h = 2, c=-3, d = ^, x = S), y = 8, find the values of the following expressions : i-fi-(i-i-a-)r, a — {x — y) — (h — c)(d~ a) ~{y — h) {x + c) ; {x + d){y + h + c) + {x~d){a--h — d) -\- {y + d){a — X — d)] {d~xy + (c + yy; (a - b) (c' ~b'x)-(c- d) (b' - a'x) + (d-b-c)(d'-a'); d — a.d-\~c_c^d-{-b c? + a d — c d — b 2. If a = 3, b = —4:, c=—9, and 2s-=^a+b i- c, find the values of the following expressions : s(s — d)(s — b) (s ~ c)] s' + (s - af + (s - by +(s~ cf ; s^ — (s — a) (s — b) — (s — b)(s — c) — (s — c) (s — a) ; 2(5 — a) (5 — b) (s — c) + a(s — b) (s - c) -\-^b(s — c) (5 — a) -f c (s — a) (s — by. SUBSTITUTION. 3. If a — 12, 3 :^ — 3, c = 1, X = 4^, find the values of the following expressions : a'-h', a' + b\ (a~b)\ (a - hf . a' + b'' a'-b'' {a + bf {a + bf' a^ + ab + b\ a^ — b\ x j 2x-J __^x--l \ x ~-l 3 4 21 (a + b)l(a + by-c'\ . dXb-c)+bXc~a)+c\a-b) W'c'-(a:'~b'-cJ ' (a- 6) (^ - e) (c-a) 4. If a — 6, ^ == 5, c = — 4:, d = — 3, find the values of the following expressions : V(^^ + ac) + V(^-' -2ac)', -^\b' + ac+ ^(c'-2 ac) \ ; 2 a - V(^' - «^) ' c^ + 2d{d''~ c') 5. If :r = 3, ?/ = 4, 2; — 0, find the values of \zx - v(^^ + fWfi^ + V(*' + f+^'n ; a,-" + f+x'; {x - yj-" + (y - z)'"" + (« - x)'"" ; 6. Fmd the values of (^ + ^ + .)-- 3(.- + y-+ z^) when (i.) x=\, y = % 2-3 (ii.) a; -2, y-3, 2-4 (iii.) a; -3, y-4, 2-5 (iv.) :t'-10, y-11, 2-12. SUBSTITUTION. 7. Given rr = 3, y = 4, z = — 5, find the values of {x + y^+ xj— 3 {x + y + z) {xij + y^ + zx) ; x'iy + z) + y'{z + r^O + 2'(:^ + y) + 2:z:yz ; x'(7j ~-z) + y\z - ^) + 2^^ ~ y) ; ihx - 4 2)^ + 9(4:r - zj - (13^ - 5^7 ; (3a; + 4y + 5^)^+(4^ + 3y + 12^)*^-(5a; + 5y+ 13^)1 8. If s = a + ^ + <^, find the values of (2s - af-\- (2s - hj - (2s + cf when (i.) a = 3, 5 ==4, ^ = 5; (ii.) a -=21, ^ = 20, ^ = 29; (iii.) a -119, ^=-120, c-:169; (iv.) a = 3, h^-\, c--=5; (v.) a =--5, Z>---12, 6' --13. 9. If a=-l, h = S, c=5, d=7, e = 9,f==-U, show that f/=(«i I+l + l+i + i^lA^iV lb he ccl de ef 2\a /y ah a^c hcd cde def 4\a^ ^7 ' ahcd hcde cdef 6\abc defy a' + h' + c' - ah -he — ca - Z>'+ c''+ c^^— Z;^ — cd~- dh — c''^ -\' d^^ -\- e^ ~ cd ~ dc — ^c SUBSTITUTION. 10. If a = l, b = 2, c = 3, d = 4:, e = 5, /- 6, g=7, show that a-\- b -{- c= icd] a-\- b -}- c -{- d= ide; a + b + c + d + e=ief; a + b + c + d+ e+f=--ifg; a^ _^_ b-'-Yc^^^ ^^('^ + '^) ; ab (a + b) «3 + j. + ,.+ ^.=.^feM+|}. ab (a + b) a^ + b'^ + c^ + d' + >?= 'ffy,? ; ab{a-{- b) ^2 I 7,2 I 2 _l ^2 I 2 I p _ /^ (y + ^) . a^j^b' + c'^{a + b + cy; oJ^ + h' + c' + d^ = (a + b + c + dy ', d + b' + c' + d' + e' = (a + b + c + d+ ef ; ^3 _!_ 53 _^ ^3 _!_ ^3 _^ ^3 j^p ^.(^a-^i_Yc^d-\-e-\-fY', ^.j^l.j^,^^^cd{c-\-d){^d-V), bc{b + e) «* + 6* + c' + ^^ = *i^±#^%^^ 6(? (0 + c) „. + .. + .. + .. + ,+y.^ .W+g(;&-i) ; SUBSTITUTION. 11. Assume any numerical values for x, y, and z, and find the values of the following expressions : (a;^ - 10a:' + ^xf + (5«* - IO3? + 1)' - {a? + Vf ; {x +Yf-2{x+ 5)' - (x + 9/ + 2(x + 11/ + (:r+12/-(a.-+16)^ {x'~yj + {2xyy-{x'^ + yj; (x' - SxyJ + (Sx'y ~ yj - {x' + ff ■ (3 «^ + 4 xy + yj + (4 a;' + 2 xy^ - (5 a;' + 4 a-y + y^)^ ; (x-yy + {y-zf+{z-xy-2.{x~y){y-z){z-x). § 1. If :r — any number (as, for example, 3), then x\== xXx)-= Zx\ 0(?{= xXx'') = Zx^\ x\= X X x^) = 2>x\ etc. Or, 3 = 0;; Zx^x^\ ?>x^ — x^\ 3^* = :r^;etc. Hence, problems like the following may be solved like ordinary arithmetical problems in "Keduction Descending." Examples. 1. Find the value of x^ — 2x--^, when x = 5. 5 5x — 2x Sx 5 I 15 -9 6 Ans. 6 SUBSTITUTION. 2. Find the value of x^ — x^ — 4:x'^ — ^^ x — o when x = o. x'-x' — 4:x'~Sx-5 3 Pi 3 ^^ -~x^ n ^x^ 3 P2 ^X^ ~4:X^ r, 2x'' 3 — 3x rg 3^ 3 p. 9 -5 r^ 4 Ans. 3. Find the value of 2a;^+12^'+6:i;'— 12:^;+10 when ^ = -5. Using coefficients only, we have 2 + 12 + 6-12 + 10 -5 Pi -10 + 12 n + 2 - 5 P2 -10 + 6 ^2 — 4 - 5 i93 20 - 12 n 8 - 5 p. -40 + 10 7*4 — 30 Ans. \ SUBSTITUTION. § 2. If the coefficients, and also the values of x, are small numbers, much of the above may be done mentally, and the work will then be very compact. Thus, performing men- tally the multiplications and additions (or subtractions) of the coefficients, and merely recording the partial reductions ^'i, ^2, ^^3) and the result ^4, the last example will appear as follows : -5)2 2 -4 8 -30 + 12 +6 -12 +10 § 3. In the above examples, the coefficients are "brought down " and written below the products ^1, ^2, Ps, Pi, and are added or subtracted, as the case may require, to get the partial reductions Ti, r^, rg, and the result r^. Instead of thus "bringing down" the coefficients, we may "carry up" the products jp^, p^, p-^, />4, writing them beneath their cor- responding coefficients, and thus get ri, i\, i\, r^, in a third (horizontal) line. Arranged in this way, Exam. 2 will aj)pear 1 -1 -4 + 3 +6 -3 + 6 -5 + 9 1 +2 +2 + 3; 4 and Exam. 3 will appear -5 2+12 -10 + 6 -10 -12 +10 + 20 -40 2 +2 -4 + 8; -30 Comparing these arrangements with those first given (Exams. 2 and 3), it will be seen that they are, figure for figure, the same, except that the multiplier is not repeated. SUBSTITUTION. § 4. When there are several figures in the value of x, they may be arranged in a column, and each figure used separately, as in common multiplication. When only ap- proximate values are required, " contracted multiplication " may be used. 4. Find the value of 3.x'^-160^^ + 344^'^ + 700^'-1910:r + 1200, given x = 51. 1 50 3 -160 3 150 + 344 -7 -350 + 700 -13 -650 -1910 37 1850 + 1200 -23 -1150 3 -7 -13 + 37 -23; + 27 64 -144 + 45 64 75.712 89.5673 - 38.0419 6.4 7.5712 8.9567 - 3.8042 5.12 6.0570 7.1654 - 3.0434 0.192 0.2271 0.2687 -0.1141 64 75.712 89.5673 - 38.0419 - 0.0036 .*. result is 27. 5. Given a; = 1.183, find the value of 64:t* - 144^' + 45 correct to three decimal places. .-.result is— 0.004. Ex. 2. Find the value of 1. x' - llo;' - 11^;*' - 13a; + 11, for x = 12. 2. ^■* + 50a;' -16a;' -16a; -61, for a; = -17. 3. 2a;' + 249a;' -125a;' + 100, for a;-: -125. 4. 2a;' -473a,-' -234a,' -711, for a; = 200. 5. x^ — 3a;' — 8, for a;:==4. SUBSTITUTION. 6. x^ - 515:r^ - 3127 :r* + b2bx^ - 2090:^^ + 3156a; — 15792, for a;-- 521. 7. 2x' + 401^* - 199 :r"' + 399^^ - m2x + 211, forr?; = — 201. 8. 1000:?;*- 81 r^, hi x = 01. 9. mx^ + nix^ - 2b1x'' - 325.^ - 50, for x = If. 10. 5x' + 497a;' + 200:r' + 196r^^ - 218:?;- 2000, for rr =- — 99. 11. 5x' - 620:?;* - 1030:?;^ + 1045^;^ - 4120:?; + 9000, for X = 205. Calculate, correct to three places of decimals : 12. .'?;' + 3:?;' -13:?; -38, for :?;=:3.58443, for :?; = - 3.77931, and for :?;= — 2.80512. 13. y*- 14y2+ y + 38, for y = 3.13131, for ?/=- 1.84813, and for y=: — 3.28319. Ex. 3. What do the following expressions become (i.) when x^a; (ii.) when x= — a? 1. x^ — 4.ax^ + 6a''x'' — 4:a^x-}-a\ 2. ^(x'-ax+a'). 3. ^{x^ + 2ax + a'). 4. (x' + ax + ay — (x" — ax + a'f. If x = 'i/ = z = a, find the value of the following expres- sions : 5- i^~y){y ~z){z~x). 6. {x + yy{y + z~-a){x + z~a). 10 SUBSTITUTION. ^{y + ^) {f + ^' — ^') +yi^ + ^) (^' + ^' - f) X y+z x+z x+y Find the value of ^ X . X 1 ahc 9. - + 7, when 07 = '• a b a-\- 10 1.1.1 — - + — -+— whenr?:=-(a-^> + c). a{b — x) b{c — x) a(x-- c) a a o — a b(b-\-a) 12. (a + ^') {h + x) — a{h -{- c)-\- x^, when r?; — — -• 13. bx -\- cy -\- az, when r^ = 6 + ^ — ^? y = c + a — b^ z^=a-\-b — c. ^^ a{l + b) + bx a _^hena; = -a. a{l-\-b) — bx a — ^bx ''• f^T- " + ^"t? .when^ = i(5-«). yx-^-bJ x — a — 2b 16. (^ — $')(^ + 2r) + (r — rr)(^ + 5'), when a7= ^ ^ — ^• 17. aX5-d?) + Z>2(^__^)_|_^2(^_^>j^ whena-^) = 0. 18. {a + b + c) (be + ca + ab) — {a + b){b + c) {c + a), when a = — b. 19. (a + Z> + c)'— (a' + b^ + c'), when a + & = 0. 20. (x + y + zy-(x + yy-(y + zy~-{z + xy + x' + y' + z\ when rz; -f 2/ + 2J = 0- 21. a^(c - b') + b\a- c') + c'(b - a') + abc{abc - 1), when Z> — a^ = 0. /7 SUBSTITUTION. 11 22. «= f«\+5^7+ l^ r^^Y- when «^ + J^ = 0. 23. Express in words the fact that (a — by = a^~2ab-\-h'^. 24. Express algebraically the fact that "the sum of two numbers multiplied by their difference is equal to the difference of the squares of the numbers." 25. The area of the walls of a room is equal to the height multiplied by twice the sum of the length and breadth. What are the areas of the walls in the following cases : (i.) length Z, height h, breadth b ; (ii.) height x, length b feet more than the height, and breadth b feet less than the height. 26. Express in w^ords the statement that (x -{- a) (x -}- b) = a^ -{- (a -\- b) X -}- ab. 27. Express in symbols the statement that ''the square of the sum of two numbers exceeds the sum of their squares by twice their product." 28. Express in words the algebraic statement, (x + yY = x^ + y^ + Sx7/(x-{- y), 29. Express algebraically the fact that "the cube of the difference of two numbers is equal to the difference of the cubes of the numbers diminished by three times the product of the numbers multiplied by their difference." 30. If the sum of the cubes of two numbers be divided by the sum of the numbers, the quotient is equal to the square of their difference increased by their product. Express this algebraically. 31. Express in words the following algebraic statement: /^3 1^ = (^ + yj — ^y. x — y 12 SUBSTITUTION. 32. The square on the diagonal of a cube is equal to three times the square on the edge. Express this in sym- bols, using I for length of the edge, and d for length of the diagonal. 33. Express in symbols that " the length of the edge of the greatest cube that can be cut from a sphere is equal to the square root of one-third the square of the diameter." 34. Express in symbols that any ''rectangle is half the rectangle contained by the diagonals of the squares upon two adjacent sides." The square on the diagonal of a square is double the square on a side. 35. The area of a circle is equal to tt times the square of the radius. Express this in symbols. Also express in symbols the area of the ring between two con- centric circles. 36. The volume of a cylinder is equal to the product of its height into the area of the base ; that of a cone is one-third of this ; and that of a sphere is two-thirds of the volume of the circumscribing cylinder. Ex- press these facts in symbols, using h for the height of the cylinder, and r for the radius of its base. Ex. 4. Perform the additions in the following cases : 1. (h — a)x ~\- (c— l>)y and {a -f l))x-\-{b -f- c)y. 2. ax~hy, (a — h)x — (a + h)y, and (a-\-h)x — (h — a)y. 3. (7j--z)a'+(z — x)ah + (x-y)h^ and (x — y) a"^ — (z — y) ah — (x ^- z) U^. FUNDAMENTAL FORMULAS. 13 4. ax + ^y + cz, hx-{- cy + az, and ex -\- ay -\- hz. 5. {a + h)x' + (h+c)f + {a + c)z\ {h + c)x' + {a + c)f + (a + h)z\ (a + c)x^ + (ct + h)f + (b + c)z\ and -(^a + b + c){x' + f + z% 6. x{a-hy + y(h-^cf + z{c-a)\ y{a-hy + z(h-cf + x{c — ay, and z{a — l)f + x(h — cf + y{c — af. 7. {a--h)x'' + {h - c)y'' + {c- a)z\ {h-c)x^ + {c-a)y'' + {a-h)z\ and {c - a)x'+{a-h)f+{b-c)z\ 8. (a + ^)^ + (^ + ^)3/ — (c + a)2;, (^ + c) 2; + (c + a):25 — {a + h)y, and (a + c)3/ + (a + ^)2; — (6 + {x + y + z){xy + yz + xz) — '^xyz [9J {taf = Sa^ + S^a:'b + 6:$abc [10] Formula [1]. Examples. 1. We have at once (x + yy -{- {x — yy = 2(x^ -{- y^) and (x + yy-{x-yy = 4:xy. 2. (a-\-h -{- c~{- dy + (a — b — c-{- dy may be written [(a + 6^) + (^ + ^) J^ + [(a + J) - (Z> + c)7, which (Exam. 1) = 2[(a+c?7 + (5 + e)^]; similarly, (a- Z> + c - cZ)'^ + (a + 5 - ^ - c^y = l(^a-d)-{}>-c)J^\{a-d)-\-{b-c)J -2[(a-cZ7 + (5-cn -\-{a-\-b-~c-dy = 2\{a^dy^q,+cy-\-{a-dy-\-{b-cy\ by Exam. 1, = ^{a'-\-b^-\-c^-\-d''). FUNDAMENTAL FORMULAS. 15 3. Simplify (^a + h+ cf — 2{a + h + c)c + c\ This is the square of a binomial of which the first term is {a + h + c), and the second —c. Hence it equals [(a + h + c) - cj = (a + h)\ 4. Simplify {a + }))'- 2 (p? + V) (a + hy + 2 (a' + b'). By Exam. 1, 2(a' + b') = (a^ + bj + (a' - by. Hence, the given expression equals (a + by-2 (a' + b') {a + bf + (a' + by + (a' - by = [(a + by - (a' + b')J + (a' - by = a' + 2 a' b' + b'^ (a' + by. Simplify : ^^- ^• 1. (:^+3y7 + (rr-3y2)^; (^a' + 'dby ~ (^a' - Sby. Show that : 2. (mx + riT/y + (nx — Tn^y = (rn^ + ^^) (x^ + 'if). 3. (mx — nyy — (n:r — rnyy = (m^ — n'^) (x"^ — y^). Simplify : 4. [(a + 3S)^ + 2(a+Sb)(a -b) + {a~ by] (a - by 5. (.T + 3)' + (:?; + 4)^ - (^r + 57, and (i^^^-2y0^-(i3/^ + 2a;7. 6. {a + b + cy + (b + cy ~ 2{b -\.c){a+b + cy Show that : 7 . {ax + byy + {ex + c^)^ + {ay - bxy + (^y - dxy = {a' + b' + c\+ d') {x' + y'). Simplify : 8. {x-Syy + {Sx'~yy-2{Sx'-y){x-^y 16 FUNDAMENTAL FORMULAS. 9. \x^ -\-xy — y^y — {x^ — xy — y^)^ (1 + 2:r + 4:?;7+ {\-1x-\- ^x^)\ 10. If a + Z> — — \c, show that + 2(2Z)-c)(2c-a) + 2(2c-a)(2a-&) = xV^'- Simplify : 11. 2(a - ^)/ - (a - 2Z))^ (a^ + \ab + Z^^)^ - (a^ + Z)^. 12. (a + Z))2-(5 + c)^ + (c + c^)^-(c^+a)l 13. {\x-yJ-\-{\y-zJ-^{\z-xJ-^2i^\x-y){\z-x) Show that : 14. (rr — yy + (y-2)' + (2;-rr)^ -2(r.-y)(.-y) + 2(y-rr)(0-^)+2(^-y)(^-:r). Simplify : 15. (1 + xy - 2(1 + x^){^. + xj + 2(1 + ^0- 16. {x^y-\-zy-{x-^y-zy~{y + z-xy-{z-^x-y)\ 17. (:r-2y + 30)^+(3^-2y)^ + 2(rir-23/ + 3z)(2y-3^). 18. (aH^'--c7 + (^'-^? + 2(Z^'-c^)(aH^'-0. 19. (r, + y)^ + (^_y)^_2(a;-y)^(:r + y)l 20. (5a + 35)'^+16(3a + 5)^-(13a + 5Z))l . Show that : 21. (3a-Z^)^ + (3Z)-c)^ + (3c-a)^-2(Z)-3a)(3Z)-c) + 2(3Z>-c)(3c-a)-2(a-3c)(3a-Z^) — 4(a + Z>+6?)'=:0. 22. If z"- = 2xy, show that (2a;' — y^ + (2;' — 2y''y + (a:' - 2zy - 2(2^;'^ - y') (z' - 2^^^) + 2 (a;2 - 2^') (z' - 2/) - 2 (a;' - 2 2') (2x' ~ f) FUNDAMENTAL FORMULAS. 17 Simplify 23. (l + x + x' + a^y + (l-~x-x' + xy + (1 - ^^ + ^-^ - ^y + (l+x~x'-- xj. 24 . {ax + hyy - 2 (a' x' + h' y') (a:t^ + hyj + 2 (a* x' + h' y') . Formulas [2] and [3]. Examples. 1. (1-2:^ + 3:^7 = 1-4:^;+ 6:r' + 4:^^^~12:r^ Ar^x' = 1 - 4:^; + 10:^'^ - 12r^ + ^x' 2. {ph-^hc^cay = a'b' + 2ab'c + 2a'bc + b'c' + 2abc' + c'a' = a'b' + Pc' + c'd + 2abc{a -\-b-\-c). 3. [(•^ + y)^ + ^^ + yT ^{x-\-yy + 2{x-\- yjix' + y^) + :^;^ + 2x'f + y^ = (^ + y)*+ (^ + 2/)' [(^ + yy + (^ - y)'1 + :r^+2:^Y + / - 2(:^; + y)^ + {x^ - yj + x' + 2x'f + / -2[(a; + yy + a;^ + y*]. 4. (a;' + :ry + 3/7 ^x' + 2x\j + 2x^y' + :r'y' + 2:^?/' + i/ =^{^ + y)'^' + ^'y' + y\^ + yj- 5. In Exam. 3, substitute b — c for x, c — a for ?/, and con- sequently b — a for ^+3/; then, since (b — of =^ (a— by, Exam. 3 gives l^a-by + (b-cy + (c-a)J = 2 [(a- by + (b- cy +(c- ay] . 6. Making tlie same substitutions in Exam. 4, we have (a' + b' + c'~ab-bc- cay = {a-by(b-cy+(b-cy(c-ay+(c-ay(a-by^ 18 FUNDAMENTAL FORMULAS. or, multiplying both sides by 4, -= 4 (a - hy (b -cf + A (h-cj {c-af+^ {c-af {a-h)\ Hence, from Exam. 5, {a--hy + {b-cy+{c-ay = 2 {a-bj (b-cf + 2 {b-cy {c-af + 2 {c-a)\a- b)\ Expand: ^^- ^• 1. (1-2^+3^^-4^7; (l-x + x'-xj. 2. {l-2x + 2x'-?>x^-~xj] {l + '^x + ^x'^ + x^. 3. \2a-b-c'~iy', (l-x + y + zy^ (i^_i2/+6^)l 4 . (x^ ~x^y + xy^ — y^y ; (ax + bx^ + c?^^ + (i:^;*)^ 5 . Show that (a' + b'' + c%x^ + y' + 2') - {ax + by + czf = (o^y — bxy 4- (-3c)(/^ + 3c-2a); (2a-Z^-3c)(5-3c--2a). 10. (^^ + /)(^^' + y')(^ + 3/)(^-y)- 11. (^' + ^y + 2/') (^' — xy + y') (^* - ^r'y" + i/). 12. (a + ^-aZ)-l)(a + Z> + a^ + l). 13. If a' = Z^' + c*, show tliat (a^ + Z^^ + c^) (Z>^ + C'- a') (c' + o^~ b') (a' + b' - c') Simplify : 14. (.^' + y'-|^y)(^' + y' + |^y). 15. (x' -^x^+^x'^-^x+l) {x' + 2x^ + ?>x^ + 2x + 1). 16. Multiply {2x — y) o? — (x -{- y) ax + 2;^ by (2^ — y) a^ -\-(x ■^y)ax — x^. Show that : 17 . {d' + b'' + c^ + ab+bc + caf - {ab -^bc-\- caf =^{a-\-b-Y 6f {a!' ^b^ -\- &), 18. (a' + Z^2 + c^ + ab-\-bc-\- caf ~ {ci^ + ah + ca- bcf = [(a + b)(b + c)y + [(b + c)(c + a)]\ 19. 4 (ab + cc^)' - (a^ + b'-c'- dj = (a+b+c-d) (a+b-c+d) {c+d+a-b) {c+d-a+b). FUNDAMENTAL FORMULAS. 21 20. Find the product of : x^ -\-y^ -\-z^ —'Ixy ^ 2xz — 2yz and :r^ + y^ + 2;^ — 2^^y — 2xz + 27/z. 21. (x' + y' + xy^2)(x'-xi/-^2+f)(x'-y'). 22. (l-6a + 9a')(i + 2a + 3a2). 23. [(771 -j- n) -\~ (p -{- q)](m — q-^-p — n). 24. 1 + :z; + ^^ ^^ + ^ — 1, ^^ — ^ + 1, and l-\-x — x"^. 25 . (a - ^^)2 (a + 5^)^ (a^ + bj (a' + 5«)l 26. Show that (x"^ + xy + y^ {x^ ~ ^y + y'^J' — ipc^iTf = {x' + x'yy + {x'f + yy. Formula A. Examples. 1. Multiply x^ — x + bhj x^ — x-~l. Here the common term \^ x"^ — X] the other terms, ~\-b and —7. Hence, the product equals {x'-xy + {~1 + b){x'-x) + {-1xb) = (x' - xy -2(x'- x) - 35 = x''~2x^-x^^2x-Zb, 2. {x — a) (^ — 3a) (r^ + 4a) {x + 6a). Taking the first and third factors together, and the second and fourth, we have the product equals {x^ + 3a^ - 4a') {x? -{-Zax- 18a') = (a;'+3a:r)'-(4a' + 18a')(:^' + 3a:r)-72a^etc. Ex. 8. Find the products of : 1. {x^ -\-2x^Z){x^ -\-2x- ^)\ {x-y-\-^z)(x~y-\-hz). 2. (^+l)(^+5)(:r+2)(a;+4); {x'-\-a--h){x'^2h-d). 3. (a'-3)(a'-l)(a' + 5)(a' + 7); (:rH^'+l)(:r*+a;'-2). 22 FUNDAMENTAL FORMULAS. 4. [(x + yj - ^xy-\ \{x + yy + hxy\ 5. (a:» + a+7)(x"-a-9); j'^ + l- l") g + | + 3 6. (wa; + y + 3)(«a; + 2/ + 7). 7. (x + a — y)(a-+a + 3y). 8. (a;'" + «''-a)(ar=» + ar"-^»). 9. (i,s*_y^ + 2)aa;*-y^-4). 11. rr-2+V2, :i;-2+V3, :t'-2-V2, .r-2-V3. 12. (:r + a + Z>) (^ + ^ — c) (:i: — a + 5) (:?; + ^ + c?). 13. ia-^l -^ c){cL-\-h -^ d)-\'{a-\- c -^ d){h -^ c -^ d) ~{p.-\-h^c-\-d)\ 14. Show that (2a + 25-6?) (2Z> + 2c -a) + (2 c+2 a-5) (2 a+2 &-c) + (2 ^>+ 2 c-a) (2 6^+2 a-5) = 9(a& + 5c+)^2 + :i^^) + (1-^)1 Show that : 10. a(a-2by-b(b--2ay = (a-b)(a + by. 11. a' (a' -2 bj + 5-^ (2 a' - Z>')' - (a' - b') (a' + bj. Simplify : 12. (x'+x7/+yy+6(x'+y')(x' + x2/+7/) + (x'-X2/+yJ, I Show that : 13. a^ (a^ + 2 bj + F (2 a^ + bj + (3 a^ bj , ={a^+1a'¥ + bj. * Simplify : 14 . (ax + byy -f ^-^ y^ + ^^ ^^ — 3 aZ>:ry (ax + ^y) . 15. What will a^ -\- h^ -{- c^ — Sabc become when I a + b+c = 0. 16. Find the value of r^:^ — / + / + 3^^3/^2;^ when i 24 fundamental formulas. Formulas [7], [8], and [9]. Examples. 1. Simplify (2x-?>yy + {^y-~bxy+{?>x-yy -?>{2x- 3y) (4y -bx){^x- y). By [9] this is seen to be [{2x-?>y) + {^y-bx) + (?>x-y)J^{0y = 0. 2. Prove that {a - hf + (b - cf + (c - af = S(a-b)(b-c)(c-a). In [9] substitute a — b for x, b — c for y, and c~a for z; for these values x -\- y -{- z = 0, and the identity ap- pears at once. 3. TvoYe (a+b + cy-(b + c-ay~(a + c-by-(a + b-cf = 24:abc. In [8] let X = b-\- c — a, y = c-\- a — b, z=^a-{-b — c', and therefore, x-\-y = 2c, y-}- z = 2a, z-{-x=^2b] and this identity at once appears. Ex. 10. 1. Cube the following : l-x + x'; a-b-c] l-2x + Sx^ -4:x\ Simplify : 2. (x'' + 2x-iy + (2x-l)(x' + 2x-2)-(x' + Sx'-l)\ Prove that : 3. xyz + (x + y)(y + z)(z + x) = (x + y-\-z)(xy+yz + zx)^ 4. (ax — byY -\- a^y^ — ¥ x^ + 3 abxy (ax — by) = (a^-b')(x^ + f). * Note that the right-hand member is formed from the left-hand one by changing additions into multiplications, and multiplications into additions ; hence, in (a; -f y -f 2) X (a: X 2/ + 2/ X z + z X a?) the signs -f and x may be interchanged throughout without altering the value of the expression. FUNDAMENTAL FORMULAS. 25 Simplify : + ?>{x-y-2z){y-z-2x)(z-x-2ij). 6. {2x' - 3y^ + 4^7 + {2^ - ^z' + 40;^ 7. {2ax-hyy + (2hy-czy + {2cz-axy -\~S(2ax-\-hy — cz) (2 hy-{~cz — ax) (2cz + arr — by). Prove : 8. (a^ + Sx'y-y^y+l^xyix + ij)]' =.[(x-yy + 9x' y] (x^ + xy + y'f. 9. 9(x' + f + z')-(x + y + zy=(4:x+4:y + z)(x-yy + (^y + 4:z + x)(y-zy + {4:z + 4:x + y){z-x)\ 10. If x-\-y-{-z = 0, show tliat x^ + y^ + 2;^ — Srrys;. 11. Ux = 2y+^z, Bhow thsit x^ -Sy" -27 z' - lSxyz = 0. Show that : 12. (x'+xy+7ff+(x'-xy+y''y + 8z^-(jz\x'+xy+y') Prove that : 13. 8(a + b + cy-(a+by-(b + cj -{c + af = ^(2a + b + c){a + 2b + c){cL + b + 2c). Prove the following : 14. (ax — byy + ^^ y^ = a^ :z;^ + 3 abxy (by — ax). 15. a'^ + b^ + c^-^abc r=^[{a-by + (b-cy + {c-ay](a + b + c). 16. (a + b+c)[(a + b — c)(b + c — a) + {b + c — a){c+a — b) + (c + a-b)(a + b-c)] =^(a-]-b —c)(b-\- c~ a)(c-{-a — b) -\-8abc. 17. (a + b + cy-?>[a(b- cy + b(c ~ay+c (a - by] =^a^ + b' + c^ + 24:abc. 26 FUNDAMENTAL FORMULAS. 18. {a + h + 1 c){a-hy + {b + c +1 a)ih - cf + {c + a+1h){c~ay=:^2{a + h + cf - bA^ahc. 19. {a + h + c)[(2a--b)(2h~c) + (2h-c){2c-a) + (2c~a)(2a-h)]=^(2a-h){2h-c){2c-a) + {201 + h - c){2h + c - a) {2c + a- h). 20. If x^ {y -{- z) =^ a^ , qf (z -{- x) = If , z^ (x -{- y) = c^ , and xyz = abc, show that a'+P + c^ + 2abc = (x + 7j)(y + z)(z + x). y Expansion of Binomials. We have, from formula [5], (a+bf = a' + Sa'b + Sab' + P', multiplying by a-]-b, we obtain (a + by = a' + Aa'b + 6a''b'' + 4:aP+b'', ■ multiplying this by a-\-b, we obtain {a + bf = o^-\-ba'b + lOa^b''+lOo?b^ + bab' + bK From these examples we derive the following law for the formation of the terms in the expansion of a-\-b to any required power : I. The exponent of a, in the first term, is that of the given power, and decreases by unity in each succeeding term ; the exponent of b begins with unity in the second term, and increases by unity in each succeeding term. II. The coefficient of the first term is unity, and the co- efficient of any other term is found by multiplying the coefficient of the preceding term by the exponent of a in that term, and dividing the product by the number of that preceding term. It will be observed that the coefficients equally distant from the extremes of the expansion are equal. MULTIPLICATION AND DIVISION. 'Zi 1. 2. 3. 4. 5. 8. 9. Ex. 11. Expand (x + yf ; {x + yj ; {x + yf ; (x + yf. What will be the law of signs, if ~y be written for y in[l]? Expand (a - hf ; (a - 2 Z^)* ; (2 Z^ - a)*- Expand (1 + mf ; (m + 1/ ; (2 m + 1/. What is the coefficient of the fourth term in (a — hy^ ? Expand (x' -y)'- (a - 2 hj ; {a' -2 bj. In the expansion of (a — by^ the third term is 66 a^^ b^ ; find the fifth and sixth terms. Show that (x + yf — x^ — y^ = 5 xy (x + y) (x^ + ^y + y^)- From [8] show that 2 [(a - bf + (b — cf + (c — af] (b-cf Hounee's Methods of Multiplication and Division. Examples. 1 . Find the product of kx^ + Ix^ + mx + n and a:i'^ + 5:?; + <^- Write the multiplier in a column to the left of the mul- tiplicand, placing each term in the same horizontal line with the partial product it gives : Jcx^ -\-lx^ -\-mx -\-n Q ax + bx akx" -\'alx^ -\-amx^ -{-ana^ pi -\~bkx^ -\-blx^ -\-bmx'^-{-bnx p^ -\-cJcx^ -\'clx'^ -\-cmx-\-C7i ps a'koi^-[-{cil'\-b'k) x^-[-(am-\-bl-\-c'k) x^-\-(an-{-bm-\-cl) x^ +(Z>n+cm).r+C7i P 28 MULTIPLICATION AND DIVISION. § 6. The above example has been given in full, the pow- ers of X being inserted ; in the following example detached coefficients are used. It is evident that, if the coefficient of the first term of the multiplier be unity, the coefficients of the multiplicand will be the same as those of the first partial product, and may be used for them, thus saving the repetition of a line. 2. Multiply ?>x' — 2af — 2rr + 3 by x" + 3r^ — 2. 1 + 3 -2 3-2+0-2+3 + 9-6+0-6+9 _ 6 +4 -0 + 4 3^6 _j_ 7^5 _ Y^^'^ -\-2x^-Zx^-\-\?>x~^ 3. Find the product of {x — 3) {x + 4) {x — 2) (:r — 5). + 4 -2 -5 4. Multiply ^' — 4r^' + 2^ — 3 by 2^' — 3. 1 -4 +2 -3 {x^ Vio?^ x^) + 12 -6 +9 1 -3 + 4 - 12 1+1 -12 -2 - 2 +24 1-1 -14 +24 -5 +5 +70 - -120 x*-Gx'- 9a;' + 94x- -120 2 2 -8 +4 -6 3 -3 2x^ — S:x^ + ^x' - 9x^ + 12^;'' — 6a; + 9 In this example the missing terms of the multiplier are MULTIPLICATION AND DIVISION. 29 supplied by zeros ; but, instead of writing the zeros as in the example, we may, as in ordinary arithmetical multipli- cation, ''skip a line" for every missing term. 5 . Multiply x' -^x' + lhj x'-x' + ?>. 1 -1 + 3 1 + 0-2 + 0+1 {x'xx' = x^) -1-0+2-0-1 +3+0-6+0+3 -?>x^ +^x^ + 3 6. Find the value of {x + 2) (x + 3) (:r + 4) {x + 5) - 9 {x+2) (^+3) {x+ 4) + 3(:^ + 2)(r^ + 3) + 77 (:^: + 2) - 85. 1 +5 -9 -16 + 3 + 4 1 -4 + 4 + 3 1 +0 + 3 -13 + -39 + 77 + 2 1 +3 + 2 -13 + 6 + 38 -26 +76 -85 x' + ba?- Ix •'+12a;- 9 7. Find the coefficient of x^ in the product of x^ — ax^ + hx^ — cx-{- d and x"^ -^px + q. 1 ~ a ~\-b ~ c -\~ d ~ap + {b — ap + q) 30 MULTIPLICATION AND DIVISION. Ex. 12. Find the product of : 1. (1 + X + x" + r^' + x') {\-x''-^x^-x'^-\-x^-x^^-\-x'''), 2. (1,+ a;') (1 — rr^ + x^) (1 + ^ + ^' + ^' + x"). 3. (ri— 5)(:r+G)(2:-7)(a;+8); (2^^-a;'^+l)(:i;^-rr + 2). 4. (:r^ + 5:^^ - \^x- \){x^ - 5x' - 16:?:+ 1). 5. {6x' - x' + 2x' - 2a^'"^ + 2^2+19^ + 6) (Sx'+ix + l). Obtain the coefficients of x^ and lower powers in 6. {^l.-f-^X -gX +yg-^ 12 8"^ )\^ 2"^ "8*^ T6^ TTS^ )' 7. Multiply 2a;^ - x' + Sx -Ahj Sx^ - 2x^ - x-1. Simplify the following : 8. (x+l)(x + 2)(x + 3) + S(x+l)(x+2)-10(x+l) + 9. 9. x(x+ 1) (x + 2)(x + S)~Sx(x+ 1) (:r + 2) — 2:r(:^+l) + 2^. 10. :^ (:^ - 1) (:?; - 2) (^ - 3) + 3 :r (^' - 1) (^ - 2) -2ri'(^-l)-2^. 11. (x~l)(x + l)(x + S)(x + 5)~14:(x-l)(x+l) + l. 12. Given that the sum of the four following factors is —1, find (i.) the product of the first pair ; (ii.) the product of the second pair ; and (iii.) the product of the sum of the first pair by the sum of the second pair : (i.) X +x' + x^^ + x^\ (ii.) x' .+ x^ + x^ + x^\ (iii.) x^ + x'' + x''' + x'\ (iv.) x^ + x' + x^^'-^-x''. 13. Given that the sum of the three following factors is equal to —1, find their product : (i.) X +x^ + x^ +x'\ (ii.) x' + x^ + x^^ + x^\ (iii.) x' + x^ + x'' +x\ MULTIPLICATION AND DIVISION. 31 § 7. Were it required to divide the product JP in the first of the above examples by ax^ -{-hx-]- c, it is evident that could we find and subtract from P the partial products p^, p^ (or, what would give the same result, could we add them with the sign of each term changed), there would remain the partial product J9i, which, divided by the monomial ax'^, would give the quotient Q. This is what Horner's method does, the change of sign being secured by changing the signs of h and c, which are factors in each term oi p^, p^, respectively. 1. —hx —c ahx^^-{al-\-hh)x*'-\-{am^r hl-\- ck)x^ -\-{an-\- bm-\- cT)x^-^{hn-\-cm)x-\-cn..P —hkx^ —hlx? —hmx^ —hnx p.^ — chx^ — clx^ — cmx — en . .p^ ax aks^ -\-alx^ +amx^ ^anx"^ .^^ kx^ -\-lx^ -^mx +n Q The dividend and divisor are arranged as in the exam- ple, the sign of every term in the divisor, except the first, being changed in order to turn the subtractions into addi- tions. The first term of the dividend (alcx^) is brought down into the line of pi ; dividing this by ax'^, the first term of the divisor, we get kx^, the first tervi of the quotient. Multiplying this term hx^ by —hx and — i, which is got by simply adding up the second column of the work, giving alx^. Dividing this second term of ^i by ax^ gives Ix'^, the second term of the quotient. Multiply Ix^ by ~hx and — x'-2a^ MULTIPLICATION AND DIVISION. 33 5. Divide 10a;'- 11:?;' - 3a;* + 20r'-|- 10^^' + 2 Arranging as in the ordinary method, we have 10- -11-3 + 20 + 10+ 0+ 2 + 3 6-3- 6 + 12 -2 -4+ 2 + 4-8 + 2 + 4 -2-4+8 5 2- - 1-2+ 4 24-12+10 Quotient = 2o(^ - x^ — 2x + 4:- 24:r'^-12a; + 10 5^'-3;r'^ + 2:z;-2' We first draw a vertical line with as many vertical columns to the right as are less by unity than the number of terms in the divisor. This will mark the point at which the remainder begins to be formed. We then divide 10 by 5, and thus obtain the first coefficient of the dividend. We next multiply the remaining terms of the divisor by the 2 thus obtained. Adding the second vertical column and dividing by 5, we obtain —1 ; we multiply by the —1, add the next column, and divide the sum by 5, and so on for the others. This method is not, however, always convenient. If the first term of the dividend be not divisible by the first term of the divisor, the work would be embarrassed with frac- tions. We may then proceed as in the following examples : 6. Divide ^^-3^* + ^' + 3^' — a; + 3 by 2x^ + x^—Zx-\-l. Let2rj; = 3/, or x^=^^' A Substitute ^ for x in the dividend and divisor, and we A have 34 MULTIPLICATION AND DIVISION. 25 2* 2' 2' 2 ' 2' 2' 2 _ y^-2x3y^+2V+2^x3y^-2\y+2^x3 2^ . y^+.V^-2x3y+2^ 2^ Dividing y^-6/+V+2V-16y + 96 by y3+y2_6y+4, by the ordinary method, and the quotient by 2^, we have y^-7y + 17 1 , 39y^-114y-28 2^ 2^ ' y^ + y^-6y + 4' Substituting for y its value 2:r, and simplifying, we get ^_7^ , 17_1 39.y^-57rr-7 2 4 "^ 8 ^ 3 8 2^' + n;2-3^ + l By comparing the dividend of A with the original ques- tion, we find that we have multiplied the successive coefii- cients of the dividend by 2^, 2\ 2^ etc., and we have mul- tiplied the successive coefficients of the divisor, omitting the first terr}i^ by the same numbers. Dividing then by Horn- er's division, we get the coefficients 1, —7, 17; and for coefficients of remainder, —39, 114, and 28. The first three of these divided by 2, 2^ 2^ are the coefficients of x^, etc. ; and —39, etc., are divided by 1, 2, 2^ Hence, the work will stand as follows : 12 4 8 x+ ^-^2x^ + x''-?>x+l 16 32 12 4 1 -6 +4 + 24 - 16 + 96 -1 -1 +7 -17 + 6 + 6 -42 + 102 -4 - 4 + 28-68 1-7 +17 -39 + 114 + 28 1 -6 -f4 MULTIPLICATION AND DIVISION. 35 Quotient >K == £^ - 1^ + 1^ - 1 ^ 2 4 8 8 39 x' lUx 28 4 (X 4 " 17 1 2x' + x''~3x + l S9x'-57x~7 7. Divi( ie5ar* + 2by 3.r^- 5a-^ 1 3 9 27 2a; + 3. 0+ 2 81 243 + 2 -9 5 10 + 20- -45- - 60 - 90 0+ 486 -280 + 225 + 1260 5 + 10-25- -140 - 55 + 1746 8 2x^ + x'-Sx+l 2^3:r^-2:r + 3 Coeffs. of Quotient = --\ - 3 3' 3'' 3' 55- ■2 +9 1746- /^ ,• . ox I ±\J X ZiO X Quotient = — - 3 9 27 3-2 + 3 55^ — 582 140 1 81 81 3^'-2a; + 3 Ex. 13. Divide : 1. 6r?;^ + 5.'^'^ - 17r^ - 6^2 + 10:^-2 by 2:^2 + 3;r-l. 2. 5^' + 6:x;^+l by r^' + 2^+l. 3. a'-6a + 5by a'-2a + l. 4. x^ — 4:x^if — d>x''if — 11 xi/ — 12y^ hj x' — ^xy — 3y\ 5. a^ — Za^x'' + 3a'x^~x^hY a^~?>a^x + 3ax^ — x\ 6. 4cx^ + ?>x^~?>x+lhY x'' — 2x + 3. *■ It will, in general, be more convenient to multiply the dividend by such a number as will make its first term exactly divisible by the first term of the divisor, and afterwards divide the quotient by this multiplier. 36 MULTIPLICATION AND DIVISION. 8. a:^ — x^y + x^if' — x^y^ + ^y^ — y^ by x^ — 3/^. 9 . Multiply x"— ix^a+Qx"" a"— 4 xa^ + a* by ^' + 2 :ra + a^ and divide tbe product by x^ — 2x^a-{- 2xa^ — a*. Divide : 10. x^ — ax* -{-hx^ — hx"^ -\- ax —Ihj x~l. 11. 6^:^+7:^;'+ 7^' +6^' + 6^ + 5 hy2x' + x+l. 12. 60(a;' + y*) + 91^y(:r2-y') by 12^^- 13^?/ + 5yl 13. 6x' - 481^^ + Idx* + 81:^^- 81:z;2 + 86:?; - 481 by ^' - 80. 14. 6x^-x' + 2x'~2x^ + 2x^+19x + 6 by Sx^ + 4:X+1. 15. a(a + 2Z>7-^(26^ + Z^/by (a-Z>)l 16. (a; + y)^+3(^ + y)^2 + 3(.r + y)z^ + 2^ hy(x + yy + 2(x + y)z + z\ 17. 10 :z;^' + 10:?;" + 10.^•^ - 200 by :?;^ + :?;'- a; + 1. 18. bmx* + (5n + cm) x^ + C7i:r^ + ahx + a(? by ^a; + c. 19. Multiply 1 +-1^^- 18:r' by l-^fx'' + ^x\ and divide the product by 1 +-2^^ — 3^^. Find the remainders in tbe following cases : 20. (x^ + ^x' + ^x + b)-r-{x-2), 21. (^'-3^' + :z;-3)-f-(:2;-l). 22. {x* + ^a^ + ^x + S)-^{x + 2). 23. {21x*-y')-^{^x-2y), 24. (3a;^ + 5:r* - ?>x^ + 7a.-' - 5a; + 8) ^ (o;^ -- 2x). 25. (5a;* + 90a;^ + 80a;'- 100a;+ 500) -^ (a;+ 17). MULTIPLICATION AND DIVISION. 37 ^ 8. The following are examples of an important use of rner's Division : 1. Arrange, o;^ — Qx^ -{-7x~~5 in powers of x — 2. 2 1 -6 2 + 7 -8 -5 — 2 2 1 -4 2 -1; -4 -7 2 1 -2; + 2 -5 1; Hence, x' -Qx' + lx-^^-- (x~2y ~5(x~2)-7; or, as4t is generally expressed, x'-6x'+7x-5 = y'-57j-7if9/=x~2. 2. Express x* + 12x^ + 4:7x'' + 66x + 2S in powers of a;+3. 1 12 47 66 28 -3 -3 -27 -60 -18 1 9 20 6; 10 -3 -3 -18 - 6 1 6 2 • -8 -3 - 9 1 3; - 7 -3 -3 1; Hence, x' + 12^^-^ + ^7 x' + 66:^ + 28 = y* - if 7/=^x-\-3. 7y^+10, After a few solutions have been written out in full, as in the above examples, the writing may be lessened by omit- ting the lines opposite the increments (—2 in Exam. 1, and 38 MULTIPLICATION AND DIVISION. 3 in Exam. 2), the multiplication and addition being per- formed mentally. The last example, written in this way, would appear as follows : 47 66 28 ^ 20 6 (10) 2 (0) --3 12 9 6 3 (0) Ex. 14. Express : 1. x^ — bx^ + 3:r — 8 in powers of ^ — 1. 2. ^^ + 3^^ + 6^ + 9 in powers of ^'+ 1. 3. x'^ — 8r^ + 24a;' — ?>2x + 97 in powers of ^ - 2. 4. r?;* + 12:?;^ + 5rr' — 7 in powers of a; + 2. 5. 3:r^ — :?;^ + 4a;' + 5:r — 8 in powers of :?; — 2. 6. x^ ~ 7^:^ + 11^' — 7:?;+ 10 in powers of r^— If. 7. x^ — 2x^ — 4r?; + 9 in powers of ^ — -|. 8. x^ — ^x'^y -{-^ xy^ ~^lf iii powers of ^ — 2y. 9. x^—bx^y-\-b xy^ — if in powers oi x — y. 10. ^x^ -\-Vlx^y -^ lOrry' + 83/^ in powers of 2:?; + y. 11. r?;^ — -|r^^ + f :r — ^ in powers of -|-a; — ^. 12. :?;* + 80;^ — 15^ — 10 in powers of ^ + 2. CHAPTER II. Symmetey. ; § 9. An expression is said to be symmetrical witt respect to two of its letters wlien these can be interchanged by substituting each for the other without altering the expres- sion. Thus, a^ -}- a'^x-{- ax^ -f x^ is symmetrical with respect to a and x, for on substituting x for a and a for x it becomes x^ -{- x"^ a ^ xa^ ^ a^ , which differs from the original expres- sion merely in the order of its terms and of their factors. So, also, x^ + cc^x + <^^ + ^^^ is symmetrical with respect to a and h, for on substituting h for a and a for h it becomes x"^ -{'If^ x^ha-{- a^ x^ which is identical with the given expres- sion. On interchanging x and a, x^-]-a?x-\~ah'\-b'^x becomes a^ -]- x'^ a-^- xh -{-l)^ a ] this is not the same as the given ex- pression, which is therefore not symmetrical with resj)ect to X and «. In like manner, it may be shown that this expression is not symmetrical with respect to x and h. An expression is symmetrical with respect to three or more of its letters if it is symmetrical with respect to each and every pair of these that can be selected. Thus, x^-^-'if + z^—Zxyz is symmetrical with respect to X, y, and z, for it remains the same on interchanging x and y, or y and z, or z and x ; and these are all the pairs that can be selected from x, y, and z. So, also, 40 SYMMETRY. and {x - a){h - cj ■\- {x-l){c ~ aj -\- {x- c){a---hy are each symmetrical with, respect to a, Z), and c, but (j^h -^-IP' c -\- & a and (:^-a)(a— Z>)' + (^ — ^)(^ — c?)' + (r?; — ^)(c— a)' are not so. oh'\-ac-\-ad-\'hc-^hd^cd is symmetrical with respect to a, ^, ^, and d, but a5 + Z>(? + , , h into c^ and c into a. Thus, a^h-^y^c-^-c^a is cyclo-symmetric with respect to the cycle (ahc), for on changing a into b, h into (?, and c into a, it becomes Z>^c? + 6'^a-f-a^&, which differs from the orig- inal expression only in the order of its terms. (a — Ijf + (^ — ef -\-{g — of is not symmetrical with re- spect to a, 5, and c, for interchanging a and h changes it into (h-ay+{a- cj +{c- bf which =-(a~bf-(b-cf — ((? — d)^, and this differs from the original expression in SYMMETRY. 41 the signs of all its terms. But this expression is cyclo- symmetric with respect to (ah c). So, also, {x-a){a-hy + {x--h){b-cy + {x-c){c~ay is cyclo-symmetric with respect to (a h c) but is not com- pletely symmetric with respect to a, b, and c. Generally/, an expression is cyclo-symmetric with respect to any set of letters, a, b, c, , h, Ic, called the cycle {ab c hlc), if it remains the same expression when a is changed into 5, b into c, , h into ^, and Ic into a. Thus, ab-\-bc-\-cd'\- da and ac + bd are each cyclo-sym- metric with respect to the cycle (abed), but are not com- pletely symmetric with respect to a, b, c, and d. Every expression which is completely symmetric with respect to a set of letters is necessarily cyclo-symmetric with respect to them ; but, as is seen by the above exam- ples, an expression may be cyclo-symmetric without being completely symmetric. Principle of Symmetry. An expression ivhich in any one form is completely symmetric^ or is cyclo-symmetric, with respect to any set of letters will in every other form be com- pletely symmetric, or be cyclo-symmetric, as the case may be, with respect to these letters. Thus, a^ + ¥ + c^ — ?>abc is symmetrical with respect to a, b, and c ; hence, it will be symmetrical when written in any other form, as, for example, in the form Again, (a — by + (b — cf + (c~ a)^ is cyclo-symmetric, but not completely symmetrical, with respect to {ab c)] it will therefore remain thus cyclo-symmetric, but not com- pletely symmetrical, under every change of form which may be given it ; for example, when it is reduced to ?.(a-b){b-c){G~a). 42 SYMMETRY. A symmetric function of several letters is frequently represented by writing each, tyjpe-ternfi once, preceded by the letter 2 ; thus, for a + Z) + (?+ + Z we write 2 a, and for ah -\- ac -\- ad -{• -\-hc-^hd-\- (that is, the sum of the products of every pair of the letters considered) we write SaZ>. Ex. 15. Write the following in fall : 1. %a^h', %{a-hy', %a(h-c)', :^ab(x-c); ^a'Pc, %{a + h) {c~ a) {c-h)- 2[(a + e)^-Z>^]; and Sa(Z) + cf^ each for a, h, c. 2. %ahc) %a^h] ^d^hc] %{a — h)] and 5 a^ (a — ^), each with respect to a, ^, c, d. Show that the following are symmetrical : 3. (x + «)(«+ ^) (^ + x)~{~ahxj with respect to a and h. 4. {a ~{~ hy -\~ {a — by with respect to a and b, and also with respect to a and —b. 5. (ab — xyy — (a-^-b — x — y) [ab (x-\~y) — xy (a + by] with respect to a and b, and also with respect to x and y. 6. a^(b — c)^+5^(^— a7+c^(a— ^)^ with respect to a, Z>, ^ and also with respect to c^ and d'^, 8. .T® + y® + 3 :ry (:r^ + xy + y^) with respect to x and y. 9. [.^'^-/ + 3^y(2:r + 3/)]^+[2/'-^ + 3:.y(2y + rt•)]^ with respect to x and y. 10. a(a + 2^/ + ^ (Z> + 2a)^ with respect to a and &, and also with respect to a and — Z>. SYMMETRY. 43 > 11. ah \[{a + c){b + c)+2c{a + h)y ~ {a- c)\h - cf\ with respect to a, h, c. 12. o? h^ ^ y^ c} -{- c^ a^ -\- 2 ahc {a + Z> + e) with respect to ah, he, ca. With respect to what letters are the following sym- metrical ? 13. xyz + bxy + 2{x' + 7f). 14. 2{d'x^ + b''y'')-2ah{xy + hy + ax). 15. {f - hy + 4^^ (/+ hy + {2fh - 2gJ. 16. {x + y){x-z){y~z)-xAjz. 17. a^h'' + h'' c" + c'a^ -2abc{a + h - c). 18. x^ - y« + 0« - 3 {x' - /) (y^ - 2^) {z" + :i-^). 19. (a + &)2 + (a + c)' + (^-^7. 20. {a + Z^)* + {a - c?)* + (/; + c)' + (« + c)\ 21. (a + ^)* + (a - cy +{b + cy + (a + of + (c ~ by. Select the type-terms in : 22. a^ + 2ab + b' + 2bc + c^ + 2ca. 23. a (b'~ c') + b (c'- a') + c (u'- P) + (a + b) (b+c) (c+a). 24. a (b + cy + b{c + ay + c{a + by- 12abc. Write down the tyjpe-terms in : 25. {x-\-yy', {x-yy-, {x + yy-x'^ — f. 26. {x + yy + ^x-yy-, {x + yy - Qc ~yy. 27. {x + y + zy-, {x-y~zy. 28. (a + Z> + c + c/y; (a^ + Z>'' + c' + c^yi 29. {aJ^by-\-ib-\-cy^(c-\-ay, 44 SYMMETRY. § 10. In reducing an algebraic expression from one form to another, advantage may be taken of the principle of sym- metry ; for, it will be necessary to calculate only the tyjpe- terms, and the others may be written down from these. Examples. 1. Find the expansion oi{a + h-\~C'\-d-\-e-{- y. This expression is symmetrical with respect to a, ^, c, ; hence the expansion also must be symmetrical, and, as it is a product of two factors, it can contain only the squares o?, 5^, c^, , and the products in pairs, ah, ac, ad, , he, hd, ; so that c^ and ah are type-terms. Now (a + 5)^ = a^ + 2 a& + 5^ ; and the addition of terms involving c,d,e, , will not alter the terms a^-\-2ah, but will merely give additional terms of the same type. Hence, from symmetry we obtain {a+h-\-c+d+e+ y-= a^ + 2ah + 2ac+2ad+2ae+ + 5' +2hc+2hd+2he + + c^ +2cd + 2ce + + c?' +2de+ + e' + This may be compactly written {%ay = %a' + 2%ah. 2. Expand (a + 5)1 (i.) The expression is of th^ee dimensions, and is sym- metrical with respect to a and h. (ii.) The type-terms are a^ a^h. Hence, {a + h)^ = a^ + h^ -\- n{a^h + Pa), where n is numerical. To find the value of n, put a = h=^l, and we have SYMMETRY. 45 3. Expand (^ + y + zf. This is of three dimensions, and is symmetrical with respect to x, y, z. We have {x + y + zy^[(:x + y) + zY^{x + yy+ = x^ -\- '^ x^ y -\r which are type-terms, the only other possible type- term being xyz. Now, since the expression contains 2>x^y, it must also contain ^x'^z\ that is, it must contain 2tx^{y-\-z). Hence, {x + y + zy= x^ + ^ x^ {y -f z) . +y' + ^f{z + x) + z'+?>z'{x+y) + n{xtjz), where n is numerical, and may be found by putting x^=^y=zz = l in the last result, giving (1 + 1 + 1)^=1 + 1 + 1 -f 3(1 + 1) + 3(1 + 1) + 3(l + l) + 72. Hence, ?^ == 6. 4. Similarly, we may show that {a + h + c + df^ a^-\-?>a^{h + c + d) + Ucd + Z)' + 3^^ \c + cZ+ a) + 6c(7a + ^•'^ + 3^^ \d^a-\-l)-\-^dab + d}-\- 3 (i^ (a + 5 + c) + 6 ahc. 5. Expand (a + Z) + ^+ J. The type-terms are a^, a^S, ahc. Expanding (a + Z> + (?/, we get o}-\-Zc^h-\-^abc-\' Hence, by symmetry, we have (:Sa)^==:Sa^ + 3^a2^ + G^«Z)^. 46 SYMMETRY. 6. Simplify {a + h- 2cy + (b + c ~2ay + (c + a- 2b)\ This expression is symmetrical, involving terms of the types o? and ah. Now, a^ occurs with 1 as a coeffi- cient in the first square, with 4 as a coefficient in the . second square, and with 1 as a coefficient in the third square, and hence 6 a^ is one type-term of the result ; ah occurs with 2 as a coefficient in the first square, with —4 as a coefficient in the second square, and with —4 as a coefficient in the third square, and hence — 6 ah is the second type-term in the result. Hence, the total result is d (a'^ -\- h"^ -\- c^ — ah — hc~ ca). 7. Simplify (x~\-y + zf -{- (x — y — zf-^- (y ~ z — xf + {z-x- y)\ This is symmetrical with respect to x, y, z\ and the type-terms are a^, 3^^y, ^xyz: (i.) a? occurs in each of the first two cubes, and ~0(^ in each of the second two cubes ; therefore, there are no terms of the type oi? in the result. (ii.) Zoc^y occurs in the ^rs^ and third cubes, and —^x'^y in the second and fourth ; therefore, there are no terms of this type in the result, (iii.) ^xyz occurs in each of the four cubes; therefore, 2^xyz is the total result. 8. Prove {a'+h'^+c'+d'') {w^+x'+y''+z^)-{aw+hx+cy+dzy — {ax — hwy + {ay — cvif + {az — dw^ + (Ijy — cxy + {hz-dxy + {cz-dy)\ The left-hand member (considered as given) is symmet- rical with respect to the pairs of letters, a and w, h and X, c and 3/, d and z ; that is, any two pairs may be interchanged without affecting the expression. As the expression is only of the second degree in these SYMMETRY. 47 pairs, no term can involve three pairs as factors ; hence, the type-terms may be obtained by consider- ing all the terms involving a, h,w,x\ these are aW, d^x^, l/w^, y^x^, —a^vf, —W-x^, —2ahwx, and are the terms of (ax—hvif, which is consequently a type- term. From (ax — hivy we derive the five other terms of the second member by merely changing the letters. 9. Prove that {x^ — yzf -f (y^ — zxf + (2;^ — xy^ — Z{p(? — yz){y^ — zx) {z^ — xy) is a complete square. The expression will remain symmetrical if {x^ — yz) {y'^ -— zx)(z^ — xy), instead of being multiplied by — 3, be subtracted from each of the preceding terms, thus giving (^' - yz) [{o(? - yzj - (y' - zx) {f - xy)^ + (3/' - ^^) w - ^^y — (^' - ^y) (^' - y^)] + (z' - xy) [(2;' — xyj - {x^ — yz) {if — zx)'\ = {x?- — yz)x(x^ + f + ^^ — ^xyz) -f = {x^ + f + z^ — 3xyz) (x^ + y^ + z^ — 3xyz). Ex. 16. Simplify the following : 1 . (a + h + cf + (a + b - cy + (h + c- ay + (c + a ~ by. 2. (a-b-cy + (b-a-cy + (c-a-by. 3 . (a + b + c~dy + (b + c + d-ay+(c + d+a-by + (d+a + b-cy. 4. (a + 5 + cy — a(b-}-c — d) — b(a-\-c~b) — c(a + b — c). 5. (x~\-y-^z-{-ny-{-(x — y — z-\-ny-\-(x-—y-\-z — ny + (x + y — z — ny. 6. (a + b + cy+(a + b-cy + (b + c-~ay+(c + a-by\ 7. (x - 23/ - 3^)'^ + {y-2z - 3xy + (z - 2x ~ 3y)\ 48 SYMMETRY. I 8. (ma -\- nb -{- rcf — (ma -\- nh — rcf — (nh + re — mdf — (re + ma — nhy. 9. a(h + c)(b' + c'- a:') + b(c + a) (c' + a' ~ b') + c(a + b)(a' + b'-c'), 10. (ab + bc + caf — 2abc(a + b + c). Prove the following : 11. (ax -\- by -\- czf + (bx -j- cy -{- azf + (ex + ay + bzj- + (ax ■\-ey -\- bzf + (ex A^by -\- azf + (bx-{-ay + ezf = 2(d' + b'' + e^)(x^+y^ + z^) + 4 (ab -{-be -\- ed) (xy + yz-\- zx). 12. (a + b + ey + (b + c-ay+(e + a — by + (a + b — cy =^4:(a' + b' + e') + 24 (a'b' + b'e' + c^a^). 13. (a + ^ + (?)^ = 2a* + 42a^^> + 6Sa^^^ + 122a^^^. 14. (:^ay = :$a' + 4c:^a'b + 6^a'b' + 12^a''bc+24:^abed. 15. (a^ + ^=^ + c^)^ + 2(ab + be + eaf -S(a' + b' + e') (ab + be + eaf =^(a' + b' + c'-Sabey. 16. (a - by(b - cf + (b- cy (e - ay + (c - ay (a - by = (a' + b^ + c^-ab-ae- bey, 17. (2a-b-cy(2b-c-ay + (2b-e-ay(2c-a-by + (2e-a-by(2a-b-ey = ^(a^ + b'' + c^- ab-bc- eay, 18. (ar^ -\-2brs -\- es^) (ax^ + 2 bxy + cy'^) — [arx + b (ry + sx) + csy'\ ^ = (ae — b'^) (ry — sxy. 19. (a^ + ab + b')(e^ + ed+d') = (ac-{-ad-{-bdy-}-(ae-}-ad-]-bd) (be~ad)-{-(be—ady. 20. Show that there are two ways in which the given product in the last example can be expressed in the form p^ +pq + 2'^ and two ways in which it can be expressed in the form p^ ~ p(]_ + (f- I THEORY OF DIVISORS. 49 21. 6(w^ + ^' + y' + z'f ==(w + xy + (w- xy + (iv + 7/y + (w- yy + {w + zy + (iv - zy + {x + yy+ {x~yy + {x+zy+{x-zy+{y+zy+{y-zy. 22. \[{a + h + cy + {a-h-cy + (h-c-ay+{c-a-hy] =^^[{a-\-h+(}f+{a-h-cy+{h^c~ay+{c-a-hy] X^[{a+b + cy+ {a-h-~ cy+ (b -c-ay+ {c-a-hy\ Theoky of Divisors. Any expression which can be reduced to the form ax'' + bx''-'^ + ex''-'' + + hx+'k, in which ?z is a positive integer, and a, b, c, , h, Ic, are independent of x, is called a Polynome in x of degree n. The expressions /(rr)**, Fixy^ <^ (a;)"*, are used as general symbols for polynomes ; the exponents n and m indicate the degree of the polynome. Theorem I. If the polynome /(xY be divided by X — a, the remainder will be /(a)**. Cor. 1. f(pcy—f{aY is always exactly divisible by x—a. Cor. 2. If f{ay = 0, f(xy is exactly divisible hj x — a; that is,/(:r)"' is an algebraic multiple of ^ — a. Cor. 3. If the polynome f(xy, on division by the poly- nome (fi (xy, leave a remainder independent of x, such remainder will be the value off(xy when <^ (xy — 0. Examples. Theorem I. 1. Find the remainder when x^ — 7x^-\-lSx^—16x''-\-9x—12 is divided by :?; — 5. The remainder will be the value of the given polynome when 5 is substituted for x. (See § 3.) 50 THEORY OF DIVISORS. 1 -7 +13 5-10 -16 15 + 9 -5 -12 20 1 -2 3 -1 4; 8 Hence, the remainder is 8. 2. Find the remainder when (x — a)^ -\~ (x ~ lif + (a + hy is divided by x-\- a. For X substitute —a, then (- 2af + {-a-hy+ {a + hy = -^a\ 3. Find the remainder when C(^ + o^ +h^ + (x + a){x + h) {a + h) is divided hj x-\- a-{-h. For X substitute — (a + J), and we obtain -{a + hj + a^ + b^ + ah{a+h)=^-2ah{a + h). See Formula [6]. 4. Find the remainder when (x" + 2ax - 2ay (x" - 2ax - 2a') + 32 (x - a)' {x + a)' is divided hj oc^ — 2al x^ — 2o? may be struck out wherever it appears. This reduces the dividend to (2 axf (- 2 ax) + 32 (^ - a)' (x + a)* = -16a'x' + 32(x'-a'y. In this substitute 2 a' for rr', and it becomes -64a« + 32a^ = -32a^ which is the required remainder. Ex. 17. 1. Find the remainder when dx'+60x^+ 54: x"" — 60 x + 58 is divided by a;+ 19. 2 . Find the remainder when pa^ — Sqx"^ + 3 r:r — s is divided hj x — a. THEORY OF DIVISORS. 51 3. What number added to 4^^+ 34:^^ + 58:^;^ + 21^^ - 123:r - 41 will give a sum exactly divisible by 2^+ 13. 4. What number taken from 10 :r^^ - 20a;«- 10:?;« - 0.89 ^r* - 8.9^' + 20 will leave a remainder exactly divisible by 10^^—11 ? Find the remainders from the following divisions : 5. (:^+l)'-:?;'-f-:^' + l; (x + a + Sy-(x-i-a + iy-^x+2. 7. (x+lf + x' + (x-iy-^x-2.. 8. (x- of {x + aj + (^^ - 2 h'y -^x^ + b\ 9 . (x^ -\-ax-\' a^) {oc^ ~-ax~\- a^) - {x'' -?>ax + 2a') {x'' + ^ax + 2a')~r- x" + 2al 10. (9a'+6a& + 4Z^^)(9a'-6a5 + 4Z>')(81a*-36a'^' + 16Z)^) -^ (3a -25)1 11. d^-ix — af -\-lP' {x—'hf ^x — a — h, 1 2 . {ax + hyy + a^ if + ^^ ^^ — 3 abxy (ax + by) -^{a + b){x + y). 13. x^ + a''^ + 5^ — 3 a5:z; -^ :r — a + ^ ; also, -^ x-\- a — h] also, ^ X — a — h. 14. Any polynome divided by :r— 1 gives for remainder the sum of the coefficients of the terms. Examples. Cor. 1. 1. a;^ + y^ is exactly divisible hj x-^ry- In ''x^—a^ is exactly divisible hj x—a,'' substitute —y for a. 52 THEORY OF DIVISORS. 2. mx^ —px^ -\- qx-\-r}i-{-p -{- q is exactly divisible hj x-{-l. This may be written {mx^ —px^ + qx^ — [m (— 1)^ —p(^—Yf-^q (— 1)] is exactly divisible by ^ — (— 1). 3. {a?-\-^xy-\- 4/)' + (:r' + 2:^3/ + 4?/')^ is exactly divis- ible by (:?; + 2y)l For (x^ + G^ry + 4y^)^ — {;—x^ — 1xy — 4^/^)^ is exactly divisible by {x^ -^ ^ xy -\- ^ y^) — (— ^'^ — 2 ^y — 4 y^) , which is 2 (a;' + 4a;y + 4y') = 2 (a; + 2y)l Ex. 18. Prove that the following are cases of exact division : 1. :r'"+' + y'"+^ -^ ^ + y ; x"-"" — y'''' ^ x -\- y . 2. :^;^^ + y^2-^:^;* + y*; ^'« + y'*' -^ a;' + y^ also, -^ ^^'^ + y'' ; also, -^x^-\- yl 3. (a:r + %)^ + (^o; + ay7 -^ (a + 5) (:^ + y). 4. {ax + % + C2;)^ — (^:?; + y^2&(3a' + ^>'0. 11. (x' + 5Z>:z; + ^>')^ + (x' -hx + bj -^2(x + h)\ 12. (a + ^y"+' + («-^r+'-2(a^ + ^0- 13. [:z;^ + 3:ry(rr-y) -y^]H [:6^^- 9^y(a,- - y) - y']' ^2(^-y)l THEORY OF DIVISOKS. 14. ^x'-5x' + 4:x-2-^x—l. 15. Any polynome in x is divisible by ^—1 when tbe sum of the coefficients of the terms is zero. 16. Any polynome in x is divisible hj x-j-1 when the sum of the coefficients of the even powers of x is equal to the sum of the coefficients of the odd pow- ers. (The constant term is included among the ' coefficients of the even powers.) Examples. Cor. 2. 1. Show that a(a-j-2by — b(2a-{-hy is exactly divisible by a-{-b. By Cor. 2, the substitution of —h for a must cause the polynome to vanish. Substituting, a(a — 2ay + a(2a — af = — a^ + a^ = 0. 2. Show that (ah — XT/Y — (a-{-b — x — T/) [ab (x + y) —xi/ (a + b)] is exactly divisible by (x ~ a){y — a), also, by (x-b){y-b). For X substitute a, and the expression becomes {ab - ayj -ib-y) {ah (^ + 3/) - ctyifL + by\ ^a\b~yy-{b-y)\a\b-y)-\^^. The expression is, therefore, exactly divisible \)j x — a. But it is symmetrical with respect to x and y, hence it is divisible by y — a\ and, as x — a and y — a are independent factors, the expression is exactly divisi- ble by {x —- d){y — a). Again, the given expression is symmetrical with respect to a and b ; hence, mak- ing the interchange of a and b, the expression is seen to be divisible by (x~b){y — b). 54 THEORY OF DIVISORS. 3. Show that 6 (a' + b' + c')-^5 (a' + b' + c') (a' + b' + c') is exactly divisible by a-\-b-{-c. For a substitute —(b-{-c), and the result, which would be the remainder were the division actually per- formed, must vanish. 6[~(b + cy+b' + c'] -~6[-(b + cf + ¥ + c'] [(b + cf + b' + c'] =-. 6[-(b + cf + b' + c'] + SObc(b+ c)(b' + bc + c'). See [1] and [6]. The expansion being of the fifth degree, and symmetri- cal in b' and c, it will be sufficient to show that the coefficients of &^ ¥c, h^ (? vanish, the coefficients of b'^c', bc^, & being the coefficients of the former terms in reverse order. Calculating the coefficients of these type-terms, we get ^\-hb'c-VdW&- ] + 30(&*^ + 2^>''c'+ ), which evidently vanishes. Hence, the truth of the proposition. 4. If a-t-^^-fc-0, \{a'-\-b'^c')^\{a'-^y' + c')^\{a'-Vb' + c% In the last example it has been proved that the differ- ence of the quantities, here declared to be equal, is a multiple oi a-\-b-\-c^ that is, in this case, a multiple of zero. Hence, under the given condition they are equal. Ex. 19. Prove that the following are cases of exact division : 1 . {ax — byj + ibx — ayf — {c^ -f b^) {x" — i/)-^a,b, x, y, a + 5, x—y. 2. ax^ — (a^ -^ b) x'^ -\- b'^ -^ ax — h. (Substitute ax for b.) THEORY OF DIVISORS. 55 ^{{ax+bijy~{a — h){x + z){ax+by) + {a~hyxz^x+y. \ (ax — hyy— {a + h){x-\-z) (ax — by) + (a-^by xz-^x-\-y. 4. Qa^x^ — 4iax^ — 10axy — ?> a^ xy -\-2 x"^ y -{-by'^ -^ 2ax — y. 5. l.2a'x~ 5.494a' ^' + 4.8a' ^' + ^.^ax' ~ x"" 6. x^ ^ x^y'^ ^x^y -^1^ ^x^ -\-y, 7. (c - d)a^ -\-^(bG -bd)a -\- "^(1? c -b^ d) -^ a -\-Zb, 8. x(x--^yf-\-y(-^x-yy-^x-y. 9. a(a + 2^)'-Z>(^ + 2a7-f-a-5, also-^a + Z>. • 10. a^— 2a'5 + a'^)' + ^'^'-2a&^''+Z>':r'-f-(a-^)(^ + a). 11. a(b - cj ^b(c ~ aj + c(a-by' -^ (a-b), (b-6), (c-a). \ 12. d(b-c) + b\c-a)-^c\a-b)-^(a-b), (b - c), {c - a). 13. a' {b-c) + b' (c — a) + c\a - b) -^ (a -b), (b - c), (c - a). 14. (a- by (c - dy + (b- cy (d - ay -(d~ by (a - cy -i- (a — b), (b — c), (c — d), (d — a). 15. [(a-by + (b-cy + (c- ay][(a-byc'+ (b- cya' + (c- ayb'] - [(a -byc + (b- cya + (c- ay by -f- (a — b), (b — c), (c — a). 16. (x + y){y + z)(z + x) + xyz-^x-\-y + z. IT. abid" - b'') + bc(b'' — c') + ca(c'' - a') -^ a+b + c. 18. (ab — be — cay — d^b'^ ~ b'^ c^ — c^ a^ ^ a+ b — c. 19. (a + 2by + (2b -Scy-(Sc~ ay + a' + Sb' - 27c' -r-a + 2b — Bc. 20. a'b' + b'c' + c'a' -Sa'b'c'' ^ ab + be + ca. 56 THEORY OF DIVISORS. Examples. Cors. 3 and 2. Find the value of 4:X^ + ^x^ — bx" + 2?>x + ^ when Since 2^^ — 3^ + 4 = 0, we have simply to find the remainder on division by 2x^ — 3:z; + 4; and, if it is independent of x, it is the value sought, Cor. 3. 4 9 - 5 23 6 3 6 9 15 - 3 -4 -8 -12 -20 4 2 2 3 5 - 1; 10 Hence, the required value is 10. 2. What value of c will make x^ — bx^-{-lx — c exactly divisible by x—2. If 2 be substituted for x, the remainder must vanish, Cor, 2. 1 -5 2 7 -c -6 2 1 -3 1-2-c bx^ + cx^—2{)x'' + l^x 1. Hence, 2 — x By Cor. 3 the remainder must vanish when the given polynome is divided by 2x'^ — ?>x-\-l. We may divide at once and find, if possible, a value of c that will make both terms of the remainder vanish ; or, we may first express cx^ in lower terms in x^ and then divide and find the required value of c from the remainder. THEORY OF DIVISORS. 57 - First Method (see page 31). 6-10 Ac -160 304 -160 3 18 24 12e + 36 36^-420 __ 2 -12 -16 -8^-24 -24^ + 280 6 8, 4c + 12, 12^-140; 28^-140 -24^+120 Hence, 28^= 140 and 24 c = 120. Both of tliese are sat- isfied by c = 5. Second Method. x' = ^x(?,x- - 1) = f a,-^ -- ^x = f(3a;-l)-|s; = 2|.a;- -|_|.X=13^^ -f .•. C3? = l^cx — fe. Substituting for cx^ in the given polynome, it becomes 6x'—5x' — 20:^;'^ + (If c + 19)x -3^-5. Divide and apply Cor. 3. 6 -10 -160 28 c + 304 -24e -160 3 18 24 36 ^420 -2 -12 - 16 - 24 280 6 8 12 -140; 28e-140 -24c + 120 We thus obtain the same remainder as by the former method, and consequently the same result. A comparison of the two methods shows that they are but slightly differ- ent in form, but the second method shows rather more clearly that c need not be introduced into the dividend at all, but the proper multiples of it found by the preliminary reduction can be added to or taken from the numerical remainder, and the "true remainder" be thus found, and c determined from it. Find the value of : Ex. 20. 1. x^ — 3x^-{-4:x'^—3x + 4:, given x^ = x~l. 2. x' - 2x' - Ax' + l^x' - 11:^' - 10, given (x- If = 2. 68 THEORY OF DIVISORS. 3. 2x'-~7x'+12x'-llx' + 2x-5, given (x-iy + 2^0. given x^'}-Sx'^—2x-\- 5 = 0. 5. 6:?;' + 9^^-16:?;* -5:^^ -12:^^-6^ + 60, given 3x^ -^ x — A = 0. "What values of c will make the following polynome vanish under the given conditions: 6. :^*+ 13^^^ + 26:^2 + 52:^ + 8^, given :?; + 11 = 0. 7. x' - 2x' - dx' + 2cx - 14, given 3:r + 7 = 0. 8. X* — 4:X^ — x"^ -{~16x-\-6c, giyen x"^ = x-\~6. 9. 2:^*— 10rr' + 4c:?:+6, given:i:' + 3 = 3:r. 10. 2x' + x^—7cx^+llx+10, given 2x = 6. 11. 4:i'* + c?:r' + 110:?;-105, given 2:^'- 5:r+ 15 -- 0. 12. 3:r^-16:i;* + e:z;'-5:i;'-^114:^;+200, given :i;' = 3.'?;-4. 13. What values of ^ and q will make x^-j- 2x^-—10x^—px-{- q vanish, if x'^ — S(x — 1)? 14. What values of ^ and q will make a}' - 5 a^« + 10 a^ - 15 a' + 29 a' -^^ pa' + q vanish, if (a'-2y = a''~S? Theorem II. If the polynome f(xy vanish on substi- tuting for X each of the n (different) values ai, a^, a^, , a,„, f{xy — A(x~ ai) (x — a^ (x — a^) (x — a^), in which A is independent of x, and consequently is the co- efficient of x"^ inf(xy. Cor. If /(xy and (j> (x^ both vanish for the same m dif- ferent values of Xjf^xy is algebraically divisible by <^0O"*- THEORY OF DIVISORS. 59 Examples. x^ + ax'^ -]-bx-}-c will vanish if 2, or 3, or —4 be sub- stituted for X ; determine a, b, c. The coefficient of the highest power of a; is 1 ; .-. :(? + ax" -^lx-\-c-=-{x-~2) (x — 3) (x + 4) .-.^--1; h = ~U; = 24:. x^ -\- hx^ -{- ex -\- d will vanish if — 3, or 2, or 5 be sub- stituted for X ; determine its value if 3 be substituted for X. The given polynome = (a; + 3) (x — 2)(x — b)\ :. the required value is (3 + 3) (3 - 2) (3 - 5) = - 12. ^ 3. aa^ 3 hx'^ -\- 2> ex -\- d will vanish if for x be substituted — 3, or |-, or 1|-, but it becomes 45 if for x there be substituted 3 ; determine the values of a, &, c, d. The coefficient of the highest power of a; is a ; H); H); If 01? -\-jpx^ -\~ qx-{-r vanish for x^^a, or h, or c, deter- mine p, q, and r in terms of a, ^, c. rz;^ +p:?;^ + qx-\-r = {x — a) (^' — 5) (a; — e) =^ x^ — {a -\- h '\- c) x'^ -\- {ah -\-hc-\- ca) x — abc. :.p=^ — (a + b + c) or — 5a, ^ = a5 + &c + ea or 5 a^, r = — abc or — 5 abc. : . aa? + ?,hx' + 2,cx + d= a{x +Z){x - .•.a(3 + 3)(3^i)(3~li) = 45; • ^y — 9 -i)(^ .-. 2x^ + Zbx^ + Zcx + cl^2{x-\-?,){x- ,•,6 = 1; c = -3i; £?=4f --i)(^ 60 THEOEY OF DIVISORS. 5. If x^ -{-px^ -\- qx -\-r vanish for x = a^ ot h, or c, deter- mine the polynome that will vanish for x = h -\- c^ or c + a, or a + S. Since x^ -\-px'^ -\- qx-\-T vanishes for a; = a, or h, or ^, x^ -—px^ -\- qx — r will vanish for x=^ — a, or ~ h, or — c, and — p =^ a-{-h -{- c. But the required polynome will vanish for X = —p — a, or — p — Z), or ~p — c ; that is, for x-{-p = — a, or — b, or —c. Hence, it is (x -{-j^T ~~P (^ +J^)'^ + ^ (^ +i^) — ^' = ^^ + 2^9^^ + (p'^ -\- q)x +pq — '?'. The following is the calculation in the last reduction. (See page 37.) -p 1 7' p !z; pq-r p p\ f+q p 1 '' 2p 6. In any triangle, the square of the area expressed in terms of the lengths of the sides, is a polynome of four dimensions ; and the area of the triangle, the lengths of whose sides are 3, 4, and 5, respectively, is 6. Find the polynome expressing the square of the area. Let a, h, and c be the lengths of the sides, and A the required polynome. 1. The area vanishes if any two of the sides become to- gether equal to the third side ; hence, if a -f- Z) = hx^ -^?icx -^ d vanishes for :^ = 2, or 3, or 4. 2. x^ -\- cx^ -\- dx ^ e vanishes for x = \\, or — 3, or \\. 3. x^ -\- hx^ + (^^ + 24 vanishes for x —.% or — 3. 4. ax^ + hx^ + c:r + 90 vanishes for ^ = 3, or — 5, or 2. 5. ax'^ -\- cx^ — ^^x -\- e vanishes for x = \\, or — 4, or 2|-. 6. ^\x^-\-^cx^-^^dx^e vanishes for x = 1 1, or — 3-|-, or 1-^. 7. ax^ + ^^^ + G-^^ — 81 vanishes for x — f , or |, or 3. 8. ax^-\- cx^-\- dx -\- e vanishes for ;r =^ 2, or \\, or —1, and becomes 14 for :r =: 1. 9. ax^ -\-cx-\-d vanishes for x == 1^1, or 2f , and becomes 49 for 57 — 3, determine its value for rr — — 3. THEORY OF DIVISORS. 65 Given that x^~px'^-\'qx — r vanishes for x^a, or b^ or c, determine the polynome that vanishes for : 10. :?; = a+ 1, or ^ + 1, or (? + 1. 11. ^ = a— 1, or ^ — 1, or ^, or cl 15. X == a{h + . 39. [5c(Z>^ + c^) + acZ(a^ + (^0](^-6?)(a-c^) + the two similar terms -j- D. I THEORY OF DIVISORS. 67 40. (h -\~ c — a — dy(b ~ c) (a—d) + the two similar terms 41. The sum of the fractions \, |-, |-, , ^, increased by the sum of their products, two by two, increased by »the sum of their products, three by three, , in- creased by their product is equal to n. 42. In any trapezium, the square of the area expressed in I terms of the lengths of the parallel sides and the diagonals, is a polynome of four dimensions ; deter- mine that polynome. 43. In any quadrilateral inscribed in a circle, the square of the area, expressed in terms of the lengths of the K sides, is a polynome of four dimensions ; find that K polynome. Theorem III. If the polynome /(^)" vanish for more than n different values of x, it vanishes identically, the co- efficient of every term being zero. Co7\ If a rational integral expression of ?^ dimensions be divisible by more than n linear factors, the expression is identically zero. Examples. ^ (x — a)(x — h)(x — b)(x — c)(x~c)(x~a) -i __n. ^ * lc~a)(c-b)'^(a-b)(a-c)'^(b-~c)(b-a)~ ' if a, b, and c are unequal ; for this is a polynome of two dimensions in x, but it vanishes for x = a, and, therefore, by symmetry, hi x = b, and for x^= c; that is, for three different values of x ; hence, it van- ishes identically. [(a + by + (c + dy](a-b)(c-d) + [(b+ cy+(a + df] (b-c)(a-d) + [(o + ay+ (b + dj] (c -a)(b-d)^ 0. 68 THEORY OF DIVISORS. Substitute b for a, and tlie expression becomes [(h + cy + {h+dy]{b-c){h~d) +[{c+hy+(h+ciy]{c~h){h-d), which vanishes; hence, the given expression is divi- sible by a~h, and consequently, by symmetry, it is divisible by (a — &), (5 — c), {c — d), (a — c), {b — d), and (a — d). But the given expression is of only four dimensions, while it appears to have six linear factors ; hence, it vanishes identically. Ex. 22. Verify the following : b'c' b'(b'-/) ^ c\c'~b') = x' + y'' + z''-b'-c\ b'c' b\h'-c') c'{c'~b') 3 x'y' (x'-d')(a'--y')z' . (x' - b') (b' ~ y') z' a' b'' ~^ if - a^) (Z)^ - a') a" "^ (&^ - z') (b' - a'') W ~^ {z^-a^){W'-z^) 1 , 1 4. (^ + a)(a-Z))(a — c) (x-\-b){b — c){b-a) 1 1 + {x ~\- c){c — a) (c — b) (x-{- a) (x -{-b)(x -\- c) 5 . be (b' - c') + ca (c' - a') + ab (a' - b') = (a + b + c)[a\b - c) + .b''(c - a) + c'(a-b)l 6 <^ + ^ I ^ + y I <^ + ^ ^(^ — y)(x-z) y(y-x)(y-z) z{z-x){z — y) a '^ xyz THEORY OF DIVISORS. 69 [^ a' (b' - c') + h' (c' - a') 4- c' (a' - b') d'{b-~c) + b\c~a) + c\a~b) =.^[(a + b+cy-a'- b' - c'l 8. (adf-\- bcf-\- bed — acc)^ + (bee + aed + aef — bdfy (a — b)(b — c) (c — a) ^^[{a-by + ib-^ey + ic-d)-^]. 10. {~x + y + z){x~y + z)(x + y — z) + x{x — y + z)(x+y — z)+y{x+y — z){~x + y+z) + z{-~ X + y + z){:x — y + z) =- 4:xyz. ^ {a' - by + (^' - cy + (g^ - g^)^ {a + b)(b + e){e + a) _ (^ _ Z,)3 + (^ _ ^y + {c~ d)\ 12. :^^(y + ^7 + y'^(2 + ^-)' + ^'(^' + y/ + 2^yK.t; + y + ^) =^2(xy+yz + zx)\ I Theorem IV. If the polynomes f{xY, ^{xy (n not less than m), are equal for more than n different values of Xj they are equal for all values, and the coefficients of equal powers of x in each are equal to one another. This is called the Principle of Indeterminate Coefficients. Examples. 1. ^l + ^1 (a - b){a -c)(a~d) {b~^ a){b - c)(b - d) -I ^ y- = 0. • (c — d)(G~h)(c — d) (d—a)(d—b){d—c) {x ~ a) (x — b){x — c) (x — d) A ^ B ^ C ^ D / X x — a X — X — c X — a in which A, B, (7, D, are independent of x. 70 THEORY OF DIVISORS. Multiply by (x — a) (x — h){x- c) {x — d). .'. x"^ =^{A + B -\- C-\~ D)x^ + terms in lower powers of x'. Now this equality holds for more than three values of X, holding in fact for all finite values of x. ,'.A + jB+C+D = 0. (/3) Again, multiply both sides of (a) by x~ a, ^ (x — b)(x — c) (x ~ d) \x — o X — c X — a J Put x = a, -J. (a — h){a~ c) (a — d) By symmetry, — — -jr^B, etc. (6 — a){b~ c) (6 — d) Adding, (I '^ {a~b){a- c){a~ dy {h ~ a){b - c){b - d) + ^1 + ^- - {c-a){c~b){c-d) {d- a) {d - b) {d - G) = A-^B-\-C+D = ^, hj(P). . 2 a\a-\-b){a + c) b\b -\- e){J> -\~ a) c\c + d){c-\- b) {a-~b){a-c) "^ {b-c){b-a)'^ (c-a)(c~b) = (a + b + cy. Assume x^ — px"^ -{- qx — r =^ {x — a){x—b)(x—c) (a) .-. x^ +px^ + qx + r = {x + a){x ~\- b) {x + c) (^) x^JrP^ + q:>? + r. ^ _A_ B ^ ^ ^ ar—px^+qx — r ^ x—a x—b x—c THEORY OF DIVISORS. 71 Multiply by x^ — px^ -\- qx — r, and equate tlie coeffi- cients of the terms in x^. In multiplying the frac- tions in the right-hand member of (y), use the factor side of (a). :.A + B+C=2p\ Multiply both members of (y) by x — a. x^x+d) {x-]-h) (x-{-c) {x—b)(x — c) = ^ + (. + 2^ + ^ + ^)(.-a). Put x^=aj (a ~ b)(a~ c) By symmetry, 2¥{b + c){b + a) p , %c^c + a){c + b) _^ {b-c){b-a) ' (c-a)(c~b) ' . a'(a + b)(a + c} . b'(b + c)(b+a) " (a-b)la~c) (b-c)(b-a) (c — a) (c — o) = (a + b + cy. 3. Extract the square root of 1 + ^ + ^"^ + ^^ + ^* + Assume the square root to be 1 -}- ax ~\- bx"^ -\- cx^ -f dx^ + /.l + x + x^ + x'+x'4- -=-(l + ax + bx' + cx^ + dx' + J = 1 + "Aax -f {a^ + '2.b)x^ + 2{ab + g)x^ + (2c^+2ac+Z)')a;*+ THEORY OF DIVISORS. .■.2a = l, .■.« = i 2b + a^ = l, *=i(l-i) = f. 2(c + ab) = l, «=i-(iXf) = T^, 2di-2ac + b'=l, <^=i(l-A-A) = - 35 - 128- .■.^{l+x + x' + -- •) = Note. As it is frequently necessary to determine the coefficient of a particular power of x, a few preliminary exercises are given on this subject. Ex. 23. Determine the coefficient of: 1 . ^* in (1 + axf + (1 + bxf + (1 - cxf, 2. x'm(li-x + 2x' + 3x')(l — x + Sx'' + x^~5x'), ' 3. x'm(l + x + 2x' + Sx^ + 4:x' + ) (l-x + x^-x^+x' — ). 4. x"^ in A{x — h) {x — c) (x — d) + B (x~ a) (x — c) (x — d) + C(x~ a){x— h) (x — d) + D {x — a) (x—h) {x— c) 5. :^* in (1 — axY (1 + axj*, 6. x^in{l + axy(l — hxf, 7. In the product (1 + ax -\- hx^ -\- cx^ -\- ) (1 — ax -{- hx'^ ~ cx^ + ) prove that the coefficients of the odd powers of x must be all zeros. Determine the value of the following expressions : 1 + ^ (a-b)(a-c)(a-d) {h - a){h - c)(b- d) {c- a){c - b)(c-d) ^ (d- a)(d-b)(d- c) THEORY OF DIVISORS. 73 I "^ + ^^ + {a~h){a-c){a-d) (b-a)(h-c)(b-d) 10. — — — — + three similar terms. {a ~b){a— c) (a — a) 11. ■; =-— — — + three similar terms. {a — o)(a — c) (a — a) 12. ; -— — — + three similar terms. (a — b){a — c) (a — a) 13. — — — - + three similar terms. (a -- o)(a — c) {a — a) 14. — > -^ -^ -(- two similar terms. (a — D){a — c) 15 • -7^ Y~-r ^ + two similar terms. {a-b)i^a-c) W a\a + h){a+c) . . ••14- 16. ^ // -^ + two similar terms. {a — o){a — c) 17. x' + 4:a^+ ^ 29. 1 - 4^ + 10^' - 20 x^ + 35^*- 56^^ + S^x\ 30. Extract the cube root of 1 + ^ to four terms. § 11. 1. Find the condition that px^ -{- 2qx -\- r and ^V -[-2q^x-\- r' shall have a common factor. Multiply the polynomials by p^ and p respectively, and take the difference of the products ; also, by r' and 7% re- spectively, and divide the difference of the products by x. p^px^ -f 2p^qx -{-pW pp^x^ + 2pq^x -{-pr^ pr^x^ -\- 2 qr^x + rr' pWx^ ■j-2q'rx-\- r'r 2 (pq^ —p'^) ^ + (P'^' —p^r) (pr' — j9 V) x-j-2 (qr' — r^q) Multiply the former of these remainders by (pr^—p^r), and the latter by 2(pq'~p'q), and the difference of the products (^r' —pWy- — 4 {pq^ ~pW) (S'^' — '^'$')- But if the given polynomials have a linear factor, this remainder must vanish, or {pr^ — pWy =^ 4:(pq^ -- p^q) (qr^ — r^q) . THEORY OF DIVISORS. 75 If the given polynomes have a quadratic factor, the linear remainders must vanish identically, or (Th. III.), pq^ -p^q = 0, pr^ —p^r = 0, and qr' ~ r^q = 0, or P = 9^ = L p q^ T 2. Find the condition that px^-^Zqx^A^^rx^s shall have a square factor. Assume the square factor to be {x — (if. On division, the remainder must be zero for every finite value of x, and consequently (Th. III.) the coefficient of each term of the remainder must be zero. Divide by (x — (if, neglecting the first remainder. a p ^q pa Sr s po^ + %qa a p pa + Sq pa 2pa^ + ?>qa p 2pa + 3q] 3(pa^ + 2ga + r) .'.pa^ + 2qa + r = 0] .'. px'^ ■j-2qx-\-r is divisible by ^ — a ; (Th. I. Cor. 2.) or, px^ + 3 qx"^ -}-Srx-{- s and px"^ -\-2qx-{~ r have a common divisor. Multiply the latter polynome by x, and subtract the product from the former, and the proposition reduces to lfpx^-{-Sqx'^-\-Srx-{-s have a square factor, ^.^^ + 23':^+^ and qx'^-\-2rx-{-s will have the square root of that factor for a common divisor. Ex. 24. 1. Determine the condition necessary in order that x^-^-px -{- q and x^ ~{-p'x-\- q^ may have a common divisor. 76 THEOEY OF DIVISORS. 2. The expression x^ + ^a^ r'+Zhx' + cx^+Mx' + '^e^ x+f will be a complete cube if 'r e d c — a^ i A M -^ a h 6a' I 3. Prove that ax^-\- hx-{- c and a-\-hx'^ -\~ ex" will bave a common quadratic factor if y c" = {c' -a'+ b') (c' - a^ + ah). 4. Prove tbat ax^ -{-ha^-]-c and a + ho^-j- ex" will have a common quadratic factor if a'})' = {a' - c''){a' - c' -\-lc). 5 . Prove tbat ax^ -^-hx^ -\- ex -\- d and a + 5^ + cx^-\- dx* will bave a common quadratic factor if (a + d)(a-dy=:(b- c) (bd - ac). 6. x^ -\-px^ -{- qx-\-r will be divisible by x' -\- ax-\-h if a^ — 2pa' + {p^ -\- q) a + r — pq = 0, and b^ — qb"^ + rpb — r'^ = 0. 7. x^ -\-px + q will be divisible by x'^ -\-ax + b if a^-^qa' -=^^ and {b' + ^) {b' - qf -^p'b\ Determine tbe condition necessary in order tbat : 8. x^-{-4:px^-]-6qx'^-\-4:rx-{-t may bave a square factor. 9. ax^-\-A. bx^ -\-6cx' + 4Ldx+e may bave a complete cube as factor. 10. x^ + 10^^^ + 10 cx^ + b dx + be may bave a complete cube as factor. CHAPTER III. DiEECT Application of the Fundamental Formulas. p- Formulas [1] and [2]. (r^:d=3/)^==a;^=b2rry+3/^ etc. § 12. From this it appears that a trinomial of which the extremes are squares, is itself a square if four times the product of the extremes is equal to the square of the mean, and that, to factor such a trinomial, we have simply to con- nect the square root of each of the squares by the sign of _the other term, and write the result twice as a factor. Examples. 1. 4:x' - 80:r^3/^ + 400y* = (2x' - SOy^ (2x' - 20y^). 2. 1 - I2x''7f + 36 ^V* = (1 - 6:r^?/^)(l -Grr^y^). 3. {a-hy+{h-cf+2{a-h){h-c). This equals {a — h +b — c){a — h -\- h — c) = {a — c)(a — c). 4. x'^ -^-y"^ -{-z^ -\-2xy — 2xz — 2yz. Here the three squares and the three double products suggest that the expression is the square of a linear trinomial in x, y, z. An inspection of the signs of the double products ena- bles us to determine the signs which are to connect X, y, z; we see that 1. The signs of x and y must be alike ; 2. The signs of x and z must be different ; 3. The signs of 3/ and z must be different. Hence, we have x-\-y—z^ or —x—y-\-z = — (x-\-y — z)j and the factors are (x -\- y — z) (x -{- y — z). 78 FACTORING. Ex. 25. 1. 9m' + 12m + 4; c'"*-2c"^ + l. 3. 9a'b'' + 12abc + 4:c''] 36^'y'- 24^/ + 4/. 6. ,^ + (cc-yy-2z(x-y)- (fJV (^^j"- 2. 7. (:r^ - yf + 2(^^ - 3/) (3/ - .^) + (y - .^)l 8. (x'^ — xyY — 2(x^ — xy) (xy — y'^) + (^y — 3/^)^. 9. (a + ^ + ^)'-2c(a+Z> + (?)+c^ ^^p^-2p^q^ + ^f 10. (3:r - 4y)' + (2:r - 83/)^ — 2(3:?; - 4y) (2:?; - 3y). 11. (:,^_,,3/ + 3/7+(a;^ + :r3/ + y7 + 2(r.^ + a;V^+y0. 12. {bx' + 2xy+1yy + {^x''+^yy - 2(4a;^ + 6y^) {bx^ + 2xy + 7y^). 1m f'lXl.n /^\m—n 14. a2 + Z>^ + c'-2a5-2Z>c + 2a(?. 15. a^ + ^* + c'-2a^^2-2a'c^ + 2Z^2cl 16. {a-by + (b- cY + {c-af + 2{a- b) (b - c) -2{a-b){c - a) + 2{b -c){a- c). 17. 4a'-12a'Z) + 9Z^' + 16a'c+16c'-245(?. Formula [4]. x^ — y'^ = {x + y) {x — y). § 13. In this case we have merely to take the square root of each of the squares, and connect the results with the sign + for one of the factors, and with the sign — for the other. FACTORING. 79 Examples. 1. {a + hY-{c + d)\ This = [(a + h) + {c + d)] [{a + h)-{c + d)] -=la + h + c + d){a + b — c — d), 2. Factor (x^ -{-bxy-{- y^y — {x' — xy ■\- y'^f. Here we have \{x^ -{- b xy -\- y^) + (x^ — xy + 3/^)] X [(2?' + 5^y + y2) — (> — :ry + ?/')] ^^ix" + xy + y'')(^Qxy) = l2xy{x + y)\ This = o? — {b — cf^ {a + b - c){a~b + 6). 4. Resolve (a^ + ^>^/ - (a^ - ^^7 - (a^ + ^^ - c^. This=:4a^^^2-(a2 + ^2-c7 =: (2a& + a^ + ^' - ^') (2a5 - a^ - Z>^ + e^). The former of these factors = {a + by-(^ = {a+b + c){a + b-c)', and the latter =^ c" - {a-by = {c + a - b){c - a + b). .*. the given expression = {a+b + c)(a + b — c){c + a-b){G — a + b). Ex. 26. 9. 81a*- 1. 10. a'~lU\ 11. a}^-¥\ 12. a'-^>2+2^>(?-cl 13. (a + 2^)2- (3:^ -4y)^ 14. {x' + yy-4Lx'y\ 15. (:r + y)'-42^ 16. {^x + by-{bx + 2>)\ 1. 49^2 _ 4^2^ 2. 9a'-\b\ 3. 81a*-165^ 4. 100:^^-36yl 5. 5a^5-205a;^yl 6. 9a;^-16y^ 7. ^c'-l. 8. ^y'-ix'z\ 80 FACTOEING. I 17. \x'y'-{x'-^f — z'')\ 18. {p(? -\- xy — y^y — {pi? — xy — y'^y, 19. {x'' — y'' + zy — ^x^z\ 20. {a + h + c + dy~{a-h + c-d)\ 21. {2+'^x + ^xy — {2 — ?>x + 4:xy. 22 . {p?+y + ^ ahy - (a' + h^)\ 23. (a^ - ^2 + c^ - d'^y -(2ac-2 hdy. 24. (a;2-3/2-27-4y'2l 25 . (a« - a^ Z)^ + bj - (a« - 5 a^ ^)^ + bj. 26. a^' - Z^^' + 6a'b' -6b^a' + 8b^a'-SaFb\ 27. (x'^ -\- y'^ + '^ — ^y — y^ — ^^y — ip^y + yz + zxy, 28. (x'' + y^ + z^ — 2xy + 2xz — 2yz) — (y + zy, 29. 2a'Z>' + 2Z>'c' + 2c'a'-a* — 5^-c\ 30. :z;* + 2/* + 2* — 2a;'3/' — 2y'2' — 22'ar^. Formula [A]. (a7+r)(:r+s) = x^-^(r^s)x-\-rs. Examples. 1. ^'«-9a;H20-(:r^-5)(r^^ + 4). 2. {x-yy-^x-y-\\^-=^{x-y-^\\){x-y~\^). 3. (a^ - a^) + ^^7 + 6^)(a' - a5 + 5^) - 4a^ + 9^>'^ - [(a^ - a5 + b^) + (2a + 3Z))] X [(a' - a5 + ^>^) - (2a - 3^)]. 4. (ar^-5^y-6(:?;^-5a;)-40 = (a;^ - 5a;+ 4) (^ - 5^ - 10). 5. {ax -\-by-\- cy — {^n — n) {ax -\-by + c) — mn = {ax + by + c — m) {ax + by + c + n). i FACTORING. 81 § 14. It will be seen that the first (or coiniiiori) term of the required factors is obtained by extracting the square root of the first term of the given expression, and that the other terms are determined by observing two conditions : I. Their product must equal the third term of the given expression. II. Their algebraic sum multiplied into the common term already found must equal the middle term of the given expression. Hence, to make a systematic search for integral factors of an expression of the form x^:^hx±c, we may proceed as follows : 1. Write down every pair of factors whose product is c. 2. If the sign before c is +, select the pair of factors whose suTu is b, and write both factors x^, if the sign before ^ is + ; ^— , if the sign before b is — . 3. But if the sign before c is — , select the pair of factors whose difference is b, and write before the larger factor x-\- or X—, and before the other factor x— or rr+, according as the sign before b is -j- or — . Examples. 1. x'^-\-^x-\-20. The factors of 20 in pairs are 1 and 20, 2 and 10, 4 and 5. The sign before 20 is +; hence, select the factors whose swtn is 9. These are 4 and 5. The sign before 9 is +; hence, the required fac- tors are (r^+4)(a; + 5). 2. x^ — d>x-\~12. Pairs of factors of 12 are 1 and 12, 2 and 6, 3 and 4. Sign before 12 is + ; therefore take the pair whose sum is 8. These are 2 and 6. Sign before 8 is — ; hence, the factors are (x — 2) 82 FACTORING. 3. ^i''^- 21a; -100. Pairs of factors of 100 are 1 and 100, 2 and 50, 4 and 25, 5 and 20, 10 and 10. Sign be- fore 100 is — ; therefore, take the pair whose differ- ence is 21. these are 4 and 25. The sign before 21 is — ; therefore, x— goes before 25, the larger factor, and the factors are (a;+4)(a;— 25). 4. x" + 12a; - 108. Pairs of factors of 108 are 1 and 108, 2 and 54, 3 and 36, 4 and 27, 6 and 18, 9 and 12. Sign before 108 is — ; therefore, take the pair whose difference is 12. These are 6 and 18. Sign before 12 is + ; therefore, x-\- goes before 18, the larger factor, and x— before 6, the other factor. Hence, the factors are (a;— 6) (a; -f 18). Note. It will be found convenient to write the factors in two columns, separated by a short space. Taking Exam. 2 above, pro- ceed thus : Since the sign of the third term is +, write the sign of the second term (in this case — ) above both columns. 1 12 (x - 2) {x - 6) Exam. 3 above. Since the sign of the third term is — , write the sign of the second term (in this case — ) above the column of larger factors, and the other sign of the pair, ±, above the other column. + - 1 100 2 50 (a; + 4) {x-2b) 5. a;' -34a; + 64. Here we have the factors 1 64 a; - 2 a; - 32 4 16 And since the last term has the sign +, and the middle term has the sign — , we write — over both columns. FACTORING. 83 6. x'+l2x-Q4:. + 1 64 2 32 X — 4 ^+16 Here, since the last term has the sign — , we write the sign (+) of the middle term over the column of larger factors, and the sign — over the other column. 7. ^*-10:x;'-144. Here we have the pairs of factors + 1 144 2 72 4 36 ^ + 8 ^^ - 18 And since the sign of the third term is — , we write the sign of the second term (in this case — ) above the column of larger factors, and the other sign (of the pair, ±) above the other column. Ex. 27. 1. x^-bx—\^\ ^2 - 9;r + 14 ; x^-^-lx-^ 12. 2. :r'-8:?; + 15; ri;'— 19a; + 84; ^'-7;r-60. 3. 4^'-2:r-20; 9:^^-150:^ + 600. 4. \x^-\-\\x-m', 25:^2 + 40:?; +15; 9 :^« - 27 :z;H 20. 5. ^x^-\-\\x^Vl\ 16 a;* -42^' -20. 6. x^-(d-\-})')x^-\-a^h'", ^{x-\-yf-^(x-^y)-m. 7. {x^ + fj - {(f - h"-) {x" + %f) - o} h\ 8. {a + hy~2c{a + h)-?>c\ 9. (:. + yy + 2{x' + f) {x + y) + {x^ - yj. 84 FACTORING. 10. {a + hy-^ah{a + b) - {p} - hy. 11. {x^ + xy + y^ + ^"^ — y^ — ^^y ~ 2y^ — 2r?;^ 12. a' -2a(h — c)-S(b— c)\ 13. (a;' + y7 + 2a'(:^^ + /) + «'-^'. 14. (.^^-10.r)^-4(ri;'^-10:i;)-96. 15. (:r^ - 14:?; + 40)^ - 25 (^-^ - Ux + 40) - 150. 16. (x' — xy + 7/y + 2xy(x'' — ri^y + /) - Sx'2j\ 17. 2^- 3^' + 2; r?;*-2:r'-3; 9x' + 9x'y' -~ 107/\ 18. c'"^ + c"^ - 2 ; ^« - a;' — 2 ; x'''' - 2x'^y'' - Sy''\ 19. ^^'"' — (a — b) x'^y"" — aZ?/^\ § 15. Trinomials of the form ax"^ -J- bx -{- c (a not a square) may sometimes be easily factored from the following con- siderations : The product of two binomials consists of 1. The product of the first terms ; 2. The product of the second terms ; 3. The algeh^aic sum of the products of the terms taken diagonally. These three conditions guide us in the converse process of resolving a trinomial into its binomial factors. Examples. 1 . Eesolve 6 x'' — 13 :ry + 6 tj\ Here the factors of the first term are x and 6^, or 2x and 3:?;; those of the third term are y and 6?/, or 2y and 3?/. These pairs of factors may be arranged : (1) (2) (3) (4) x 2x y 2y ^x ^x 6y 3y FACTORING. 85 Now, we may take (1) with (3) or (4), or (2) with (3) or (4) ; but none of these combinations will satisfy the third condition. If, however, in (4) we inter- change the coefficients 2 and 3, then (2) and (4) give 2x 3y and ?>x 2y, where we can combine the " diagonal " products to make 13, and the factors are and '^x — 2y. The coefficients of (2), instead of those of (4), might have been interchanged, giving the same result. 2. 6^^~15^y + 63/^ Here, comparing (2) and (3), Exam. 1, we see that their diagonal products may be combined to give 15, and the factors are 2x — y and 3 a; — 63/. 3. 6a;' — 20a;y + 6yl Here, again referring to Exam. 1, we see at once that it is useless to try both (2) and (4), since the diag- onal products cannot be combined in any way to give a higher result than l?>xy. But comparing (1) and (4), we obtain, by interchanging the coefficients in 4, ^_^y and 6a;— 23/, which satisfy the third condition. Or, we might interchange the coefficients of (3), and take the resulting terms with (2), getting 2a; — 6y and 3 a;— y. 86 FACTOEING. Here the large coefficient of tlie middle term shows at once that we must take (1) and (3) together. Inter- changing the coefficients of (1), we have ^x— y and x-\~Qy. The same result will be obtained by interchanging the coefficients of (3). Qx^ — l^xy ~ 6y", 6x'^-{~bxy — 6y'^. b6x'' + 562xy + 20y\ b6x'' — 122xy + 20y\ 56x'' — 102xy — 20y\ b6x''-229xy + 20y\ 56^'-94:ry + 20yl 18. 56x' — 276xy-207/. S6x'-S3xy-157/. 20. 72x^~19xy-A0y\ § 16. Generally, trinomials of the form ax^ + bx-^-c (a not a square) may be resolved by Formula [A] ; thus, Multiplying by a we get a^x^ + hax + ac. Writing z for ax, this becomes z^-{'hz-\- ac. Factor this trinomial, restore the value of z, and divide the result by a. Examples. 1. 6:^-' + 5^ — 4. Multiplying by 6, we get (6:r)2+5(6^)-24 or z''+5z-24:. Factoring, we get (z — 8) (2; + 8) ; hence, the required factors are |(6a;- 3) (6x + S) = (2^-1) (3^ + 4). Ex. 28. 1. 6x'-S7xy + 67f. 11. 2. 6af + 9xy-6y\ 12. 3. 56x'-16xy + 20y\ 13. 4. b6x''-d6xy-20y\ 14. 5. 56a;^-1121a:y + 203/1 15. 6. b6x'-6Sxy + 20y\ 16. 7. 56x'-b5Sxy--20y\ 17. 8. 66x'' + Se>xy-20y\ 18. 9. b6x'-67xy + 20y\ 19. 0. 56x' + Sxy-20y\ 20. FACTORING. 87 Factoring 2;^ — 13 2;?/ + 36 y^, we get (2; — 4y) (2; — 9y) ; hence, the required factors are i(6^-4y)(6x-93/)-(3a;-2y)(2a;-32/). 3. 33 — 14:^-40 rrl Factoring 1320 - 14^ — z\ we get (30 - 2;) (44 + z) ; hence, the required factors are ^V(30- 40^) (44 + 40a;) -(3 -4a;) (11 + 10^). Note. The factors may conveniently be arranged in two columns, each with its appropriate sign above it. - + Exam. 1, above : 1 24 ■ 2 12 J(6a; - 3)(6a; + 8) = {2x - l)(3a; + 4). Exam. 2, above : 1 36 2 18 3 12 • J(6a;-4)(6a;-9) = (3a;-2)(2aj-3). Another method of factoring trinomials of the form ax^ -\-hx -\- c is as follows : Multiply by 4 a, thus obtaining 4 a''^a;2 + 4a5a; + ^ac. Add h^ — b^, which will not change the value, 4a^a;^ + 4:abx + 6^ — 6^^ + 4ac ; by [1] this may be written (2 ax + bf — (b'^ — 4ac). Factor this by [4], and divide the result by 4 a. Example. Factor 56 x"^ + 137 a;- 27,885. Multiply by 4 x 56 or ^2 X 112, 1122 a;2 + 2 X 137 X 112a; - 6,246,240. Add 137^ - 137^ then, 1122 a;2 + 2 X 137 X 112 a; + 137^ - (137^ + 6,246,240) = (112a; + 137)2-6,265,009 = [(112a; + 137) + 2503] [(112a; + 137) - 2503] = (112a; + 2640)(112a; - 2366). We multiplied by 4 X 56 ; we must, therefore, now divide by that number. Doing so, we obtain as factors (7 a; + 165) (8 a; — 169), FACTORING. Ex. 29. 1. 10a;^ + rr-21. 8. 15a' +lSa'b' -20b\ 2. 10^*'-29:z;-21. 9. 12^2-^-1. 3. 10^' + 29:r-21. 10. O^r^y' — 3rry' — 6y*. 4. 6:r' — 37^ + 55. 11. 40;^ + 8;2;3/ + 3yl 5. 12a' -5a -2. 12. 6 5^ :r^ - 7 ^^^ - 3 rz;^ 6. 12^^ — 37;r+21. 13. 6x' - x'f -S5y\ 7. 12rr2 + 37a; + 21. 14. 2^* + ^^-45. 15. 4:x'-S1x''f + 9i/\ 16. 4(^ + 2/-37^(^ + 27 + 9r?;^ 17. 6(2x + 3yy + 5(6;r^+ 5:ry - 6f) -6(Sx~ 2y)\ 18. 6(2:?; + Zyf + 5 (6:?;' + 5xy — ^y^J -^{?>x- 2y)\ 19. ^{x''+xy+yy+l?>{x'+x^y''+y')-^^5{x'-xy+y'')\ 20. 21 {x" + 2xy+ 2^^)'- 6 (a;'- 2a:y + 2/)'- 5 {x^+4:i/). Extended Application of the Formulas. § 17. The methods of factoring just explained may be applied to find the rational factors, where such exist, of quadratic multinomials. Examples. 1. Kesolve 12:?;2 — xy — 20^/' + 8^ + 41y - 20. In the first place we find the factors of the first three terms, which are ^x-\- 5y and ?>x — 4:y. Now, to find the remaining terms of the required fac- tors, we must observe the following conditions : FACTORING. 89 1. Their product must = — 20. 2. The algebraic sum of the products, obtained by multiplying them diagonally into the y's, must = il y. 3. The sum of the products, obtained by multiplying them diagonally into the x^, must = ^x. We see at once, that —4, with the first pair already found, and + 5, with the second pair, satisfy the required conditions ; and hence the factors are 4rr+53/ — 4 and 307 — 4?/ + 5. . p' + 2pr-2q^ + lqr-?>r''+pq. Here the factors of^^ -\-P9. — 2^^ are p + 2q and P~ ^^ Now find two factors which will give — 3 r^, and which, multiplied diagonally into the ^'s and ^-'s respec- tively, will give 2pr and 7 qr ; these are found to be — r, taken with the first pair, and + 3 r, taken with the second pair. Hence, the required factors are p-\-2q — T and p — q + ^r. The work of seeking for the factors may be conveniently arranged as follows : 3. x^ + xy~'2.y'' + 2xz + lyz-^z\ Keject : 1. The terms involving z ; 2. The terms involving y ; 3. The terms involving x ; and factor the expression that remains in each case. 90 FACTORING. 1. x^^xy-'2.y'^(x — y){x-\-'2.y). 2. x^-\-1xz-'^z^ = {x + Zz){x-z). *3. -2y^ + 73/z-3.^=(--y + 3^)(23/-.). Arrange these three pairs of factors in two sets of three factors each, by so selecting one factor from each pair that two of each set of three may have the same coefficient of x, two may have the same coefficient of 2/, and two the same coefficient of z (coefficieyit includ- ing sign). In this example there are x~y, x + ?>z, -y + ^z, and x-\-2y, x—z, 2y — z. From the first set select the common terms (including signs) and form therewith a trinomial, x — y -^-Sz. Eepeat with the second set, and we get x-{-2y — z. .'. x'^ + xy- 2y''+2xz + lyz - Sz' ^(x-y + Sz)(x + 2y~zy 4. 3x''~8xy-~3y'' + S0x + 27, 1. Sx''-8xy~Sy'' = (Sx + y)(x~dy). 2. Sx'' + 30x + 27 = (3x + ^)(x + 9). 3. -Sf + 27 = (y + 3)(-Sy + 9). .'. the factors are (3^ + ?/ + 3) (r?; — 3y + 9). 5. 6a'-7ah + 2ac~20b' + QUc-A8c\ 1. 6a^-7ab~20b''=:(2a~bh)(Sa + 4:h). 2. 6a' + 2ac--^Sc' = (2a + 6c)(Sa-Sc). 3. -20h' + e>4:hc-4:Sc' = (-5b+6c)(4:h-8c). .-. the factors are {2a-5b + 6c)(Sa + 4cb~-8c). To find, where such exist, the factors of ax"^ + bxy + cxz -\- ey^ + gyz + hz^. Multiply by 4 a : 4 a^ :r^ + 4 abxy + 4 acxz + 4 aey"^ + 4 agijz + 4 ahz^. FACTORING. 91 Select the terms containing x, and complete the square : thus, 4a^;r^ + 4:abx7/ + ^acxz + b'^if + 2bcxz + c'^z'^ - (b' - ^ae)f -2(hc- 2ag)yz - (c^ -4:ah)z^ = {2ax + by + czy - [(^^ - 4ae)y' + 2{hc- 2ag)yz + {c" - ^ah)z^\ If the part within the double bracket is a square, say {r)iy + nzy, the given expression can be written (2 ax -\- by -{■ czy — {my + nzy, which can be factored by [4]. Factor and divide the result by 4 a. If the part within the double bracket is not a square, the given expression cannot be factored. If h and c are both even, multiply by a instead of by 4 a, and the square can be completed without introducing fractions. If e be less than a, it will be easier to multiply by 4e instead of by 4 a, and select the terms containing y. A similar remark applies to h. This method can evidently be extended to quadratic multinomials of any number of terms. I] I Examples. 1. Eesolve x^ -\- xy '\- 2xz — ^y"^ -\- lyz — ^z^ into factors. Multiply by 4 : 4a;2 + 4.xy + ^xz - ^f + 2^yz - 12^1 Complete the square, selecting terms in x : A:x' + 4,xy + Sxz + y'' + ^yz + 4:z^--'^y'' + 2^yz-l^z^ = {2x + y + 2zy - (3y - 4^)^ = [{2x+y+2z) + {^y~^z)][(2x+y+2z)-{2>y-^z)-\ -= {2x+^y-2z){2x-2y+Qz) - ^{x+2y-z){x~y+Zz). .'. the factors are {x -{- 2y — z) {x — y -{■ ^ z). 92 FACTORING. 2. 6a'~7ah + 2ac-20b' + 64:bc-4:Sc\ Multiply by 4 X 6 = 24 : 144 a^ - 168 aS + 48a^ - 480 ^^ + 15365(7 - 1152 c^ = (12a - lb + 2cy - 529 5^ + IbQUc - llbQc' = ll2a-7b + 2cy-(23b-d4:cy = (12a+ 165 - 32c)(12a - 305 + 36c) = 24(3a + 45-8c)(2a-55 + 6c). .*. the factors are 3a + 45 — 8c and 2a — 55 + 6^?. 3. x'' + 12xi/ + 2xz + 262f — S7/z-9z^ =-- (x'+ 12xy+2xz+36y'+ i2yz+z')-10f-207/z-l 0/ = (x + e^ + zy-[(y + z)^-10f = [x+(6+^10)y + (^10 + l)z] x[:.+ (6-viO)y-(VlO-l)4 4. 3a' + 10r^5 - Uac + 12ac?- 85- - Sbd+Sc' -8crl Multiply by 3, not 4x3, since the coefficients of the other terms in a are all even : 9a' + 30a5 - 42ac + 36ac^ - 245' - 245^+ 24^'- 24:cd. Select the terms containing a, and complete the square : (3a + 5b-7c + 6dy -A9b'+70bc-S4:bd-25c' + 60cd-36d' = (3a + 5b-7c + 6dy-(7b-5c + 6dy = l3a+12b-12c + 12d)(Sa-2b-2c) = S(a + Ab~4:C + Ad)(3a-2b-2c). .'. the factors are a + 45 — 4c + 4(i and 3a — 25 — 2c. Ex. 30. 1. 7a;'-:i;3/-6y'-6:r-20y-16. 2. 20:r'-15:ry-53/' — 68.^ — 422/ -88. 3. 3:z;* + a;'y'-4y' + 10a;'-17y'-13. 4. 20^'-20?/' + 9a:3/ + 28.r + 35?/. FACTORING. 93 5. 72^'-8y' + 55^y+123/-169a; + 20. 6. x^ — xy —\2y'^ — bx — Iby. 7. ^x^+l^xy + ^y'' + 2xz~z\ 8. ^x^ + ^y''-l?>xy-^z'-2yz + ^xz. 10. Ibx' - IGy* - 22^'y'' + Ibz' + \^fz^ + mx'z\ 11. ^a^-\h¥ — ^ab-'2X&~?>Uc-^ac. 12. a^ + ^' + c'-2a^Z^^-2Z)^c^-2c^al § 18. Trinomials of the form ax^ -\-hx^ -\- c can always be broken up into real factors. If a and c have different signs, the expression may be factored by § 16. If a and c are of the same sign, three cases have to be considered : (i.) Z>-2V(«^)- (ii.) Z)>2VM. (iii.) h<2-^{ac). Case I. h = ^ ^{ac). This case falls under § 12, For- mula [1], where examples will be found. Case II. Z> > 2 -yj(ac). This case falls under § 16, where examples will be found. The following additional examples are resolved by the second method of that section : Examples. 1. ^x' + bx^y''+y\ Here we see that {^y'^y will make, with the first two terms, a perfect square, and we therefore add to the given expression, (f y^)^ — (|■3/^)^ 94 FACTORING. The expression then becomes = (2x^ + ^yJ-^y^ = (2x' + 2f)(2x^ + ^f) = {x' + y')(Ax' + f). 2. 3x'+6x' + 2. Here, multiplying by 4 X 3, and completing the square as in Exam. 1, we have 36a;* +72^^62 + 24 -6^ -(6:^^ + 6)2-12 = l6x' + 6~ V12) (6^2 + 6 + V12), which, divided by 4 X 3, gives the required factors. 3. ax* + hx'^ + c. Proceeding as in Exam. 2, we have, by multiplying by 4 a, ax^ -}-bx'^ -\- c = (4:a''x' + 4:abx''+h^ - 5^ + 4a(?) -^ 4a ^ [2ax''+b+'y/(b'-Aac)] [2ax''+b-^(b'-4:ac)] -^4a. Ex. 31. 1. x^ + lx' + l] ^x' + Ux' + l. 2. x*+7x''y'' + y'] 3x' + 5x''y^ + y\ 3. 4:x' + lOx'' + S] Six + yy + bz'ix + yy + z', 4. x'+7x'^y' + S\y'; x' + 7x'i/ + S^7j\ 5. 4:x' + 9x^y' + {iy'] 4:(a + by +10c'(a + bf + 3c\ 6. Sx' + Sx'^y'+^-^y'] 36:r* + 96r^ + 55. 7. 5:r* + 20:c'^ + 2; 4a*+12a2 + l. 8. 4:(x + 7/y+12(x + yyz' + z'] 5x' + 20x'y' + 2y\ FACTORING. 95 9. 9x'+Ux' + 4t; 2x' + 12x'(7/ + zf + lb(y + z)\ 10. 2:r*+12r?;^ + 15; 7x' + ^0x'' + 4:5. 11. 8:r* + 36r?;'3/' + 293/^ 7^* + 20:?;'3/'- 20?/*. 12. 7(a-^)* + 16(a-^7c^+5c^ fa* + 3a^^>^ + ^^ 13. Sx' + 6x'f + 2y'', S(ia + by + 6(a^~bJ + 2(a-by. 14. 49a*-84a^Z^^ + 22&^ 25m* + 60m^7i^ + 27?i\ , 15. 49(m + ny-84(m2-n7 + 22(m-7z)*. i ^ Case III. b < 2-y/(ae). This case may be brought under § 13. The following examples illustrate the process of reduc- tion and resolution : 1. a:^-7a^ + l. Examples. ^ We have to throw this into the form d^ — b'^: I ' x'-7x''+l = (x' + iy-9x^ =- (^^ + 1 + Sx) (x' + l- Sx). 2. 9x' + Sx'f + 4y* = (Sx' + 2y7 - 9x'y'^ = (3:r' + 2y' - dxy)(Sx' + 2y'+3xy). 3. :t^* + 3/* = (:r^ + y7~2a;^3/^ .^ (^2 _^ ^2 _,_ ^^ y2) (x^ + y^ - ^3/ V2). == (^' + y' + f ^y) (^' + 3/' - f ^y). 5. ax' + bx' + c= (x'^a + V^)'- (2 V^- ^) x" = lx'^a+-^c~C<2V^c-b)xl § 19. It is seen from these examples that we have merely to add to the given expression what will make with the first and last terms (arranged as in Exam. 5) a perfect square, and to subtract the same quantity. In Exam. 2, I 96 FACTORING. for instance, the sq uare root of 9x^ =^3 x^, the square root of 4 3/^^=2 y'^; hence, ^x^-\-2y'^ is the binomial whose square is required; we need, therefore, 12x'^y^] but the expression contains 2>x'^y'^\ hence, we have to add and subtract 12x^y'^ — ?>x''y^=9x^y\ Hence, we derive a practical rule for factoring such expressions : 1. Take the square roots of the two extreme terms, and connect them by the proper sign ; this gives the first two terms of the required factors. 2. Subtract the middle term of the given expression from twice the product of these two roots, and the square roots of the difference will be the third terms of the required factors. 6. x' + ^x''f + y\ Here -y/:r* = x^, y/y^ ~ y^ and the first two terms of the required factors are x^ + y'^ ; twice the product of these is +2ri;^y^, from which, subtracting the mid- dle term, -j^^^y^, we get -f-|-:r^3/^ ; the square roots of this are ± \xy. Hence, the factors are 0? -\-y'^ ^ \xy. Note that, since ^y'^ =+3/^, or — y^, it may sometimes happen that while the former sign will give irrational factors, the latter .will give rational factors, and con- versely. 7. x^ —Wx^y'^ -\-y^. Here, taking +3/^, we have ^^ + 3/^ + ^y V^^ ^^^ x^ -\- y'^ — xy y'lS. But, taking — y^, we have x^ — y"^ + ^ocy and x^ — y'^ — Z xy. Sometimes hoth signs will give rational factors. FACTORING. 97 8. l^x'-llx^tf + y\ Here we have (4^;^ + y^ + 3 xy) (4:r^ + 3/^ — 3 xy), and also (4 x^ — y^ + 5 xy) (4a;^ — y^ - - 5 xy). tEx. 32. 1. x' + 2x^y'' + 9y'] x' ~ x'' y'' + y' ; x' + x^y'' + y\ 2. x' + 4:y'] IQx' + y'-xy') \x' + y\ 3. .2;*+l; ^' + 9y^ 1-12^^ + 16/. 4. :r*-7:r2 + l; :r* + 9; ^^r^ + y'- 30:^1 5. y'-x'+llxHf', ^^4/; ^* + 4^2 + 16. 6. ^x' + y'-^\x'y''] x' + y' -^x'^y'' ) 4:x' + l. 7. ^'"^+64/"*; ^*"* + 43/'"*; i^^ + i^y* — 5f :r'?/\ 8. 4^'-8;r2 + l; 7^^' - i^r* - 36/ ; x^ + aUj\ 10. 16a;*- 2507^ + 9; 4:r' - 16^^;^ + 4 ; 13:i;y - 9:r*-4y^ 11. 4^*-12f|a;'y2 + 9/; x' + ^x' + ^b. 12. a* + 5* + (a + ^y; l + a* + (l + a)\ 13. {x + yy-1z\x + yy + z\ 14. (a + ^)* + 7c' (a + 5)2 + g\ 15. 16a* + 4(5-c)*-96i'(5-c)^ 16. 4(a + 5y + 9(a--5)*-21(a2-52)^ 17. {x' + f-xyy-1{x' + yj + {x + y)\ 18. (a^ + a5 + hj + 7 (a^ - Z>^)*^ + (a - ^>)*. 19. 16a* + 4a2 + l; r^*-41:r2+16. 20. (a2 + iy + 4(a2 + l)2a2 + 16a^ I {x+lf + ^^x' - ly + ^(x-l)\ FACTORING. §20. We can apply [4], § 13, to factor expressions of the form ax^ + ^^^ + ^^^ — '^'^<^- This may be written, a (x^ — r'^) -{- bx (x'^ -{- r) =^ [a (x^ — r) + hx'\ {x^ + r). Examples. 1. 6:^' + 4^^+12;r-54 ^^{x'-^) + 4:x{x'' + ?>) = (^^ + 3)[6(:r2-3) + 4:r] = (^2 + 3)(6a;^ + 4:z;-18). 2. 11^*+ lOa;^- 40^27 -176 = ll(^^-16) + 10:^(:r^-4) -=(:r^-4)[ll(^^ + 4) + 10:i;] = (:r^-4)(ll^'^ + 10:r + 44). 3. ^Ox' + ?>0x^ + mx -im = 10(4:r* - 16) + lbx{2x'' + 4) = {2x^ + 4) [10 (2a;^ - 4) + Ibx'] =:(2r^ + 4)(20a;^ + 15^-40). Note. To determine r, take the ratio of the coefficient of x^ to the coefficient of x. Ex. 33. Kesolve into factors : 1. x^ + 23(^ + ^x-^. 6. 10^* + 6:2;'+ 30^ -360. 2. 2:?;* + 2a;' + 6^ -18. 7. i.r* + 20:?;' + 4:?;- y^. 3. ri;* + 3^'+12:2;-16. 8. 25:?;*- 40^' + 8^- 1. 4. 3:i;* + ^'-4rr-48. 9. ?>1^x'-~^0x^+^^x-^Q, 5. 5:r* + 4:^' -12.^-45. 10. 63ri;*-39a;'+52a;-112. 11. d>\0x' + ^-^x^ + ^x-2^. 12. 2i2a;*- 33a;' -3:^-2. FACTORING. 99 13. \x' + i-^x'~^x~^\. 14. ^0x'~?>2a^y + e)4:xij — ^20y\ 15. 24:x'-l2x^y + ?>0xf~imy\ 16. 2x' + \x^y-^xf-bl2ij\ 17. ll:?;*+10:r'-12.'?;-15|l. 18. 40 :t'^ + 30 r' + 60^' - 160. 19. I?>x'-l2x^y+12xf-^^d>y\ 20. ?>x' + dx^y+12xi/-4cd>t/. 21. bx^ + 4^^y — 120;?/^ — 45?/^ 22. 4:2.-* — 14:r> + 28a;/-16/. 23. a;* + 80:?;V+16^y3-2Vy'. 24. 2x^~x^y + ^x2/-12y\ § 21. Formulas [1] and [4] may sometimes be applied to factor expressions of the form ax'^ + hx^ + cx^ + rhx + r^ a. This may be put under the form a (^'* + t") + hx (x' + r) + ex' = a(x'^ + ry + ^:z;(^^ + r) + {c-- 2ar)x', which can sometimes be factored. Examples. 1. :t'* + 6 r'^ + 27:?;' + 162:^+729. ^4_f_ 729 + ^x{x' + 27) +27:r' = {x" + 27)' + ^xix" + 27) + 9^' - 36:^' = (V + 27 + 3:r)' - 36 a;', which gives the factors x''-Zx-\- 27 and :?;' + 9^ + 27. 100 FACTORING. '= (x' + 5/ + 4:x(x' + 5) - 6x' = (x' + 5)^ + 4:x(x' + 5) + 4:X' ~ 10^^ = lx' + 5 + 2x- ^Vl(^) (^'' + 5 + 2^ + ^z^VlO). Ex. 34. Kesolve into factors : 1. x'-6x^ + 27x^' — 162x+729. 2. x' + 2x' + ^x'' + 8x+16. 3. ^* + ;r' + ^' + ^-f 1. 4. :z;* — 4:?;^ + ^?;^ — 4:?;+ 1. 5. 4^*-12^^-6^^-12:r + 4. 6. x'+Ux'-25x'-70x + 25. 7. 16:r* — 24:r^-16:r'+12:r + 4. 8. a;* + 5^'-16;r' + 20:?;+16. 9. ^* + 6:?;^-ll:z;'-12:?; + 4. 10. ^* + 4^'y + ^'y' + 12:r/ + 93/*. 11. x^ + 6x'-9x'" — 6x + l. 12. r^* + 4:z;^?/ — 19:?;^2/' + 4rry^ + y*. 13. 4:r* + 4:?;V-65^'^3/'-10:r/ + 25y*. 14. :z;* + 6:?;'y-9:r'y'--6:r/ + 3/\ 15. rr* + 6^V+10:^V' + 12^3/3 + 4/. 16. 9x' + 18x''7j-b2x''7/-12xf + A7/. 17. ll^* + 10:?;'y + 393^^V2+205;/ + 44y*. Factoring by Parts. § 22. To factor an expression which can be reduced to the form a X F(x) + b Xf(x). (L FACTORING. 101 When the expression is thus arranged, any factor com- mon to a and h, or to F{x) and /(a;), will be a factor of the whole expression. The method about to be illustrated will be found useful in cases where only one power of some let- ter is found. Examples. 1. Y^^qXqy acx^ — ahx — hc^x-^lp-c. Here we see that only one power of a occurs, and we therefore group together the terms involving this letter, and those not involving it, getting a {cx^ — hx^ — bc^x -{-Ifc = ax (ex — b)~be (ex — b)~ (ax — be) (ex — b). 2 . Factor m^ x"^ — mna'^ x — mnx -f- ri^ a^. Here we observe that a occurs in only one power (a^). Therefore, w^e have — a^ (mnx — n^) + I'n? x^ — mnx = — no!- (nix — n)-\- mx (mx — 71) =. (mx — n) (mx — na^). 3. Factor 2:^' + 4aa; + 3Z'^ + 6a^. Here we observe that the expression contains only one power of both a and b. We may, therefore, collect the coefficients in either of the following ways : a(4.x + Qb) + (2x' + ?>bx), or 5 (3 a; 4- 6a) + (2x^ + 4.ax). Now the expressions in the brackets ought to have a common factor ; and we see that this is the case. Hence, a(4:X + 6b) + (2x'+Sbx) = 2a(2x+3b) + x(2x + Sb) = (2x + 3b)(x + 2a). 102 FACTORING. 4. abxy + IP'y^ -\- acx — & — aihxy + ex) + Z^^y^ — & -=^ ax(hy + ^) + ihy + ^) (% — c) ^ {hy + c){ax + hy — c). = — 5 {y' - 2ay + a') + / - 2ay' + a?y -=-h{f ~2ay + a') + y{y'^ ~2ay + a') = (y~'b){y-a)\ 6. 2^^y + 2 hx^ ~ bx^y + 4 ahx'^y — ^^y^ + 4 a^y^ — 2 ahxy'^ — 2 ay^ = h (2x^~x^y + ^ax^y ~ 2axy'^) + 2ri;^y — ^r^y^ + 4a:ry^ — 2a3/^ = Z>a;(2:r^ -- ^r^y + ^axy — 2a?/^) + y (2rr^ — :?;^y + 4 axy — 2 ay^) = (y + hx) (2x^ — x'^y-{-4: axy — 2 ay"^). And 2a;^ — :?;^y + 4aa;y — 2ay'^ = a(4:xy — 2y^) + 2c(^ — ^^y — 2ay {2x — y) + ^'^ i2x~y) = {2ay + x'){2x-^y). 7. ^'+(2a~^)a;'-(2a^-a'):r-a'^ = ^ (— rr'^ — 2 aa; — a^) + ^r^ + 2a:2;^ + a?x =^ — h{x-\-ay + x(x-\- ay -={x-h){x + a)\ 8. px^ — {p — q)x'^-\-{p — q)x + q = q(x^~x + l)+px^~px'^+px = q(x'^ — x+ l)+px(x'^~x + l) = lpx+q)(x''-x+l). Ex. 36. 1. x'^y — x'^z — y'^ + yz. 3. :r^2;^ + a:r^ — a^z;'^ — a'. 2. abxy + b'^y'^ + acx — c"^. 4. 2^;^ — aa; — 4^a; + 2a5. FACTORING. lOS 5. x' + 2hx + Zax + Qab. 8. ^x'+l2ax+lQihx+lbab. 6. x'~h''x'-a'x + o?h\ 9. a^ + {ac~h'')x'+hcx\ 7. x''-o?x''-h''x^ + o^h\ 10. a^ + {ac~h'')x''-hcc(^. 11. ahx^ + (ac — hd)x'^ — {af+cd)x + df, 12. j)^'^ — (p + q)x^ -\- {p-{- q) X — g'. 13. d' + ab + 2ac~2b''+lhc -~^c\ 14. r'+(a+l).T' + (a + l)r?; + a. ^ 15 . Ttipx^ + (m^' — 77p) x^ — {inr + ^$') x + n?\ 16. :l^^ — (a + ^ + c?) o;^ + (ab -{-bc-\- ac) x — abc. 17. x^ + {a — b~c)x^~ {ab — bc + cd) x + abc. 18. x^ -{-{a-\-b — c) x^ — (5c? — ca — ab) x — ahc. 19. a!' 01? — a^x^y — a^xy + c^y'^— ax^yz + 0:^2; — ^^2; + ay'^^;. 20. a^ bx^ + a^^^^y + acdxy + 5c(iy^ — aefxz — ^e/ya;. 21. a^x^ — a(b — c)x'^ + c{a — b) x + c^. 22 . ??ia;^ — n:r^ y + r:r^ 2; — mxy'^ + ny^ — ry^ z. 23. amri;^ + (m5y — nay + mc2;) :r — nby'^ — ^ic^ya;. 24. {am — &cm) r^^ + {am — ben) x-\- an-{- nax. 25. db'^ & — y^&xy — d^&yz + c^xy'^z — a^b'^zx-]- b'^x^yz -\-a^z^xy — x^y'^z^. 26. x^—m'^x^—{n—n^)x^ + {m^n—m^n^)x^ — a{x'^-\-n^—n). 27. 1 - (a - 1):?; - (a - ^) + 1)^;^ + (a + 5 - c?)a;^ — (5 + c) a;* + cx^. 28. a^x^ — a^{b — c+d)x'^y — {abc — abd-\-acd)xy^+bcdy^. 29. m^npx^ — {n^p — m^?^'^ — m^pq)o(? — {n? + 71^5' — 77^^ 92^) :z7 — n^q. 30 . m'^/)^ :?;^ + ni^p'^ x^ — {p'^n^ — (f m?) x^ y^ — {p^ n^ — (f m^) x^ y^ — (7^^ ^^ + ri'^ (fx) y^. 104 FACTORING. §23. Sometimes an expression whicli does not come directly under the preceding form may be resolved by first finding the factors of its parts. Examples. 1 . ahx^ + ^i^^y^ — ^^ ^y — ^^ ^y • Here, taking ax out of the first and third terms, and by out of the second and fourth terms, we have ax (bx — ay) — by (bx — ay)^ and hence {ax — by) (bx — ay). 2. x^ — (a + b)x^ + {d'b + ab'')x~a^h\ Here, taking the first and last terms together, and the two middle terms together, we have (x^ + ab) {x^ — ab) — {a + b)x^ + ab {a + b)x — (x^ + ab) (x^ — ab) — {a-\-b) x(x^ — ab) = (x^ — ab) [x'^ + ab~(a-\-b) x\ = (x^ — ab) (x — a)(x — b), 3. X^'^ — 4:X'^ + 3 ^ ^m|<^2m _ 1) _ 3(^»* _ 1) = x'^lx"' + l)(x'^ — 1) — S (x""' — 1) == (^rf — 1) [^™ (^"* + 1) — 3]. Ex. 36. 1. a^ — ab ~\- ax — bx. 7. a^ — b'^-j-ax—ac — bx~{-bc. 2. abx^-\-Wxy—(j^xy—ab\f. 8. a^ + (1 + a) ab + V^, 3. x^ + «^^ — dx — a*. 9. x^ -\-^xy {x^ — y'^) — y*. 4. a^x-{-2a'^x'^-\-2ax^-{-x^. 10. x^ — y^-\- x"^ -{-xy-^y'^. 5. acx' + (ad-bc)x-bd. 11. 2b + (b' - 4:)x-2bx\ 6. 25x'-6x' + x''-l. 12. x' + 3x^-4:. FACTORING. 105 13. p'-p'q — 2pq^ + 2q\ 20. a^ — 4:ah'' + ?>h\ 14. a^ + a' — 2. 21. a''" — 3 a"^ c'^ + 2 c?-^ 15. ?>a^b'-2ah''-~l, 22. ax^ - (a^ + h) x" + h\ 16. if — 2>y + 2. 23. 35a;'^— Ga'^r*^— 9a\ 17. 2a'-a'Z)-a^' + 2Z)l 24. a' ^)2+ 2 a^c^— a' c^- 5^ cl 18. 5'"* + Z)''" — 2. 25. arn?--ah'' + h''m — m\ 19. y^"— 23/'"2"— 2?/V"+c'^ 26. i— 6a2 + 27a^ 27. (:i'-3/7 + (l-n^ + y)Cr-y)2;-2l 28. 24m' — 28m'n + 6mn' — TtiI 29. rr'"+" + :^;"3/'* + a:"*3/"' + 3/"'+^ 30. rr'* + 207^2/ — o^o(^-{- x^'if' — 2aa;3/^ — y*. Application' OF the Theory of Divisors. § 24. By Theorem I. we prove that x^ — a"^ is divisible by a; — a always^ x^ — a" is divisible by :^ + a when n is even^ x"^ + a** is divisible by rr + a when 9^ is odd. By actual division we find in the above cases : ^— !^ r= x''-' + x^-'a + + m"-' + a«-^ (1) 5!+_^ = :r"-^ - x^'-'a + - xa""-' + a""^ (3) Examples. 1. Resolve into factors x^ — y'. Here x — y is one factor, and by (1) the other is x' + xy + y''. 106 FACTOEING. 2. Resolve a^ + (b — cf. Here a + (^ — (?) is one factor, and by (3) tlie other is a'-a(b-c) + (h-c)\ 3. Resolve rr^^+1024y^^ This equals (x^f -\-[(2yyY, one factor of which is x^ -{- (2y)^ and by (3) the other factor is (xj - ixy{4:2/) + (xji^yj - x\AyJ + (Aff = x'' - 4:x'y' + IQx'y' - 64.x' f + 256/. 4. Resolve (x — 2?/)^ + (2^ — 3/)^ into factors. Here, by (3), we have = {x-27/y-(x- 2y) (2a; - y) + (2a; - y)l .*. the factors are S (x — y) (7 x"^ — 13:ry + 7y^). 5 . Resolve rr^ + rr^y + ^^3/^ + ^^y^ + ^y* + 3/^. By (1) we see that this equals x^ — y^_ {x' + 'i/'){x^ — 'i^) x—y ^~y = (^+y) (^' - ^y+y'') (^' + ^y+y')- 6. Resolve r^;" — rr^^a + r^^a^ — x^a^ + ^^a* — :r®a^ + ^^a^ — x*'a}-\- x' a® — o;^ a® + :ra^® — a}^. This equals :i; + a x-{-a _ (x^ + g') (^' - ^') (^ + a') x-\-a FACTORING. 107 Ex. 37. Factor the following : 1. x^-y^', T'~\\ rr' + 8; 8a'-27:r^ 8 + a'a;^ 2. a;^-a^^ 27a^-64; a^^ - Z;^ rr^«-32y^. 3. Find a factor which, multiplied by a' + a'^ + a'Z^' + ab^ + ^^^ will give a^ - Z^^ 4. By what factor must x^— 4:x'^y-{-16xy'^~(j4:y^ be mul- tiplied to give x^ — 256 2/^ ? 5. Factor x'^ + x^y + x^y"^ + ^*y^ + ^^y* + ^^3/^ + ^3/^ + 3/^- Find the factors of the following : 6. (By'~2xy~(Sx'-2yJ; a'~l6b\ 7. x^ — y^ — x(x'^ ~ y'^) + 3/ (a; — y)^ 8. Z> (x^ — a^) + <^^ (^^ — «0 + <^^ (^ — «). 9. b {m^ + a^) + am {m^ — a^) + a^ (m + a). 10. x^-~y^ + 2xy(x' + x'y'' + y'). 11. (a' — Z)c)' + 8Z>^c^ rr*"* — a'». 12. rr^-3a2;2 + 3a=':r-a' + Z>l 13. (ri;' + Sf)(x + y) — Qxyix" — 2^y + 4/). 14. 8:r' — 6a:y(2rr + 3y) + 27yl 15. l — 2x+4:x^-^x\ 16. a^ + a^Z^c + a^b'^c' + a^b^& + ab^'c' + ^^^c^ §25. The principles illustrated in Chap. II. may be ap- plied to factor various algebraic expressions, as in the following cases : 108 FACTORING. Examples. 1. Find tlie factors of {a + h + c) {ah + hc + ca) — (a + h) (h + c)(c + a). 1. Observe that the expression is symmetrical with respect to a, 5, c. 2. If there be any monom>ial factor, a must be one. Put- ting a = 0, the expression vanishes ; hence, a is a factor, and, by symmetry, h and c are also factors. Therefore, ahc is a factor. 3. There can be no other literal factor, because the given expression is of only three dimensions, and ahc is of three dimensions. 4. But there may be a num^erical factor, m suppose, so that we have {a-Yh-\-c) {ah-\-hc-\-ca) — {a-\-h) (h+c) ( = 0, or a = b ; substituting b for a, the ex- pression becomes zero. Therefore, a — b is a factor; by symmetry, b — c and c — a are factors. Now the product of these three factors is of three dimen- sions, while the expression itself is oifour dimensions. There must, therefore, be another factor of one dimen- sion. It cannot be a monomial factor, for the expres- sion has no such factors. It cannot be a binomial factor, such as a-\-b, for then, by symmetry, b-\-c and c+a would also be factors, which would give an ex- pression of six dimensions. It cannot be a trinomial factor, unless a, b, and c are similarly involved. For instance, if a — b-\-c were a factor, then, by sym- metry, b — c-\-a and c — a-\-b would also be factors, and the dimensions would be six instead oifour. The other factor must therefore be a-{-b-\-c. Hence, ». a\b - c) + b\c - a) + c\a - b) = 7n(a — b) (b — c) (c — a) (a -{- b -{- c). To find m, put a — 0, 5 = 1, and c=2, and we have — 6 = 6 m ; therefore, m — — 1. Hence, the factors are —(a—b)(b — c)(c—a){a-{-b-{c), or (a — b){a — c)(b — c)(a-{-b-^ c). Prove that a' + b' + c' + S(a + b)(b + c)(c + a) is exactly divisible by a + 5 + ^, and find all the factors. Let a -f 5 + c = 0, or a — — (b-\-c)', substituting this value for a, we have -{b + cf + b' + c' + 3bc(b + c), OT-(b + cy + ib + c)\ which = ; and therefore a + b -\- c is sl factor. FACTORING. Ill As before, we find that there are no monomial factors. Since a-^b-^c, the factor already obtained, is of 07ie dimension, the other factor must be of two dimensions, and as it must be symmetrical with respect to x, y, and z, it must be of the form m (a^ + b^ + c^) + n(ah + bc-\- ca), in which rfi and n are independent of each other, and of a, Z>, and c. To determine their values, put c = 0, so that o? + h^ + c^ + ?>{a + h){h + c) {c + a) = {a + h + c) [m{o? + ^' + C") -\-n{ah + hc + ca)] becomes a? + h^ + ^ah{a + h) = {a + h)[m{a'' + h'') + nah\ But a^-\-P + Sah{a + h) == (a + h)\ .-. (a + hf = {a + b) [m {p? + }f) + nah\ :.(a-\- by = m (a^ + b"^) + nab. That is, ^2 _j_ j2 _|_ 2ab = 771(0" + b') + nab. Now this is true for all values of a and b. .*. m— 1 and n = 2. .-. a' + ¥ + c' + 3(a + b)(b + c)(c + a) = (a + b + c)[a'' + b' + c' + ^lab + Z^(? + ca)] = (a + b + c){a + b + cy ==(a + b + cf. 7. Simplify a(b + cf+b(a + cy + c(a + by--(a + b)(a-c)(b~c) - (a-b)(a-c)(b + c) + (a-b)(b- c)(a + c). Let a = 0, and the expression becomes be' + cb' + ^^(5 - c) - Z>c(^ + ^) - ^^(^ - c), which equals zero ; therefore a is a factor ; by sym- metry, b and c are also factors. 112 FACTOEING. The expression is of three dimensions, and abc is of three dimensions, there cannot therefore be any other literal factor. Hence the expression equals mahc. To find m, let a = Z> = c = 1, and we have 4 + 4 + 4 = m; m = 12. .'. the expression = 12ahc, In the preceding examples the factors have been linear, but the principle applies equally well to those of higher dimensions. (See Th. II. Cor.) 8. Examine whether x'^-\-l is a factor of Let x^ +1^=0, or ^'^ = — 1, and substituting, the ex- pression vanishes ; therefore, :?;** + 1 is a factor. 9. Examine whether a^ + ^^ is a factor of 2a'' + a'h + 2a^h'' + ah\ Let o? -{-h^ = 0, or a^ = — H^. Substituting, we have 2b'-ah'-2h' + ah\ which = 0, and therefore o? + Z>Ms a factor. 10. Prove that o? + ^^ is a factor of w'+a'h + 0^1^ + 0.^1)^ + ah' + h\ Let a^ + h^ — 0, or a^ = — h^. Substituting, we have -a'P- al' - 5^ + a^Z>^ + ah' + l\ which = 0, and therefore d^ + Z>Ms a factor. Ex. 38. Kesolve into factors : 2. hc(b — c) — ca{a — c) ~ ah(h — a). FACTORING. 113 5. {a + hy--{h + cf + {c-a)\ 6. a(h — cy + h{c-ay + c{a~h)\ 7. {a + h -\- c) (ah -{-he -{- ca) — ahc. 8. a' (c - h') + ^-^ (a - c^) + ^' (^ - a^) + ahc (ahc - 1). 9. a\h + c)-\-h\c + a) + c'(a+h) + 2ahc. 10. (a — ^) (c - h) (c - XO + (^ - c) (a — h) (a — ^) + (^-a)(Z)-A)(^-^). 11. x^y"" + ^r'y* + :r'2' + x'z' + 7/'^' + ?/'2* + 2x'y'z\ 12. (a-Z>)^ + (^)-c7 + (c-a)l 13. ah(a + h) + hc(h + c) + ca(c + a) + (a' + ^' + c')- 14. a' (c - P) + h\a- c') + c'(h~ a') + ahc (a' h' c'-l). 15 . x'(y'~ z') + 7/ (z' - x') + ^'^ (:?;^ - y'). 16. !^* + y* + ;s* ^ 2:r'?/' " 27/ z' - 2z'x'. 17. (^ — c) (x — 5) (rt' — c)-{-(G — a)(x — c) (x — a) -{- {a — h){x — a){x — h). 18. {a + hy + (h + cy + {G + af + '^{a + 2h + c){h + 2c + a){c + 2a + h), 19. Show tliat a^ -\-d!'W — ah'^ — ¥ has a^ — 5 for a factor. 20. Show that (x + ?/)^ — x'^ — / = 1xy{x + 7j) (x" + xy + y^. 21. Examine whether x^ — 5 a; + 6 is a factor of 22. Show that a — 5 + c is a factor of a\h + c)- h^c-i-a) + c\a + h) + ahc. 114 FACTORING. 23. Show that (2^ + 3^ is a factor of and find the other factor. 24. Find the factors of a^ {h — c) + h\c— a) -\- c^ {a—h). Factoring a Polynome by Trial Divisors. § 26. To find, if possible, a rational linear factor of the polynome x^ + hx""-^ + ex''-'' + -{-hx + h in which h, c, , A, h, are all integral, substitute succes- sively for X every measure (both positive and negative) of the term h, till one is found, say r, that makes the poly- nome vanish, then x — r will be a factor of the polynome. Examples. 1. Factor;r^ + 92;'+16:ir + 4. The measures of 4 are ±1, ±2, and ± 4. Since every coefficient of the given polynome is positive, the pos- itive measures of 4 need not be tried. Using the others, it will be found that —2 makes the poly- nome vanish. Thus, 1 9 16 4 -2 -14 -4 -2 17 2; Hence, the factors are {x -\- 2) (x^ -\-l x -{- 2). The labor of substitution may often be lessened by ar- ranging the polynome in ascending powers of x, and using the reciprocals of the measures of Ic instead of the measures themselves. Should a fraction occur during the course of the work, further trial of that measure of h will be needless. FACTORING. 115 Factor x^ -I0x^-^?>x + 60. The measures of 60 are ± 1, rb 2, i 3, ± 4, zh 5, etc. Neither + 1 nor — 1 will make the polynome vanish. Try 2 ; thus, 1 60 -63 30 -10 1 2 30 -16} A fraction occurring, we need go no further. — 2 will also give a fraction, as may easily be seen. Next try 3 ; thus, 60 -63 -10 1 20 20 14i A fraction again occurring, we may stop, also give a fraction. Next try 4 ; thus, - 3 will 60 -63 -10 1 1 15 -12 4 15 -12 -5i - 4 will also give a fraction. Next, trying 5, we find it fails, and we then try — 5 ; thus, 1 60 -63 -12 -10 15 1 -1 5 12 -15 1; The remainder vanishes. The factors are, therefore, {x + b){x'-lbx+l2). § 27. When k has a large number of factors, the number that need actually be tried can often be considerably les- sened by the following means : 116 FACTORING. For X substitute successively three or more consecutive terms of the progression , 3, 2, 1, 0, —1, —2, —3, Let , h^, ^2, ^1, ^, ^-1, ^-2, ^-3, , denote the correspond- ing values of the polynome ; and let r denote a measure of Ic positive or negative. The substitution of r for x need not be tried unless r — 1 measure Ic^, r — 2 measure ^2, , and also r + 1 measure 1c_i, 7- + 2 measure lc_2, If no measure of h fulfil these conditions, the polynome will have no linear factor. If p denote a positive or arithmetical measure of k, the preceding criterion may be conveniently expressed as follows : 1. The substitution of +p for x need not be tried unless jo — 1 measure \, p — 2 measure Jc^, , and also p + 1 measure k.-^, p+ 2 measure p_2, 2. The substitution of — ^) for x need not be tried unless p + 1 measure ^j, p + 2 measure k^, , and also p — 1 measure k_i, p — 2 measure k_2, In trying for measures, the signs of ^2, Jci, k, , may evidently be neglected. If kt vanish, t positive or negative, then x— t will be a factor of the polynome, and should be divided out before proceeding to test for other factors. Examples. 1 . Eind the factors of ^' — 10 ^' — 63 ^ + 60. Here ^=-60, ^i=— 12, ^2-=-98, /;_i = 112, ^_.2-138. Tabulating the trial measures, we get k; 98 1, 9 12 2 3, 4, 60 3, 4, 5, 6, 112 4, 7. 138 10, 12, FACTORING. 98 7, 12 4, 6, 60 3, 4 5, 6, 1 112 4, 138 1, 3, 117 10, In the upper or positive table, no measure of 60 gives a full column ; hence, no positive integer substituted for X will make the given polynome vanish. In the lower or negative table, 5 is the only measure of 60 that gives a full column ; hence, — 5 is the only negative integer that need be tried for x. Sub- stituting — 5 for X, the polynome vanishes ; hence, ^ + 5 is a factor of x^ — lOx'^— 63rr + 60. In constructing the above tables it is evident that 12 is the highest measure of 60 that need be tried in the upper table, for the next measure, 15, would give 14 as a trial-measure of 12 (the absolute value of ^_i), and higher measures would give higher trial- measures. Similarly, 10 is the highest measure that need be tried in the lower table. Since it can make no difference in the full columns which of the lines of measures is made the basis from which to construct these columns, it will be found best to construct the tables by the measures of that one of the ^'s which has the fewest number of them. Find the factors of x' + 12:?;^ - iOx' + 67 x - 120. ^--120, ^i--80, h = -^^, yl'_i=--238. Selecting the measures of 34 for trial-measures, and tabulating, we get 34 1 80 2 120 238 3 2, 17, 34, 17, 84, 16, 15, 14, 118 FACTORING. Here, in the only column that is full, 15 stands in the line of 120, the absolute value of h, and as the col- umn is decreasing the sign of the 15 must be minus; hence, the only measure of k that need be tried is — 15. On substituting — 15 for x, we get -120 67 -40 12 1 -1 8 -5 3 - 1 15 -8 5 -3 1; Hence, the only linear factor of the given polynome is X + 15, and, as is seen from the substitution, the other factor is x^ — '^x;^ -\-bx — 8. Factor x"^ — 21 x^ +I4cx+ 120. ^-120, ^1-108, ^2 = 56, h_, = SO. 56 1, 2, 4, 7, 8, 14, 28, 56, 4, 7, 8, 14, 28, 56, 108 2, 3, 9, 3, 6, 27, k; 120 3, 4, 10, 2,5, 80 4,5, 1,4, The positive or increasing columns give 3 and 4 to try ; the negative or decreasing columns give —2 and —5. Using these in order, we get x—'^ is a factor. 120 14 ■ -27 1 1 40 18 - -8 - -1 8 40 18 -8 - -1; 1 10 7 1 4 10 7 1; -1 -5 -1 2 5 1; -4 is a factor. rr+2 is a factor, and there remains :r+5, a factor. Hence, the factors are {x — Z){x — 4) (x + 2) (^ + 5). FACTOHING. 119 4. Factor x*" —px^ -\- {q —1) x"^ -\-px — q. Jc=—q,Ici =l~p + (q-l)+p — q = 0, h_, = l+p + {q-l)--p-q^O. Since both k^ and k_i vanish, the polynome is divisible by both X —\ and x-\-l. 1 1 -p 1 -^3 + 1 p i-'P -q !Z 1 1 -p+l -1 a-p +p -1 1 -p ?; Hence, the other factor is x^ —px-{- q. 5 . Factor x' + 2 ax^ + {a" + a)x'' + 2a^x + a\ h ^ a\ h = 1+ 2 a + {o?+ a) + 2 a^+ a' = (a+l)\ The positive measures of Ic are 1, a, a^, a^. Of these 1 may be rejected at once, since neither Jci nor ^2 van- ish ; and a^ and a^ may also be rejected, since ki or (a+ 1)^ is not divisible by either a^ ± 1 or a^ rb 1. But Jci is divisible by a-\-l, and Jc^i is divisible by a—l] thus, we need try the substitution of only — a for X. 1 2a a' + a 2 a' a? — a -a' -a' -a' 1 a — a a -a' 1 a; acto] 'S are ^x- + a) ^(xH - a ). 6, Factor ^^ — (a + 6') x'^ + (^ + ac) x — be. Ic =-~ he, Jci == l—(a-\-e)-\-(b-\-ae)—he = l—a-j-b—e-^-ac—he, Jc^i = — l~(a-]-c)—(h^-ae)—be = — (l+a+5+c+«. 19. 6. x'+^x'+l0x'+l2x+^. 20. 7. x^-^x + 2. 21. 8. x^ + 2x^ + ^. 22. 9. m^— 3 w?n + 4 mv?— 2 n^. 23. 10. x^ + 2x^ + 2. 24. 11. m^— 5 w?n + 8 mn^~ 4 nl 25. 12. h^+h''c + 1hc'+Z%c\ 26. 13. m^~4:mn^ + Sn^. 27. 14. a'-7a'b+2SaP-16b\ 28. 29. x'-lSx'+USx'- •30. ^*-9:r'y + 20 x''^/ :r^- 11^^ + 39:?; -45. x^ + bx''+7x + 2. a^-3a^- 193a +195. p' - Sp' - 6j9 + 8. a^-Qa'"" +110"^ -6. a'~4:la'P + 16b\ a'-a'b'-2ab^ + 2b\ f - ^f + 6^-4. y*-5y^ + 8y^-8. a*-2a'+3-a'-2a + l. a^-\-a^W'~\-ab'--?>b\ 2a^«_a'«_a" + 2. 288a; +252. -392:y« + 18?/^ FACTORING. 121 §28. To find, if possible, a rational linear factor of the poly nome ^^n ^ ^^n-i _^ ^^n-2 _^ + hx + Jc, in which a, h, c h, h are all integral. First Method. Multiply the polynome by a"~\ {axy+h(axy-^+ac{axy~^'+ +aJ'-Vi{ax)+a''-^k] or, writing y for ax, yu _|_ lyn-Y _|_ ^^yn-1 _j_ _|_ ^n-IJ^y _j_ ^n-1^ Factor this polynome by the method of the last article, replace y by ax, and divide the result by a''"^ Example. Factor 3 ^* + 5 ^' - 33 x^ + 43 :r - 20. Multiply by 3^, and express in terms of 3 x. (3 x)' + 5(3 xj - 99 (3 xj + 387 (3 x) - 540 ; or, y^ + 53/^ - 99y^ + 387y - 540. Here ^^=-540, h =-1 + 5-99 + 387-540 = -246, Z;_i == 1 - 5 - 99 - 387- 540 = - 1030. 6, 41, 82, 123, 246. 246 540 1030 1, 9 2 3, 3, 4, 3, 41, 246 3, 6, 41, (Trying by factors of 246, 540 2, 5, instead of by factors of 1030 1, 540, for convenience.) The only factors of 540 in full columns are 4 in the upper table and 2 in the lower one ; hence, we need try only the substitutions 4 and — 2. 1 -540 387 -135 -99 63 5 -9 1 -1 4 - 135 63 -9 -1; 122 FACTORING. Hence, 3/ —4 is a factor. The substitution of —2 need not now be tried, since we see tliat 135 is not a mul- tiple of 2. The other factor is, therefore, y' + 9y'--63y + 135. Replacing y by 3r^, and dividing by 27, 2V(3^ - 4) (27r^ + 81:^^ - 189^ + 135) -= (3^ - ^){x^ + Zx^ - 7^ + 5), which are the factors. §29. Second Method. Write 7)i for "a measure of a," and r for ''a measure of Ic, positive or negative " : For X substitute every value of t-'^th till one, say r'-^m', be found which makes the polynome vanish ; then on'x — r' will be a factor. Should a fraction be met with in the course of substitution, further trial of that value of r -^ m will be useless. Should Ic have more factors than a, it will generally be better to arrange the polynome in ascending powers of x and use values of m -^ r instead of r -v- m, making r positive and on positive or negative. To reduce the number of trial measures, for x substitute successively three or more consecutive terms of the pro- gression , 3, 2, 1, 0, —1, —2, —3, , then denoting the corresponding values of the polynome by , h^, h^, h^, Ic^ The substitution of r for x need not be tried unless r— ??i measure ^1, r— 2?7i measure Jc^^ , and also T-\-in measure ^_i, rH-2m measure lc_2, If p denote a positive or arithmetical measure of h, this criterion may be expressed as follows : 1. The substitution oi +p for x need not be tried unless p — 77i measure h-^, p — 2m measure k^, , and also p + m measure k^^, p f 2 m measure ^ ..^, FACTORING. 123 2. The substitution of — ^ for x need not be tried unless |)+m measure k^, p-\-2m measure k^, , and also p — m measure ^_i, p — 2m measure. ^_2, It must be remembered that here m may be either positive or negative, as may also be one or more of the quantities, p -\r'm, p — m^ p -\- 2m, p — 2m, etc. Examples. 1. Factor36:r'^+171a:^-22r?; + 480. ^'-480, k =-665, h -1408, k_, - 637, Ic_, = 920, and 7)1 may have any of tlie values, ±1, ±2, ±3, ±4, ±6, ±9, ±12, ±18, ±36. In forming the table of trial-measures, write out the measures of 1408, that is, k^ ; they are 1, 2, 4, 8, 11, 16, 22, 32, 44, 64, 88, 128, 176, 352, 1408. Taking each of these in succession, add to it each value of 7)1 separately. Should the sum appear among the measures of 665, that is, Ic^, which are 1, 5, 7, 19, 35, 95, 133, 665, enter these measures of k.2 and Jci in a column in the table, writing above them the value m used. How- ever, should the sum not be a measure of 665, another value of m must be tried. When all the values of 771 have been tried with one measure of 1408, another measure must be taken till all have been used. This having been done, proceed to test which of the col- umns can be filled up with measures of 480, 637, and 920, respectively, these being the values in this case of Ic, k__i^ k^2- 124 FACTORING. The table will then appear thus : 4-4, +6, +3, +1, +3, +3, +3, ^; 1408 1, 1, 2, 4, 4, 16, 32, 665 5, 7, 5, 5, 7. 19, 35, 480 8, 6, 10, 637 7, 13 920 8, 1113 -2, -6, -3, -9, -3, -9, -1, -3, _9, -4, -6, -9, -3, 1408 1, 1, 2, 2, 4, 4, 8, 8, 8, 11, 11, 16, 22, 665 -1, -5, -1, -7, 1, -5, 7, 5, -1, 7, 5, 7, 19, 480 -3, -4, - -16, -2, 6, 2, -10, 3, -1, -2, 16, 637 -7, -1, -1, -7, 13, 920 -10, -4, -5, 10, ]113 -7, 7, There still remain five full columns, while the given polynome, being of the third degree, cannot have more than three linear factors. To reduce the num- ber of these columns, and, as a consequence, the number of trial-measures, extend the table by calcu- lating ^_3 and the corresponding column-numbers for the full columns. ^_3 = 1113, and the column- numbers are 9, —13, —7, —9, and 7. Of these, 9, — 13, and —9 must be rejected, not being measures of 1113. This leaves only —7 and 7, to which correspond Q Q —— and ■— — as the values of — to be tried in substi- 2 16 r tution for -. (See table above.) Making trial of these two, the polynome is found to vanish for -— - but not for — —• 16 2 The actual work of substitution will be as follows. FACTORING. 125 Arrangement in ascending powers of x : 480 -22 171 36 - 720 1113 - 1926 ^3 2 -3 16 240 480 -371 — 22 -90 642; 171 21 -945 36 -36 30 — / 12; Hence, tlie factors are Sx-{-16 and 12:?;^ — 7^ + 30. The latter factor cannot be resolved, for it does not contain 3x-\-16, and the only other factor, viz., 3r^ + 2, left for trial by the tables above, has been tried and has failed. 2. Factor 10 a;^ - x' (15 ?/ + 4 2) - x' (40?/^ -63/^) + x(60f + Uy'z) - 24:y^z. Here on = ± 1, ± 2, i 5, or i 10. Jc = -24:y'z. y^i - 10 - (153/ + 4^) - (40y^ - 67/z) + (60y'+16y'z)-24:y'z = 10--152j-4:0f + 60y' -2z(2~3y-Sf + 12y') --(5-20)(2-3y-8/ + 123/^). ^_,-(5 + 2.)(2 + 3y-8y^-122/0, as may easily be found by making the calculation. We get at a glance 2:2 a factor of Z:, 22; — 5 a factor of Jci, and 22+ 5 a factor of Ic_i ; hence, taking 7?i = 5, 9 7 we are directed to try the substitution — for x. •^ 5 2z 10,- -(15y+4.),- Az - (40y2-6y2), (60/+16y^;2), -24y^2; -^yz -l^y'^z 2^fz -Sf 12/; 126 FACTOEING. Sence, 5x — 2z is one factor, the other being 2x' - ^yx'y - 8^/ + 12/. Seeking to determine the factors of this, we obtain m = ± 1 or ± 2, ^' == 12, l^ = 3, h, = 0, X^_i = 15, /j_2--0. The vanishing of Jc2 shows that a; — 2 y is a factor; and the vanishing of Jc__2 shows that ^ + 2y is also a fac- tor. Dividing these out, the remaining factor is found to be 2ii; — 3y; so that the proposed poly- nome resolves into (5x-2z)(x-2 y) {x-\-2y) {2x~Zy), The factor ^x — 2z might easily have been got by the method of § 23, but the present solution show^s we are inde- pendent of that section. It may also be obtained by re- arranging the polynome in terms of y. Factor: Ex.40. 1. 2c(^-~2^x^^mx~2^', x^ -1 x'^y + l(Sx2f -I2f. 2. \2x' + bx''y + xy'' + ?>y''', Sx^-l4:X + (S. 3. ^x^ ~\bax + a?x~ba^\ 2x^ + ^x''y +1 xy'' ~-?>^. 4. 2h'~1h^c-Wc' + hc'~4.c'- Iba^ + A^ o?h + l?>ah'' -\2Jy\ 6. 150:?;* - 725 r> + 931 x'^y'' + 920 xy^ - llb2y\ 7. 36 x'- 6(9 -7y)a;^-7(9 +Uy)x''y+ 3(49 -40y)^/+180y'. 8. I0x^~-a^{lbij+^z)+x\^0y''+^yz)+x{m'i^-l^yh)~-2^''z. § 30. If the polynome a:r"+ Z>:r**~^+ -\- hx -{- Jc, in which a, h, , h, Jc are all integral and n greater than 3, have no rational linear factors, it may have rational quad- FACTORING. 127 ratio factors. Let m denote a positive measure of a, and r denote a measure, positive or negative, of Tc. Tlie rational quadratic factors of the polynome, if there be any, must be of the form r)ix^ ~\- qx — r. To determine such factors we may proceed as follows : For X substitute successively three or more consecutive terms of the progression , 3, 2, 1, 0, — 1, — 2, — 3, , and denote the corresponding values of the p)olynome by , ^^3, h^, hi, k, Jc_i, Jc_2, Jc_s, Let r^ denote "a measure of y^s, positive or negative"; rg denote *'a measure of ki, positive or negative " ; etc. Then mx'^ -{- qx—7' need not be tried as a factor of the polynome, unless an arithmetical progression with q as common difference can be formed from among the values of , 9m+r3, 4m+^2, ^>^+^i, ^', ^>2'+r_i, 4??i+'^-2, 977i+r_3, , in which the coefficients of vi are the squares of the terms of the series , 3, 2, 1, 0, - - 1, -2,-3, Examples. 18. h_ ■ 78. 1. Factor :27* — 3^'- 13^^' + 36^-, m =1, k =-~ 18, ki = 3, 7^2 == — 6, Z^a = — 9, Ic^i = -63, Ic^,=- 102, /l^_3---81, Trying for rational linear factors as by § 28, it will be found there are none. We therefore proceed to seek for rational quadratic factors. To do this, we first tabulate the arithmetical values of ^3, rg, 7\, 9m 4m 9 1. 3, 9, 6 1 '^ 3, 6, 3 1, 3, 18 1 2 3, 6, 9, 18, 63 1, 3, 7, 9, 21, 63, 102 1, 2, 3, 6, 17, 34, 81 1, 3, 9, 27, 81, 78 1, 2, 3, 6, 13, 26, 51, 102, 39, 4 m 9 m 16 m m = 1 128 FACTOEING. Taking these both positive and negative, we next tabu- late tlie values of 9m + r3, 4m + r2, m + ri, This done, we then proceed to select and arrange in col- umns any arithmetical progressions that are found to run completely through the table, one term of the progression in each line of the table in regular order, thus : r; — 0, 6, 8, _9 1, 2, -2, 0, 2, -18, -9, -6, -^2, -20, -8, -98, -47,- -30, -72, -18, 0, -62, -23,- -10, 10, 12, 18, 0, 12 3, 5, 6, 7,10, 2 7 4, 4, 2 -3,-2,-1, 1, 2, 3, 6, 9,18, 6, -3 -6,-2, 0, 2, 4, 8,10,22,64, 8, -8 -13,-2, 1, 2, 3, 5, 6, 7,10,21,38,55,106, 10,- -13 6, 8, 10,12,18,36,90, 12,- -18 3, 10, 13, 14, 15, 17, 18, 19, 22, 29, 42, 55, 94, 14,- -23 There are two columns of progressions : in the first, r = 6 and q or the common difference is 2, giving the trial factor x^ -\-2x --^] in the second column, r=— 3 and 5' = — 5, thus giving the trial factor x^ — bx-\-?>. On actual trial, it will be found these are the factors of x' - ^x^ - 13:r2 + 36a; - 18. Factor 6^^ - 53 .r^ + 179:r^ - 299 2;' + 260^ - 96. Here m may be 1, or 2, or 3, or 6. ^ = -96, ^1 =-3, Z:,=.4, ^3 = -9, h = Jc_, - _ 893. The factors of k, Jci, , are : 32 9 4 3 96 893 32, \J1. A/, A/1, , cXiC . 2, 4, 8, 16, 32, I6m 3, 9, 9m 2, 4, 4m 3, m 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96, 19, 47, 893, m FACTORING. 129 I As this table yields no complete column in aritlimetical progression, the given polynome has no rational linear factor. (§ 29.) Forming the table for vi=l, it will be found that it also does not yield any trial divisor. The table for m = 2 is : 0, 16, 24, 28, 30, 31, 33, 34, 36, 40, 48, 61, 28 9, 15, 17, 19, 22 27. 19 4, 6. 7, 9, lo! 12, 10 -1, 1, 3, 5, 1 -96, -48, -32, -24, -16. -12, -_8, -6, -4, -3, -2, -1,1,2,3, -8 -891, -45, -17, 1, 3, 21, 49, 895, -17 This gives the complete column set out at the right. In it, r = — 8, and the common difference is — 9 ; hence, we have 2x'^ ■—9x-{-S as a trial divisor. On actual trial it will be found to be a factor of the given polynome, the co-factor being 3.r^ — 13:r'^ + 19^— 12. Ex. 41. Factor : 1. x'~12a^ + 4:7x'-66x + 27, 2. x'~6x^-2x' + S6x~-24:. 3. x'-2x'-2bx' + lSx + 24:. 4. :?;5-31^^ + 186r?;-180. 5. 1 - 45a;^ + S2x' + 281 x' - 518^^ + 252^«. 6. a' - 38 a'f + 2SaY + 345 d'y' - 664:ai/' + 180/. 7. 2:?;*- 5^^- 17:^'^ + 53^- 28. 8. 6x'- 5Sx' + 83a;* + 45^;^ - 257^^ + 32:r + 15. 9. 6-47y + 108y^-74y^ + 12/. 10. 6:r^ - 17a;* + 5:^^^ + 13:^^-2^-2. CHAPTEK IV. Measures and Multiples. § 31. When one quantity is to be divided by another, the quotient can often be readily obtained by resolving the divisor or dividend, or hoih, into factors. Examples. 1. Divide a' - "lab + ^^ - c^ + 'lcd~~- d'' hj a- b + c -d. Here we see at once that the dividend = {a-hy-{c~d)\ and hence quotient — a—b — (c~d) = a—b — c~\-d. 2. Divide the product of o? -\- ax-\- x^ and c^ + x^ by a*+a^^^ + ^^ Here o^ -\- x^ ^= {a -\- x^ {o? — ax -\- x^), and the divisor Hence, the quotient i^ a-\- x. 3. Divide a^ + a^b + o?c~ abc — b'^c — b& by d^ — be. The dividend is a {c^ — bc)-\-b {c^ — be) + e (a^ ■— be). Hence the quotient ^=^ a-{-b ~\- e. 4. {a^ + b^ — c^J^^abe)-^{a + b-c). J)Wi^Qn^ = a^ + b^+?>ab{a+b)~e^~^ab{a + b)+^abG = {a + by-c^~2> ab {a + b-c), which is exactly divisible by a + ^ — e. Quotient = a^ -{- b"^ + e"^ — ab -\- be + ea. MEASURES AND MULTIPLES. 131 5. Divide x^ — x^y + x^y"- — 29 if' + xy^ — y^ by x^ — y^. The dividend is (§ 24) evidently {x^ — y^)^{x-\-y^^ and this divided by x^ — y^ =-- (^' + /) -^ (^' + y) = ^'' — ^y + y'- 6. Divide h(^x^-\-c^)-^ax{x^—dF)-\-d'{x^a>) by (a+Z>)(.r+a). Striking the factor x-\- a out of dividend and divisor, we have h {pf — ax -\- o?) + ctx (x — a) + a^ = h{x'^ ~ ax + a^) + a (x^ — ax -\- a^) = (a + ^) (x^ — ax + a^). Hence, quotient =^ x^ — ax-\- al 7. Divide apx^ + x^ (aq + ^j9) + x'^ (ar + ^^7 -\-pc) + :?; (^^ + br) + c?r by a:r^ -i-hx-\- c. Factoring the dividend (§ 22), we have (ax'^ -{-bx-i- c)(px'^ -\-qx -{-r). Hence the quotient equals the latter factor. 8. Divide 6x' - ISax' + ISa'x' - 13a\x~5a' by 2x'^ — Sax — a^. This can be done by § 15. The divisor is 2x'^ — a^ — Sax, and we see at once that 3^^ + 5a^ must be two terms of the quotient. Multiplying diagonally into the first two terms of the divisor, and adding the products, we get -i-7a^x^; but + 13 a^^^ is required. Hence, + 6a^x'^ is still re- quired, and as this must come from the third term multiplied into —Sax, that third term must be — 2 ax. Therefore, the quotient is 3^^ + 5a^ — 2 ax. Note. By multiplying the terms —2 ax, —Sax, diagonally into the x'^'s and a^'s, respectively, we get the remaining terms of the divi- dend. It is, of course, necessary to test whether the division is exact. 132 MEASURES AND MULTIPLES. 9. J)Wid.Q2a'-o?b~l2a^h''~bah^+Whjo?-~h''-2ah. Here, as before, one factor is a^ — H^ — 2ah\ hence two terms of tlie otlier factor are 2o? — Ab'\ Multiply- ing, as in the last example, we get — Qa'^h'^] but — 12 a^ b'^ is required. Hence, ~^a?b'^ is still needed, and +3aZ> is the third term of the required quotient, which is therefore '2ia^ — 4:h^-{-?>ab. 10. Prove that (1 + X + x' + + x''-^) (l~x + x^~ + x''-^) = 1 + ^' + ^* + + :^'"-l Product ^l—x"" ^l+x" ■ X ■ \~~ x l-\- X 1 - x"'' = l + x' + x^+ + a;2"-l 11. J)Wid.Q{a^ — bcy + U^c^hjo? + bc. =: (a' - bcf + (2 bcf -^ (a' -be) + 2 be = (a^- bey - {a? - be) X2be + (2bey = a'-4:a''be+7b'e\ 12. Divide 1 + 2,357,947,691 2;^ by 1 - 11:?; + 121 ^^ Dividend ==1 + (11^)' - [1 - (11^)^ + (11^)''][1 + (11^)']. Divisor = [1 + (11 rr)'] -^ (1 + 11:^). .-.quotient- [1- (11:^7 + (11^)^(1 + 11^). Ex. 42. Find the quotients in the following cases : 1. 1 — x-\-x'^ — x^ -i-1 — X. 2. l-2x' + x^-^x' + 2x^ + 1. MEASURES AND MULTIPLES. 133 5. I~4:x''+l2x^-^x'^l + 2x-?>x\ 6. (a' - 2ax + x'') (a''^ + ?>a'x + Zax'' + x"") -^ a' - x\ 7. x^ — y^ + 2;^ + 3 :r?/2; -^ :i; — y + 2;. 8. 6a'~^a'b + 2a'b' + ISab' + 4:b'-^2 a' ~ 3ab + Ab\ 9. 4:x'~x^f + 6x7/ — 9f-i-2x' + Sf~xy. 10. a' + b'-c'-2a'b'---a'-b'~c\ 11. 21a'^-16a-^6 + 16a'Z)'-5aZ>^ + 2Z»*^3c^*^-a5 + 5l 12. 2a^-7a'-46a-21--2a' + 7a + 3. 13. [a'(b -c) + P(c -a) + c'(a-h)] ~a + b + c. 14. x^ — dax'^ + Sa^x — a^-{-b^~7-x~a + b. 15. x' -y' + z' + 2x''z' — 2y' - 1 -^ x' ^f + z''-l, 16. :?;* — (a + c) x^ + (^ + ac) :r^ — bcx -^ x — c. 17. r'^ + ^'y + :r?/' + ?/-'^--.'^ + y. 18. x'' — ^^y + rr^y' - x'y^ + x^7/~-x''y'^ + xy^-~y'^~^x^-{-7/\ 19. a* + Z)* — 6'^^ - 2a' 5'^ _ 2c' - 1 -- a' ~ b'~c' - 1. 20. a' - ab^ - ac^ - 2a'^ + 2Z>^ + 2bc' + ?>a^c-?>¥c-?>c' -^a + ?>c — 2b. 21. a'Z> — ^a;' + a'^ — r?;^-^ (:r + ^)(a — :r). 22 . a{b ~ cf + b{c — aj + c{a -by -^ o? - ah -ac + be. 23. a' 5' + 2 a5c' - a' c" ~ b'' d" -^ ab + ac - be. 24. ^•'+y^+3:?;y~l^:r + y-l. 25. x^ — x^ — 2-^x'' — x+l. 134 MEASURES AND MULTIPLES. 26. a* — 29a' — 50a— 21 -^- a' — 5a— 7. 27. (2x — yjof' — (^ + y)'a':r' + 2(^2; + y)ax^ — x^ -^ {2x — y) a^ -]- {x -\- y) ax — x^. 28. (^^-l)a^-(:u' + ^^'-2)a'+(4^'+3:i'+2)a-30r+l) -^(:?;-l)a'-(:2;-l)a + 3. § 32, The Highest Oommon Pactor of two algebraic quan- tities may, in general, be readily found by factoring. The H.C.F. is often discovered by taking the sum or the differ- ence (or sum and difference) of the given expressions, or of some multiples of them. Examples. 1. Find the H.C.F. of (h — c)x'+ (2ah -2ac)x + a'b - a'c and (ab — ac -{- h"^ — he) x -{- a"^ c -\- alf — a^h — ahc. Taking out the common factor h — c, we get (h — c){x'^+2ax-\-ah) and {h — c)\{a—h) x — a^+ah'\. Therefore, h — c\^ the H.C.F. of the given expressions. 2. Find the H.C.F. of 1 — x + y + z — xy + yz — zx — xyz and \ — x — y — z + xy-\-yz-\-zx — xyz. Their difference is 2y + 2z — 2xy— 2zx = 2(l — x){y + z). Their sum is 2 — 2:r + 2yz — 2xyz = 2 (1 — x) (1 + yz). Therefore, the H.C.F. is (1 - x). 3. Find the H.C.F. oix> + ?>x'- 8;r' - 9a; - 3 and a^ - 2x' - e>x^ + 4:r^ + 13:r + 6. The annexed method of finding the H.C.F. depends on the principle that, if a quantity measures two other quanti- ties, it will measure any multiple of their sum or difference. MEASURES AND MULTIPLES. 135 1 1 + 3 — 2 -6 -8 + 4 -9 + 13 -3 + 6 («) (b) (c) [= (d) if) -45 -45 5 + 6 -12 — 22 -9 -(a)~(b)] 2 1 + 6 -2 -6 -16 + 4 -.18 + 13 -6 + 6 (a) X 2 (b) 3 + 4 -6 -12 -5 -27 -25 -2 ' 15 15 C.F.= + 18 + 20 — 2 -36 -30 -6 -66 -60 -6 (.)X3 (£^)X5 1 25 27 + 3 + 30 + 36 + 3 -60 -54 + 1 -110 -108 (e)x5 (c;)x9 2 -6 -6 -2 (9) H. 1 +3 ^{x + iy. + 3 + 1 The coefficients are written in two lines, (a) and (h). They are then subtracted so as to cancel the first terms, (a) is next multiplied by 2, and added to cancel the last terms. If (c) and (d) had been the same, their terms would have been the coefficients of the H.C.F. Since they are not, we proceed with them as with (a) and (h) till they become the same. When (a) >nd (b) do not contain the same number of terms, it is more convenient to find only (c), and then use this with the quantity containing the same number of terms. The general rule is to operate on lines containing the same, or nearly the same, num- ber of terms. 136 MEASURES AND MULTIPLES. 4. Find the H.O.F. of 3 a;' + 2 :^' - 14 ^ + 8 and 6 ^' - 1 1 ^'^ + 13 ^ - 12. 8 +2 -14 +8 6 -11 +13 -12 («) id) 6 +4-28+16 (a)x2 15 -41 +28 (5-7)(3-4) H.C.F. = 3a;-4. (b)~(a) If (a) and (h) have a common factor, its first term must measure 3 and 6, and its last term must measure 8 and 12. () (a*'^ — b^^ — c^ (in + 7i) (m^ — 7i^), and (a — 5) (a^ + b^) ^ c^{m~ n) (m^ + n^). The H.C.F. of the last expressions is c*(m^ — n^) ; hence, the L.C.M. = cHm'-n')^^^;^^'^^^' I^ (x — ay measures x^ + qx + ?-, find the relation be- tween q and r. Let :?; + m be the other factor ; then x^ -}- qx -i- r = (x — ay(x + m) ^=x^ -{- (m — 2d) x'^ -{- (a? — 2 am) x + nid^. Equating coefficients, m— 2a ==0, a^—2am--=q,ma^ — r. r ft r"^ or a = — • 4 Hence, m = 2a, and •. «^ -4a^ = ^, 2a' and a^ = _ 9 8' or <:/ = -- 1^ ; and a' = Therefore, 4~~ 27' ^27 = 0. 138 MEASUEES AND MULTIPLES. Or thus : Dividing x^ ^ qx -\~ r by (x — a)'^ we find the remainder (q -\- S a^) x -\- r — 2a^, and as this will be the same for all values of x, we have, by equating coefficients, ^ + 3 a^ ^ 0, and r - 2a' = 0, or q^ — 27 a^ and r^ = 4:a^ ; therefore — + ^ = 0, as before. Ex. 43. Find the H.C.F. of the following: 1. 2x' + 3x'+bx' + 9x~S', Sx'-2x'+10x'-6x + d. 2. x'+(a + l)x''+(a+l)x + a] x'+(a-l)x''-(a-l)x+a. 3. px^—(p—q)x'^-\-(p—q)x-{-q ; px^—(p-\-q)x'^-\-(p+q)x—q. 4. ax^—(a-h)x^-(b-c)x-c] 2ax^+(a+2b)x''+{b+2c)x+c. 5. l-^^x-S^x^+^x'-x'] l-lji^x~3x' + lj\x'+x\ 6. ac'« + ^c?'* + (a + 5)c«+^ a'c'' + a'c' + c^'b' + b'c\ 7 .. a V + a^-2 abx" + Z> V + a^b^ - 2 a'b and 2 aV - 5 aV + 3 a« - 2 Z> V + 5 a'bV - 3 a'b\ 8 . (ao; + byy ~{a — b) (x + 2;) (ax + Z^y) + (a — Z>)^ ^2; and {ax—byf — (a + 5)(ri; + z){ax — Sy) + (a + hfxz. 9. a(Z>^-c^) + Z>(d^-aO + c(a^-&0 and a {ly^ -&)-\-b {c' - a') + ^ («' - ^0- 10. a + a'"* + 6^ + 1 and a'^ - a'"* + a^'^ — 1. 11. \i X? ^r (^^^ -\-bx-\- c and x^ + a':?; + Z*' have a common factor of one dimension in x, it must be one of the factors of ()^ + (b - cf + (0- a)' and (a' - bj + (b' ~ cj + (c' - d^. MEASUEES AND MULTIPLES. 139 13. Find the H.O. F. of and 3(y' - 4y' + 5y - 2)x' 14. If x^ -\-px + q and x"^ + '^^ + "^ have a common linear factor, show that (n — qy + '^{'^ ~~ pT = 7n{m — p)(n — q). 15. Find the L.O.M. of :r'- 3.^2+3:^-1, x^-c^ — x+l, x'-2o(? + 2x-l, 2.ndix'~2x' + 2x' — 2x+l. 16. Find the L.C.M. of x^-\-^x'' + llx+Q, x'' + 1x'' + Ux + ^, x' + Sx"- + 19^ + 12, and x' + dx' + 2Qx + 24. 17. Find the value of y which will make 2(y' + 3/)^' + (lly - 2):^ + 4 and 2(f + y')x' + {ll\/--2y)x' + (f + 5y)x+5y-l have a common measure. 18. The product of the H.C.F. and L.C.M of two quanti- ties is equal to half the sum of their squares ; one of them is 2x^ — llri;^ + 17:^: — 6 ; find the other. 19. If :^ + a and x — a are both measures of x^ -\-px'^ -{- qx -[~T, show that pq — r. 20. li x^ -\- qx -}- r and x^ + mx + n have a common meas- ure (x — ay, show that q'^n^ = rn?r^. 21 . If the H.C.F. of x^ -{-px + q and x^ + mx -\-nhQ x-\-a, their L.C.M. is x^-\- (m — a) oi^+px^-^ (a^+ mp) x-\-a(m — d) (a^+p) . 22. li x^ -\~ qx+1 and x^ -\-px^ + 5'rr + 1 have a common factor of the form x-^-a, show that (p-l)-'-^(p-l) + l = 0. 140 FRACTIONS. 23. If x^ -j- px^ -}~ q smd x'^-\-mx-\-n have x -\- a for tlieir H.O.F., show that their L.C.M. is rr*+ (m — a +p) 0(^ +p (jri — o^x"^ -\- c^ {a—p)x + a? (a —p) (m — a). 24. If x'^ +px + 1 and a^ -\-px^ -^ qx-^X have x — a for a common factor, show that a = \-q 25. Find the H.C.F. of {a^ -V^J -\-{})' - &y ^{c"- - a^J 2.Tidd'{h — c)-\-l\c~d)Arc\a~h), 26. If a be the H. C.F. of I and c, p the H.C.F. of c and a, y the H.C.F. of a and ^>, and S the H.C.F. of a, ^>, and d?, then the L.C.M. of a, Z>, and c is 27. If ^+^ be the H.C.F. of the x'+ax+h and x''+a^x+h\ their L.C.M. will be x^ -\-{a -\- o! — c)x^ + {aa^ — &^x-\- {a — c){o} — c)c. 28. Show that the L.C.M. of the quantities in Exam. 2 (solved above) will be a complete square if X =^ y^ -\- z^ — 'if z^. 29. Find the H.C.F. of x^ + 2x^ + 3^* - 2x'' + 1 and ^x^ + x' + 11 x'' -1x^-2. Fractions. § 33. When required to reduce a fraction to its lowect terms, we can often apply some of the preceding methods of factoring to discover the H.C.F. of the numerator and denominator. Examples. 1 ac + hy + ay + he __ c{a + h)-\-y(a+h) __ c + y af+2bx + 2ax+bf ~ f{a+h) + 2x{a+b)~ f+2x FRACTIONS. 141 a^ _ ha' - a'h'' + ah\ _^ a\a' + h'~ah{a + b)'\ a' - ba' - a¥ + h' ~ a{a' - h') -h{a'- h') _ g (g + h) {a — hy _ a ~{a-h){a'-h') " a^ + h" ■ x^ — x^y-\'0(? y^ — x^ y^ + xy^ — y" Here the numerator is evidently (x^ — ?/) -^ (:r — y), and the denominator is • ^. The result is therefore ^• {x-\-yy' — x' — ^_ hx'^y + lOrg'y' + Wi x^ y^ -\- h xy^ {x + yy-\-x'^y'~ {x^yy-x'y'' + {x? + y''y-x'f _^ hxy\x? -\-f-\- 2xy {x + y)] (^' + y' + ocy) [{x + yf + xy + x^ + y''- xy] bxy{x + y) {x^ + xy+y'') _ bxy{x + y) ^{x' + xy + yy 2{x' + xy + yi x''~\2x + ?>b 5. x^~\Ox' + ?>lx — ^0 Here we see at once that the numerator =^{x~b)(x — l) ; and it is plain that :?: — 7 is not a factor of the denomi- nator ; we therefore try x—b (Horner's division), and find the quotient to be o;^ — 5 ;r + 6. Hence, the result = — — -• X ~DX-\-0 x'^2x^ + ^ x' — ^x'+Sx-^l The factors of the numerator are at once seen to be x^ -\-2x-{-?> and ^^ — 2 a; + 3, of which the latter is one factor of the denominator, the other being (Hor- ner's division) x^ — 2x — 7. Hence, the result is —-^ ^t_^. x'-2x-l 142 FRACTIONS. Ex, 44. Reduce the following to their lowest terms : 1 ^^ ~~ "^ ^ + ^ ' 3 xy~ — 13 rry -f 14 ^ x'-2x'-Sx-m' 73/^-173/^ + 6?/ „ x^-\~ aV + a'^ . x^ -\-x~12 x^ + ax^ — a^x — a^ x^ — bx"^ -{-7 x —3 a^-3x + 2. x' + 2x'' + 9 x^ + 4:x' - 5 x^- 4.r' + Ax' — 9 2 + bx . x' + 2x'' + 2x 2b + (b'-A)x~2bx'' x' + 4:x a^x + 2aV + 2 ax^ + ^* 25^* + 5^^ — :r — 1 rz:*^ — x^y + ^^y'^ — ^r^ + a^y^ — ^^y^ + xy^ — 3/"^ ri;^ + x^y + :i;^3/'^ + ^y + x^y^ + rr''^3/^ + xy^ + 3/^ 3aV~2aa;^-l y^ a) ^ ^ "^ • 4aV-2aV-3a^^ + l' ^. + g __.^Yr3/-3/^' g d'(b -c) + b\c ~a)-\-c\a~ b) abc (a — b)(b — c){c — a) <, {a + h + cr . a\b - c) + b\c~a) + c'(a - b) 10. From Exam. 4 (solved above) show that {a-bY+{b~cy+{c-aY _ {a-by+{b-cf+{c-af {a-bf+{b-cy+{c-af b{a-b){b-c){c-a) 11. ' (^ + y)'-^-y' . {x + yf -x^-y'^ FRACTIONS. 143 12. Show that {a~hy + {h - cy + {c ~ ay (ci-hf + {h - cf + {g ~ af = i^[{o.-by + (h-cy + {c-ayi § 34. In reducing complex fractions it is often convenient to multiply both terms of the complex fraction by the L.C.M. of all the denominators involved. Examples. 1. Simplify ^^-- + ^^)--^^---X Here the L.C.M. of all the denominators involved is 12 ; hence, multiplying both terms of the complex fraction by 12, and removing brackets, we have 6a: + 8 — 8 + 6^ ;^ 12^; _ Zx 21-4rr-17 4-4.T \-x a — b a ~ -_ l + ah -J , a{a~b) 1 + ab Here, multiplying both terms by l-\-ab, we get a{l + ab')-a + b _ b {a' + 1) _ ^ l + ab + a{a-b) a' -f 1 1 1 + 4 — ^ Here, multiplying both terms of the fraction which follows :?; — 1 by 4 — a;, the given fraction becomes 1 at once ; 1 I 4 — ^ 144 FRACTIONS. and now, multiplying both terms by 4, we have 4 _ 4: 4^ — 4 + 4 — ^ 3x It may be observed that when the fraction is reduced to the form 7-^3, we may strike out any factor com- mon to the two denominators, and also any factor common to the two numerators ; it is sometimes more convenient to do this than to multiply directly by . the L.C.M. of all the denominators. A a- T-r fa + h , a — h\ fa^ + h^ a^ — h' 4. Simplify ' ^ ' » -^ ' ^ a-b a + hj \a'-~b' a^ -\-h\ Here the numerator of the first fraction is {a + by + {ct — b)\ and the denominator is a^ — b'^\ the numera-: tor of second fraction is (a^-\-b'^y — (o^ — b'^y, and the denominator is a^ — b^\ the former denominator can- cels this to a^ + 5^ which, of course, becomes a multi- plier of the first numerator. Hence, we have ^^'Y?^tt'Jt^V^^^ = ^4±ff Occasionally, we at once discover a common complex factor ; strike this out, and simplify the result. 5. — ^ ^ Here the denominator a^ y^ c^ ab a b) c^ \a b cj\a b c and cancelling the common factor, we have , and, multiplying by abc^ this a ' b c _ abc be -\- ca— ab FKACTIONS. 145 I Ex. 45. Simplify the following : a+b . a~h l_i[l_|(l_^)]' gj^b a~b a—b a+b XX _1^ 1_ ^ x^y x — y \~a 1 + a /&, ' ^ j I 2.T a , 1 x^ — if' 1 — a 1 + a 1 1 + 1 + a 1 + a 2a^ a^ + ^\ a ab^ 4 ' 2b'' d' + b'' ~^b a-\-b . a~-b ^ j , b'' —7—, + — :j a + b + - ^ c -\- a c — a , a o. , . a-Vb , a — b , j . a' _IE^ + _^ a+b+ — c — a c -\- a b x—1 , y — 1 , 2 — 1 Sxt/z X y z ' yz — zx — xy -'- i -^ i -'- X y z 2 2 2 d + ¥-\-c' ^ d b-" c' dh'c' a A b c be ac ab 146 FRACTIONS. o. a — b o? — hy \a + b a^ -{-h' 10. <^__b+J_(i ^^+1 1 IV 2.bc 11. 12. 13. 14. a b + c 2(1- x) (1 - xy ^ / x - a V / a; + a l-x ^\l-xj^ x^-aV fx+a x + aj'^^'^ o / a + ^ \ __ o f a + ^ V"^ I fa + b x^ ~ x^ y -\- x^ y^ — x^ 1^ -\- xy^ -- If' _ fx — y x^ + x^x + x^y"^ + x^if + xy^ + y^ ' \x + y^ 16. Find the value of — — 1 y wlien X = } (a + b). 2na — 2nx 2nb — 2nx 17. Find the value of V[l - V(l ~ ^)] FRACTIONS. 147 18. Find the value of ^(a + hx)+^{a-hx) ^^^^ ^ _ 2ac V(« + bx) - V(« - bx) ^ (1 + c") § 35. When the sum of several fractions is to be found, it is generally best, instead of reducing at once all the fractions to a common denominator, to take two (or more) of them together, and combine the results. Examples. 1. Find the sum of x~\-y y — X _ o^j—j^ 2x-2y 2x + 2y x' + y''' Here, taking the first two together, we have {x + yy + {x~ yy ^ x" + f . ^{x'-f) x'-y''' now add this to — x' — y^ x' + y' . ^ (x' + yy - (x' ~ yj 4 x' y' and we get ^ ^ '^ \ — ^ ^^^ ^ -^-^^ x^ — y* x^ — y 2. Find the sum of 1 + ^ . AiX ■ 8;r \ — X \~x \-\-x^ l-Yx' \ + x Here, taking the first and the last together, we have (1 + xy - (1 - xj ^ ^x . 1-x' l-x'^ taking this result with the second fraction, we have ,1 + x'' l — xy 1 — a;* now take this result with the remaining fraction, and we get 1 , 1 \ 16^ l + x'J 1-a^ 148 FHACTIONS. x"' X''^ 1 , 1 O. 1 ; — — — -j- - 1 X''+l ^^—1 ^^ + 1 Taking in pairs those whose denominators are alike, we have ^3n . 1 ^2n 1 ^" ^ 1 X^'+l The work is often made easier by completing the divi- sions represented by the fractions. 4. Findthesumofl+-^^±i~l^^- 2(07- 1) 2^ + 2 By dividing numerators by denominators, this 2:r-2 2^ + 2 2.r-2 2:i; + 2 X , X — 9 X -{- 1 x — 8 5. 1 * X — 2 X — 7 X — 1 X —6 We have, by division, 2 .1- 2 _i. X - 2 x~7 x—1 x—6 2 2 2 2 :?; — 6 ^ — 7 a;--l _ 2(2^-8) 2(2.T-8) (x-2)(x-6) (x~l)(x~7) :^ax-m( ^ ^ "^ -^^^-8^ + 12 x'-8x+7j = (80 - 20 :r) -^ (x' - 16x' + S3x'- 152:?; + 84). [Denominator = (x'' - Sxf + 19 (x'' ~8x) + 84.] FRACTIONS. 149 6. Find the value of ^+|^ + ^^ when x = ^. By division, 4« . + l + _i^_2 + 4f-^^+- x — 2a x — 2b \X'—2a x—2b^ but the quantity in the brackets ^j^+P^^zA^ 0, since {a + b)x = ^ab. (x-2a)(x~2b) y ^ J Hence, the value of the given expression is 2. Ex. 46. Simplify the following : x — a ■ cc^^ax + a^ x^ — a^ 5 x-\- a x^ — a^ 2 a^ + b^ a'~^a^b + ^ab''-b^ a(a-b)-b(a~b) a'-ab + b' a'-b' a'+ab + b' 3 / 1 . 1 . 2a \/ 1 1 2x \a-\-x a — 4. ^^ + :?; a^ ~\-x'^J \a-\-x a~x cr -\- x' b ab , ab 5. a + b a — b ab — V^ d^ + ab 3 + 2^ _ 2 -3a; , l^x-x^ 2-x 2 + x x'-4: ' e. :r..^.+ ' ^ lf Sx + 2y \ l fSx~2y ■ 2\3x-2yJ 2>^3^+2y, 8X —\~ ^ «^ ^ -L -L O^ I it/ , • 7^ 7~7; — TT 7^ ^7-T+ /. 9 ^TT + 9. 2a;-l 2a;+l a;(l-2a;) a;(4a;^-l) :^(16a;*-l) 1 4 9 x-1 2a; + 2 ^• + 2"^2(.r + 3) (x + 2)(x + 3J 150 FRACTIONS. ^{x + y) 2{y-x) ^{x'-f) , ^{x' + y') 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. x — y x + y ^ + f ■r )' {x+^y] -^2 X -{-a x-\- h^ fa + X 4: ax Sd^x a — x\ \a~x d^-j-x"^ a^-\-x^ a + xj ■ \d'-x^~^ a' + x^ d' + xy' 5x-4:.12x+2_ 10:r + 17 9 11^-8 a , a , 18 a' 2 a' ■ab' a' + b' a'-b' (a-b)(d' + b') 12 :r + 10a ^ 117a + 28^ _ ^g a'-b' 3x-\-a 9a + 2x -3 5x-A 4:x-17 8:t'-30 . 10:^- x-4: 2a;-72:?7-5 x-l' -n- -I XT, 1 n a + b + 2c , a + b + 2d Find the value of ' ' -| ' ' , a + b — 2c a-\-b~-2d when a-\-b- c + d a^"" yn^^ r f ^n_yn x^ + y'^ x^'—^f x^' + y'' {a-by^ {a~by^ 1 ^a-bf-l {a-bf+1 {a-by-l {a-by+l __1 + I I {a'-h'){x'+b') ^ (b'-a%x'+a') {a^+a%x'+b') l+x , l-x _ 2 _2^ l_^3-r^_^^3 ]L-a;^ x^ + \ o?+o?b + ab'+b' (a+by-Sab (g-bf-a'+b' a'-d'b-ab' + b' (a-bf+Sab (a + bf-a'-b' FRACTIONS. 151 § 36. The following are additional examples in which a knowledge of factoring and of the principle of symmetry is of advantage. Examples. ' {x + zy~^f~^{y + xf-z^^{z + yy-x' Cancelling the common factor x — y -{- z in the two terms of the first fraction, there results ~ ; x + y + z hence, by symmetry, the denominators of the other two fractions will be x-{-y -\-z, and the numerators will be y-\-z~x,z-\-x — y. Hence, the sum of the three numerators =^x-\-y -\- z, and the result =^ 1. o o- yn cib I bc^ . ca impiy (^_^)(^__^)-l-(^__^)(^_^)"^-(^__^)(^_^y The L.O.M. of denominators is evidently (a — h)(b — c) (c — a). This gives for numerator of first fraction — ah{a ~- h) ; and, by symmetry, the other numerators are — hc{h-—c), — ca{c — a). Hence, we have - ah{a-h) + bc{b - c) + ca{c- a) {a — o){o~ c) (c — a) _ (a — h)(h — c) (a — c) __ -t (a — b)(b — c) (c — a) 3. Keduce the following to a single fraction : {a — h)(a — c) {x — a) {b — a) (b — c){x — b) + 2 -. (c — a){c — b) (x — c) 152 FRACTIONS. Here the L.C.M. i^{a—h){h—c){c—a){x-a)(x~h){x—c)\ the numerator of the first fraction is — a(h — c)(x — h) (x — c), and, therefore, by symmetry, that of second is — h(c — d)(x — c)(x — a), and that of third is — c(a — h) (<)c — a)(x — h); and their sum is — [a(h — c) (x — b) (x — c) -\-h(c — a)(x-~ c)(x ~ a) -}- c(a — h)(x — a)(x — b)]. This vanishes if a = b; hence, a — b is a factor, and therefore, by symmetry, b — c and c ~ a are also factors. Now the product of these is of the third degree, while the whole expression rises only to the fourth ; hence, x'^ cannot be involved. The other fac- tor must therefore be of the form mx + n, in which in is a number. To determine n, put x = 0, and the expression becomes abc (a — b-\-b — c-\-c~-a)==0] hence, ?2 — 0, or the other factor is mx. To determine m, put a = 0, b = 1, c = — 1, and m will be found to be 1. The numerator is, therefore, x(a — b)(b — c)(c — a)y and the result is (x — a)(x — b) {x — c) Simplify a-\-b . b + c , c -\- a {b — c)(c — a) (c — a) (a — b) (a — b)(b — c) L.C.M. of denominators is (a — b)(b — c) {c — a)\ hence, first numerator is a^ — b'\ and, by symmetry, second numerator is b"^ — &, and third numerator is c^ — c^ \ the sum of these = 0, which is the required result. FRACTIONS. 153 4. Keduce 2 ,2^ 2 , {x-yy + {y~zY + {z-xy 1. X ^-y y-^ ^-^ {^-y){y--^){.^-^) Here the numerator becomes 2{y-z){z-x) + 2{x-y){z-x) + 2{x-y){y-z) + {x-yy+{y~zY+{z--xY, which is evidently [{x-y) + {y-z) + {z-x)J = Q. Observe that the denominators become the same by changing the sign between the fractions, and that the expression is symmetrical with respect to a and b. The numerator of the first fraction is a'' + 6 aW + 12 a'b' + 8 a'b', and, by symmetry, that of the other is ~b'''~6 bV - 12 b'a' - 8 ¥a\ Their sum is, therefore, a}-^ -h'' + 6 aW (a« - ^«) - 8 aW (a« - b^) = (a' - b') (a' + b' + 6 a'b' - 8aW) = (a^ - b') {a' - bj = (a' + ¥) (a^ - bj, and since the denominator of the given expression is (a^ ~ b^y, therefore the result is a^ + 5^ Ex. 47. Simplify the following : x-\-y ) \x + y ) \ a — b J \b — a J a-\- b I b -}- c I c-{- a (b ~ c){c~ a) (c ~ a)(a— b) (a — b)(b — c) 154 FRACTIONS. • + {a - h) (a -c) (b -a)(b-cy (c- a) (c - b) a — b . b — c . c — a , (a — b)(b ~ c)(c ~ a) a-^b b-^ c c-\- a {gl -^ b^{b ^ c){c -\- a) + • (a-\-b){a-\-c)(^x-^ay {a-^b){b~c)(x-\-b) & 9. (a -\- c)(b — c) {x-{- c) (x-y){x-z) {y-x){y-z) {z-x){z~y) (a — b){a — c) (b — a)(b — c) (c — a)(c — b) 1.1.1 i-oe-') (;-')(;'') e-')(i-' 10. x^ 'x'~-2y'\\ ,f2x'-f 11. 1 (^) + c - 2 a) (c + a - 2 Z^) (c + a - 2 ^) (a ,+ ^> - 2 ^) + 1 13. (h + cy'^(c + ay^(a + by a^ ^ P (a — b){a — c) {x — a) (b — d)(b — c) {x — b) 4 {c — a){c — b) (x — c) (:r-3/)(2;-:r) {y-z){x-y) {z-x){y-z) RATIOS. 155 {a+hY + (b-cf + {a + cY 2_. 2_ 2 (a + ^) (^ — + 7ZC? ) - -p ma + y^g _ W W ^ m6 + nd "i^^b + n(i 7?i6 + nd a c A very important case of this is 7n = l, n = ±:l; hence, a c __a-\- c a — c b d b+d b~d (5) Also^^^ (6) a+b e+d ^ ^ For, by (2) and (5), a _b _ a~b _ a-\-b . a — bc — d e d c — d c + d' ' ' a + b c + d Or thus : a — b__b d e — d a + b a_j_^ c_j^^ c + d b d 156 RATIOS. Generally, to prove that, if 7 = -, any fraction whose a numerator and denominator are homogeneous functions of a and h, and are of the same degree, will be equal to a similar fraction formed with c instead of a, and d instead oih. Express the first fraction in terms of -, and for - substi- c ho tute its equivalent -, and reduce the result. By (2), the fractions may be formed of a and c^ and h and d. jr-a c e ma -\- nc -\- pe ace r>^\ 11- = -==-, !- ^^^-— = - or - or — (7) b d f mh + nd+pf h d f ^ ^ mbl -) + nd mu -|- nc + 2^^ _ W v^ mb + nd -f- pf "rnb + nd +pf {mh^nd-^pf)- b a ^^(7) nih^nd-\-pf h Tc a c ^ m p If -— - and — — -C-, b d n q ma z^pc pa rb mg mxi pa ^r>\ nb zh qd qb ±1 nd nb qb T^ ma pc^ ma_±^ 1 /p-x ' nb qd nb ±qd pa m£ pa rb mc qb nd qb ± nd But — r==-^, hence the equality stated in (8). nb . qb ^ J w h d f n q s ma :^pc ±1 re _ pa zhrczt me _ _ ^^ _ /q\ nb ±2 qddzsf qb zt sdzhnf nb RATIOS. 157 If an upper sign be taken in a numerator, the corre- sponding upper sign must be taken in tlie denominator ; if a lower sign, the corresponding lower sign ; otherwise, all the signs are independent of each other. Examples. 1 . if - ^ -, show that —- ^ -— • b d 1a + 5h 7c + 5d The given fraction = 5^-4 5^-4 _^ ^_d ^ 5c-4:d 7^ + 5 7^ + 5 7c. + 5cZ' b d 2. If - ^ -, show that ^-,^--^3 ^ 3-,^-^, Dividing the given fraction by b^, we have 4+4 3^-4 and this becomes, on substituting for - its equal — , b d ^3 2 c} ± I- 3 — ^d''^ d' _ 2c'+Sc'd 3_^_4 Sc'd-4:d'' d' 3. If 3a = 25, find the value of ^' +^' a'b - ab' — - + 1 j ^ [ — — - J [by dividing both numera- tor and denominator by h^\ But, from the given relation - = -, we have, by substituting for ■-, (^ + l)-^(|-|) = 35^(-6) = -5f 158 RATIOS. li - = -, prove that , ' ,, X -= — ^ ) • We nave - —-j= — —.' and this multiplied by -gives — = ( ' \ • ^ ^ d^ d" \c + dj x^ — ax^ -{-bx-\-ccr — ax^b a Multiplying both terms of second fraction by x, it be- a^ -f~ aa^ — hx . x^ — ax'^ + bx now each of the given fractions difference of numerators c -, difference of denominators c Hence, x"^ -{- ax — b =^ x"^ — ax -\- b or 2ax = 2b. b Therefore, x = - a Iff^g^g show that f + "^ + ^^ ^ ^' + "' + < b d f bd+df+fb b' + d'+f y ac ce __ eg _ ac -\- ce -{- ea ^^ bd ~ 'df~ fb ~ bd+ df+fb By (7) making m=^n =^p = 1. Also^' = ^-^ = ^-+^^i^. By (7). But 7-:==—, hence the required equality. bd b^ The problem is a particular case of (9), with all the signs +, and a for m, b for n, c for p, etc. . RATIOS. 169 If the fractions given equal to one another have not monomial terms, instead of seeking to express the proposed quantity in terms of one fraction, and then substituting an equivalent fraction, it is often better to assume a single letter to represent the common value of the fractions given equal, and to work in terms of this assumed letter. 7 Tf ^"1~ ^ — ^ + ^ _ c-\-a ' ?>{a-b)~ ^{h - c)~ b{c - a) prove that 32 a + 35 5 + 27 c == 0. Assume eacli of the given fractions = x, so that a-\-h = ^{a — h)x, h-\-c=^^(b — c)x, c-\-a='b{c — a)x, a-\-h . h -\- c , c-\-a , 7,7 , x c\ or — ^ 1 ^ 1 ~=^x{a—b-\-o — c-\rC—a)^\). O 4b O Hence, adding these fractions, we have 32a + 35^ + 27^ = 0. This example might also be worked as a particular case of (7); thus, aAr'b _ h-\-c _ c + a Z{a-b) 4:{b-~c)~b{c-a) _ 2Q{a+h) + lb{h + c) + l2{c+a) _ ?>2a + ?>bh+21c m{a-~h)+m(b-c)+mlc-a) o Hence, 32a + 35^> + 27g ^ X ^ + ^ =0. ?>{ar-h) Transposing terms, etc., we have h" bd d'-'^ f df'^ d}~ ' "(f-|)"+(7-|)'-»^ 160 RATIOS. That is, the sum of two essentially positive quantities = ; therefore each, of them must = 0. Hence, we have a c r. J e c r, ---=0,and^-- = 0; hence, f = ^=^ therefore, g-' = ^ ' + ^' + < Also f = P^-, tence, g = (p^^)\ ; b b + d+f 'b-' \b + d+f) therefore, ('^+^Y= f + f + f ■ Ex. 48. 1. 11 - = —, prove that — ' b d'" ab-ib' cd~4:d' 2. If- = —, prove that prove /a + cV 6 cZ'^ W-d'^ \b-d) 3. Given the same, show that each of these fractions \\b-'+d-'J 4. If 2:27 = 3 y, write down the value of 2x'~x'y + f ^^^ ^^ x'-Sx^y + 2y\ x^y + xy^ + 2 y^ (a;''^ — y^)^ 5. If- = -=:-, show that- = -— , ^ / b a f mo — nd —pj 6. From the same relations prove that — — / o^ — ^g— '^^ Y h^ \h — md—nfj 7. Ifl±^^^fl+£i±4\then^ = (&-a)H-(5 + a). RATIOS. 161 8. If -^^^^ — ^ — / ^ \y 1 = a, prove that x — •\/(a + x)—-^{a-—x) 1 + 0^ ^ jn mx -j- a-j- b mx — c — d ,i, b — c 9. li ■ ■ — = -, prove that x = 10. If nx-{-a-\- c nx — b — d' n — m a~b b — c c — a __ a-\- b -}- c ay -{-bx bz-\- ex cy -\- az ax -\- by -{- cz then each of these fractions = , a~\-b -\- c not being zero. a — b 2{b — c) 6(c — a) 12. If V«+V(«-^) _ 1_ 3i,ow that ^^^ = (^^\ -\Ja —\/{a — X) a a \l-{- aj rp^ oi'y 01 'Y'^ 13. If —- — ^—- = -^ -, and x, y, z be unequal, show that each of these fractions is equal to x -{- y -{- z. 14. If ^' + 2^ + 1 ^ y^ + 2y + 1 g^Q^ ^^^^ ^^^^ ^f ^^^g^ :r^ — 2;r + 3 y^ — 2y + 3 fractions = {xy — 1) -=- (xy — 3). 15. If 25^--16 ^ y-4) ,3i,,^that^+l=§. 10a; + 8 2rc-4 a; +5 5 •^ 6 + c y-2b y~2o 17. If Y^!+i.VV^^±^1 = Y^^^^\ prove that 25a^ + 276' + 22c^ = 0. 18. Tf a' _ ^' _ c' ^ o;^ — yz y — 2;;r z"" — a;y show that a^x + Z>^y + /(aaM^ (1) c^'^aM^^ (2) Dividing (1) by (2), -== -77 ; hence, ac^ = db^. c as Also, b' = 9aMi (3) Dividing (3) by (2), - = 3a ; hence, b' = Sac. c 7. Find the relations subsisting between a, b, c, d, e, when ax*' + bx^ + cx'^ -j- dx -\~ e is 3i> complete fourth power. Assume ax^ + bx^ -{- cx"^ ~\- dx -\- e = (a^x + e^y = ax^ + 4 ah^x^ + 6 cae^x"^ + 4 a^e*.?; + e. Equating coefficients, we have ^ — 4 ai^i, c = 6 aM, c?=4a^ei; whence, 5c?— 16ae (1) bc:= 24:a^ei = 6a X 4aiei=:: 6 ad (2) cc?--24al^--6c X4aic4^65e (3) 168 COMPLETE SQUARES. 8. Show that x^ -{-px^ + qx"^ -{-rx -\- s can be resolved into two rational quadratic factors if s be a perfect square, negative, and equal to p^ — ^q Since — 5 is a perfect square, let it be ?^^. Assume ^* -{-px^ + ) be a com- plete square, then a = 3^, or Z> = 3a. 11. Find the simplest quantity which, subtracted from aV +4 a5^+ 4 acx + 5 5c + b'^c'^, will give for remain- der an exact square. 12. x^ — 4: x^ —- x"^ -\- 16 X — 12 is resolvable into quadratic factors of the form x"^ -f mx -{-p and x"^ -{- nx -]- q ; find them. 13. Find the values of m which will make x'^ -f max + a^ a factor of x* — ax^ -f a^x"^ — a^x + a*. 14. Show that if ^*+ ax^-\-hx'^-\~cx-\-d be a perfect square, the coefficients satisfy the relations 8c = a(4:b — a^) Sind64:d=(4b-a'y. 15. Investigate the relations between the coefficients in order that ax"^ -f 5y^ + cz^ + ^^y + ^3/^; -\-fxz may be a complete square. 16. If r^^ + ax"^ -i-bx-\- che exactly divisible by (x + dy, show that i (b' -d'')=- = d{a-2 d). 17. Determine the relations among a, b, c, d, when * ax^ — bx^ -\- ex — d\^ 2. complete cube. 170 COMPLETE SQUARES. 18. The polynome ax^ -{'?>bx^ + ?>cx + d is exactly divis- ible by {a — xy ; show that (ad - bey = 4 (ac - h') (hd - c"), 19. Find the relation between^ and q when x^ -{-px'^ -\- q is exactly divisible by (x — ay. 20. If x"^ + nax + a^ be a factor of x^-\- ax^-\-aV-\- a^x -\- a*, show that 71^ — n — 1 — 0. 21. If x^ + ax^ + hx'^ -{- ex '\- dhoi the product of two com- plete squares, show that (4Z)-a7 = 6H {^h'-d')a = ^c, a-yy(Sa'-2h) = Sb. 22. Prove that ri;"^ + »^^ + 5':^:''^ + ro; -f s is a perfect square if p'^s = r^ and q =^+ 2y^s. 23. If aa^ -}-Shx'^-{-Scx-\-d contain ax"^ -{-2hx-{-c as a fac- tor, the former will be a complete cube, and the latter a complete square. 24. If mV -f ^rx; -\- pq -{- q"^ he a perfect square, find p in terms of m and q. 25. Find the relation between p and q in order that x^ -\-px'^ -{- qx-{-r may contain {x + 2)^ as a factor. 26. li x^ -\-px^ -{- qx-\-r be algebraically divisible by 3:^;^ + 2px + q, show that the quotient is ^ +^- o CHAPTER V. Linear Equations of One Unknown Quantity. § 38. Preliminary Equations. Although the follow- ing exercise belongs in theory to this chapter, in practice the numerical examples should immediately follow Exer- cise I., and the literal examples Exercise HI. Like those exercises, this one is merely a specimen of what the teacher should give till his pupils have thoroughly mastered this preliminary work. But few numerical examples are given, it being left to the teacher to supply these. Ex. 60. What values must x have that the following equations may be true ? 1. x-b-=0', x-?>\ = 0] x~a = 0] x + ?>^0. 2. x + ^ = 0-^ x + a = 0\ a; + 3==6; :r — 4 = 6. 3. x~a=^h] x-\~a^=^c) x~h = ~c] 6 — a: = 3. 4. 8-37... 10; 5 + a;--ll; 9 + a; = 4; 1-x^-b. 5. 8 + :?;-= — 6; a — x=^'^h', 2a^x + 2>h] ^a = bh—x. 6. 2x — ^-=^] ?>x + d>-=20] ax = a^; mx^hm. 7. Sx = c; ax = 6; ax=^0] (a -{- b) x = b -{- a. 8. (a~b)x = b-a] (a+b)x=(a+by] (a~b)x=a^-b\ 9. {a + b)x=.b'-a'] (a'~ab + b')x = a' + b\ 10. (a'~b')x^a~b', (a' ~ b')x =^. a + b ; (a' + b')x=l. 172 LINEAR EQUATIONS. 11. a-\-x — b = a-{-b] x — a-{-b = b ~ x-\-a. 12. 2a — x = x — 2b] ax-{-bx = c; ax — b^=cx. 13. ax~b = bx — c] ax — ab = ac. 14. ax — a^ = bx ~ b"^ ] ax — a^ = bx — b^. 15. ax~a^ = P--bx] ax -{- b -{- c = a -{- bx -}- ex. 16. a~bx — c = b-~ax-\- cx] a-\-bx-\-cx^-=ax—b-^cx^. 17. bx ~ cx^ -\- e =^ ex — b - -ex^] 3X 18. 6 c ax^-—' 19. a5:r=f + ^ lcx^^ + ^. b a be 20. \x = b] 1^ = 8; 0.5^ = 2; 0.3x = 0m. 21. 0.02rr-:20; 0.3:r==0.2; 0.4:?;= 0.6. 22. 0.i8a; = 1.8; --5; ^ - c. a b nc. ci^ h ^ _ <^^ _ z. 6 e a+b a+b 25. 26. 29. & — a a — b' a+ J 5+a a-\-b a — e ^ a-\-e'^ a + b' 1_1. 2_3 a; 2' a; 5' 1_1_. l_a. ^ ab' X b' a b, 7_1 1_ x~ c' X 3 4 3 , 4 _33 20 "^50; 5x 5; - + -=0. 3 x^ c 5 -f^ 7 ■ 5-7, 9 27. -=4; --=T 28. -7 3a; -4 4-3:r LINEAR EQUATIONS. 173 30. (x-4:)-{x + b) + x=^3; 2x-(x-b) ~(4:-Sx)==5. 31. 2(S-x)+S(x-S)=0', 2(3a;-4)-3(3-4rr)+9(2-^)=10. 32. a(l~2x)—(2x — a)^l', x — 5(a~ x) == bx — 5a. 33. mx(3a~4:) + Smx — Sa+l = 0. 34. a(bx — c)-{-b(cx — a)-\-c(ax — b) = 0. 35. a (ax ~b)-\-b (ex — c) -\- c (ex — a) — 0. 36. a (bx — a) -\- b (ex — b)-\-e (ax — c) — 0. 37. a(x-2b) + b(x~2c) + c(x-2a)==^a'' + b' + c', 38. 3J3[3(3:r-2)-2]-2|-2=:l. 39. 9J7[5(3:r-2)-4]-6|-8==l. 40. i^i[iab + 2] + 2) + 2] + 2| = l, 41. 4?|[iab + 2] + 4) + 6] + 8| = l. 42. iii[ia^-i)-i]-ii-i=o. 43. m[m^-H)-H]-m-n=o, 44. llk\[^(f[|^ + 4] + 8) + 12] + 20| + 32-68. 45. tJf[ia[^^ + 7]-3) + 6]-l} = 4. 46. r\q[p(n[mx-~a] — b) — c] — dl~e = 0. 47. (l + 6:r)^ + (2 + 8a;)^==(l + 10;r)l 48. 9(2ri: - 7)^ + (4a; - 27)^ =: 13(4:^;+ 15)(a: + 6). 49. (3-4rr)^ + (4-4^7-:2(5 + 4^)^ 50. (9 - Ax)(9 - 5x) + 4(5 -^)(5 - 4^) == 36(2 - xf. § 39. In solving fractional equations, the principles illus- trated in the sections on fractions may frequently be applied with advantage, as in the following cases. When an equation involves several fractions, we may take two or more of them together. 174 LINEAR EQUATIONS. Examples. 1. Solve ^ + g-^=-i-. Here, instead of multiplying througli by the L.C.M. of the denominators, we combine the first fraction with the last, getting at once 7:^-3 7 1 6ri; + 2 14 2 7x ~3~3x-{-l, and x=l. 2 Solve ^^ + ^ - 1^^-^ +g^!g-£zLl^,. 9 17a; -32 ^3 12 36 In this case, taking together all the fractions having only numerical denominators, we get 8^ + 34+12^-21^ + rg+16 ^ 13.r-2 . 36 17^-32' 0T — = '^^^-^ , .-. 425 r^; - 800 = 234^^^ - 36 ; 18 17^-32 hence, ^ = 4. It is often advantageous to complete the divisions repre- sented by the fractions. 3. Solve 4^-17 __3|_-22a; 9 33 X\ 54: J Here, completing the divisions, we have 4^_17_1 , 2^^ _6 X 9 9 9"^ 3 " ''^ :r"^9' 9 9 X n Therefore, — 2 = — -, or rr = 3. X LINEAR EQUATIONS. 175 ax -\- h . ex A- d , 1 1 ^ — = a-\- c. X — TYl X — n am~\-h .\a + 6. 7. cn + d + c + —^ — — a-\- c. X — 771 X — n .*. (ar)x + Z^) (x — 72) + (en + d) (x — m) = 0. .'. (am + ^ + <^^ + <^) ^'^^ — (<^ + c) 7)in + Z)7Z + dm. 5. Similarly may be solved ax + h ex - ■ d ex^ +fx — g -a-\- c -{- e. x ~vi ' x — n ' {x -— Tu) {x — n) am, ~{~ h ^ en -\- d [e (m + n) +/] x — emn — g X — 'in ' x~n ' (x — 7Yi) {x — n) .'. (am, + h) (x — n) -{- (en -}- d) (x — m) + [e (?7^ + n) +/]^ — eTnn — ^ — 0. .•. [(a + c)m + ^ + ((? + e)7i + d-\-f]x = (a~\-b -{- e) mn -\-bn-\- dm + g. = 0. 132^7+1 8:y + 5 . 3:r+l :^-l .-.44- 43 + 8 + 52. 13 3:1' +1 .•.39.T+13-=43r^ 25 -i x-\ 43, and x - = 52. or 13 43 1 3a;+l 14. M^±M^5- 23 x+l ' Sx + 2 ' ' a;+l Taking the last fraction with the first, and multiplying the resulting equation by 15, we have 240:r + 63_^,^ , 5^-30 ;.80- 3x + 2 97 :75- x + l - 75 + 5 - Sx + 2 97 _ 35 3:^7+2 x+l' 35 x+l X = 27, and x = 3f . 176 LINEAR EQUATIONS. 8 ^~<^ , X —h , X — C b -{- c a-\- c b -\- c . x — a 1 I x~b 3. l + ^JH^-^l ^0. 9. 10. b + c a + c b-\- a ' ^~(<^ + ^ + g) . X — (a -}- b -{- c) . rg— (a+^ + g) _/ b -}- c a-\rC b ■\- a which is satisfied by a; — (a + ^ + ^) = ; .*. a; = a + Z> + ^. m . n _ m -{-n x — a x — b X — c 7n(x ■— c) , n(x — c) , .*. — ^^ ^ -) — ^^ —^ = m-\-n. x — a x — b .Ma^^^njb-cl^^ (See Exam. 4.) X — a x — b :. \m{a — c) + n(b — c)]x = mb (a — c) -{- na(b — c). 3Vv-5 77 9\7/-lOj 18 12 2 5 . 1 **3 ' 3/-5 ' 6 ' 3/-7 9 3/-IO 18 2/-5 y--7 3/- 10 2/ - 5 3/ - 7 .-. ^:i^ + -^=^ = 0. 2/-5 y-7 .•.-82/ + 50-0. .•.y = 6i. Ex. 61. 10:r + 17 12a; + 2 _ 5a; - 4 18 13a; -16 6a; + 13 9a; + 15 . o^ 2a;+15 15 5a;-25'^ 5 4. 13. LINEAR EQUATIONS. 177 7a;+l _ 3 5 fx + A\ , o 1 4:X-7 , 2-14:X d^ + x ^ 10 -3f^ __ 19 2:z:-9 7 14 2 21* 2x-{- a , Sx — a __ ^^^ 3(^^^2(^Ta)" ^' ^-4 , 3^-13 1 6^ + 5 18:2;-6 3 3:g+ 1 ^^-- 11 _ -1 2:?:-15 2:?;-10~ 8. ^-^t^ — 5^2. 12 _^ .r — 4 ^ 2 -f. .T-7 ^-12 ,^ 3a:- 19 , 3a;-ll n 10. : :— = D. X - 13 x+7 2:z:+l"^3(:r-3)~6 12. ^ + 1 I ^^ + 4 _ 9 4(:i,'+2) 5a; + 13 20 5(2.^ + 3) 5-7.-^^^_3^ 2a;+l 2a;-5 14. -^ + . 1 4 a: — 7 a; — 9 a; — 8 j^g 7 a: + 55 3>r__g 3a;'^ + 8 2a; + 5 2 2a:-4' 16 1^ I 15 _ 32 07-16 ~^a:-18 a;-17* 178 LINEAR EQUATIONS. l-25a ; S-2^x ^ 28-5.^- 10:?;- 11 x 15 14(^-1) 3 30 "^3* 18. 1 2 + 2i^^-^r.^ _,^ 2_^ :r — 2 6 — 6r^ + ;r^ :r — 5 ^^ 30 + 6a; ■ 60 + 8^ _ 48 ^^ 20. :?;+l ^+3 x+1 5x^ + x-S ^ 7x''-3x~9 5x — 4: 7.^—10 Q - 5/ • X o X ~Y~ J- I ^ O x~2 x—7 x~l ^—6 22 x'-Sx-9 x'-7x-17 ^ 2(x'~6x-lb) ^ X — 6 x — 9 x — S 23. 24. 25. 26. 27. 4^ + 7 ■ 4:r + 9 ^ 4:r + 6 . 4rg+10 4a: + 5 4;r+7 4:r + 4 4^+8* 2:?;-3 2^-4 2:i;-7 2a; — 8 2:r-4 2x-b 2:?;-8 2:r — 9 7a; + 6 _ 2:r + 4|- , x_^ Ux _ a;--3 28 23:?;- 6 4 21 42 :?; — 5,:?; — 11 X'-7 , x—9 X — o X -12 :?;-8 :?;-10 ^-125 2 — 6;r ^ 5:r-^(10-3a;) 2 13 ~^ 39 10 — 17;r . 1 + ^ , '7 ^ 1 ' 27(:?;^-:?;+l)"^2(a;^ + l)"^54(:?;+l) 9(0:2+1)* 2x''+x~30 , :r2+4:r-4 _ :?;'^-17 . 2;2;2+7a;-13 2a;-7 :r~l a;-4 "^ 2a;-3 30. _a . x — h __ (g — 5y __ 2(a — 3;) ■b x — a (x — a)(x — b) a-{- x LINEAR EQUATIONS. 179 i 3^ 12.r+10a 28x + 117a _j^Q 32 ^H^-^ 131:^-11 ^13 ^:^-7 ■ IS jrX-d 33. 1 X 16x 2(x~iy 2(:^-l) *2(:r^ + l) (x~l)(x' + l) 7|- — X X (^ 34. KI- + 4)-^V^=|^^-lJ- ^^ 3:r 81:?;'- 9 q 3/2:^'-1\ 57-3;r. 35. — • — -^=6x ( 1 2 (3:^;-l)(:r+3) 2^^ + 3 7 2 36 1 + 2^+1 _ 4:^ + 5 ^ x^-~x-\-'l _ . 2(:r-l) 2(a;+l) :r'^-2^ + l 7:^;~3Q 5a;-7 2-21:y 10| i-;r-3 21 .^42:y-171_-^Q , 2:^ — 9 _i/4_,7^>v 33^ 18:^-22 1+6.,^ 101-54^ , 13-2:^'^ ^ 8 ^ 8 37. 39. 40. 4-9:r 5-12:r o 240:^ 1-3^ 7-4:r 7-25:r + 12:r' 8:r + 25 16^+93 __ 18:r + 86 6.^ + 26 2.x' + 5 2.r + ll 2:r + 9 2x-\-n' 41. — i 1 L_+ 1 ^0. x-^a-\~h X — a + h x + a—h x — a — h § 40. The results deduced in tlie sections on ratios may often be applied with advantage. 180 linear equations. , 7 Examples. ax -\- _7n ex -\- d n .^ {axArl)d-{cx-\-d)h ^ md-nh .p ^^^. ' {cx+ d)a — {ax-\-h)c na—mc 7)id — nh :.x = ♦ na — mfic ^ ax^ -\-hx -\- c a /O. =1 • rax -\- nx -{-p m {ax^ -{-hx-\- c) — ax^ __ a (vix^ -\-nx -]- p) — rrix^ w. bx4-c a , .*. ' — = — , etc. nx -\-p m Sx+7 Sx~13 3. 4. (Page 155) 07 + 4 X — 4: By (5), page 155, eacli of these fractions difference of numerators difference of denominators "8 x + 4c 2: + 4'^^2 ^ + 4'"^ mx ~\- a-{-b _ mx -{- a-{- c nx — c — dnx — h — d mx + a + h nx — c — d i. /o\ ^KK . . ; T = 7 — :^; or by (2), page 155, mx + a + c nx — b — d mx + a + b _ nx — c — d ^ b — c b — c 01 (n — m)x^^a-\-b-{-c-{-d\ .'.x= 1,2,3 6 6a x-{-da x — 2a x- LINEAR EQUATIONS. 181 Transposing, -— x — oa + 2 :i^ + 3a 6 3 x — a x — 2a Sx-9a _ 3:^-9 ^ ^ \.r.rK\r.4^^Q^ " x'-Sax-lSa' x' -3a:r + 2a^ 2Qa^^' ^'^^''''^" .\Sx~9a = 0. .-. x--= 3a. Ex. 52. 1. 1 + :r _ 1 1—x a 8. X -{-7)1 a + b x—m a—b 2. x~a 9. a + b a — b l-\- ex 1 — ex 3. ax-\-b _m ax — b n 10. a ~\~ bx _ c -{- dx a + b c + d 4. a + x ^ b + 2x 11. a + bx e + dx a — b e — d 5. a{h + x)_^ a — x 12. a—x a+x b — X b + X 6. a b 13. 2x^-~bx+9 _x''-1x+b a—x b —X 2x^-1x+?> x'-Sx+2 7. a + x __a + b a—x a—b 14. ax + b — e {b — eY ax-b + e {b + ey 15 - a^x a'-b' 2e o = —2cx. a — b a^ + ab + b'' 16. 2x-^1 _x+1 2^-3 :?;+ll 19. 23:^: + 54 36a;- 7 115^-29 180^+23 17. 4:x-b _10:?;-32 2x+l0 bx — Bi 20. 210^-73_21:r+7.3 310:z;-66 31:?; + 8 18. 57^-43 ?>9x- _7 21. mx—a—b __ rax — a—c 19:?;+ 13 13 a; +25 mx—c—d nx—b—d 182 LINEAR EQUATIONS. x^ -\- ax" — hx -\- c _ x^ ^ ax — h x^ — ax^ -^hx -\- c x^ ■— ax ^ b gV + a^hx'^ — acx + d _ dx^ + abx -- c aV — a'^bx'^ + acx + d aV — abx + c 8x' + Ux' — Sx + 5 _ 4:X' + Qx — 4 : ' Sx'-l2x' + Sx + 5 4:X'-6x + 4:' be ac ab \a b c J aoc o a6c \ b^ X. 28. ^ • ^^ (a + 5)^ ' (a -5/ (a^-Z)^)(a + Z)) {a -by „ :g + g ^ — 5 . x + c {a — b)(c~ a) (a ~b){b ~ c) (b — c)(c— a) a-\- c {a --b)(b — c) (c — a) fx + 2a\ , fa + 2xV ^ 30. x[ — ' \ A. a [ — ' p=2a. \x — a J \ a — X J x + a x — a a^ x'^+ax + a^ x^ — ax + d^ x (a;* + a V + a^) 32 x + ct ^ C^^ + ct + c ^ ' x + b \2x + b + c)' 33 {x+df-b'' {x+by-a^ :^-{a+by ' x''-{a-by \ X+l X+4: y \ X+2 ^ X+Z J LINEAR EQUATIONS. 183 § 41. Sometimes a factor independent of x can be dis- covered and rejected. Examples, ?>ahc hx , c^}? 1. a-^h a {a + hf Transpose — and factor ; then, a ah a + h Q , ah 36'-' {a + h) ah ] "a + h x+a x—h {a —h)(c — a) {a — h)(h — c) (h — c)(c — a) h + c (a — h)(h — c) (c — a) Add, term by term, the identity (Tb. III., page 67), x—a . x—h . x~c _ (a -h)(c - a) (a~h)(h ~ c) (h - c) (c - a)~ . 2x ._ h + c ' ' (a~hy(c — a) {a~-h){h — c){c — a) ■Hm 184 LINEAR EQUATIONS. 3. {x + a + by + {a + by -{x + bf - (x + af + x^ -\- a^ -\- b^ = abc. The left-hand member vanishes for ^ = 0, and hence, by symmetry, for a = and ^ = ; therefore, it is of the form Tuabx, in which tji is numerical. Put x = a=^b, and m is found to be 6. Hence, the equation reduces to 6abx ^= abc, and .'.x=^^c. 4. /x — ( \x — I - aY_x ~2a + b b) X — 2b -{-a Let X — b = 7)1, X — a = n, and hence, 7n — n = a — b; then we have n^ n — {m — n) 2n — m m^ ■m + (r)i — n) 2 m — n .*. 2mn^ — n^= 2m^n — m^. .'. m^ — n^ — 2m7^ {m^ — rf) ^= 0. .*. (7n^ — 7f) {m} -\-n^ — 2 mn) = 0. .-. (m + n) (m — ny = 0. But 771 — n = a — b, and rejecting this factor, which does not contain x, w,-\-n = 0. But 7n-\-n^=2x — a — b. .'.2x-a — b = 0. .'.x = ^(a+b). Ex. 53. 1 . a(b — x) + b (c — x) — b (a ~ x) -{- ex. 2. (a + bx){a — b) — (ax — b) = ab(x + l). 3. (a-b)(x — c) + (a + b)(x + c) = 2(bx + ad). 4. (a - b)(x - c)-{a + b) {x + c) + 2a{b + c) = 0. LINEAR EQUATIONS. 185 5. {a—h){a — c){a'\- x) + (a + b){a + c) (a — x) — 0. 6. {a — h){a--c + x) + {a + h){a+c — x) = 2a^. Solve in (x — c). 7. (m + a) (a + ^ — ^) + {<^ — '^) (^ — x) = a(m-\- h). 8. m(a + 6 — :r) = n(.:z;— a — ^). 9. (?7^ + 7z) (m — n — x) -{- r)i{x —7i) — n (x—m) = m 10. ^ — ^ J ^ — ^ , i^ — ^ _^3 m n p ' a^b ~ X Ij^c — X , c^a — x n\ a b c - fv €C X I U X I o ^ r\ ^T I ^ i — jr~^- DC ca ab 1 ~ ax ■ 1 — bx ■ 1 — g^ _ pv be ca ab Deduce tlie solution from that of No. 12. -. a — bx , b — ex , c ~ ax ^ 14. — - — + + j-=-0. be ea ab - J, / I 7 t \ a'^ -{-b^ 2 abx , fa-\-b\ a ~ 6 a + b \a — bj 16 3a5g ■ a^b'' {2a-^b)V'x ^ {b + ?>ac)x a+b {a + by'^ a{a + bY a ,^ 10 , 4 9 , 2 Q , . 1 17. f-- = - + -. Solve m - X ^ X 6 x 18 71^ 23-a; 7 1 * re 3~ 3:^ 12 4^* 19 7 13__ 2(5:r-12) 17 ■ 10 '3 5^ Zx %) x' 20 ^Q~^ j_ IB + rg ^ 7:g + 266 _ 4:?;+17 3 "^ 7 .T + 21 ~ 21 * 186 LINEAR EQUATIONS. 21.^+ 3 1 22. x + 3 2(x + 3) 2 2(x + 3) 6:^ + 5 1 + 8^ ^ 1 — :r x-S 3^-15 " 15 3 5 ' 1 1 a—- X—- 23. ^-^^ 2-^. 24. ,1a; ,1a A ^ X 25. (:^-l)(^-2)-(ri;-3)(rr-4) = 3. 26. (r^-3)(:i;-4)-:(:r-2)(:i'-6). 27. 2(^-4)(3:r + 4) + (2:^-3)(3^ + 2) -6(^-2)(2:^-3)=.0. 28. (a — :r) (Z) — a;) ==: 2:1 30. {a — x){h-{-x)=^h'' — x\ 29. ((2 — :?;)(rr— 5)^ri;' — c'. 31. (x — a){x — h)=zx'' — o?, 32. (a + rr) (5 + r^:) = (a — rz;) (h — x). 33. (a^ + ^) (Z):^ + a) = (5 — a:?;) (a — Z^rr). 34. (<2 — ^) (Z> — :r) + (a — c — a;) (r^ — Z) + (?) = 0. 35. (a — r?;) (Z> — x) — {c — x){d — x) = (c + d)x — cd. 36. (:x; --a)(x — h) — {x- c)(x — d) = {d — a) (d—b). 37. [(a''-b')x-ah][a-(a + b)x] + 2a¥x = [(a + byx + ab][b — (a-b)x]. 38. (:^ + l)(ri; + 2)(:^ + 3) = (^-3)(:2; + 4)(r^ + 5). 39. (x + l)(x + 2)(x + S) = (x-l)(x--2)(x-S) + 3(x+l)(4:X + iy 40. (:^;+l)(:^: + 4)(:^+7) = (a; + 2)(:^ + 5)^ 41. (x+2)ix + by = {x + 3)\x + 6). 42. (2; - l)(:r -- 4:)(x-6) -x(x -2)(x-9)-= 136. LINEAR EQUATIONS. 187 43. 44. 45: 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. (a + x){b+ x) (c+x) — (a~x)(b — x) (c - x) = 2(x' + abc). (x — a)(x—h)(x—c) — (d—a)(d—b)(d—c)__ :-d (x~dy. x{x — ay — {x — a + b)(x — a + c){x~b — c) ^{o} + bc){b + c). {x - a + b){x — b + c){x - c + d) - x^(x ~-a + d) = bc(d— a). (x — a-{-b)(x — b-{-c)(x — c-\-d) — x(x~a + c)(x — c-j-d) =bc(d—d). (x~2a) (x-2b) (x-2c) - (x-a~b) (x-b-c) (x~c~a) = (a+b+c) (a'W+c')-9abc. x^~(x-~a-{~b)(x~b-]-c)(x — c-\~a) = (a-i^b-{-c)la'+b'+c')~2(a'b + b'c+c'a)~Sabc. \ xj\ x]\ x) X^ X {x + a)(x-\-b)-^{x^c){x-^a) = {x'^b){x-\-d) J^{x-^d){x-\-c), (ax + b) (ax — c) ~ a(b — x) (ax + b) =^a^(x — c)(x — b) — a (ax ~ c) (c — x). 2x-?> ?>x-2 _ bx''~2^x — 4 : x-4: x-^ ^'^-12:^;+ 32* bx-l ?>x + 2 _ x^ — ?>0x + 2 3(:^+l) 2(x~\) 6^2-6 3:r-7 3(a:+l)_ ll:r + 3 2^-9 2(x + ^) 2x^-Zx-2i 10r?;+7 _g 7rr — 5 , 8:?; — 7 ?>x-2 ^x-\ '^x'-'dx-\'2 2x+1 3:r-6 b(x-l) _ ^x-2 ■ bx-S 2^+2 ?>x-1~^2x~b 9:1—25 2:i— 5 9a;-25 3rr-7' 188 LINEAK EQUATIONS. 58. 59. 60. 61. 62. 63. 4:X~S 3 4:X'+2 66. 67. 68. 1 + x l~x x' - X— a x-h x — m x — n '-^ 1 00 I h-' 4 l-2x 4 4 a . ex _ = 0, tlie given equation will hold irrespective of the values of ^— a and x—b, and therefore of the values of x ; but \i a — b be not zero, then must either X — a^=0, ov x~~b ^=^0. .'. x^= a, or X = b. 4. Solve 221^^-5^-6=.0. Here we have the factors 17:?: — 3 and 13 a: + 2 ; hence, the equation is satisfied by 17^: — 3 = 0, or x^^, and also by 13rr + 2 = 0, or x = — ■^^. 5. ^o\YQ{x-af + {a-by + {b~xy = 0. The expression is equal to 2>(x ~ a) (a — b')(b — x), and therefore vanishes for x — a = 0, or x=^ a; and for X — b =^0, or x = b. Ex. 54. 1. If an equation in x have the factors 2r?:— 4 and 2.r — 6, find the corresponding values of x. 2. If an equation give the factors 2x — \ and 3:?:— 1, what are the corresponding values of :r ? 3. If an equation give the factors ^x^~12 and 4.r — 5, find the corresponding values of x. Find the values of x for which the following expressions will vanish : 4. x''-2x+l] 4:x''--l2x + 9. 5. 9x^-4:] x'~(a + by; x''~2ax + a\ LINEAR EQUATIONS. 193 6. x'-9x + 20] 4^' -18:?; + 20. 7. x' + x-6; x^-x-Vl', 9:r^ - 9a;- 28. 8. 6^-^-12:r+6; 6:r^-13^'+6; 6:r^-20:r + 6. 9. ^x^-~hx-^\ 6:?;^-37^ + 6; 6^^ + :?; — 12. 10. A certain equation of the fourth degree gives the fac- tors x^—x — 2i and 4 a;^ — 2 a; — 2. Find all the values of rr. Find the values of x in the following cases : 11. x^~'2.'bx^-Z'b''x=^{). 12. x^ ~ ax^ — o?x-\- a^ ^=^^. 13. x''-Zx-\-'l=^^. 14. a;' — 2a^ + 2a'^ — a^ = 0. 15. x^-\-(h + c)x'-hcx-})'c-hc^ = ^. X — a j^x — h ^ (a — b)'^ x^ — a^ X — b X — a (x — a)(x — b) {x — a){x — b) 17 . x^ - bx" - a^x + a^b = 0. 18. Zx^' + babx'-^ a^y'x - 4 aW = 0. 19. x\a~b) + a'(b-x) + b\x-a) = 0. 20. (^-^)(^-g) I (x-c)(x-a) ^-^^ (a —b)(a — c) {b — c)(b — a) fx--2aW X — + a \x + a J "Ax — a 21. x[ ^ ■ l + af , \=^x^-a\ x-\- a J 22 . {x + a + bf - x^ - o? -b^ =^ {x + a) (a^ - b""). 2o ctb . bx , ax (b—a) (x— a) (x—a) (a—b) (a—b) (b—x) a—b 24. Form the polynome which will vanish for a; = 5 or — 6 194 LINEAR EQUATIONS. 25. Form tlie polynome which will vanish for x^=a or 4a or 3(2 or —4 a. 26. Form the equation whose roots are 0, 1, —2, and 4. § 43. Employing the language of Algebra, the principle illustrated in the preceding section may be stated as fol- lows : Definition. Any quantity which substituted for x makes the expression fix) vanish, is said to be a root of the equa- tion f{x) = 0. Thus, if a be a root of the equation /(:?;) = 0, then /(a) = 0. By Th. I., if a; — a is a factor of \h.Q polynome f {xY , then f{ay = 0, and a must be a root of the equation f(xy = ; hence, in solving the equation, we are merely finding a value, or values, of x, which will make the corresponding polynome vanish. Suppose f(xy =(x — a) {xY~^ = 0, we are required to find a value, or values, of x which will make (x~ a)cf> (xy~^ vanish. The polynome will certainly vanish if one of its factors vanishes, whether the other does or not, and will not vanish unless at least o?ie of its factors vanishes. Hence, (x — a){xY~^ will vanish if x — a = 0, quite irre- spective of the value of <^ (5;)"~\ Also, if c^ {xY''^ = 0, the polynome will vanish, irrespective of the value of x — a. It follows, therefore, that iif{xY can be resolved into two or more factors, each of these factors equated to zero will give one or more roots of the equation /(:r)** = 0. "When there can be found two or more values of x which satisfy the conditions of given equations, they are some- times distinguished thus : Xi^ x^^ ^3, etc., to read "one value of :r," "a second value of ^," ''a third value of a;," etc. Thus, if ^ ^^ _ ^^ ^^ _ ^) (^ _ ^) _ Q^ .*. Xy = a, ^2 — ^, ^3 == c. LINEAR EQUATIONS. 195 Examples. Solve 2x^ - l^x" + 21 X- 18 = 0. Factoring, (^x - 2) {x — ?>){2x-^) = 0. , . X^ Zi^ X2 ' — ■ Oj ^3 ■^^* x"^ — (a + Z>) a; + (0^ + ^) ^ = (<3^ + <^) ^. .-. x'~(a + h)x + (a + c)(b-c)== 0. . • . ^r^ - [(a + c) + (5 - c)] a; + (a + c) (^> - c) = 0. .-. [:r - (a + c)] [^ - (^ - c)] = 0. .*. :z:i == a + a — 5 (3; — a){x~ b) , (g; — 5) (:g — c) __ -j (c — a) (c — b) {a -~b)(a — c) 16. 196 LINEAR EQUATIONS. Subtract term by term from the identity (see page 67) (x — a)(x — h) . (x — h)(x — c) , (x — c)(x — a) _ -. (c — a) (c - b) {a — b){a — c) (b — c) (b ~ a) :.{x~c){x — a)=^0. :.Xi^=c, 2^2 = a. 6. Find the rational roots of x^— 12:^^+51^'^— 90:?; + 56 == 0. Factoring the left-hand member by the method of § 27, {x - 2) (2; - 4) (;r' — 6^ + 7) =- 0. :.Xi = 2, ^2 = 4, or ^^ — 6^ + 7^0. Since x^ — ^x + 1 cannot be resolved into rational fac- tors, we know that it will not give rational roots ; therefore, ^1 = 2, a^g = 4 are the only values that meet the condition of the problem. In order that two expressions having a common factor may be equal, it is necessary either that the common factor should vanish, or else that the product of the remaining factors of one of the expressions should be equal to the product of the remaining factors of the other expression, and it is sufficient if one of these conditions be fulfilled. In symbols this is If {x — a)f(x) — (x~a)(f> {x), r.Xi — a OTf(x) = <^ (x). 7. ^+- = a^ — X a 1 1 X — a X — a :.x — a ax 1 ax .'. x~— a = 0, or ax=^\. /.Xi = a, X2--= — a (x + a+bXx+b + c) = (x — Sa + b)(2x-Sa + 2h-c). . x + a + b _ 2x~da + 2b — c _ x — 4:a+b — c x — Sa-\-b x-}'b-{-c Sa-\-c Page 155, (5). LINEAR EQUATIONS. 197 2(x — a-\-b) __x — a-\- b X — 3a + b 3a + (? .'. Xi=^ a — b. i(^2 — 3a + 5) — 3(2 + ^. :. X2=^9a — b -\-2c. 9. {x~2) (x-b) (x-6) (x-9) + (y+2) (y-4) (y-5) (y-11) + (z+l)(z+5)(z+8)(z+12) = x(x-A) (x-7) (^-ll) + (y+l) (y-1) (y-8) (y-10) + (.+2) (.+3) (.+10) (.+11). Let:?;'-=:i;'— ll:r, y'=f-97/, and .'==.' + 13.. .•.(^'+18)(:r'+30) + (y'-22)(y'+20) + (.'+12)(.'+40) - x' (x'+ 28) + (y '-10) (y'+ 8) + (.'+ 22) (.'+ 30). .•.^^'' + 48:r'+540+y'*'^-2y'-440 + .'2 + 52.'+480 =::2;''+28rr' + y'^-2y'-80+.'^+52.' + 660. .-. 20:?:'-0. :i;'-lla;=-0, ^^i = 0, rr.^ll. Ex. 55. What can you deduce from the following statements ? 1. ^ . ^-0. 3. (a-b)x = 0. 2, A ' B ' C=0. 4. 12^y==0. 5. What is the difference between the equation (:^;-5y)(:r-4y+3) = and the simultaneous equations a; — 5?/ = and:^; — 4?/ + 3 czr 0? What values of x will satisfy the following equations ? 6. x(x—a) = 0. 11. x(x' — a'')=^0. 7. ax(x + b) = 0. 12. a'x^^b'^x. 8. (x~a)(bx- c)=^0. 13. x' + (a — xf =- a\ 9. ax^ = 3ax. 14. ^' + (a — 2;)' = (a -2^)1 10. x' = (a-\-b)x. 15. (a-:r)'+(:z;-^)'-:a'+6l 198 LINEAR EQUATIONS. 16. (a — x)(x — b) + ab = 0. 17. (a - xf --(a-x)(x-b) + (x- bf = a' + ab + b\ 18. x^ — {a-b)x — ab^^. 19. x^-~{a-\-b-\-c) x^ + {ab + be + ca) x — abc = 0. If X must be positive, what value, or values, of x will satisfy the following equations : 20. (x-5)(x + 4:) = 0. 23. 3:^2-10r?; + 3-:0. 21. :?;' + 29:?;-30==0. 24. x' -ISx^ + ^6 = 0. 22. x'-17x-84:-=0. 25. :i;'-2^' — 5a;+6 = 0. Solve the following equations : 26. (a~xy + (x-by = (a-by. 27. (a — xy - {a- x){x — b) + {x - by = (a- by. 28. a'(a~xy = b'(b-xy. 29. a'(b-xy = b\a—xy, 30. (:r-a)^ + (a-^>)' + (^-:r/-:0. 31. {x~iy = a(x'-l). r.n a — x x — a ^„ a-\'b — x a — c + x X — b c -\- X a~c — X a-\-c — x 34. {x — a + b){x — a + c) = (a— by — x\ 35. (x-ay — b'' + (a + b-x)(b + c~x) = 0. 36. (a + b + c)x''~(2a + b + c)x + a=0. ^^ a + b~x_a~]-b — c c X 38. (a~xy + (a~by=(a + b-2xy, 39. a;(a + ^>— :r) + (a + Z> + c)c = 0. 40. (n—p)x'^-\'(p~7ri)x-]'m — n^=0. ,- aa;^ — 5:r + c c? --, ax"^ — bx4-c a~b -{- c 41. ! = -. 42. ! — = — rnx — nx-{-p p mx^ — nx-\'p m — n-\-p LINEAR EQUATIONS. 199 43. ^x^ + a'-h''-2{a + h)x={a~x){h+x)~{a+x){h-x). 44. (2a -h- xy + ^{a~ by = (a + b~ 2x)\ 45. {2a + 2c — xy = {2h + x){?>a-h + ?>c-2x). 46. {^a-bh + x){pa-?>h-x)^(la-b-^x)\ 47. {2>a-h + x){Za + h-x) = (ba + U~Zx)\ 48 . a{a — h)-~b{a — c)x + c(h — G)x^=^0. 49. {ab+bc + ca){a:' + x+l) + {a-by = (2ac + b')(x' + x+l) + (a-cfx. 50. (x+l)(x + S)(x-4:)(x-1) + (x-l)(x-S)(x + 4:)(x + 1) = 96. 51. (x-l){x + S)(x-b)(x + 9) + (x+l)(x-3)(x + 5)(x-9) + 18 = 0. 52. x + l = 3i. 56. ^-^.^-^ IB X X — b a — X 6 ,1 a-{-b , a — b ^^ a—x b-{-x m 7i 53. x-\ — = — '—•-I -• 57. ' — X a — b a-{-b b-\-x a—x n m lab ^^ a , X m , n 54. X = - — -• 58. _-(-_=: 1 X b a X a n m 55^ a+_^ , Mif^2i 59. (^-^)' + (^^-^)' ^^ b -\- X a + x ^' {a—x){x—b) 2 60. 61. 62. 63. {x + ay + {x -by _^ d' + b'' {x + ay-lx-by 2ab ' (a-xy-(x-by _ 4:ab (a — x)(x — b) a^ — b"^ (a — xf + {a — x){x -b) + {x - by __4:9 (a - xy -la-x){x-b) + (x - by 19* 2a' + a(a- x) + (a + ^)' _ 3 2a' + a(a + x) + (a-xy 2 200 LINEAR EQUATIONS. 64. (b--xy + (2-xy = i7. 65. (x-ay + (a-by + (b-xy = x'-a\ 66. (a-xy + (x~hy--=(a-by. 67. (x + ay - (a + by + (b~ xy =(x + a)(x + b) {a + b), 68. x^-{x~by-{x-a + by - cc" + {x~ ay + (a - by + b^ = {a-b)c\ 69. {x + ay -{x + by -(x- by ~ (2ay +(x- ay + (a + by + (a~ by - (a' - b')c. 70. (x-a+by~(x~ay + (x~by~x' + a'-(a-by:^-b\ ^- (a — xy + (x — byo^^/ 7 X 73. 2(a- xy - 9 (a - :z;)^(:^^ - ^>) + 14 (a - xy(x - 5)^ - 9(a - ^)(:r - ^>y + 2(x - Z>/ = 0. 74. 4(6^ - :?;)* - 17(a - :r)^(^ - ^)^ + 4(:z;- Z))* = 0. Find tlie rational roots of the following equations : 75. a;*-12^^ + 49a;2-78^ + 40---:0. Letz^x'-Qx, 76. x'-6x' + 7x' + 6x~8 = 0, 77. :r* - 10:r' + 35a:' -50:^; + 24 = 0. 78. 32:r'-48:2;'-10a;' + 21^+5 = 0. 79. ^.-3- 6:1;' + 5a: + 12 = 0. 80. lla:^ + 10a;^-40a; = 176. 81. 5 4 9 4 ^ 5 _Q^ X X — a x--2a x ~Sa x — 4:a /?_ LINEAR EQUATIONS. 201 82. 14 , 5 4 __ 14 :r + 20 x+b x — 4: a; — 55 x — AO x-2b g 2x+5a x-\-8a . x _ x — a ^4- 5a , 2x—ba X x—a x — 2a x — '^a x~4:a x~ba 84 ^-^4 ■ x + 2 . x + 4: _^ x + ^ x-1 x — S x-]-2 X X ~1 x — 2 x—3 X — 5 X x — 1 x—2 x~-3 x — 4: X'-5 x — 6 86. T' - f"^^] x' - ^2^^'- = X + ^« - ^^ ^1 - '■^' a + bj 1-j- ex V^ + ^J \1 + ^^- g^^ 4:x' + 4:a'-33xV ^ , .^^, _ ^^,^ + 9^a2 _ 2a'\ 2x + a ^ ^ 88. ^ • ^ - ^*^ 89. :r*^ - 11 :r + 28 x' - 17.T + 70 x' - 14^' + 40 8 ■ 8 ^ ^^ x' — Qx + 5 ^' - 14a^ + 45 x' — 10:r + 9* 90. Q(a~ xy - 25 (a - xf(x -h) + 38(a - xf^x - by - 25(a - n^)(:^; -by + 6(x~ by = 0. CHAPTER VI. Simultaneous Linear Equations. § 44. There are three general methods of resolving simultaneous linear equations : first, by substitution ; second, by comparison ; third, by elimination. The last is often subdivided into the method by cross-multipliers, and the method by arbitrary multipliers. In applying the elimination method, the work should be done with detached coefficients, each equation should be numbered, and a register of the operations performed should be kept. Ex. Resolve u + v + x -\- y + 2= 15, U + 2V+ 4:X+ 8y + 16z= 57, U + SV+ 9a; + 27 y + 8X2:= 179, u + 4:v + 16x+ 64y + 256z = 453, u + 5v-\-26x+ 1252/ + 6252 = 975. u V X y z Register. 1 1 1 1 1 = 15 (1) 12 4 8 16= 57 (2) 1 3 9 27 81 = 179 (3) 1 4 16 64 256 = 453 (4) 1 5 25 125 625 = 975 (5) (2)-(l) 13 7 15= 42 (6) (3) -(2) 1 5 19 65 = 122 (7) (4) -(3) 1 7 37 175 = 274 (8) (5) - (4) 1 9 61 369 = 522 (9) 7) -(6) . 2 12 50= 80 (10) 8) -(7) 2 18 110 = 152 (11) (9) -(8) 2 24 194 = 248 (12) 11) -(10) 6 60= 72 (13) 12)-(11) 6 84= 96 (14) (14 -(13) 24= 24 (15) it(13)- 60(16)] 1 =2 (17) H(10)- [12(17) + 50(16)]} 1 =3 (18) (6) -[3(18) + 7(17) + 15(16)] . . .1 =4 (19) (D- (19) + (18) + (17) + (16)] .1 =5 (20) SIMULTANEOUS LINEAR EQUATIONS. 203 An examination of the Register will show how easy it would have been to shorten the process ; thus, (10) is (7) - (6), which is (3) + (1) - 2(2) ; similarly, (11) is (4) + (2) - 2 (3) ; therefore, (13) is (4) + 3 (2) - 3 (3) - (1), etc. Ex. 56, Solve the follow^ing systems of equations : 1. 2:,^ + 3y = 41, 4. i«-¥2/ = l. 3:t-f 2y=-39. |.r-|y + 5 = 2. 5^+7y=17, 5. iy = i«-i. 7x~5y= 9. \y = ^x-\. 3. ix + y = e>, 6. l.bx-2y = l, 3.T-4y = 4. 2.5a; -3?/ = 6. 7. 3.5x + 2^y = = 13 + 4 \x-?..by, 2|rr + 0.83/ = = 22i + 0.7^-3|y. 8. 1 , 1_5 X y 6 11. y X y 6 Ux *^-4 = 2. y 9. X y ' 12. 8 + y *- 15 4_ X y"" 6 y ^ 0. 1.6 2.7 x~ y '■ 13. 5a; 0.3_g 0.7^ y 0.8 , 3.6 . =5. X y 10x^9_3j 7 y 204 SIMULTANEOUS LINEAB, EQUATIONS. 14. ^x-i(y+l) = l, j^ ^ + 2y + l _o i(^+l)+f(y-l) = 9. ■ f-^+] 5 7 «-y+3 15. x + 2y 2x + y ^^ a; + 3y+13 _^q 7 _ 5 ' 0.4a; + 0.5y- 2.5 3a;_2~6^' 0.8a: + 0.1 y + 0.6 _1 5a;+3y-23 2' 16. il±l^ = 8 19 ^+1 .y+2_ 2(a;-,y) «~y ' "345 3y-5 4 3^ 2a;-y + 3 a:-2y + 3 _, 3 4 3ar-4,y + 3 , 4a:-2,y-9 _. 4 "^ 3 ■ 21. 20(a;+l) = 15(y+l) = 12(a: + 2/). 22. (a:-2):(y + l):(a; + 2/-3)::3:4:5. , 23. (a;-5):(2/ + 9):(a: + 2/ + 4)::l:2:3. 24. ^+§=^±8, a;+l 3/+5 2a;-3 __ bx-& 2(y+l) 52/+ 7 25. (^-4)(2/ + 7) = (a;-3)(y + 4), (a; + 5)(y-2) = (a; + 2)(y-l). 26. (a;-l)(52/-3) = 3(3a;+l), (a;-l)(4y + 3) = 3(7a;-l). 27. (a;+l)(2y+l) = 5a;+ 9y+l, (a; + 2) (By + 1) = 9a: + 13y + 2. SIMULTANEOUS LINEAR EQUATIONS. 205 28. (3^-2)(5y+l) = (5:t'-l)(3/ + 2), (3^-l)(y + 5) = (a; + 5)(73/-l). 29. x + y = 2>l, 37. x + y + z = ?>, 3/ + 2 = 25, 2^.+ 4y + 8^ = 13, 2; + a; = 22. 3:r + 9^ + 27 ^ =- 34. 30. 2x + 2y=1, 38. :r + 2y + 32; = 32, 7:r + 9^ == 29, 2:r + 3y + z = 42, y + d>z=Vl. ?>x + y + 2z-=M). 31. l.?>x~\.9y-=^l, 39. :r + y+22 = 84, 1.7y-l.lz-=2, x + 2y + z = 33, 2.92; - 2.1 :i: - 3. 2a; + 3/ + == 32. 32. 5rr + 33/ + 22 = 217, 40. Sx + Sy + z = 17, 5^: — 3y == 39, 3:r + y + 3 2 = 15, 3y - 22 = 20. a: + dy + 3z = 13. 33. ^x-^y:=0, 41. a; + 2y — 2; = 4.6, irr-i^^l, 3/ + 22-r?7=10.1, ^z-y = 2. z + 2x-y^b.7. 34. li^r + liy^-lO, 42. a; + 2y-0.72: = 21, 2ix + 2f 2r =20, Sx + 0.2y — 2 = 24, 3iy + 3f 2 = 30. 0.9a; + 7y - 2^ = 27. 35. x + y — z = n, 43. a; + y--=l|2 + 8, y + z~x=lS, 3/ + 2=2|y-14, z-\-x — y = n. 2 + x = 3f a; - 32. 36. a; + 2/ + z = 9, 44.la; + i2/ + i2 = 36i a; + 2^ + 42=15, i^- + iy + i2 = 27, ar+32/ + 92 = 23. i^-+iy + J-2=i8. 206 SIMULTANEOUS LINEAR EQUATIONS. 45. "^-^=2, 49. --^:^1 ' + ? = 4. X z 46. :i:i-L^^2, 50. ?+l + 5=:4, ~ X y z X y z 9 12_10^^ . X y z ^y -^ x + y b y^ ^1^ rr + 2; 6 zx __ 1^ 48. ^+^ = 2, 62. —^^==20, 43/ — 3:r 15, y + 1 '-'y y + 2 4 ^ + 1 ^ + 3_ 1 rr + 1 2 ^x + y — 9 z+l ^y+z. — x-^\ ^> Zz^x — 2 y + 1 x-^y _ 10 y-z ■ •*•'-'> X + z_ -Q x — y ■ ^j y + z - 1 x + b x + S_ -2 y + z ^1 3/ + 3_ -1 x+ z -*-> z + d_ 1 47. r:_L^ = io, 51. 2x-Sz y^ ^12. a; + y 2 42/ — 5z 53. (a: + 2)(2y + l) = (2a;+7)y, (ar- 2)(30 + 1) = (a: + 3)(3z- 1), (2/ + l)(z + 2) = (y + 3)(z+l). 54. (2a;-l)(y+l) = 2(a:+l)(y-l), (a; + 4)(2 + l) = (a; + 2)(2 + 2), (y-2)(2 + 3) = (y-l)(2 + l). SIMULTANEOUS LINEAR EQUATIONS. 207 55. {x + l){by-^) = {1x+l){2y-?>), G/ + 3)(. + 2)^(3y-6)(3.-l). ' 56. 21^ + 31y + 422 = 115, ^(2x + y)^^{^x + z) = 2{y + z). 57. lb{x-2y)=.b(2x-Zz) = ?>{y + z), 21a^+31y+41z-135. 58. ^x{y + z) = 4:y(z + x)=:'^z{x + y), '- + W--9. X y z ■ 59. ?>x + y + z^20, 60. i^ + 2 + 8y == 30, ?>u + x + 4.y^m, bu + y + z-=lO, 3 w + 6:r + 2 = 40, 4:u + x + z = l0, 5i^ + 8y + 32; = 50. '^u + x + y=^lO. § 45. The principle of symmetry is often of use in the solution of symmetrical equations. For, from one relation which may be found to exist between two or more of the letters involved, other relations may be derived by symme- try ; also, when the value of one of the unknown quantities has been determined, the values of the others can be at once written down, etc. Examples. 1- {x + y){x + z) = a, (^ + y) (y + ^) = ^. {x+ z){y + z)-=c. Multiply the equations together and extract the square root. .\(x + y){y + z){z + x)-=^{ahc). 208 SIMULTANEOUS LINEAR EQUATIONS. Divide this equation by the third. ,\x-{-y^= ^^ ^ ; and, therefore, by symmetry, a b Hence, we get ah — hc-\- ca ^ 2-^/{ahc) whence y and z may be derived by symmetry. 2. x + y + z^O, (1) ax + hy + cz = 0, (2) hex + cay + ahz + (a — b){h — c) (c — a) = 0. (3) cx(l)-(2) gives (c~a)x + (c-b)y = 0. Hence, y — ^ ^, and similarly, b — c (a-b)x b-c Substitute in (3) these values of y and z, and reduce ; then, x{a--b)(c — a)~{a — b) (b — c)(c — a), or, x = b — c. Hence, y = c — a, z=^ a — b. 3 . a(yz —zx — xy) = b(zx —xy— yz) = c(xy —yz— zx) == xyz. Divide the first and last equations by axyz ; then, -, and hence, by symmetry, I a X y z' 1_1 1 1 b y z X 1_1 c z 1 X 1 y SIMULTANEOUS LINEAE, EQUATIONS. 209 Therefore, - + - =^ — -, and, by symmetry, c 2 X c a 2 y a 2 4. ax + hy + cz^l, ' (1) o?x + hhj + c'z=l, (2) o?x + bhj + ch=-l. (3) '" cX{l)~ (2) gives a{c ~ a)x + b{c'''b)y^c - 1. (4) cX{2)-(?>)giYe^a\c-a)x + b\c-b)y = c—l. (5) ^ X (4) — (5) gives ab{c—a)x—a\c—a)x—b{c—l)—{c—l), or a(a — b) {a — c) X ^ (c '— 1) (^ — 1). Iherefore, x = -^ f^ \ ', a{a—-b){a — c) whence y and z may be derived by symmetry. 5. Eliminate x, y, z, u (which are supposed all different) from the following equations : X = by -\- cz + du, y z= CZ -\- dic-[- ax, z = du -\- ax -}- by, u=- ax '\-by ■\- cz. Subtracting the second equation from the first, x~y^=by — ax, or {1 -\- a) x =^ (1 -{- b) y , which, by symmetry, = {l + c)z = (l + d)u. These relations may also be obtained by adding ax to both members of the first equation by, to both mem- bers of the second equation, etc. 210 SIMULTANEOUS LINEAR EQUATIONS. Now divide the first equation by these equals. 1 _ h , c J d And since =1 , we have 1 + a l + (2 1 a , h , c . d i 1+a 1+^ l+c l+d Ex. 67. 1. Given ax-[-hy =^ c, and that x = — -— -, a^x ~\-})hj ■= c\ derive the value of y. 2. Given hx = ay, , and that x = -~ -~\ be — ad dx-{-7nd=cy-{-nd, derive the value of y. 3. Given ax ~{- by -}- cz -— d, and that x = — ^ -^ (, a (a—b) (a—c) c^x-\V'y^&z = (i^ write down the values of c^x-\-Wy-\-c^z ^= d^, y and z. 4. There ia a set of equations in x, y, z, u, and lu, with cor- responding coefficients (a to x^ etc.), a, b, c, d, and e ; one of the equations is x)= by -{- cz -\- du -{- ew, write down the others. , Solve the following equations : X , y y , z T X , z m n n p vi p 6. X -\- ay -{- bz = VI, y ~{- az -{- bx = n, z-\- ax~\-by =p. 7. x-\- ay -=^1, y ~{-bz-=^m, z -\- cu = n, u-\- dw ==^p, w -\- ex-=^r. SIMULTANEOUS LINEAR EQUATIONS. 211 8. Eliminate x, y, z (supposed to be all different), from the following equations : x=^hy -{- cz, y =^ cz-\- ax^ Z'^ ax-\- by. 9. Eliminate x, y, z from X y J z I y+^ z+x x+y 10. Having given 1^, X — by -}- cz -\~ du + ew, y ^=^ cz -{- du -\- eiv -\- ax^ ^ z =^ du -\- ew -{- ax -}- by^ u = ew -{- ax -\- by -}- cZj w — ax -{- by -}- cz + du, show that -^ +-A_ +^ +_iL. . ^_ :^ 1. 1 + a^ 1 + 6^1 + ^^1 + 6^^1+6 § 46. Resolution of Particular Systems of Linear Equations. Examples. 1. x + y + z = a, (1) y + z + u^-b, (2) z -\-u+x = c, (3) u+x + y = d. (4) (l)+(2)+(3)+(4) S(u+x+y+z) = a + b + c + d, (5') 3(1) 3(:r+y + z)=.3a, (6') i[(50-(6')] ^ = ^(-2a + 5 + . + c^). The values of x, y, and 2; may now be written down by symmetry. The following is a variation of the above method, appli- cable to a much more general system. Assume the auxiliary equation u + x + y + z = s. (5) 212 SIMULTANEOUS LINEAR EQUATIONS. Hence, (1) becomes s — u~- a, (6) (2) becomes s — x = h, (7) (3) becomes s — y=^c, (8) (4) becomes s — z = d. (9) (5) + (6) + (7) + (8) + (9),45-. + a + 5 + ^ + c^. Therefore, s = l(a + h + c + d). s is now a known quantity, and may be treated as such, in (6) giving u = s~~a, in (7) giving x = s~ b, in (8) giving y = s-c, in (9) giving z=^ s — d. 2. yz = a{y^z), (1) zx^l{z-^x), (2) xy=-c{x~\-y). (3) (X)~ayz, _ + - = _, y z a ^-' ^- = \ z~^ X b' {?>)^cxy, 1 + 1=1 X y c This may now be solved like Exam. 1, using the recip- rocals of a, b, c, X, y, and 2, instead of these quantities themselves. (2) ^ bzx, aiu + bi(x + y + z) ^ Ci, (1) a^x -]- b^(y -j- z + u) = c^, (2) a-sl/ + h(z + u -i- x) = c,, (3) aiZ + hi(u + x + y) = Ci. (4) Assume the auxiliary equation, w + a; + 2/ + 2 = s. (5) (1) becomes biS — (bi — Wi) m = Ci. Therefore, — ^s-w=— ^J— . (6) -«1 SIMULTANEOUS LINEAR EQUATIONS. 213 f Similarly, from (2), —^ s~x = —^ — (7) ^ O2 — ^2 02 — a^ Similarly, from (3), -A_ s~y= — ^^ • (8) t>3 — <^3 o^ — as : Similarly, from (4) , --^ s~z^ —^^ • (9) (5) + (6) + (7) + (8) + (9), ^1 — ai Z)2 — aa ^3 — <^3 ^4 — «4. ^L+^i__. (10) b^~ai ^2 — ^2 ^3 — ^3 b^ — a^ From (10) we can at once get the value of 5, which may therefore be treated as a known quantity in (6), giving hiS — Ci h — ai and the values of x, y, and z may be obtained from (7), (8), and (9), or they may be written down by symmetry. ax + b(i/ + z) -^-c, (1) ai/ + b(z + u)=d, (2) az -j- b (u~{- x) = e, (3) au+b(x + y)=f. (4) Assume m + a; + y + z = s, (5) (l) + (2)+(3) + (4), (a + 2b)s=c + d+e+f. (6) Hence, 5 is a known quantity, and may be treated as such. From (1) and (5), bs — bu -}- (a — b)x = c. Therefore, bu — (a — b) x = bs ~ c. (7) Similarly, from (2) and (5), bx — (a — b)y =^bs — d, (8) 214 SIMULTANEOUS LINEAR EQUATIONS. 5. From (3) and (5) , by-~{a — h)z = hs- e. (9) From (4) and (5), hz-(a~ h)u= hs -/. (10) &(7) + (a-^)(8), h^u — (a — hyy = ahs — he — {a — b) d. (11) S(9) + (a-Z>)(10), Vy - (a - Vfu = ahs -he- (a, - h)f. (12) 5^(11)+ (a -5)^(12), _6[J'^(c-c^) + («-6y(e-/)]. (13) The values of x^ y, and z may now be written down by symmetry. cc" + o^x + ay + 2; = 0, ^3 _|_ ^2^ _j_ ^^ _!_ ^ ^ 0, & ^ c^x -\- cy -\- z ^^ 0. The polynome i^^ ■\- xf -\- yt -\- z vanishes for t^=^a, t=^h, Therefore, by Th. II., page 58, for all values of t, f' + xe + yt + z = {t — a) (t -h)(t~ c) = f — (a + b -{- c)f + (ab + he + cd) t — ahe. Therefore, by Th. III., page 67, x^ — ^a + b + c), y ^= ab -\- be -{- ca^ 2 = — ahe. 6. x + y + z + u = \, (1) ax + hy-\-cz + du = 0, (2) a^x + b''y + ch + d''u = 0, (3) a'x + ^V + ^'^ + ^'^ == 0. (4) Employing the method of arbitrary multipliers, (4) + Z(3) + m(2) + n(l), u = n (5) a' x+ ¥ y+ C z^d^ + Id' + lh' + h? + ld' -\-ma + mh -\-mc + md + n +.n + « + n SIMULTANEOUS LINEAR EQUATIONS. 215 To determine x, assume h^ + lh' + mh + n^ 0, (6) c^ + lc^+mc +71=^0, (7) cZ^+ ld''+ md+ n^O. (8) Therefore, x= -— — (9) a^ + la^ + ma + n But the system (6), (7), (8) has been solved in Exam. 5, from which it is seen that l = — (h -\-c-\- d), 7n = hc-\~ cd-{- db, n = -- bed, and a^ + a^l + am -j- n = (a — b)(a — c)(a — d). Hence, using these values in (9), — bed x = : — (a — b) (a — c) (a — d) The values of y, z, and u may now be written down by symmetry. ^-^-^ +r7^„=l. (1) 1, (2) m — a m — b m — c n — a n • b n — c X y z + _f_=l. (3) p— a p — b p — Assume 1 ^ t — a t — b t~-c __ f + Bf + Ct + D ,4^ (t-a)(t-b)(t-c) ^ ^ But in virtue of equations (1), (2), and (3), the first member of (4) vanishes for t = m, t-=n, and t=p', and hence, f + Bf^ + Ct-\- D vanishes for the same values of t] and therefore, by Th. II., page 58, f + Be + Ct + n = (t -m){t- n) (t -p). 216 SIMULTANEOUS LINEAR EQUATIONS. Therefore, (4) becomes 1 _ _x_ y_ z__ _ (t ~ m) (t — n)(t — p) t-a t-h t-c~ {t-a)(t — b){t~~c)' To obtain the value of x, multiply both sides of this equation by (t — a). t-a — x- y^^~^^ - ^(^-^) = (t-m)(t-n)(t-p) t—b t-c {t-h)(t-c) Now t may have any value in this equation ; let t — a. Hence, x = {a-m){a--n){a-p) ^ , {a — b){a — c) The values of y and z may now be written down by symmetry. 8. ^ + ^ ^ 3/ + ^ ^ ^ + ^ n\ p q r Ix + my + nz = s^. . (2) By § 37, x-{- a y ~{-b _ z-\~ c lx-{-7ny -{-nz-j- la-j-mb -j- no p q r Ip -\- "rriq + nr ,o\ s^ + ^a + "mb -\-nc r> (2) = — \ ' ' = B, say ; tp + mq-\-nr therefore, x=^pR — a, y = qR — ^, z^=rJR~c. 9. yz + zx + xy — {a + b-\-c) xyz. (1) yz + zx __ zx + xy ^ xy + yz ^2) a b c (l)^xyz, \jr\ + \ = a+b + c. (3) xyz 1+1 1+1 1+1 {2)^ xyz, ^_JL=y—l=l-Jl. (4) SIMULTANEOUS LINEAR EQUATIONS. 217 10. § 37 and (3), (4) and (5), . . - + - — za X y 1 2 2 2 X y z a-[-b + c :2. y z 21), Z X (3) -(6) X^ c a+b X X ^= a — b-{- c, - = a + b- y -c, 1 a-\- c ^ I y-<^ . a—c a—b :2. (1). . x-\-c ' 'a + b -1 = 1 y + ^ a + c , x — a -b + c a + c — b — y a + b a + c Similarly, from (2), x — a -b + c a—b + c—y a — c a — b (3) and (4), x — a — b + c = —+-(a — b + c a + c^ a — ^ + ^-y)- But unless ^~^ = a — c > (5) (6) -a+b + c. (1) (2) (3) (4) If a+c a—b this cannot be the case except for a — b + c — y=^0, in which case x — a — b + c^=^0 also, giving X = a + b — c and y — a — b + c. a + b a — c ... 7 2 2 2 I _ • --2 . ^2 _ ^2 _ ^2^ a+c a—b b'-c''~-= 0, or (b + c)(b-c)==: 0. Therefore, b = c or b = — c. (5) (6) 218 SIMULTANEOUS LINEAR EQUATIONS. But, if ^ — + c or — c, (1) and (2) are one and the same equation ; hence, if (1) and (2) are indepen- dent, (6) cannot be true, thus leaving only the alternative (5). 11. 2ax = {h + c~d){y^z\ (1) 2hy = {c-\^a-h)(z-\-x), (2) {x^y^zy + x^-^fA-z'^^{a'^y'-\-&). (3) (1) and page 155, (5), _ yArZ ^ x-\-y-\-z (4) (5) h-^c — a 2a h-\-c-\-a (2) and page 155, (5), y _ x+z ^ x+y+z c-{- a — h 2 b c-{- a-[-b (4), (5), and page 155, (5), . x-\-y + z _ ^ _ y _ g a-\-h-\-c h-\-c — a c-\-a — h a-\-h — c . x'' " (h + c-af jx + y + zy + x' + y' + z' {a+h + cy+{h+ji-ay+{c-^a-by-{-{a + h~-cy Keduction and (3) ^il+l+lLt^ + f+A^l, Therefore, x^ = {b-]- c — of. Ex. 68. 1. ax-\-by~ c, 3. ax + by = c, mx + ny = d. mx + ny = c, 4. ? + ^-l 2, ax + by = c, ah mx — ny ^= d. x -\-y ^= c. SIMULTANEOUS LINEAR EQUATIONS. 219 5. :r , ?/ 1 ^^ X a a b y b X _.y 1 X -\-71l G b a y -\-n d 6. ^ + f = l, a b b a 7. ax -\~bc=^by -\- ac, x + y = c. Q a b X y b , a -4- - = 71. X y (a-{-c)x —(a — c)y = 2ab , (a-{-b)y — (a — b)x = 2ac. 10. a 12. x+y_a+b+c y+l a-b+c y~l a—b~c x + 1 a-\-b-^c 13. x — a-\-c __b y — a + b c y+b c+a x+c b+a 14. x + c 1 y + b _c, a-\-b a-^c x-b y-c__c, a~ c a — b 15. ^ 1 y __ m~a m-~b " + y^i. y~c b X — y = a~b. 16. x + y + z^O, (b + c)x + {a + c)y + {a + b)z^O, box + acy + abz — 1. 17. x-\-y-\rZ = l, ax-^-by + cz = m, ^ \ y \ ^ ^l I — a l — b l~ c 220 SIMULTANEOUS LINEAR EQUATIONS. 18. 19. X — a y ~h z — c l(x ~ a) -{- mi^ — b) -\- n{z — c) ^= \. X ~a y ~h z — c Ix + my ■\-nz--=^\. 20. a{x — a)=^h{y — h)=^c{z~c)^ ax -\~ by -\- cz = w}. 21. x-{-y-{-z^=^a-\-b-\-c, bx -\- cy -]- az = a^ -{- b"^ -{- c^, ex -^ ay -{- bx = a} -{- b'^ + c^. 22. x + y-^z = a + b + c, ax '\- by -{- cz =^ ab -\- be + ca, (b — c) X -}- (c — a) y -\- (a — b) z =^ 0. 23. X -{- y -\- z = on, 24. ax~\-by -{- cz = r, X : y : z = a : b : c. nnx — ny, qy =^pz. 25. xy -{• yz -{- zx = 0, ayz + bzx + cxy — 0, bcyz + acxz + a^rry -\-(a — b)(b — c) (c — a) xyz == 0. 26. (a + 5)a; + (^ + c)y + (c + a)z = ab + bc-{- ca, (a + c)x-{-(a + b)y-\-(b-}~c)z = ab + ac-{-bc, (b + c)x + (a + c)y + (a +Z>)2 -: a^ + b' + c^ 27. mx + ny + pz + qu = r, ^ _ y _ ^ _ ^ SIMULTANEOUS LINEAR EQUATIONS. 221 28. ^(3/ + ^) ^ y (^ + ^) ^ ^(^ + 3/) , a b c --\ \--=^a + b + c. X y z 29. {a — h) {x + c) — ay+hz = (c~a){y'{-h)--cz-\-ax — 0, x + y + z^2(a + h + c). 35. l + l-l^l y z X a z X y b' 1 1_1__2 X y z ~ c 30. ax-\-by — 1, by -\- cz = 1, cz -\- ax= 1. 31. ly +mx = n, nx -\~ Iz = rti, 7)iz-\- ny =^ I. 32. X -\~ y = a, y + ^=^, X -{- Z '= c. 33. , Tnn y+z x= ^ In z-\- X — y =^ — ' Im x + y-z^-. 34. W-2a, y 2 Z X X y 36. (a+b)x+(a~b)z = 2bc, {b +c) y +{b—c) X = 2ac, (c+a)z +(c—a)y = 2ab. V z 37. x+Y = a, o c y-\ = ^, ^ c a , X y Z^ T = C, a b ' 38. ^_ + -X_ = 5-a, b -{- c c-\-a X . z b -{- c a + b ^= a — Cj -JL-^.^- = c-l. c -\- a a-^ b 222 SIMULTANEOUS LINEAR EQUATIONS. 39. X -\- y — z=^ a, 40. u-\- v — x ^= a^ y + z —v = h, V -\- x~y =^h, z -{- V — x-= c, X -\- y — z ^= c, V -{- X — y ^= d. y -\- X — u =■ d, z -j-u—v = e. Ex. 59. Resolve : 1. (a + h)x + (a~b)y = 2(0" + b''), (a - h)x + (a + Z>)y -= 2(a' - b'). 2. x + y=^a, 4. ^a~b)x+(a+b)y = a+b, X^ — y"^ z=b. _j^^ y - 1 3. 2x — Zy=m, a + 5 a~-b a-\-b 2x^ — Sy'^ = 71^ + rry. e/ z>\i a + ^ + 1 a~ £ g -f~ ^ ~ <^ 7. x~y b — c X — y —\ ^+A = U+A. ^ + y + 1 _ ^^ a + & a + c a; + 3/— 1 y — a a + b x~y + l a — 1 X y d? -\-b^ X — y — \ \ — b SIMULTANEOUS LINEAR EQUATIONS. 223 11. ^ - y + 1 - a^ IS- {ct--h){x+c)-ay+hz=0, ' ^ + y—^ ' {c~a){y+h)-cz+ax=0, ^ + y + ^ ^l^ X + y + z = 2{a + h + c). x~y — 1 19. -^ + ^=a + h, 12. —-- + .^J—=a + h, a + b a-h y ^ ^ ^.h \ c, X , y o ^+^ ^-^ - + ^==2a. a z . X . — — + - =c + a. a-f- b — c 13. {a-\-c)x-{-(a—c)y=^2ah^ (a+b)y—la--b)x=2ac. 20. — — | — '^ ^-=0, b-\-c c — a a — b 14. o} + ax + y = 0, ^- V- f--A_:^0, ^2_|_^^_|_^_0. ^-^ ^-^ ^+^ 15. y-\-z~~x=-a^ 6 + c c— a a+^ 2; +07 — 3/ = ^, z 21. re + _^_^4__i_^l, 16. 7:.+ lly+. = a, | + _X_+_^^1, 7y+llz +x = b, 7z -{-llx + y~c. c c — 1 (? — 2 17. ^ + ^-?^2a5, 22. -^ = a x ' y z x + y ' c a b ^ zx z X y z+x a b c c a c a b a b c 224 SIMULTANEOUS LINEAR EQUATIONS. 24. X __y z u abed 31. X +ly =.a, y + mz= b, £ . ^ . ^ . !f _1 z -\- nu = c, m n p q r u + pv = d, 25. ax=^by ■='cz=^ y^ — '^ — x — u. du, 32. V -\- qx = e. 26. y + z=-au, x-\-z = bu, x-\-y ^= cu, 1 — x a \-y b y + z+u=b, z + u + v = c, V -{- X -\- y =^ e. 27. x + y = m, y -\- z =^n, z -}~u=^ a, u — x = b. 33. X — y -\- z ^= a, y — z+u = b, z —u-{-v = c, u — v-\-x = d, 28. llx+9y + z- -u = a, V — a; + 3/ = 6. \ly-\-9z + u—x = b, \lz+9u + x — y=^c, 34. x + y + z—u = a, llu+9x + y~z = d, yj^^j^u-v-=^b, 29. x+ay+d'z-\-d'u+a^=0, z + u + v-x = e, x+by+bh+Pu+b* =0, u + v + x — y=^d, x+cy+ch+c^u+c* --0, v+x + y — z=e. x+dy+^z+(^u+d'== 0. 35. x-j-y + z — u — v—a, ^ ' y + z + u-v-x = b, y + z=b, z + u^c, ,+u + v-x~y^e, u+v = d, u + v + x-y-z = d, V + x = e. v + x + y — z—u — e. SIMULTANEOUS LINEAR EQUATIONS. 225 36. 2x — y — z + 2u — V = Sa, 2y — z — u -{- 2v — X =^ Sb, 2z — u -- V -{- 2x — y = 3c, 2u — V — X ~\' 2y — z = 3c?, 2v — X — y '\- 2z — u = Se. 37. V ~ 2x -{- Su — 2y -{- z = a, x-2y + 3v — 2z + u^b, y — 2z + 'dx ~ 2u + V — c, z-2u + ?>y~2v + x = d, u — 2v + Sz — 2x -\- y = e. n — c — n OHAPTEE VII. PuEE Quadratics. § 47. (A) If an equation reduces to the form (j}ix 4- ny = c^, then (mx + n)^ — c^ — 0. Hence, (mxi ~\-n) — c=^0, and therefore Xi = or (mx2 -}- n) ~\- c = 0, and therefore X2 = IIV {B) If an equation reduces to the form fmx + ^Y_ o? \px + q)~ h''' tlipn X - ^^-^^ _ -qa-nh tilt; 11 jji — - , jj2 ^ • mb—pa inh-\-pa (See Exams. 4 and 5 below.) Examples. ^ x + ?>{a — h) _ a(?>x + ^a-lh) x — 3(a-h) b(Sx~7a + 9b) Apply, if !^^-£, therefore !?L+^:^^£±f. n q 7n — np — q Hence ^ 3^ 3r^(a + ^) + 9a^ - 14a& + 9^^ ' S(a-b) Sx(a~-b) + 9(a'-b') Dividing the denominators by S(a — b), ' x[x + S{a + b)] = Sx(a + b) + da" - Uab + 9b\ Therefore, x' = 9a''-Uab + 9 b\ PURE QUADRATICS. 2. 2a + 4:by_bx—da + 3b Apply, if — =-2, therefore = I — ^, n q n p and factor the numerator {x + ^a- 2by -{x — 2a + Uy- . l'2{x+a + b){a-b) _ I2{a~b) {x + ^a-2by bx + ?>a-9b . x + a + b _ x + 4:a — 2b _ S(a~b) " x + 4:a — 2b 6x + Sa-9b 4:X~a—7b' by taking difference of numerators and difference of denominators. To the first and third of these fractions apply, if ^=■2, therefore ^^=^. n q n — m q~p x-\-a-\-b 3(a- ' 3(a-^) ^x — ^a-^b :A[x'-{a + by']'-==9{a-b)\ .'.x'^\[^{a + by + 9{a-hyi {x + 2y ^a x^ — 2x b {x+2y = ^ (1) m{x-{'2y -\-n{x'^ — 2x) ma+nb But {B) can be applied if m and n are so determined that m{x-\-2y -\-n (x^ — 2:^) is a square. This requires that 4 m (m + n) = (2 m — nf. .'. 4 m^ + 4 mn = 4m^ — 4mn + n\ .'. 8m = n. 228 PURE QUADRATICS. Assume m = 1, then n = 8, and (1) becomes, on substi- tution and reduction, {x + 2y _ a _ 2 (3;r-2)'~^r+86"''' '^^• .„_2{l+r) 2(r-l) (^+ir a " {x' + l){x^-2x+l) b Yov x^ -\-l write ^r^;. . {xz + 2xy ^ a^ . {z + 2y ^ a xz{xz~2x) b z(z — 2) b This equation was solved in Exam. 3, hence z may be treated as known. But^^ii-- . x' + 2x + l _z + 2 "x^-2x + l z~2 2 + 2 -, a form solved in {E). Ex. 60. 1. {x + a + b){x~-a + b) + {x + a — b){x — a — b)^0. 2. {a+bx){b — ax) + {b + cx){c—bx)-\-{c+ax){a—cx) = 0. 3. (a + 5a;') {ax — b) + {b+ ex) (bx — c) + (c + ax) (ex — a) = i(a^ + b' + e'), 4. (a + x)(b-x) + {l + ax)(l-bx)^(a + b)(l + x'). 5. (a + x)(b + x)(e ~ x) + (a + x) (b ~x){e + x) + {a — x){b + x) {c + x) + {a — x) (b — x) (c + x) + (a — x)(b + x) (e — x) + (a + x) (b — x)(e — x) — babe. PURE QUADRATICS. 229 6. {a + x) (b + x) {c + x) + {a + x)(b + x) (c — x) -{- (a -\- x) (b ~ x) (c + ^) + (^ — ^) (^ + ^) (<^ + ^) -]- (a -{- x) (b — x) (c — x)-\-(a — x) (b + x) (c — x) + (a — a;) (b — x) (c -{- x) -\- (a ~ x) {b — x)(c — x) = Sx\ 7. {a + bb + x){ba+b + x)=-?>{a + b+x)\ 8. {a + l1 b + x)(\l a + b + x) = ^(^ci + b + x)\ 9. {9a—1b + ^x){^b-1a + ^x)-=-(2>a + Zh + x)\ 1 Q?^ _[_ cd _ Q - _ g — a: _ b — x o?—h^x^ c^ — d'^x^ ' 1 — ax 1 — bx 11 ^^ — Q^ I x-\- a _cy ^n. x-\-a+'2b _ b — 2a-\-2x ' .r+1 x-l^"^' ' x+a—2b~ b + 2a—2x -2 a + x _ x + b a+^b+x _ ?>b — a+x a — x x — b a — 4:b+x ?)b-\-a — x ^„ ax -\-b _ ex ~\- d -J. x-\-ba-{-b x—a-] -b a-\-bx c-\-dx x~Sa^b a—x^3b 14 <^ — ^ _ 1 — bx a — 7 b + x a-j-bb-j-x 1 — ax b — x la — b—x ba-\-b-[-x 20. 21. 22. 23. 24. ?)a—'b^x bb — ^a-\- X a — -3b + x ba — db + x 3 a -2b + Sx ■_ x — a + 2b a --2b + x 3x~Sa + 2b 3a ~2b-\~3x _ x—7a + 8b a -2b + x 3x-- ba + 4:b 5a -6b + x_ __3a~-5b + Sx a -{- X a + b-}-x a + b~x _S(a-b + x) 3a~ b — Sx a — bb -{-X 230 PUHE QUADRATICS. 25 "^ a + h — X _ 3 (g — /; -f x) ba + ?>h — ?>x~ a~llb + x' 2g ba-h + x ^ 2{2a~h + x ) ' ■'2(ia4-2b-x) a+llb~x ' „,- 7 a — h -\- X __ a(a -j- bb -\~ x) 1b-a + x~b{ba + b+x) 28 ^ + <^ -— ^ _. a{x-{- a-\-bb) X ~ a + b b {x -{- b a -{- b) 29. 30. 31. 32. r ba-U + xV _^ 1a-^b + ?>x \bb~?>a + x)' lb--'da + ?>x 'a + bb + x\_a + 11b + x ba-\-b -\- xj lla-\-b-\-x b + x\__l1a + b-x /7a a + xj VJb-{-a — x l1a + b-x ^ o?(a + l1b + x) a+llb-x U'{Vla + b + x) 33 {x -\rl a + b){x — a-\-b) __ x — ba~\-b {bx + ?>a-llb){x-a+llb)^ bx+la-b2b 34 (l + 3^+5.2:^)(^^ + 3:r+5) _9 {l + 2x + ^x'){x' + 2x-[-?>) 4* 35. ~ I + 1 = ^^ :?;"^- 11a; + 28 x'-llx+IQ x^--l4:x + ^0 36. ^ + ^ = ± x'-^x + b x' - Ux + 45 x'- llo; + 10 37. x\b - a') + a\x - b') + Z>X« - ^') + a^^(«^^ - 1) = (a-x')(b'-a'). QUADRATICS. 231 > Quadratic Equations and Equations that can be resolved as quadratics. § 48. ((7) If an equation appears under the form (a — x) (x~h) =z c, (1) then ^1 = |- (a + 5 + r), x.2=^^(a~{- b — r), in which ?^^ = (a — by — 4 c. From the identity (a — x) -{- (x — b) = a — b W9get (a-xy + 2(a-x)(x-b) + (x-by--=(a~by, (2) (2)-4(l) {a~xy -2(a-x)(x-b) + (x -by = (a — by — 4tc = r^ say. Then, [(a- x) - (x - b)J -r' == 0] hence, [(a—Xi)—(xi—b)]-]-r = 0, and .\Xi = |-(a-t-Z>+r) ; or, [(a-~X2)~-(x2—b)]—r = 0, and .'..X2=' ^(a+6— r). Examples. 1 . x' + {ab + ly - {ciS+ b') (:i;*^ + 1) + 2 (a' -b')x+l. .■.x' + a'b' = (a' + b') x' + 2 (a' -b')x + (a- by. /.x'+2abx'+a'b' = (a+byx' + 2(a:'-b')x + (a—by. .-. x^ + ab = ±[(a + b)x + (a — b)], or x'^ ^ (a -}- b) X -{- ab = dz (a — b). .'.y ^{a+b)x + \{a + by = \{a-by^{a- b). .'.x-^^{a-^b) = \^[{a-by±^{a-b)l 2. ax-^b mx — n bx-\- a nx — m Add and subtract numerators and denominators, (g + Z^) (.r + 1) _, (m + n) {x — 1) (a — ^) (:?; — 1) (m — n) (:^ + 1) 232 QUADEATICS. f^±l\'= {a-b)(^^ + n) ^ ,._ 3^^, "\x~lj (a -j-b)(m — n) t + 1 _s-l S—1 5+ 1 3. (a~xy + (h-xy = c. In tlie identity (u + vy=::u' + v' + 4:(u + vfuv - 2t^V let u = a — X, V = X — b. , .'.u-\-v—a — b and u^-{-v^^c. /.(a-by = c+4(a~by(a~-x)(x-b)-2(a-xy(x--by. Write z for (a — x)(x~b). .'. z' -2(a- byz + (a- by = ^[c + (a- by] = f, say. .•.[z-(a-byy=:f. .'.hj(JBy z, = (a~by-t- z, = (a-by + t. .'. z is known. But (a — x)(x~b) = z. .•.by((7), x,^^(a + b + r); :r^ - |(a + 5 - r), (1) in which r^ = (a~by — 4:Z = (a~by-4:[(a-by-t]--=At-S(a-by\ .ox 0T(a-by-A[(a-by+t] = -4:t-^(a~by) ^ ^ SindLf = ^[c + (a-byi (3) Hence, x is expressed in terms of a, b, and r ; r is expressed in terms of a, b, and t ; t is expressed in terms of a, b, and c ; and the expressions for r and t are cases of (A). 4. (a -:t') (b + ^^ + (a - a;/ (5 + x)=ab (a' + ^^'). Let a — x^=n~~ z and 5 + ^ — '^ + 2;. .■.w = |(a + J). (1) QUADRATICS. 233 The equation reduces to (n^ - z') [(n + zf + (n- zf] = ah (a' + ^'). .-. (n' - z') (2n' + 6nz') = ah (a' + h'). .-. (n' - z') (V + Sz') = ah (a' - ah + h'). '£■ may now be found by (C), and from (1) x=^\{a—h)-\-z, Zz^^\{a-lf or \{\^ah-a^-h''). :,x = ^, ova-h, ov ^{a-h)+\^{?>0ah-'S>a^—2>h^), 5. rr*-4 = ^-±^; .-.x^-^x'-bx'-U^O. x^-2 Find the rational linear factors of the left-hand member by the method of § 27, page 116. .\{x-2){x + 2){x' + 2x^ + ^) = 0. :,x-2 = 0, or rr + 2^:0, or :r* + 2.^^ + 3 = 0. The last of these equations may be solved as a quad- ratic, giving :i;'-:-lzb2V-2. .-.rr =±l±V-2. :.x, = 2- x, = -2) x,= l + -yJ-2- x,= l~-yJ-2) ^5 = -l+V-2; x, = -l--^~2. Note. In solving numerical equations of the higher orders, the rational linear factors should always he found and separated^ as dis- junctive equations, before other methods of reduction are applied. Such separation may always be effected by the methods of §§ 26-29, and, unless it is done, the application of the higher methods may. actually fail. Thus, if it be attempted to solve as a cubic the equa- ^^^^ 0,-3 -9a; -10 = 0, ^ the result is x = {5 -\- y/- 2)h + (5 - V" 2)3, which can be reduced only by trial. The left-hand member can, however, be easily factored by the method of § 27, and the equation ^^^^^^^ ^0 (x + 2){x^-2x~d)^ 0, which gives .t = 2 or 1 ± y/6. 234 QUADRATICS. 6. (x-2y~x' + 2' = 0. Factor (see Exam. 20, page 113), rejecting constant fac- tors, .-. x(x — 2) (x^ — 2x + 47 = 0. ...x^O, or x — 2 = 0,OY x''-2x+4:^0. The last equation gives r^ == 1 ± V~ ^• Ex. 61. Solve the following equations : 1. (x + a + bf = x' + a' + b\ 2. (x-i-a + hf = x' + a' + b\ 3. (a — h)x' + (b-x)a' + (x-a)b^ = 0. 4. (a — b) x^ -j-(x — b) c^ ■\-(x-\-d) b'^ — 2abx. 5. {x-~af + {a-by + (b — xy^O. 6. {x - aj + (a - by + {b - r?;)^ = 0. 7. (a^ - Z^) :r* + {x^ - a) b' + (P -x)a' = abx (a^b'^x' - 1). 8. {x ~ a){x — b) {a — b)-\r{x — b) (x — c)(b — c) + (x — c)(x ~ a) (c ~ a) = 0. 9. ?l_i=:0. 11. ± = 0. x—1 x^—1 10. ^ = 0. 12. - = 0. x^ —1 x*—l 13. x' + 5a^~16x'' + 20x~16^0. (See § 21.) 14. x*-Sx^ + 5x''-\-6x+4: = 0. 15. (x-ay + x' + a' = 0. 17. rr (r?; - 2)^ (re + 2) - 2. 16. 2a^ = {x~6y. 18. (4rr^-17):r+12 = 0; 19. :r*+ (a^> + l)'^ - (a' + b')(x' + 1) + 2(a^- ^^^) a: + 1. QUADRATICS. 235 20. x'' {x - 169)2 -f- 1 7 .r = x'' - 3540. 21. Qx{x'' + Vf + {2x'+by=lb0x+l. 22. 2;r(:r-l)2 + 2-=(.^'+l)^ 24. 5^;^= 12.x^^ + 1. 23. .X'*=12:i; + 5. 25. {x + ^f ='^{2x ~-l)\ 26. ^ + _^+_^ + _^ + _L.+_^___0. ^' .r — 1 X — 2 X - 3 .^' — 4 x — b 27 (^+iy _m 3Q {x'+l){x'+V) _on {x' + l){x-iy' n ' (x'-l){x'-l) n 28. (^+1)' ^^ 31. (^'+l)C-^^+l) _m 2^ vV + l)(rr'+l) _m 32 (^' - 1)'^ _m {x + l){x'+l) n ' x(x' + l)(x-iy n 33. 34. 35. 36. 37. 38 x{x-\-\y' n(ji — m) {x" + 1) (x - Yf ~ 2m{27n - n) {x^+iy _ 4m^ X {x^ — 1)^ m^ — n^ {x-l){x'' + Vf _2{m-n)\ {a^ — 1) {x + 1)^ mn x^—1 _ 2m (a:+l)(:r^-l)~2m-7i' (a;^ — 1) (a; + 1)^ _ m + w (a;^ + 1) (r?; — 17 ~ m — 92 (x+ 1) (a;' + 1) _ m + 71 (a;- -l)ix*- -1) r>i~n 39. r'-f -^. 41. ;.^-f-^ 6>:r — a 40. .r'_«^-^ bx~a 42. .^...-«^'+^^ + ^ a + So; + cx^ 236 QUADRATICS. 43. x' = (x-iy(x' + l). 44. aV = (a-xy(a'--x''). 45. x'' = (x-ay(x'-l), 46. 7)i(x -\- TTb — n) (x — 7n -{• 7 ny = n(x — m -\- n) (x -}- 7 m — rif. 47. 7n^ (x-\-7n-\- 17 n) (x — 7n — 5 ny = n^(a7+ 17 m + n)(x — 5m -\- ny. 48. 7n? (x -{- on -}- 17 n) (x — on -\- 7 ny = n^(x-{-17m + n)(x-}~7m — ny, 49. _4-_=r_. 61. L_^ — =c. X a n x^~ax + a^ x"^ — ax-\-o? ' {x-\- ay a—x.x—h m 53. x~ b a — X n g^ 2a? + a{a- x) + {a + xy _ c + 1 2a^ + a{a + x) + {a — xy c—1 55. x'^-{-(a — xy = c. 56. x^ + {x-^y^S2. 57. {a~xy+{x-hy=-e, 58. :?;H(a — ^)^=-a^; ^^+(6 -^7=:= 1056. 59. (a - rr)^ (a; - hy + {a- xy (x - by - a'h' (a - h). 60. {x-a+hy-{x-ay + {x-hy-:(^ + a''-{a-hy--b^ = {a-h)c\ {a-xy + {x-hy _ a' + h' {a-xy + {x-hy a^ + h' ^^ (a-xy + {x-hy _ a' + b' (a - xy + {x-- by QUADRATICS. 237 63. {a-xy + {x-hy a'-b' 64. (a-xY (b~xf_a' P b — X a — X b a 65. a — X . X — b a b {x-by (a-xy b' a' 66. (a-xy + (x-by_ a' + b' (a + b-2xy (a + by 67. (a-xy + (x-by_ a^-b' (a + b-2xy (a + by 68. (a-xy + ix-by ^ ^ 69. (a-xy~(x-by__ (a-b)c (a — x) — (x — b) (a — x)(x — b) 70. (^_^)5 + ( 5)5 ^ 71 {a-xf^ix-Vf c (a - xy + {x- by ' {a-x){x- b) 72. (l+xJ^{x'-Zy. 73. ^'+-^ ^xix'+l) b ^4 {x + \y{x^ + V) _ {x-iy{x^-x+i) {x'-x + iy~~b ' (x+iy{x' + i) b ^^ (x' + iy _a x(x + iy b 78 (^+^y ^^ x(x' + l) b 79 •'^(^+1)' — ^ (x - ly b 80. 81. 'x''+x+r T x'-x+T \_ o. L(.T+i)'^jLcr-i)^J 'b x^ — x^ -\-\ _a {x^-iy ~~ b 82 ^(^' + 1) — ^ 238 SIMULTANEOUS QUADRATICS. 33 ( x+l)(x' + l) _a gg ix±iy^a (a;-l)(r^-l) /; ' x' + 1 h \x-\){x'-\-\) b ' x'^+l h § 49. Quadratic Equations Involving Two or More Variables. 1. {x + y){x' + f) = a, (1) x^y + xy"^ = c, (2) (l) + 2(2), .•.(a; + y)' = a + 2c. ■.x + y = -^{a + 2c). (3) (Any one of the three cube roots.) (l)-f-(2), a^ + y' __ ft . . /'^^ly Y_ f^^Il^?^. By (3), :,-y ^ y(«-2.X ^^^ ^ ^(« + 2c) : + y/ a + 2c l£) . Also, x + y =2li2±M. ^^ ^(a + 2c) . V(a + 2c) + V(«-2c) 2^(a + 2c) ■ ._ V(« + 2g)-V(a-2c) ^ 2^(a + 2c) Not any one of the six sixth-roots of a+2c may be used indifferently in the denominator, but only any cube-root of whichever square root of a + 2c is used in the numer- ator. Thus, if the radical sign be restricted to denote merely the arithmetical root, if k be defined by the equa- tion F — Jc-\- 1 = 0, and if to and n indicate any integers whatever, equal or unequal, the value of x may be written. SIMULTANEOUS QUADRATICS. 239 2. Sx' — bxy + ^y''-='^{x + y), (1) 11 :t^^ - d>xy + by^ = 13(:^ + y), (2) First Method. Eliminate (x + y). .'. lO^x"" - 65:ry + SOy^ - 99.r' - 72^7 + 45^/1 .\bx' + 1xy — ^y'' = 0. .•.(5^-3y)(a; + 2y)^0. .•. ^ = f y or — 2y. Substitute these values for x in (1), .-. 72y^ -: 360y, or 45y2 - - 9y. .•.y==0, or 5, or —J-, and x^=0, or 3, or f. Second Method. Take the sum of the products of (1) and (2) by arbitrary multipliers Ic and I. Tc(Sx^ - bxy + 3y0 + ^(llo;^ - ^xy + by'') = {^h+l?>l)(^x + y). (3) Determine 1c and Z so that the left-hand member of (3) may, like the right-hand member, be a multiple of a; + y- This may be done by putting x=^ — y in (3), from which 16^ + 24^ = 0. .'.2Tc = -^l. .•.if/^=3, l = -2. Substituting these values in (3), it becomes 2x'' + xy — y'^ —-x-\-y. .' . {x + y^i^x - y) =^ x + y, or" (x -{-y){2x — y — V) =^0. :. either x + y =^0, ov 2x — y —1=^0, ,'. y^= — x, or 22^ — 1. 240 SIMULTANEOUS QUADRATICS. Substituting these values for x in (1), it becomes 16^2 ==0, or 10^'-7rr + 3=-272;-9. .*. x = 0, or 3, or -|, and 3/ = 0, or 5, or — -|-. 3. x^ + xy + 7/' = a' + ah + b\ (2) (a' + by-a'b' a^y + xi/^ _ a^b + ab^ {x^ + yy - ^y ~ («' + ^')' - «'^' Write z for a;^ + y'^ ci^ + b^ (3). . 2 y^ l-z' l~¥ \z = k or — -• fc xy ab c? + b^ \ • ^ =— — or !— — . (3) x"^ + y^ d^ + b'^ — ab xy ab a^ + b'^ .*. sz = or ■ x'^ + xy + 3/^ a^ + ab-\- b^ a^ — ab + b'^ (2), .-. xy = aJ or (a' + J^ "'^ + ''^ + ^[ - (4) a — ab -}- V[(2) + (4)], \a^ + ab + b^ ^(2a^^ab + 2b^)^^ ab + b' SIMULTANEOUS QUADRATICS. 241 V[(2)-3(4)], and X — 7/ = ±:(a — b), '' + ab + b' i^(2a' + ab+2b')J^ -ab + b' .'. X = ±:a,±: bj or i[-y/(2a'-ab + b') 3/ = ± 5, =b a, or -. i[^(2a'-ab + b') ix + i/)(x* + ^') = b. (2) Put 2=^^- x' + f . 1 — z a 1-22^ b .■.2az'-bz-(a-b) = 0. :.4taz = b± y/{%a? - Sab + b'') = b +r, say. ._x^_^b±r .g. _. /g + y __|_ I 2ffl + 6 + r x — y \2a — 6 — r . a; _ V(2a + & + r) + V(2 « - 6 - r) ■> V(2« + 6 + »-)-V(2«-&-'-) _ r V(2a + & + r) + V(2a - 5 - r)? .4. 2(6 + r) ^^ 242 SIMULTANEOUS QUADRATICS. /OX / N5/4a+2Z)+2rV4aV/4a-Z>-rV , . .0 fy\ p2a^(2a+b+r)(Aa-b-ry- \ _ , \i/J[32(2a + b + r)(4:a-b- rfj /.x _ \^(2a + b + r) + ^(2a-b-r)r ^ ^' ^ 1024(2a + b + r)(4a - ^ - r)^ . V(2a + ^ + r) + V(2a-^>-r) 2^[(2a + ^» + r)(4a-6-r/] ' in which r = ± V(^ a^ — 8 a5 + Z>^). The value of y may be derived from that of x by the first form in (4). 5. x^ = ax~ by, (1) y' = ay- bx. (2) ^X(l)-yX(2), yX(l)-xX{2), xy {x^ — 'if)=^b{x'^ — y'^). .*. either x — y = Qy from which :r = 3/ = 0, or -^/(a — ^), (3) or :r* — r^V + ^V + ^Z + y^'^^C^ + y)' (4) and xy {x" + xy + 3/') = b{x + y). (5) (4) + (5), {X + 3/)X^^ + 3/^) ^{a + b){x + y). (6) (5), (^ + yy - {x'^ + ff = ^b{x + y). (7) V[a)^ + 4(6)], (^ + yy + (o;^ + yy '-=2t{x + y\ (8) in which t = y/[(a + by + 4: b']. (9) I SIMULTANEOUS QUADRATICS. 243 i[(7) + (8)], .■.ix + yy = {2b^t){x + y). :.{x + yf = 2h + t. .■.{x + y) = ^{2b + t). (10) (6) -^(10) :.x' + y''= ^t^ • (H) ^ ' "- ' ^y -^(26 + ^ ^ 2(ll)-(10)^ 2a — i! ^(25 + ,.^_ V(2«-0. y ^(26+0 (10) and a; + y = Yl^f + ^X ^ ' -^ ^(26 + . ^_ V(2& + ^) + V(2a-<) ^(26 + and .^V(2^ + 0-V(2a-0^ in which < == V(«' + 2a6 + 5 6'). 6. a;* — c* = w(a;+y)*, (1) y* + e* = n(x-yy. (2) Let x~-y .•.2 + l=_^and2-l=-?^. (3) a;-y a;- y (l) + (2), x' + y* = m(x + yy + n{x-yy. .: (z + 1)* + (z- ly = 16 (mz* + w). .-. (8m-l)2*-62^ + (8n-l) = 0. 244 SIMULTANEOUS QUADRATICS. . _ / 3+V[9-(8m-l)(8n-l)1 • z= / 3-r VI o — yuv/^— xyi^u/i.— -l;| ... \ 8m -1 ■ ^' (2) and (3), (z - !)*(» - y)* + 16 c* = 16 w (a; - y)^ 2c and and ^[16«-(z-l)^] 2cz >[16n-(2-iy]' c(z+l) _ c(z+l) (5) y=T ^[16w- (z - 1/] ■g) {a + h + cy Write r^ for 2 (a' + ^' + ^' - 3 ahc). (9) ...:, + 3/ + ,^_^.^. (10) Returning to (8), . , , X2 2(a?-\-h'^+c^—ah—hc—cd) /qn a+6 + c (4) 2Cr^+y-+.^-rry-y.-.^-) =. ^'^+^+'^' - (11) i[(8) + (ll)],a;^ + y^ + z' = '^' + f + < (12) (5) and (10), x' + f + z' ^f— = a. a-\- b-\- c :. rx^{a+h + c) {x^ + y'' + z^) -a(a + h + c) (12) = o? + h'' + c" - o.{a + h + c) ^y^j^c'-aih + c), (5), (6), and (7) are symmetrical with respect to {xyz\ahc) ; (10) shows this substitution does not aiFect r, and conse- quently the values of y and z may *be written down at once from that of x. SIMULTANEOUS QUADRATICS. 251 Ex. 62. 1. ^[{1 -xy + f] = l^{l -x)y', x' + 4.y = f + 4., 2. I0x'-^y'' = 2x^', 8^2-63/^ = 130;. 3. xy^(^^-xy = {2-y)\ 4. ri:' + y2=-8:r + 9y=144. 5. x' + f = x + y + l2- xy + ^ = 2{x + y). 6. x + xy + y = b) x' + xy + y''^1, 7. x^ + y^^1xy = 2^{x + y). Q 2 , ,2 35 28 X + 3/ ^ry 9. x' + xY + y'=^l?>^\ x^y + xY + xif = 11^. 10. (rc + y)(:r'+3/0 = 17^y; {x-y){x^ -y'') = %xy. 11. 25(:^^ + y^) = 7(^ + 3//-175a;y. 12. 2a;2 - 3/2 = 14 {x'' - 2^^) = 14 (r^ - 3/). 13. 2x'-Zxy=:^^{x-^y)] ?>{x'~?>y'') = 2{2x^-~^xy). 14. 2:r2-:ry+5y2=10(a;+y); x'+^xy+'^y' =^l^x+y). 15. (2a;-3y)(3:?: + 4y)-=39(:r-2y); (3 :^ + 2y) (4 :r - 3 y) -: 99 (:?; - 23/). 16. {x+2y){x+?>y) = ?>{x+y)', {2x+y){^x+y) = 2^(x+y) . 17. r?; + 3/-=8; :r* + 3/* = 706. 18. :?; + 3/ = 5; 0:^ + 3/^ = 275. 19. :z; + y = 2; 13(a;^ + y^) = 121 (:?;^ + 3/=^). 20. rrr + y = 4; 41 (rz:^ + 3/') = 122(a;* + y*). 21. r^2__ 5^y _|_y2 _^ 5 ._ Q . xy =^ x-\-y —1. 22. :?;2 _|_ y — 5 (^.^ __ y^ . ^^ + 3/^ = 2 (a; — 3/). 252 SIMULTANEOUS QUADRATICS. 23. ?>{x' + y) = ?>{x + y'') = l?>xy. 24. 10 {x" + y) = lO{x + y^) -- 13 (x" + y% 25. x^ + y = ^', ^ + y2^2^. 26. 9(:r^ + 3/)-=3(ri7 + y2) = 7. 27. ^ + ^y + y'=5; r?;*'^ + ^y + y^ = 17. 28. :?; + 3/ = 2; (:r + 1)^ + (y ~ 2/ -= 211. -■ ^(-i)&+i)=*wi)(^i): |g±-;=i@^)- 30. a; + y~ — ; x — y ^= xy. xy 31. a; + 3/ + l--0; :r^ + / + 2--0. 32. x + y^l; 3 (;r' + y') =- 7. 33. 4.t/==5(5-.t); 2(^^ + y')-=5. 34. 27^77 -IT; 9(:r^ + 3/^)=:~8. 35. (.1- + yj + ^xY = 5 - 12y ; y {x" + y^) _^ 3 =, q. 36. X -{-y =^xy\ x^ -{- y"^ =^ x^ -\- 'jf. 37. oc'y + xy' = h. 56. x^y-\-xy'^ = a{x^-\-y'^)\ x'^y~xy'^~h{x^-~y'^), 58. r^^ + y2 — ^^2^2 __ ^^ ^^ _|_ ^^^ 59. a^^y = a (^^ + 3/^) — Z> (a; + 3/)^. 60. rry (:?; - [- y) — a ; r?;^ (x^ + ?/^) = Z). 62. ^r* + 3/* = m (r?;2 + y"^) ; o;^ + a-y + y^ = 92. 63. a^>(rr + y)--^y(a + ^>); r^' + y^ -= a' + 5^ 64. x^ + f = a{x + y)] x^ ^y"" = h{x + yy. 65. :^' + 3/2-:a; ^^ + y^ -- 5 (a;' + 3/^). 66. xy^=a\ x^ -{-y^ ^=h{x^ -^y^). 254 SIMULTANEOUS QUADRATICS. 67. {x~y){x' + f) = {a-h){a^ + ¥)', a^-y'^ = a'~h\ 68. x^ — y^ = a\ x^ + i/ =.h{x — y). 69. x + y^a\ x^ + y^-=h. 70. x + y = a\ x'-\-y'='b. 71. x-\-y =.a\ x^ + y"^ ^= SVy^. 72. x-\-y=-a + h] {a-hy{x' + y') = {x~yy{a} + h'), 73. x + y = a', c{x^ + y^)^ xy{x^ + y^), 14., {x + yy = a{x'' + y'')\ xy = c{x + y). 75 . x^y + xy"^ = a^ ; c^ (x^ + y^) = x^y^. 76. r?;^ = a (x^ + y^) — cjt^/ ; y^ -= c (x^ + y^) — axy. 78. o;'^ — 2/* = a^^y ; (:r^ + y'^y == ^^ (r^;^ — 3/^). 79. (a; + y) x^y'^ — a; x^ -{-y^ = h. 80. (a; + y) ^y = <^ ; x^ -{-y^ =^ hxy. 81. a;* + y* = a(a; + y)'; r?;^ + y^ =r ^ (:r + y)l 82. x^ + ii^^y'^ + y* "^ <^ ; ^^ — ^y + y^ — 1. 77. x"^ — y'^-=^a^\ x^ — 'i^ = c^ -y ,. x\l + xY )_ar i+xy '\ y\l + xyy b\l-xy] x' + y''=.a{x + y)\ x' + y^ = h{x^ + y'). x^ + y^^a] {x + y){x^ + y') = b{x'' + y''). {x" + y') {o(^ + y') = axy ; (a; + y) (^* + y*) = ^>rry. (^ + y)^(a;^ + y^) - a ; (:r^ + f)\x' + yO - b. (x — y) (:?;'^ — y^) (ri;* — y*) — 4 aa:y ; {x + y) (o;^ + yO (o;^ + y^) = 5(^ - y). x'^y + ^y* = <^ {oc^y + ^y^) = 5 (rr* + y*). 83. {x' + f)xy-- 84. 85. 86. 87. 88. 89. SIMULTANEOUS QUADRATICS. 255 90. a{r' + y^) = ah{x + y)=^hxy{x' + y^). 91. X^ _ y3 ^3 _ ^3 ^ X" — ^ 92 x^ + f _ a^ + h\ x' + y' __ a'+b'' x^ — ?/^ a^ ~ h^ ' x^ — 3/^ o? — h^ 93. x^ ^^2ax — hy ] 3/^ — 2 ay — 5^. 94. {x + y) (^^ -f- y^) = a ; {x — y) (o;^ — y^) — b. 95. (^+y)'(^'+^y+y') _ 3^2 . (x'+y')(x''+xy+y') 96. (:?; + y) (^^ + y^) = aa;y ; (x — y)(ci^ — y^) = hxy. 97. (:^+y)(^^+y^) = a(:r^+y^); (a;-y)(^^-y^) ^ ^(:r^+y^). 98. (^+3/)'(^+y') _ ^2 . (^-y)X^ -/) _. ^2 (x'+xy+y^Xx'^+y') ' (a;''^~^y+y'')(a;'+y') 99. (^'+3/')(^+y)' _,o^2. (^'-f)(^-yy _2b' x'^ + xy + y'' ' x' — xy + y'' 100. (^' + 3/')(^ + 3/y _3^2. (^'-3/')(^-yy _352 (:r' + rz:y + y'7 ' {x" - xy -\- y'-J 101. a:y(a; + y)(^' + y') = o^; xy{x — y){x^-f) = b. 102. :^(^+y)(^+2y)(^ + 3y)-a^ (:^.+y)2+(a;+2y)^= ^. 103. (:r+l)(y-l) = a(:r-l)(y + l); (^^ + l)(y-l/^^^(:.-l)Xy^ + l). 104. x + y = a(l+xy)', {x + yf =-¥{!+ xY). 105. a: + y==a(l + a;y); ^^ + y^ = Z)'(l +ri;y). 106. (a; + y)(y-l)-a(:r-l)(y+l); {:^-V){y-l)^b\f-l){x-l). 256 SIMULTANEOUS QUADRATICS. (1 + ^)(1+.V) ^ {l + x)\l + yr ^ • {l-x)(l-y) ' (l-^)(l-3/) • 108. (^ + ^)(g + y) _„. (c^ + ^0(g^ + yO _^_ (c — «) (o — y) ' (c* — a;*) (e* — y*) 109. (^ + ^)(y + ^) -a. (^' + mO(y* + wO _^ (a; — 7n){y — n) ' (x — m)\y ~ n)* 110. 116. (a;-l)(y-l) o' (:>fi-l)(f-l) c 111. ^Il±i^) = a;^!Ii+^ = J. x(l + /) ' x\l + y') 112. 1+5==. I?; ^Ili5+^==5. x(l+y') ^a + f) (x + y)(xy+l) _ a' + h\ x(y'+l) _ a- {x — y){xy — V) 2ab ' y(a^— 1) a-\-b 117 {x + y){l + xy) ^ ,.„ _ ,^ . a:(l- ,yO ^ , 118. (^ + .y)(l + ^y) _„. (x' + ?/)(l+a;y) _^_ (a;-y)(l-a;y) ' {x'-f){l-xY) 120 ( ^'+.v^)a + y + z){x + y + z)+{xy + yz + zx)=^Q>l, {^x — y + 2z){x + y + z)—b(xy + yz + zx) = b. 4. ^^-y2==0, 5. (^' + 3/' + 2'/ + (^ + 3/)'=-31, ^' + y' + 2' = 21. 6. x + yz = l^, ' y + zx=- 11, z-\-xy = 10. 7. x + y = 8z, x' + y' = lS4:z\ ^ + y' + 2'=-134. 8. X +y =5z, x^ + y'' = S9z, x^ + y'=105z\ 9. a; +y =72;, af + y' = 25z\ x' + y' = 674cz'. 258 SIMULTANEOUS QUADRATICS. 10. V3/ V :?;^ + 2/" = 20,2720. 11. x-^y : y^z : 2;+^ : : a : ^ : c, (a+^+c?) ^72; = 2. 12. x^y : y+2; : 2;+:?: \\a\h\c, (a-\-b-{-c)xyz ■= 2 (^'+y+2:). 13. (a7 + y — 2;):r = a, 15. (y + z)(2x + y + z) = a, (x-y + z)y = b, (z + x)(x + 2y + z) = b, (—x + y + z)z=-c. (x+y)(x + y + 2z) = c. 16. X (y -{- z) : y (z -{- x) : z (x -\- y) = b -j- c : c -\~ a : a + b, xy -{- yz + zx = (a + b + c) (x -\- y -\- z). 17. (a + b)x+(b + c)y+(c+a)z = {a+b + c)(ix+y+z), (^ + y)^ + (y + .^)^ + (^ + ^)^==4(a^ + ^^+c0. 18. c(5; + 2/) + ^(^-2;)-a(y + 2) = 0, b(x — z)~(a — c) y, x'^ + y' + z'^a' + b' + c^ 19. ^ + y — a2; = :r — Z>3/ + 2; = — c^ + 3/ + 2; = xyz. 20. (a + 5 + c) (:r - y) + a(^ + 0) - ^(y + z) = 0, (a + ^ + c) (ix — z) + alx + y) - c(y + z) = 0, (b + cy'^(c+ay~^(a + by ' , X , V 7 , z 21. a:v + - = «, 2/21+^ = 6, zx+- = c. ^ z X y 22. 3/ + 2: : 2; + ^ -^ + 3/ : • ^ + ^^ ^ + <^ • » + ^, (^ + y + 2) (0:3/2;) = (a + 5 + c) (:ry + 3/2; + zx) . 23. x'^ — yz= a, y'^ — xz — b, z^ — xy ^ c. 24. a;^+(y-z)' = a^ y^+(2;-a;y-:^>^ z2 + (a;-y)^ = c^ SIMULTANEOUS QUADRATICS. 259 25. x^ '\- xy ■\- y'^ =^ a^ y'^ + 3/2; + 2^ ^ Z>^, 2^- -\- zx -\- x^ = cl 26. X ■\- 1^ -- 'i' -\- Sxyz — a(x -\-y — z), ^3 _ ^3 _j_ 2;3 _[_ ^xyz — h{x — y^z), -x'-\-f-\-z' + Zxyz-=-c{~x-\-y + z). 27. a; + y + 2a2 = 0, 34. (:^ - t/)^ = a2;(^ + y), a;' + y"- - 25V =0, a;^ - / = hz{x + y)^ ^n _|_ ^n _j_ ^n ^ C". {x ~ yj = Cz(x' + /). 28. 29. 30. 31. 32. 33. X (y --l) = a(z~l), 39. xy = uv = a^, x^y^-l) = h\z^ - 1), x^y-\-u-\-v = l, x'^f-X) = c^(/- 1). ^^ + y' + 1^^ + 'y' = &. x + y — az = 0, y{x^ + f) = h\ 35. x-y = a, u--v = hj x^ + f = c\ xy = uv, x{y-l){z-l) = 2a, x'~y^+u'-v^ = c(a+b), x\f-l){z'~l)=Uz, x\f~l){f-l) = Qcz\ 36. X +y =a, oc {y—l) = a{z — l), x' + u' = c\ x\y^-l) = h\z'~ll f-^^v'=e\ x^{f-l)=^c\z'-l). x{y — l) = a{z-l), x\f-l) = h\z^~l), x\f-l) = c\z^-l). 37. xy ^=uv = a^, x-\~y + u-]-x = hy x\y^~l)^h\z'-ll 38. xy = uv == a^ x + y + u+v = h, x'{y'+l) = c\z'+l\ x' + y* + u^ + v'^c\ 260 SIMULTANEOUS QUADRATICS. 40. xy — uv = a^, 47. x^ + y^ — <^^ x + y + u + v-=h, ^2 ^ ^2 ^ ^2^ \ ^ J ^ yi^ ^ ^ ux-\- vy = c^, 41. xy = uv — (^, vx-{-uy = n^, X -\-y + u-\-v = h, {x + uy + {y + vy = c\ 48. x + y + u + v = a, xy -\-uv =^ Jy^, 42. xy^=uv, ^j^y + u+v=a, ^' + f = ^\ x^ + y'' + y? + v' = h\ u^ + v'' = n\ x^ + y^ + u^ + v^ = &. 49. y{\-\-x'-)--=-'^x, x+y+u + v-=a x' + y' + u' + v' = h\ V 1 y 4,4,4,4 4 ^(1 +^0 =2-^. 44. :^y — i^^;, 50. x-{-y^-u-j-v = a, X +y +u +v =a, (^^^yyj^(u + vy = h\ x' + y'' + v} + v' = h\ , , .2 , / , V x^ + f + u^ + v^ = c\ {x + vy + {y + uy = e\ 45. xy — uv, x+y+u + v=a, x' + y' + u' + v'^b', x^ + y^ + u'' + v' = c\ 61. X "la- — u y + ^ a — 2u y - 2h — u z^x h- ■2u z _2c- — u 46. xy — uv = 0, xu-\-yv = a', a;» + 2^» + M»+t,' = (r'. a? + f + z' = e'. SIMULTANEOUS QUADRATICS. 261 52. = a, 53. 54. 1 + x + x'' i + y + 3/' l + x + f x+1 fx~~\ ' —-a 56. 55. 61. 62. 63. 3/ + 1 X^ -\-X-\-\ _ yi(x_ y' + y+i' 1 x-\'y __ a l-\~xy b -{- c • ^~y ^ — ^ 1 — a;y a 57. - — '-^- = • > 1 — xy of- — a^ x~y _^ 2hp l + xy^b'' — /?2' (l-:r)(l-y) I- a (l + x){l-y) _, l + h (l--x)(l + y) l-h X -\-y c^ — a? 58. l + xy ■y^ 2a l + xy a^ + a^ x~-y _ h''~~^' 1-xy b' + l3'' 60. 2ax^(b + c~ 2by = (c -\- a - (x+y + zy+x' X — •1_ a — 1 y- 1 b- -1' ar'- -J., _(l_ -1 x-\-y 1 -j- xy m 1 — xy . x — y __ 2b x — y 1—xy n ^^ y(l + x'') x{l+y') ■ «) (y + ^). + f+z^ = ^{a- + b' + c% 64. ^3 + y3=-^, y^ 1 ¥ xy — xy x + y x'^-\-xy-\-y'^ _ x^-^^f xy x^—xy-\-y'^ a b x^ + x'^y^ -{-y^ ^ a, x"^ -}- xy -{- y'^ = b. 65. xy -\-- = a(x^ — ?/^), xy =^ b (x'^ + y^). 66. a^ = a(x''+7/) — bxy, y^ =zb (x^ + y^) — a^y- 262 SIMULTANEOUS QUADRATICS. 67. 4:c(x' + l)=:(a + b)(x~y)\ ^ciy'-l) = {a-b)(x-yy. 68. x'-n'^^(x' + xy + f)(x + 7j), ^^^n' = -^(x' + xy + f)(x + y). 69. X 4- xi^ i 2=<^^ y + ^ 70. = h. x + y x^ -^-1? — ^^==a, xy xy 71. ^(3/ + 2) = a, y{z-\-x) = l, z{x+y) = c, 72. {x + y){x + z) = a, {z + x){z +y)^c, 73. x{x + y + z)-=-a — yz, y{x + y-{-z) = h~zx, z{x + y + z) = c — xy. 78. Find the real roots of the system of equations, x^ -\- w^ -\- v^ =^ a^ vw -{- u{y + z) — he, w'^ + y'^ +v? — h^, wu-{- V (z + x) = ca, v^ -\-y? + z^ = a/' a/' ^^^•> ^^^ ^7 ^^^^ writers restricted to indicate the arithmetical or absolute roots, sometimes called the positive roots. Thus, V4 = 2, but 4i - ± 2. Also, v[(-2)^]-V4-=2. Sy27 = 3, but 27* - 3 or 3 ("^-i-^^^Y .•.8i = (li)-^27. -^16 - 2, but 16i = ± 2 or ± 2^. .-. 16i=:(li)-^16. 264 INDICES AND SURDS. With this restriction, the general connecting formula would be In the following exercises this restriction need not be observed. Ex. 64. 1. What is the arithmetical value of each of the following: 36*; 27*; 16^; 32*; 4*; 8*; 27*; 64*; 32*; 64*; 81*; (3|)* ; (b^f ; (l^\y ; (0.25)*; (0.027f ; 49''-^ 32«-^; 8P'^? 2. Interpret a-2 ; a^ ; cf \ (a^)''; a^"'; a \ (a~^) ; 3. What is the arithmetical value of 36"* ; 27"*; (0.16)""^' ; (0.0016P; (ip; (^\p ; (^^^)"* ; (5,^)-* ? 4. Prove {dry = (ar)"^ ; (a"^)« = (a«)"* ; a""^ ^ (a"^)"^ ; and express these theorems in words. 5. Simplify a* X a* ; c* X c?* ; m^ X m"^ ; rfi X n~"^^ ; (7i)*x(2|)*x(3i)*; a\ c\ d\ e-^'\ .* . .02^^^c6^^i . (i^-^i 6. Remove the brackets from (a«)*; (5)"*; (c^^ (a«)*; (5)-*; (.')"^ (^V*; (^"¥ ; (/"^)"^ INDICES AND SURDS. 265 7. Remove the brackets and simplify (^*-bV"*)V"¥; #-««.-»-«« #-i)^ ■ 8. Simplify -x\x~''^\-xY^f; «[(-a;)~^(-a;)-'']~*; 9. Determine tlie commensurable and the surd factors of 12*; 24*; 18"*; (-81)*; 12^; 64*; (yV)* L (6f)"'. I The surd factor must be the incommensurable root of an integer. 10. Simplify 8* + 18* -50*; 72* + (A^)^ - (^1^)"* ; [(6 + 2*) (6 - 2*)]* ; (2* + 3*)^ + (2* - 3*)^ ; (2* + 3*) (4* + 9* - 6*) ; (7* - 3*)*(7* + 3*)* ; \\{a-\-x)(x^h)-f-\{a-x){x-h)-f\'- [a^ + (a' - o^ff X [a* - (a' - a;')*]*. Express as surds : 11. a^ .^/*; 0-*; K'^K 12. ri;"+i; 3/-"+§; a'-'^ &-*^+^. a TO— 3 13. {ax-Vf\ {x' — ^x-^Xy^] {p — qxy-l. Express with indices : 14. -{/a^] ^c^; yx^] ^/y"-"; -y/iax)] ^a-\ 266 INDICES AND SURDS. 15. ^(c^ + lf)^ ^(a' + hj; [^ia' + h')J; -^[(a-h)x]; 16. {Jf; (ry*; (6~V'; (^V'; («'^)~*; Simplify tlie following, expressing the results by both notations : 17. aXoT^ ] a^X a~^ ; a^ X a'^ ; a X a~^ ; a~^ X -y/a ; ahK'^ X ah'K^d ; a*^^ X a" W*. IS ^' -^- y^' :^' i^- il?i^. c(ahy — ac hc — c^alSf 1 _^ 3 _3 3n 3» a + a '^ . a^ — a ^ . a" 2 — a2 a^ + 1 + a 19. «*_«-*' «*_«-*' «-? + «?' «+! + «-' 11 20. Divide x — y\ijxn — yn\ x^ + a^ + ^Hy / + a^ + a* ; ^ + y + ^ - 3 a; V^* ^y ^* + y* + 2^ ; 2a5 + 256? + 2m - a^ - h^ - c^ by a* + Z^* + el Ex. 66. 1 . Express the following quantities (i.) as quadratic surds ; (ii.) as cubic surds ; (iii.) as quartic surds : a; 3a; 2a'; c^x\ x"" ) ?/*; a""*; -; mx~~p\ 0.1; 0.01; \Xx\ INDICES AND SURDS. 267 2. Reduce to entire surds : x^X', «-^'a; Z>^^5^ 3-{/3; 4-^2; V^; i-\/4; 4^9; 3^i; V 7)1 3. Reduce to their simplest form : V12; V8; V^O; a/i^; 4^'0.250; vi; A^i; V^i 5^(-320); -.t/(i-^); v^^^ V(«'^0; V^'\ V(«^); V^"^'; -V^""^"; V^'"+'; -v/^'"-'; vK^ + «'); -^(a^ + 2 a*^ + aV) ; -^\{x -^ 1) (:r^ - 1)] ; ^[(a^ + 2 a^ + a:^) {cc' + rr^] ; ^[(:r^ - a^)\x - a)] ; V(4.T^-8ri;^ + 4:r); V(S^'-16^ + 8); ^[(rc^ - 2 + a;-^)(^* - 2.'r^ + 1)] ; |/2^-2+2^^\ . \ nx^-^x^-\-Zx\ . //(a^-a^)H4a^" \ V :^+2+ri;- V ' \V2 7^'+ 18:^+3; ' \V «-^ 4. Compare the following quantities by reducing them to the same surd index : 2: VB; 2:^9; V^ : a/3 ; VIO^a/^O; 2 V2 : -^22 ; <(':^(r\ ^x-.-^y; ^x'.yy- ^x^-.^x^- ^a:-^h:-^C) 268 INDICES AND SURDS. 5. Reduce to simple surds with, lowest integral surd index : V(»;-v/(a/^); a/(Vc); ^(V^'); a/(a/^') ; ^(^^"); ) ' -^(x + g) - VCar - g) - V(a; + &) + V(a^ - b) . V(a; + g) + V(«- «) + V(* + *) + V(^- ^) ' yg , ^ . | /a + a; \ _ l/ g — x \ . j g+Va; . | Va;-V.y . | a + V(g^-l) . 1 1 V^_V^ _L . _L' v^+Vf* V^ Vy V^ V^ 272 INDICES AND SURDS. 10. Find the values of the following expressions for n =: 1, 2, 3, 4, 5, respectively : 1 • (2 + V6)''+^-(2+V6) _ (2-V6)''+'-(2-V6) " 2V6L 1 + V6 1-V6 J 11. Show that 2(^^sb+v(^-i)r-'+b-v(^'-i)r-'=F2] is a square for n ^ 1, 2, or 3, respectively. 12. Extract the square roots of: ^ + y — 2-y/(a;y) ; a-}- c-]-e-{-2-y/(ac -{-ce); a + 2c + e + 2 V[(a + c){c + e)]', 2a + 2-yJ{p? - c^) ; 2[a2 + h''- V(a* + a'^' + h')] ; x-2 + x-^', V^+2 + V^"'; x + ^x^ + x^ + 2x^x + 2x'^x] x^—xy + \y''+^{^a^y~SxY + xy^)', 2x+^{Zx^+2xy-y'')\ 5-2V6; 10 + 2V21; 9 + 4V5; 12-5V6; 70 + 3V451; 4-V15; 4-V15; 7+4V3; 9 + 2V6 + 4(V + 2V3); 15.25- 5 V0.6. 13. Find the value of: ^^-— ^ — V-T, given X ^ — 7-^ — ~ and y ■= — , v • V(^' + 2/'). given X ■= ^(aV), y -= ^(aV) ; Va + ^)-V(l-"), given ^ = -2£L ; V(i+^) + V(i-^) «' + *' COMPLEX QUANTITIES. 273 14. If ^(x + a + h)+-^{x + c + d) = -y/{x + a-c) + -^(x-b + d), .\b + c = 0, 15. Simplify Kl+V5)^-2 ^ Kl-V^)^-^ . Complex Quantities. Quantities of the form a-{-b^—l, in which neither a nor Z> involves V~^' ^^^ called Oomplex Quantities. The letter i (or j) is frequently used as the symbol of the diten- sive unit V~-^' ^^ ^^^^ a-{-h^—l would be written a~j~hi. So also V~^ = i^x, ^—x X V~ y = ^^ VC^y) ~ ~ V^cV» and i^ = — i. Ex. 66. Simplify the following, writing i for -r^— 1 in any result in which the latter occurs : 1. v-4; V-S6; V-81; V-8; V-12; V-^^; ^-8; V-^xV-6; V-6XV-8; V-8XV12; V-8x-) and i(- 1 - zV3) satisfy the equation = ; that (:?; + (oy + co'^2;7 ::=:r^+y^+ 2;^+ 3 (:r+ toy) (2/+ 0)2;) (2; +a):r) and that (a;+y+2;)(a;+a)3/+w^2;)(a;+(o^3/+(o2;) =^^^+3/^+2;^— 3r?:y2;, in which w represents either of the preceding complex quantities. Hence, prove that : (i.) \_^a-h~c-\-{b~6) iV37 = \2h-G-a\{c-a) % V3]'-: [2c-a-5+ (a-^>) i^ZJ ; (ii.) 16^ + "^^ + '^^^ ~ 3 uvw -: (a' -\-h''-\-&~ Zabc) {x'' + y' + 2' - 3:2:^0), \i u^= ax -\- hy -\- cz, v = ai/ -{- bz -{- ex, lo ~az-\-hx-\- cy, or if u=^ax -\-cy -\- bz, v = cx-{-by-{- az, w = bx-\- ay + cz. 12. Prove that i [ V^ + ^ + ^V(10 " 2 V^)] satisfies the equation — ^t_ = 0. Writing o) for the preceding complex quantity, prove that (7 + 0) + o)^ + 3a>^)(7 ~io'- oy'-Sio') - 71, and (^ + y + 2;) (a; + cD^y — iiih) (x — (i)^y—ioz) (x—wz + u)*z) X (x+(ti'^y+o^'z) = x^+y^ + z^—5x^yz + 5:ryV. Prove that [Aa + (b - c) (-y/ b - 1) + (b + c)i^ (10+2^ 5) f = K«+^)[-i + ^V(V5 + 2)] + («- ^)[V5 + MV5 - 2)] X — x^ a — h " ^-x' {a-hf 2 ^ ^{a + hy + {a-hY ^ o? + ah + b' "^ {a + by + ^{a~by a'-ab + b'' m V(l + x)-n V(l - a;) = VK + ^')- (1) Square both members and reduce, .-. (m^ - n") X - 2mnV(l - ^) = 0. (2) Transfer the radical term and square both members, .-. {pv? - nyx'' - 4mW(l - x"). (3) .•.(m2 + n7:i;'=:4mV. (4) ± 2??in TTh^ + n^ (5) The above follows the usual mode of solving equations involving radicals; viz., make a radical term the right- hand member; gathering all the other terms into the left- hand member, square each member; repeat, if necessary, until all radicals are rationalized. This method is con- venient, but it does not explain the difficulty that only one of the values of x in (4) satisfies (1); viz., -f 2mn The other value, , satisfies the equation m^ + TV TYl V(l +x)^ny/0--x)= ^{m' + n'). SUED EQUATIONS. 279 The explanation is simple. Squaring both members of (1) IS really equivalent to substituting for (1) the conjoint equation, [m V(l + x) - nV(l ~x)- VK + ^')] [m^(l + x) + n^(l-x)--y/{m' + n')] = 0, (6) which reduces to (2) above. Treating (6) or (2) by transferring and squaring is equiva- lent to substituting for it the equation [m V(l + x)~ n V(l - ^) - V(^' + ^')] X [mV(l + ^) - n^(l -x) + -y/im' + n')] X [m V(l + x) + ny^(l-x)~ V(^' + n')] X [m V(l + x) + ^ V(l - ^) + V('^^' + ^')] = 0, (7) which reduces to [(m^ — 92^) :u — 2mn^(l — x"^)] [(m''-n')x + 2mnV(l - ^')] ^ 0, (8) which further reduces to (3). Thus the whole process of solving (1) is equivalent to reducing it to an equation of the type A = and then mul- tiplying the member A by rationalizing factors. Thus, instead of solving (1) we really solve (7), that is, a conjoint equation equivalent to four disjunctive equations. (See page 191, § 42.) Now the values given in (4) will satisfy (7), the positive value making the first factor vanish, the nega- tive value making the third factor vanish, while no values can be found that will make either the second or the fourth factor vanish. Hence, if one of such a set of disjunctive equations is proposed for solution, the conjoint equation must be solved; and if there be a value of x which satisfies the particular equation proposed, that value must be retained and the others rejected. 280 SURD EQUATIONS. This process is the opposite to that given in §§ 42 and 43 ; there a conjoint equation is solved by resolving it into its equivalent disjunctive equations. The two processes are related somewhat as involution and evolution are. Further, it should be noticed that just as there are four factors in (7) while there are only two values in (4), it will in general be possible to form more disjunctive equations than there are values of x that satisfy the conjoint equation, and consequently it will be possible to select disjunctive equations that are not satisfied by any value of x, or, in other words, whose solution is impossible. This will perhaps be better understood by considering the following problem : Find a number such that, if it be increased by 4 and also diminished by 4, the difference of the square roots of the results shall be 4. Keduced to an equation, this is V(^ + 4)-V(^-4)-4. (9) Kationalizing, this becomes [4-V(^ + 4) + V(^-4)] x[4-V(^ + 4)-V(^-4)] x[4 + V(a^ + 4) + VG-e-4)] x[4+V(«+4)-V(-^-4)]=0, (10) whicli reduces to [24 -^ 8 V(a^ + 4)] [24 + 8 V(^ + 4)] = ; that is, 9 — (a; + 4) — 0, or a; = 5. Now x^b satisfies (10) because it makes the factor 4-V(^ + 4)-V(a--4) vanish, and it is the only finite value of x that does satisfy (10), or, in other words, there are no values of x which will make any of the factors SURD EQUATIONS. 281 4- V0^ + 4)t- VC^-4), 4 + V(^ + 4) + V(^'-4), or 4+V(^ + 4)-V(^'-4), vanish. There is, therefore, no number that will satisfy the conditions of the problem. It will be found that as x increases -\/{x + 4) — ->/(:?; — 4j decreases ; hence, as 4 is the least value that can be given to X without involving the square root of a negative, the greatest real value of ^{x + 4) — -yjix - 4) is ^8, which is less than 4. We see by this that our method of solution fails for (9) simply because (9) is impossible. 5. ^[{a + x)(b + x)]-yj[{a-x){b-x)] - V[(^ -^)(J> + ^)] - V[(« + ^) (^ - ^)]. (1) Collecting the terms involving -y/^a + x) and -y/(a — x), respectively the equation becomes [^{a+x)-^{a-x)] [ y/{h-\~x)+^{b - x)] - 0. (2) This is satisfied if either -yjia + x)~. V(a ~ ^) = 0, (3) or V(^ + ^) + -y/ip - ^) = 0. (4) The rational form of (3) is {a + x) — (a — x) — 0, which is satisfied by a; = 0, and this also satisfies (3). The rational form of (4) is {b -\- x) — (b — x) = 0, which requires x = 0] but this does not satisfy (4). Hence, the second factor of the left-hand member of (2) can- not vanish. Therefore, the only solution of (2), and hence of (1), is x=^0, derived from (3). 282 SURD EQUATIONS. 6. ^{^a + x)+^{a-x) = ^{2a). Cube by the formula, {u + vf = u^ + v^ + '^uv{u-^ v). .-. {a + x) + (a- x) + 3 {/[2a(a' ~ x')] = 2a. .\2a(a''-x') = 0. ,\ x = it a. Both these values belong to the proposed equation. The rationalizing factors of -^(a + x) + -^(a-x)-^(2a) = are -{/(a + x) + o) ^{a - x) - co^ -^(2 a), and -Y/(a + ^) + w^-\/(<^ — ^) — «-\/(2a). See Exam. 11, page 276. The remarks on Exam. 4 will apply mutatis mutandis to equations of this type. 7 a/(^ + xy + ^{a' - x') + ^{a - xj _ ,^' • ^{a + xj - ^(a^ - x') + ^{a - xf ''' ^ ^ Assume -^{a {-x^ =u and ^(a —x) = v. j.u^-\-v^ = 2a 2ind u^ -v^ = 2x. u - V _x ' ' u^ + v^ a Also (1) becomes V? -\-uv-\- v^ u — v (2) (3) Multiply both numbers by u^ - v^ u~v • • ^= c • u^ ~\-v^ u-{-v u-{-v SURD EQUATIONS. 283 .-. by (2), f = ,?i^". -(4) a u-\-v Again, adding and subtracting denominators and numer- ators in (3), uv c—1 Adding and subtracting 2 (denominators) and numera- tors in this, V? — 2uv -\- v^ _ 3 — c ( u — v .'. substituting by (4), — = ^^ dc-1 8. [^(x + a) + ^X^-a)J[^{x+a)--^(x-a)] = 2c. (1) Assume u = -y/^x -\- a) and v — -^/(x — a) , and (1) becomes (u + vy(7i ~ v) ^ 2c, or (u + v)\u'-v'') = 2c. (2) Also ii^—v^ = 2a or (u' + v^)(u'-v') = 2a, (3) and u^ + v^ = 2x. (4) From (2) and (3), (u - v)\u' -v') = 4:a~2c. (5) 284 SURD EQUATIONS. .-. (2) X (5), (u' - vj(u' - vj or (u' - vj = ^c(2a-c), (6) Also(3y + (6), [(u' + vj + (u' - vj] {v? - vy =^^(o? + 2ac-c^) or {u' + v') {u" - vy -=-2{a' + 2ac- c"). Substituting by (4) and (6), 2x -yj{2ac - c') = 0" + 2ac - c\ 9. [^(a+x)+ ^(a-x)J[^{a+x)+-^(a-x)] ^2cx. Divide the terms of the identity -^(a + xy - -^(a - xy = 2x by the corresponding terms of the equation. • 4l/^' + ^\ __ g + 1 "\\a-x) c-i , a + x __f c+i y a — x \c—\J 10. ^{a - xf + -^[(a -3-)(b- x)-] + ^{h - xy ^-^{a' + ab + b''). Divide the terms of the identity i/{a - xf - -i/{b + xy = a-b by the corresponding terms of the equation. a — b .■.-^{a-x)-^{b-x)- ^{a^+ab + b'') SURD EQUATIONS. 285 Cube, using the form {u — vy = u^ — v^ — Suv(u — v). {a — x) — (h — x) -^^[{^OL~x)(h-x)] ""-^ a' + ab + h'' ^ o} + ah + b' ah .'. (a — x){h — x) ^{o} + ah + by a'b' (p} + ab + by a form solved in (C), page 231. 11. (g - ^ y v( a - x)-\-{x- br^{x - b ) _ ^^ ^ {a — x)^{a ~x)-\-{x- b)-y/{x — b) Write a~-b in the form {a — x) -\- {x — b), and multi- ply by the denominator of the left-hand member. .-. (a - xy-^{a -x) + ix- bf^Xx - b) = (a-xy^(a-x)+(x - by-^(x - b) + {a — x){x — b)[^{a — x) + V(^~*^)]- .-. {a - x) {x - b)[^{a - x) + V(^' - ^)] = ,'.{a — x) =^0, or X ~ b — 0, or ■y/(a-x) + ^(x ~b)^0. Xi — a, x^i — b. The equation -^/{a — x) -\- ^{x - b) = has no solu- tion, for the sum of two positive square roots cannot vanish. The solution x ^= i (a-}- h) belongs to the equation V(a - ^0 - - V(^ - ^) ^ 0. 286 SURD EQUATIONS. 12. b-\-x _ Square both members, subtract 4, and extract the square root. 3-\-x _ :±V(^^-4). a —07 =- i [c ± V(^' ~ 4)] = e, say. 'Z) + o; (^-^) . a + 5 1 + ^^ ,\x= t (a -^)+(« + 5)l^] c'-Sc-2 Or thus, cube both members. ^4"^ a — a; . (a - xY + {h + xy (a — x)(b -{- x) . V {h + x)-{a-x) "[(h + x) + {a--x) a + h 6-1^0 + 2 Prove that c^-3c + 2 l-e" _c+l \ c-2 l + e" c- l f g-2 We + 2 if 2e = c?iV(^'-4). SURD EQUATIONS. 287 Ex. 67. 1. ^(x + 4:) + ^X^-S)^7, 2. V(3^+l) + V(4^^' + 4)-l. 4. -y/^inx) — -yj(nx) — 7)i — n. 5. -yj{hx) + -^(ab + hx) = ^x. 6. V^ + V(^+3) ^ 7. ^{ax + x') = (l + x). 8. ^(17:r~26)-| \x 10. 6 + :r— VC^' + ^O^^'- 12. V(2ri;-27a)--9V^- V(2^)- 13 5V(2.2;-l) + 2V(3.T-3) __^,, * 4V(2a;-l)-2V(3^-3) '^' 14 V2a;+V(3-2r^) _3 V2-^- V(3-2a;) 2 15 2^(3a; + 3) + -^(7:. + 8) _g • 2^(3a; + 3)-^(7:i; + 8) 16. 33 [13 - 2 V(^ - S)] == 3 [13 + 2 V(^ - 5)]. 18. v(^ + g) + v^ ^ v^ + yg V(a; + c) - V^ V^ ~ V 'a 288 SURD EQUATIONS. 19 V^- + 28 _ v:r + 38 V^ + 4 ~' v^ + ^* ^2:i' + 9 -^2:^+15 21 V^W-2a _ Va; + 4a ■y/X+b ^x+3b 22 3^-1 __ 1 + V3r^ V3^ + l 2 23. V ^--V(^-^)^^. Va -f ■ V(<^ — ^0 24. V^^+ V ^-^ 25. ^><-' + 1 + V(q^'^' - 1) _ ^^ a^+ 1 — ^(aV - 1) 26. l-Vri-- + :i'')-- V^+ V^' 45. VC^'^' - 3<^' - 2x'') + V(3a' - 3^>2 - x'')'-^a + x. 46. [ V(a + a;) + V(« - ^)] [ V« + V(«' + ^')] ^ 2a;. 47. Vi^' + ^-^)+^(^'-^^^)--^jJ^' "■ >l(^)-^(-$-;-^)-v(.-»-). 49. V[(2« + ^y + b'] + V[(2a - a-y + 6''] = 2a. 50. rV(«-^)+V(:r-&)T^ _ V(« — x) ~ V(^ ~ ^) 51. 52. 53. 54. V(i+^^)-V(i-^'0 * ^(l + x'^-^a-x') b ^(l + x')+-^(l^x') _a ^{l + x')--^(l-x?) h 290 SURD EQUATIONS. ^{l + x')--^{x'-l) b V(^''^ + l)-V(^^^-l) ^ 57. V(4^ + ^ " ^^) - 2 V(^ + b~-2x) = -^h. 58. V(3^-2^ + 2ri;)-V(3«-2^-2:r)-^2va. 59. V(2« - ^ + 2a:) - V(10a - 9 ^ - 6:?;) -= 4 V(« - ^)- 60. V(3^ -^b + bx) + VG^ - a) -: 2 V(^ + a). 61. V(^^ - 4^ + bx) + V(^' - a) -= 2 V(2^ — 2b). 62. V(5^ - ^<^ + 4^) + V(^^ — 3a — 4^))-=2 V(^' + a). 63. V(2^*+H2^)+ V(10«+9'^-6^) ^ 2 V(2a+^-2a;). 64. 2 V(2a+H2:r)+ V(10a+5-6:^) = V(10^+9^-^^)- 65. V(2a-13^+14a:)+ V[3(^-2a+2:r)] =- 2V(2a-5+2:^). 66. ^[^{1 a + b + x)]- ^\a+1 b - x) ^ 2-y/{1 a + b - x). 67. v[(^^ + ^)(^ + ^)] + V[(« - ^0(^ - ^)] -= 2 V(«^). 68. V[(« + ^)(^* + ^)J - V[(^ - ^)(^ - ^)] -= 2 V(^^). 69. -^/(aa; + x^) — y/{ax — o;^) — ^{2 ax — a^). 70. ^{ax - ri;') + y («^ + ^') "= V(2 «^' + «')• ^2 x-^^{ax) a + -^J{ax) __ x — a a — ^{ax) X — -y/iax) a 73 Vr {a + :^0 (^ + ^)1 + V [(a - ^) (^ - ^ )1 .__ \a ' ^\(a + x)(x + b)]-^[{a-x){x~b)] \b 74. V[(a + ^0 {x + ^^)] - V[(« - ^) (:^ - ^')] \^ | 3a- \3a- j 3a - 2^> + 2a : _., [ Va + V(2a - 2^>)f 26-2a: 2^)-a SUED EQUATIONS. 291 75. ^(a + x) + -^(a-x) = 2-^a. 76. i/(a + xf - ^{a' - x') + ^(a - xf = -^a'. ^(1 + xy + ^(1 - X') + ^(1 - xY _ .„ 78. ^(i+xy + ^(i- xy = 2 J ^(1 - x'). 79. ^(3 + a;) + ^(3 -a;) =^6. 80. ^(1 + xy + ^(1 -xy = 5 [-{/(I + x) + ^(1 - x)]\ 81. s/(14 + a:)^ - -{/'(ISe - :^-) + a/(14 - ^y = 7. 82. [^(9 + x)+-^{9--x)]-^{8l-x') = 12. 83. [^(14+a;y-^(14-a;)^][^(14+a;)— ^(14-^a;)] = 16. 84. [^(57+x)'+^'(57-a;)^][^(57-a;)+^(57+a;)] = 100, 85. 5[) _^ ^^ V(a — :^) + V(-^ — ^) 125 V(^ - Q^) + Vfa + (i) — V(2 g) _ 4 /^ + g 126 VC^ — ^) + V ^ — ij^LH^, ■yy(x~b)+-y/c \a;-6* 127. ^(g - xy - ^[(g - :r) (a; + b)] + -{/(a; + &)^ = -^(d'-ab + b'). 294 SURD EQUATIONS. 128. \ {/(a -- xY ~ ^[(a - x) (x - ^)1 + ^(x -- bJl' 129. [i/(a-xy+^(b^xyf - (a+b)\{/(a x)+^(b+x)l 130. ^(a ~ x) + -^(x - b) - ^c. 131. ^(a + xf - ^(a - xf = ^{2 ex). 132. ^(a - ^0' + ^ -^r^. .•.n + r2-23, and ^/(nrs) = -i/. .•.r==i[23±V(23^-^W^)] = -i-(23± 1^^6303). /.x=S or - 1 + CO ^(11-1 + -jV V6303) + o,^-^(lli-^iV6303), in which 0) = 1 or -|(1±V3). 4. Find the cube roots of ~-10 + 9i^3. Assume i (y + a V^) = (- 10 + 9 i V^)* = a,^(-10 + 9V3), (1) in which and (0 = 1 or-i(l+zV3) or-|-(l-zV3), and therefore (0^ - 1 or - 1(1 - ^V3) or - 1(1 + i^S). .•.X(y_aV^)--o)^^(-10-9iV3). (2) 302 CUBIC AND QUAETIC EQUATIONS. a) + (2), y = a.^(-10+9/V3)+^(4 + 43iV5)-coV(4-432V5). (8) (Sy, az^z^ 86zV5 - 63aV2. 63 + 2; .\a^Zi=2i^5, ^ 86iV5 ^ 43^V5(5 + 4V15) "V^2 10-8V15 25-240 --^(VS + 4V3), Substituting in (1) these values of y and a-y/z, i(3/i + «V^i) = 4 + iV^, i(3/. + aV^2)-i(-4+V15-V^-4V3) i(y3 + aV^3) = -i(l-^V3)(4 + ^VS). Ex. 68. Solve tlie following equations : 1. :r^ — 3:i;' + 9:r-5--0. 7. x^ + 3x^+x+l = 0. 2. x^-3x^ + 9x-9 = 0. 8. 3a^^ + 27:r'^-9:i'+41 = 0. 3. 2r'^~6a;2+18.r+17 = 0. 9. Sx' + 27x''-9x+4:=0. 4. :i^^-3^'-15a;-13 = 0. 10. ;r^-18:^-33- 0. 5. x'~3x'-l5x~25=-0. 11. n-^-9a;-12 = 0. 6. :^' + 9:r'+9:r+15--0. 12. x^~6x'' + l0x-l=-0. 13. x' + y^ = 6, and :i;' + y' = l. 14. ^'^ + y^ = 6 rn/n, and :r^ + 3/^ = ?nn (27?i — n). 304 CUBIC AND QUARTIC EQUATIONS. 15. x'-Sx' + 9x + (k-8){l + Jc-') = 0, 16. x'-Sx'--15x + {h + 216)(l + Jc-')-200=0. 17. x' + 9x' + 9x-S(k+ 24) (1 + k-') + 16 = 0. 18. x^-2x-5 = 0. Find the cube roots of: 19. -8 + 6V3. 20. -55-126iV3. 21. 5 — 7iVS. Solve ax^ -{-Shx'^ -{- Sex -{- d=0, given : 22. hd=c\ 23. 2a'hd= a'c^ + h\ 24. a'bcd = a'c' (b' + ac) - b\ 25. JcG + H^ = ¥, h being arbitrary. 26. Show that H=0 is the condition that ax'^ + 2bx-\-c shall have a square factor ; and that A = is the condition that ax^ -\-?>bx'^ + Scx -\- d shall have a square factor. 27. Show how to solve the cubic by assuming ax^ + Sbx^ + Scx+d =-m{ax + b + t,y + (l — m){ax + b + t,)\ and determining m, ^i, and ^2- 28. li ti= (xi-\- (00:2 + oi^x^y and 4 '= (^i + w^^2 + wr^s)^, where w^ + w + 1 = 0, find z^i + 4 and z^i 2^2, and apply the result to solve the cubic. 29. If Xi, X2, and x^ be the roots of a cubic, express (xi — x.2y(x2 — x^y(x3--Xiy m terms of the coeffi- cients. 30. Prove that if all three roots of a cubic are real and unequal, A will be negative ; but, that if two of the roots are complex, A will be positive. cubic and quartic equations. 305 § 53. The Quartic. Let the general quartic equation be ax' + Ux^ + ^cx' + ^dx + e = Q. (21) Assume a{ax' + ^hx^ + Gar' + 4,dx + e) = {ax'' + 2hx + c + 2ty- {2^rx + s)\ (22) Expand and equate coefficients of like powers of x. :.r = at~(ac~¥), (23) s-yjr =--2ht-{ad- he), (24) s'' = ^f-\-4:ct-{ae- c^). (25) :.[at- {ac - If)] [4:f + 4:ct-(ae- c')] = [2bt-(ad-bc)]\ .\4:f-~(ae-Ud+3c')t + (ace + 2bed- ad' - eb' - c') = 0, (26) which may be written 4:t'-It + J=0, (27) in which I=ae-4:bd+3 e\ (28) J= aee +2bed- ad'' - eb"" - c\ (29) Selecting any one of the three values of t determined by the cubic (27), the corresponding value of r may be found by substitution for t in (23), and then that of s by substitu- tion in (24), or if r — 0, in (25) ; and the quartic in (22) may then be resolved into the quadratic factors, « ax' + 2{b- -^r)x + e + 2t-s, and ax' + 2{h + 'y/r)x + e + 2t-\-s. (30) Each of these factors equated to zero will give a pair of the roots of the quartic equation (21), which will thus be completely resolved. (31) 306 CUBIC AND QUARTIC EQUATIONS. The equation (27) is called the Reducing Cubic of the Quartic (21). Examples. 1. Solve x'-Ux' + Alx'-QQx + 27 = 0. (See Ex. 41.) Let x' - 12^;^ + ^7 x' - 66x + 27 = fx'^~6x+^ + 2 1\~ (2^rx + sf = x'- 12a;^ + /'se + ^+ 4 ^5 - iAx' - (94. + 2it + As^7^) x+^ + -^^t + 4:e - s\ Equating coefficients of like powers of x, 47 = 36 + — + 4^-4r, 66 = 94 + 24^5 +.4 5 V^, 27 = ^ + -%U + Ae-s\ OD or and .■.6r = 6t+7, s^r = -(6f + 7), 36 s' = 144 i? + 1128 i! + 1237. .-. (6^+7) (144 <^+ 1128);+ 1237) = = 216(6i!+7)l .-. 6< + 7=0, 144 («+ 1128;!+ 1237 = 216(6 i! + 7). .-.144;;'- 168^-275 = 0. .-. (12i-25)(12<+ll) = 0. . , _ 7 ,25 . . 11 = """12' r> 13 .-.r, =0, '"2 = -4' »"3" 1 = 4' si' = i^. s., =-3V13, S3-- = -3. CUBIC a:nd quaktic equations. 307 n , IIV 13 ^ 2j 4 = (x'~6x+ I2f - 13 (^ - 3)^ = (x'-6x+6y-(x- sy = 0. The last gives :?;' — 7a; + 9=-0 or :r' - 5:r + 3 = 0. .\x^-^(1 ±:-y/lS) or ^ = l(5=fc VIS)- 2. Solve9.^*-54:i;H60^'-72:z; + 16 = 0. Here a -9, b = -?^, c=10, d=~lS, e = 16. .'. r-=9t- (90 - 182i) = 2i(4:t + 41), 5 V^ - - 27 ^- (- 162 + 135) - - 27 (^^ - 1), = 4:(f + 10t- U) = 4:(t + ll)(t-l). .'.9(4:t + 41)(t + ll)(t~l)=.129(t-l). or (4^ + 41)(^+ll)-81(^-l). If ^-1, .-.r-lOli and 5 = 0. .-. (9x'' - 21 X + 12)^ - 405a;^ = 0. .-. 3a;^-(9 + 3V5)^- + 4 = 0, or ^x^ - (9 - 3 V5)^ + 4 = 0. .-. x = l[9 + 3 V5 ± V(78 + 54 V5)], or x=^i[9- 3 V5 ± VC^S - 54 V5)]. Ex. 69. Solve tlie following equations : 1. x'-Qx^~2x'' + ?>Qx-2^ = Q. 2. :^*-2a;^-25^' + 18:x; + 24 = 0. 308 CUBIC AND QUARTIC EQUATIONS. 3. 2:r*-5:r'-17:^' + 53:r-28-30. 5. :r*- 12a;- 5 = 0. 4. x'+14:x' + 4:8x + 4:9 = 0. 6. a;*-12:r-17-:0. 7. x'-8x'-12x' + S4:x-6^ = 0. 8. x' + 2x'-S7x'~S8x+l = 0. 9. 121r^* + 198a;^-100^^-36a; + 4-=0. 10. x'+y^li, x + y'' = 2i. § 54. The cubic in (27) will, in general, give three values of t. Let them be denoted by ti, t^, 4- Also let the corre- sponding values of r and of s be denoted by ri, r^, r^, and Si, 52, 53, respectively. Let Xi, x^, x^, and x^ denote the roots of the quartic. Then, by (31), ax'' + 2(b- ■^n)x + c + 2t,-s, = (32) will furnish a pair of the roots of the quartic (21), say a;i,a;2, ^^^ ax' + 2(b + ■^r,)x + c + 2t, + s, = will furnish the complementary pair, x^, x^. So also ax''+2(h — ^r,)x + e + 2^ - ^2 = (33) will furnish a pair of roots different from either of the above pairs, say Xi, x^, and ax' + 2{b+ ^r,)x+c + 2t, + s, = will furnish the complementary pair X2, x^. Finally, ax' + 2(b - ^r,)x + c+2t, -s, = will furnish a pair of roots different from any of the pre- ceding pairs. These must therefore be either Xi^ x^, or else X2, x^. Then ax'' + 2(b + ^r,)x+c + 2t, + s, = will furnish the complementary pair ; that is, either x^, x-s, or Xij x^, as the case may be. CUBIC AND QUARTIC EQUATIONS. 309 Let y^ = 1, then y may be so determined that ax' + 2{h-y^o^,)x + c + 2U-ys,^0 (34) will furnish the pair of roots Xi, x^, and then ax' + 2(b + y-yyr,)x + c + 2^ + 753 = will furnish the complementary pair X2, x^. ■Bj(32),x, + x, = ~^(b-^n). By (33), x, + x, = ~-(b-^r,), CL By (34), x, + x, = -^(b ~y Vra). (35) 45 t By (21), x^ + x^ + x^ + x^ = a .'. axi + b = v^i + V^'2 + r V^3 ; ax2 + b = Vri — Vra — y -y/r^ ; arr3 + Z> = — Vn + V^2 - r V^3 ; (36) ax^ + b = ~- Vn - ^n + r V^3. Also, from the first three equations of (35), 7 V^^i V^2 V^'3 = [^+2 a(a;i+:r,)][5+ Ja(^i+^3)][5+^ a(:?;i+:r4)] = P+iab'(Sx,+x, + x, + x,) + ia'b[(x,+x,)(x, + x,) + (^1+^3) (^i+^4) + (^i + ^4) (^1+^2)] + i a^ (xi + 3:2) (:?;i + x^) (xi + rr^) ==: F + iab\Sxi + x,^ + X3 + x^) + ia'5 [3^i' + 2^1 (^^2 + ^-3+^^4) +^2^3 + ^^2^4+^3^'4] + I a^[xi^ + x;' (x^2+^3+Xi)+Xi(x2X^+x\Xi+x^x,)+x.,x,x,] ^53+ia5^(2^,-l^) + ia^5(2.,^-^^,+^^) + ia'(- — x,'-~)==~b'+^abc-ia'd \ a a J ^ = -i(a''d-Sabc + 2b'), 310 CUBIC AND QUARTIC EQUATIONS. That is, y-yJn-yJi\^n = -iG. (37) [See (5), page 297]. Therefore, y = + 1 or - 1 (38) according as G is negative or positive. Hence, by (4), (5), (21), (23), (27), (28), (29), (36), and (38), if ax' + 4 bx' + 6 ex' + Adx + e^O, then shall Xi=-(-b + V^"i + V^^2 + r V^'3), x^=-(~b + v^'i — V^'2 — y V^s), Xs=-(-~b~ y/r, + V^2 - y^/r^), (39) x^ = -(~b - Vn — V^'2 + y V^'3), in which ri = ati — H, T^^at,-R, (40) r^ = at^ — H, ti, 4, 4 being the roots of the equation, 4:f-It+J=0, (41) JH:=^ ac - b\ I ^ae-Ud+?>c\ J = ace + 2 bed — a(^^ — ^^^ — c^, Q=a^d-Zabo-\-'lb\ and y = + 1 or — 1 according as G is negative or positive. § 55. The roots given in (39) may also be expressed in terms of any one of the three values of t, as follows : By (40), ri + r, + r3-a(?!i + «52 + 4)-3^ CUBIC AND QUARTIC EQUATIONS. 311 But (41), t^ + t, + t^ = 0. .•.n + r2 + r3--3^. Now, v^2 + r V^3 = V(^'2 + ^3 + 2y V^2 V^3) = V(- n - 3 ^ ^\ See (37). In like manner it may be sliown that Eeplacing r^ by ati — ^, see (40), and solving (41), the result becomes If ax^ + ^hx^ + ^cx" + ^dx + e = 0, then .. =_1 1 -._V(«^-^) f ^[-«^-2^+;;^J } .(42) in whicli t = - i^'d ^+ ^,^3 V(27/^- P)] SML" ' 3V3 ■i\l[^-3^v(^^'^-^>] (^^) ZTr::. ac - h\ I =^ae-Ud+Zc\ J — ace + 2 hcd — ac?^ — e^^ — c^, G=a^d~?>ahc+2h\ 312 CUBIC AND QUARTIC EQUATIONS. Ex. 70. 1 . Keduce the quartic ax^ -{- 4:bx^ -{- 6 cx"^ ~\- 4: dx -\- e = to the form y' + 6irf + 4:G7/+ a' I- ^II' = 0. 2 . Show that the two quartics x^ + 6 Hx^ =b .4 (?^ + K^ have the same reducing cubic. Solve the quartics : 3. r^^ - 24a;' ±32:r- 132-0. 4. a;^-6:^'±208a;-321-=0. 5. rr^-e:^;' =4= 16:^ — 33=^0. 6. x''-^x^±:\^x-\-m = ^. 7. :r*-6:r'±48:?;— 117==0. 8. :t^* + 6^' — 60^ + 36 =-0. Show that, if x^, x^, ^3, x^ be the roots of a quartic, 9. A.^H=--%{x^-xi)\ 10. '2.^aU^%{x,-x^\x^-x:)\ 11. oA (jT ^^^^ ztz {^Xi~\-X2 X^ ~^4)\^1\'^3 ^i ^2yV'^l~l *^4 •^2 "^S/- 12. 432 a^J^ [(xi — x^) (x^ — x^) — (x^ — x^) (x^ — ^^2)] X [(xi — Xs) (^4 — ^2) — (^1 - ^4) (^^i - ^^3)] X [(xi — x^) {x^ — Xs) — (xi — x^) {x^ — X,)]. 13. 4.ir'-a'Iir+a'J+G' = 0. 14. Prove that, if 27J'^ = /^ the quartic has a pair of equal roots. 15. Prove that, if /= J=0, the quartic has three equal roots. 16. Prove that, if a^I=12JI^, and a^J=8IP, the quartic has two distinct pairs of equal roots. CUBIC AND QUARTIC EQUATIONS. 313 Solve the quartic x' + G Ha:' + 4 Gx + ci'I- SH'^O: 17. By reducing it to the form (x' + 2^7/x + ^0 (x' - 2^yx + z,) = 0. 18. By reducing it to the form y2 (^' + yix + zf - yi {x' + y.,xy = 0. 19. By reducing it to the form {x'J^yy~{z,x-\-z,J = ^, 20. By assuming the roots to he of the form in which a^ = 1,^^ = 1, y^-=l. 21. By assuming the roots to be of the form in which z"-^ + 1 = 0, and n is integral. 22. Apply the method of Exam. 20 (Euler's Method) to solve the quartics x'- 6 Ix^ ± 8 V(^+ ^?^'+ ^'-3 Imn) :r-3(4 mn -I') = 0. 23. Reduce the quartic to the form y*+ 6 Cy^ + ^ — 0, by assuming x= ^ ''^^, and suitably determining z^ and Z2. 24. Reduce the quartic to the form y* + 4 %' + 6 (7/ + 4 % + 1 = , by assuming ^ — 2:1 + ^2^, and suitably determining Zi and Z2. 25. Make the same reduction as in the last question, by assuming x = Zi-{~ z^y'^, and suitably determining Zi and z-2. 26. Eliminate x between x'+6irx'' + AGx+a'I-Sir'' = smd x'^-{-2yx-{-z^=0, and so determine y that the resulting equation may reduce to the form z' + e>Cz'' + U==0. 314 CUBIC AND QUARTIC EQUATIONS. If Xi, X2, ^3, ^4 denote the roots of the quartic x' + 6JIx' + 4.Gx + a'I-3IP = 0, form the cubic whose roots are : iOO. XyX^ "I" *^3'^4) ^1^3 1" X^X^^ X^X^ ~\~ x^x^. ^ 00 ^1^2 ^3^4 ^1*^3 X4a;2 XiX^ ^ 2**^3 Xi ~Y' X2 X^ ■ X^ Xi ~p X^ X^ ■ X2 X\ ~Y' X^ X2 X^ 30. (X1X2 — ^3^4) (^1 + ^2 — ^3 — ^4), etc. 31. (x^-x^y^x^—x^y, (xi—x^y^x^—x^y, (xi-x^y^x^-x^y, 32. (xi — X2) (X2 — Xs) (x^i — Xi) (Xi — ^1), (^1 ^3) V^3 ^i) (.^4 "" ^2J V'^2 ^1/j (:ri — :r4) (0:4 — X2) (X2 — x^) (x^ — ^1). 33. Show how to solve the quartic, knowing the roots of any of the above cubics. 34. Reduce each of the cubics in Exams. 27 to 32 to the standard form Ay^ + Cy + B^O. Form the equation whose roots are : 35. The squares, 36. The cubes, of the roots of ax^ + 3 bx'^ + 3cx -{- d=0. Form the equation whose roots are : 37. The squares, 38. The cubes, of the roots of ax'' + Ux^ + 6cx^ + 4:dx + e = 0. 39. Form the equation whose roots are the squares of the differences of the roots of a cubic. 40. Form the equation whose roots are the squares of the differences of the roots of a quartic. CHAPTER X. Determinants. I. Definitions and Notation. § 56. The symbol ai h^ denotes the expression aiZ>2 — a^x, which is called a Determinant of the Second Order. The symbol denotes the expression a, ^2 (?2 —a-i «1 h Ci ^2 b. C2 ^3 k C:i bi Ci ^3 ^3 + «3 I ^1 Ci \ h c. which is called a Determinant of the Third Order. The symbol a. h, Cl h (22 K C2 ^3 h. Cz h denotes the expression «! ^2 C, h -^2 h ^1 h bs Cs ^3 ^3 h K Cn K hn Cn "k + ---+(-l)"^'a„ h -1 f^n- which is called a Determinant of the nth Order. 316 DETERMINANTS. 3. a~\-h a — h a — h a-{-h Examples. = {a + hy ~ {a~hy = 4:ah, \ X y 1 x^ y' 1 a;" y' T y' x^' 3/" X y U" y" + X y x^ y^ - x^y^^ — x^^y^ — xy^^ + x^^y + xy^ — x^y. 12 3 = 13 4 -2 2 3 + 3 2 3 2 3 4 4 5 4 5 3 4 8 4 5 - 1(15-16)-2(10-12) + 3(8-9) -l+4-3-=0. ^ a h c a z y b z X c y X = — a a h c + h a h c —c a h c z X z y z y y X y X z ri = + a V — abxy — cazx 4" ^y —abxy —hcyz -hcyz — cazx • gy — a V + Z^y + c^2;^ — 2 abxy — 2 ^(^y^; — 2 caz^r Ex. 71. Expand the following, i.e., write them in ordinary alge- braic notation : 1. a y 2. a b ^, y a 4. b a b X y ^ X b X y ma y mh X 6. ma my b X 7. a m.y b 7)IX 8. a y mb mx a + ' h + ' / y V X ] 10. a + I mh y + m,x ) X 11. a, - a\- -b ~b' DETERMINANTS. 31^; 12. x"^ + a^ ah ah x"^ + I 13. ab -a' -be 14. a — b, — 2a I ~2b,b~a\ 15. C(: a^ «3 b. ^. ^3 Ci Ci ^3 16. Ch h Cs a^ ^2 Ct a, b. Ci 17. X y z z X y y z X 18. 1 1 1 X y 2 a b e 19. 1 1 1 a h e \a^ b' c' 20. a X y :?; a z 3/ z a 21. a b -^c a b b c-\-a 22. 1 1 Y+x 1 1 1+2/ 23. mai m^i m(?i I a, ^2 '^•i as ^^3 ^3 24. ai + 7/?.a3 b^ + mb^ e^ + mc^ a.i h, ^2 r/3 ^3 25. «1 ^1 e^ d^ ^2 ^2 2 2 c^ d^ e-^ f^ gi\, determine: 28. A,, 6. 32. A. 29. A,, 5. 33. A,,,. 30. A, 34. A. 31. 35. 2,6- •^*^* -^3, 5- *'■*• ^^3,7- *^»^* -^5, 2- 36. Prove that a determinant will not be changed in value by any permutation of the rows and the columns which merely changes the order of the elements of either diagonal, without changing the elements themselves. § 59. Theorem III. If A denote any determinant of order n, ^^^^ the determinant formed from A hy strilcing out the ipth roiu and the qth column, and a^^ q the (p, q)th element of A, then will :(-l)^-^K,Ai,,-a2,,A,, ^ + + (-1)^ a, Cor. 1. A=:ai,^J.i,^ + a2,^^2,3+«3,^^3,3+ 'p,^^p,q\ J- - a^^ \Ap^ 1 -j- ap^ 2-^i>, 2 r (^p> 3^p i + + ««,n^ Examples. 1. Let A ~ I ai h^ c-s \ and q = S. a, h, Ci Ci a, h, 1 ^2 1>-1 C'l --^{-ly C2 a^ Z>2 a-, h c. c-i a^ b's {-~iy{' 1 ^2 h ~ -C2 cti h, a., b-s + c. ai bi ^2 ^2 = (— l)XeiAi,3 - C,A2,3 + ^3^3,3). 326 DETERMINANTS. 2. Let A = I ai Z)2 ^'3 (i^ e-^ | and ^ = 4. .-. A==(— 1)'| c?i a2 ^3 c, e-^\ "=■'■ (— 1)' (<^l I ^2 ^3 6'4 ^5 i — C?2 I «! ^3 <:?4 ^5 I + (^3 I «! Z)2 <:?4 ^5 I - ^4 I Cbl h C3 +^ / d e 9 a b c h h I m n P =(-iry a b c h h I 7)1 n p Cor. 3. If the elements on one side of the principal diag- onal of a determinant be all zero, the determinant will be equal to the product of the diagonal elements. If the elements on one side of the secondary diagonal be all zero, the determinant will be equal to the product of the secondary diagonal elements multiplied by (— l)5"("-i>, n beiyig the order of the determinant. Examples. a b c d e 0/ d e 0/ adf. 00 a = —a Q b = -ab d 00 h c d e a h Od ef g h k ah hi ■ abdg. DETERMINANTS. 327 Cor. 4. The order of a determinant may he raised with- out altering its value by prefixing a column of zeros, and superposing a roiu of elements, the first of which raust he unity, hut the others may he any finite quantities whatever. ai — X hi- EXAMPLE. 1 X ai — X ao — X ^2 — y Theorem IV. If each element of a row of a determi- nant consist of two terms, the determinant may he resolved into the sum of two determinants, the first of which is got from the original determinant hy striking out one term of each of the elements in question, and the second, hy restoring these and striking out the others. Conversely : The sum of any numher of determinants which are alike, except as regards the nith row in each, is equal to a determinant which is like the given determinants except that each element of its mth row is equal to the sum of the corresponding elements of all the given determinants. Examples. a-{-x d g = a d g + X d g h—y e h h e h — yeh c+z f k c f k z f k a h c d e f g ~\-m h k — n a b c de f g h h + a h c d e f m — n \ «i ^-2 c^d^\ + \aih,e^d^\ + \ a^ h, g^ d^ \ =r |ai ^2 (^3 + ^3+^3) d^\ 328 DETERMINANTS. Theorem V. If each elevient of any roiv of a determi- nant he multiplied {or divided) hy the same factor^ the determinant will he m^ultiplied (or divided) hy the said factor. Cor. 1. If all the elements of any row he divisihle hy a common factor, such common factor may he struch out of these elements and ivritten as a coefficient outside the hars of the resulting determinant. Cor. 2. If the sign of every element of a row he changed, the sign of the determinant will he changed. Examples. Za h c Sd e f 3g h k = S a h c de f g h k 4 8 6 9 15 20 -=2.3.5 1 2 4 12 3 13 4 30 X 1 = 30. Here the common factor 2 is struck out of the first row of elements, 3 out of the second row, and 5 out of the third row, and their product is written as coefficient of the result- ing determinant. 3. he 1 a ca 1 h ah 1 c ahc ahc a c^ ahc h h"^ ahc c c^ 1 a a^ 1 h h' a h — a c -h ~c a h a h -c = (-iy :(-iy -a -h a — c h c —a h a c h c DETERMINANTS. 329 But this is the original determinant, say A, with its columns written as rows ; ,\A = (-iyA', .-. A=:0. Theorem VI. Ani/ determinant can always he trans- jormed into a determinant of the same order in which the non-zero elements of any one row or 07ie column are all unity. Examples. «1 ^1 Ci ^2 h C2 «3 bz C2 1. Let Multiply each element of the first column by hiCi, each element of the second column by CiGi, and each element of the third column by aibi ; .\a,'b,%'A-- aibiCi h^c^ai ajJxGx b^c^ai chbiCi hc^a^ 1 a^biCi a^biCi = aibiCi Ciaibi Czaibi c^aibi 1 b.2Ciai bsCiai 1 c^aibi CsCiibi = aJ)iCiA\ say ; .-.A: aihiCi 2. Reduce 3 -5 6 4 -2 7 4 4 3 7 3 5-2 6-3 5 to an equivalent determinant having the elements of its second column all unity. The least common multiple of the elements of the second column is 12, and the quotients of 12 by these elements are — 6, 3, 4, and 2, respectively. Multiply the first row by 330 DETERMINANTS. — 6, the second by 3, the third by 4, and the fourth by 2, and divide the determinant by — 6x3x4x2:= — 144. The result is J^ I - 18 1 - 42 - 24 12 -15 1 9 21 24 1 20-8 8 1-6 10 1 -18 12 -42 - -24 144 -15 12 9 21 24 12 20 -8 8 12 -6 10 Ex. 74. Expand the following determinants : 4. a h c d e 9 h\ 1 a' 1 V" 1 c e a b c d e 9 h 5 41 3 7 3 4 5 6. ^1 yi ^1 02 ^'3 ya ^3 5 4 0-2 4 B| a, ftj as a, b^ h C2 Cj c, e, X y 1 X y X V y X a e X b Of X k c X d X g h I X 10. «1 hi Ci di Ci h^ C2 d^ 62 hO C?3 ^40 d,0 h (4 ^5 Show that : 11. 12. ai ^1 = ai 5i X ^1 yi ^2 h, a^ h 0C2 yi as h ^1 yi ^4 ^4 ^2 2/2 X f ^ ^ -\ X y^ -1 :r y' -1 ^ a: y -y X y -y rr y -y ^ DETERMINANTS. 331 Resolve the following determinants into determinants with monomial elements : 13. Cli ^1+1 Ci'l a^ h.2 + x ^2 «3 ^3 + ^ ^3 1 14. a-i hi Ci + x^ «2 ^2 ^2 + ^^ «3 ^3 C^ + X 15. cci + x + y hi Ci ih--x-\-y h^ c^ ch + x — y ^3 (?3 16. ai+x hi + y Ci a^ — X h^'\-y i Cl = ^1^1 CL'l ^2 ^2 (22 ^2 ^3 ^3 ^3 aa^sO + ai ■y ^2 I + 2 1 16 -y c (i I I ^ y Cl + C3|0 J, la, ai 5, ^1 di = (22 ^2 c, d. «3 ^3 c, d. (24 Z)^ c, d. hi Cl di + (2i ^2 4 ^4 C^4 a^ h^ (?4 +i. Cl di ^'i i bi Ci a.i Z>2 ^2 ^^^2 ^>>2 <^2 a3 ^3 c^ mb^ b^ c^ ai + mbi + ?z^, which = 1. x-[~ (ay -{- ii^z (ay oi^z x-{- lay -\- o)^z X coy X + o)y + (joi^z iii^z X / . X -\- lay -\- (a^z is a factor of A. Operate with w^ instead of co, and therefore with w in- stead of co^, and X + co^y + idz will be seen to be a factor of A. .*. ^ =^ m {x -\- y -\- z) {x -\- (oy -{- oy^z) (x + la^y + loz) in which 771 is numerical. The principal diagonal of A is x^, and the factors give 77ix^, .'. m — + 1. .-.A ^-(x + y + z)(x + o}y + lah) (x + oy'y + (oz) = x^ -{- y^ -j~ z^ — Zxyz. Ex. 75. Evaluate : 3. 5. 1 3-5-8 4 7 2-6 3 10 12 6 -9 1 13 19 2. -1-1 1 -1 1 1 - 1 1 1 1 1 1 1 1 1 -1 -5 9 5-5 9-15 19 23 5 19-10-15 - 5 - 23 15-25 4. 1 14 15 4 8 11 10 5 12 7 6 9 13 2 3 16 17 24 1 8 15 23 5 7 14 16 4 6 13 20 22 10 12 19 21 3 11 18 25 2 9 6. 1111 a h c d a' h' c' d' a' ¥ c' d' 338 DETERMINANTS. 1111 abed a' b' & d'' a' b' c' d' 1111 abed a' F c' d' a' b' e' d' 11. 1 1 1 1 a' b' c' d' a' b' (? d' a* b* c* d* (a+bf e' e' a' (b+ef a' b' b' (e+ay 10. 12. 111 1 a' b' 1 a' e' 1 b' c' by-\-ez bx ay cz-j-ax az bz ex ey ax+by 13. a b c a b c a' V c' a'b' c< 14. X a b e c X a b b e X a a b e X 15. X a b e d d X a b c e d X a b b e d X a a b e d X 16. X y y y y y X y y y y y X y y y y.y ^ y y y y y X 17. X X y X y y y X y X X X y X y y\ 18. {b+cy b-" e" {c+df a' a} {a+by 19. a -\- b -\- e -\- d a — b — e-\- d a — b -{- c — d a — h — e-\- d a -\- b -\- c -{- d a-{-b — c~ d a — b -{- c ~ d a-\-b — e~ d a •{- b -\- c -\- d 20. {b+c+dy a'' a'- {e + d^ay {d+a+by d-" d' d'' {a + b + ey DETERMINANTS. 339 § 61. Any determinant of tlie third order may readily be evaluated by the following method, called The Method of Sarins. Let the determinant be ai b, Ci a^ b, c. % b^ Cs Repeat in order the first and second rows below the determinant (or the first and second columns to the left of it) ; thus, ai bi Ci or, ai bi c^ a^ b^ \ / \,X X /, a^ 62 (?2 «2 ^2 G'l ^2 ^2 X, X /.X X ^z. ^3 ^3 Cz «3 ^3 ? i>i ^\, q \ ^p, 2 ^2, q \ ^p, 3 ^S,p \ ^tC. Before forming the product as above, Aj or A2 may either or both of them be transformed by rearranging the rows or the columns, or by changing rows into columns. The pro- duct of the same two determinants will therefore appear DETERMINANTS. 343 under different forms depending on the arrangement of its factor-determinants, but these forms will all have the same value. If one of the determinants to be multiplied together be of a lower order than the other, its order must be raised to that of the other. (Cor. 4, Theor. III.) Examples. «2 ^2 I I ^\ 3/2 tti bi X Xi X2 = (22 ^2 yi y-i «! a.2 X Xi X.2 yi y2 =^ X ^1 yi ^2 y2 ^^^ CtiXi ~y~ 0^X2^ ^2*^1 l~ ^2*^2) «i^i + ^lyi, «2^^ + ^2yi, «i^i + «2yi, ^1^1 + ^2yi, biXi -\- 1)2X2, cciVi + ^iy2 «2yi + ^2y2 «i^2 + ^iy2 «2^2 + ^2y2 a^X2 + a2y2 ^l^'2 + ^2y2 chVi + ct2y2 %i+%2 ai ^1 Ci X X^ X2 x^ ^2 ^2 3 i ^1 di a2 &2 3 >1 l+l Czdx\ I ^3 Z>2 l + l C3 C?2 I I I a3 ^4 l+l ^3 (^41 K^ll + k^C^il |a4^2| + k4<^2| l^l^sl+k^C^sl This is a skew symmetrical determinant for (I ai Z^2 1 + I ci d, I) + (I ^2 ^1 1 + I ^2 ^1 1) = 0, by Theor. II. ; and the same holds for every other pair of conjugates. § 64. If from A, a determinant of order n, there be erased m rows and m columns, the determinant formed from the remaining rows and columns taken in order, is called a Minor of A of order m — n. The minors obtained by erasing one row and one column of any determinant are called the Principal Minors of that determinant. Two minors which are so related that the rows and col- umns erased in forming one of them are exactly those not erased in forming the other, are called Complementary Minors. Thus, I ai &2 1 ^2 ^5 I I «3 C?4 I + I ^1 ^3 e^ — I ^1 ^3 ^5 I I <^2 C?4 I + I ^1 ^4 ^5 — I ^2 ^3 2 2 C2 d^ C2 d^ c/2 -hi ■Ci - di ■1 ■Ci ■di -1 «! hi Ci di ai hi Ci di cti hi Ci di Ch h c,. d. a, b. C; d. ^2 ^2 ^2 ^2 1 - 2x x' u^ v^ w^ 1 -^■I/f U V lU 1 -2z z' 1 1 1 348 DETEUMINANTS. 22. 1 —3a 3a' -a' 1 -3b Sb' -b' \ -3d 3(P -d' and deduce therefrom that 9 (a - by (a - cy (a - df (b - cj (b - a' I' c' d? a' b' m + 6? — 0. Multiply the latter equation by m, and subtract the product from the former. .-. hvi^ -\-2cm. + d=0. Combining this equation with the second of preceding, the condition required is found to be that ax"^ -\-2bx-\- c ^=0 and hx^ -\-2cx + d=0 shall have a common root, and the condition for this has been found in No. 4 above; viz., - 0. a 2h c a 2b c b 2c d h 2c d 352 DETERMINANTS. 6. To find the condition that the expression • ax^ + hy^ + cz^ + 2fyz + 2gzx + 2 hxy may be the product of two linear factors. Let the factors be a^x + piy + jiZ and a^x + ySgy + 722;. Multiply these together, and equate the coefficients of the product with those of like powers of the variables in the given expression. 2/ = Ay2 + A71 2 ^ = ai72 + a27i 2 A =- ai/52 + a2^i 1 a h g hbf 9 J ^ aia2 + a2ai aj^a + ^2^ ^i72 + ct27i P,a, + P,a, p,p, + p,P, A72 + A71 7i<*2 + 72«i 71/^2 + 72^1 7172 + 7271 71 72 I ^2 A 72 «! ^1 7i :0. Hence the required condition is that :0. a h g h b f 9 f <^ 7. If and then will y = a,X+(3,Y+y,Z (1) Z--^a,X+l3.,Y+y,Z ax'' + by' + cz' + 2fyz + 2gzx + 2 A^y - ^X^ + J^Y' + (7^2 + 2i^F^ + 2GZX+2JIXY, (2) A H G HBF O F C ai /5i 71 ^2 ^2 72 as A 73 a h g hbf 9 f ^ DETERMINANTS. 353 Substitute for x, y, and z in (2) their values in (1), and equate coefficients of like powers of X, Y, and Z. .'. A = aa.^ + ha} + C(xi + 2/a2a3 + ^ga^pii + ^ha^a^, B - a^,' + h^.} + e^i + 2/A/?3 + 2^ft^i + 2 A^A ^ (7 - ay,^ + ^y/ + ^y/ + 2/y,y3 + 2^y3yi + 2 Ay^y, F - aySiyi + Z^^.y^ + ^^^373 +7(^73 + A72) + ^ ( A71 + A73) + ^ ( A72 + A71) 6^ :=: a7iai + Z^yatta + Cygag +/(72a3 + 730^2) + 9 (ystti + 71^3) + ^ (71^ + 72^1) H^ aa^Pi + 5a2/S2 + Ca-Sz +f{a^l3, + as^a) + 9 (a,^i + aiA) + A (a A + a^ft). Now ai A 71 ' ^2 /^2 72 ^3 ft 73 i a h g hh f 9 f ^ tti 02 03 iSi ft ft a h g h h f «! ^1 71 ^2 ft 72 71 72 73 9 I ^ as ft 73 aai + Aaa + ^tts Aai + ^a2+/a3 ^ai+Zaj + Cttg «ft+Aft+^ft A/?i+^ft+/ft ^^i+/ft+cft a7i+A72 + (773 hy^ + hy^+fy-^ ^7i+/y2 + <^73 AUG H B F G F Q ch Pi 71 ^2 ^2 72 as ft 73 8. Eliminate ^, y, and 2; from the equations a;r' + %' + ^2' + 2/y2 + 2gzx + 2 hxy = 0, (1) ^1^ + ky + "^^^z = 0, (2) Jc.^x + 4y + '^2^! = 0. (3) 354 DETERMINANTS. Fii^st Method. Let X^ and Ag be homogenous linear func- tions of X, y, and z, such that Ai(2) + X2 (3) = (1). .-. ax^ + hy'^ + cz^ + 2/y2; + 2^2:0; + 2 A^y = (V + Ziy + miz) Ai + (^2^^ + 4?/ + 77122;) A2. .-. ax+hij + gz==^ Jc.X^ + ^2^2, (4) hx + by -{- fz = liXi + ^2^2, (5) ^^ + /y + ^^ ^ ^Ai + ^2^2. (6) Now eliminate x, 3/, 2;, Ai, A2 from (2), (3), (4), (5), (6) by the method exhibited in No. 3, page 350. a h g Ici IC2 = 0. h h J li 4 g f c mi m2 ^1 li TTli X'2 4 ^2 Second Method. Multiply (2) by :r, y, and 2;, succes- sively, and (3) by x and by y. .*. kix'^ + mi2;a; + 4^y = 0, 4y' +miy2; +^ia;y=-0, m^z^ + 4y2; + ^i2;:z; — 0, Jc2X^ + m22;:x; + 4^y = 0, 4y' +m2y2; +^2^y = 0. Now eliminate x'^, y^, 2;^ y2;, 2;:^, :ry from these five equa- tions and (1), by the method of No. 3, page 350. a h c 2/ 2g 2h =0. k, mi 4 4 mi ki m, 4 h ^2 7n2 I2 I2 7n2 ^2 DETERMINANTS. 355 9. To solve the simultaneous equations a^x" + 2 h^xy + hiif = m^, (1) a^x"^ + 2 h-^xy + ^23/^ = '^2- (2) Write them in the form (aiX + hi7/)x + (hiX + %)y = mi, (a2^' + ^23/)^' + (/hx + h{y)y^ m^. Let Y aix + h^y Kx + hiy a^x + h^y h^x + b^y vy = — (<^2^ + ^2y) '^^1 + {chx + Aiy) ^2. .'. (v + |^i^2|)^ + |^im2|?/ = 0, (3) — I ai ma |:?; + (V — i Ai ??i2 1) y =^ 0. (4) /. (v' — 1 Ai ma p) + I <^i ^2 1 I ^1 ma I = 0. .*. V = ± V(l ^1 ^2 r — I «i ^2 1 I ^1 ma I). Hence v i^^y be treated as known, and then by (3), y- I K "^h Substitute this value of y in (1), and there will result a pure quadratic in x, from which the value of x may be immediately obtained. 10. To solve the simultaneous equations aix"" + bi7/ + 2 hixy = mj, (1) «2^ + hy = mj. (2) This may be treated as a particular case of the preced- ing, or otherwise as follows : 356 DETERMINANTS. Let (3) Write tlie given equations in the form a^x + h^y — wi^ V ==-- {cLxX + Ky) ^2 — ( Ai'o; + h^y) a^. Vy == — «2^>^^i + (<^i^ + hiy) m2. ( V + Aima) X + ^i^22y — ^2'^^! == 0, — aimao; + ( V — K'ith) y + aami = 0, a^x + ^22/ — '^2 — 0. -0. V + Aim2 h{m2 ^2^^i «2 ^2 ^2 (22 ^2 '^2 which pure quadratic gives at once the two values of v, which may consequently be treated as known. Then from (2) and (3), a^x + ^23/ = m2, (a.1^2 — hia2)x + (hih^ — hiO^ = V ; two linear equations from which to find x and y. 11. Eliminate x from the simultaneous equations x^ —px^ + qx — r = 0, (1) y = ai + h,x + cix\ (2) DETERMINANTS. 357 Multiply (2) by x, and in the result substitute the value of x^ given by (1). .-. x7/ = c^r + (ai -- c,q) x + (b^ + c^p) x^ = ^2 + h^x + c^x"^, say. . (3) Eepeat with (3) and (1) instead of (2) and (1). .-. x'y -= c.,T + (a., — c^q) ^ + (^2 + c^p) x" fe =: ^3 + Z^g^r + c^x'', say. Eliminate x and x^ from (2), (3), and (4). -=0, ai — y h. C\ a^ h - y Ci «3 h c^ — y which, on being expanded, gives a cubic in y. 12. To find the condition that U = ax^ + hx^ + c^^ + dx^ + ^^ +/ and V=ax^ + I3x'+ yx +8 may have a common factor, and to find that factor, apply the method of elimination exhibited in Example 4, page 350. The result is : If ahcdefOO^O, OahcdefO 00 a bcdef a^ySOOOO Oa/Sy8000 OOa^ySOO OOOa^yS 0000al3yS U'and Fwill have a common factor w^hich, to a constant multiplier, will be 358 DETERMINANTS ax + b a ax + /3 a c d e f b c d e f y S /? y 8 a /? y 8 a y^ y 8 If this determinant vanish identically, i.e., if the constant multiplier be zero, ^and "Fwill have a common quadratic factor which, except as to a constant multiplier, will be ax' -\-bx-\- c d e f aa;'+;8a;+r 8 ax + li y 8 a /3 7 8 If this determinant vanish identically, ^and F will have a common cubic factor which will necessarily be "For V divided by a constant. Example. Let it be required to find the common quadratic fac- tor of 6^^ — x^--x^+ lOx^ + l^x - 40 and 2x^ + x^ — x+l0. Following the above-described method, it is found to be ^x''~x-l 2x' + x- 1 2 10 10 -1 1 14 -40 10 -1 10 :10 = 100 6x' — x+7 2a;' + a;-l 2^-f 1 - 2x'+ x-1 14 10 1 10 -1 10 15 10 DETERMINANTS. ]59 -1500 2x''~-x+2 2a;' + a;-l lb00{2x^-^x + D), Rejecting from this the constant multiplier, 1500, the common factor is 2a;^ — 3.r + 5, as may be proved either by actual division or by evaluation of the determinant for a linear factor. Ex. 78. Apply determinants to solve the following equations : 1. ?>x+1y = d>, 42; + 9?/ = ll. 2. 2x + 5y=^20, Sx-4:y=---7. 3. 3:r-5y + 42 = 5, 1x+2y~?>z=-2, 4:x + ^y— z=^7. 4. X —y+ z = 6, 7x-%y + llz=--64:, 23x-21y + 24:z = lbi. |(^ + 2y)=i(3y + 4z) = i(6z + 5a;), x + y-z =126. ■ 6. l+^u + ^z + y + ^z = 0, \ + lu + ^x + iy + ^z = 0, 24 15 2« + 3y Sx + iz 30 , 37 7. 3z + 4z 222 + Sy + Qz 8 = 2, = 3, by + 9z 2a; + 3y 8. 115(13 - x) + 719 (y - 19) - 590(37 - z) ■■ ■27, 5(13-a--)+2_37-z y-19 y-21 = 4. 360 DETEEMINANTS. 9. u-\-b — a(x-\-y), 10, (a -{-b -j- c)x = ay +bz -^c, X -j- b = {a -\- 1) (y -\- z), (a+d-\-e)y — ax -\- dz -j- e, y + b==(a + 2)(z + u), {b + d+f)z = bx + dy+f, z + 5 = (a + 3) (^^ + ^)- Generalize. 11. Given a:i = %i, y2 = «i^'i + 3/i, ^2 = %2 + %, 2/3==«2^2 + y2, ^3 = %3 + ^2, 2/4 = ^3^3 + 3/3, prove that ^4 = — ^1 -^4 -1 1 — Os -1 1 -h -1 1 — ftz -1 1 -h -1 1 — «! -1 1 -h. 12. Given x=-bi/{ai+y), y=b^/{a^, + z), z = h^/{a^+u), and u^bi^/a^] prove that x = bi a, -I -^ bz ^3 -1 b^ a' «! -1 ^2 CL2 — 1 ^3 as — 1 b^ a^ (Take for variables x, xy, xyz, xyzu, and eliminate the last three.) Solve 13. ax-\-by — cz = 2ab, 14. {c + a)x—{c—a)y~2bc, by + cz — ax=2bc, (a + b)y — (a — b)z=2ac, cz -{-ax—by=2ac, (b + c)z — (b — c)x — 2ab. DETERMINANTS. 361 15. {z-\-x)a — {z — x)b = 2yz, {x-\- y)h— (x- y)c = 2xz, {y+z)c-{y- z)a-=2xy. 16. X + ay + a^z -\- a^u + a* = 0, x + hy + hh + h^u + ^' = 0, X -{- cy + c^z -{- &u + c* =: 0, x-\-dy-\- dh+ d^u+ d'= 0. 17. x-{- ay + o?z + a^u=^ d, ^-\-^y + ^^2; + h^u — a, X + cy + c^z -{- chi — 5, x-\- dy+ dh+ d^u= c ; and if a, b, c, d, are the roots of the quartic V* —piv^ +P2V' —p;V +p^ = 0, determine x, y, z, u, in terms of pi, p2, ps, p^. 18. u-\- ^+ y+ z^=k, au-\- bx-\- cy -{- dz — I, a^u + b^x + c^y + c?^2; = m, a^t^ + ^^^ + ^V + ^^^ "^ '^• 19. Show that either of the following systems of equations can be reduced to the other : (1) Xi+ X2+ Xs = Ui, (2) 3/1 + a?/2 + a'^y, = v^, axi + bx2 + cxs = U2, 3/1 + by^ + ^'3/3 = 1^2, a^x^ + ^2:^2 + c'rr3 = ih ] yi + ^3/2 + ^^3 == -^s- Generalize. 20. There is a certain rational integral expression whose value depends on that of x, and into which x enters in no degree higher than the third. Its value is 4 when x=0, is 9 when x = l, is 20 when x—2, and is 49 when x — S. Find the expression. 362 DETERMINANTS. Solve 21. y^ a + Tc h -\- k c + k d -\- k = 1, + : y I ^ + a-\-n b -\- n c-\-n d ~j- n :1, 1, 1. 22. (g — ^>) (g — g) _ Q^ 23. 24. 25. bz + cy c^ + as; c — a c ^ (c-a)(b-c) _Q^ ay-\-bx a — b \ X X X =0. X 1 c b X c 1 a X b a 1 1 1 1 1 (a^ + bj a' b' 1 x' (x' + bj b* 1 x' a' (x' + ay -0. a—bx X X X b — € X X X X c — d X X X X d — a = 0. 26. a? b' c' (a + xf (b + xf (c+xf (2a + xy (2b + xy (2c + xf = 0. DETERMINANTS. 363 27. Determine a, 5, and c so that the two systems of equations ax + by — cz — /, a^x + ^{y + -^^z = l^, ax — by + cz = m, a^x + ySay + y^z — mj, — ax + by + cz=^n; a-^x + fty + 732: = Ui ; may be satisfied by the same values of Xj y, z. Apply to the case ax-^by — cz— 4, 2x— 7/+ 3 2; = 9, ax~by-\-cz— 8, 3:r + 2y — 2^ ==^ 1, — ao; + 5y + (72; = 16, — r?; + y + 2; =^ 4. 28. Solve 2a; + 3.y-42; _ 3rg + 4y-2^ _ 4^ + 2y-3^ :r + 5 5a; 4a:— 1 _^ x^y-z 6 29. Eliminate x, y, and 2; from {a^x + b.,y + ^>22; + c^)/u = {b^x + agy + b^z + ^2)/'^ = 1 — CiX — c^y — c^z ; ux -\- '^y -\- '^'z ^=^ 1. 30. Determine a, given ^ + y + 2; + -u; — 0, ax + by + cz-{-dw = 0, a^b^c^e ' a'^ c' 31 . Solve li (liX + m^y + Wi2;) =^ am/ + bui^, 4 (4^ + "^hy + ^22;) ^ am2^ + J7^2^ 4 (4a: + m-sy + 7232;) == ama^ + bn^^ 44 + ^zmg + 712^3 — 44 + ^3^1 + ^3^1 = 44 + ^iTTZa + 71x^2 =^ 0. 364 DETEEMINANTS. 32. If Xi, 3/1 ; X2, ^2 ; ^3 J 3/3 are the values of x and y that satisfy each possible pair of the equations aiX + biy + Ci = 0, «2^ + hy + ^2 = 0, «3^ + % + ^3 = 0, prove that I 1 ^'23/3 I = 1 <^A^3 P-^ l\ CcA I I <^253 I I CtA \]' 33. The equations x + S7/ + 5z + Su = S4:, x+27/ + 5z + 4:u = S6, x+ y + 2z+ u = lS, x + 3y + 8z + 5u = 51, have for sole solution x = l, y-—2, z = S, t^ — 4, but . on attempting to find the value of u by indetermi- nate multipliers, on adding together the equations multiplied respectively by 1, a, 13, y, and equating to zero the coefficients of x^ y, and z in the resulting equation, we obtain the incompatible equations, 1+ a+ 13+ y-O, 3+ a + 2i8 + 3y-0, 5 + 2a + 5i8 + 8y-0. Explain the paradox. 34. Eliminate x, y, and z from ax -{- by -{- cz — 1 = hiX -{• a^y — z + c = CiX — y-\-aiZ + h=-~x-{-ciy-{-biZ-\-a = 0, 35. Eliminate u, v, w, x, y, z from aiU + biV + CiW = 0, ttir?; + /3iy + yiZ = u, a^u + bia,+ Cia3, afi^+b.^.^+cSz. a.,y^+hy>,+ C-,y^ |. DETERMINANTS. 365 36. Eliminate u, v, and w from au-\~ hv -{- gw = \u^ hu + hv +fw ~ \v, gu +fv + ciu — Xw ; and w, V, w, x, y, z from the three preceding equa- tions combined with the three following : ax -{- hy -{- gz — u ~ \x, hx+by +fz =v ~Xy, 9^ +fy +cz^w — \z] and reduce the two resultants to the same /orm. 37. Eliminate first, x, y, z, second,/, g, h, from aw -\- hy — gz =0, hw -\-fz — hx = Oj cw -\- gx —fy = 0. 38. Show that <^l<^2 + ^1^2 + «2^ + V + <^2^ + ^2^' <^2% + ^2^3 + ^2^3 + ^2'^3 agai + ^gJ^ + CgCi + cZgC?!, a3a2 + &3&2 + ^3^2 + <^3^2» Generalize. agHV+CgHc^a^ 40. Let Ai = |ai^2C3l, A2=|ai^2y3|, and AjAa^ l^i^aC'sl, then will ao ^o 7o <^0 ^1 -^2 -^3 &0 -^1 -^2 -^3 (?0 v^l ^2 ^3 366 DETERMINANTS. State this tiieorein in tlie cases r. Ai = A2, ao--ao, Po=--ho, yo=Co] 2°. A, = A2, ao = ao, l3o=--bo = yo = Co=^0. 41 . Given Ui = UiX + biy + CiZ -f di = 0, U2 E= a-^x + b^y + C22; + r:?2 = 0, ^^3 = a3:^; + % + (?32; + (^3 = 0, 1^ (1) U4, = a^:?; + b^y + C42; + c?4 = 0, 2^5 = a^x + 553/ + c^z + J5 --- 0. 1°. Determine tlie value of x that will satisfy ttlt^l + a2W2 + <^3^3 + Cl4^i + <^5'^5 "^ ^) ^ CiUi + ^2^2 + ^3'2^3 + (5s+:r^) + c(5;9S + 10^r+yi5)+ jt and evaluate the resulting determinant. [Omit the first three equations, — this eliminates a, e, and d\ in the remaining three, take for variables 5 + cp, cq, and Z):?: + cy +/•] 48. If-^ +^2; + (7y + i)z^ = 0, Az-B + & + i)i; =- 0, Jw+^v + C^^--^ ==0, then will ^2 ^2 - w^ -\-2iXvw l — u^—y^—w^-^-^uyiv Q2 JJ l-~u'^ — v'^—z^-\-2uvz 1- 49. Eliminate u, v^ and w from w^i + vyi + 'w;2;i = 0, 0. -y'^ — z^ + 2 xyz a h 9 u h b f V 9 f c w u V w DETERMINANTS. 369 50. Show that the system of equations a(b — c) . b (c — a) . c(a — h) __r. a — a — p c — y a — a 13 — b y — c is satisfied by either a — b-{-l3 — a — b — c + y — l3 — c — a-\-a — y, or a^y = aby = a/Sc. Tf ^^ + cty — cz _ cx — b7/ + az __ — ax + cy + bz d' + h' ~ d' + d' ~ b'' + c^ then will X y 2, c a b b c a 0. 52. Eliminate x, y, and z from a(.^-l) + 5(y-l) + c(.-l)^0, x + y + z = l, -0. X y z c a b b c a [Reduce the determinant to the form ax -\-by + cz x + y -\- z =0.] ab -\- be -\- ca a-\-b -\- c 53. If Ix -{- TYiy -\- nz =0, ax — by — cz = u^ — ax-\-by — cz =v, — ax — by-\-cz = w, aux + bvy + cwz = 1, then will m' + n' _ n' + P I 771 n \ = 0, X y z \ P + m' (bn + cmy (cl -\- any (am + biy == x"^ -\- ?/ + 2;^ and 4 d'b'cXx'+y'+zy-(a'+b^+c'')(x'+y'+z')+l= 0. 370 DETERMINANTS. 64. Exhibit in a single equation the result of eliminating u, X, y, z from ax -\- hy -{- gz = aiii + XiX, hx + by +fz = b,u + Ajy, 9^ +fy + cz = Ciu + Aiz, ai^+ h^y-\- (?i2; — ; and V, Xi, 3/1, Zi from axi + hyi + gzi = aiV + Xj^, hxi + Z>yi + /2;i = b,v + Ajy, ^^1 + fyi + G^\ = OiV + X2Z, ctiXi + b,yi + (?i2;i -- 0. 55. If ai + biy + CiU = a2V + b2+C2Z = asX+biW + Ci—s, and wo; — vy = wz = l^ then will «! — s 5i a. ^3 /x +/i/ = w/x,, ^^ +/)". + (?v -= uv, iix-[-lw , uy-\-r}iw _^ uz + nw _ • + - ■ + ; 0. 69. (a-u)f-gh (b-u)g-fh (c-u)h~Jg Eesolve a system of three equations in three unknowns of which one equation is quadratic and two are linear. 70. Apply the method of Example 9, p. 355, to resolve x'-\-f^z'--^\ UiX + hiy + CiZ = UX, DETERMINANTS. 373 a^x + h^y + c>2Z = uy, «3^ + hy + 3y^2; + 3 c^xz!- -f 3 (^z^^^^ + 6 (frrya; ^ 0. 73. Eliminate x, y, and z from ^2 _|_ ^2 _ 2^:r?/ = 0, and assuming the resulting relation to hold among g, A, and Ic, find the H.C.F. of the functions m' + 1;' - - 'iguv -(1- -/). ?;^ + w'' - - 2 At/'ty -(1- -A'O, wHw'- - 2 fooM -(1- -^'0. 74. Eliminate x, ?/, and 2; from (/i — '^0 y^; + gxzx + /^i-ry =- 0, f^y^ + (^2 -- '^) ^^ + ^^^2^^ = 0, f^yz-\-g^zx-\-{h^-u)xy =0. 374 DETERMINANTS. 75. Eliminate x, y, and z from a,o^ + biy''+c^z^ + 2{f^-u)yz-\-2g^zx+2h^xy = 0, a^jXp- + b^y"^ + c<]2^ + 2/23/2; -}-2(g2 — i^)zx-}-2 h^xy = 0, a3x'' + b,y^+c.,z'' + 2fsyz + 2y^zx + 2(hs-u)xy = 0. 76. Expand x' ~ a' 7/ z' x' f - b' z' x^ y^ z^ — c^ -0. 77. Expand a'~f-z' y- z' x" b''-x^-z^ z' x^ y^ & — x^ — y^ = 0, and reduce the expanded equation to the form av z>y , c\^ = a^-s^ h'-s' c'-s' s'^ = x"^ + y^ -\- z\ 78. If a, (3, y be the values of u satisfying x-{- a — u y z X y -{- b — u z X y z -]- c — u and if th( d — y — z y z X b — z — x z X y c — x m will (cT-a)(cT~l3)(cT-y) = 0, in which < T = a + (3 + y-~a — b — c. = 0, 0, 79. Solve of' + 2x'y + 2a;y (y - 2) + y' - 4 = 0, x' + 2xy+2f~5y + 2=-. 0. DETERMINANTS. 375 ' 80. Solve 81. Solve x^ + Sxy^ — 6x7/ ~x-{-y^ ^-=0, 3 :?; V - 3 ^' + y' - 3 y^ - 3/ 4- 3 - 0. -6. 1)^ (x + l)(y+l) [Transform by u = x-{- 7/, v= xy, and eliminate u.'] 82. Solve ^^ + a;y + / = 17, x + xy + y^b, [Transform by u=^x-\-y -\- xy, v =^ {x-{- y) xy']. 83. Solve x' + y'' + z' = (a + u)\ (a-xy + f + z' = (/3 + u)\ (a, - xf + (h, ~ yy + z' - (y + u)\ (a, - xj + {h, - 7/)^ + (6', - zj - (8 + u)\ 84. If a = ^iX + a^/^ + <^3^j ^ = ^A + ^2)^ + ^si', ax^ + )8a;i' + y:ri + 8 = 0, a:r3^ + )Sa;3^ + ya;3 + 8--0, in which x^, x^, and ^^a are the roots of x^ -\-px^ + qx -\-r^=^0, show that 1 p g r =0, d, d. What does this equation become if o^i = ^^2 ? What does it become if x^ = x^ = x^? 1 ? 9 «! ^1 Cl a. b. C2 (h 6a Cs 85. If aiXi^ + h^i^ + <^i^i + di _ a^x^ + h^x^ + c^x^ + 6^2 aa^l^Z^ + h^'i + ^1^2 + <^2 <^2^2^ + ^2''^2'^ + 3 C^ d^. 86. If u,^+ vi' = u^ ■\-v^--=-\, then both { a' {x - u^^ + h\y — v^f ] (xu^ + yv^ — 1)' -=-\a^{x—u^''-\^'b\y-v:^''\{x%ii-^yvi-Vf and \ a^(x — Uif + h^(y — ViY \ { (xv2 — yu^'^ — {x — Wa)^ -={a\x— u^y + bXy - v^f | { (xvi — yu^f ~{x-u,y-(y — Viy\ are satisfied by X y I Ui Vi 1 U2 V2 1 = 0. 87. If rAh (h,' + P)= r, A3A1 (A^^ + P) = r.hA (V + P). then will -0. Ai A2A3 ^2^3 h. hA r-,i\ h hji2 i\r2 88. If ?' + ^' = ^ + ^^ xxi ■ y.Vi ■ g^^ a'^ "^ />^ "^28,^ ^2 I .V2 1 a^ "^ 6'^ ' 2 81^ \X2 , y,y., , ct _ 1 ^2 ^ ^2 ^28' a' DETERMINANTS. 377 show that j 1 X y _ ahaaia^ 1 -^1 Vi ~ ~2MA 89. Given that aia;' + 2 Z>ia.- + ^i--0 and a^,x' + 2h,x + c.,-=-0 have a common root, determine it. Apply to case of I 990^' -441 5; -5390 = 0. 1 825 x^ - 428 X - 4620 = 0. 90. Given that ax^ -\-?>bx'^ -\-?>cx-{- d has a square factor, find it. Apply to 2940 x^ + 812 x"" - 8385 x - 6300. 91. Determine the condition that ax^-\~?>bx'^-\-?>cx-{- d=^0 and ax^ -\-2px + y = shall have a common root, and find it. Apply to I 30^' + :^' + 35^ + 204-0, 1 110^' -23a; -357 = 0. 92. Determine the condition that ax^ + 4 hx^ -{-Qcx^ + ^dx-{- e and aa;' + 3^.r' + 3y^ + S shall have a common linear factor, and find the common factor. 93. Determine the condition that ax^ + 4:bx^ -\- ^ cx'^ + 4l dx + e and ax' + ^Px^+Q>yx'+4:^x + € may have a common quadratic factor, and find the common factor. Apply to r 60 .r* - 4ru^ + 37ri;^ - :r + 28, I ^Qx' + ?>x^ + Mx" + 22^; + 56. 378 DETERMINANTS. 94. Determine the conditions that aiX^ + 2 biX ~{- Ci = 0, a2x'^-\- 2K^x + c,i = 0, a-iX'^ + 2 b^x + c^ — 0, shall have a common root. 95. -Determine the condition that ax' + Ux' + Qcx' + 4:dx + e = shall have two equal roots. 96. Determine the conditions that ax' + Ux' + 6cx' + 4:dx + e = may have three equal roots. 97. Determine the conditions that ax^ + 5hx'+10cx' + 10dx'' + bex + g=:0 may have three equal roots. 98. Determine the conditions that ax^hy may be a com- mon factor of a^x^ + 2 h^xy + c{y'^ and a^x^ + 3 h^jX^y -f 3 cc\ __ S(be-cd) 2{ac-b') S(ad-bc) ae + 2bd-Sc' ^ 2(ce-d') ^ be — cd and prove that these equations are equivalent to 7-7-0. 112. Show that the conditions that the quartic in 111 shall have three equal roots are the same as the condi- tions that ax'' + 2bx + c=^0, bx'' +2cx + d=0, ex' + 2dx+e=0, shall have a common root, and express them as determinants. (See problems 94 and 96 above.) DETERMINANTS. 381 113. Find the relation that must exist between g and h in order that z^ - 10^ V + 12 h'z + 5^-^-0 may have a pair of equal roots. For what values of z will each of the following equa- tions have a pair of equal roots ? 114. x^ ■\-{z- \)x' -\-iz- 8)x -6(z-2) = 0. 115. x' + 2zx' + (z' -5z- 75)x - 5(z' + 5^ - 50) - 0. 116. x' + (3^ - 2)x' - (6z + 16)x- 4:bz = 0. 117. x' — 2(z + 2)x' + {8z + 3)x-6z = 0. 118. x' + (z + (j)x' + (4;^ + 11)^; + 3 (^ + 2) = 0. 119. x'-~^x + 2z = 0. 122. x'+^zx-(l-zy-4:z' = 0. 120. x'-^x' + z' =0. 123. 2a;^-3^;r + / + l = 0. 121. x''~ozx + z'-=0. 124. x'—9x' + ^x + z = 0. 125. X' + 4:X' + 4:4:X'-9QX + Z = 0. 126. \x'-(l-z)x'lb'-^'z(l-zy = 0. 127. x' + x^ + x'' + z = 0. 128. Find the relation that must hold among the coeffi- cients that x' + 6bx' + 156?:?;' + 20dx' + 15 ex' + 6fx + g' may break up into the cubic factors x^ + 3 ^1^^ -{-obiX-\-g and x^ -{-3 a^^x' + 3 h^^x + g. 129. Show that the discriminant of a {x^ + 3/^) + ^ {yx^ + v~^ y^) xy + c (v^x + v~'^ y) ^y is a rational integral function of a, h, c, and (t'^+'y"^) and of the second degree in the last of these. 382 DETERMINANTS. 130. Prove that (A^ - ah) {ax' + hf + 2fy + 2gx + 2 hxy) --aP--hg' + 2fgh can be resolved into linear factors, and find them. 131. Show that if a + h + c:==^0 and a^b + b'^c + c^a + ^mabc = 0, then will ax' -\- hy' -\- cz' -\-2 {mc -\- d)yz-\-2 {ma + h)xz + 2 (m^ + c) xy be the product of linear factors. Find them. 132. li x' + 2 hxy -\-y' — bx—*ly-\-Q has linear factors, find them. 133. For what values of A will 2{x' + by' + z' -yz-1xz + 2xy) ~X{x'' + ^y'' + 2z') be resolvable into linear factors ? Find the factors in each case. 134. For what values of A will 80:?;' + 8y' — ^z' - ?>yz + bzx + 33rry - A {x' + y' + z') be resolvable into linear factors ? Find the factors in each case. 1 35. If {l^x + m^y + niz)' + {l^x + m^y + n^zf + {kx + m.,y + n.,zf ^ :r'' + y' + z\ show that 80.r' + 8?/ - 4^2 - 32/0 + bzx + 33:ry = a {l^x + mi2/ + 7ii2;)'^ + p {l^x + w^y + n.,zy + y {kx + m^y + n.iz)\ in which a, p, and y are the values of A found in the preceding problem. DETERMINANTS. 383 136. Find the condition that {x'+yz){h — cXl + Be) + (y' + zxXc - a)(l+ Jcca) + (z^ + xyXa - ^)(1 + kab) may break up into linear factors. 137. Find the condition that + (as:?;' + h^y'' + c.,2;' + 2f^yz + 2 (/..^z + 2 h^xy) shall be resolvable into linear factors. 138. Determine k so that 4:x'' - 9/ — 2^' - ^yz + 2^:2 + ?>xy + h{x-?>y-\-z){x + y-~bz) may be resolvable into a pair of linear factors. 139. Find the condition that Ic {a^x^ + 3 hix'^ + 3 Ci^r + d^ + ^(^2^'* + 3 h,x^ + ?>c^x + d^) shall have a square factor, and this condition being fulfilled, find the square factor. 140. Given xyd' + yyb'^-zyc'^o, Ix + my + 712; = 0, find the condition that the ratios x:y\z shall be each single-valued. 141. Given fV + gY + hh'' - 2ghyz - 2fhxz - 2fgxy = and Ix + "iny + nz = 0, find the condition that the ratios x\y\z shall be each single-valued. 384 DETERMINANTS. 142. If ax-\-l^l/ + yz = 0, and f/x + g/y + h/z = 0, and if the ratios x \y \ z are each single-valued, then will 143. Find the condition that if ax" + hy'' + cz" + 2fyz + 2gxz + 2hxy-= 0, nd a h g Q ' z ~y h h f -z X g f c y —X ^ -2 y 1 I z -X I m -y X Q I n I m n P + m' + ri' le ratios x \y \z shall be each single-valued. 144. Given ax^ + by^ + Ax'-^Bx+ C=0 and 2y' + 2xy'' + x''y + ?>Ay + B = 0, and show how to apply the resultant to obtain a solution of the quartic in x. 164. Given p=h ~a, f3i=-- c —h, /3., = d—c, fis^e—d, y=l3,~l3, yi-A-A, y.-P.-P.. S ==yi— y, 81-= y2— yi, 7/ == 81—8, 161. 162. 163. show that a ^ y P y 8 y 8 7; 165. Show that if a o b c c d I h c c d d e 0, then w411 {ac — 1)^) j ax"^ + 2hxy -f cy"^ bx^ + 2cxy + dy"^ \ba?-{-2 cxy -j- dy'^ cx^ -j- 2 dxy~\- e^f ax + Z>?/ bx + ^V ^o; + (?y ex -\- dy DETERMINANTS. 389 . Given c XqX^ + 4 aiX^ + 6 a^x"^ + 4 ag^r + a^ = YoU + 7i^ + y^w = 0, -0, show that ao ai a2 ^0 /?! A 7o yi 7-2 -27 ao «! as «! ^2 as a2 as a^ 1 67. If Ai = a'" + yg"^ + y"^ + 3"^ + etc., in which a, /3, y, 8, etc. are the roots of aox"" + aio;""^ + agO;''"^ + + a,^ = 0, show that /^-(-ao-'X ai ao 2 a2 «! ao 3 as a2 ai «0 ma^ a^_i a^ and that - a - ^~ ^ ^^"^ Ix2x3....m A^i 1 aS, /Si 2 '^'s ^2 -^1 3 ^m '^TO-1 ^m-2 ^/n-3 -^-^1 168. If Sn = a'' + ^'^+c^* + etc., then will -^(a-bf, = :^(a-by(a~cy(b~c)\ s„ >s\ =: ] Si S-: So Si S-i Si s, s. S, 8, s,\ jSq Si S2 S-s S, jS, s, s\ S-i Ss /S4 s^ S-i Si S^ Sg Generalize. = :^(a-by(a~-cy(a~dy (b-cy{b-dy{c-dy. 390 DETERMINANTS. 169. If ai, ^2, <^3> <^n ^^Q the roots of f(xY=^ 0, show that K^r- X «! ai ai ai X a^ a^ a.^ a^ X a-s as as as ^ an a^ an a^ ai ai ^2 a., ^3 as a^ a, X an c c 170. If B^ ' + ar + a-r + + ar, show that 1 X x"^ Oq ^1 ^n 0„_i bn btn-\ X X ai a, ai X a^ a, a, X 171. Resolve into factors, -^0 Ol bn~\ Si S2 Sn S\ Ss cy o ai «i a^ ^2 as as X «n 1 1 1 1 8o y 8, t 8. f 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8\ 8, 8, in which 8,^ i = a''+b" + c" . Show that 8, 1 8, 8, 8, 8., 2 8i 3 ... ... ... 8,. 8„. 1 &L-, isL. ... m— 1 ... 8, 8„. b'ln DETERMINANTS. 391 -(-i)nx2x3 but = 0, if m > ??, in which Sr = a/ + az"" + a/ + r being any positive integer. (fi — l)naia2 a„, ii 771 = 71, 173 . If u^ ■= aitti"* + ^202"" + asttg"* + a^a^"" + c(5a-J^, prove that = 0. Uo Ui u. Us Ui U2 Us u. u. U, u^ u, 21, u. ?/5 u^ U, U-o Ue u, W'5 Wfi u. u^ u. u. u. Ue Ue U^ ^7 u. Us u. U9 Uio 174. Obtain G in determinant form by eliminating x between and ax -{- b — 1/ — 0. (See § 62, p. 297.) 175. Obtain I in determinant form by eliminating x between ax' + Abx' + 6cx^ + 4:dx + e=^0 and ax -\~b -~ 'i/ = 0. (See Ex. 70, prob. 1, p. 312.) 176. Express IT, A, /, J, and P — 27 J^Mn determinant form, given as data the propositions stated in prob. 26, Ex. 68, and probs. 14 and 15, Ex. 70. 177. Express Us, Is, Is, G^s of the equation Sox'+4:S,x'+6S,x'+AS.sX+jS,=:^,Xx+ny^O in terms of H, /, J", G of the equation aox' + 4 ttiX^ + 6 (22^^ + 4 a^x + a^ = 0, of which Ti, r2, r^, r^ are the roots. 392 DETERMINANTS. 178. Express Hs, Zs, Js, (^s, and // — 277"/, as functions of tlie differences of the roots of the quartic. 179. Express A^ as a function of the differences of the roots of the cubic. 180. If x = XiU + /xi'y, y = X^u + iL^v, transforms ax^ + 2 hxy + hy'^ mto Au' + 2Suv + Bv\ find the value of J, H H, B 181. If y ^\u-\- ii.p + v^w, transforms ax" + by'' + cz^ + 2fyz + 2gxz + 2hxy into Av? + Bv" + Cw' + 2 i^i;i^ + 2 ff i^z^; + 2 Hm, find the value of A H -^ a h g H B F h b f G F C 9 f c 182. If ^ = {Ky + i^iz)/{X,y + ii^,z), ayX^ + 2hiX -\- Ci = (Ay + 2 B,7jz + C,z^)/(Ky + /.,.)^ and a2^^ + 2 ^2^4-^2 = (A,f + 2 ^23/^ + C,z')l{\,y + ^2^)^ then will ^1 2A a ^1 2^1 a ^2 2^2 a ^2 2^2 a «! 25i 2 ^2 DETERMINANTS. 393 183. If the quartic (a, h, c, d, e){x, 1)* = be transformed by the homographic transformation ^ = (Ky + /^i)/(\y + /^2), then will in which J/= \ /xi [Jf is called the modulus of the transformation.] 184. Find Ay/A^. for the homographic transformation of the cubic {a,b,c,d) {x, Vf — 0. 185. If K-{p'-q')/{p'+q% f^i = 2pq/(f+q'l A, — 2pq/{f + q'), /., = (/- qy(f+ q'\ X ~ XiU + /xif , y = X^^ + /^2'i^, then will x'^ + y^ = u^ + v^, [A transformation that changes x^-\- a;./+ ^^3^+ + x,^^ into u^ + u,^ + u^ + + u^ is termed an orthog- onal transformation of the nth order.] 186. Form an orthogonal transformation of the third order and determine the value of its modulus. (See prob. 181 above.) 187. Form an orthogonal tranformation of the fourth order. 188. Show that H, I, J", G, P-21J^ ^yq the same for both the quartics a^^ + 2 h(ficy + c^y"^ aiX^ + 2 h^xy + c^y^ = 0, aiX^ + 2 bixy + c^y"^ a^x'^ + 2 h^xy -j- c^iy"^ Gqx'^ + 2ain;3/ + ^2^^ ^0^^ + 2 ^i.t?/ + h.^y"^ I — 0. box'^ + 2 Z^i^ry + b^y"^ CqX^ + 2 Cjo:?/ + cx + d)\ If a, p, and ttj, ySi are the roots of o,x'^ -\- 2bx -{- c '-= and a^x^ + 2 ^i.r + cx + d=0 and a^x''+2b^x + c^-=0 into a product of the differences of the roots of the two equations. By eliminating g, A, Jc, and / from a = {ga + h){]ca-Vl), ^ =^ (gb + h)(B + 1), y^(gc+h){kc + l), S^(gd+h)(kd+l), DETERMINANTS. 399 = 1 ah+cd a/3 + y8 1 be -{- da Py + 8a 1 ac-\-bd ay + pS prove that 1 a a aa I h P bp \ C y cy 1 d S d8 ^(a-/3Xb-cXy-S)(d-a) ~{a~b)(P-y){c~d){8~a) = (a-y)(c-d)(S~P){b~a) -(a-c)(y-8)(d-b)(l3~a) ^(a~-8)(d-b)(/3-y)(c-a) -(a-~d)(8-l3)(b--c)(y-a). 216. Similarly, prove that 1 a aa 1 b bp 1 c cy and that =(^a-b){^-y)c-{a-^){b-c)a, 1 a-\- a aa 1 b+p b/3 1 c -{- y cy ^(a-b){l3-y)(c-a) + io^-(3)(b-c)(y-a). 217. a, (3, y, 8 and ai, /3i, yi, 8, being the roots of two quartics, prove that if T^Ji^ = Ii^J^\ then also will :0. 1 a cti tttt] 1 P /8. /8^. i y yi rri 1 8 88, 218. If u, V, w denote the roots of aV — aIx-{-2J~0, and Ui, Vi, lUi the roots of ajV— aiIiX~{- 2 Ji — 0, a, p, y, 8 the roots, and / and JT the invariants, of (a, ^, c, d, e) (rr, 1)*==0, and ai, ^i, yi, 8i the roots, and Ii and f/i the invariants, of (ai, &i, Ci, c?i, ^i) (a;, 1)* — 0, then will 400 DETERMINANTS. 1 a aj aai 1 /5 A PP. 1 y yi yyi 1 8 8i 8Si -4jl 1 1 u Ui V Vi 10 IV, a /8i tttti I I 8S, 219. If a, y8, y, S be the roots of (a, 5, e, c?, e) {x, ly = 0, and ttj, /?i, yi, 8i those of (ai, ^i, Cj, Jj, Ci) (:r, 1)* = 0, form the equation whose roots are the twelve dif- ferent values of 1 1 1 1 220. If (a - aO (/? - ^0 + (a^ - ^) (^ - a) - 0, in which a and p are the roots of ax^ + 2 Z>a; + c= 0, . and ttj and Pi are the roots of aiX^ + 2h^x-\-Ci — 0, show that aci — 2hh,-\- cai = 0, and that aci — 2hhi-\- cai is a factor of the invari- ant J of the quartic (ax' + 2bx+c) (a.x' + 2 b,x + c,) = 0. 221. Reduce \x^ + y'-(l-Jc)gx + 7n''\' = il + Jcy\/i^(x' + f)-gyi to the form \x'' + f + Ax + B\' = ax'' + hx + c, and show that h'' — ^ABh + ^A'c = 0. 222. So determine h and I in terms of a, h, and o, that (x''+ 7/'+ zy+ 2 ax'+ 2 h7/'+ 2 cz'+ 2 Jcx+ 1 = may for y — assume the form (x' + z' + 7n){x' + z' + n)--=0; and for z — 0, the form /? DETERMINANTS. 401 ■ 223. If a, p, y, 8, four of the roots of the quintic {a,h,c,d,ej){x, 1)' -- 0, be connected by the relation a + yS == y + 8, show- that €, the fifth root, will be given by the equations 224. If a, /?, y, 8, four of the roots of the quintic (a, h, c, d, ej) (x, If = 0, be connected by the relation S(2a-^-y)(2^-y-a)(2y-a-/3) = 0, then will €, the fifth root, be determined by z=^ae + b, z' -SSz+lQG^O. 225. If (a-^)(y-8) + (^~y)(8-a)-0, in which a, ^, y, 8 are four of the roots of the quintic asf + b = c{x+Vf, then will j^^ - 2' 3 VJT + 2^^ 3'iy = 0, in which J = b'c' + cV + a'b' -2abc(a + b + c), K= a^b'^c' (be + ca + ab), L = a'b'c\ 226. Reduce (a, b, c,d,e,f) (x, yf to the form K (^ + Kyf + ^2 {pc + k{yf + ^3 (^ + %)^ and hence prove that ( ) (a;, 1)^ = can be re- duced to the form l{x^ Xf — mx^ — n = 0, into linear factors. . Kesolve 1 1 1 1 c^ b' 1 c' a' 1 b' a' 402 DETERMINANTS. 228. , P, y being express the roots of - 2)X^ + qx — r = 0, a /? y -4- 1 1 1 y y y^ 1 y /^ fi y u a 1 y a a /^ a i (^ a in terms of ^9, 5', and 229. Find the value of 230. a b a c b c c b a Show that a^ h a' Y b' / & ^' a' 1 1 1 1 a ^ 1 a Z> 1 b ^ 1 c c a 231. If = 16^^ (;?^ - aa) {f - bp) (p' - cy) 2^' = aa + b/3 + cy. ^0, 1 1 1 1 1 a' /3-^ t 1 a^ c' 1 fi' c' 0' a' 1 / V a? then will (a^ + ¥ + c^ + a^ + p'' + y'^)(aV + b^^P" + c^^) = 2 a^a\a'+ a") + 2 ^^^9X6*^+ /?^)+2 c V(c^+ y') + a^)8V + bya^ + c'^a^yS*^ + a'b''c\ 232. If a'+/?' + a^ = C^ then will DETERMINANTS. 4(J5 1 1 1 1 -0. 1 a' P' y'^ 1 a' c' ^^ 1 1^' c' a' 1 / h' a' 233. If x^ + if -2 i.xy = -^^ and 1 r li = 0, f + z^- ~2pyz = = «^ y 1 a 1 Z^ +X'' - 2yzx - = ^', P a then will 1 1 1 1 = 0. 1 x'' y' ^2 1 x' c*^ ^'^ 1 f c' a^ 1 z^ h' OJ" 234. Prove that X y z ; = (a:r - -by + cz)\ — X c b ~y - — c a ■— z - -h — a and generalize the theorem. 235. Evaluate X y z y X y z Z X z and use the result to prove that u^ -\~ v^ -\- w^ — ?> uvw = {a^ + h^ + c^-?> ahc) {x^ + f + z^- Zxyz), wherein u = ax-\-by-^cz, v^=cx-\-ai/-\-hz, w = bx-{-cy-\-az. 236. Evaluate and use the result to prove that 1 - (2x' - ly ~ (2y'-iy - {2z~Vf + 2{2x' ~l){2f -l){2z' -V) = -4c(l-x^-y''-z'+2xyz){l~x''-y''-z'-2xyz). 1 X y X 1 z y z 1 404 DETERMINANTS. 237. - c -d h a d a b d c -b a a + ib — c-\-id ■ id id = (a' + b' + c' + dy, i' + l = 0. Apply this identity to prove tliat the product of the sum of four squares by the sum of four squares can be reduced to the sum of four squares. 238. Prove that a pb qc pqd — b a —qdj qc — c pd a ~pb — d —c b a = (a^ +^&^ + qc^ -\-pqdy. Hence prove that (a^^pb''-\-qc^-^pqd''){A^^pB^\qC''-^pqB') = (a A +pbB + qcC+pqdny +p (- aB + bA- qcB + qdC)' + q{- aC+pbD + cA~ pdBy +pq(- aD - bC+ cB + dA)\ 239. Show that a^ "lab b' aa ap + ab bp a' 2 a/? p-" = {ap + ab)\ and generalize. If u = {x-- ai) {x — a.2) (x — aa) (x ~ a„) = x"" -^1^;"-^ +i?2^'"~' i~TPn, show that 240. 1111 1 X a^ \ ai X 1 «! «2 1 «! a.j_ Cla a^ «3 Cii X a^ ^3 X ---nx''-^ + {n—l)p^x''-''-{n-2)p.,x''-^+ DETERMINANTS. 405 241. 242. 111 1 ai X X 1 X a^ X 1. X X ct^ =-( — iyinx''-'^-(n~l)p,x''-''+ \. X a^ a^ Oi X as (X\ C(/2 X = w + S X — Gr 243. Show that a-}- X X X X P -{- X X X X y-{~ X X X X X X X h-\- X = apyZx{/a +/^ +/y +/8 +/x). "ove that 1 1 c 1 a 1 a 1 a -(o- a — b -by 1 1 1 b b b c b b a c b a a c and gener alize. 245. Writing /(:r) for (ci~x)(a2 — x) (Cn — x), show that Ci a a a _af(b) — bf{a) b c^ a a ~ a — b b b Cj, a ... b b b Ca ... 246. Show that a^ a^ a^ % ^1 a^ a^ a^ bi b^, a^ ^5 ^1 ^2 ^3 % bi b.2 Z>3 b^ = bia^a^a^a-^ + b^h^a^a^a^ + bf^^Jj^a^ar, + b^^Jj^a^. 406 DETERMINANTS. 247. 248. 249. 250. a b c h a a h h c c h h h h c c c c a a a a a b h b c b c c a a a a a c a b b b b b b a \(ia-b){b~c){c~a)\'' X 1 - + - c — a a~b c b^c 1 1 a b c b a a b b c c c a a a a a a a b c b = n\{a-b){b~c){c-a)\''-'' X {a' +b''+c'~bc ~ ca - ab). Prove the two following identities, and generalize them : a — X b c b c ~~ X a c a b--x ■-.{x~B,){x'-SA\ /S'„ = a + a>"Z> + a>'"c, 0)2 + 0) + 1 ^ 0. a — X b c d b c — X d a c d a—x b dab c — X = {x-8,){x-8,){x'-8A), 8n = a + i^'b + i'^'c + i'% i' + 1 = 0. DETERMINANTS. 407 251. (S-uy x' f z" u^ {8-xy y" z^ u^ x^ iS-yf z^ u" x" y" (S-zf = 2 S'uxyz (/u +/x +/y +/z - 4:/S) wherein S= u -{- X -{- y -\~ z. 252. 1 1 1 1 Q/+zy f 1 x^ (x + zj z" 1 x" y' {x+yy ~{x + y + zyiix''+y'' + z^-2yz-2xz~2xy). 253. 1111 1 {8-uy x' f z" 1 1^ " " ' " 1 y? 1 u' (s-xy r x' (S-7jy z' x' f (s~zy ^S'\uXS~2u)+x\S-2x) + 2/(S-2y) + zXS-2z)+2ux7/z(/u+/x+/y+/z)l S=u + x + y-{-z. Prove the four identities next following, in which U= (x - 2ai) (x ~ 2a,) (x - 2a„). 254. (x ~- a J a.' as a^' (x — tta)^ ai^ ai a^ {x — a^y 255. I x — 2a^) 1 1 1 1 {x — a^'^ a^i ai 1 a^ {x — a^^ a^ 1 a^ a^' {x~a-^^ = — nUx'"- ^ 5 \ ^ X~2ara 408 DETEEMTNANTS. 256. 257. I x~2a^) «! a^ as ai {x—a^y a^a^ o^^a^ a^ a^a^ {x — a^ a^a^ = - ^ri;*^-^ S a; — 2a^ 258. X — Ai a2 as a\ X -^2 ^3 (Xj (X2 ^ -^3 =a;(^-/^)--\ 259. If /(a, ^>) = «! (22 <^3 o: — ^ y(S'=ai + a2 + + ««, -4m = >S'— a, I , a — "TYi , a~n b — k b ~ I b ~ 7)1 show that /(a, a) /(5,a) /(., a) f{d^ a) ^ 0. /(a, 5) /(5,5) /(.,^) f{d,b) f(a, c) f(b, c) f(c, c) f(d, c) f(a,d) f(b,d) f(c,d) f(d,d) 260. Expand 261. Expand A — c 2a Sa 2a 2b X — c 2a Sa Sb 2b X-~c 2a 2b Zb 2b \-c a b c d+X b c d — X c d+^X e e f f 9 d~\X DETEEMINANTS. 409 262. Show that if (x — a){x — b) (x — c) ^x^- a + b — c 4a 6a •px 6b 45 a + b- 4:b 6b c 4a a + b- 4:b qx- 4:a 6a c 4:a a-{-b - ■r, p q r V — 4: p q r —16 p q -4: p 263. If a-\-b-\-c-—0, ab-\-bc-\-ca— q, abc~r, then will X-h'c' ab(a' + c') ac(a' + b') ab (b' + c') X - aV be (a' + b') aclb' + c') bcia' + c') k-a'b' = \'-Xq (q' - 2r') + q\ 264. Show that X-~2bp — 2cy aP -}- ab ay-\-ac I a^ -{- ab \ — 2aa — 2cy by-{- jSc ay+ac by + ^c \—2aa-2bp\ = [(X + aa+bp + cyy -{a'+b''+c'){a^+p''+y'')\(X-2aa-2bl^-2cy). 265. Show that X + aa + a'a' ap + a'yS' ay + a'y' 5a + Z>V m + cV X + b/3 + b'p' cf3 + c'l3' by+b'y' X+cy + c'y' = X--'{X'' + X(A + ^) + AB-CDl ; wherein n is the order of the determinant, and A = aa' + bb' + cc' + ■B = aa'+PI3'+yy+ C = aa^ + bf3'+cy'+ I)=aa' + pb'+yc'+ 410 DETERMINANTS. 266. 267. 268. 269. 270. Show that 1 a a a^ I IS p p^ 1 y y' yy' 1 8 8' 8S' 1 a I 1 /^l 1 1 1 a + yS y+8' S + y' ap yS' By' and generalize the proposition. Show that a, h, Cy d, a. h C2 d. a^ h a^ K ^ i a., hAx\ a-\-b q c -\- d x + y a + b q c -{- d c -\-d x-\-y xi x{yx y{ yiZy^ 2;/ 2:1:^1 X2 x%y% y-i y^iZi z-i z-ix^ ^i ^33/3 2/3' 2/32^3 2:3' 2:3.^3 x^ X{y^ 3/4 3/42^4 2^4 2^4:^4 ^5 ^52/5 2/5 y^^h 2^5 Z^X^ ^<' ^62/6 yi 2/62^6 '^i ^&x& -^23 -^56 ~ ^23 -^56 ^23 ^56 ^^ -^23 ^56 ^23 -^56 -^23 ^56 ■^'i^^&l^^U -^61 ^U^&\ •^34'^61 ^34-^61 -^^^34^61 412 DETERMINANTS. •^ m, n ^^^ ^m'^n '^iv^mi ^m, n = ^myn -^nym- 274. If a, 5, c, d, e,f denote the six determinants that can be formed from the array I a, 13, y, 8, then will ad -}- he -}- cf = 0, 275. Prove that I ^1^2^/4 t^ I I ^2^3/4 4 1 — I aid.,e.J^ t^ I I b^c^ej^ t^ | = 1 aAesf^ t^ I I Cid^eJ^. in |. 276. If An, Bn, Cn are the inverse elements of | a-J)^c^ \ with respect to a„, 6„, c„, show that 1 1 Ai A2 as A, + 1 1 1 + bi 62 63 £, -B2 ^3 111 Ci C2 ^ 9 f c I n rti I d 287. If I ai K, Cs I = 0, then will 288. If X ai hi X Ci Gi — X bi Ci y a., \ y C% CL^ y ^2 ci z az h z Ci a-i z h., Cs he a square. a b c d cti bi Ci di a^ Z^2 Ci d^ Q a b Q c d ai bi Ci di a^ bi C2 di = 0, then will a c d — a b d b c d ai Ci di ai bi di bi Ci c/i a^ Ci di cii bi di bi Ci di = 0. 289. Show that b c d + a c d e 2b d e f Sc e f ff + a b c b c d d c d e 2e d ef Zf a c d b b d e c 2c e f d Sd f y = 0. + a b d b c c e c d 2df d e Se y 416 DETERMINANTS. 290. Given A^ = \ knhp^Y "^ W ^2^3! 1 4^*1 | k'^n^i 292. A^, = \km^p^\ A, = I lim2ps Wlim^l 14^4 1 \hnii I As = I hn^p^'' -^ \\ km., I 1 4^4 1 I l,vi, 1 1 , WliTu^l |4^3| 14^1 II,. find the value of AJi + A^k + A^l^ + ^44. 291. Given Un =aiX + hiy + Ci2;, 1^22 = h^ + <^2y + t/i^^, Un =biX + ^22/ + e^z, 1^23 = ^2^ + c?iy + d^z, Wi3 = Ci.r + e^y + C22;, ^^33 = ^2^ + <^23/ + <^32:, U21^^ W12, 1^31 = Wi3, I632 = U23f Ui = XUn + 3/W12 + 2:2^13, U2 = XU21 + 3/W22 + 2:2^23, Us = a;w3i + 2/W32 + 2:^33, C/'= xui + ywa + 2:^3, then will I Un, ^22, '^^33 1 2;^ = 4 t^n 2/12 t^i +QU\ Un U22 \ . U21 U22 U2 Generalize. Un Uu U, U21 U22 U2 Ui U2 «! a2 + Clo <^3 <^4 <^5 <^6 <^7 «2 «3 + <^1 <^4 + <^0 CLb «6 <3',7 «3 «4 + «2 «5 + <^1 t)^6 + <^0 <^7 <^4 «5 + «3 «6 + <^2 «7 + <3^i «0 as ag + <^4 <^7 + «3 <^2 <^1 «0 ag «7 + <^5 «4 c?3 a2 ai ^o ai ae as is divisible by <^2 — CCo ^3 Cti ag — aj a4 — ao as a^ — a2 as ~ ai a^ — ao as — ag ag — az ay — ai ag — a4 ai — a^ — at <^7 ~ a^ — a4 — as as a2 ai a4 aa a2 ai c^o ^5 ag a, ■ ai a2 «6 — «7 a; ao «i -ao Generalize the result obtained. DETERMINANTS. 417 293. Simplify CtiiQj22 ^12 ^^11^32 ^13^12 ^Il<^n2 <^ln<^12 in which Op^ = a^^. 294. If v(a, 6, (?)2„+i = a ^ c a h c a h c a cab Oca then will V (a, ^, (?)2n+i = «V («^ — 2 be, ^^ c^)„. 295. Also v(^,l,l)2n = vG^^-2,l,l)n+v(^'-2,l,l)„_i. 296. lff{vi,n,p) prove that 1111 ^m pm ^m, gm a" ^ y'* 8^ a^ ^V yP ^P (a+^+y+S)/(^/^ n,^) - a^yS/(m-l, n-1,^-1) =f{m+l, n,p)+f(m, n+l,p)+f(m, 7i,p+l). 297. If A = i a„ b,, c„ K\ and A ('^' ^' denote the I, 7n, ) result of replacing the Ith, mth, columns of A by a^, fii, , a^, (3m, , respectively, show that Ly 771) cy 7vy X A (^. /^. I, m, )' the determinant on the left of " = " being of the order r. 418 298. DETERMINANTS. 1 1 1 1 -_ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Show that the value of a determinant formed like the above, but with m units and n zeros in each line, is 771 if m. be prime to n, but is zero if m be not prime to n. 299. If (a, h, cj, g, h)(Ocl>, O + cfy, If = 0, and (a, b, cJ, ^, A)(c^x. <^+X» 1)' = 0, and show that if a = a, /3=b, y=:c, K=f, \=g, ii=h, then will ac + h' + 2hg~^fh=:0. 300. The minors of order 2n—l of a skew symmetric determinant of order 2n are divisible by the square root of the determinant. V