UC-NRLF B 4 E5D M37 SSSBESwiviin.. I ^gwjgsgaHKOT'^ , 4 ELEMENTARY THEORY OF EQUATIONS / BY LEONARD EUGENE DICKSON, Ph.D. CORRESPONDANT DE l'iNSTITUT DE FRANCE PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF CHICAGO NEW YORK JOHN WILEY & SONS, Inc. London: CHAPMAN & HALL, Limited Copyright, 1914, BY LEONARD EUGENE DICKSON Printed in U. S. A. Stanbope ipress P. H.CILSON COMPANY BOSTON, USA. ^_3j^ PREFACE The longer an engineer has Ijecn separated from his alma mater, the fewer mathematical formulas he uses and the more he relies upon tables and, when the latter fail, upon graphical methods. Although graphical methods have the advantage of being ocular, they frequently suffer from the fact that only what is seen is sensed. But this defect is due to the kind of graphics used. With the aid of the scientific art of graphing pre- sented in Chapter I, one may not merely make better graphs in less time but actually draw correct negative conclusions from a graph so made, and therefore sense more than one sees. For instance, one may be sure that a given cubic equation has only the one real root seen in the graph, if the bend points lie on the same side of the a;-axis. Emphasis is here placed upon Xe^^'ton's method of solving numerical equations, both from the graphical and the numerical standpoint. One of several advantages (well recognized in Europe) of Newton's method over Horner's is that it applies as well to non-algebraic as to algebraic equations. In this elementary book, the author has of course omitted the difficult Galois theory of algebraic equations (certain texts on which are very erroneous) and has merely illustrated the subject of invariants by a few examples. It is surprising that the theorems of Descartes, Budan, and Sturm, on the real roots of an equation, are often stated inaccurately. Nor are the texts in English on this subject more fortunate on the score of correct proofs; for these reasons, care has been taken in selecting the books to which the reader is referred in the present text. The material is here so arranged that, before an important general theorem is stated, the reader has had concrete illustrations and often also special cases. The exercises are so placed that a reasonal^ly elegant and brief solution may be expected, without resort to tedious multiplications and similar manual labor, ^'ery few of the five hundred exercises are of the same nature. Complex numbers are introduced in a logical and satisfying manner. The treatment of roots of unity is concrete, in contrast to the usual ab- stract method. Attention is paid to scientific computation, })oth as to control of the limit of error and as to securing maximum accuracy with minimum labor. An easy introduction to determinants and their application to the solu- tion of systems of linear equations is afforded by Chapter XI, which is independent of the earlier chapters. m61?34 IV PREFACE Here and there are given brief, ])ut clear, outlooks upon various topics of decided intrinsic and historical interest, — thus putting real meat upon the dry bones of the subject. To provide for a very brief course, certain sections, aggregating over fifty pages, are marked by a dagger for omission. However, in compensa- tion for the somewhat more advanced character of these sections, they are treated in greater detail. In addition to the large number of illustrative problems solved in the text, there are five hundred very carefully selected and graded ercercises, distributed into seventy sets. As only sixty of these exercises (falling into seventeen sets) are marked with a dagger, there remains an ample number of exercises for the briefer course. The author is greatly indel^ted to his colleagues Professors A. C. Lunn and E. J. Wilczynski for most valuable suggestions made after reading the initial manuscript of the book. Useful advice was given by Professor G. A. Miller, who read part of the galley proofs. A most thorough read- ing of both the galley and page proofs was very generously made by Dr. A. J. Kempner, whose scientific comments and very practical sugges- tions have led to a marked improvement of the book. Moreover, the galleys were read critically by Professor D. R. Curtiss, who gave the author the benefit not merely of his A\'ide knowledge of the subject but also of his keen critical ability. The author sends forth the book thus emended ^\ith less fear of future critics, and with the hope that it will prove as stimulating and useful as these five friends have been generous of their aid. Chicago. Februury, 1914. CONTENTS Chap. Page I. The Graph of an Equation 1 II. Complex Numbers 18 III. Algebraic and Trigonometric Solution of Cubic Equations 31 IV. Algebraic Solution of Quartic Equations 38 V. The Fundamental Theorem of Algebra 47 VI. Elementary Theorems on the Roots of an Equation 55 VII. Symmetric Functions 63 VIII. Reciprocal Equations. Construction of Regular Polygons. Trisection of an Angle 81 IX. Isolation of the Real Roots of an Equation with Real Coefficients 93 X. Solution of Numerical Equations 109 XI. Determinants. Systems of Linear Equations 127 XII. Resultants and Discriminants 150 Miscellaneous Exercises 167 Answers 177 Index 183 THEORY OF EQUATIONS C-1,4) CHAPTER I The Graph of an Equation 1. For purposes of review, certain terms already familiar to the reader are defined here. Through a point 0, called the origin, draw a horizon- tal straight line OX and a vertical straight line OY. These lines are called the axes of coordinates; in particular, OX is called the x-axis. Choose a convenient unit of length. Consider any point P in the plane and let Q be the point of intersection of the x-axis with the vertical line through P. By the abscissa x of the point P is understood the number of units of length in OQ in case Q lies to the right of the origin 0; but, in case Q lies to the left of 0, x is the negative of the number of units of length in OQ. Similarly, the ordinate y of the point P is the length of PQ when P is above the x-axis, but is the negative of the length of PQ when P is below the x-axis. For the point P in Fig. 1, x =+4, y =—11. The real numbers x and y which a point determines in this manner are called its coordinates. Conversely, any pair of real numbers determines a point. Figure 1 shows the points which represent various pairs of values of x and y, satisfying the equation (1) ?/ = x2-6x-3. For example, the point P represents the pair of values x = 4, y = —11, and is designated (4, —11). Since the value of x may be as- signed at pleasure and a corresponding value of y is determined by equation (1), there is an infinitude of points representing pairs of values 1 (6,-3) THEORY OF EQUATIONS 'Cr. 1 satisfying the equation. These points constitute a curve called the graph of the equation. In Fig. 1, the curve intersects the x-axis in two points; the abscissa of one point of intersection is between 6 and 7, that of the other point is between — 1 and 0. The x-axis is the graph of the equation y = 0. Thus the abscissas of the intersections of the graph of equation (1) and the graph oi 7j — arc the real roots of the quadratic equation (!') x2-6x-3 = 0. Hence to find graphically the real roots of the last equation, we equate the left member to y and use the graph of the resulting equation (1). For other methods, see §§ 16-18. EXERCISES Find graphically the real roots of .x^ — 6 x + 7 = 0. Discuss graphically the reality of the roots of x- — Q x + 12 = 0. Obtain the grapli used in Ex. 1 by shifting the graph in Fig. 1 ten units upwards, leaving the axes OX and OY unchanged. How may we obtain similarly that used in Ex. 2? 4. Locate graphically the real roots of x^ + 4 .r- — 7 = 0- 2. Caution in Plotting. If the example set were (2) y = S x^ - 14 x^ - 9 .t2 + 11 X one might use successive integral values of x, obtain the points (-2, 180), (-1, 0), (0, -2), (1, -6), (2, 0), (3, 220), all but the first and last of which are shown (by crosses) in Fig. 2, and be tempted to con- clude that the graph is a U-shaped curve approxi- mately like that in Fig. 1 and that there are just two real roots, — 1 and 2, of (2') 8x'* - Ur* - 9x2-Mlx -2 = 0. But both of th(>se conclusions would be false. In fact, the graj^h is a W-shaped curve (Fig. 2) and the additional real roots arc^ \ and |. This example shows that it is often necessary to employ also values of x which are not integers. The purpose of the exami)le was, however, not to point out this o])vious fact, but rather to emphasize the chance of serious error in sketching a curve § 3] THE GRAPH OF AN EQUATION through a number of points, however numerous. The true curve between two points below the a:-axis may not cross the x-axis, or may have a peak actually crossing the a;-axis twice, or may be an M-shaped curve crossing it four times, etc. For example, the graph (Fig. 3) of (3) y = x^ -\- 4:X" — 11 crosses the .x-axis only once. But this fact can not be concluded from a graph located by a number of points, how- ever numerous, whose abscissas are chosen at random. We shall find that correct conclusions re- garding the number of real roots can be de- duced from a graph whose bend points (§ 3) have been located. We shall be concerned with equations of the form aox"" + aix"-! + + an-iX + a„ = (ao ^ 0), in which ao, Oi, . . . a„ are real constants. The left member is called a 'polynomial in x of degree n, or also a rational integral function of x^ and will frequently be denoted for brevity by the symbol f(x) and less often by /. Fig. 3 3. Bend Points. A point (like M or M' in Fig. 3) is called a bend point of the graph of y = f{x) if the tangent to the graph at that point is horizontal and if all of the adjacent points of the graph lie below the tangent or all above tlie tangent. The first, but not the second, condi- tion is satisfied by the point of the graph of y = x^ given in Fig. 4 (see § 6). In the language of the calculus, /(x) has a (relative) maximum or minimum value at the abscissa of a bend point on the graph of y = fix). Let P = (x, y) and Q = (x -\- h, Y) be two points on the graph, sketched in Fig. 5, oi y = f{x). By the slope of a straight line is meant THEORY OF EQUATIONS [Ch. I the tangent of the angle between the line and the a:-axis measured counter- clockwise from the latter. In Fig. 5, the slope of the straight line PQ is Y-y _ f(x-hh) -fix) h h (4) Y .r / Y-y h X Fig. 4 Fig. 5 For equation (3), fix) = x^ + 4 x^ — 11. Hence fix + h) = ix + hy + 4 (x + /i)2 - 11 = x' + 4 x2 - 11 + (3 a:- + 8 x)h + (3 x + 4)/r + h\ The slope (4) of the secant PQ is here 3x- + 8.'c+ (3.'c + 4)/i + /i2. Now let the point Q move along the graph towards P. Then h approaches the value zero and the secant PQ approaches the tangent at P. The slope of the tangent at P is therefore the corresponding limit 3 a-- + 8 a: of the preceding expression. In particular, if P is a bend point the slope of the tangent at P is zero and hence a: = or x = — f. Equation (3) gives the corresponding values of y. The resulting points il/= (0, -11), iir = (-!, -H) §4] THE GRAPH OF AN EQUATION 5 are easily shown to be bend points. Indeed, for re > and for x between -4 and 0, x"^ {x + 4) is positive, and hence /(a;) > -11 for such values of X, so that the function (3) has a relative minimum at a; = 0. Similarly, there is a relative maximum at a: = — |. We may also employ the general method of § 8 to show that M and M' are bend points. Since these bend points are both below the a;-axis, we are now certain that the graph crosses the a:-axis only once. The use of the bend points insures greater accuracy to the graph than the use of dozens of points whose abscissas are taken at random. 4. Derivatives. We shall now find the slope of the tangent to the graph of ?/ = f(x) , where f(x) is any polynomial (5) fix) = aox" + aia:"-i + • • • + an-ix + a„. We need the expansion of f{x + h) in powers of x. By the binomial theorem, ao{x -\- h)" = aox"" + naoa;"-'/i + -^^ aoX"--h^ -\- . . . ^ a,(x + h)"-' = a^x"-' + (n - l)a,x^'-% + ^"^ ~ ^^j,"' ~ ^ K.x"-^-' + • • • , a„-2(a: + h)- = a„-2X- + 2 an-^xh + dn-ih^, Qn-liX + h) = Qn-lX + ttn-lh, The sum of the left members is evidently f(x + h). On the right, the sum of the first terms (i.e., those free of h) is f(x). The sum of the coefficients of h is denoted by f'{x), the sum of the coefficients of § h- is denoted hy f"{x), • • • , the sum of the coefficients of h/^ 1-2 •• A; is denoted by /(^'^ (x) . Thus (6) fix) = naox"-' + (n - l)aiX--'~ + • • • + 2 a„_2a: + a„_i, (7) fix) = nin - 1) ao.x"-2 + in - l)(n - 2)ayX-' + • • • + 2a„_2, etc. Hence we have (8) f(x + A) = fix) +f'(x) k +/"Wj^ +/"'Wr:|:3+---+/'"'Wi-T2^- 6 THEORY OF EQUATIONS [Ch. I This formula (8) is known as Taylor's theorem for the present case of a polynomial f{x) of degree n. We call f'{x) the (first) derivative of f{x), and fix) the second derivative of f{x), etc. Concerning the fact that f"{x) is the first derivative of f'{x) and that, in general, the A:th derivative /(*■") (a^) of f{x) equals the first derivative of /^'■'"'K^); see Exs. 6-9 of the next set. In view of (8), the limit of (4) as h approaches zero isf'{x). Hence f'{x) is the slope of the tangent to the graph of y = f{x) at the point {x, y). In (5) and (6), let every a be zero except a^. Thus the derivative of aox" is naoX"~\ and hence is obtained by multiplying the given term by its exponent n and then diminishing its exponent by unity. For example, the derivative of 2 x^ is 6 x~. Moreover, the derivative of /(.r) equals the sum of the derivatives of its separate terms. Thus the derivative of a;^ + 4 x^ — 1 1 is 3 x- + 8 a:, as found also in § 3. 5. Computation of Polynomials. The lalwr of computing the value of a polynomial J{x) for a given value of x may be much shortened by a simple device. To find the value of a:^ + 3 ^2 - 2 .r - 5 for X = 2, we note that x^ = x-x- = 2 x^, so that the sum of the first two terms is 5 x"^. This latter equals 5 • 2 x or 10 a;, adding this to the next term —2 x, we get 8 a; or 16. The final result is therefore 11. Write the coefficients in a line. Then the work is: 1 3-2 - 5 ^ 2 10 16 1 5 8 11. In case not all the intermediate powers of x occur among the terms of f{x), the missing powers are considered as having the coefficients zero. Thus the value —61 of 2 a;^ — x^ + 2 x — 1 for a: = —2 is found as follows: 2 0-1 2-1 1 -2 -4 8-14 28 -60 2-4 7 -14 30 -61. For another manner of presenting this method see Ch. X, § 4. THE GRAPH OF AN EQUATION EXERCISES 1. The slope of the tangent to ?/ = 8 x^ — 22 a:^ + 13 a; — 2 at {x, y) is 24x2 _ 44a; + 13. The bend pomts are (0.37,0.203), (1.46, -5.03), approxi- mately. Draw the graph. 2. The bend points of y = r' - 2 x - b are (.82, -6.09), (-.82, -3.91), approximately. Draw the graph and locate the real roots. 3. Find the bend points of ?/ = x^ + 6 x^ + 8 .r + 8. Locate the real roots. 4. Locate the real roots of /(.c) = x* + x^ — x — 2 = 0. The abscissas of the bend points are the roots of f {x) = 4 .r^ + 3 x^ — 1 = 0. The bend points of y = /'(x) are (0, —1) and ( — ^, — f), so that /'(x) = has a single real root (it is just less than ^). The single bend point of y = /(x) is (^, — f J), approxi- mately. 5. Locate the real roots of x^ — 7 x^ — 3 x- + 7 = 0. 6. /"(x), given by (7), is the first derivative of /'(x). 7. If /(x) = /i(x) +/2(x), the ^th derivative of / equals the sum of the A:th derivatives of /i and f^. Use (8) . 8. f^'^^i^x) equals the first derivative of f'^^~^\x). Hint: prove this for / =ax"'; then prove that it is true for / = /i + /e if true for /i and /2. 9. Find the third derivative of x'' + 5 x^ ]:)y forming successive first derivatives; also that of 2 x^ — 7 x' + x. 10. The derivative of gk is g'k + gk'. Hint : multiply the members of g{x + A ) = gi^) + f7'(-c) h + • • • and k{x + h) = ^(x) + k'{x) h + • • • and use (8) for f=gk. 6. Horizontal Tangents. If (x, y) is a bend point of the graph of y = fi^)> then, by definition, the slope of the tangent at (x, y) is zero. Hence (§ 4), the abscissa a; is a root of f'(x) = 0. In Exs. 1-5 of the preceding set, it was true that, conversely, any real root of f'(x) = is the abscissa of a bend point. However, this is not always the case. We shall now consider in detail an example illustrating this fact. The example is the one merely mentioned in § 3 to indicate the need of the second requirement made in our definition of a bend point. The graph (Fig. 4) of ?/ = x^ has no bend point since x^ increases when X increases. Nevertheless, the derivative 3 x- of x^ is zero for the real value a; = 0. The tangent to the curve at (0, 0) is the horizontal line ?/ = 0. It may be thought of as the limiting position of a secant through which meets the curve in two further points, seen to be equidistant from 0. When one, and hence also the other, of the latter points ap- proaches 0, the secant approaches the position of tangency. In this sense the tangent at is said to meet the curve in three coincident points, their abscissas being the three coinciding roots of x^ = 0. In the 8 THEORY OF EQUATIONS [Ch. I usual technical language which we shall employ henceforth, x^ = has the triple root x = 0. The subject of bend points, to which we recur in § 8, has thus led us to a digression on the important subject of double roots, triple roots, etc. 7. Multiple Roots. In (8) replace a: by a and h by x — a. Then (9) /(a:)=/(a)+/'(a)(a;-a)+r(a)-^^f^+r'(a)^'^~"^' ' 1-2 ' •' ' M • 2 ■ 3 ' Thus the constant remainder obtained by dividing any polynomial f{x) by a; — a is /(a), a fact known as the Remainder Theorem. In par- ticular, if /(«) = 0, f{x) has the factor x — a. This proves the Factor Theorem: If a is a root of /(x) = 0, then x —a is a factor of /(a:). The converse is true: If x — a is a factor of /(.t), then a is a root of f{x) = 0. In case f{x) has the factor {x — a)-, but not the factor {x — aY, a is called a double root of /(x) = 0. In general, if f{x) has the factor {x — a)'", but not the factor {x — a)"'+\ a. is called a multiple root of multiplicitij m oi f{x) — 0, or an ?»-fold root. Thus, 4 is a simple root, 3 a double root and —2a triple root of 7{x-4:)ix-3y-(x-\-2y = 0. This algebraic definition of a multiple root is in fact equivalent to the geometrical definition, given for a special case, in § 6. The second member of (9) is divisible by (x — a)- if and only if /(a) = 0, /'(a) = 0, and is divisible by (x — a)^ if and only if also /"(a) = 0, etc. Hence a is a double root of f(x) = if and only if f(a) = 0, f'(a) = 0, /" (a) ?^ 0; a is a root of multiplicity ni if and only if (10) f{a) = 0, f'{a) = 0,/"(a) = 0, • • • , /("-')(«) = 0, /("')(«) ^ Q. For example, zero is a triple root of x* + 2 x' = since the first and second derivatives are zero for x = 0, while the third derivative 24 x + 12 is not. If /(x) and/'(x) have the common factor (x — a)'"~^, but not (x — a)'", where m = 2, then a is a root of f{x) = of multiplicity 7n. For, a is a root of multiplicity at least ni — 1 of both /(.r) = and /'(x) = 0, so that the equalities in (10) hold; also /("■)(«) ^ holds, since otherwise a would be a root of l)oth /(x) = and /'(x) = of multiplicity m or greater, and (x — a)"" would be a common factor. Hence if f{x) andf'{x) have a greatest common divisor g{x) involving x, a root of g(x) =0 of multiplicity § 8] THE GRAPH OF AN EQUATION 9 m — \ is a root of f(x) = of multiplicity m, and conversely any root of f{x) = of niulti-plicity m is a root of g(x) = of multiplicity m — 1. The last fact follows from relations (10), which imply that a is a root of f'{x) = of multiplicity m — 1, and hence that f{x) and f'{x) have the common factor {x — a)"'"^, but not (x — a)'". In view of this theorem, the problem of finding all the multiple roots oi f{x) = and the multiplicity of each multiple root is reduced to the problem of finding the roots of g{x) = and the multiplicity of each. For example, let J{x) = x^ — 2 x^ — 4 x + 8. Then /'(x) - 3 x2 - 4 X - 4, 9/(x) = /'(x) (3 x - 2) - 32 (x - 2). Since x — 2 is a factor oi fix) it may be taken to be the greatest common divisor of /(x) and/'(x), as the choice of the constant factor c in c(x — 2) is here immaterial. Hence 2 is a double root of /(x) = 0, while the remaining root —2 is a simple root. EXERCISES 1. x^ — 7 x^ -f 15 X — 9 = has a double root. 2. X* — 8x^+16 = has two double roots. 3. X* — 6x2 — 8x — 3 = ]^jjg a triple root. 4. Test x^ - 8 x» + 22 x2 - 24 X + 9 = for multiple roots. 5. Test x^ — 6 .r^ + 11 X — 6 = for multiple roots. 8. Inflexion and Bend Points. The equation of the tangent to the graph oi y = f{x) at the point {a, jS) on it is y=na){x-a)-V^ [^ = /(«)]• For the abscissas of its intersections with the graph oi y = f(x), we have, from (9), r(a)^^j^ + r'(a)^^;^'+- • • =0. If a: is a root of multipHcity m of this equation, the point (a, /3) is counted as m coincident points of intersection of the tangent and the curve (just as in the example in § 6). This will be the case if and only if * (11) f"ia) =0, f"'{a) = 0, . . . , /(-i)(«) = 0, /(-)(«) ^ 0. For example, if /(x) = x* and a = 0, then m = 4. The graph oi y = x* is a. U-shaped curve, whose intersection with the tangent (x-axis) at (0, 0) is counted as four coincident points of intersection. * If m = 2, only the last relation of the set is retained: /"(a) 5^ 0. 10 THEORY OF EQUATIONS [Ch. I If m is even, the points of the curve in the vicinity of the point of tangency {a, /3) are all on the same side of the tangent and the point (a, /3) is, by the definition in § 3, a bend point. But if ni is odd (wi > 1), the curve crosses the tangent at the point of tangency {a, jS) and this point is called an inflexion point, and the tangent an inflexion tangent. To simplify the proof, take {a, /3) as the new origin of coordinates and the tangent as the new a;-axis. Then the new equation of the curve is y = cx"^ + dx'^+i + • • • {c^0,m = 2). For X sufficiently small numerically, y has the same sign as ex"" (§ 11). Thus if m is even, the points of the curve in the vicinity of the origin are all on the same side of the a:-axis. But if m is odd, the points with small positive abscissas lie on one side of the a:-axis "and those with numerically small negative abscissas lie on the opposite side. For example, (0, 0) is a bend point of the graph of y = x*. But (0, 0) is an inflexion point of the graph (Fig. 4) of y = .T^ and the inflexion tangent y = crosses the curve at (0, 0). Here/"(0) = 0, /'"(O) = 6, so that ?« = 3, in accord with the evident fact that a;^ = has the root zero of multipUcity 3. We have,' therefore, in the evenness or oddness of m in (11) a practical test to decide which roots a of f'{x) = are abscissas of bend points and which are abscissas of inflexion points with horizontal inflexion tangents. EXERCISES 1. If fix) = 3 x^ + 5 x3 + 4, the only real root of fix) = is x = 0. Show that (0, 4) is an inflexion point, and thus that there is no bend point and hence that fix) = has a single real root. 2. x^ — 3 x^ ■]- 3x + c = has an inflexion point, but no bend point. 3. x^ — 10 x' — 20 x^ — 15 .r + c = has two bend points and no horizontal inflexion tangents. 4. 3 x^ — 40 x^ + 240 x + r = has no bend point, but has two horizontal inflexion tangents. 5. Any function x^ — 3 ox^ + • • -of the third degree can be written in the fonn/(x) = (x — a)^ + ax + 6. The straight Hne having the equation y = ax-\-b meets the graph oi y = fix) in three coincident points with the abscissa a and hence is an inflexion tangent. If we take new axes of coordinates parallel to the old and intersecting at the new origin («, 0), i.e., if we make the transformation x = X-\-a, y=Y, of coordinates, we see that the equation /(x) = becomes a reduced cubic equation X^ + pX + q = (cf. Ch. III). 6. Find the inflexion tangent to y = x^ + 6 .r- — 3 x + 1 and transform x^ + Gx^ — 3x + l =0 into a reduced cubic equation. § 101 THE GRAPH OF AN EQUATION 11 9. Real Roots of a Cubic Equation. It suffices to consider J{x) = x' - 3lx + q (1^0), in view of Ex. 5 above. Then /'= 3 {x'~ - I), f"= Qx. in< 0, there is no bend point and the cubic equation f(x) = has a single real root. U I > 0, there are two bend points (Vl, q-2l VI), {-Vl,q-^2lVl) and the graph oi y = f{x) is evidently of one of the three types: If the equality sign holds in the first or second case, one of the bend points is on the a:-axis and the cubic equation has a double root; the condi- tion is that q2 _ 4 p = o. The third case is fully specified by the condition g2 < 4 P, which implies that Z > 0. Hence x^ — 3 Ix -\- q = has three dis- tinct real roots if and only if q- < 4: P, a single real root if and only if q- > 4 Z^; and a double root {necessarily real) if and only if q^ = 4: P. Fig. 8 EXERCISES. Apply the criterion to find the number of real roots of: 1. a:3 + 2a; -4 = 0. 2. a;^ - 7x + 7 = 0. 3.z^-2x-l = 0. 4. x^ - 3 X + 2 = 0. 5. x^ + 6 x2 - 3 X + 1 = 0. 6. The inflexion point of ?/ = x^ — 3 /x + g is (0, q). lO.f Trinomial Equations. For m and n positive odd integers, m > n, let fix) = X"' + px" + q (p ^ 0). 12 THEORY OF EQUATIONS tCn. I Here x = is a root of /'(x) = only when n > I and then tlio tangent at (0, q) is the horizontal inflexion tangent ?/ = 0, /' has no real root except x = 0. Thus there is no bend point and hence a single real root of /(x) = 0. If p < 0, there are just two bend points, their abscissas being b and —b, where b is the single positive real root of 6™~" = —np/m. The bend points are on the same side or opposite sides of the x-axis according as m = q + pb"(l - ~^, fi-b) = q - pb" (l - ^) are of like signs or opposite signs. The number of real roots is 1 or 3 in the respec- tive cases. Hence there are three distinct real roots if and only if the positive number exceeds both q and —q, i.e., if -p5"{l-^ -p-b" > ^• m m — n The first member equals ft", so that its (m — ?i)th power is the ?«th power of ^m-n — —np/m. Hence the conditions are equivalent to -(f +1 -'"'- )n — n EXERCISES t l.t x^ + px + 7 = has three distinct real roots if and only if (Sf o>(:;)+.. 2.t If p and q are positive, x-"' — p.v-" + q = has four distinct real roots, two pairs of equal roots, or no real root, according as f^y"_f_i^r">0,=0,cr<0. \ m I \m — n/ 11. Continuity of a Polynomial. Hitherto we have located certain points of the graph of y = /(x), where fix) is a polynomial in x wdth real coefficients, and taken the liberty to join them by a continuous curve. § 12] THE GRAPH OF AN EQUATION 13 The polynomial /(x) in the real variable x shall be called continuous at X = a, where a is a real constant, if the difference D=fia + h)- /(a) is numerically less than any assigned positive number p for all real values of h sufficiently small numerically. We shall prove that any polynomial f{x) with real coefficients is con- tinuous at X = a, where a is any real constant. The proof rests upon Taylor's formula (8), which gives Z)./'(a)„+0|),.+ ...+_|^,,. Denote by g the greatest numerical value of the coefficients of h, h?, . . . , h". For h numerically less than k, where k < 1, we see that D is numerically less than g{k-\-k''-\- - ■ ■ +k-) P. In fact, f{x)=x^^{ao-h), = ^ + |+ ••• +^- By the last result in § 11, the numerical value of ^ is less than that of Qq when l/x is positive and less than a sufficiently small positive number, say l/P, and hence when x > P. Then Oo + has the same sign as Qo, and hence f(x) the same sign as Oox". The last result holds also when a: is a negative number sufficiently large numerically. For, if we set x = —X, the former case shows that /( — A") has the same sign as ( — l)"aoA'" when A' is a sufficiently large positive number. We shall therefore say briefly that, for x = +00, f(x) has the same sign as oo; while, for x = — 00 , f{x) has the same sign as ao if n is even, but the sign opposite to ao if n is odd. EXERCISES 1. x^ + ax^ -\- bx — 4: =^ has a positive real root [use .r = and x = +^]- 2. x^ + ax^ + 6x + 4 = has a negative real root [use x = and x = — xj. 3. If oo > and n is odd, oox" +•••+«« = has a real root of sign opposite to the sign of a„ [use x= —00, 0, +00]. 4. X* + ttx^ -\- bx^ -\- ex — 4: = has a positive and a negative root. 5. Any equation of even degree n in which the coefficient of x" and the con- stant term are of opposite signs has a positive and a negative root. 14. The accuracy of a graph of y = f{x) can often be tested and important (;on(;lusions drawn from it by use of the Theorem. No straight line crosses the graph of y = J{x) in more than n points if the degree n of the polynomial f{x) exceeds unity. A vertical line x = c crosses it at the single point (c, /(c)). A non- vertical line is the grai)h of an equation y = mx + h of the first degree, and the abscissas of the points of crossing are the roots of mx + 6 = f{x). The proof may now be completed by using the next theorem. § 15, 16] THE GRAPH OF AN EQUATION 15 15. Theorem. An equation of degree n, f(x) = aox" + aix"-i + • • • + a„ = 0' (oq ^ 0), cannot have more than n distinct roots. Suppose that it has the distinct roots ai, . . . , «„, a. By the Factor Theorem (§ 7), x — ai is a factor of /(x), so that f{x) = {x- ai) Q(x), where Q(x) is a polynomial of degree n — 1. Let x = 0:2. We see that ^("2) — 0, so that as before Q(x) = {x - 02) Qi(x), f{x) = (x - ai){x - ai) Qi(x). Proceeding in this manner, we get f{x) ^ ao{x - ai)(x - 0:2) ... (x - a„). For the root a, the left member is zero and the right is not zero. Hence our supposition is false and the theorem true. EXERCISES 1. The curve in Fig. 3, representing a cubic function, does not cross the x-axis at a second point further to the right, nor does the part starting from M' and running downwards to the left later ascend and cross the x-axis. 2. The curve in Fig. 2, representing a quartic function, has only the four cross- ings shown. 3. Form the cubic equation having the roots 0, 1,2. 4. Form the quartic equation having the roots ±1, ±2. 5. If Qqx" + • • • = has more than n distinct roots, each coefficient is zero. When would the theorem in § 14 fail if n = 1? 6. If two polynomials in x of degree n are equal for more than n distinct values of X, they are identical. 7. An equation of degree n cannot have more than n roots, a root of multiplicity m being counted as rn roots. 16. Graphical Solution of a Quadratic Equation. If (12) x2 - ax + 6 = has real coefficients and real roots, the roots may be constructed by the use of ruler and compasses, i.e., by elementary geometry. 16 THEORY OF EQUATIONS [Ch. I Draw a circle having as a diameter the Une BQ joining the points B = (0, l)and Q = (a, 6); the abscissas ON and OM of the points of intersection of this circle with the x-axis are the roots of (12). The center of the circle is (a/2, (6 + l)/2). The square of BQ is a^ + (b — iy. Hence the equation of the circle is (^-5 %/,_^ + i- 0, set = X VI; we get x* + x^ - ax + 6 = 0. We now employ the graphs of y = x* + x^, y = ax — h. EXERCISES Solve by each of the two methods 1. x2-5x + 4 = 0. 2. x2 + 5a; + 4 = 0. 4. x2-5x-4 = 0. 5. x2-4x + 4 = 0. Solve graphically the cubic equations x2 + 5 x - 4 = 0. x2 - 3 X + 4 = 0. 7. x3 - 3 X + 1 = 0. x3 + 2 X - 4 = 0. 9. x' - 7 X + 7 = 0. 10. Find graphically the cube roots of 20, -20, 200. 11. State in the language of elementary geometry the construction of Fig. 9 and prove that OC = TQ = b, TD = OB = 1, chord BN = chord DM, ON = MT, ON + OM = a, ON • OM = OC'OB = b. Why are OM and ON the roots of (12)? 12. Any reduced cubic equation x^ = px -\- q can be solved by use of a fixed parabola x^ = y and the circle x^ + y^ = (/x + (p + l)y. (Descartes.) 13. x* = px- + qx + r can be solved by use of a fixed parabola x^ = y and the circle x- -\- y"^ — qx -\- {p -\- \)y -\- r. (Descartes.) 14. Solve the cubics in Exs. 7-9 by the method of Ex. 12. 15. Solve X* = 25 x^ - 60 X + 36 by the method of Ex. 13. i 181 THE GRAPH OF AN EQUATION 17 18. t Tlie approximate values of the real roots of a cubic equation 2^ + p2 + g = may be found by a graphical method due to C. Runge.* We assign equidistant values to z. For each z, we have a linear equation in p and q which therefore represents a straight line when p and q are taken as rec- tangular coordinates. On a diagram showing these lines we may locate approximately the line (and hence the values of z) corresponding to assigned values of p and q. The method applies also to any equation involving two parameters linearly. For the solution of a numerical cubic equation by means of the slide rule (and an account of the use of the latter), see pp. 43-48 of the book just cited. * Graphical Methods, Columbia University Press, 1912, p. 59 (also, Praxis der Gleichungen, Leipzig, 1900, p. 156). Earlier by L. Lalanne, Comptes Rendus Acad. Sc. Paris, 81, 1875, p. 1186, p. 124.3; 82, 1876, p. 1487; 87, 1878, p. 157, and in Notices r^unies par le Ministere des travaux . . . exposition univ. Paris, 1878. CHAPTER II Complex Numbers (For a briefer course, this chapter may be begun with § 5.) 1. 1 Vectors from a Fixed Origin 0. A directed segment of a straight fine is called a vector. We shall employ only vectors from a fixed initial point 0. The sum of two vectors OA and OC is defined to be the vector OS, where S is the fourth vertex of the par- allelogram having the fines OA and OC as two sides. In case A coincides with 0, the vector OA is said to be zero; then OS = OC. A force of given magnitude and given dir- ection is conveniently represented by a vector. By a fundamental principle of mechanics, two forces, represented by the vectors OA and OC, have as their resultant a force represented by the vector OS, as in Fig. 10. Thus if two forces their resultant is represented by the sum of Fig. 10 are represented by two vectors, the vectors. When referred to rectangular axes OA" and OY, let the point A have the coordinates OE = a, EA = h, and the point C the coordinates OF = c, FC = d. Draw AG parallel to OA' and SGH perpendicular to OA". Since triangles OFC and AGS are equal, AG = c, GS = d. Hence the coor- dinates of the point aS are OH ^ a -\- c and HS = h -{- d. The sum of the vectors from to the points (a, h) and {c, d) is the vector from to the point (a -\- c, b -\- d), whose coordinates are the sums of the corresponding coordinates of the two points. Subtraction of vectors is defined as the operation inverse to addition of vectors. If OA and OS are given vectors, the vcK'tor OC for which OA -\- OC = OS is denoted l)y OS - OA, and is determined by the side OC of the parallelogram with the diagonal OS and side OA. 18 §2] COMPLEX NUMBERS 19 2.1 Multiplication of Vectors. Let A be a point \r,6l with the polar codrdinatcs r, 6. Then r is the positive number giving the length of the line OA, while 6 is the measure of the angle XOA when measured counter- clockwise from OX, as in Trigonometry. Let C be the point \r', d'l with the polar coordinates r', 6'. The product OA • OC of the vectors from to A = \r, 6} and to C = Ir', e'l is defined to be the vector from to P = \rr', d -{■ d'L Fig. 11 To construct this product geometrically, let U be the point on the X-axis one unit to the right of 0. Let the triangle OCP be constructed similar to triangle OUA, such that corresponding sides are OC and OU, CP and UA, OP and OA, and such that the vertices 0, C, P are in the same order (clockwise or counter-clockwise) as the corresponding vertices 0, U, A. Then OP : r' = r : 1, so that the length of OP is rr'. The angle XOP, measured counter-clockwise from OX, equals d + 6', and may exceed four right angles. Hence the product of the vectors OA and OC is the vector OP. If OC = OU, then OP = OA, and Ot/ • OA = OA. Hence vector OU plays the role of unity in the multiplication of vectors. Division of vectors is defined as the operation inverse to multiplication of vectors. If OA and OP are given vectors, the vector OC for which OA-OC = OP is denoted by OP/OA. If A = \r,d\ smdP = In, d^] then C = \ri/r, di— 61 . Division except by zero is therefore always possible and unique. EXERCISES t Lt Vector addition is associative: {OA + OC) + OL = OA + (OC + OL). 2.t Vector multiplication is associative: {OA • OC) • OL = OA > {OC • OL). 3.t Draw the figure corresponding to Fig. 12, when OA is in the third quadrant and OC in the first quadrant. 20 THEORY OF EQUATIONS [Ch. II S.f Symbol for Vectors from 0. We consider only vectors starting from the fixed point 0. Such a vector OA is uniquely determined by its terminal point A = (a, h) and hence by the Cartesian coordinates a, b of the point A referred to fixed rectangular axes OX and OY. We may therefore denote the vector OA by the symbol [a, b]. Then (1) [a, b] = [c, d] if and only if a = c, b = d. By the definition of addition and subtraction of vectors (§ 1), (2) [a, b] + [c, d] = [a + c, 6 + d], (3) [a, b] - [c, d] = [a-c,b- d]. As our definition of the product of two vectors was made in terms of polar coordinates, we must now express the product in terms of Cartesian coordinates. By Fig. 11, we have a — r cos 6, b — r sin 6. Similarly, if the point (c, d) has the polar coordinates r', Q', c = r' cos 6' , d = r' sin d'. Hence the definition (§2) of the product of two vectors gives [a, b] [c, d\ = [rr' cos (6 + d'), rr' sin {B + 6')], the final numbers being the Cartesian coordinates of the point with the polar coordinates rr' and ^ + 0'. But rr' cos {d + e') = rr' (cos 6 cos d' - sin d sin d') ^ ac - bd, rr' sin {6 + 6') = rr' (sin 6 cos 9' + cos d sin d') = be + ad. Hence, finally, (4) [a, b] [c, d] = [ac - bd, ad + be]. Given a, b, e, f, we can find solutions c, d of the equations ac — bd = e, ad -\- be ^ f, provided a- -\- b"^ 9^ 0, viz., a and b are not both zero. Then [a,b][c,d] = [e,f] determines [c, d], its expression being ,,. k/] ^ V ae + bf af - b cl ^^^ [a,b] La' + ^" a^ + fe^J Hence division, except by the zero vector [0, 0], is always possible and unique. §4] COMPLEX NUMBERS 21 4.t Introduction of Complex Numbers. Giving up the concrete in- terpretation in § 3 of the symbol [x, y] as the vector from the origin to the point (x, y), we shall now think abstractly of a system of elements [x, y] each determined by two real numbers x, y, and such that the sys- tem contains an element corresponding to any pair of real numbers. While the present abstract discussion is logically independent of the earlier exposition of vectors, yet we shall be guided in our present choice of definitions of addition, multiplication, etc., of our abstract symbols [x, y] by the desire that the vector system shall furnish us a concrete representation of the present abstract system. Accordingly, we define equality, addition, subtraction, multiplication and division of two ab- stract elements [x, y] by formulas (l)-(5). In particular, we have [a, 0] ± [c, 0] = [a ± c, 0], [a, 0] [c, 0] = [ac, 0], -^ = [^-0]' provided a 5^ in the last relation. Hence the elements [x, 0] combine under our addition, multiplication, etc., exactly as the real numbers x combine under ordinary addition, multiplication, etc. We shall there- fore introduce no contradiction if we now impose upon our abstract system of elements [x, y], subject to relations (l)-(5), the further condi- tion that the element [x, 0] shall be the real number x. Then, by (4), [0,1] [0,1] = [-1,0]= -1. We write i for [0, 1]. Hence i^ = —1. Then [x, y] = [x, 0] + [0, ij]=x-{- [y, 0] [0, l]=x + yi. The resulting symbol x + yi is called a complex number. For y = 0, it reduces to the real number x. For y ^ 0, it is also called an imaginary number. The latter is not to be thought of as unreal in the sense that its use is illogical. On the contrary, x + yi is a convenient analytic rep- resentation of the vector from the origin to the point (x, y), and the sum, product, etc., defined above, of two such complex numbers then repre- sent those simple combinations of the two corresponding vectors (§§ 1, 2) which are constantly used in the applications of vectors in mechanics and physics. Since these vectors from are uniquely determined by their termi- nal points, we obtain a representation (§8) of complex numbers by points 22 THEORY OF EQUATIONS [Ch. ll ill a piano, a representation of great importance in mathematics and its applications. If in (l)-(5), we replace the symbol [a, h] by a + bi, etc., we obtain the formulas given in § 5. 5. Formal Algebraic Definition of Complex Numbers. The equa- tion x^ — — 4: has no real root, but is said to have the two imaginary roots V — 4 and — V — 4. We shall denote these roots by 2 i and —2 i, agree- ing that I is a definite number for which i~ =—1. Similarl}', we shall write Vs i in preference to V — 3. If p is positive, Vp is used to denote the positive square root of p. If a and b are any two real numbers, a + bi is called a complex number and a — bi its conjugate. Two complex numbers a -\- bi and c + di are called equal if and only if a = c, 6 = d. Thus a + &i = if and only if a = 6 = 0. Addition of complex numbers is defined hj (a + bi) + (c + di) = (a + c) +(6 + d)i. The inverse operation, called subtraction, consists in finding a complex number z such that (c + di) -\- z = a -\- bi. In notation and value, z is (a + bi) - (c + di) = (a - c) + (6 - d)i. Multiplication is defined by (a + bi){c + di) = {ac - bd) + (ad + bc)i, and hence is performed as in formal algebra with a subsequent reduction by use of i^ = — 1. If we replace 6 by —6 and d by —d, the right member is replaced by its conjugate. Hence the product of the conjugates of two complex members equals the conjugate of their product. Division is defined as the operation inverse to multiplication, and con- sists in finding a complex number q such that (a + bi)q = e -\- fi. Mul- tiplying each member by a — bi, we find that q is, in notation and value, e +fi ^ (e +/0(a - bi) ^ ae + bf af - be . a + bi a2 + 62 a'-^ + fe' a' + &'^" Since a- + 6- = implies a = b = when a and b are real, division except by zero is possible and uni(iue. § 6, 71 COMPLEX NUMBERS 23 6. The Cube Roots of Unity. The roots of x^ = 1 are unity and the numbers for which ^^^a:2 + a; + l =0, {x-\-^y=-l x + h = ±^VZi. X — 1 Hence the three cube roots of unity are 1 and EXERCISES 1. Verify that co' = u-, wcc' = 1, w- + co + 1 = 0, w^ = 1. 2. The sum and product of two conjugate complex numbers are real. 3. Express as complex numbers 3 + 5i a + bi 3 + V^ 2-3i' a-bi' 2 + V^' 4. If X, y, z are anj^ complex numbers, x-\ry = y + X, {x + 2/) + 2 = .T + (y + 2), xy = yx, {xy)z = x{yz), x{y -{- z) = xy + xz. What is the name of the property indicated by each equation? 5. If the product of two complex numbers is zero, one of them is zero. G-t Deduce the laws in § 5 from those in § 4. 7. Square Roots of a + hi found Algebraically. Given the real num- bers a and h, b 9^ 0, Ave seek real numbers x and y such that a -\-bi = {x -\- yi)" = a;- — 7/- + 2 xyi. Thus x^ — y- ^ a, 2xy = h, (x2 + ,/)2 = (:,2 _ yoy + 4 ^2y2 = «2 + 52. Since x and y are to be real and hence x~ + y' positive, x~ + ?/2 = Va^ + b-, the positive square root being the one taken. Combining this equation with x^ — y^ = a, we get , Va" + 62 _^ a , Va^ -\- ¥ - a a;2 = ., y2 = 24 THEORY OF EQUATIONS [Ch. ll Since these expressions are positive, real values of x and y may be found. The two pairs x, y for which 2 xy = b give the desired two complex num- bers X + yi. It is not possible to find the cube roots of a general complex number by a similar algebraic process (Ch. Ill, § 6). EXERCISES Express as complex numbers the square roots of 1. -7 + 24i. 2. -11 + 60/. 3. 5-12i. 4. 4cd+(2c2- 2(^2)1. 5. c2 - ^2 _ 2 V- c^d^. 8. Geometrical Representation of Complex Numbers. Using rec- tangular axes of coordinates, we represent* a + bi by the point A = (a, 6). The positive number r = Va- + b^ giving the length of OA is called the modulus (or absolute value) of a -{- bi (Fig. 11). The angle 6 = XOA, measured counter-clockwise from OA", is called the amplitude (or argument) of a + bi. Thus (6) a-\-bi = r(cos 6 + i sin 9). The second member is called the trigonometric form of a+ bi. If c + di is represented by the point C, then the sum of a + bi and c -{- di is the complex number represented by the point *S (Fig. 10) determined by the parallelogram OASC. Since OS = OA -\- AS, the modulus of the sum of two complex numbers is equal to or less than the sum of their moduli. For example, the cube roots of unity are 1 and I = cos 120° + i sin 120°, 0^= -i-iV3i = cos 240° + i sin 240°, Fig. 13 and are respresented by the points marked 1, w, w- in Fig. 13. They form * It wiU be obvious to the reader who has not omitted §§ 1-4 that the present rep- resentation is essentially equivalent to the representation oi a -\- bi by the vector from O to the point (o, 6). § 9. 10] COMPLEX NUMBERS 25 the vertices of an equilateral triangle inscribed in a circle of unit radius and center at the origin 0. 9. The product of the complex number (6) by r'(cos a -{- i sin a) is rr' [cos (0 + a) + i sin {6 + «) ], since (7) (cos 6 -\- i sin 9) {cos a -{- i sin a) = cos (6 -{- a) -^ i sin {d -{- a). The latter follows from cos 9 cos a — sin 9 sin a = cos {9 -\- a), cos sin a + sin 9 cos a = sin {9 -^ a). Hence the modulus of the product of two complex numbers equals the product of their moduli, and the amplitude of the product equals the sum of their amplitudes. The product may be found geometrically as in Fig. 12. For the special case a = 9, (7) becomes (cos 9 -\- i sin 6)- = cos 2 + t sin 2 9. This is the case n = 2 of formula (8). In particular, we see why the amplitude of w^ is 240° when that of co is 120° (end of § 8). 10. De Moivre's Theorem. // n is any positive integer, (8) (cos 9 -\-i sm.9y = cos n9 + i sin n9. This relation is an identity if n = 1 and was seen to hold if n = 2. To proceed by mathematical induction, let it be true if n = m. Using (7) for a = m9, we then have (cos + 1 sin 0)'"+^ = (cos 9 -\- i sin 6) (cos 6 + i sin 9Y = (cos + i sin 9) {cos m9 + i sin m9) = cos (m + 1)0 + i sin (m -{-1)9. Hence (8) is true also if /i = m + 1. The induction is thus complete. Since cos 9 -{- i sin e represents the vector from the origin to the point ll, ol, given in polar coordinates, its nth power represents (§2) the vector from to the point 1 1, ndl and hence is cos nd + i sin nd. 26 THEORY OF EQUATIONS [Ch. ll 11. Cube Roots. To find the cube roots of a complex number, we first express it in the trigonometric form (6). For example, 4 a/2 + 4 V2 i = 8 (cos 45° + i sin 45°). If it has a cube root of the form (6), then, by (8), r^ (cos Se + isinSd) =8 (cos 45° + i sin 45°). Their moduli r^ and 8 must be equal, so that the positive real number r equals 2. Since 3 6 and 45° have equal cosines and equal sines, they differ by an integral multiple of 360°. Thus 6 = lo° + k' 120° (k an integer). Since in (6) we may replace dhy 6 -^ 360° without changing a + hi, we ob- tain just three distinct cube roots (given hy k = 0, 1, 2): 2 (cos 15° + i sin 15°), 2 (cos 135° + i sin 135°), 2 (cos 255° + i sin 255°). EXERCISES 1. Verify that the last two numbers equal the products of the first number by w and w*, given at the end of § 8. 2. Find the tliree cube roots of —27; those of — i. 3. Find the three cube roots of — § + ^ Vs i. 12. nth Roots. Let p be a positive real numljcr. As illustrated in § 11, it is evident that the nth roots of p (cos A + i sin A) are the prod- ucts of the 7ith roots of cos A -{- i sin A by the positive real nth root of p. Let an nth root of cos A -}- i sin A be of the form (6). Then, by (8), r"(cos nd + i sin 7id) = cos A -\- isiriA. Thusr" = 1, r = 1, and nO = A + A- • 360°, where k is an integer. Thus n distinct nth roots of cos A -\- i sin A are given by ._. A-\-k- 360° , ■ ■ A+k' 360° ,, ^ , ., (9) cos h^sm ■ (A: = 0, 1, . . . , w— 1), n n whereas k = n gives the same root as ^ = 0, and k = 7i + 1 the same root ask = 1, etc. Hence any number 9^ has exactly n distinct wth com- plex roots. EXERCISES 1. Find the five fifth roots of - L 2. Find the nine ninth roots of L Which are roots of .r' = 1? 3. Simplify the trigonometric forms of the four fourth roots of unity. Check the result by factoring x'' — 1. 131 COMPLEX NUMBERS 27 13. Roots of Unity. (10) By (9) the n distinct nth roots of unity are (/c = 0, 1, . . . , n- 1), 2kT , . . 2 kit cos h I sin n n where now the angles are measured in radians (an angle of 180 degrees equals tt radians, where tt = 3.1416, approximately). For k = 0, (10) reduces to 1, which is an evident nth root of unity. For /b = 1, (10) is (11) 27r , . . 27r COS \- I sni — . 71 n By DeMoivre's Theorem (§ 10), the general number (10) equals the A;th power of r. Hence the n distinct nth roots of unity are (12) r, r2, r*, . . . , r"-i, r" = 1. I The n complex numbers (10), and therefore the numbers (12), are rep- resented geometrically by the vertices of a regular polygon of n sides inscribed in the circle of radius unity and center at the origin with one vertex on the a:-axis (Fig. 14). Fig. 14 Fig. 15 For n = 3, the numbers (12) are co, or, 1, shown in Fig. 13. For n = 4, we have r = cos 7r/2 + i sin t/2 = i. The fourth roots of unity (12) are i, i- = — 1, i' = —i,i^ = 1. These are represented by the vertices of a square inscribed in a circle of radius unity (Fig. 15). EXERCISES 1. For 71 = 6, r = — w^. The sixth roots of unity are therefore the three cube roots of unity and their negatives. Check by factoring x^ — 1. 2. From the point representing a + hi how do you obtain that representing — (a + hi)l Hence derive from Fig. 13 and Ex. 1 the points representing the six sixth roots of unity. 3. Which powers of a ninth root (11) of unity are cube roots of unity? 28 THEORY OF EQUATIONS ICh. II 14. Primitive nth Roots of Unity. An nth root of unity is called primitive if no power of it, with a positive integral exponent less than n, equals unity. Since only the last one of the numbers (12) equals unity, the number r, given by (11), is a primitive nth root of unity. For n = 4, both i and —i are primitive fourth roots of unity, while 1 and —1 are not. Just as i^ = —1 and i* = +1 are not primitive fourth roots of unity, so r* is not a primitive wth root of miity if k and n have a common divisor d {d > 1). Indeed, n k (r^)d = (r")T= 1, whereas n/d is a positive integer less than n. But if k and n are relatively prime, i.e., have no common divisor exceeding unity, r^' is a i3rimitive nth root of unity. To prove this, we must show that (r*)' ?^ 1 if Hs a posi- tive integer less than n. Now, by De Moivre's Theorem, 2klT , . . 2klTr n' = cos- H I sin 71 n If this were unity, 2 klir/n would be a multiple of 2 r, and hence kl a multiple of n. Since k is relatively prime to n, the second factor I would be a multiple of 7i, w^hereas < I < n. Hence the primitive nth roots of unity are those of the numbers (12) whose exponents are relatively prime to w. EXERCISES 1. The primitive cube roots of unity are w and ur. 2. For r given by (11), the primitive ?/th roots of unity are (i) for n = G, r, r^; (ii) for 71 = 12, r, r^, r^, r". 3. For n a prime, any 7ith root of unity, other than 1 , is primitive. 4. If r is a primitive 15th root of unity, r'', r^, r', ?•'- are the primitive oth roots of unity, and r^, r'"' are the primitive cul)e roots of unity. Show tliat their 8 prod- ucts by pairs give all of the primitive 15th roots of unit}'. 5. If n is the product of two primes p and q, there are exactly (p — l){q — 1) primitive nth roots of unity. 6. If p is any primitive nth. root of unity, p, p-, p^, . . . , p" are distinct and give all of the ?ith roots of unity. Of these, p'^ is a primitive ?ith root of unity if and only if k is relatively prime to 7i. 15. Imaginary Roots Occur in Pairs. The roots of x^ + 2 ex + d = are (13) -c+Vc'-d, -c-Vc"- d. § 16] COMPLEX NUMBERS 29 If c and d are real, these roots are both real or are conjugate imaginaries. The latter case illustrates the following Theorem. If a and b are real numbers, 6 5^ 0, and if a -\- bi is a root of an equation with real coefficients, then a — hi is a root. Let the equation be f(x) = 0. Divide /(.r) by (14) (x - a)2 + 62 = (a: - a - bi) (.c - a + bi) until we reach a remainder rx -\- s oi degree less than the degree of the divisor in x. Evidently r and s are real. If the quotient is Q{x), we have f(x) = Q{x) 1 (x - a)2 + 6^1 -\-rx-\-s, identically in x (Ex. 6, p. 15). Let x = a -}- bi. Since this is a root of f{x) = 0, we see that = r(a + bi) + s, = ra + s, = rb. Since b ^ 0, we have r = and then s = 0. Thus/(.r) has the factor (14), so that/(.r) = has the root a — bi. 16. t Generalization of the theorem in §15. The sum of the roots (13) oi x~ -\- 2 ex -\- d = equals the negative of the coefficient 2 c of a;, and their product equals the constant term d. It follows that 2 + i and — 2 are the roots of z^' -iz- 4-2 i = 0, and that 2 — i and — 2 are the roots of z^-\-iz-4:-^2i = 0. We have here an illustration of the following Theorem. If a and b are real nu7nbers and if a -\- bi is a root off(z) = 0, then a — bi is a root of g(z) = 0, where g{z) is obtained from the -polynomial f{z) by replacing each coefficient c + di by its conjugate c — di. Consider any term (c + di)z'' of f(z). Replace z hy x -\- yi, where x and y are real. The term {c + di)(x + yi)^ of f(x + yi) has as its conjugate imaginary the product (c — di)(x — yi)'' 30 THEORY OF EQUATIONti [Ch. ii of the conjugates of the factors of that term (§ 5). But the new product is a term of g{x — yi). Hence the latter is the conjugate A — Bi of j{x -\- iji) ^ A -\- Bi, where A and B are polynomials in x and y \\ith real coefficients. Take x = a, y = h. Then A=B = by hypothesis. Hence g{a—hi) = 0. EXERCISES l.f The theorem in § 15 is a corollary to that in § 16. 2. Solve x^ - 3 .c2 - 6 .c - 20 = 0, with the root - 1 + V^ 3. Solve .r* -^x^-\-bx'-2x-2 = 0, with the root 1 - i. 4. Find the cubic e(iuation with real coefficients two of whose roots are 1 and 3 + 2i. 5.t Given that x^ + (1 — O-i-' + 1=0 has the root i, find a cubic equation with the root —i. Form an equation with real coefficients whose roots include the roots of these two cubic equations. 6. If an equation with rational coefficients has a root a + v 6, where a and h are rational, but V6 is irrational, it has the root a — \/h. [Use the method of § 15.] 7. Solve x^ - 4 x^* + 4 .r - 1 = 0, with the root 2 + Vs. 8. Solve x3 - (4 + V3).c2 + (o + 4 V3)x - 5 Vs = 0, with the root Vs. 9. Solve the equation in Ex. 8, given that it has the root 2 + i. 10. What cubic equation with rational coefficients has the roots |, ^ + V2 ? CHAPTER III Algebraic and Trigonometric Solution of Cubic Equations 1. Reduced Cubic Equation. If in the general cubic equation (1) x^ -\- hx^ -\- ex -\- d = 0, we set X = y — 6/3, we obtain a reduced cubic equation (2) y' + py + q = 0, where (3) P = ^-3' ^ = ^~3+'27'* A geometrical interpretation of this process was given in Ex. 5, p. 10. We shall find the roots yi, tjo, y^ of (2). Then the roots of (1) are (A\ h h h (4) Xi = yi- ^, X2 = y2- ^y X3 = yz - ^• 2. Algebraic Solution of Cubic Equation (2). We shall employ a method essentially that given by Vieta * in 1591. We make the substi- tution (5) y = .-l in (2) and obtain 3 Multiplying each member by z^, we get (6) z' + qz'-^ = 0. Solving this as a quadratic equation for 2', we obtain (7) ^=-|±Vg, « = (fj+(f * Opera Math., IV, published by A. Anderson, Paris, 1615. 31 32 THEORY OF EQUATIONS [Ch. ill By Ch. II, § 11, any number has three cube roots, two of which are the products of the remaining one by (8) co=-i+iV3z, a;2=-l-iV3 2. Since (_| + VS)(-|-VS)=(-|J. we can choose particular cube roots (9) A = \/-^+VR, b = \/-'^-VR, such that AB = — p/3. Then the six values of z are A, 01 A, 0)^ A, B, a:B, co25. These can be paired so that the product of the two in each pair is — p/3: AB = -p/3, uA ' orB = -p/S, oi-A ' uB = -p/3. Hence with any root z is paired a root equal to — p/{S z). By (5), the sum of the two is a value of y. Thus the three roots of (2) are (10) yi = A+B, y. = co.4 + oi'-B, y^ = o>-A + co5. These are known as Cardan's formulce for the roots of a reduced cubic equation (2). The expression ^4. + 5 for a root was first published by Cardan in his Ars Magna, 1545, although he had obtained it from Tartaglia under promise of secrecy. EXERCISES 1. For ?/ -loy- 126 = 0, y = z + o/z and 28 - 1262^ + 125 = 0, 2^ = 1 or 125, 2 = 1, co, 0,2, 5, 5 «, 5 «'. The first three 2's give the distinct 7/'s : G, to + 5 co-, oj- + 5 w. 2. Solve 1/ - 18 2/ + 35 = 0. 3. Solve x^ + 6 x^ + 3 x + IS = 0. 4. Solve y^-2y + 4: = 0. 5. Solve 28 x' + 9 x^ - 1 - 0. 6. Using w^ + o) + 1 = 0, show from (10) that yi + 2/2 + j/3 = 0, 7/i?/2 + 2/12/3 + 2/22/3 = p, 2/12/22/3 = —q. 7. By (3), (4) and Ex. 6, show that, for the roots of (1), Xi + X2 + X3= —b, X1X2 + X1X3 + X2X3 = c, X1X2X3 = — d §31 CUBIC EQUATIONS 33 3. Discriminant. By (10) and w^ = 1, yi-y2= (i-co)(A-co25), yi-yz= -co^ (1 - a))(A - coB), ?/2 - ?/3 = CO (1 - C0)(A - B). To form the product of these, note that co^ = 1 and, by (8), (1 - co)3 = 3 (co2 - co) = -3 VS Z. Since the cube roots of unity are 1, w, co^, we have x^ — 1 = (x — l)(.r — aj)(x — co^), identically in x. Taking x = A/B, we see that (11) A'-B^= (A - B){A - co5)(A - co25). The left member equals 2 VTi by (9). Hence (12) (2/1 - y^) (yi - 2/3) (2/2 -ys)=QV3VR i. The product of tlie squares of the differences of the roots of any equation in which the coefficient of the highest power of the variable is unity shall be called the discriminant of the equation. Thus the discriminant is zero if and only if two roots are equal, and is positive if all the roots are real. In view of (12) the discriminant A of the reduced cubic equation (2) has the value (13) ^ A = -108/^ = -4793-2752. By (4), Xi — X2 = yi — yo, etc. Hence the discriminant of the general cubic (1) equals the discriminant of the corresponding reduced cubic (2). By (3) and (13), (14) ^ = 18 bed -4:¥d-{- b^c^ - 4 c^ -27 d\ It is sometimes convenient to employ a cubic equation (15) ax^ -}- bx^ + C.T + d = 0, in which the coefficient of x^ has not been made unity by division. The product P of the squares of the differences of its roots is evidently derived from (14) by replacing 6, c, d by b/a, c/a, d/a. Thus (16) a^P - 18 abed - 4:¥d -\- fcV - 4 ac^ - 27 aW. This expression (and not P itself) is called the discriminant * of (15). * Some writers define — ^V «*^ to be the discriminant of (15) and hence —2V ^ as tkat of (1). On this point see Ch. IV, § 4. 34 THEORY OF EQUATIONS (CH.m 4. Theorem. A cubic equation with real coefficients has three distinct real roots, a single real root, or at least two equal real roots, according as its discriminant is positive, negative or zero. It suffices to prove the theorem for a reduced cubic equation (2) in which p and q are real. First, let A = 0. By (13), R = 0. Using (8), we find that the roots (10) are (17) A+B, -HA-{-B)±UA-B)V3i. But .4 and B, in (9), may now be taken to be real, since R = 0. li R > 0, A 9^ B and A + B is the only real root. U R = 0, then A = B and the roots are real and at least two are equal. Next, let A > 0, so that R <0. Since —lq+ ^/R is an imaginary num- ber it has (Ch. II, § 11) a cube root of the form A = a -]- jSi, where a and 13 are real and /3 ?^ 0. Then (Ch. II, § 16) 5 = a — ^i is a cube root of — 2? ~ \^R. For these cube roots, the product AB is real and hence equals — p/3, as required in § 2. Hence 2/1 = 2 a, y.2= -a- ^VS, 2/3 = -a + /3 VS. These real roots are distinct since A f^ 0. EXERCISES Find by means of A the number of real roots of 1. ?/3 - 15 y + 4 = 0. 2. 7f - 27 y + 54 = 0. 3. x^ + Ax^ - llx + 6 = 0. 4. Using A = (.Ti — XiY (xi — XsY {x^ — XsY, show that, if .ri and .T2 are con- jugate imaginaries and hence X3 real, A < 0; if the x's are all real and distinct, A > 0. Deduce the theorem of § 4. 5. Deduce the same theorem from Ch. I, § 9. 5. Irreducible Case. When the roots of a cubic equation are all real and cUstinct, R is negative (§4), so that Cardan's formulae present their values in a form involving cube roots of imaginaries. This is called the irreducible case.* We shall derive modified formulae suitable for numer- ical work. Since any complex number can be expressed in the trigono- metric form, we can find r and d such that (18) -^,q-\-VR = r (cos d + i sin 6). * This term is not to be confuted with "irreducible equation." §61 CUBIC EQUATIONS 35 In fact, the conditions for this equahty are Hence (19) — 1 g = r COS 6, R = —r^ sin^ 9. J.2 = J.2 (cos2 d + sin2 d) =12^ - R = ^, = V/^' -^ = i^-\/i? Since i2 is negative, p is negative and r is real. Since R < 0, the value (19) of cos d is numerically less than unity. Hence d can be found from a table of cosines. The complex number conjugate to (18) is (20) - 1 g - V^ = ^(cos d - i sin 6). The cube roots of (18) and (20) are d + m- 360'^ _^ . . e + m- 360° ] , ^ . _, 5 it I sm 5 (m = 0, 1,2). -^[cos For a fixed value of m the product of these two numbers is — p/3. Hence their sum is a root of our cubic equation. Thus if R is negative, the three distinct real roots are (21) 2Y-^cos ^ (m = 0, 1, 2). EXERCISES 1. Solve the cubics in Exs. 1, 2, page 34. 2. Solve ?/ - 27/ - 1 = 0. 3. Solveif- 7y + 7 = 0. 4. Find constants r and s such that y^ + py + q = tzt ^^ (2/ + '^^^ - s(y + rfl identically in y. Hence solve the reduced cubic equation. 6.t Algebraic Discussion of the Irreducible Case. Avoiding the use of trigonometric functions, we shall attempt to find algebraically an exact cube root x -\- yi oi a -{- hi, where a and b are given real numbers, b 7^ 0. We desire real numbers x and y such that (x + yiy = a + bi, 36 THEORY OF EQUATIONS [Ch. Ill whence a;^ — 3 xy- = a, 3 x^y — y^ = b. Thus 2/ 5^ and we may therefore set x = sy. Hence {s^-Ss)y' = a, (3s2-l)?/3 = 6 Eliminating y^, we get Set s = t -\- a/b. We obtain the reduced cubic equation The R of (7) is here — k-. Thus Cardan's formute for the roots i involve A = \/~ k + ki = \/^ . 0. In the irreducible case we may avoid Cardan's formula^ and the simplifications in § 5. The same final results are now obtained by a direct solution based upon the well-known trigonometric identity cos 3 a: = 4 cos^ x — S cos x. * H. Weber and J. Wellstein, Encyklopddie der Elementar-Mathemalik, I, cd. 1, p. 325; ed. 2, p. 373; ed. 3, p. 364. §71 CUBIC EQUATIONS 37 This may be written in the form ^ — I z — lcos3x = {z = cos x). To transform cubic (2) into this one, set y = nz. Thus n^ nr The two cubic equations are identical if — n3 = \/4^. eos3x = :i«.^ 2 ■ V 2f Since /2 < 0, p < and the value of cos 3 a: is real and numerically < 1. Hence we can find 3 x from a table of cosines. The three values of z are then cos X, cos {x + 120°), cos {x + 240°). Multiplying these by n, we get the three roots y. Example. For if — 2y — 1 = Q, we have n2 - 8/3, cos 3 X - V27/32, 3 x = 23° 17' 0", cosx = 0.99084, cos (x + 120°) = -0.61237, cos {x + 240°) = -0.37847, y = 1.61804, - 1, - 0.61804. EXERCISESt Solve by the last method 1. ?/ - 7 2/ + 7 - 0. 2. x3 + 3 x2 - 2 X - 5 = 0. 3. x3 4- .x2 - 2 X - 1 = 0. 4. x3 + 4 x2 - 7 = 0. 5. The cubic for Mn § 6 has three real roots; in just three of the nine sets of solutions X, y, both are real. CHAPTER IV Algebraic Solution of Quartic Equations 1. Ferrari's Method. Writing the quartic equation (1) x" + bx^ + cx^ -\- dx + e = in the equivalent form (x'' + i bxy = (i ^2 _ c).^2 -dx-e and adding {x- + | bx)2j + I ?/^ to each member, we get (2) (x2 + Pa; + i yY ={lb'~-c + y)x' + {\hy - d)x + | y^ _ g. We seek a value yx of y such that the second member of (2) shall be the square of a linear function of x. For brevity, write (3) ¥-4.c + 4:y, = t\ We here assume that t f^ (c/. Exs. 3, 4, p, 40). We therefore desire that (4) i ex' + (1 by, -d)x^-\ yi^ - e = (| te + iMnl^l The condition for this is that the terms free of x be equal: 6^ — 4 c + 4 ^1 Hence yi must be a root of the resolvent cubic equation (6) ?/ - ci/2 -\. (bd- 4. e)y - 6^6 + 4 ce - d^ = o. After finding (Ch. HI) a root yi of this cubic equation, we can easily get the roots of the quartic equation. In view of (2) and (4), each root of the quartic equation satisfies one of the quadratic equations ,. [x-' + lib- t)x + hy,- {h by, - d)'t = 0, ^ ^ U- + Hb + t)x-{-hyi + {h by, - d)/t = 0. 38 §2.31 QUART I C EQUATIONS 39 EXERCISES 1. For x^ + 2 x' - 12 a;2 - 10 X + 3 = 0, show that (6) becomes 2/^ + 12 2/2 — 32 ?/ — 256 = 0, with the root iji = —4, and that (7) then become x2 + 4 X - 1 = 0, x2 - 2 X - 3 = 0, with the roots —2 ± V5; 3, —1. 2. Solve x^ - 2 x3 - 7 x2 + 8 X + 12 = 0. 3. Solve x^ - 8 x^ + 9 x2 + 8 X - 10 = 0. 2. Relations between the Roots and Coefficients. Let Xi and X2 be the roots of the first quadratic equation (7), Xs and Xi those of the second. The sum and product of the roots of x^ -{- Ix + m = are —I and m respectively (Ch. II, § 16, or Ch. VI, § 1). Hence .-^1 + 0:2= -hQj - t\, X1X2 = 1 2/1 - (^ hyi - d)/t, (8) \xz-\- Xi= - \ (6 + 0; ^32:4 = I yi + {\ hiji -d)/L Using also (5), we find at once that (9) xi -j- Xi + X3 + Xi = -b, XxXiXzXi = \y^ - (| y^ -e) = e, (10) X1X2 + a;irr3 + 2:1X4 + 0:2X3 + X2X4 + X3.T4 = X1.T2 + (xi + X2) (xs + X4) + X3X4 = c, (11) X1X2X3 + X1X2X4 + X1X3X4 + X2X3X4 = XiX2(x3 + X4) + X3X4(xi + X2) = — d. It follows from Ex. 3, p. 40 that (9)-(ll) hold also when there is no root ^1 for which t ^ 0. For any quartic equation (1), the sum of the roots is —b, the sum of the products of the roots two at a time is c, the siwi of the products three at a time is — d, the product of all four is e. A proof based upon more fundamental principles is given in Ch. VI, § 1. 3. Roots of the Resolvent Cubic Equation. These are (12) yi = XiXo + X3X4, 2/2 = X1X3 + X2X4, 2/3 = X1X4 + X2X3. The first relation follows from (8). If, instead of yi, another root of (6) be employed as in § 1, quadratic equations different from (7) are ob- tained, such however that their four roots are Xi, X2, X3, X4, paired in a new way. This leads us to expect that 2/2 and 2/3 in (12) are the remaining roots of cubic (6). To give a formal proof, note that, by (9)-(ll), 40 THEORY OF EQUATIONS ICh. iv (13) 2/1 + 2/2 + 2/3 = C, 2/i2/2 + 2/12/3+ 2/22/3 = (a:i+X2 + 0:3 + X4) (0:13:22:3 + • • • + 0:20:30:4) - 4 0:10:20:30:4 = 6d — 4 e, 2/12/22/3 = (0:10:20:3 +•••)" + o-iO-20-30-4K^i + • • 0^ - 4 (0:10:2+ " • •)! = d2_^e(6- - 4 c). Hence by Ex. 7, p. 32, or by Ch. VI, §1, yi, 1/2, 2/3 ^re the roots of (6). EXERCISES 1. Why is it sufficient for the last proof to verify merely the first two relations (13)? 2. In Lagrange's solution of quartic (1), we begin by sho\\dng that the num- bers (12) are the roots of cubic (6) by using (13) and the theorem of § 2. Let a root iji be found. Then we obtain .riX2 = Zi and .r3X4 — Zj as the roots of 2^ — yiZ + e = 0. Next, Xi + xz and .T3 + Xi are found from (xi + Xi) + (xa + X4) ^ -b, Z2{xi + X2) + Zi{x3 + Xi) = -d. Hence Xi and X2, Xs and Xi are found by solving quadratic equations. Give the details of this work. 3. If the t corresponding to each root of (6) is zero, equation (1) has all its roots equal. For, by (3), the y's all equal c — lb-. By (13), 3 ?/i = c, 3 yi^ = bd — 4:6. Hence c = §6^, 5? b* = bd — 4:6. Eliminating e between the latter and {I ¥Y = yi^ = ¥e — 4ce -{- d^, which follows from yi = c — Ib^ and (13), we get (t\ ¥ -d)^ = 0. Then (1) equals {x + { b)' = 0. 4. Prove that Ex. 3 is true by showing that t- = (xi + X2 — X3 — X4)-. 5. Solve :i^ -\- px + q = ip 9^ 0) by choosing c so that the quartic (x - c)i.r' + px + q) = shall have as its resolvent cubic (6) one reducible to the form z^ = constant. Here (6) is y^ - py- + c{cp + ^q)y - c-p"^ — 2 cpq — q- -{- <^q = 0. To remove the second term, set ?/ = z + p/3. We get z^ -\- Az -\- (?q — I C'p- — cpq — ([• — ^-y p' = 0, where A = p& -\-2tcq — \ p^. We are to make .4=0; thus §41 QUARTIC EQUATIONS 41 since c^ + cp + 5 = 36 cR/p'^. Our quartic has the root c and hence by (81), with b replaced by —c, also the root Kc + — c, where i^ = c^ — 4 p + 4 y. Hence the given cubic has the root i(i-c) = Vz- ip + ic^-lc, which may be reduced to Cardan's form (Atner. Math. Monthly, 1898, p. 38). 4. Discriminants. Replacing y by F + c/3 in (6), we get (14) Y'-{-PY-\-Q = 0, in which (15) P = 6d - 4 e - i c2, Q = -¥e + lhcd-\-^ ce-(P - /f c^. Hence (Ch. Ill, § 3), (1/1 - y^YiUi - ysKy^ - y.Y = -4^^ _ 21 Q\ By (12) Vi - yi = {xi - Xi){x2 - xz), (16) yi - tjz = {xi - a;3)(a:;2 - Xi), 2/2 - 2/3 = (xi - a;2)(x3 - 3:4). The discriminant A of the quartic (1) is defined to be (17) A = {xi — 0:2)2(3:1 — xsYixi — Xiy{x2 — X3y(x2—Xiy{x3—X4y. It therefore equals the discriminant of (14) : (18) A = -4P3_27Q2. Any quartic equation and its resolvent cubic have equal discriminants. Some writers define the discriminant of (1) to be A/256 and that of a cubic to be —A/27. In suppressing these numerical factors, we have spared the reader a feat of memory, simplified the important relation between the discriminants of a quartic equation and its resolvent cubic, and moreover secured uniformity with most of the books to which we shall have occasion to refer the reader. Finally, we note that in applications to the theory of numbers, the insertion of the numer- ical factors is undesirable and in special cases unallowable (c/. Bull. Arner. Math. Soc, vol. 13, 1906, p. 1). EXERCISES 1. For ax* -{- bx^ -{- cz^ -\- dx -\- e = 0, P = p/a'^, Q = q/a^, where p = bd — 4iae — c^/3, q = — 6^e + 5 bed -\- f ace — ad^ — ^j c*. The discriminant is defined to be a® A; it equals — 4 p' — 27 q^. 42 THEORY OF EQUATIONS ICh. IV 2. If X and y are interchanged in f = ax* + hxhj + cx-if- + c^.f//^ + ei/, a function is obtained which may also be derived from / by merely interchanging a with e, and b with d. Show that the latter interclianges leave p, q and the dis- criminant unaltered. 3. Since the sum Fi -f F2 + Fs of the roots of a reduced cubic is zero, Fi = HFi - F2) + § (Fi - F3), . . . , and any root and hence any function of the roots is expressible as a function of the differences of the roots. Thus P and Q in (15) are functions of Fi — F2, etc., and hence of yi — y^, etc. Using (16), show that p and q equal polynomials in the differences of Xi, . . , , X4. 4. When x is replaced by x + ty, let / of Ex. 2 become /' = a'x" + h'xhj + • • • + e'y\ Show by Ex. 3 that p and q equal the corresponding functions V' = h'd' - 4 a'e' - c'V3, q' = -b'¥ + • • • . 5. The results in Exs. 2 and 4 are special cases (used in a short proof) of a gen- eral theorem : When x is replaced by Ix + my and y by rx + sy, let / become /'. Then, using the notations of Ex. 4, we have p' = D*^p, q' = D^q, where D = Is —mr. Hence p and q are called invariants of /. Verify the theorem for the case when x is replaced by Ix, y by y. 6. The discriminant is an invariant and the factor is D^'^. 7. Using oax* + 4 aix^y + 6 aixh/ + 4 asxy^ + a^y* in place of the former /, show that p=— 4/, g=16J, where / = 0004 — 4 OiOs "h 3 02", J = 000204 + 2 010203 — aoOa" — 01-04 — 02^. In (14) set F = 2 z/a; then z^ — Iz + 2J = 0. The discriminant is 256 (P-27J2). 5. Descartes' Solution of the Quartic Equation. Replacing x bj' 2 — 6/4 in the general quartic (1), we obtain a reduced quartic equation (19) s!" -j- qz" + rz -\- s = 0, lacking the term with ^. We shall prove that we can express the left member of (19) as the product of two quadratic factors * (22 -\- 2 kz -\- l){z'' - 2kz + m) = z' + (I -{- m - 4: k'-)z- -{- 2 k{m - l)z -f Im. * If the coefficients of z be denoted by k and —k (jxs is usually done), the expres- Bions (23) for the roots must be divided by 2. But the identification with Euler'a Bolution is then not immediate. §61 QUARTIC EQUATIONS 43 The conditions are I -\- m — 4:k^ = q, 2 k {m — l) = r, Im = s. If k 7^ 0, the first two give Then Im = s gives (20) 64 A:« + 32 qk^ + 4 (g^ - 4 s)k^ - r^ = 0. The latter may be solved as a cubic equation for k"^. Any root k^ 9^ gives a pair of quadratic factors of (19) : (21) 22±2A;2 + i5 + 2A;2T^. The 4 roots of these two quadratic functions are the 4 roots of (19). If q = r = s = 0, every root of (20) is zero and the discussion is not valid; but the quadratic factors are then evidently z'^, z^. EXERCISES 1. For 2^ - 3 ^2 + 6 2 - 2 = 0, (20) becomes 64 A;« - 3-32 k^ + 4-17 k"" - 36 = 0. The value A;^ = 1 gives the factors 2- + 22 — 1, 2^ — 22 + 2, with the roots -1±V2, 1±V^. 2. Solve 2* - 2 22 - 8 2 - 3 = 0. 3. Solve 2^ - IO22 - 2O2 - 16 = 0. 4. Solve x-* - 8 x^ + 9 x^ + 8 X - 10 = 0. 6. Symmetrical Form of Descartes' Solution. To obtain this sym- metrical form, we use all three roots A;!^, A'2^, k-^ of (20). Then k^ + ki + ^32 = -1 5, k^k-^-k^"- = rV64. It is at our choice as to which square root of k^ is denoted by -\-k\ and which by — fci, and likewise as to -^.k-i, ±ikz. For our purposes any choice of these signs is suitable provided the choice give (22) kxk'.kz = -r/8. Let ki 5^ 0. The quadratic function (21) is zero for k = ki if 44 THEORY OF EQUATIONS (Ch. iv Hence the four roots of the quartic equation (19) are (23) /vi + ^-2 + A-3, ki-k2-kz, -/u + A-2- A-3, -k.-ko + kz. Writing ¥ = y, we see that, if yi, y-i, y^ arc the roots of (24) Uy-" + 32g!y2 + 4 (r/ - 4s)?/ - r^ = 0, then the roots of (19) are the four values (25) z = V^i + Vy<> + V?73, obtained by using all of the combinations of the square roots for which, by (22), (26) V^ VJ^> VFs = - V8. We have deduced Euler's solution (Ex. 1) from Descartes'. EXERCISES 1. Assume with Euler that quartic (19) has a root of the form (25). Square (25), transpose the terms free of radicals, square again, and show that 2« - 2 (i/i + 2/2 + yz) 2- - S 2 Vyi Vy. Vys + (^/i + ^2 + I/s)" - 4 (i/i7/2 + yiys + yiyd = 0. From the relations obtained by identifj'ing this with (19), show that yi, y^, ys are the roots of the cubic (24) and that (26) holds. 2. Solve Exs. 1-4 of the preceding set by use of (23). 3. In the theory of inflexion points of a plane cubic cuiwe occurs the quartic equation z* — Sz- — t Tz — i\ S- = 0. Show that (24) now becomes (-^■-' -(0-(f)' and that the roots of the quartic are where the signs are to be chosen so that the product of the three summands equals + T/Q. Here lo is an imaginary cube root of miity. 4. The discriminant A of the quartic equation (19) equals the quotient of the discriminant D of (24) by 4^ For, the six differences of the roots (23) are 2 {ki ± A-2), 2 (A-i ± /:3), 2 (^-2 ± k,). Thus A - 4« L, where L = (A;,2 - ki^Yiki'' - ki^Yiki- - kz^y = {y, - y^y-{yi - yzfiy^ - yz)\ By definition, D = 64^1. Hence D = 4« a. §7] QUARTIC EQUATIONS 45 5. Give a second proof of Ex. 4 by setting y = 2/4 in (24) and then z =Y — 2 q/3. We obtain (14), in which now b = 0, c = q, d = r, e = s. The discriminant of (14) equals A. Hence A = (^i — Z2)- • • • = 4^L = Z)/4«. 6. If a quartic equation has two pairs of conjugate imaginary roots, its dis- criminant A is positive. Hence, if A < 0, there are exactly two real roots. 7. Theorem.* A quartic equation (19) with q, r, s, real, r ^ 0, and with the discriminant A, has 4 distinct real roots if q and 4 s — 5^ are negative and A > 0, no real root if q and 4 s — 5^ are not both negative and A > 0, 2 distinct real and 2 imaginary roots if A < 0, at least 2 equal real roots ?/ A = 0. Since the constant term of the cubic equation (24) is negative, at least one of its roots is a positive real number. Let, therefore, yi > 0, so that 2/22/3 > 0. Thus ki = Vyi is real. There are four possible cases to consider. (a) 7/2 and yz positive. Then each kj = V^/yis real and the roots (23) of the quartic equation are all real. (p) y-i = yz < 0. Then A-o = ±^-3 is a pure imaginary. If k2 = kz, the first two roots (23) are imaginary and the last two are real and equal. If A'2 = — kz, the reverse is true. (c) 2/2 and 2/3 distinct and negative. The roots (23) are all imaginary. {d) 2/2 and yz conjugate imaginaries. Then ki is imaginary and conju- gate with either A'3 or — As, so that one of the numbers A'2 + A3 and A2 — A:3 is real and the other imaginary. Just two of the roots (23) are real. Now, if A = 0, at least two ?/'s are equal by Ex. 4 of the last set. Thus we have case (h) or a special case of (a). In either case, the quartic has at least two equal roots, by (17), and they are real in both cases. Henceforth, let A ?^ 0. By the same Ex. 4, A has the same sign as the discriminant D of the cubic equation (24). If A < 0, we have case {d). Finally, let A > 0, so that yi, yo, ys are real. If q is negative and q^ — 4: s is positive, equation (24) has alternately positive and negative coefficients and hence has no negative root, so that we have case (a). But if q and 4 s — g- are not both negative, the coefficients are not alter- nately positive and negative, so that the roots yi, y-i, ys are not all posi- tive,** and we have case (c). * Proved by Lagrange by use of the equation whose six roots are the squares of the differences of the roots of (19), Resolution des equations numeriques, 3d ed., p. 42. ** The coeflacients are — (//i + 2/2 + ys), yiU'i + 2/i2/3 + UzUi, — yiyiUz- 46 THEORY OF EQUATIONS [Ch. iv EXERCISES 1. Apply this theorem to the quartic equations in Exs. 1-4, p. 43. 2. Verify that a quartic equation (19) with two pairs of equal imaginary roots has r =• 0. Deduce the last case of the theorem. 3. Why does the theorem imply its converse? t CHAPTER V The Fundamental Theorem of Algebra l.f Theorem. Evenj equation with complex coefficients (1) Kz) = 2» + a,2«-i + . . . + a„ = has a complex {real or imaginary) root. For n = 2, 3, or 4, we have proved this theorem by actually solving the equation. But for n = 5, the equation cannot in general be solved algebraically, i.e., in terms of radicals. We shall first treat the case in which all of the coefficients are real. Relying upon geometrical intuition, we have seen in Exs. 3, 5, p. 14, that there is a real root if n is odd, or if both n is even and a„ is negative. But, as in the cases of certain quadratic equations and s'* + 2^ + 5 = 0, an equation of even degree may have no real root. No proof of the theorem for all cases has been made by such elementary methods. The proof here given of the theorem that any equation with real co- efficients has a complex root is essentially the first proof by Gauss (1799 and simphfied by him in 1849). We are to prove that there exists a complex number z = x -\- yi such that f(z) = 0. We may write (2) Kz) = X + Yi, where X and Y are polynomials in x and y with real coefficients. We are to show that there exist real numbers x and y such that (3) X = 0, Y = 0. For example, if fiz) = 2" - 4 s-'' + 9 2' - 16 2: + 20, then X = x" - 6 xY + ?/ - 4 a;3 + 12 xif + 9 x^ - 9 ?/ - 16 a; + 20, hY = 2xhj -2 X7f - Qxhj + 2 ?/^ + 9 x?/ - 81/. The graph of F = is the x-axis {y = 0) and the graph (indicated by the dotted curve in Fig. 16, asymptotic to the lines x = 1 and ?/ = ± x) of 2 (x ^ 1)7/ = 2x3 - 6x2 + 9x - 8. 47 48 THEORY OF EQUATIONS [Ch. V Note that there is no real y for x between 1 and 1.73. Since X = is a quadratic equation in y"^, its graph is readily drawn. There is no real y ior x = 0.05 and 1.6 Fig. 16 and the intermediate values. Cases in wliicli the values of y- are positive and rational are X -4 - 2 - 1 2 3 y2 5, 148 2.5, 54.5 2,25 4,5 1,8 1,26 The graphs cross at the points (0, 2), (0, —2), (2, 1), (2, —1), and the roots of f{z) = are z = ±2 t, 2 ± {. We shall employ also the trigonometric f onn of z : (4) z = r(cos 6 -\- i sin d), 5 11 FUNDAMENTAL THEOREM OF ALGEBRA 49 where ^6 <2t. Set t ^ tan ^ 6. Then 2t _ 2tan^ rT^~ sec2 1 ( 2f tan Thus - = 2 sin ^ ^ • COS ^0 = sin cos 9 sin 9 tan 6 1 + ^2 1 -f' r (1 + tiy z = 1 -{-f Hence by (1) and (2), (H-/-)"(X+yO=r"(l+^0'"+«i^"~Hl + ^0'"~'(l+^^)+ • • • + an (1+^2)". Expanding the terms on the right by the binomial theorem, we get (5) X = ,M^, Y= ^(^) Avhere F(t) is a polynomial in t of degree 2 n, and (7(0 a polynomial in t of degree less than 2 /?, each with coefficients involving r integrally. Each point {x, y), representing (Ch, II, § 8) a complex number z — X -\- iji having the modulus r, lies on the circle x"^ -{- y^ = r^ with radius r and center at the origin of the rectangular coordinate system. To find the points on this circle for which A" = or F = 0, we solve F{t) = or G{t) = (in which r is now a constant), and note that to each real root t corresponds a single real value of sin 6 and a single real value of cos 6, consistent with that of sin Q, and hence a single point {x = r cos d, y = r sin 0) . But an equation of degree 2 n has at most 2 n distinct roots (Ch. I, § 15). Since the degree of G{t) is less than that of the de- nominator of Y in (5), the root ^ =oo of F = must be considered in addition to the roots of G{t) = already examined; for t =cc, 6 = ir and the point is ( — r, 0). Thus neither X nor F is zero for more than 2n points of the circle with center at the origin and a given radius r. By proper choice of r, this circle will have an arc lying within any given region of the plane. Hence neither X Jior Y is zero at all points of a region of the plane. From (4) and DeMoivre's Theorem (Ch. II, § 10), we have ^k — yk (^(jQg ^.^ _J_ I gjjj ]^QY Hence, by (1) and (2), F=r" sin n^ + air "-1 sin (n— 1)0 + aor"-^ sin (71 — 2)0+ • • • +a„_irsin0. 50 THEORY OF EQUATIONS Let g be the greatest of the numerical values of aj, . . 1 D I denotes the numerical value of the real number D y=r»(sinn0 + Z)), \D\^g(^l + ^,+ • • provided r > 1. If c is a positive constant < 1 and if r > 1 + g/c, then I D I < c. Hence for all angles 6 for which sin nd is numerically greater than c, Y has the same sign as its first term ?"' sin 7i6 when r exceeds the constant 1 + g/c. In our example, we have Y = ?•■» sin 4 9 — 4 r^ sin 3 + 9 r- sin 2 — 16 r sin e. The limit 1 + 16/c for r exceeds 17 and is larger than is convenient for a drav/ing. But for r S 10, 4 16 Y = r*{sin4d + D), \D\= - + -+ — = 0.4 + 0.09 + 0.016. sin 30° 24', let C be the number of radians in 7° 36'. Thus c = sin 4 C. The positive angles d (d < 2 tt) for which sin 4 d exceeds sin 4 C numerically are those between C and | tt — C, between J tt + C and h TT — C, between ^ ir -{- C and f tt — C, , . . , between iir -\- C and 2 w — C. For any such angle 6 and for r = 10, F has the same sign as sin 4 e and hence is alternately positive and negative in these successive intervals, the solid \+ _/ arcs in Fig. 17. Denote by 0, 1, 2, . . . , 7 the \ / points on the circle with center at the origin and •• \ /^ TT ^ TT 4 TT ■^ ""^v.-- 4.^" ' radius 10 whose angles d are 0,-, — -, . . . , — ?-» ^- — ,S>^ 4 4 4 ^ respectively. Fig. 17 In the general case, denote by 0, 1, 2, ... , 2 n — 1 the points with the angles TT 2 X (2 n - 1) TT u, , , . . . , n n n on the circle with center at the origin and radius a constant r exceeding the above value 1 + g/c. Let nC be the positive angle < 7r/2 for which sin nC = c. We define the neighborhood of our A;th point of division on Taking c = 0.506 = sin 30° 2 / / / 1- 1 A 4I 1 1 10 § n FUNDAMENTAL THEOREM OF ALGEBRA 51 the circle to be the arc bounded by the points whose angles are kTr/n — C and kir/n + C. In Fig. 17 for our example with ri = 4, each neighborhood is indicated by a dotted arc. In the successive arcs (marked by solid arcs) between the neighborhoods, Y is alternately positive and negative, since it has in each the same sign as sin nd. It is easily seen that sin ^, sin 2 ^, . . . , sin nd are continuous functions of 6 (a fact presupposed in interpolating between values read from a table of sines). Since r is now a constant, Y is therefore a continuous function of 6, and has a single value for each value of d. But Y has oppo- site signs at the two ends of the neighborhood of any one of our points of division on the circle. Hence (as in Ch. I, § 12), Y is zero for some point \vithin each neighborhood, and at just one such point, since Y was shown to vanish at not more than 2 n points of a circle with center at the origin. We shall denote the points on the circle at which Y is zero by ^0, ^1, P2n- For our example, these points Po, . . . , Pi are given in Fig. 18, which shows more of the graph of F = than was given in Fig. 16, but now shows it with the Fig. 18 scale of length reduced in the ratio 4 to 1 (to have a convenient circle of radius 10). We have shaded the regions in which, as next proved, Y is positive. 62 THEORY OF EQUATIONS [Ch. v Let the constant r be chosen so large that X also has the same sign as its first term r" cos nd, for 6 not too near one of the values 7r/(2 n), 3 tt/ (2 n), 5ir/{2n), . . . , for which cos w^ = 0. Since these values correspond to the middle points of the arcs (01), (12), . . , , no one of them lies in a neighborhood of a division point 0, 1, ... . Now cos n0 = +1 or — 1 when 6 is an even or an odd multiple of ir/7i, respectively. Hence X is positive in the neighborhood of the division points 0, 2, 4, . , . , 2 n — 2 and thus at Po, -P2, P4, • • • , but negative in that of 1, 3, 5, . . . , 2 n — 1 and thus at Pi, P3, P5, • • • • We saw that Y is not zero throughout a region of the plane. Hence there is a region in which Y is everj^vhere positive (called a positive region), and perhaps regions in which Y is everywhere negative (called negative regions), while Y is zero on the boundary lines. In Fig. 18 for our example, there are three positive (shaded) regions, the two with a single point in common being considered distinct, and three negative (un- shaded) regions. Consider that part of the boundary of ^2^30 which lies inside the circle. At every point of it, Y is zero. Now X is negative at Pz and positive at P2 and hence is zero at some intermediate point a on this boundary. Hence at a both X and Y are zero, so that a represents a complex root (in fact, 2 i) of m = 0. To extend the last argument to the general case, let R be the part in- side our circle of a positive region having the points P2 h and P2 h+i on its boundary. The points of arc Po hPi h+\ may be the only boundary points of R lying on the circle (as for P2P3a and P^Pid in Fig. 18), or else its boundary includes at least another such arc P2 ;tP2 k+i (as shaded region PiP^bPePTC in Fig. 18). In the first case, X and Y are both zero at some point (a or d) on the inner boundary, since X is negative at P2A+1 and positive at P2 h and hence zero at an intermediate point. In the second case, a point moving from P2 h to P2 h+i along the smaller included arc and then along the inner boundary of R until it first returns to the circle arrives at a point P2A; of even subscript (as in the case of P4P56P6). In- deed, if a person travels as did the point, he aaiII always have the region R at his left and hence will pass from Po k to Po k+i and not vice versa. Since X is negative at P2 /,+i and positive at P2 k, it (as also F) is zero at some point b on the part of the inner boundary of R joining these two points. Hence b represents a root of f{z) = 0. Thus in either of the two pos- sible cases, the equation has a root, real or imaginary. §21 FUNDAMENTAL THEOREM OF ALGEBRA 53 2.t It remains to prove that an equation F{z) = 0, not all of whose co- efficients are real, has a complex root. By separating each imaginary coeffi- cient into its real and purely imaginary parts, we have F(z) = P -{- Qi, where P and Q are polynomials in z with real coefficients. Let G{z) = P — Qi. The equation F{z) . G{z) ^P^-\-Q^ = has real coefficients and hence has a complex root z = a -\- hi. If this is a root of F(z) = 0, our theorem is proved. If it is not, then G(a + bi) = 0. Then by Ch. II, § 16, F(a — hi) = 0, and the given equation has the root a — hi. EXERCISES l.f For z^ = 11 -\- 2{, draw the graphs of A' = 0, F = and locate the three roots of the cubic equation in z. Fig. 19 2.t For z^ — 4i z — 2 = 0, Y = r^ sin 5 e — 4 r sin 6. Using polar coordinates, show that the graph of F = gives the boundaries of the regions in Fig. 19: first plot the horizontal line corresponding to sin 6 = 0, and then, using various angles e {e i?^ 0, tt), find by logarithms tlK corresponding positive r from 4 sin 6 r* = sin 5 d 54 THEORY OF EQUATIONS |Ch. v To fiiKl the points on these boundaries (F = 0) for whicli also X =r^cos5d — 4ircosd — 2 = 0, replace r* by the earlier expression. We get ^/•„ r ^ -rxo-r sin 59 4 ?-(sin d cos 5 d — cos d sm 5 9) = 2 sni 5 0, r = ^ • 2 SHI 4: Comparing the fourth power of this fraction with that for r*, we get sin^ 5 = 64 sin 9 sin^ 4 0, which holds for 6 = 85° 21' 30" or its negative. We then get r and therefore the roots €,5 = 0.11679 ± 1.43851. On the horizontal line are three real roots, best found by methods of approxima- tion given later: a = 1.518512, = -0.5084994, 7 = -1.2435964. (H. Weber and J. Wellstein, EncyJdopddie der Elementar-Mathematik, ed. 1, I, p. 212, p. 296.) 3.t Other References. For proofs of the fundamental theorem by Gauss, Cauchy and Gordan, see Netto, Vorlesungen fiber Algebra, I, p. 25, p. 173. The shortest proofs are by the use of the theory of functions of a complex variable, and may be found in texts on that subject. For an algebraic proof resting upon the theory of functions of a real variable, see Weber, Lehrbuch der Algebra, 2d ed., vol. 1, pp. 119-142. See also Monographs on Topics of Modern Mathematics, 1911, p. 201, edited by Young (article by Huntington). In the Amer. Math. Monthly, vol. 10 (1903), p. 159, Moritz has pointed out hidden assumptions in various in- complete proofs. CHAPTER VI Elementary Theorems on the Roots of an Equation 1. Relations between the Roots and the Coefl&cients. Given an equation in x of degree n, we can divide its members by the coefficient of a:" and obtain an equation of the form (1) /(.r) =x^-{- pia:"-i _[_ p^_^n-i + . . . ^ p^ = 0^ By the fundamental theorem of algebra (Ch. V), it has a root ai, and its quotient hy x — ai has a root ao, etc. Thus (2) Kx) ^(x- a,){x - ao) . . . (.r - a,), identically in x. Since the polynomial has n linear factors, each having one root, we shall say that the equation has n roots. These may not all be distinct; exactly w of them equal ai, if ai is a root of multiplicity m, i.e., if exactly m of the linear factors in (2) equal x — oti. Next, {x — ai)(x — ai) ^ X- — (ai + a2)x + ai«2, {x — ai) (x — 0:2) (x — as) =x^ — (ai + 0:2 + az)x'^ + (aia-y + «ia3 + ccza^jx — aia2(Xz. Thus for n = 2 or 3, we see that the product (2) equals (3) re"- (ai + • • • + a;„).-C"-i + (aia2 + Uia^ + aoaa + • • • + an-ian)x"-' — {aia2a3 + aia2ai-\- • • • -\-an-2an-ian)x"~^-{- • • • +( — l)"aia2 - ' • a„. Multiplying this by x — an+i, we readily verify that the product is a function which may be derived from (3) by changing n into n + 1. It therefore follows by mathematical induction that (2) and (3) are identical. Hence (1) and (3) are identical, so that «1 + ^2 + • • • + «„ = — Pi, ai(X2 4- 0:10:3 + • • • + a;n-io:,i = P2, (4) 0:10:20:3 + o:iavQ;4 -f • • • 4-a:„_2Q:„_iQ:„ = —ps, aia2 • • • q;„-iq;„ = (— 1)"2}„. For n = 3 and n — 4, the complete formulae were given and proved other\vise in Ex. 7, p. 32 and Ch. IV, § 2. 55 56 THEORY OF EQUATIONS ICh. VI In an equation in x of degree n, inwhicJLtJ]£^^^cicntofjt^Lisjunity, the sum of the roots equals the negative of the coefficient of x"~'^, the sum of the products of the roots two at a time equals the coefficient of x'^~'^, the smyi of the ■products of the roots three at a time equals the negative of the coefficieyit of x"~^, etc.; flnaUy, the product of the roots equals the constant term or its nega^ live according as n is even or odd. For example, in a cubic equation having the roots 2, 2, 5, the coeffi- cient of X equals 2- 2 + 2- 5 + 2- 5 = 24. Given an equation a^x'^ + aio:""^ + • • • =0, Ave first divide by ao and then apply the theorem to the resulting equation. Thus the sum of the roots equals — ai/ao. EXERCISES 1. Find the quartic equation having 2 and —2 as double roots. 2. Find the remaining root in Exs. 1, 3, p. 9. 3. If a real cubic equation x^ — 6 .c- + • • • = has the root 1 + "^—5, what are the remaining roots? 4. Form by the theorem the equations in Exs. 3, 4, p. 15. 5. Given that .r" - 2 x^ - 5 .r^ - 6 .r + 2 = has the root 2 - V3, find another root and, by using the sum and product of the four roots, form the quad- ratic equation for the remaining two roots (avoid division). 6. Find, by use of (4), the roots of .r^ - 6 .^;3 + 13 x" - 12 .c + 4 = 0, given that it has two double roots. 7. Solve x^ — 3 X- — 13 .r + 15 = 0, with roots in arithmetical progression. 8. Solve 4 x^ — 16 .c- — 9 .c + 36 = 0, one root being the negative of another. 9. Solve x^ — 9 .c- + 23 x — 15 = 0, one root being triple another. 10. Solve .r^ — 14 x- — 84 .t + 216 = 0, with roots in geometrical progression. 11. Solve x* — 2 .r^ — 21 x- + 22 x + 40 = 0, with roots in arithmetical pro- gression. Denote them by c — 3 6, c — 6, c + &, c + 3 6. 12. Solve .r* - 6 x^ + 12 x^ - 10 x + 3 = 0, with a triple root. 13. Find a necessary and sufficient condition that /(x) = x^ + ^hx- + IHX + Pa = shall have one root the negative of another. Note that (a2 + «3) (ai + "3) (ai + "2) is obtained by substituting x = — pi in (2). 14. If for n = 4 the roots of (1) satisfy the relation aiao = azai, then pi^Pi = ps'^. Note that (4) gives — Pa = a\a2{az + aj) + aiai{ai + a^) = — piaiao. 15. What is the coefficient of y"~^ in the equation ?/"+••• =0 whose roots are ai — /i, • • - , a„ — h, when the as are the roots of (1)? For what value of §2,31 THEOREMS ON ROOTS OF EQUATIONS 57 h is this coefficient zero? Hence to remove the second term of an equation by replacing x by y -\- h, what value of h must we take? Check by the binomial theorem. 16. Find the equation whose roots are the roots of .r^ — 6 x^ + 4 = each diminished by 3. Remove the second term by transformation. 17. Prove the binomial theorem by taking the as all equal in (2) and (3) and counting the number of terms in each coefficient of (3). 18. Using (1) and (2), show that (1 - «i2)(l - a,') • • • (1 - an') = il+P2 + Pi+ • --y- (pi + 2^3 + P5 + • • •)^ (1 + ai2)(l + ao=) •••(! + a,r) = (1 - p, + p, )2 + (p, _ p, + p,- ... )2. 19. Since .ri, . . . , X4, determined by relations (8) of Ch. IV, give the correct values of the sums (9)-(ll), they are the roots of the quartic equation. Why does this give a new solution of the quartic? 20. Using Ex. 6, p. 32, make a similar argument for the cubic. 2. Upper Limit to the Positive Roots. For an equation fix) = aox" + ai.x"-i 4- . . . + a„= (ao 5^ 0) with real coefficients, we shall prove the Theorem. If ao, ai, . . . , a^-i are each^O, while Ok < 0, and if G is the greatest of the numerical values of the negative coefficients, each real root is less than 1 + ^G/oq. For positive values of x, f{x) is numerically greater than or equal to aox"" - Gix"-'' + a;«-'^-i + • • • + x + 1) „ „ /a;"-^+^-l\ _ a;»-^-+Mao(a;^ - x''-') -Gl +G — aoX (t I - 1 — ; . \ x — I / X — I But, if X > 1, x^ - x^-i = (x - 1)K Hence if a;=l + VgJo'o, OoCa;^ - x'^-') ^ G, f(x) ^ 0. 3. Another Upper Limit to the Roots. If the numerical value of each negative coefficient he divided by the sum of all of the positive coefficients which precede it, the greatest quotient so obtained when increased by unity gives an upper limit to the positive roots of the equation. If the coefficient of x'^ is positive, we replace x"" by (x - l)(a;'"-i + a;'«-2 + . . . + a; + 1) + I. 58 THEORY OF EQUATIONS [Ch. VI The argument will be clearer if applied to a partif^ular case: f{x) = jhx'' - pix^ + ]hx^ + psx- - PiX + P'o = 0, where each pi is positive. Then/(.c) is the sum of the terms Po{x — l)x* + Pq{x — l)x* + po(x — l)x- 4- po(x — l)x + po{x - 1) + po — piX* p2(x — l)x- + 2>2(a: — l)a; + P'zix — 1) + P2 7)3(3: - l).r + Psix - 1) + ps — PiX Pa. The sum of the terms in each column will be positive, if a: > 1 and PO {X - 1) - Pl> 0, {po -\-p2 + PZ){X - 1) - P4 > 0, since only in the first and fourth columns is there a negative part. These inequalities both hold if Po 7>0 + P2 + P3 EXERCISES Apply the methods of both § 2 and § 3 to find an upper limit u to the roots of 1. 4x^ - 8x4 + 22x5 ^ 98_j.2 _ 73^ + 5 = 0. By §2, w = 1 + 73/4. By § 3, u= 3, since 1+8/4 = 3, 1 + 73/124 < 3. _ 2. x^ + 4a;'»-7x2-40.r + l = 0. By § 2, w = 1 + ^^40 = 4.42. By§3,M = 9. 3. x4-5a;3 + 7x2 -8x + 1 = 0. 4. x^ + 3 xs - 4 x5 + 5 X* - 6 x3 - 7 .x2 - 8 = 0. 5. x7 + 2x5 + 4x4-8.t2-32 = 0. 6. If A is the greatest of the numerical values of Oi, . . . , a„, each root is less than 1 + A/ao. In the proof in § 2, set ^- = 1 and replace G by .4. 7. A lower limit to the negative roots of /(x) = may be found by appljnng the above theorems to/(— x) = 0. To obtain a lower limit to the positive roots consider /(I /x) = 0. 8. Find a lower limit to the negative roots in Exs. 3, 4. 9. Find a lower limit to the positive roots in Ex. 5. 4. The Term *' Divisor." In certain te.xts it is stated that the relation ai a2 . • • «n = ±Pn in (4) implies that "every root of an equation is a divisor of the absolute term." This statement is either trivial or else is not always true. It is trivial if it means merely that the absolute term can be divided by any root (that root being a complex number), yielding a quotient wliich is a complex num- ber. For, in this sense division is always possible (except when the divisor is zero), and a root not zero is a divisor of any numl^cr whatever. The statement quoted was certainly not meant in this trivial sense, with no special force. The only other sense, familiar to the reader, in which a constant is said to be a divisor 5 5, 6] THEOREMS ON ROOTS OF EQUATIONS 59 of another constant is the following: An integer r is a divisor of an integer p if p/r is an integer, so that p = rq, where q is an integer. For example, 4 is a divisor of 12, but not of 6. In this reasonable sense of the term divisor in such a connec- tion, the statement quoted becomes intelligible only when modified to read: every integral root of an equation with an integral absolute term is a divisor of that term. But this is not always true. The integral root 6 of x' — -3°- x + 4 = is not a divisor of 4; the root 2 of .c- — ^ a; — 3 = is not a divisor of —3. The correct theorem is that next stated. 5. Integral Roots. For an equation all of whose coefficients are inte- gers, that of the highest power of the variable being unity, any integral root is a divisor of the constant term. In certain texts, we find a correct statement of this theorem, but an erroneous proof. When ai and Pn are integers and aia2 . . . an = ± Pn, it is falsely con- cluded that ai is a divisor of />„. But 12 • 3 • j = 9 and 12 is not a divisor of 9. Also the examples at the end of § 4 show the falsity of this argument and, indeed, of any argument not making use of the hypothesis that all of the coefficients are integers. A correct proof is very easily given. Let d be an integral root of equa- tion (1), in which now pi, . . . , /)„ are all integers. Then (5) d- + p,d"-' + Pid"-^ + • • • + Pn-ld + Pn = 0. Since d obviously divides all of the terms preceding the last term, it must divide p„. Hence if there be integral roots of an equation of the specified type, they may be found by testing in turn each positive and negative divisor d of the constant term p„. The most obvious test is to compute (by the abridgment in Ch. I, § 5) the value of f(d) and note whether or not this value is zero. We may shorten the work very much by various methods, and most by a combination of these methods. Evidently it is unnecessary to test a value of d beyond the limits of the positive and negative roots. 6. Newton's Method for Integral Roots. Consider an equation (1) with integral coefficients. Let d be an integral root. It is a divisor of pn and we may set Pn = dq„-i. By removing the factor d from each term of (5), we get d"-i + pid«-2 + . . . -f p,^_^d 4- p„_i + g„_i = 0. 60 THEORY OF EQUATIONS [Ch. vi The left member is divisible by d, and hence 7J„_i + g„_i = dqn-2, where g„_2 is an integer. Then d"-- + pid"-' + • • • + Pn--d + Pn-2 + qn-2'= 0, Pn-2 + qn-2 = dqn-3, where qn~z is an integer, etc. Conversely, if such a relation holds at each step and if, finally, 1 + go is zero, then df is a root, and the quotient of f{x) by a; — d is x"-^ — qix"-- — qoX"-^ _ . . . _ q^_^x - g„_i. Indeed, in the product of the latter by x — d, the coefficient of .r"~' for i > is dqt-i — qt and this equals pt by our relations. Corollary. If d is an integral root of an equation f(x) = x" -\- • • • = with integral coefficients, the quotient of /(.r) by a: — d is a polynomial with integral coefficients. This process is a modification of synthetic division (Ch. X, §4). Example, fix) = .r* — 9 x^ + 24 x- — 23 .r + 15 = 0. Since evidently there is no negative root, and since 10 is an upper limit to the positive roots, we have only to test the divisors 1, 3, 5 of 15. Now /(I) = 8. For d = 3, the work is as follows: 1 - 9 24 - 23 15 ^ _6 -^ _5 - 3 1-8 - 18 Here we have divided 15 by 3 and placed the quotient under — 23. Adding, we get —18, whose quotient by 3 is added to 24, etc. Since the last sum is zero, 3 is a root. The (luotieiit has as its coefficients the negatives of the numbers in tlie second line (see the first line below). We test this quotient for the root 5: 1 -6 6 -5 -1 1 - 1 -5 5 Hence 5 is a root and the quotient is x^ — x -\- 1. The latter does not vanisli for X = ±1. Hence 3 and 5 are the only integral roots and each is a sim])le root. If we had tested a divisor —3 or 15, not a root, a certain (luotient would not be integral and tlie work would be stopped at that point. 7. Another Method. A divisor d is to be rejected if c^ — m is not a divisor of /(m), where 7ii is any chosen integer. § 81 THEOREMS ON ROOTS OF EQUATIONS 61 For, if d is an integral root of f(z) = 0, fix) ^(x-d) Q(x), where Q(x) is a polynomial with integral coefficients (§ 6, Cor.). Then f(7n) = (m — d)q, where q is the integer Q{m). In the example of § 6, /(I) = 8 is not divisible by 14, so that 15 is not an integral root. Consider the new example J{x) = x^ - 20 .1-2 + 164 X - 400 = 0. There is no negative root and 20 is an upper limit to the roots. The positive divisors of 400 less than 20 are 1, 2, 4, 16, 5, 8, 10. The last three are excluded since /(I) = —255 is not divisible by 4, 7, or 9. Also 16 is excluded since /(2) = — 144 is not divisible by 14. Incidentally we have excluded the divisors 1 and 2. The remaining divisor 4 is seen to be a root either by Newton's method or by computing /(4) . In case there are numerous divisors within the limits to the roots, it is usually better not to begin by listing all of the divisors to be tested. For, if a divisor is found to be a root, it is preferable to proceed with the quo- tient, as was done in the Example in § 6. EXERCISES Find all the integral roots of 1. x^ - 10.c2 + 27.r- 18 = 0. 2. .r" - 2 x^ - 21 x2 + 22 .c + 40 = 0. 3. .t5 + 47 x* + 423 x^ + 140 .c^ + 1213 x - 420 = 0. 4. x^ - 34 x^ + 29 .r2 + 212 x - 300 = 0. 8. Rational Roots. A7nj rational root of an equation with integral coefficients, that of the highest power of the variable being unity, is necessarily an integer. Let a/b be a root, where a and b are integers with no common divisor greater than unity. Set x = a/b in (1) and multiply the members of the resulting relation by 6"~^ We get ^ + Pia"-^ + P2a"-"b + • • • + Pn-\ab"-'~ + p.b'^-^ = 0. All of the terms after the first are integers. Hence b divides a". Unless b = ±1, 6 has a prime factor which divides a" and hence also a, contrary to hypothesis. Thus a/b = ±a is an integral root. 62 THEORY OF EQUATIONS [Ch. vi The rational roots of any equation with rational coefficients can now be readily found. If I is the least common denominator of the fractional coefficients, we multiply the members of the equation by I and obtain an equation aoi/" + ai7/"-i + . . . + a„ = 0, where ao, . . . , a„ are integers. Multiply the left member by ao"~^ and set Qoy = X. We obtain an equation (1) with integral coefficients, that of ic" being unity. To any rational root ?/i of the equation in ?/ corresponds a rational root ao?/i of (1), which must be an integer, in view of the theorem just proved. Hence we need only find all of the integral roots of the new equation (1) and divide them by ao to get all of the rational roots y of the original equation. Frequently it is sufficient (and of course simpler) to set ky = x, where k is a suitable integer less than ao. EXERCISES Find all of the rational roots of 1. y' - ¥ y' + ^F f~ - 40 ?/ + 9 = 0. 2. 6?/- 11 2/2 + 6 i/- 1 = 0. 3. 108 2/3 - 270 2/2 _ 42 2/ + 1 = 0. [Use k = 6.] 4. 32 2/3 - 6 2/ - 1 = 0. [Use the least k.] Form the equation whose roots are the products of 6 by the roots of 5. a;2 - 2 X - i = 0. 6- a;^ - i x^ - i x + I = 0. CHAPTER VII Symmetric Functions 1. S-polynomials ; Elementary Symmetric Functions. A polynomial in the independent variables Xi, Xo , . . . , Xn is called symmetric in them if it is unaltered by the interchange of any two of the variables. For example, xi'' + x~^ + x^^ + 2,Xi + 2>X2^Zxz is" a symmetric function of xi, Xo, x^. The sum of the first three terms is denoted by Srcr and the sum of the last three by 3 S.Ci. In general, if t is a product of powers ol Xi, . . . , Xn, whose exponents are integers S 0, Sf denotes the sum of this term t and all of the distinct terms obtained from it by permutations of the variables. Since such a S-polynomial 2^ is unaltered by every permutation of the variables, it is unaltered in particular by the interchange of any two variables and hence is a sym- metric function. For example, if there are three variables a, /3, y, Xa'^IS^y = a~^~y + cx^y'^ + ^^y'~a, 2a2/33^ = a'^^'y + ^-c^y + a~y^(3 + y'^a^^ + jS^^a + y^^^a. Just as in the case of the initial example, any symmetric polynomial is evidently a linear combination of S-polynomials with constant coefficients. The S-polynomials, of the first degree in each variable, (1) Ei = Sa:i, E2 = l^XiXo, Ei = IiXiX-^Xs, . . . , En = XiXo . . . Xn-ix^ are called the elementary symmetric functions ol Xi, . . . , Xn. Frequently we shall employ the notation cki, . . . , q;„ for the indepen- dent variables. By Ch. VI, § 1, ai, . . . , q;„ are the roots of an equation of degree n, (2) f{x) = a;" + pio;"-! + pox""'^ • • • -h p„ = 0, in which —j)i, po, —pi,..., ( — !)"?)„ equal the elementary sj^mmetric functions of the roots. It is customary to make the latter statement also for an equation whose roots are not independent variables. 63 64 THEORY OF EQUATIONS (Ch. VII But in the latter case it is preferable to saj' that —pi,P2, . . . equal the elemen- tary symmetric functions formed for the roots, thus indicating that we have in mind the values of certain functions of arbitrary variables Xi, . . . , .t„ for Xi = ai, . . . , Xn = an- It may happen that the resulting polynomials in ai, . . . , a„ are not sj^mmetric in ai, . . . , «„. For example, if the three roots are a, /3, /3, we have — pi = a + 2 /3, p2 = 2 a/S + /S', —p3 = CC0-, which are the values of Xi-\- Xi + Xs, etc., but are not themselves symmetric in a, /3, /3, being altered by the interchange of a and ;8. However, this point will give no trouble in the exercises below, since the roots are given distinct notations andjnay, if it is desu'ed, be regarded as independent variables. 2. Products of S-polynomials. It is a fundamental theorem that any symmetric polynomial in the roots is expressible rationally and integrally in terms of pi, pz, . . . , Pn and the coefficients of the symmetric poly- nomial. To prove this, it suffices to show that any 2-polynomial is ex- pressible rationally and integrally in terms of the elementary symmetric functions. Postponing the general proof, we shall now treat several special cases and assign others as exercises. Example 1. If a, /3, 7 are the roots of x' + ?^-i'" + qx + r = 0, p'-= {a + l3+yy- = a- + fi- + y'- + 2 (a/3 + ay + fiy) = Sa^ + 2 q, Sa2 = p2 _ 2 g, _ pg = 2a . Sa^ = Sa^^ + 3 a^y, Zoc^^ = 3 r - pq, Xc^py = pr, Sa2^2 ^ CEapy - 2 a^yZa = q^ - 2 pr. The student should carry out in detail the steps here indicated. Example 2. The student should learn how to express a product like Sa • Xa0 in Ex. 1 as a sum of 2-functions without writing out their expansions, since the latter method is very laborious in general. To obtain the types of 2-functions in the product, it suffices to use a single term (called leader) of one factor, saj'' a. Then if we use any term of lla^ which contains a, we get a term of Z)a-/3; while if we use any terra not containing a (hence 0y in this example), we get a terra afiy. It remains to find the coefficients of these 2-functions I2a-i3 and a0y. To get 0^13, we must take the term a of ^a and the terra a0 of "Eaff, so that 1,a'0 has the coefficient unity. To get a/37, we maj^ take a or /3 or y from 2a and the com- plementary factor I3y or ay or a/S, respectively, from 2a/3. Hence 2a . 2a^ = 'Ea'^13 + 3 a0y. 3 3 6 As a check, we have marked under each 2 the number* of its terms. Then the total number of terms is 3 X 3 = 6 + 3. * Found by the theory of combinations in Algebra, and not by writing out in full the S-functions. S3] SYMMETRIC FUNCTIONS 65 Example 3. To find the product of the S-functions of a, (3, 7, 5, we use the leader a.^ of the first. To obtain the four types of 2-f unctions in the product, we first use a term of s containing both a and /3; second, a term of s containing a- but not /S; third, a term with a but with neither o^ nor /3; fourth, a term free of a and /3. The respective types are those in 2a,S • Za^3 = 1 ^a^0' + 2 Za^/Jy + 2 "^a^^y"- + 3 ^cc^y'b. 6 12 12 12 12 4 The coefficient of any S-function on the right is obtained by counting the num- ber of ways its leader can be expressed as a product of terms of the 2-functions on the left. The coefficient of ot^t is 2 since we must take either a/3 or fiy from 'Zafi (for, we must take a or y, since s does not have a term with two exponents equal to 2; while if we take ay, the complementary factor a^y is not in s). To obtain afiy-b, we must take a term from s with 7- and a or /3 or 5. The first and second coefficients are evidently correct. EXERCISES If a, /3, 7, 5 are the roots of x^ + px^ -\- qx- -\- rx -{- s = 0, find 1. 2a-/3-. [Square 2a/3.] 2. 'LoL^0. [Use Sa^ . 2a/3.] 3. Sa". [Square Za\] If a, /3, 7 are the roots of x^ + p-C" + g.r + r = 0, find the cubic equation with the roots 2 2 2 4. a^, ,32, 7'. 5. a/3, a7, /37. 6. -, -, -• a p 7 By multiplying 2,ri by a suitable S-function, express in terms of functions (1) 7. Sxi2 (if n > 1). 8. 2.C12.C2 if (n > 2). 9. 1^x^X2 (if n = 2). 10. Sxi" (if 71 > 2). 11. 2.C1--* (if n - 2). 12. llxiHiXz. 13. For equation (2) with n > 4, show that 2ai2a2a3a4 = — P1P4 + 5 Ps, llctx^at^az = 3 /:»i7)4 — Jiip^ " 5 Ps- 14. For equation (2) with n > 5, show that 2ai2a2^a3a4 = p2p4 — 4 piPs + Pe, 2ai2a2'''a3^ == P3^ — 2 ^2^4 + 2 piPs — 2 J^e- 3. Fundamental Theorem on Symmetric Functions. Any polynomial symmetric in Xi, . . . , .r„ equals a polynomial in the elementary symmetric functions Ei, . . . , En of the x's. The proof, illustrated in Exs. 1 and 2 of §4, tells us just what elemen- tary symmetric functions should be multiplied together in seeking the expression for a given symmetric polynomial in terms of the £^'s and hence perfects the tentative method used in the earlier examples. 66 THEORY OF EQUATIONS [Ch. vii It suffices to prove the theorem for any homogeneous symmetric poly- nomial *S, i.e., one expressible as a sum of terms h = axi^^Xi'i . . . Xn''" of constant total degree k = h + ki -\- • • • + fc„ in the x's. Evidently we may assume that no two terms of *S have the same set of exponents ki, . . . , kn (since such terms may be combined into a single one). We shall say that h is higher than the term hxi^x-,^'^ . . . xj'n if ki > U, or if ki = li, k'i > k, or if fci = ^1, A-2 = k, ks > k, . . . , so that the first one of the differences ki — h, ko — k, ks — k, . . . which is not zero is positive. If the highest term in another symmetric polynomial S' is h' = a' x^^' X-]"'' . . . a:/"', and that of S is h, then the highest term in their product SS' is hh' = aa'a;/i+^'' . . . a:„'^"+'^""'. Indeed, suppose that SS' has a term, higher than hh\ (3) cxi''+''' . . . xj-+^-', which is either a product of terms t = bxi^' . . . Xn'", t' = h'xi^^' . . . xj» of S and S' respectively, or is a sum of such products. Since (3) is higher than hh', the first one of the differences I -\- il — A"l — a"i , . . . , („ "T tn "n '^"n which is not zero is positive. But, either all of the differences h — ki, . . . , In— kn are zero or the first one which is not zero is negative^ since h is either identical with t or is higher than t. Likewise for the differences h' — ki, . . . , In — kn'. We therefore have a contradiction. It follows at once that the highest term in any product of homogeneous symmetric polynomials is the product of their highest terms. Now the highest terms in Ei, Eo, E3, . . . , En, given by (1), are Xi, 3-'iX2j ^'l^2'r3, . . . , XiXi . . . Xn, respectively. Hence the highest term in Ei'^^E-f"- . . . £'„"" is §4] SYMMETRIC FUNCTIONS 67 We next prove that, in the above highest term h of >S, kl = A^2 = ft'S • • • — kn- For, if ki < k^, the symmetric polynomial S would contain the term axi^'^xj'^Xi^^ . . . xj"", which is higher than h. If ko < h, S would contain the term aXi'^^Xo'^^Xs''^ . . . xj^", higher than h, etc. By the above result, the highest term in a = aEi'^^-^^E'i'^-''^ . . . ^„_/-" '-^•"£;/« is h. Hence Si = S — a is a, homogeneous symmetric polynomial of the same total degree k as S and having a highest term hi not as high as h. As before, we form a product cri of the E's whose highest term is this hi. Then S2 = Si — ai is a homogeneous symmetric polynomial of total degree k and with a highest term /12 not as high as hi. We must finally reach a difference St — (Ti which is identically zero. Indeed, there is only a finite number of products of powers of Xi, . . . , Xn of total degree k. Among these are the parts h' , hi, h^' , ... of /i, hi, ho, . . . with the coeffi- cients suppressed. Since each hi is not as high as hi-i, the h', hi, h^ , . . . are all distinct. Hence there is only a finite number of hi. Since St — at ^ 0, S = (J-\-Si = (T-\-iyi-{-So= ' • • = ff + CTl + (T2 + • • • +0"^ Hence *S is a polynomial in Ei, Ei, . . . , En. 4. At each step of the preceding process, we subtracted a product of the £"s multiplied by the coefficient of the highest term of the earlier function. It follows that any symmetric polynomial equals a rational integral Junction, with integral coefficients, of the elementary symmetric func- tions and the coefficients of the given polynomial. Corollary. Any symmetric polynomial with integral coefficients can he expressed as a polynomial in the elementary symmetric functions with integral coefficients. Instances of this important Corollary are furnished by the results in all of our earUer examples and in those which follow. 68 THEORY OF EQUATIONS [Ch. VII Example 1. li S = "^x^x^xz and w > 4, we have «7 = EiEz = 5 + 3 2x12x2X3X4 +10 2x1X2X3X4X5, Si = S — a = —3 2X1^X2X3X4 — 10 2X1X2X3X4X5, (Ti = — 3 EiEi = — 3 (2x1^x2X3X4 + 5 2x1X2X3X4X5), 'S2 = Si — 0-1 = 5 2x1X2X3X4X5 = 5 £'5, *S = cr + jSi = ff + tri + *S'2 = £'2£'3 — 3 £'i£'4 + 5 £"5. Example 2. If *S = 2x1^2X3 and n > 4, a = Ei^Ez = El (2x1^x2X3 + 4 2x1X2X3X4) = 2x1^x2X3 + 2 2x1^X2^^X3 + 3 2xrX2X3X4 + 4 (2Xi2x2.C3X4 + 5 2X1X2X3X4^-5), Si = S — (X = —2 2x12x2^X3 — 7 2xilr2X3X4 — 20 2x1X2X3X4X5. Take o-i = — 2 £'2£'3 and proceed as in Ex. 1. Remark. The definition of a 2-polynoinial in § 1 may be extended to 2-func- tions in general. For instance if there are three variables a, 0, y, V- = - + - + -. V- = - + - + - + - + - + -• ^a a /3 7 ^a a a P y y EXERCISES If a, /3, y, 8 are the roots of x* + px^ + qx- + ?'X + s = 0, 1. y— -• 2. V^ = V«.Vl-4 = ^-^'-4. Aa S A/ + rif + si/ = 0, obtained by replacing x by 1/y in the former quartic equation. + y + d ^-p - « . x^l = > = -4.-P 2,-' _V^ o V 7 _ 3 r - p(7 12 7. 12 4 8. 'O «' + /S" 10. SYMMETRIC FUNCTIONS 69 12. Prove that the degree in any single x of a homogeneous symmetric poly- nomial *S' is the total degree of the equal polynomial in the E's. Hints: First show that no term of *S' has an exponent > A;i, so that the degree of S in any single x is ki. Next, o- is of total degree A;i in the £"s. Set h = a'xi'^ .... Then <7i is of total degree A-i'( = A:i) in the £"s and not every exponent in o-i equals the corre- sponding exponent in a. Thus a is not cancelled by o-i, ai, . . . . 13. Given a pol.ynomial in the £"s of total degree d, show that the equal func- tion of the .t's is of degree = d in any single root. 5. Sums of Like Powers of the Roots. If ai, . . . , «„ are the roots of (2), we write Si = 2ai, So = Soii^, and, in general, Sk = ^ai = a-^ + oi'i' + • * * + oCt!'- The factored form of (2) is (4) j{x) = {x - a^{x — a^^ . . . {x - an). In this identity in x, we may replace x hy x -{■ h. Thus J{x + h) = (x + /l — ai){x -\-h — a'^^ . . . (x -\- h — an). In the expansion of f{x + h) as a polynomial in h, the coefficient of the first power of h isf'{x), by the definition of the first derivative oi f{x) in Ch. I, § 4. In the right member, the coefficient of h is (x - a-i)(x — as) . . . {x — a„)+ • • • +(x — ai)(x — a-z) . . . (x — q:„_i). Here the first product equals /(a;) -^ {x — ai), by (4), etc. Hence (5) /'(;,) ^JM_ + JM_+ . /(^) X — ai X — a2 X — an If a is any root of (2) , /(a) = and f(x) _ f(x) - /(«) _ a:" - g" ^.n-i _ ^n-i x — a ~. „ I / 1 i~ * * ■ \ yn—l X — a X — a X — a X — a x — a = x"--^ + cxx"-~ + a'^x"-^ + • • • + pi(a;"-2 -f ocx'''^ + • • • ) (6) ^^ = x^-' + (a + pi)x"-2 + (a2 + p,a + po)^:"-^ + • • • + (a'' + pia*^-i + poQ:*-2 + . • . + p,^_,a -\- Pk)x''-''-^ + ' • ' . Taking a to be ai, . . . , «„ in turn and adding the results, we have by (5) f'{x) = wx"-i + (si -f npi)x"-^ + (s2 + piSi -\- np2)x''~^ ■ ■ • -\- {Sk -\-piSk-i-\- P2Sk-2 + ' ■ • + PA;-iSi + np;t)a:"-*-i+ • • • . 70 THEORY OF EQUATIONS ICh. vil By Ch. L, § 4, f'{x)=nx"-'-\-{n-l)piX"-^-\-{n-2)p2X"-^-{- • • • -\-{n-k)pkX--''-'-{- • • •» Since the coefficients of like powers of x are equal, we get (7) si + pi = 0, So + piSi + 2 p2 = 0, . . . , Sk + ViSk-i + P2Sfc-2 + • • • + P;fc-iSi + kpk = (A; = 1, 2, . . . , w - 1). We may therefore find in turn Si, So, . . . , s„_i : (8) Si = -2h, S2 = pi^ -2 p-i, S3 = -pi^ + 3 pipo - 3 p3, . . . . To find s„, replace a; in (2) by ai, . . . , a„ in turn and add the resulting equations. We get (9) Sn + p-.Sn-l + P2Sn-2 + ' ' " + Pn-l^l + lipn = 0. We may combine (7) and (9) into a single formula: (10) Sk + piSk-i + PiSk-o + • • • + Pk-iSi + kpk = Q (k =1,2, . . . ,n). To derive a formula which shall enable us to compute the Sk for k > n, we multiply (2) l)y a;^"", take x = ai, . . . , x = an m turn, and add the resulting equations. We get (11) Sk + PiSk-i + PiSk-i + • • • + PnSk-n = (k > n). Relations (10) and (11) are called Newton's form uIob. They enable us to express any Sk as a poljoiomial in pi, . . . , Pn- EXERCISES 1. For a cubic equation, S4 = 2)1'' — 4 pi-p2 + 4 pips + 2 7)2^. 2. For an equation of degree n S 4, S4 = pi^ — 4 j^rpo + 4 ]h]h + 2 7J2- — 4 ^4. 3. If we define Pn+i, Pn+2, ... to be zero, relations (10) hold for every k. Hence if .pi, p2, . ■ . are arbitrary numbers unlimited in number, and if o-i, 0-2, .. . are computed by use of <^fc + Pi 0. If 6 > c, for 7n as above. Since a -\-b > c, a -{- c > b, mllafa-^a-f= Sa {SbSc — Sh+c) — {Sa+bSc — Sa+6+c) — (Sa^-cS^ — Sa+h+c), (13) '^afa'^a:^ = — (SaSbSc — SaSb+c — SbSa+c — ScSa+b + 2 Sa+b+c) (6 > c). But if 6 = c, we have where r = 1 if a > 6, r = 3 if a = 6. Hence (14) 'Eai'^a'J'as^ = K^aSb' - S„S2 6 - 2 SbSa+b + 2 Sa+2 6) (tt > 6), (15) Zai^aa^as" = h(sj - 3 s^Sa a + 2 S3 a). The fact that any ' "Z-polynomial can be expressed as a polynomial in the functions Sk is readily proved by induction. We have Sa^ai^ai" . . . ar^ = f^afao^a^' . . . a^+i" + ti'Eai^+^ao'' . . . ar" + • • • + tri:,a^a2'' . . . 0:/+^, where t is a positive integer, and ti, . . . , tr are integers s (for example, ry= -%lw-]- ■ . . +a/)r r=l r=l 00 ■, if y is sufficiently small in absolute value to ensure the convergence of each of the series used. The coefficient of y^" in (piy + • • • + Pny'^Y may be found by the multinomial theorem. Hence, after dividing r = ri + • • • + r„ into the multinomial coefficient, we get ,,.,, ^ (-! )'•■+ •••+'-..A;-(ri+ • • • + r„ - 1)! ^ ^ (10 s,^^ ;,! ,,!... ,„! T'^'P^' • • • P'^^ * Edward Waring, Misc. Analyl., 1762; Meditationes Algehraicce, 1770, p. 225, 3d ed., 1782, pp. 1-4. No hint is given as to how Waring found (17); his proof was in effect by mathematical induction, being a verification that Sk, sa— i, . . . , Si satisfy Newton's formulip. But (17) had been given earUer by Albert Girard, Invention nouvelle en I'alghbre Amsterdam, 1G29. §91 SYMMETRIC FUNCTIONS 73 where the sum extends over all sets of integers ri , . . . , r„, each S 0, for which (18) ri + 2 r2 + 3 rs + • • • -{- nVn = k. Here r! denotes 1 • 2 • 3 . . . r if r S 1, and unity if r = 0. 9. Elementary Proof of Waring's Formula. Divide each member of (16) into the negative of its derivative; we get (19) -pi-2 pojj - ■ ■ ' - npnV''-^ _ ai _,_... _^ 1 + Pi2/ + • • • + PnU" 1 - aiy 1 - any In the identity (20) T^^i + Q + Q"+ • • • +Q'-'+ ^^ l-Q" i ^ : ^ . ■ ^ ' l-Q set Q = agy and multiply the resulting terms by ag. Hence the second member of (19) equals (21) . + ,.,+ ...+ ..,/-. + .^^,^f/^'.^^^^. , the polynomial (f)(y) being introduced in bringing the fractional terms ai^+V(l - «i2/), etc., to the common denominator (16). In (20), we now set Q = —piy — • • • — Pn?/". Thus ! + ,..+ '••+P.r - 2; (- mp^y +■■■+ v.ry + YT^-' r=U where \p{y) is a polynomial. Expanding this rth power by the multino- mial theorem, we see that the left member of (19) equals ^y(-l)r,+ ...+r„ + l (^l+ '".^"''• pi'-i • • • p/nyn+^r,+ ..-+nr„_^E ^^ ' ri! • • • r„! (d = Pi+2p22/4 ), the sum extending over all integral values ^ of ri, r2, . . . , rn such that ri + • • • + ^re < fc, while ^^ is a fraction whose denominator is 1 + P\y + • • • and whose numerator is the product of y^ by a polynomial in y. In the expansion of the part preceding E, the terms with the factor y^ may be combined with E after they are reduced to the same denominator 74 THEORY OF EQUATIONS [Ch. vil as E. The resulting expression* is now of the same general form as (21), so that the coefficient of /"' must equal Sk. This coefficient is the sum of ^^_iy.+ • • • 4-.„ + i (^^ + • • • +'"^- p/-+ V- . . . P/" (ri + 2r2+ • • • +nrn = k- 1), 2;^(-l)n+ ■ • • +r„+i (^^ + • • • + ""^- p/.p/.+ i . . . pj. (ri + 2 ro + • • • + nr„ = A; - 2), 3^(_l)n+...+.„+i ^^'^+ • • • +J^'^- p,r.pj^-p/,+ i . . . p„r„ (n + 2 ro + . . . + nr„ = A; - 3), In the first sum employ the summation index Vi + 1 instead of ri; in the second sum, r-z + 1 instead of ry] etc. We get (ri + • • • + r„ - 1) ^ ri!r2! (rs — 1)! . . . r„! where now (18) holds for each sum. Adding these sums, we evidently get the second member of (17). Example 1. Let ?t = 3, A; = 4. Then ri + 2 r2 + 3 ra = 4 and (ri, r2, r3) = (4, 0, 0), (2,1,0), (1,0,1), (0,2,0), /3' 2' 1' 1' \ = ]h* - 4 pi-p2 + 4 PiP3 + 2 P2'. * The difference between it and (21) is an expression of the form (21). Suppose there- fore that an expression (21) is identically zero. Taking y = 0, we get 6i = 0. The quotient by y is identically zero. Then 62 = 0, etc. § 10] SYMMETRIC FUNCTIONS 75 Example 2. Let n = 2 and write p for —pi, q for p2, r for Vi. Then n = k — 2r. If /c is the largest integer = k/2, the suni of the A;th powers of the roots of X- — px -\- q = \s ^ {-lYk-jk-r-iy. ,_^_ ^^^A ik-2r)lrl ^ ^ r=0 ^ -^ 10. t Certain Equations Solvable by Radicals. Regarding p as a vari- able and g as a constant, denote the polynomial in the preceding Ex. 2 by F(p). The equation F(p) = c, where c is an arbitrary constant, can he solved by radicals. Indeed, if x is a particular root oi x^ — px -{- q = 0, the second root is q/x, and Sk = a;^ + p • \x/ This expression in x is therefore the result of replacing p by x -\- q/x in F(p), as shown by the quadratic equation. Hence F{p) = c then becomes x'' -\- (-] = c, x-'^ — cx^ + 5^ = 0. Solving this as a quadratic equation for x^, we get ^•---y/'^--^ ^ 2=^V4-5 Since the product of these two expressions is q^, definite values i+v/f-'A '=\7i-V3-«' can be chosen so that pa = q. Hence if e be a primitive kth root of unity, the 2k values of x can be separated into pairs pe™, (re*=~^ (w = 0, 1, . . . , k — I), such that the product of the two in a pair is pa = q. Now x + q/x is a value of p. Hence the k roots p of F{p) = c are p^m _|_ ^^k-ni (fn = 0, I, . . . , k - 1). Thus F{p) = c can be solved by making the substitution I Q p = x -\- -' X For fc = 3, the equation is, p^ — Z qp = c and the present method be- comes that in Ch. Ill for solving a reduced cubic equation. 76 THEORY OF EQUATIONS [Ch. vil EXERCISES l.f Solve DeMoivre's quintic /;^ — 5 qp^ + 5 q-p = c for p. 2.t Solve 7;^ — 4 qp'^ + 2q^ = c for /; by this method. 3.t Write down a solvable equation of degree 7. Solve it. 4.t Solve 7/^ + 10 // + 20 7/ + 31 = 0. 11. Polynomials Symmetric in all but One of the Roots. If P is a polynomial in the roots of an equation f{x) = of degree n and if P is sym- metric in n — I of the roots, then P equals a polynomial in the remaining root and the coefficients of P and f{x) . For example, P = 3 ai + ao- + as'- + • • • + a,c is such a polynomial and P — 2a;i- + 3 ai — ar = pr — 2 p-z + 3 ai — ar. If a is the remaining root, P is symmetric in all of the r(3ots of the equa- tion (6) of degree n — 1, whose coefficients are polynomials in a, pi, . . . , Pn. Hence (§ 3) P equals a polynomial in a, pi, . . . , pn and the coeffi- cients of P. Example 1. If a, /3, 7 are the roots of /(x) s x^ + px- + qx -\- r = 0, find X or + /3- _ «- + 0' . «- + 7- , jS ' + 7' a + 13 a + /3 " + 7 /3 + 7 Since ffi- + y- = p- — 2 q — a-, (3 + y = —p — a, ^a- + /3'- ^;;--25-a' ^/ ,2r/\ o.o^l But a -{- p, -\- p, y -\- p are the roots jji, y^, y% of the cubic equation obtained from j{x) = by setting x + p = y, i.e., x = y — p. The resulting equation is if _ 2 py2 _^ (^j2 _^ ^^),y -I- ,. - pq = 0. Since we desire the sum of the reciprocals of ?/i, 7/2, 7/3, we set 7/ = 1/z and find the sum of the roots Zi, Zi, za of 1 - 2 7^2 + (/j2 + ^)22 _,_ (,. _ p^),3 = 0. Hence ^a-jf^ _ 2q- -2 phj + 4 pr + /3 pq-r " + P -^^i ^^' pq-r' ^ + g/2/ = 0. Thus the desired substitution is 2 = — p + 3 q/y, y = 3 q/{z + />). 18. Hence find the discriminant of the reduced cubic equation. 124 Cauchy's Method for Symmetric Functions. If Xi, . . . , Xn are the roots of (2), any polynomial P in Xi, . . . , Xn can be expressed as a polynomial in X2, . . . , Xn, Pi, . . . , Pn, in every term of which the expo- nent of Xo is less than 2, the exponent of X3 less than 3, . . . , the exponent of Xn less than n. To this end, we first eliminate Xi by using 2.ri = —pu Then we eliminate Xo'^'ik = 2) by using the quadratic equation satisfied by X2 and having as coefficients polynomials in Xs, . . . , x„. This quad- ratic may be obtained by dividing f{x) by {x — X3) . . . (x — Xn), or by noting that Xl '^ X> = —pi — X3 — • • • — Xn, 3:10:2 = Po- {Xi + X2) (.T3 + • • • + Xn) - XzXi - ... - Xn-\Xn. Next, we eliminate xz''{k ^ 3) by using the cubic equation obtained by dividing /(x) by {x — x^ . . . {x — a:„). Finally, we eliminate a:„*(A; S n) by using j{xn) = 0. Ex-\MPLE. To compute by this method the discriminant A = (xi - .r2)''(.ri - .r3)-(.r2 - x^)- of f{x) 3 x^ + px -\- q = 0, we note that Xi and X2 are the roots of -^^ = Qix) = x2 + XX3 + X32 + p = 0. x — X3 Since Sxi = 0, (xi - X2)- = {-2x2- XzY' = 4 Q(X2) - 3 .1-3- - 4 p = - 3 X3- - 4 />, (xi - Xi)(.C3 — X2) = Q{x3) = 3 X32 + p, A = (-3 X32 - 4 p)(3 X32 + pY = -27(.C3^ + pxj)- - 4 p', A= -21q^-^p\ §131 SYMMETRIC FUNCTIONS 79 We can now easily prove the fundamental theorem of § 3 : if P is sym- metric in xi, . . . , Xn, it equals a polynomial in pi, . . . , pn- For, P = A -{- Bxo, where neither A nor B involves Xi or X2. Since P is unal- tered when Xi and Xo are interchanged, A-\-Bx2 = A+ Bxi. If Xi 7^ Xi, then 5 = 0; and, by continuity, B — even when Xi = Xz. Hence P = C-^Dx, + Ex^, where C, D, E do not involve a^i, X2 or x-i. Since P is unaltered when X3 and Xi are interchanged, or when x^ and X2 are interchanged, the equation = C-P-\-Dy + Ef- has the three roots Xi, X2, X3. Hence if Xi, X2, Xs are distinct, D = E = 0, P = C, and by continuity these relations hold also if two or all three of these a;'s are equal, so that P is free of Xi, X2, X3. Similarly, P can be re- duced to a form which is free of each Xi. 13. t Tschirnhausen Transformation. We can eliminate x between (2) and (22) X = uo + Uix + ihx" + ■ • • -i-Un-iX"-^ and obtain an equation in X of degree n. First, from the expressions for X^, X^, . . . , we eliminate x", x"-+^, ... by use of (2) and get (23) X^ = W20 + U21X + U22X- -\- • • • -^ ih n-i^;" S X" = UnO + UnlX -\-t('n2X^ + • ' ' + ^^. n-lX"'^, where the w,-,- are polynomials in Uo, . . . , Un-i and the coefficients of (2). In any one of these equations (23) we set x = a;i, . . . , x = Xn in turn and add the resulting relations. If Xi, . . . , X„ are the values of X for X = Xi, . . . f X — 2^nj set Sfc = ^xi\ Sk = SXi^ Then Si = niiQ + UiSi + M2S2 + • • • + Un-lSn-X, (24) S2 = WW20 + U21S1 + 1*22-52 + • • + U2 n-lSn-h Sn = nUnd + UnlSl + W„2S2 + • ' • + Itn n-lSn-l' 80 THEORY OF EQUATIONS tCn. vil Since the elementary symmetric functions of Xi, . . . , Xn arc expres- sible in terms of >Si, *S2, . . . , ^S^ (§ 6), we can find the coefficients of the equation having the roots Xi, . . . , Xn'. (25) X" + PiX«-i + P2X"-2 + . . . + P„ = 0. Another method of forming this equation is given in Ch. XII, § 9. If we seek values of Uo, ih, . . . , w„_i, such that Pi, P2, . . . , P* shall all vanish and therefore Si = S2 = • • • = Sk = 0, by Newton's identities (7), we have only to satisfy a system of k equations [see (24)] homogeneous in «o, . . . , Un-i and of degrees 1, 2, . . . , k, respectively. In partic- ular, Si = enables us to express Uo in terms of Ui, . . . , so that (26) x=..(.-^)+4-l)+ •••+.»-.(-— ^> Example. For n = 3, X = ui(x — I si) + uoix- — ^ §2), S2 = 2X1^ - Ui%S2 - i Si^) + 2 UMSZ - I S1S2) + U^KSi - i 82^). Thus S2 = gives (3 S2 — sr) ui = (siSo — 3 S3 + V — 3 A)«2, A = S0S2S4 + 2 S1S2S3 — S0S3" — 81^84 — S2^ Hence the cubic equation is reduced to X'' + P3 = by the substitution X = (.si.so - 3 .S3 + V-3A) (3 X - si) + (3 .S2 - .s'i-)(3 x^ - S2). By Ex. 6, p. 158, A is the discriminant of the cubic equation. EXERCISES l.f For 71 = 4, take 2^3 = in (26) and find the cubic equation for Ui/1/2 which results from P3 = (i.e., *S3 = 0, since *Si = 0). The new quartic equation X^ + F2X' + P4 = may be solved in terms of square roots. 2.t For n = 5, the condition for »S2 = is that a certain quadratic form q in Ui, . . . , Ui shall vanish. Now q can l)e expressed as a sum of the squares of foiu- linear functions Ly of Ui, . . . , u-i. Taking Li = 1X2, L3 = iLi, where {2 = — 1^ we have Si = 0. By means of tlie r(!sulling two linear relations between Ui, . . . , W4, we may express S3 as a cubic function of «i, 112, for example. We must therefore solve a cubic equation in Ui/mo to find the u's making also S3 = 0. Tlie new quintic equation is X^ + PiX + P5 = 0. If Pi ^ 0, set X = ?/ \^i. Then ?/ + ?/ + c = 0. (Bring, 1786; Jcrrard, 1834.) CHAPTER VIII Reciprocal Equations. Construction of Regular Polygons. Trisection of an Angle 1. For certain types of equations, such as reciprocal and binomial equations, there exist simple relations between the roots, and these relations materially simplify the discussion of the equations. An equation is called, a reciprocal equation if the reciprocal of each root is also a root. Apart from possible roots 1 and —1, each of which is its own reciprocal, the roots are in pairs reciprocals of each other. For example, the equation fix) = (x-l)(x2-ix + l) = is a reciprocal equation having the roots 1,2, 5. If we replace x by 1/a; and multi- ply the resulting function by x^, we get —f{x). Here (1) holds for n = 3 and for the minus sign. In general, if fix) ^x^-i- ' • ■ +c = is a reciprocal equation, no root is zero, so that c 5^ 0. If r is any root of f{x) = 0, 1/r is a root of f{l/x) = 0, and hence of ../(i).i + + ex"" = 0. Since the former is a reciprocal equation, it has the root 1/r. Hence any root of the former equation is a root of the new equation. Thus, by (1) and (2) of Ch. VI, the left member of the latter is the product of f{x) by c. Then, by the constant terms, 1 = C". Hence c = ±1 and (1) x"/Q = ±/(x). Thus if p,- a:"""* is a term of /(x), also ± p,.^' is a term. Hence (2) f{x) = x" ± 1 + pi(a:"-i _t. _^) _|_ p2(x"-2 ± x^) + • • • . 81 82 THEORY OF EQUATIONS [Ch. vill If n is odd, n = 2 t -\- 1, the final term is and a: ± 1 is a factor oif{x). In view of (1), the quotient has the property that Hence Q{x) = is a reciprocal equation of the type (3) a;2' + 1 + ci(.x2'-i + x) + c^ix'-'-'- + a;^) + • • • + Ctx' = 0. Indeed, the highest power x-' of a: has the coefficient unity and the con- stant term is unity, so that it is of the form (2) with the upper signs. If n is even, n = 2t, and if the upper sign holds in (1), we have just seen that (2) is of the form (3). Next, let the lower sign hold in (1). Then Pt = 0, since a term ptX^ would imply a term —pix^. The final term in (2) is therefore Hence /(x) has the factor x- — \. As before, the quotient is of the form (3). In each case we have been led to a reciprocal equation of type (3). The solution of the latter may be reduced to the solution of an equation of degree t and certain quadratic equations. To prove this, divide the terms of (3) by xK Then (4) (,.+^) + ,,(,.-. + _L^) + ,,(,.-. + _^ + • • • +c,_i(.r + ^) + c, =0. To reduce this to an equation of degree t, we set (o) X + - = 2. Then a;2 + -, = 22 _ 2, x^^\ = z" -?>z, . . . , X- .r' while the general binomial in (4) can be computed from (6) ^ + S = K"""+?-)-(^"'+^} § 11 RECIPROCAL EQUATIONS 83 For example, a;4 + 1 = 2(^3 _ 3 2) _ (^2 _ 2) = ^4 _ 4 22 -I- 2. However, we can obtain an explicit expression for x'' + l/x'' by noting that it is the sum of the A;th powers of the roots x, 1/x of 2/' - (x + ~jy + X .- = ?/- 2?/ + 1 = 0. The sum of the kth powers of the roots oi if — py -^ q — was found in Ex. 2, p. 75. Taking p = z, q = I, wc have /^x : , 1 , 1 , , Hk-3) , , k(k - 4)(fc - 5) , , , (7) a;* + ^ =z' - kz'^-'- + \^2 ^ ~ 1.2.3 ^ + ' " ' fc(fe-r-l)(fc-r-2; . . . (fc-2r+l) ^ ^ + ^ ^^ 1 . 2 . 3 . . . r ^ ^ Hence (4) becomes an equation of degree t in z. From each root z we obtain two roots x of (3), which are reciprocals of each other, by solving the quadratic equation x- — zx -{- 1 = 0, equivalent to (5). Example. Solve x^ — 5 x* -\- 9 x^ — 9 x"^ -{- 5 x — 1 = 0. Dividing by a; — 1, we get X* - 4 x^ + 5 .c2 - 4 X- + 1 = 0. Thus x2 + -, - 4 (x + - ) + 5 = 0, z~-4z + 3 = 0, z = 1ot3. x^ \ x/ For 2 = 1, x2 - a; + 1 = 0, x = ^ (l ± V^). For 2 = 3, x^ - 3x + 1 = 0, X = § (3 ± V5) . These with x = 1 give the five roots. EXERCISES Solve by radicals the reciprocal cciuations 1. x^ - 7 x" + x3 - x2 + 7 X - 1 = 0. 2. x^ = 1. 3. x« = 1. 4. x^ + 1 = 0. 5. Find the 2;-cubic for x'' = 1. 6. Find the s-quintic for x^^ = 1. 7. The 2-quartic for x^ = 1 is 3" + 2^ - 3 2^ - 2 2 + 1 = 0. It has the root - 1 since the 2-equation for x^ = 1 is 2 + 1 = 0. Verify that, on removing the factor 2+1 from the quartic, we get the 2-cubic 2^ — 3 2 + 1 = for (x^ — 1) /(x^ — 1) = 0. 8. What are the trigonometric representations of the roots of the 2-equations in Exs. 5 and 6? Hint: if x = cos + i sin e, 1/x = cos d — i sin e. 84 T I! FA) RY OF EQUATIONS [Ch. VIII 2. Binomial Reciprocal Equations. A reciprocal equation with only two terms is of the form x" ± 1 = 0. Its roots were expressed in terms of trigonometric functions in Ch. II. But now we wish to use only alge- braic methods.* We might proceed as in § 1, first ** removing the factor a: ± 1 (if n is odd) or a;^ — 1 (if n is even and the lower sign holds), and then applying substitution (5) to obtain the ^-equation. Except for special values of n (as those in Exs. 2-6, § 1), there is a more effective method, leading to auxiliary equations of lower degree than the 2-equation. For in- stance, it will be shown that x" — 1 =0 can be solved in terms of square roots; it is only a waste of effort to form the z-equation of degree 8. 3. The new method will first be illustrated for x' — \ = ^ since it then differs only in form from the earlier method of treating reciprocal equations. Removing the factor x — 1, we have (8) x^-\-x^-\-x^ + x'' + x'-\-x + l = 0. If r is a particular root of (8), its six roots are (Ch. II, § 13), (9) r, r-, r\ r\ r\ r\ By the substitution (5), we obtain the cubic equation (10) 2^ _[_ ^2 _ 2 2 - 1 = 0, whose roots are therefore (11) zi = r + - = r -{- r^, Zi^ r^ + - = f- + r^ 23 = r^ + -^ = r^ + ?'■*• r r- 7"* The new method consists in starting with these sums of pairs of the six roots and forming the cubic equation having these sums as its roots. Since r is a root of (8), 22i = r + r- +•••+?•«=- 1, ^z^zi = 2 (r + • • • + r*') = -2, 2i2o23 = 2 + r+ • • • +r'^= 1. Hence Z\, Zo, zz are the roots of (10). If a root Z\ be found, we can obtain r from the quadratic equation r^ — z^r -\- \ = 0. * It is an important fact, not proved or used hevo, that .c" rfc 1 = is solvable by radicals, namely, by a finite number of applications of the operation extraction of a single root of a known number. Cf. Dickson, I ntroduction to the Theory of Algebraic Equations, John Wiley & Sons, pp. 77, 7S. Note that it suffices to treat the case n prime, since .r''^= .1 is equivalent to the chain of equations y'' = A, x^ = y. ** If n = pq, we may remove the factors x^ ± 1 if p is odd. See Ex. 7, §1. §4.51 RECIPROCAL EQUATIONS 85 We can, however, find r by solving first a quadratic equation and after- wards a cubic equation. To this end, set (12) Ui ^ r -\- r" + r*, y-i ^ r^ -{- r^ -\- f^. Then 2/1 + ?/2 = - 1, yiV'i = 3 + r+---+r« = 2, so that i/i and y^ are the roots of 2/2 + 2/ + 2 = 0. Then r, r~, r* are seen to be the roots of p3 - 2/ip- + yop -1 = 0. 4.t The Periods. We now explain the principle discovered by Gauss by which we select the pairs from (9) to form the periods Zi, Z2, Zs in (11), and the triples to form the periods yi, y^ in (12). To this end we seek an integer g such that the six roots (9) can be arranged in the order (13) r, r», ro\ r<}\ /■'/', r»'\ each term being the ^th power of its predecessor. The choice ^ = 2 is not permissible, since the fourth term would then be r^ = r. But we may take g = S, and the desired order is (14) r, r^, r^, r^, r^, r^, each term being the cube of its predecessor. To form the two periods yi and 7/2, each of three terms, we take alternate terms of (14). To form the three periods Zi, z-y, 23, each of two terms, we take any one of the first three terms (as r^) and the third term after it (then r^). 5.t Solution of x^'^ = 1 by Square Roots. Let r be a root 9^ 1. Then r" — 1 [ ± _ ,,16 4- ^15 _^ . . . ^ ^ 4_ 1 ^ 0. r — 1 As in § 4, we may take gr = 3 and arrange the roots, r, . . . , r^^ so that each is the cube of its predecessor: M mO y*V /V»1U ,y»lo ^5 ^15 <^ 1 1 -y»lb ,y<14 y%^ lyi ^4 yt\Z ^2 v-z = r'* + — , = 2 cos —=■• 1/ 1/ r* 17 Hence Vi > V'> > 0, and therefore Zi > 0. Similarly, ,,1,,,1 _, 6x,_ 10 TT „ Gtt ^ 7t^„ t^i = r^ + - 4- r^ + -. = 2 cos v^ + 2 cos — -=- = 2 cos v=r — 2 cos — r > 0, r^ r^ 17 17 17 17 „ Gtt , - IOtt , ^ 12 7r , _ 14 7r ^,, 2/2 = 2cos— + 2cos-Yy- + 2cos— y- + 2cos--y-< 0, i 6] CONSTRUCTION OF REGULAR POLYGONS 87 since only the first cosine in y-y is positive and it is numerically less than the third. But ijiijo = -4. Hence t/i > 0. Thus (15)-(17) give 2/2 = ^-^17-1), yi = \ Wvi - 1), ^\ = \ yi + Vl + i7/r, We now have the coefficients of (18) and know that Wi > ^J2 > 0. These results are sufficient for the next problem. Of course, we could go on and obtain the explicit expression for Vi, and that for r in terms of square roots. 6.t Construction of a Regular Polygon of 17 Sides. In a circle of radius unity, construct two perpendicular diameters AB, CD, and draw tangents at A, D, which intersect at S (Fig. 20). Find the point E in AS for which AE = I AS, by means of two bisections. Then AE ^l, OE = \ Vl7. Let the circle with center E and radius OE cut AS at F and F'. Then AF =^EF -EA=OE-\ = l y„ AF' = EF' -{-EA =0E + \= -^y^, OF = VOA' + AF' = Vl + 1 y{', OF' = VTTIy?- Let the circle with center F and radius FO cut AS at H, outside of F'F; that with center F' and radius F'O cut AS at H' between F' and F. Then AH = AF + FH = AF + OF = \y, + VTThA' = ^i, AH'= F'H' - F'A = OF' - AF' = wi. 88 THEORY OF EQUATIONS [Ch. viri It remains to construct the roots of equation (18). This will be clone as in Ch. I, § 16. Draw HTQ parallel to AO and intersecting OC pro- duced at T. Make TQ = AH'. Draw a circle having as diameter the line BQ joining B = (0, 1) with Q = (21, lOi). The abscissas ON and OM of the intersections of this circle with the a;-axis OT are the roots of (18). Hence the larger root Vi is OM = 2 cos 2 7r/17. Let the perpendicular bisector LP of OM cut the initial circle of unit radius at P. Then 2t cos LOP — OL — cos 17 LOP^^- Hence the chord CP is a side of the inscribed regular polygon of 17 sides, constructed with ruler and compasses. 1. For 71 = 5, EXERCISES 2, the periods are r + r^, r- + r^. Show that they are the roots of the z-quadratic obtained in Ex. 2, p. 83. 2.t For n = 13, find the least g, form the three periods each of four terms, and find the cubic havhig them as roots. 3. For ?( = 5, E.\. 1 gives r + /■■* = 2 cos 2 tt/o = 5 ( V 5 — 1). In a circle of radius unity and center draw two perpendicular diameters ADA', BOB'. With the middle point .1/ of OA' as center and radius MB draw a circle cutting OA at C (Fig. 21). Show that OC and BC are the sides sio and ss of tlie inscribed regular decagon and pentagon rcspectivel3\ Hints: MB - I V5, OC = I {VE - 1), BC = Vi + OC- = h ^/lo -2V5, Sio = 2 sin 18° = 2 cos ^ = OC, o si" = (2 sin 3G°)2 = 2 [1 - cos^-j = i (lO - 2 V^), s, = BC. 7. t Regular Polygon of n Sides. If n be a prime such that n — 1 is a power 2'' of 2 (as is the case when n = 3, 5, 17), the n — 1 imaginary nth roots of unity can be separated into 2 sets each of 2''~^ roots, each of these sets subdivided into 2 sets each of 2''~- roots, etc., until we reach the 8 81 CONSTRUCTION OF REGULAR POLYGONS 89 sets r, 1/r and r-, l/r-, etc., and in fact * in such a manner that we have a series of quadratic equations, the coefficients of any one of which depend only upon the roots of quadratic equations preceding it in the series. Note that this was the case for n = 17 (§ 5) and for n = 5. It is in this manner that it can be proved that the roots of x" = 1 can be found in terms of square roots, so that a regular polygon of n sides can be inscribed by ruler and compasses, provided n be a prime of the form 2'' + 1. If n be a product of distinct primes of this form, or 2^ times such a prod- uct (for example, n — 15, 30 or 6), or if n = 2'" (w > 1), it follows readily that we can inscribe by ruler and compasses a regular polygon of n sides. But this is impossible for other values of 7i. This impossibility will be proved for n = 7 and n = 9, the method of proof being applicable to the general case. 8. Regular Polygons of 7 and 9 Sides; Trisection of an Angle. For brevity we shall occasionally use the term " construct " for " construct by ruler and compasses." If it were possible to construct a regular poly- gon of 7 sides and hence angle 2 7r/7, we could construct a line of length 2 cos 2 7r/7, the base of a right-angled triangle whose hypotenuse is of length 2 and one of whose acute angles is 2 t/7. Set Then 27r , . . 27r r = cos-=- -f- 1 sm -=-• 1 27r . . 27r , 1 ^o 27r - = cos — I sm _- > r -\- - = 2 cos -^ • r ^7 7 ^ r 7 Hence 2 cos 2 7r/7 is a root of the cubic equation (10). This equation has no rational root. For, if it had a rational root, it would have (Ch. VI, §8, §5) an integral root which is a divisor of the constant term — 1, whereas neither + 1 nor — 1 is a root. Hence we shall know that it is im- possible to construct a regular polygon of 7 sides by ruler and compasses as soon as we have proved (§10) the next theorem. * See the author's article "Constructions with ruler and compasses; regular poly- gons," in Monographs on Topics of Modern Mathematics, edited by J. W. A. Young, Longini^ns, Green and Co., New York, 1911, p. 374. In addition to the references there given (p. 386), mention should be made of the book by Klein, Elementarmathematik vom Hoheren Standpunkle aus, Leipzig, 1908, vol. 1, p. 125; and ed. 2, 1911. 90 THEORY OF EQUATIONS ICh. vm Theorem. It is not possible to construct by ruler and compasses a line whose length is a root of a cubic equation with rational coefficients but having no rational root. This theorem shows also that it is not possible to construct a regular polygon of 9 sides and hence that it is not possible to construct the angle 40° by ruler and compasses. Indeed, if r = cos 40° + i sin 40°, then r + 1/r = 2 cos 40° is a root (Ex. 7, p. 83) of 2^ - 3 2 + 1 = 0. The same equation follows also from the identity cos 3 A = 4 cos^ A — 3 cos A by taking A = 40°, replacing cos 120° by its value — ^, and setting 2 = 2 cos 40°. Since neither divisor 1 nor — 1 of the constant term is a root of the 2-cubic, there is no rational root. Corollary. It is not possible to trisect every angle by ruler and compasses. Indeed, angle 40° cannot be constructed, while angle 120° can be. 9. Duplication of a Cube. Another famous problem of antiquity was the construction of a cube whose volume shall be double that of a given cube. Take the edge of the given cube as the unit of length and denote by X the length of an edge of the required cube. Then a;^ — 2 = 0. Since no one of the divisors of 2 is a root of this cubic equation, the theorem stated in § 8 implies the impossibility of the duplication of a cube by ruler and compasses. 10. t Cubic Equations with a Constructible Root. It remains to prove the theorem in § 8 from which we have drawn such important conclusions. Suppose that (20) x^ + a.T2 -^ j3x-\-y = (a, /3, y rational) is a cubic equation having a root Xi such that a line of length Xi or — .Ti can be constructed by ruler and compasses. We shall prove that one of the roots of (20) is rational. The construction is in effect the determination of various points as the intersections of auxiliary straight lines and circles. Choose rectangular axes of coordinates. The coordinates of the intersection of two straight lines are rational functions of the coefficients of the equations of the two § 10) GEOMETRICAL CONSTRUCTIONS 91 lines. To obtain the coordinates of the intersection of the straight Une y = mx + b with the circle (.T - pY + (y - q)- = r2, we eliminate y and obtain a quadratic equation for x. Thus x, and hence also y, involves no irrationality (besides irrationalities in ni, b, p, q, r) other than a square root. Finally, the intersections of two circles are given by the intersections of one of them with their common chord, so that this case reduces to the preceding. Hence the coordinates of the various points located by the construction, and therefore also the length ± Xi of the seg- ment joining two of them, are found by a finite number of rational operations and extractions of real square roots, performed upon rational numbers and numbers obtained by earlier ones of these operations. If Xi is rational, (20) has a rational root as desired. Henceforth, let Xi be irrational. Then .Xi is the quotient of two sums of terms, each term being a rational number or a rational multiple of a square root. A term may involve superimposed radicals as r = VlO - 2\/5, s = VlO + 2 V5, ^ = \/4 - 2 V3. But t equals Vs — 1 and would be replaced by that simpler value. As a matter of fact, r is not expressible rationally * in terms of a finite number of square roots of rational numbers, and is said to be a radical of order 2. A term having n superimposed radicals is of order n if it is not expressible rationally in terms of radicals each with fewer than n super- imposed radicals. In case a:i = 2 r — 7 s, we would express Xi in the form 2 r — 28 Vs/r, involving a single radical of order 2; indeed, rs = 4 Vs. If Xi involves Vs, Vs and Vlo, we replace Vl5 by Vs • Vs. We may therefore assume that no one of the radicals of highest order n in Xi is a rational function with rational coefficients of the remaining radicals of order n and radicals of lower order, that no one of the radicals of order n — 1 is a rational function of the remaining radicals of order n — \ and radicals of lower order, etc. Let Vk be a radical of highest order n in a;i. Then a + bVk Xi = 7=> c-^dVk * That is, as a rational integral function with rational coefficients. 92 THEORY OF EQUATIONS [Ch. viil where a, . . . , d do not involve Vk, but may involve other radicals of order n. U d 9^ 0, Vk 9^ c/d, in view of the preceding assumption. Thus we may multiply the numerator and denominator oi Xihy c — d Vk. Hence, whether d 9^ or d = 0, we have Xi = e+fVk (f^O), where neither e nor / involves Vk. Since Xi is a root of (20), we have A -\- B Vk = 0, where A and B are polynomials in e, f, k, a, /3, 7. li B 9^ 0, we could express Vk as a rational function —A/B of the remaining radicals in the initial Xi. Hence B = and therefore ^4. = 0. But the result of substituting e — f Vk for x in the cubic function (20) is evidently A — B Vk. Hence X2 = e — f Vk is a new root of our cubic equation. The third root is X2 = —a — Xi — Xo = — a — 2e. Now a is rational. If e is rational, X3 is a rational root of (20), as desired. The remaining case is readily excluded. For, if e is irrational, let Vs be one of the radicals of highest order in e. Then, as above, X3 = (j-\-hV~s {h 9^ 0), where neither g nor h involves Vs, while g — h Vs is a root 9^ Xs of (20), and hence identical with Xi or X2. Thus e ± / Vk = g — h Vs. Now Vs and all the radicals appearing in g, h, s occur in .T3 and hence in e. But Vk is not expressible in terms of the remaining radicals of Xi. We have now proved that if the constructible root .ri of (20) is irra- tional, there is a rational root x^. 11. t Problems such as the triscction of any angle can often bo solved by means of certain curves. We note, however, that there exists no plane curve, other than a conic section, whose intersections b}^ an arbitrary straight line can be found by ruler and compasses.* * J. Peterson, Algtbraische Gleichungen, p. 169. CHAPTER IX Isolation of the Real Roots of an Equation with Real Coefficients 1. Method of RoUe.* There is at least one real root of f'{x) = be- tween two consecutive real roots a and b of f(x) = 0. For, the graph oi y = f{x) has a bend point between a and b. Corollary. Between two consecutive real roots r and s of fix) = 0, lies at most one real root of f{x) = 0. For, if there were two such real roots a and b of the latter equation, the first theorem shows that/'(^) = would have a real root between a and b and hence between r and s, contrary to hypothesis. Now f{x) = has a real root between r and s if /(r) and f(s) have oppo- site signs (Ch. I, § 12). Hence the Corollary gives the Criterion. If r and s are consecutive real roots off'{x) = 0, thenf{x) = has a single real root between r and s if and only if f{r) and f{s) have opposite signs. At most one real root of f{x) = is greater than the greatest real root of f'{.x) = 0, or less than the least real root of f'{x) = 0. The final statement follows at once from the first theorem. Example. For f{x) = 3 x^ - 25 .c^ + 60 a; - 20, tV/'(x) = x" - 5 a;2 + 4 = (.r^ - l)(a;2 - 4). Hence the roots of /'(.c) = are ±1, ±2. Now /(_^)=_^, /(_2)=-36, /(-l)=-58, /(1) = 18, /(2)=-4, /(+^)=+^. Hence there is a single real root in each of the intervals (-1,1), (1,2), (2,+^), and two imaginary roots. The 3 real roots are positive. 2. The first theorem of § 1 is a special case of RoUe's Theorem. Between two consecutive roots a and b of f(x) = 0, there is an odA number of real roots of f'{x) = 0, a root of multiplicity m being counted as m roots. * Traite de I'alyebrc, Paris, 1G90. Hudde knew the method in 1659. 93 94 THEORY OF EQUATIONS [Ch. ix We may argue geometrically, noting that there is an odd number of bend points between a and h, the abscissa of each being a root ol J'{x) = of odd multiplicity, while the abscissa of an inflexion point with a hori- zontal tangent is a root of J'{x) = of even multiplicity. To give an algebraic proof, let fix) = {x- ay{x - hYQix), a for x = h, and hence vanishes an odd number of times be- tween a and b (Ch. I, § 12). But, in the left member, (x — a)(x — b) and f{x) remain of constant sign between a and b, since f{x) = has no root between a and b. Hence /'(.r) vanishes an odd number of times. Corollary. If f{x) = has only real roots, f'{x) = has only real roots distributed as follows: an (m — l)-fold root equal to each m-io\d root oi f(x) = ioT 7n ^ 2; a single root, which is a simple root, between two consecutive roots of /(.r) = 0. For, if the roots of /(.r) = are a, b, c, . . . , arranged in ascending order, of multiplicities r, s, t, . . . , respectively, then a, b, c, . . . are roots of f'{x) = of multiplicities r — 1, s — 1, t — 1, . . . , and between a and b lies at least one real root of /'(.r) = 0, etc. The number of these roots of /'(.r) = is thus at least (r-l) + l + (.s-l) + l+(i-l)+ . . . =r + s + t+ ■ ■ ' -1 = 11 -I, if n is the degree of /. But /' is of degree n — 1 and hence has only these roots. Thus only one of its roots lies between a antl b. EXERCISES 1. .r^ — 5 .c + 2 = has 1 negative, 2 positive and 2 imaginary roots. 2. x* + •?■ — 1 = has 1 negative, 1 positive and -i imaginary roots. 3. x^ — 3 .r'' + 2 X- — 5 = has two iinaginarv roots, and a real root in each of the intervals (-2, -1.5), (-1.5,- 1), (1, 2). 4. fix) = 4 .T^ — 3 .r-» - 2 .1-2 + 4 .r - 10 = has a single real root. Hint: Fix) = I fix) = 5 .r" - 3 .r^ - .r + 1 = has no real root, since F'ix) = ha.s a single real root and for it F is positive. 5. If /(^"^X-r) = has imaginary roots, fix) = has imaginary roots. 0. U fix) = has exactly r real roots, tlie number of real roots of /(.r) = is r + 1 or is less than r + 1 by an even number, a root of multiplicity ?/i being counted as m roots. 3] ISOLATION OF REAL ROOTS 95 3. Sturm's Method. Let f(x) = be the given equation with real coefficients, and f'{x) the first derivative of f{x). The first step of the usual process for seeking the greatest common divisor of f{x) and f'(x) consists in dividing/ by/' until we obtain a remainder r{x), whose degree is less than that of /'. Then, if qi is the quotient, we have / = qj' + r. We write /2 = —r, divide /' by/2, and denote by/3 the remainder with its sign changed. Thus / = (Zi/' - h J' = (hh - h h = q^f-s -fi,.... The latter equations, in which each remainder is exhibited as the nega- tive of a polynomial fi, yield a modified process, just as effective as the former process, for finding the greatest common divisor G oi f{x) and/'(x) if it exists. Suppose that —fi is the first constant remainder. If /» = 0, then /a = G, since /s divides fi and hence also /' and / (by using our equations in reverse order) ; while conversely, any common divisor of / and /' divides /2 and hence also fz. But if fi is a constant 5^ 0, / and /' have no common divisor involving x. This case arises if and only ii f{x) = has no multiple root (Ch. I, § 7), and is the only case considered in §§4-6. Before stating Sturm's theorem in general, we shall state it for a numerical case and illustrate its use. Example. f{x) = x^ + 4 x^ - 7. Then /' = 3 .c^ + 8 x, f={\x+t)f' -f„ /2 = «^x + 7, r = (Hx + fwv)/!-/!, f^ = m\. For x = \, the signs of /, /', fo, fz are h + +, showing a single variation of consecutive signs. For x = 2, the signs are + + + +, showing no variation of signs. Sturm's theorem states that there is a single real root between 1 and 2. For X = —00, the signs are 1 h, showing 3 variations of signs. The theorem states that there are 3 — 1 = 2 real roots between —00 and 1. Similarly, X Signs Variations - 1 -2 -3 -4 1 ++ 1 + + 1 1 111 + + + + + 1 2 2 3 Hence there is a single real root between —2 and —1, and a single one between —4 and —3. Each real root has now been isolated since we have found two num- bers such that a single real root lies between these two numbers or equals one of them. 96 THEORY OF EQUATIONS ICh. ix 4. Sturm's Theorem. Let f{x) = be an equation with real coejficients and without multi'ple roots. Modify the usual process for seeking the great- est common divisor of f{x) and its first derivative* fi(x) by exhibiting each remainder as the negative of a 'polynomial fi: (1) / = Qlfl - fh /l = (hh - h /2 = ^3/3 -fi,..., fn-2 = Qn-lfn-l " /n, where ** /„ is a constant 5^ 0. If a and b are real numbers, a < b, neither a root of f{x) = 0, the number of real roots of f{x) = between a and b equals the excess of the number of variations of signs of (2) fix), f,{x), f,{x), . . . , fn-,(x), /„ for x = a over the number of variations of signs for x = b. Terms which vanish are to be dropped out before counting the variations of signs. For brevity, let V^ denote the numl^er of variations of signs of the num- bers (2) when a; is a particular real number not a root of /(.r) = 0. First, if Xi and x^ are real numbers such that no one of the continuous functions (2) vanishes for a value of x between Xi and Xo or for x = Xi or X = X2, the values of any one of these functions for x = Xi and x = X2 are both positive or both negative (Ch. I, § 12), and therefore F^^ = V^^. Second, let p be a root of /,(.r) = 0, where \ ^ i < n. Then (3) f^-.{x) =5-/.(-r)-/.+i(.r) and the equations (1) following this one show that fi-i(x) and fi(x) have no common divisor involving x (since it would divide the constant /„). By hypothesis, fi(x) has the factor x — p. Hence fi-i{x) does not have this factor x — p. Thus, by (3), /,_i(p) = -/.+i(p) ^ 0. Hence, if p is a sufficiently small positive number, the values of fi-iix), /.(.t), fi+i(x) for X = p — p show just one variation of signs, since the first and third values are of opposite signs, and for x = p -\- p show just one variation of * The notation /i instead of the usual /', and similarly /o instead of /, is used to reg- ularize the notation of all the /'s, and enables us to write any one of the equations (1) in the single notation (3). ** If the division process did not yield ultimately a constant remainder ?^ 0, / and/i would have a common factor involving x, and hence f{x) = a multiple root. § 41 ISOLATION OF REAL ROOTS 97 signs, and therefore show no change in the number of variations of sign for the two values of x. It follows from the first and second cases that F„ = V^ ii a and (3 are real numbers for neither of which any one of the functions (2) vanishes and such that no root of f(x) = lies between a and /3. Third, let r be a root of f{x) = 0. By Taylor's Theorem (8) of Ch. I, Kr-p) = -vnr) + hvT{r)- . . . , fir + v)= Pf'ir) + i pT'ir) + . . . . If p is a sufficiently small positive number, each of these polynomials in p has the same sign as its first term. For, after removing the factor p, we obtain a quotient of the form ao + s, where s = aip -\- a2p^ + . . . is numerically less than ao for all values of p sufficiently small (Ch. I, end of § 11). Hence if /'(r) is positive, /(r — p) is negative and /(r + p) positive, so that the terms f{x), /i(x) = f'{x) have the signs \- for X = r — p and the signs + + for x = r + p. If /'(r) is negative, these signs are -\ and — — respectively. In each case, f(x), fi{x) show one more variation of signs ior x = r — p than for x = r -{- p. Evidently p may be chosen so small that no one of the functions fi{x), ...,/„ vanishes for either x = r — p or x = r -\- p, and such that fi{x) does not vanish for a value of x between r ~ p and r -\- p, so that f{x) = has the single real root r between these limits (§1). Hence by the first and second* cases, /i, . . . , fn show the same number of variations of signs ior x — r — p and X = r -{- p. Thus, for the entire series of functions (2), we have (4) Vr-, - Vr+, = 1. The real roots of f{x) = within the main interval from a to 6 (i.e., the aggregate of numbers between a and b) separate it into intervals. By the earlier result, Vx has the same value for all numbers in the same interval. By the present result (4), the value of V^ in any interval ex- * The argument in the second case when applied for i = I requires the use of /o = / and hence does not indicate the variations in a series lacking /. To avoid the necessity of treating this case i = 1, we restricted p further than done at the outset so that fi{x) shall not vanish between r — p and r + P- This necessary step in the proof is usually overlooked. Moreover, we have not adopted the usual argument based upon the continuous change of x from a to b, in view of the ambiguity of Vx when a; is a root of /(x) = 0, etc. 98 THEORY OF EQUATIONS fCa. IX ceeds the value for the next interval by unity. Hence Va exceeds Vb by the number of real roots between a and b. Corollary. If a < 6, Va— Vb. EXERCISES Isolate by Sturm's theorem the real roots of 1. .T^ + 2 X + 20 = 0. 2. x^ + X - 3 = 0. 5. Simplifications of Sturm's Functions. In order to avoid fractions, we may first multiply /(.t) by a positive constant before dividing it by Ji{x), and similarly multiply /i by a positive constant before dividing it by f2, etc. Moreover, we may remove from any Ji any factor ki which is either a positive constant or a polynomial in x positive for * a = x ^ h, before we use that fi as the next divisor. To prove that Sturm's theorem remains true when these modified functions/, Fi, . . . , Fm are employed in place of functions (2), consider the equations replacing (1): fi = kyF,, c,f=q,F,-hF,, c,F, = q,F,-hF,, CiF-l = 53^3- kiFi, . . . , CmF,n-2 = qm-\F m-l — k„,F,„, in which Cs, Cs, . . . are positive constants and F,„ is a constant ^ 0. A common divisor (involving x) of Fj-i and F,- would divide Fi-o, . . . , Fi, Fi, f, /i, whereas f(x) = has no multiple roots. Hence if p is a root of Fi{x) = 0, then F,-i(p) t^ and Ci+iFi_i(p) = -/vi+i(p) Fi+i{p), c,>i > 0, ki+i{p) > 0. Thus Fi-i and Fi+i have opposite signs for x = p. We proceed as in § 4. Example 1. If f{.v) = .r^ + 6 x — 10, /i = 3 (x- + 2) is always positive. Hence we may employ / and Fi = 1. For x = — oo, there is one variation of signs; forx= +x, no variation. Hence there is a single real root ; it lies between 1 and 2. Example 2. U fix) = 2 x^ — 13 x- — 10 x — 19, we may take /i = 4 x^ - 13 X - 5.. Then 2f=xfi-h h= 13x-+15x + 38= 13(x+U)^+HI^ * Usually we would require that ki be positive for all values of x, since we usually wish to employ the limits — oo and -{-cc. § 6] ISOLATION OF REAL ROOTS 99 Since jz is always positive, we need go no further (we may take F2 = 1). For X = —, U = -2 p.c - 3 g, 4 fj, == (-6 2;x 4- 9 ci)U - h U = -4 p' - 27 q% so that /s is the discriminant A (Ch. HI, § 3). Let [p] denote the sign of p. Then the signs of/, /i, ft, fs are - 4- + [p] [A] for X = -cc, + + - [p] [A] for X = 4-^. For A negative there is a single real root. For A positive and therefore p negative, there are 3 distinct real roots. For A = 0, /2 is a divisor of /i and /, so that X = — 3 g/(2 /;) is a double root. 8. If one of Sturm's functions has p imaginary roots, the initial equation has at least p imaginary roots. (Darboux.) 6. Sturm's Functions for a Quartic Equation. For the reduced quar- .tic equation /(2) = 0, / = z'^ -\- qz"^ -{- rz -\- s, (5) • f,^4:z'^2qz + r, /o = — 2qz' — 3rz — 4:S. Let q 9^ and divide cffi by /o. The negative of the remainder is (6) U = Lz- 12 rs - rq^, L = Sqs-2q^- 9r\ Let L 5^ 0. Then f^ is a constant which is zero if and only if / = has multiple roots, i.e., if its discriminant A is zero. We therefore desire f^ expressed as a multiple of A. By Ch. IV, § 4, (7) A = -4 P3 _27 Q^ P = -4 s - |, Q = § gs - r^ - ^^^ ^. 100 THEORY OF EQUATIONS [Ch. IX. We may employ P and Q to eliminate (8) 4s=-P-|', r=== -Q-igP-Zzg'. We divide L'^f^ by (9) f, = Lz + 3rP, L^9Q + 4:qP. The negative of the remainder is (10) 18 r2gP2 - 9 r'^LP + 4 sL^ = r/A. The left member is easily reduced to q~\. Inserting the values (8) and replacing L^ by L(9 Q + 4 qP), we get - 18 qQP'- - 12 g2p3 _ y ^4 p2 _^ 2 qP'~L + S r/PL - 3 q'-QL. Replacing L by its value (9), we get q-1. Hence we may take (11) /. = A. Hence if qLl 7^ 0, we may take (5), (9), (11) as Sturm's functions. Denote the sign of q by [g]. The signs of Sturm's functions are + - - {q\ - \L\ [A] for X = - X, + + -[?] {lA [A] for .r = + X. First, let A > 0. If g^, and that P is negative by (7), so that each term of (10) is = 0, whence A > 0. Hence, if gLA 5^ 0, there are four distinct real roots if and only if A and L are positive, and q negative; two distinct real and two imaginary roots if and only if A is negative. See Ex. 5 below. EXERCISES 1. If f/A 5^ 0, L = 0, then /s = 3 rP is not zero and its sign is immaterial in determining the number of real roots: two if 5 < 0, none if g > 0. By (10), 5 has the same sign as A. §0] ISOLATION OF REAL ROOTS 101 2. U rA 9^ 0, q = 0, obtain —fs by substituting z =-- — 4 s/(3 i) in /.. .Show that we may take/s = rA and that there are just two real roots if A < 0, no real root if A > 0. 3. li A ^ 0, q = r — 0, there are just two real roots if A < 0, no real root if A > 0. Since A = 256 s^, check by solving z* -\- s = 0. 4. If A 5^ 0, qL = 0, there are just two real roots if A < 0, no real root if A > 0. [Combine the results in Exs. 1-3.] 5. If A < 0, there are just two real (distinct) roots; if A > 0, g < 0, L > 0, four distinct real roots; if A > and either g = orL = 0, no real root. [Com- bine the theorem in the text with that in Ex. 4.] 6. Apply the criterion in Ex. 5 to Exs. 2, 4, 6, p. 99. 7. Apply to Exs. 1-3, p. 39, and Exs. 1-4, p. 43. 8. Show that the criterion of Ex. 5 is equivalent to the theorem in Ch. IV, § 7. If A > 0, L > 0, g < 0, then 4 s - g^ < by (6). Conversely, if A > 0, g < 0, 4 s - g2 < 0, then L > 0. For, if L = 0, 9 Q = -4 gP < 0, since P < by the value (7) of A. Thus 81 Q^ S 16 q'~P'-, A = 8, where 5 = -4 P3 _ i_(i g2p2 = 4 P2(- P - 1 g2) = 4 P2(4 s - g2) < 0, — P having been replaced by its value in (7). Thus A < 0, contrary to hypothesis. The two criteria for four real roots are therefore equivalent. The criterion for 2 distinct real and 2 imaginary roots is A < in each theorem. By formal logic the criteria for no real root must be equivalent. 9. If a, /3, 7 are the roots of a cubic equation f(x) — 0, Sturm's functions* /) /i> h, /s equal products of positive constants by {x-a){x-l3){x-y), ^{x-l3)(x-y), ^^{a - ^X - y) , {a - py{a- yY{0 - y^. Why is it sufficient to prove this for a reduced cubic equation? Take / as in Ex. 7, p. 99. Proof is needed only for the third function. In it the coefficient of x equals 2 2a- — 2 2a/3 = — 6 p, while the constant is — Sa^Y + 6 a^y = —3 g — 6 g, by Ex. 1, p. 64. Thus the third function equals 3/2. 10. Sturm's functions for any equation with the n roots a, fi, . . . , ir, oo equal products of positive constants by (X - a) ... (.C - co), S(.C -0) . . . (X- co), ^{a - /3)2(x - y) . . . {x - c.), 2(a - py-ia - yYiff - yYi^ - 8) . . . {x - c.), . . . , (« - ^)2 . . . (^ _ „)2. Verify this for n = 4, using § 6. A convenient reference to a proof for any n is Salmon's Modern Higher Algebra, pp. 49-53. 11. There are as many pairs p of imaginary roots as there are variations of signs in the leading coefficients of Sturm's functions, i.e., p = F+«,. Hints: Of any two consecutive Sturm's functions, the one of even degree has the same signs for X = — 00 and x = + 00 , while the one of odd degree has opposite signs. * In Exs. &-12, it is assumed that there are ?i + 1 Sturm's functions for the equatiou of degree n. 102 THEORY OF EQUATIONS [Ch. IX Hence, for the two £uiiGtions, F-c« + V+m = 1. There are n pairs of coiisecu- tive Sturm's functions. . Hanoe F-to + V.+x — n, the degree of the equation. Subtract V-co — I +«= = '''> ^'^^ number of real roots. Thus 2 r+oo = n - r = 2 p. 12. B}' Exs. 10, 11, the number of pairs of imaginary roots is the number of variations of signs in the series 1, n, 2(a - /3)^ !(« - /3)-(a - 7)^/? " 7)', • • • , provided no one of these sums is zero. 7.t Sturm's Theorem for the Case of Multiple Roots. Let*/„(.r) be the greatest common divisor of f{x) and /i = f'{x). We have equations (1) in which /„ is now not a constant. The difference Va — Vi is the num- ber of real roots between a and b, each multiple root being counted only once. If p is a root of Ji{x) = 0, but not a multiple root of /(x) = 0, then fi-i{p) 7^ 0. For, if it were zero, x — p would by (1) be a common factor of/ and/i. We may now proceed as in the second case in § 4. The third case requires a modified proof only when r is a multiple root. Let r be a root of multiplicity m, m ^ 2. Then/(r), /'(r), . . . , /('«-')(r) are zero and, by Taylor's Theorem, fir + p) = ^ . ^ ^'". . ^^^ /^-KO + • • • , These have like signs if p is a positive number so small that the signs of the polynomials are those of their first terms. Similarly, f{r — p) and f'{r — p) have opposite signs. Hence / and /i show one more variation of signs for x = r — p than for x = r -\- p. Now {x — r)"""^ is a factor of /and/iandhence, by (1), of/2, . . . , fn- Let their quotients by this factor be (f>, 4>i, . . . , (i)n- Then equations (1) hold after the /'s are replaced by the 0's. Taking p so small that i(a:) = has no root between r — p and r -\- p, we see by the first and second cases in § 4 that i, . . . , „ show the same number of variations of signs for a; = r — p as for a; = r + p. The same is true for /i, . . . , /„ since the products of (^1, ... , ^n by {x — r)""^ have for a given x the same signs as !..., <^„ or the same signs as — (/>i, . . . , —„. But the latter series evidently shows the same number of variations of signs as ^i, . . . , <^„. Hence (4) is proved and consequently the present theorem. * The degree of /(x) is not n, nor was it necessarily n in § 4. §81 ISOLATION OF REAL ROOTS 103 EXERCISES l.f For/ = x' - 8 .1-2 + IG, U = -i"' - ^ -i-, h = -c- - 4, /i = a/2. Hence w = 2. Then F_oo = 2, F* = 0, and there are only two real roots, each a double root. 2.t / = (.c - \Y{x - 2). 3.t {x - \Y{x + 2)^ 4.t x^ - a;^ - 2 x + 2. S.f Budan's Theorem. Lef a and b be real numbers, a < b, neither a root of f{x) = 0, ayi equation of degree n with real coefficients. Let Va denote the 7iumber of variations of signs of (12) fix), fix), f"{x), . . . , fi"){x) for X = a, after vanishing terms have been deleted. Then Va — Vb is either the nu7nber of real roots of f{x) = between a and b or exceeds the number of those roots by an even integer. A root of multiplicity m is here counted as m roots. In case Va — Vb is or 1, it is the exact number of real roots between a and b. In other cases, it is merely an upper limit to the number of those roots. While therefore the present method is not certain to lead to the isolation of the real roots, it is simpler to apply than Sturm's method. Indeed, for an equation of degree 6 or 7 with simple coefficients, Sturm's functions may introduce numbers of 50 or more figures. The proof is quite simple if no term of the series (12) vanishes for re = a or for X = 6 and if no two consecutive terms vanish for the same vdue of x between a and b. Indeed, if no one of the terms vanishes for Xi ^ X ^ X2, then Vx, = F^-,, since any term has the same sign for x = Xi as for X = Xi. Next, let r be a root of f^'^(x) — 0, a < r < b. By hy- pothesis, the first derivative f^'~^^^(x) of f-'^ix) is not zero for x^r. As in the third step (now actually the case i = 0) in § 4, /^'K^) and /'^'+^)(^) show one more variation of signs for .r = r — 7? than for x = r -\- p, where p is a sufficiently small positive number. If i > 1, /(') is preceded by a term /(i-i) in (12). By hypothesis, f'-'~'^'>(x) 5^ for x = r and hence has the same sign for x = r — p and x = r -\- p when p is sufficiently small. For these values of x, /(*K-^) h^-s opposite signs. Hence f^'~^^ and /(*') show one more or one less variation of signs for x = r — p than for X = r + p, so that /^*~'^ f^'\ /^'+^^ show two more variations or the same number of variations of signs. Next, let no term of the series (12) vanish for a: = a or for a; = 6, but let several successive terms (13) /('X^), f^'+'Kx), . . . , f^'+^--'Kx) /(i) /(i+1) /(H h (-1)/ (-l)-l (-1) h + + + 104 THEORY OF EQUATIONS [Ch. ix all vanish for a valiio r of x between a and h, while /('+'^ (r) is not zero, say positive.* Let U be the interval between r — p and r, and lo the interval between r and r -\- p. Let the positive number j) be so small that no one of the functions (13) or /('+^)(x) is zero in these intervals, so that the last function remains positive. Hence /('+'~')(a;) increases with X (since its derivative is positive) and is therefore negative in /i and positive in h. Thus Z^'+'^-K^) decreases in /i and increases in 1-2 and hence is positive in each interval. In this mamier we may verify the signs in the following table: . . . /(''+J-3) /('+/-2) /(*•+/-!) f(i+i) ... - + - + . . . + + + + Hence these functions show j variations of signs in 7i and none in /o. If i > 0, the first term of (13) is preceded by a function /('"^^(x) which is not zero for x = r, and hence not zero in /i or I2 if p is sufficiently small. If J is even, the signs of /('~^) and/(') are + + or 1- in both /i and L2, showing no loss in the number of variations of signs. If j is odd, their signs are /i + - or /2 + + - + SO that there is a gain or loss of a single variation of signs. Hence show a loss of j variations of signs if j is even, and a loss of j ± 1 if j is odd, and hence always a loss of an even number = of variations of signs. If i = 0, /(^) = / has r as a j-fold root and the functions in the table show j more variations of signs for a: = r — p than ior x = r -}- p. Thus, when no one of the functions (12) vanishes for x = a or for .r = b, the theorem follows as at the end of § 4 (with unity replaced by the mul- tiplicity of a root) . Finally, let one of the functions (12), other than /(.r) itself, vanish for x = a or for x = b. If 5 is a sufficiently small positive number, all of the N roots of f{x) = between a and 6 lie between a + 5 and b — 5, and * If negative, all signs in the table below are to be changed; but the conclusion holds. §9] ISOLATION OF REAL ROOTS 105 for the latter values no one of the functions (12) is zero. By the above proof, Va+s - Vo-' = N-{-2t, where t, j, s are integers ^ 0. Hence Va — Vb = N -\- 2 (t -\- j -\- s). Example. For/ = x^ — 7 x — 7, /' = 3.^2-7, /" = 6.r, f"' = (j. There is one variation of signs for x = 3, but none for x = 4, so that just one real root lies between 3 and 4. For / /' /" /'" -2 - 1 — 1 +5 — 12 +6 3 variations — 1 — 4 — 6 +6 1 variation. Thus there are two real roots or no real root between —2 and —1. The former is the case. The reader should isolate the two roots by finding an intermediate value of X for which the series shows two variations of signs. EXERCISES Isolate by Budan's theorem the real roots of l.t x?-x'^-2x + l= 0. 2.t x^ + 3x'^-2x-5 = 0. 3.t If /(a) y^ 0, Va equals the number of real roots > a or exceeds that number by an even integer. 4.t There is no root greater than a number making each of the functions (12) positive, if the leading coefficient of /(.r) is positive. (Newton.) 5.t Divide /(x) = x" + aix"~^ + • • • by x — a; then fix) =(x--a)l.c"-i + .i-"-V(«)+ • • • + gn-xicc)l+f{a), where gi{a) =« + «!, giia) = «- + cha + a2, . . . . If a is chosen so that ^i(a), . . . , Qn-iiot), /(a) are all positive, no positive root of f{x) = exceeds a. (Laguerre.) 9. Descartes' Rule of Signs. The number of positive roots of an equation with real coefficients either equals the number V of variations of signs in the series of coefficients or is less than V by an even integer. A root of multiplicitij m is here counted as m roots. For example, x^ — Sx'^-\-x-\-l = has either two or no positive roots, the exact number not being found. But — 3 x^ -{- x + I = has exactly one positive root. 106 THEORY OF EQUATIONS ICh. ix Consider any equation with real coefficients f{x) = aox" + aix"-^ + • • • + a„_i.r + a„ = 0, with a„ 5^ 0. For x = the functions ( 1 2) have the same signs as dn, dn-l, . . . , fll, Go, SO that Vo = V. For a: = + x , the functions have the same sign (that of ao) . Thus Vo — V^ — V is either the number of positive roots or exceeds that number by an even integer. Next, the theorem holds if /(O) = 0, as shown by removing the factors x. Corollary. The number of negative roots of f{x) = is either the number of variations of signs in the coefficients of f( — x) or is less than that number by an even integer. Thus x^ — S x^ -\- X -\- 1 =0 has either two or no negative roots, since x^ — Z x^ — x -\- 1 =0 has two or no positive roots. EXERCISES 1. x^ — 3x + 2 = has one negative root and two equal positive roots, 2. 2^ -\- a^x + 6^ = has two imaginary roots if 6 ?^ 0. 3. For n even, x" — 1 = has only two real roots. 4. For n odd, x" — 1 = has only one real root. 5. For n even, x" + 1 = has no real root; for n odd, only one. 6. x** + 12 x^ + 5 X — 9 = has just two imaginarj'^ roots. 7. x** + aV + 6'X — c- = (c 9^ 0) has just two imaginary roots. 8. To find an upper limit to the number of real roots of /(x) = l:)ctween o and b, set a + by X = 1+2/ \ ■" b-x, multiply by (1 + ?/)", and apply Descartes' Rule to the resulting equation in y. 10. t Fourier's Method. If Budan's Theorem gives a loss of two or more variations of signs in passing from a to a larger value b, and hence leaves in doubt the number of real roots between a and b, we may employ a supplementary discussion. First, let /, /', /" show two variations of signs at a and no variation at b, while the series beginning wnth /" shows no loss in variations (as in the Example in § 8). Then /" is of constant sign between a and b, and the § 101 ISOLATION OF REAL ROOTS 107 graph oi y = f{x) has a (single) maximum or minimum point between a and b, according as/" is negative or positive. If the sum m _ m of the subtangents at the points with the abscissas a and 6 is > h — a, the tangents cross before meeting the x-axis and the graph does not inter- sect the X-axis between a and h, so that there are two imaginary roots in view of Budan's Theorem and (14) 71 = v_^-v^ = (7_.^ - Va) + (Va - V,) +:(n- VJ. In the contrary case, we examine the value half way between a and h, etc. Clearly the case of imaginary roots will disclose itself after a very few such steps. Next, in the general case, we shall find, after a suitable subdivision of the interval, three consecutive functions /(/), fU+i)^ /(;+2) showing two variations of signs at a' and no variation at b', while the later terms of the series show no loss in variations of signs. We may therefore decide as in the first case whether there are two real roots of fU) = in the interval [a', b'] or not, and in the latter alternative conclude that/ = has two imaginary roots.* Example. Let f{x) = x' - 5x* - 16 x^ + 12 x^- - 9 .c - 5. Then fix) = 5 x" - 20 x3 - 48 .1-2 + 24 X - 9, -i-/"(x) = 5 x3 - 15 x2 - 24 X- + 6, tV/'"(.c) =5.t2- lOx-8, jhf""{x) - X - 1, PK.C) = 120. There is just one real root in each of the intervals (— 3, —2), ( — 1,0), (7, 8). The interval (0, 1) is in doubt, the signs being - - + - - + for x- = 0, — — — — — + for X = 1, where is read — . The ./ of the text is here 1 . Now nil _ I'M = -48 . JL = 3 3 /"(I) /"(O) 4 (-28) ^4(6) 7'^8 ' * For further details, see Serret, Algcbre Superieure, ed. 4, I, pp. 305-31S. 108 THEORY OF EQUATIONS [Ch. IX so that we must subdivide the interval. For x = ^, the signs are the same as for x = 1. Thus the loss in variations of signs occurs in the interval (0, ^). Now ni) _ r(0)^ - UtI- 1 3 1 rU) /"(O) 4(-9i)"^8 2 Hence there are two imaginary roots. EXERCISES l.t x^ - 3 or* + 2 x^ — 8 x2 + 3 .r — 25 = has 4 imaginary roots. 2.1 X® + x^ — X* — x^ + X- — X + 1 = has G imaginary roots. CHAPTER X Solution of Numerical Equations 1. Newton's Method. To find the root between 2 and 3 of a:3 - 2 a: - 5 = 0, Newton * replaced re by 2 + p and obtained p3 + 6 p2 ^ 10 p - 1 = 0. Since p is a decimal, he neglected** the first two terms and set 10 p— 1 = 0, so that J) = 0.1, approximately. Replacing p by 0.1 + g in the preceding cubic equation, he obtained f/ + 6.3 g2 + 11.23 g + 0.061 = 0. Dividing —0.061 by 11.23, he obtained —0.0054 as the approximate value of q. Neglecting (f and replacing q by —0.0054 + r, he obtained 6.3 r2 + 11.16196 r + 0.000541708 = 0. Dropping 6.3 r-, he found r and hence rr = 2 + 0.1 - 0.0054 - 0.00004853 = 2.09455147. This value is in fact correct to the seventh decimal place. But the method will not often lead as quickly to so accurate a value of the root. The method is usually presented in the following form. Given that a is an approximate value of a real root of fix) = 0, we can usually find a nearer approximation a + ^ to the root by neglecting the powers h^, h^, . . . of the small number h in Taylor's formula /(a + /0=/(a)+/'(a)/i+r(a)|+ . . . and hence by taking f(a)+r(a)h = 0, h = ^j^' We then repeat the process with a + h in place of the former a. * Isaaci Newtoni, Opuscula, I, 1794, p. 10, p. 37 [found before 1676], ** At this early stage of the work it is usually safer to retain also the term in p^ and thus find p approximately by solving a quadratic equation. 109 110 THEORY OF EQUATIONS Thus in Newton's example, wc have, for a = 2, [Ch. X h = /'(2) =I0' «' = « + ^ = 2.1, -/(2.1 )^ -0.061 ^ _ f'(2.1) 11.23 0054 2. Graphical Discussion of Newton's Method. Using rectangular coordinates, consider the graph oi y = fix) and the point P on it with the abscissa OQ = a (Fig. 22). Let the tangent at P meet the x-axis at T Q T T, Fig. 22 Fig. 23 and let the graph meet the x-axis at S. Take h = QT, the subtangent. Then QP=Kn 2 and 7 (as sliown by the above examination of the terms in //•'' and y''). ExAMPLK 2. Find the root between —4 and —3, (correct to seven decimal places, of the equation in E.x. 1. Using the multipliers -4, +0.6, +0.008, we find that .r = -4 + 0.608 + y where ?/ - 6.176 if + 7.380992?/ - 0.004556288 = 0. Thus y just exceeds 0.0006. The sum of the terms in y^ and y- is -0.000002 to six decimal places. Carr^nng tlie division of 0.004558 by 7.3S1 until the sign of the remainder is in doubt, on account of the doubt in the sixth decimal place, we 5 5, 6] SOLUTION OF NUMERICAL EQUATIONS 119 get y — 0.0006175, with the shght doubt due to the approximate value of the divisor and that of the y- term. Since the cube of 6.176 is just less than 235.6 (as shown by logarithms), the sum of the terms in y^ and y'^ is —0.000002356 to nine decimal places. Cariying out the division of 0.004558644 by the exact coefficient of y, we get y = 0.0006176, correct to seven decimal places. Hence X = -3.3913823. EXERCISES 1. Find to 7 decimals the root of .i-^ + 4 .r- — 7 = between —1, —2. 2. Find to 7 decimals all the roots of .c^ — 7 x — 7 = 0. Find to 5 decimals all tlie real roots of 3. x^ + 2 a; + 20 = 0. 4. a;^ + 3 x^ - 2 a; - 5 = 0. 5. x^ + x'-2x- I = 0. 6. x^ + 4: x^ - 17.5 x^ - 18 a; + 5S.5 = 0. 7. x* - 11727 x + 40385 = 0. (G. H. Darwin.) 8. Find to 8 decimals the root between 2 and 3 of x^ — x — 9 = by making only three transformations. 5.t Without the intermediation of the idea of division by x — h, we may show directly that the process of § 4 yields the correct transformed equation. For simplicity, we take a cubic equation f(x) = ax^ -{- hx^ -\- ex -\- d = 0. Our process was as follows: a b c d 1^ ah ah" + hh ah^ + fe/i2+ ch ah + h ah ah"^ + 6/i + c 2 ah" + bh ah^ + bh" -\-ch-\-d = f{h) 2ah-{-b ah 3 a/i2 + 2 6/i + c = f'(h) a 3ah + b = hf'Qi) Hence the transformed equation is \r{h)v' + \f"(ji)f ^nh)p +/(/o = 0. The terms of the left member, read in reverse order, are those of Taylor's formula for the expansion of f{h -\- p). Hence the above process yields the equation obtained from f(x) = by setting x = h -{- p. 6.t Numerical Cubic Equations. After finding a real root r 5^ of fix) = x^ -\- bx"" -\- ex ^ d = 0, 120 THEORY OF EQUATIONS [Ch. X we may o})tain the remaining roots ri and r^ from ^1 + »'2 = —h — r, Tiro = = r' -j- br -\- c. We have (2) (ri - ro)^ = (ri + ro)^ - 4 rira = &2 _ 4 c - 2 6r - 3 r^. Thus ri — ro is either real or a pure imaginary. Making use also of ^1 + fo, we shall have the real or imaginary expressions of ri, Vo. As it would be laborious to compute the right member of (2), we may make use of a device. We have (ri-ro)2 = 62_3c_/'(r). The value of f'{r) for the approximate value of /• obtained at any stage of Horner's process is the coefficient preceding the last one in the next transformed equation (§5). Example. Let/(.r) = x^ + 4:X- — 7. By Ex. 1, p. 117, /'(1. 164) = 13.376688. If we continue Horner's process, using the multiplier m = 0.000248, and retaining only six decimal places, we see that we must twice add 7.492 m = 0.001858 to the preceding /' to get /'(?•) = 13.380404, r = 1.164248. But this continuation of Horner's process is unnecessary. Using J"'{x) = 6 and the work on p. 118, we have fix + m) = fix) + mf'ix) + 3 m^, ^/"(1.164) = 7.492, /'(r) = 13.37 . . . + 2 m (7.492) + 0.0000002 - 13.3804042. Hence we get (r, - r,)2 = 2.6195958, n - ro = 1.6185165, n + r, = -5.1642479, n = -1.7728657, r. = -3.3913822. Tf. Numerical Quartic Equations. Let fix) = X' + bx^ + cx2 + f/x + e = have two distinct real roots r and s. When these arc found approximately by Horner's process, we get at the same time f'{r), f'is), approximately. Call the remaining roots ri and ro. Then >'i + ^2 = — 6 — r — s, TiTi = c — ir -^ s)ii\ + fo) — rs = c -{- bir -\- s) -\- r- + rs + s^, in - r.,)- = 6'' - 4 c - 2 b(r + s) - 3 r^ - 2 rs - 3 .s^, in - r,)'~ib + 2 r + 2 s) = - in - r,y'i2 n + 2 r, + b) = 6^ - 4 6c - 8 c(r + &)+ 6( - 7 r- - 10 rs - 7 s-) - G r^ - 10 r's - 10 rs~ - 6 s\ §81 SOLUTION OF NUMERICAL EQUATIONS 121 To the second member add the jDroduct of 10 by H + r^s + rs^ + s^ + 6(r2 + rs + s-) -\- c(r -^ s) + d = -^MjIliM = q. r -- s Hence (ri - r.Yib + 2r + 2s)=63-4 6c + 8 d+fir) + /'(«)• From this equation we get n — r^ and then find Vi and ro, approximately. EXERCISES t 1. After finding one of the real roots of the cubic equations in Exs. 2, 3, 4, 5, 8, p. 119, find the remaining roots by § 6. 2. Treat the quartic equations in Exs. 6, 7, p. 119, by § 7. Find two and then all of the roots of 3. x^ + 12 X- + 7 = 0. 4. x' - 80 x3 + 1998 x^ - 14937 x + 5000 = 0. S.f Graffe's Method. First, let all of the n roots a:i, . . . , .r„ be real and distinct numerically. Choose the notation so that Xi exceeds X2 nu- merically and X2 exceeds X3 numerically, etc. In (3) 2.rr = ..-(l+|, + |=+-- each fraction approaches zero as m increases, so that Xi" is an approxi- mate value of Sxi"* if ?w is sufficiently large. Similarly, (4) 2xi"'X2- = x^'^xr ( 1 + ^ + — +••• + -^^^ + Xi'"- Xi'" Xi"'X2" so that Xi^Xi"^ is an approximate value of liXi^x-f^ for m large. Now x{^, . . . , Xn"" are the roots of (5) ?/" - So'i'" • y"-i + Sa;i'"X2'" • 2/""^- • • • = 0. As illustrated in the examples below, it is quite easy to form this equation (5) for values of m which are the successive powers of 2. After obtaining the equation in which m is sufficiently large, we divide each coefficient by the preceding coefficient and obtain approximate values of the nega- tives of Xi'", X'f, .... Indeed, the coefficients are approximately 1, — X{^, X{^X-f, —Xi'^X-^^^Xii^, .... Example 1. For x^ -\r x"^ — 2 x — \ — Q, we first form the cubic equation whose roots are the squares of the roots x\, x^, xz of the given equation. To this 122 THEORY OF EQUATIONS [Ch. X end, we transpose the terms x"^, — 1, of even degree, square, replace x- by y, and get* if - 5y'- + Gij - 1 = 0, whose roots are v/i = xr, ?/j = .r2-, //a = x-^-. Repeating the operation, we get ^-'^ - 13 z- + 26 2 - 1 = 0, y^ - 117 ?;- + 050 y - 1 = 0, with the roots Zi = y{', . . . , and Vi = zr, .... Hence the roots of the y-cubic are the 8th powers of .ri, X2, X3. By logarithms, the Sth roots of 117, ff?, sh (the approximate values of Xi^, x-i^, Xs^) are 1.81.3, 1.239, 0.4-150, which are therefore approximate numerical values of .ri, x-z, x-i. The next step gives the equation w' - 12389 ic- + 4222GG 10 -1=0. The 16th roots of 12389, etc., are - 1.80225, 1.24676, -0.44504, to which the proper signs have now been prefixed (their product being positive and sum being —1). Instead of repeating the process, we may now obtain as follows the values of the roots correct to five decimal places. We had the logarithms of the last approxi- mations to the roots and hence see at once that {xs/x-iY^ affects only the Sth decimal place and that (.r3/.ci)^^is still smaller. The coefficient of 10 is 2xi^®.r2>^ whose expression (4) involves only the first three terms. Hence .nisrois = 422266, correct to 7 decimal places. The reciprocal is .r3'^ whence Xs = —0.44504 to 5 decimal places. By the approximate values of .ri and .r-. from the it'-cubic, (xi/xiY^ = 0.002751. Thus 1.002751 .ri's = 12389 = S-ri", whence Xi = —1.80194 to 5 decimal places. By the displayed equations, ., 422266 X 1.002751 12389 .T2 = 1.24698. We have now found each root correct to five decimal places. As a check, note that the roots are (Ch. VIII, § 3, § 8) 27r ^ Air ^ 67r 2 COS—-, 2 COS-—, 2cos-;r- 7 7 7 The above process requires modification if several of the largest roots are equal or approximately equal numerically. If Xi and Xo are approxi- mately equal, but sufficiently different from x^, . . . , x^, numerically, an approximate value of .ri™ is 2 ^-Yi'"- Next, consider a cubic equation with two conjugate imaginary roots * We may use symmetric functions: 2(/i = 2xi- = (Sxi)- — 2 S.C1X2 = 5, etc. § 8] SOLUTION OF NUMERICAL EQUATIONS 123 X2 and xz, whose modulus (Ch. II, § 8) is r, and a real root Xi numeri- cally greater than r. Then the real number xf^.x^ is numerically less than or equal to the sum ^ /mod. X2 [ ±Xi of the moduli of its two parts, and hence approaches zero as m increases. Thus, by (3), an approximate value of Xi'" is 2xi"'. Example 2. For .c^ - 2 a; - 2 = 0, Xi > 1.7, X2X3 = r^ = 2/a;i. Since 2 < (1.7)^ r < 1.7 < Xi. Forming the equation whose roots are the squares of the roots of the x'-cubic, that whose roots are the fourth powers, etc., we get ?/3 - 4 ?/2 + 4 7/ - 4 = 0, z'-Sz''- IGz- 16 = 0, v^ - 96 i;2 - 256 = 0. Thus .ri is approximately \/96 = 1.7692 . By two more steps, we get a-i = ^^85032960 = 1.769293, correct to six decimal places. For a cubic equation in which Xi < r, we employ the equation in X obtained by setting x = l/X. Its root 1/xi exceeds numerically the mod- ulus 1/r of the imaginary roots l/Xi, l/xs. Hence the equation in X is of the type last discussed. EXERCISES t 1. The equation whose roots are the Sth powers of the roots Xi, X2, X3 of x^ — 4 x''^ — .r + 3 = is w^ - 74474 w^ + 46213 iv - 6561 = 0. Dividing the negative of each coefficient by the preceding coefficient and extracting the Sth root of each quotient, we get 4.06443, 0.94, 0.78. The first is a good approximation to Xi. The last two are approximately equal and hence not good approximations to —X2, X3. To avoid this inconvenience, add unity to each root (i.e., replace x by X — 1). Treat the equation in X and so obtain good approxi- mations to Xi, Xi, X3. 124 THEORY OF EQUATIONS [Ch. x Treat by the present methods 2. r' - 2 a; - 5 = 0. 3. x'' - 2 x- - 2 = 0. 4:. t^ + ix'- - 7 = 0. 5. a;3 + 2 X + 20 = 0. For further details on the determination of imaginary roots by this method, see Encke, Crelle's Journal, vol. 22 (1841), p. 193; and examples by G. Bauer, Vorlesungen ilber Algebra, 1903, p. 244; and C. Runge, Praxis der Gleichungen, 1900, p. 157. 9.t To determine the imaginary roots of an equation f{z) = with real coefficients, expand /(a: + yi) by Taylor's Theorem; we get Kx) +f'{x)yi - fix) t^- r{x) ,^^^+ • . . =0. Since x and y are to be real, and y 9^ 0, (6) m - fix) ^ + r'ix) ^ . 2^ 3 . 4 - • • • =0, f(x)-nx)j^+fi^Kx)f,- • • • =0. By eliminating y^ between these two equations, we obtain an equation E{x) = 0, whose real roots x may be found by one of the preceding methods. In general the next to the final step of the elimination gives y- as a rational function of x, so that each real x which yields a positive real value of y^ furnishes a pair of imaginary roots x ± yi of f{z) = 0. But if there are several pairs of imaginary roots with the same real part x, the equation in y^ used in the final step of the elimination will be of degree greater than unity in y-. Example. For f{z) = s-* — 3 + 1, equations (G) are x< - X + 1 - 6 xhf + y^ - 0, 4 x^ - 1 - 4 xy^ = 0. Thus ^' "^ ^' ~ 4^ ' " ^''' "^ ''■' "^ IB " ^• The cubic equation in x^ has the single real root x2 = 0.528727, x = ±0.72714. Then y^ = 0.87254 or 0.184912, and z = x + yi = 0.72714 ± 0.43001 i, - 0.72714 ± 0.93409 i. 10] SOLUTION OF NUMERICAL EQUATIONS EXERCISES t 125 1. For the quartic equation in Ch. V, § 1, eliminate i/ between equations X = 0, Y — 0, corresponding to the present pair (6), and get x{x - 2) (16 x' - 64 x^ + 136 x- -lUx + 65) = 0. Show that the last factor has no real root by setting 2 x = w -{- 2 and obtaining («'2 + l)(iv^ + 9) = 0. Hence find the four sets of real values x, y and hence the four complex roots x + yi. 2. If r and s are any two roots of f{z) = and we set r -\- s y 2i we have r = x -{- yi, s = x — yi, so that f{x ± yi) = 0. Hence E{x) = has as its roots the ^ n(n — 1) half-sums of the roots oif{z) = in pairs. If, however, we eliminate x between equations (6) and set — 4 y^ = ty, we obtain an equation in w whose roots are the § /t(n — 1) squares of the differences of the roots of/(2) = 0. 10 1- Lagrange's Method. The root between 1 and 2 of x^ + 4 a;2 - 7 = may be expressed as a continied fraction. Set x = 1 + 1/y. Then * -2i/-\-nif + 7y-{-l = 0. Since —2i/-\- 11 y- must be negative, we have y > 5. We find by trial that y lies between 6 and 7. Set y = 6 + 1/z. -2 11 -12 7 - 6 1 [6 6 -2 - 1 -12 1 -78 7 -2 -13 -12 -77 •25 2_25_77 z" z"- Z Since 7 2^ - 77 ^^ > 0, 2 > 11. Now a; = 1-f 7 2' - 77 22 - 25 2 - 2 = 0. The value of z lies between 11 and 12. 1 72 + 1 6 + 1 62 + 1 * We may of course first set x = 1 + rf, find the cubic equation in d, by our earlier method, and then replace d by \ly. 126 THEORY OF EQUATIONS ICh. X Using z = 11, we find that x is just smaller than 1.1642. But z is in fact just greater than 11.3. Using z = 11.3, we find that Hence a: = 1.1642 to four decimal places. There is a rapid method of evaluating a continued fraction and a means of finding the limits of the error made in stopping the development at a given place. For an extensive account of the theory and applications of continued fractions, see Serret's Coum d'Algcbre Superieure, ed. 4, I, pp. 7-85, 351-368. CHAPTER XI (1) Determinants; Systems of Linear Equations 1. In case there is a pair of numbers x and y for which \aix + hiij = ki, \ch.x^- hiij = ki, they may be found as follows. Multiply the members of the first equa- tion by 62 and those of the second equation by —61, and add the resulting equations. We get (0162 — a.<>h\)x = k^o — kihi. Employing the respective multipliers —ao and ai, we get (ttihi — a2hi)y = 01^2 — CL2ki. The common multiplier of x and y is (2) 0162 - aihi, which is called a determinant of the second order and denoted by the symbol * (2') 1 ai 61 1 ao 62 The value of the symbol is obtained by cross-multiplication and substrac- tion. Our earlier results now give (3) a2 62 X = ki 61 ^-2 62 1 ai hi a2 62 y = ai ki 02 /C2 • We shall call ki and ki the known terms of our equations (1). Hence, if D is the determinant of the coefficients of the unknowns, the 'product of D by any one of the unknowns equals the determinant obtained from D by substi- tuting the known terms in -place of the coefficients of that unknown. * The symbol for an expression should show explicitly all of the quantities upon whose values the value of the expression depends. Here these are ai, 61, a<>, h%. The advantage of writing these in the symbol (2') in the order in which they occur in the equations is that the symbol may be written down without an effort of memory by a mere inspection of the given equations. 127 128 THEORY OF EQUATIONS [Ch. XI Example. For 2 a; — 3 y = —4, 6 x — 2 ?/ = 2, we have 14x = 14, x = l, = 28, y = 2. 2 -3 -4 -3 6 -2 X = 2 —2 142/ = 2 -4 6 2 EXERCISES Solve by determinants the systems of equations 1. 8x-?/ = 34, 2. Sx + iy = 10, x + 82/ = 53. 4x + y =9. 3. «.c + ^i/ = a^ bx — ay = ab. 4. Verify that, if the determinant (2) is not zero, the values of x and y deter- mined by division from (3) satisfy equations (1). 2. Consider a system of three linear equations QiX + biy + Ciz = ki, (4) 023: + bojj + c^z = ki, CisX + fcg^ + CzZ = ks. Multiply the members of the first, second and third equations by * (5) 62C3 — hsCo, 63C1 — &1C3, 61C2 — 62C1, respectively and add the resulting equations. We obtain an equation in which the coefficients of y and z are found to be zero, while the coeffi- cient of X is (6) ai&2C3 — aih^C2 + a^b^Ci — a-ybic-i + ad>iC-z — a^h'^Cu Such an expression is called a determinant of the third order and denoted by the symbol (6') The nine numbers Oi, . . . , C3 are called the elements of the determi- nant. In the symbol these elements lie in three (horizontal) rows, and also in three (vertical) columns. Thus a-i, 62, c^ are the elements of the second row, while the three c's are the elements of the third column. * A simple rule for finding these multipliers is given in § 3. fll &1 Ci 02 62 Ci as 63 Cz DETERMINANTS 129 The equation (free of y and z), obtained above, is Gi 61 Ci /Cl 61 Ci 02 62 C2 a; = /v2 62 C2 as &3 C3 ^-3 &3 Cg since the constant member was the sum of the products of the expres- sions (5) by ki, ki, ks, and hence may be derived from (6) by replacing the a's by the A;'s. Thus the theorem of § 1 holds here as regards the value of X. 3. Minors. The determinant of the second order obtained by eras- ing (or covering up) the row and column crossing at a given element of a determinant of the third order is called the minor of that element. For example, in the determinant D given by (6'), the minors of ai, ao, az are A, 62 C2 hz Cz A^ = hi Ci h Cz &2 C2 respectively. The multipliers (5) are therefore Ai, —A2, Az. Hence the first results obtained in § 2 may be stated as follows : (7) D = aiAi - a^A-i + azAz, (8) 6iAi - 62A2 + Ms = 0, CiAi - C2A2 + C3^3 = 0. The minors of 61, &2, &3 in this determinant D are -Si = a2C3 — a3C2, B2 = ttiCz — azCi, Bz = a^c-z — aoCi. Multiply the members of the equations (4) by — -Bi, B2, —Bz, respectively, and add. In the resulting equation, the coefficients of x and z are seen to equal zero: (9) -ayBi + a^B. - azBz = 0, -CiBi + CoBo - CzBz = 0, while the coefficient of y is seen to equal the expression (6) : (10) D = -hBi + 62S2 - bzBz. Hence the theorem of § I holds here for the variable y. The reader should also verify that, if he uses the multipliers Ci, —C2, Cz, where C, is the minor of Ci in D, he obtains an equation in which the co- efficients of X and y are zero : (11) aiCi - a2C2 + azCz = 0, hd - 62C2 + 63C3 = 0, while the coefficient of z equals the expression (6) : (12) D = CiCi - C2C2 + C3C3, and then conclude that the theorem of § 1 is true as regards z. 130 THEORY OF EQUATIONS [Ch. XI 4. Expansion According to the Elements of a Column. Relations (7), (10), (12) are expressed in words by saying that a determinant of the third order may be expanded according to the elements of any column. To obtain the expansion, we multiply each element of the column by the minor of the element, prefix the proper sign to the products, and add the signed products. The signs are alternately + and — , as in the diagram. + - + - + - + - + 5. Two Columns Alike. A determinant * is zero if any two of its columns are alike. This is evident for a determinant of the second order: c c dd = cd — cd = 0. To prove it for a determinant of the third order, we have only to expand it according to the elements of the column not one of the like columns and to note that each minor is zero, being a determinant of the second order \vith two columns alike. EXERCISES Solve by determinants the systems of equations (expanding a determinant having two zeros in a column according to the elements of that column) : 1. x+ )j-\- z = n, 2. X + y + z = 0, 2X-&IJ- z = 0, x + 2y + 3z= -1, Sx + 4y + 2z = 0. x + Sy -\-Gz = 0. 3. Noting that ^i, .42, As of § 3 do not involve ai, ^2, a^, we may obtain the first expression (S) from (7) by replacing each ni by 6,-, and the second expression (8) from (7) by replacing each Ui by Cj. Hence (8) are the expansions of = according to the elements of the first column. 4. Prove similarly that (9) and (11) fohow from § o. * Here and in §§ 6-11 we understand by a determinant one of the second or third order. After determinants of higher orders have been defined, it will be shown that these theorems are true of determinants of any order. Ih Ih Cl Cl h Cl bo 62 ("2 = 0, Cl b2 C2 bs 63 C3 Cz 63 C3 §6, 7] DETERMINANTS 131 6. Theorem. A determinant having ai + gi, a^ -\- qi, . . . as the ele- ments of a column equals the su7n of the determinant having ai, Oo, . . . as the elertients of the corres'ponding column and the determinant having qi, q^, . . . as the elenierits of that column, while the elements of the remaining columns of each determinant are the same as in the given determinant. For determinants of the second order, there are only two cases: ai + gi 6i a2 + g2 &2 = tti hi a2 h-2, + qi hi q2 hi hi tti + qi bo 02 + q2 = hi ai 62 ao. + hi qi hi qo For determinants of the third order, one of the three oases is «! + gi hi ci 02 + g2 ho C2 az + qi 63 C3 ai hi Ci gi hi ci = ao bo Co + qo. 62 C2 03 &3 C3 g3 bz Cz To prove the theorem we have only to expand the three determinants according to the elements of the column in question (the first column in the first and third illustrations, the second column in the second illustra- tion) and note that the minors are the same for all three determinants. Hence ai + gi is multiplied by the same minor that ai and gi are multi- plied by separately, and similarly for a^ -\- qo, etc. 7. Removal of Factors. A common factor of all of the elements of the same column of a determinant may he divided out of the elements and placed as a factor before the new determinant. In other words, if all of the elements of a column are divided by n, the value of the determinant is divided by n. For example. nai hi = n «! &1 noo 60 a2 62 ai nhi Ci ai 61 Ci a-i nhi Co = n ao ho Co ai nhz Cz az hz Cz Proof is made by expanding the determinants according to the elements of the column in question. 132 THEORY OF EQUATIONS [Ch. XI 8. Theorem. A determinant is not changed in value if we add to the elements of any column the products of the corresponding ele7nents of another column by the same number. For example, ai + ^'^i &i 02 + nb2 bi as follows from the first relation in § 6 tti 6i a2 bo Similarly, by the third, fel 6i Ci h bo Co fo3 h^ Cz Oi + nbi bi Ci Oi bx Ci ao + nbo 62 Co = 02 62 C2 + n 03 + ri&3 bs Ci 03 63 C3 in which the last determinant is zero by § 5. In general, let Oi, 02, . . . be the elements to which we add the products of the elements 61, bo, . . . by n. We apply § 6 with gi = nbi, 52 = w&2, .... Thus the modified determinant equals the sum of the initial determinant and a determinant having bi, 62, • • • in one column and nbi, nbo, ... in another column. But the latter determinant equals (§7) the product of n by a determinant with two columns alike and hence is zero (§5). ExL\MPLE. Multiplying the elements of the last column by 2 and adding the products to the elements of the second column, we get = -44. For the next st(;p, we have nmltiplied the elements of the third column hj' —1 and added the products to the elements of the first column. Expanding the third determinant according to the elements of the third column, we note that two of the minors are zero (having a row of zeros) , and hence obtain the determinant of the second order written above. The last step is simplified l^y use of § 10. 9. Interchange of Rows and Columns. A determinant is not altered if in its symbol we take as the elements of the first, second, . . . rows the elements (in the same order) which formerly appeared in the first, second, . . . columns: 1 -2 1 1 1 1 -2 8 1 2 3 = 1 8 3 = _2 8 3 = 3 10 6 4 3 6 10 3 3 10 3 D^ tti 61 02 &2 Oi bi Ci Oo 62 C2 03 &3 C3 Ol O2 61 &2 Ol 02 O3 by bo 63 Ci C2 C3 = A. § 10, 111 DETERMINANTS 133 bi 63 ai 03 fli as + &2 - C2 61 63 Ci Cs Ci C-i The proof is evident by inspection for the case of determinants of the second order. For those of the third order, we expand A and find that its six terms are those in the expansion (6) of D. 10. Expansion According to the Elements of a Row. To prove that determinant D, given by (6'), may be expanded according to the elements of any row (say the second *) : D = —a^Az + 62^2 — C2C2, with the same rule of signs as in § 4, we note that (§9) D = A= —02 since A can be expanded according to the elements of its second column. After interchanging the rows and columns in these three determinants of the second order, we have the minors A2, B2, Co of 02, bo, Co in D. Example. The third determinant in the Example of § 8 is best evaluated by expanding it according to the elements of its first row, since two of its elements are zero. Indeed, we obtain +1 multipHed by its minor. 11. Theorem. A determinant is not changed in value if we add to the elements of any row the products of the corresponding elements of another row by the same number. We shall show that D, given by (6'), equals ai &i Ci D'= ao + nai 62 + nbi c-i + ncy a.3 &3 C3 Now D = A, where A is given in § 9. By § 8, ai 02 + nai as A = 61 62 + nbi 63 Ci Co + nci C3 Interchanging the rows and columns of A, we get D'. Hence D' = A = 2). * While for concreteness we have here (and in § 11) treated but one of several cases, the proof is such that it applies to all the cases. 134 THEORY OF EQUATIONS ICh. xi EXERCISES 1. Evaluate the numerical determinant in § 8 by removing the factor 2 from the second column and then getting a determinant with two zeros in the second row. Solve the systems of equations (by removing, if possible, integral factors from a column and reducing each determinant to one with two zeros in a row before expanding it) : 2. x-2y+ z=12, 3. 3 x - 2 y = 7, :r + 2 iy + 3 2 = 48, 3y-2z = G, 6a; + 47/ + 3z = 84. 32-2.c=-l. Factor a single determinant, and solve 4. X+ y + z = 1, 5. ax -{- hy -{- cz = k, ax-\- hy -\- cz = k, a-x + h~y -\- ch = k^, a-x + b-y + 0^2 = k'\ aiv + ¥y + c^2 = k*. 6. Obtain in its simplest form the value of x from ax -{■ y -\- z = a — 3, x-\- ay + z = -2, X -\- y -\- az = —2. 7. Deduce the case n = 2 of § 7 at once from § 6, by taking qi = a,-. 8. Give the proof in § 10 when the third row is used. 9. Give the proof in § 11 for a new^ case. 10. A determinant of the third order is zero if two rows are alike. 11. Hence prove that Z)' = Z) in § 11 by expancUng D' according to the elements of its second row. 12. Prove the theorem about rows corresponding to that in § 6. 13. From Ex. 12 deduce Ex. 11. 12. Definition of a Determinant of Order ii. In the six terms of the expression (6), which was defined to be the general determinant of order 3, the letters a, h, c were always written in this sequence, while the sub- scripts are the six possible arrangements of the numbers 1, 2, 3. The first term ai&2C3 shall be called the diagonal term* since it is the product of the elements in the main diagonal running from the upper left hand corner to the lower right hand corner of the symbol for the determinant. The subscripts in the term —aibzCo are derived from those of the diagonal term by interchanging 2 and 3, and the minus sign is to be associated with the fact that an odd number (here one) of interchanges of subscripts were used. To obtain the arrangement 2, 3, 1 of the subscripts in the * Sometimes called the leading term. § 12] DETERMINANTS 135 term -\-a2hzC1 from the natural order 1, 2, 3 (in the diagonal term), we may first interchange 1 and 2, obtaining 2, ^1, 3 and then interchange 1 and 3; an even number (two) of interchanges of subscripts were used and the sign of the term is plus. EXERCISES 1. Show that a like result holds for the last three terms of (6). 2. Discuss similarly the two terms of a determinant of order 2. While the arrangement 1, 3, 2 was obtained from 1, 2, 3 by one inter- change (2, 3), we may obtain it by applying in succession the three inter- changes (1, 2), (1, 3), (1, 2), and in many new ways. To show that the number of interchanges which will produce the final arrangement 1, 3, 2 is odd in every case, note that any interchange (the possible ones being the three just listed) changes the sign of the product P = {Xi- X2)(Xi - Xz){X2 - Xi), where the a:'s are arbitrary variables. Thus a succession of k interchanges yields P or —P according as k is even or odd. Starting with the arrange- ment 1, 2, 3 and applying k successive interchanges, suppose that we obtain the final arrangement 1, 3, 2. But if in P we replace the subscripts 1, 2, 3 by 1, 3, 2, respectively, i.e., if we interchange 2 and 3, we obtain — P. Hence k is odd. Consider the corresponding question for n variables. Form the prod- uct of all of the differences Xi — Xj {i < j) of the variables: P = (Xi- XojiXi - Xs) . . . {Xi - Xn) • (.X2 — X3) . . . (X2 — Xn) ' \Xn~l Xn)- Interchange any two subscripts i and j. The factors which involve neither i nor j are unaltered. The factor di(xi — Xj) involving both is changed in sign. The remaining factors may be paired to form the products zt{Xi - Xk)(xj - X,,) {k ^ 1, . . . , n; k 9^ i, k 9^ j). Such a product is unaltered. Hence P is changed in sign. Suppose that an arrangement ii, U, . . . , in can be obtained from 1, 2, . . . , n by using a successive interchanges and also by 6 successive interchanges. Make these interchanges on the subscripts in P; the 136 THEORY OF EQUATIONS [Ch. XI resulting functions equal { — lyP and ( — 1)^P, respectively. But the resulting functions are identical since either can be obtained at one step from P by replacing the subscript 1 by d, 2 by 12, . . . , n by i^. Hence i-iyp^{-iyp, so that a and b are both even or both odd. We define a determinant of order 4 to be (13) «! bi Ci di a2 62 C2 di as 63 Cz dz cii bi Ci di 2 ± aqbrCA, (24) where q, r, s, t is any one of the 24 arrangements of 1, 2, 3, 4, and the sign of the corresponding term is + or — according as an even or odd number of interchanges are needed to derive this arrangement q, r, s, t from 1, 2, 3, 4. Although different numbers of interchanges will produce the same arrangement q, r, s, t from 1, 2, 3, 4, these numbers are all even or all odd, as just proved, so that the sign is fully determined. We have seen that the analogous definitions of determinants of orders 2 and 3 lead to our earUer expressions (2) and (6). We will have no difficulty in extending the definition to a determinant of general order n as soon as we decide upon a proper notation for the ti^ elements. The subscripts 1, 2, . . . , n may be used as before to specify the rows. But the alphabet does not contain n letters with which to specify the columns. The use of e', e", . . . , e^"^ for this purpose would conflict Avith the notation for derivatives and l)esides be very awkward when exponents are used. It is customary in mathematical journals and scientific books (a custom not always followed in introductory text books, to the distinct disadvantage of the reader) to denote the n letters used to distinguish the /) columns by d, €2, . . . , €„ (or some other letter with the same subscripts) and to prefix (but see § 13) such a subscript by the subscript indicating the row. The symbol for the determinant is therefore en ^12... ein (14) D = C-21 e-22 €nl en2 e2n e„„ § 131 DETERMINANTS 137 Bj^ definition * this shall mean the sum of the n! terms (14') (-iyc,,ic,,2 . . .ei^^n in which ii, ii, ■ ■ - , in is an arrangement of 1, 2, . . . , n, derived from I, 2, . . . , 71 hy i interchanges. For example, if we take n = 4 and write ay, 6y, Cy, dj for eyi, ey2, ey3, Cji, the symbol (14) becomes (13) and the general term (14') becomes (— 1)' a,, hi^ Ci^ d,-,, the general term of the second member of (13). EXERCISES 1. Give the six terms involving (h in the determinant (13). 2. What are the signs of aJb^C'dxeA, ihhiCzd^ei in a determinant of order five? 3. The arrangement 4, 1, 3, 2 may be obtained from 1, 2, 3, 4 by use of the two successive interchanges (1, 4), (1, 2), and also by use of the four successive inter- changes (1, 4), (1, 3), (1,2), (2,3). 4. Write out the six terms of (14) for n — 3, rearrange the factors of each term so that the new first subscripts shall be in the order 1, 2, 3, and verify that the resulting six terms are those of the expansion of U in § 13 for n = 3. 13. Interchange of Rows and Columns. Determinant (14) equals D' = en en . . . e„i 6l2 C22 • • • 6„2 e\n e2n • • • enn Without altering (14'), we may rearrange its factors so that the first subscripts shall appear in the order 1, 2, . . . , n, and get { — I)'eik,e2k2 ■ • • enk„- Since this can be done by i interchanges of the letters e (corresponding to the i interchanges by which the first subscripts ii, . . . , in were derived from 1, . . . , n), the new second subscripts ki, . . . , kn are derived from the old second subscripts I, . . . , n by i interchanges. The resulting signed product is therefore a term of D'. Hence D = D'. * We may define a determinant of order n by mathematical induction from n — 1 to n, using the first equation in § 17. The next step would be to prove that the present definition holds as a theorem. ' 138 THEORY OF EQUATIONS [Ch. XI 14. Interchange of Two Columns. A determinant is changed in. sign by the interchange of any two of its: coliinms. Let A be the determiniint derived from (14) b}- the interchange of the rth and sth columns. The expansion of A is therefore obtained from that of D by interchanging r and 6- in the series of second subscripts of each term (14') of D. Interchange the rth and sth letters e to restore the second subscripts to their natural order. Since the first subscripts have undergone an interchange, the negative of any term of A is a term of D, and A =-D. 15. Interchange of Two Rows. A determinant D is cha7iged in sign by the interchange of any two rows. Let A be the determinant obtained from D by interchanging the rth and sth rows. By interchanging the rows and columns in D and in A , we get two determinants D' and A', either of which may be derived from the other by the interchange of the rth and sth columns. Hence, by §§ 13, 14, A = A'= -D' = -D. 16. Two Rows or Two Columns Alike. A determinant is zerojf any two of its rows or any two of its columns are alike. For, by the interchange of the two like rows or two like columns, the determinant is evidently unaltered, and yet must change in sign by §§ 14. 15. Hence D = -D, D = 0. 17. Expansion. A determinant can he expanded according to the ele- ments of any row or any column. Let Eij be the minor of e,/ in D, given by (14). Thus E',y is the deter- minant of order n — I obtained by erasing the ith row and the jth column (crossing at Cij). We first prove that D^CnEn-eoiEn + e.iEn- • ■ ■ + (-l)"-'e„i£J„i, so that D can be expanded according to the elements of its first column. The terms of D with the factor en are of the form (-l)*eiie,-,2 . . . e,„n, where i2, . . . , i'„ is an arrangement oi 2, . . . , n derived from the latter by i interchanges. Removing from each term the factor en, antl adding the quotients, we obtain tlie {n — 1)! properly signed terms of En- § 181 DETERMINANTS 139 Let A be the determinant obtained from D by interchanging the first and second rows. As just proved, the total coefficient of 621 in A is the minor ei2 ei3 ... ein C32 633 ... Csn en2 era Cnr, of 621 in A. Now this minor is identical with E21. But A = —D (§ 15). Hence the total coefficient of e-^ in D equals —£'21. Similarly, the coefficient of 631 is -£"31, etc. To obtain the expansion of D according to the elements of its kth col- umn, where k > 1, we consider the determinant 5 derived from D by moving the kth column over the earlier columns until it becomes the new first column. Since this may be done by A; — 1 interchanges of adjacent columns, 8 = (— l)*^~iZ). The minors of the elements en,, . . . , e„fc in the first column of 8 are evidently the minors Eik, . . . , Enk of ei^, . . . , €«& in D. Hence, by the earlier result, (15) {k = 1, . . . ,n). y=i Applying this result to the equal determinant D' of § 13, and changing the summation index from j to k, we get (16) (i = 1, . . . , n). jt=i This gives the expansion of D according to the elements of the jth row. One decided advantage of the double subscript notation is the resulting simplicity of the last two expansions. Of course the sign may also be found by counting spaces as in § 4. 18. The theorems in §§ 6-8, 11 now follow for determinants of order n. Indeed, the proofs were so worded that they now apply, since the auxiliary theorems used have been extended (§§ 13, 16, 17) to determinants of order n. 140 THEORY OF EQUATIONS (Ch. XI EXERCISES 1. Prove the theorem of § 15 b}' the direct method of § 14. b -{- c c -\- a a -{- b bi + Ci Ci + (h ai + bi 62 + C2 f 2 + 02 02 + 62 By reducing to a determinant of order 3, etc., prove that a b c «! h Ci 02 b. C2 2 -1 3 -2 1 7 1 -1 A' 3 5-5 3 — — ^i 4 -3 2 -1 b c d b' C2 d^ C3 d^ = abcd(a — b) b' c* d^ ae + bg af + bh ce + dg cf + dh 4. 1 1 1 2 1 1 3 4 6 10 10 20 = 1. abcdia — b){a — c){a — d){b — c){b — d){c — d). a b e f c d g h [use § 6]. aici + bid + CiCz aji + h/n + Ctfs oigi + big2 + Ci^s 0261 + &2S2 + 0263 02/1 + 62/2 4- C'fs a-igi + 62^2 + Cog 3 a-iCi 4- 6362 + CsCs aa/i + 63/2 + C3/3 Osgri + 63(72 + ^3^3 ai 61 Ci 02 62 C2 • 03 ^3 C3 Ci /i fifi 02 /2 !72 es /a f73 Write out only the 6 of the 27 determinants (§ 6) which are not necessarily zero. 8. Hence verify that the product of two determinants of the same order (2 or 3) is a determinant of like order in which the element of the rth row and cth column is the sum of the products of the elements of the /Ih row of the first determinant by the correspond in^j; elements of the cth column of the second. 9. Express (a^ + 6^ + c^ + d-){e- 4"/" + ^" + h'') as a smn of 4 squares by writing e + fi g + hi a-\-bi c-{- di — c-\-di a — bi ■g + hi e — fi as a determinant of order 2 similar to eacli factor. §'19, 201 DETERMINANTS 141 10. If Si = a' + ^' + tS 1 1 1 3 Si S2 Sl S2 S3 S2 S3 S4 11. Using the Factor Theorem and the diagonal term, prove Ex. 5 and 1 1 Xi Xi X\ X2" 1 Xn » n(n-l) where P is given in § 12. 12. With the notations of § .3, and using (7)-(12), prove that A, -A2 A3 B, B, -Bz • c\ -C2 Cz O] 61 Ci a^ &2 C2 03 63 Cz \D = D ; D Hence the first determinant equals D-. 19. Complementary Minors. The determinant D of order 4 in (13) is said to have the two-rowed complementary minors M = az &3 M' = C2 (h Ci di since either is obtained by erasing from D all the rows and columns having an element occurring in the other. Similarly, any r-rowed minor of a determinant of order n has a definite complementary- (n — r)-rowed minor. In particular, any element is regarded as a one-rowed minor and is complementary to its minor. 20. Laplace's Development. Any determinant D equals the sum of all the products ± MM', where M is an r-rowed minor having its elements in the first r columns of D, and M' is the minor complementary to M, while the sign is -{- or — according as an even or odd number of interchanges of rows of D will bring M into the position occupied by the minor Mi whose elements lie in the first r rows and first r columns of D. 142 THEORY OF EQUATIONS ICh. XI For r = 1, this development becomes the known expansion of D according to the elements of the first column (§ 17); here Mi = Cu. If r = 2 and D is the determinant (13) of order 4, Oi 6i Ca dz Oi 61 Ci rf2 ( Oi 61 C2 d2 D = . • + . 02 h C4 di 03 63 Ci d. 04 64 C3 dz 02 &2 Cl di 02 62 Cl rfl 03 &3 Cl di + — • + 03 63 C4 d, 04 &4 C3 C^3 04 64 1 C2 d^ The first term of the development is MiJ//; the second term is — il/.l/'(in the nota- tions of § 19), and the sign is minus since the interchange of the second and third rows of D brings this M into the position of Mi. The sign of the third term of the development is plus since two interchanges of rows of D bring the first factor into the position of My. If D is the determinant (14), then Ml = en en eir err Ml' = Cr+lr+l • er+\, en r+\ 6n n Any term of the product Miilf / is of the type {-lyei.xe^ ei^r ' \ i-J Ci^_|.,r+1 • . • ei^fi) where r'l, . . . , ^V is an arrangement of 1, . . . , r derived from 1, . . , , r by I interchanges, while iV+i, • • • , in is an arrangement of r + 1, • • • , n derived by j interchanges. Hence ii, . . . , tn is an arrangement of I, . . . , n derived by i -jr j interchanges, so that the above product is a term of D with the proper sign. It now follows from § 15 that any term of any of the products ± MM' of the theorem is a term of D. Clearly we do not obtain in this manner the same term of D twice. Conversely, any term t oi D occurs in one of the products ± MM'. Indeed, t contains as factors r elements from the first r columns of D, no two being in the same row, and the product of these is, except per- haps as to sign, a term of some minor M. Thus t is a term of MM' or of —MM'. In view of the earlier discussion, the sign of t is that of the corresponding term in ± MM', where the latter sign is given by the theorem. § 21. 22j DETERMINANTS 143 21. There is a Laplace development of D in which the r-rowed minors M have their elements in the first r rows of D, instead of in the first r columns as in § 20. To prove this, we have only to apply § 20 to the equal determinant obtained by interchanging the rows and columns of D. There are more general (but less used) Laplace developments in which the r-rowed minors M have their elements in any chosen r columns (or rows) of D. It is simpler to apply the earlier developments to the de- terminant ± D having the elements of the chosen r columns (or rows) in the new first r columns (or rows). EXERCISES L. a b c d e f g h a b J k j k c I ' I m I m I a b c d 1 2 e f g a b c e f g h d h ^ a b e f c g d h - a c e g • b d a d e h • = 0. 3. Check § 20 by showing that the total number of products of n elements is Cr" ' r\{n — r)\ = n\, where Cr" is the number of combinations of n things r at a time. For Laplace's development of many special determinants, see Ch. XIL 22. Product of Determinants. The important rule (Ex. 8, p. 140), for expressing the product of two determinants of order n as a determi- nant of order n is found and proved easily by means of Laplace's develop- ment. For brevity we shall take n = 3, but the method is seen to apply for any n. We have ai 6i Ci ei /i g\ a-2 hi C2 . 62 h g2 az 63 Cz ez fz gz ai &i Ci Ch ^2 C2 Ci-i 63 Cz -1 Ci /. gl -1 62 h Qi -1 ez h gs 144 THEORY OF EQUATIONS (Ch. xi In the determinant of order 6, multiply the elements of the first column by ei, fi, gi in turn and add the products to the corresponding elements of the fourth, fifth and sixth columns, respectively (and hence introduce zeros in place of the present elements Ci, /i, ^i). Then multiply the ele- ments of the second column by 62, f-i, 92 in turn and add the products to the corresponding elements of the fourth, fifth and sixth columns, re- spectively. Finally, multiply the elements of the tliird column by 63, /s, gs in turn and add as before. The new determinant is ai 61 Ci aiei+6ie2+cie3 ai/i+&i/2+ci/3 01^1+ 61^2 +C1C/3 02 &2 C2 0261 + 6262 + C2e3 02/1+62/2+02/3 02^1 + 62^2 + C2C/3 as h C3 0361+ 6362 +£363 03/1+63/2+03/3 azgi-\-hzg2+Czgz -1 -1 -1 By Laplace's development (or by expansion according to the elements of the last row, etc.), this equals the 3-rowed minor whose elements are the long sums, and written in Ex. 7, p. 140. 23. Systems of Linear Equations. l\\ the n equations OuXi + ai2X2 + • • • + oi„a:„ = ki, (17) On 1X1 + ClniXz + • • • + dnnXn = A:„, let D denote the determinant of the coefficients of the n unknowns : D = Oji ai2 . . . oi, Onl 0„2 • • • On, Let Aij be the minor of a,; in D. Multiply the members of the first equa- tion by All, those of the second equation by — A21, . . • , those of the /ith equation by (— 1)"~^ A„i, and add. The coefficient of Xi is the expansion of D according to the elements of its first column. The coefficient of Xz is the expansion, according to the elements of the first column, of a deter- minant derived from D by replacing an by an, • . . , o^ii by a„2, so that this determinant has the first two columns alike and hence is zero. In this manner, we find that (18) DXi = Ki, Dxi = K2, . . . , Dxn = Kn, 5 24) DETERMINANTS 145 in which (see (|3) of § 24) Ki is derived from D by substituting ki, . . . , kn for the elements an, . . . , Uni of the tth column of D. Another proof of (18) follows from Dxi = anXi an ttnlXi a„2 ain ann anXi + an].Xi-\- ■ ■ + ai nXn Oi„ = Kx. We have now extended to any n results proved for n = 2 and n = 3 in §§ 1-3. If D 9^ 0, the unique values of a;i, . . . , Xn determined by division from (18) actually satisfy equations (17). For instance, the first equation is satisfied since kiD — anKi — anKi ainKn = ki an ai2 . . ain h an ai2 . . ai„ ki 021 ^22 . • 02 n fCn flnl a„2 . . a„„ as shown by expansion according to the elements of the first row; and the determinant is zero, having two rows alike. 24. Rank of a Determinant. If a determinant D of order n is not zero, it is said to be of rank n. In general, if some r-rowed minor of D is not zero, while every (r + 1) -rowed minor is zero, D is said to be of rank r. For example, a determinant D of order 3 is of rank 3 if Z) f^ 0; of rank 2 if D = 0, but some two-rowed minor is not zero; it is of rank 1 if every two-rowed minor is zero, but some element is not zero. It is said to be of rank if every element is zero. In the discussion of the three equations (4), five cases arise: (a) D of rank 3, i.e.,/) ?^ 0. (/3) D of rank 2 {i.e., D = 0, but some two-rowed minor ^ 0), and K,= ki bi Ci k2 62 C2 ^"3 bs Ci K2 = Ol h Ci a2 h Ci aa ki C3 Ki = ai bi h 02 62 h as h h not all zero. 146 THEORY OF EQUATIONS [Ch. XI (7) D of rank 2 and K\, K->, K3 all zero. (5) D of rank 1 {i.e., every two-rowed minor = 0, but some element ^ 0), and {I, j chosen from 1, 2, 3) di ki bi ki d ki I dj kj bj kj cj kj I not all zero; there are nine such determinants K. («) D of rank 1, and all nine of the determinants K zero. In case (a) the equations have a single set of solutions (§ 23). In cases (ff) and (5) there is no set of solutions. In case (7) one of the equations is a linear com- bination of the other two; for example, if (hb-y — a-ibi 9^ 0, the first two equations determine x and y as linear functions of z (as shown by transposing the terms in z and sohang the resulting equations for x and y), and the resulting values of x and y satisfy the third equation identically as to z. Finally, in case (e), two of the equations are obtained by multiplying the remaining one by constants. For (/3) the proof follows from (18). For (7), (5), (e), the proof is given in § 25. The reader acquainted wnth the elements of solid analytic geometry will see that the planes represented by the three equations have the foUo\ving relations: (a) The 3 planes intersect in a single point. (fS) Two of the planes intersect in a line parallel to the third plane. (7) The 3 planes intersect in a common line. (S) The 3 planes are parallel and not all coincident. (e) The 3 planes coincide. 25. Fundamental Theorem. Let the determinant D of the coefficients of the unknowns in equations (17) he of rank r, r < n. If the determinants K obtained from the (r + l)-roived minors of D by replacing the elements of any column by the corresponding known terms ki are not all zero, the equa- tions are inconsistent. But if these determinants K are all zero, the r equa- tions involving the elements of a non-vanishing r-rowed minor of D determine uniquely r of the variables as linear functions of the remaining n — r vari- ables, and the expressions for these r variables satisfy also the remaining n — r equations. For example, letr = n —\. Then D = and the K's arc the Ki, . . . , Kn of § 23. Hence, by (18), the equations are inconsistent unless Ki, . . . , Kn are all zero. This affords an illustration of the following Le.m.ma 1. If every (r + l)-rowed minor .1/ formed from certain r + 1 rows of D is zero, the corresponding r -f 1 equations (17) are incon- sistent if there is a non-vanishing determinant K formed from any M by replacing the elements of any column by the corresponding known terms ki. 251 DETERMINANTS 147 For concreteness,* let the rows in question be the first r + 1 and let ctii . , . ttif ki K = 5^0. dr+l 1 . . . dr+l T kr+i Let di, . . . , dr+l be the minors of ^i, . . . , kr+i in K. Multiply the first r + 1 equations (17) by di, —di, . . . , { — ly dr+i, respectively, and add. The right member of the resulting equation is ± K. The coeffi- cient of Xs is an ... air ai , dr+l 1 • . . dr+l and is zero, being an M. Hence = ±iv. dr+l Lemma 2. If all of the determinants M and K in Lemma 1 are zero, but an r-rowed minor of an M is not zero, one of the corresponding r + 1 equations is a linear combination of the remaining r equations. As before let the r + 1 rows in question be the first r + 1. Let the non-vanishing r-rowed minor be dii . . . ai, (19) dr+l = drl 5^0. Let the functions obtained by transposing the terms ki in (17) be Li = diiXi + di2X2 + • • • + dinXn — ki. By the multiplication made in the proof of Lemma 1, diLi - doLi -\- ■ • • -\- {-l)'dr+iLr+i = ^K = 0. Hence Lr+i is a linear combination of Li, . . . , Lr. The first part of the fundamental theorem is true by Lemma 1. The second part is readily proved by means of Lemma 2. Let (19) be the non-vanishing r-rowed minor of D. For s > r, the .sth equation is a linear combination of the first r equations, and hence is satisfied by any set of solutions of the latter. In the latter transpose the terms involving Xr+i, . . • , Xn. Since the determinant of the coefficients of rci, . . . , Xr is not zero, § 23 shows that Xi, . . . , Xr are uniquely determined linear functions of Xr+i, . . . , x„ (which enter from the new right members). * All other cases may be reduced to this one by rearranging the n equations and relabeUing the unknowns (replacing Xi by the new .Ci, for example). 148 -^ [theory of equations [Ch. XI EXERCISES 1. Write out the proof of the theorem in § 25 for the cases (7), (5), (e) in § 24. Discuss the following systems of equations: 2. 2x-\- y + Sz = l, 3. 2x+ ?/+ 32= 1, 4:x + 2y- z=-S, 4.r + 27/- 2 = 3, 2x+ ?/-42 = -4. 2x+ ?/-42 = 4. 4. X- 3y+ 42 = 1, 5. x- 3y + 42 = 1, 4x- 12?/ +162 = 3, 4x-- 12;/ + 162 = 4, 3x- 9y-\-12z = 3. 3x- 9y + 12z = 3. 6. Discuss the equations in Exs. 4 and 5, p. 134, when two or more of the num- bers a, b, c, k are equal. 7. Discuss the equations in Ex. 6, p. 134, when a — —2. 26. Homogeneous Linear Equations. When the known terms A:i, ... , kn in (17) are all zero, the equations are called homogeneous. The determi- nants K are now all zero, so that the n homogeneous equations are never inconsistent. This is also evident from the fact that they have the set of solutions a;i = 0, . . . , .T„ = 0. By (18), there is no further set of solu- tions if D 5^ 0. If D = 0, there are further sets of solutions: if D is of rank r, there occur n — r arbitrary parameters in the general set of solu- tions (§ 25). A particular case of this result is the much used theorem: A necessary and sufficient condition that n linear homogeneous equations in n unknowns shall have a set of solutions, other than the trivial one in which each unknown is zero, is that the determinant of the coefficients he zero. 27. The case of a sj^stem of fewer than n linear equations in n un- knowns may be treated by means of the Lemmas in § 25. In case we have a system of more than n linear equations in n unknowns, we may first discuss n of the equations. If these are inconsistent, the entire system is. If they are consistent, the general set S of solutions may be found and substituted into the remaining equations. There result conditions on the parameters occurring in »S\ and these linear conditions may be treated in the usual manner. Ultimately we get either the gen- eral set of solutions of the entire system of equations or find that they are inconsistent. To decide in advance which of these cases will arise we have only to find the maximum order r of a non-vanishing r-rowed determinant formed from the coefficients of the unknowns, taken in the regular order 28] DETERMINANTS 149 in which they occur in the equations, and ascertain whether or not the (r + l)-rowed determinants K, formed as in § 25, are all zero.* 28. An important case is that of n non-homogeneous linear equations in n — 1 unknowns Xi, . . . , Xn-u By multiplying the known terms by a:„ = 1, we bring this case under that of n homogeneous linear equations in n unknowns (§ 26). Then (18) gives Dx^ = 0, Z) = 0, so that the given equations are inconsistent if D ?^ 0. There is no set of solutions of the n equations aiio^i + ai2X2 + + ai„-iXn-i — ki, "T" O/n n—lXn—l "'/i, an ani dln-l ki an n— 1 l^n 9^0. EXERCISES Discuss the following systems of equations: 1. x+ y + 3z = 0, 2. 2x- y+ 4:2 = 0, x + 2y + 2z = C x+ 'iy - 22 = 0, x + oi/- 2=0. x-lly +Uz = Q. 4. 6 X- + 4 ?/ + 3 2 - 84 WJ = 0, x + 2 7/ + 3 2 - 48 w = 0, X -2y + 2 - 12 w = 0, 4x- + 4?/- 2-24t/; = 0. 6. 2 X + 1/ + 3 2 = 1, 4x + 2y - 2 = -3, 2x+ y-4:z= -4, 10x + 5y-Gz= -10. 3. .T - 3 ?/ + 4 2 = 0, 4a; -12y + 162 = 0, 3.C - 9y + 12z=--0. 2.C+ 3y- 42+ 5 iv = 0, 3 X + 5y — 2 + 2 li; = 0, 7x+ny- 92 + 12i« = 0, 3x+ 4?/- 11 2 + 13 a; = 0. 2 X - y + Sz = 2, X + 72/+ 2=1, 3x + 5y - 5z = -3, 4X-32/ + 22 = 1. 8. Obtain a consistent system of equations from the system in Ex. 7 by replac- ing the term —3 by a new value. 9. In three linear homogeneous equations in x, y, z, w, the latter are proportional to four determinants of order 3 formed from the coefficients. * For an abbreviated statement, the concepts matrix and its rank are needed. Cf. Bocher, Introduction to Higher Algebra, p. 46. CHAPTER XII Resultants and Discriminants 1. Introduction. If the two equations ax + 6 = 0, cx-\-d^O {a9^0, c^O) are simultaneous, i.e., if x has the same value in each, then X = = , R = ad — be = 0, a c and conversely. Hence a necessary and sufficient condition that the equations have a common root is i? = 0. We call R the resultant (or eliminani) of the two equations. The result of eliminating x between the two equations might equally well have been written in the form 6c — ad = 0. But the arbitrary selection of R as the resultant, rather than the product of R by some constant as —1, is a matter of more importance than apparent at first sight. We seek a definite function of the coefficients a, 6, c, d of the func- tions ax + h, ex + d, and not merely a property /? = or i? 5^ of the corresponding equations. Accordingly, we shall lay do^^^l the definition in § 2, which, as the reader may verify, leads to R in our present example. Methods of elimination which seem plausil)le often yield not R itself, but the product of R by an extraneous function of the coefficients. This point (illustrated in Ex. 3, p. 156) indicates that the subject demands a more careful treatment than is often given. We may even introduce an extraneous factor zero. Let a 5^ 0, /(x) = .T- — 2 aX — 3 a^, g{x) — X — a. From / subtract (x + a)g. Multiply the remainder, — 2a(x4-a), by x — 3 a and add the product to 2 a/. The sum is zero. But the resultant is —4:a^ (the value of / for X = a) and is not zero. As we used g only in the first step and there, in eflect, replaced it by x- — a^, we really found the resultant of the latter and /. The extraneous factor introduced {cf. Ex. 7, p. 152) is the resultant oi x -\- a and / and this resultant is zero. 150 §21 RESULTANTS AND DISCRIMINANTS 151 2. Resultant of Two Polynomials in x. Let ( /(a:) - aoa:"' + aix'"-! + • • • + am (ao ?^ 0) ^ ^ ^ ^(a:) = box- + 6ix"-i + • • • + 6„ (6o ^ 0) be two polynomials of degrees m and /i. Let cci, . . . , a™ be the roots of f{x) = 0. Now q:i is a root of (j{x) = only when g{ai) = 0. The two equations have a root in common if and only if the product g{ai)g{a-z) . . . g{am) is zero. This symmetric function of the roots of f(x) = is of degree n in any one root and hence is expressible as a polynomial of degree n in the elementary symmetric functions (Chap. VII, §3), which equal —ai/ao, 02/00, .... To be rid of the denominators Oo, it suffices to multiply our polynomial by ao". We therefore define (2) Rif, g)= c'o"g{ai)g{a2) . . . g{lying the elements of the last three rows by a^. To the eleniouts of the new fourth row add the products of the ele- ments of the 1st, 2nd, 3rtl, 5tli, 6th rows by —60, —b\, —62, «i, «2 respectively Icorresponding to the formation of the third function (S)]. To the elements of the fifth row add the products of the elements of the 2nd, 3rd, 6th rows by —60, bi, ai respectively Icorresponding to the second function (8)1. Finally, to the ele- ments of the sixth row add the products of the elements of the third row by —bo [corresponding to ai^ — bof]. llcnce ao «! Oo 03 ao fll 02 as oo «1 fl2 fls bo b, ^2 63 bo ^1 b2 63 bo 61 b2 h RESULTANTS AND DISCRIMINANTS 161 Oo Ol 02 as Oo Ol 02 as Oo ai aj as (0063) (0163) (0263) (0062) (0063) + (0162) (0163) (ao6i) (Oo?>2) {(lobs) ao' R = so that R equals the 3-rowed minor enclosed by the dots. The method of Bezout therefore suggests a definite process for the reduction of Sylvester's determinant of order P. n (when m = n) to one of order n. Next, for equations of different degrees, consider the example / = Qqx'^ + ttix^ + aox- + Qzx + 04, g = 60^^ + 6ix + 62. Then aox^g — bof, (aox + ai)x^g — (&oX + &i)/ equal respectively (oofoi) x^ + (00^2) X" — ttsbox — aA, (oofeo) x^ + I (flifeo) — Qs&ol a;2 — '0361 + 0.460 1 X — 0461. The determinant of the coefficients of x^, x^, x, 1 in these and xg, g, after the first and second rows are interchanged, is the determinant of order 4 enclosed by dots in the second determinant below. It is the resultant R{L g) by § 4. As in the former example, we shall indicate the corresponding operations on Sylvester's determinant Oo ai 02 as 04 R u Oo ai fl2 03 04 60 61 b-i bo b. 62 bo &1 &2 bo &1 &2 Multiply the elements of the third and fourth rows by oo. In the resulting deter- minant Ofl^i?, add to the elements of the third row the products of the elements of the first, second and fourth rows by —60, — &i, Oi respectively. Add to the elements of the fourth row the products of those of the second by —bo. We get 162 THEORY OF EQUATIONS ICh. XII Oo Oi 02 as a* ao Oi ao as Ga (0062) (0162) - 0360 — 0301 — (libo — 0461 (oobi) (00^2) —Uzbo — 0460 bo 61 62 60 61 62 Of? R = Hence i2 equals the minor enclosed by dots. EXERCISES t 1. For m = S, n = 2, apply to Sylvester's determinant R exactly the same operations as used in the last case in § 8 and obtain (0062) (0162) — 0360 —0361 R = (ao6i) (0062) — a36o 60 bi 62 2. Hence show that the discriminant of Oax^ + Uix"^ + a^x + 03 = is 2 0002 0102 + 3 O0O3 2 01O3 Oi 2 02 3 03 3 Oo 2 Oi 02 = 18 a<)Oi0203 — 4 Oo02^ — 4 0/03 + 01^02- — 27 00^03-. For in = n = 4, reduce Sylvester's R (as in the first case in § S) to (0061) (0062) (ao&s) (ao&4) (0062) (flo^s) + {(hb-i) (0064) + (0163) {(iibi) (ao&a) («o64) + (0163) (0164) + (0263) (0264) (0064) {aibi) (0264) (0364) 4. For / and g of degree n, the ith function (8), when ^VTitten as a determinant of the second order, is seen to equal diix"-'^ + di2X"-- + • • • -\-din, where Then dij = (oo6,+;-i) + (oi6,+y_2) + n(>i-l) 2 D, D R-i-l) On 1 • . • djiii This D is called the Bezout determinant of / and g. Show that dji = da. + (oi_i6,). . dm 9) RESULTANTS AND DISCRIMINANTS 5. Hence verify for m = n = 5 that R can be derived from 163 (ao6i) (00^2) («o63) (0064) (aoh) (0062) (00^3) {(lobi) (ctah) (aA) (0063) (0064) ioJ>b) (aA) (aA) (0064) (0065) (aA) (aA) (aA) (0065) (aih) (a-A) (a-A) (ciA) by adding to its nine central elements the elements of (aifeo) (0163) (0164) (aA) (aA) + (aA) (aA) (aA) (aA) {asbi) 6. If R(f, g) = 0, we obtain a consistent set of equations by omitting one of Bezout's equations. Hence they determine x. If m = /i = 2, find x. If 7n = n = 3, find X. 7. If m = 71, set gi{x) = x'"~"g{x). Then Rif, 9) = R(f, 9i) -^ (-l)'"("'-")a^— ». 8. If m = n, R{cf + dg, sf + tg) = ±{ci - dsY'R{f, g). [Find the new {aih,).] 9. Express as a determinant of order m the resultant of /(x) = and x"* = 1. [Multiply/ by x and reduce by x"' = 1; repeat.] 9.t Without employing the results of §§3, 4, we may give a direct proof that the determinant (5) is the resultant of/ and g, given by (1). While the method is general, we shall present it only in the case m = 3, n = 2. In the equation (9) tto Oi ^2 a^ — z Oo Ql tti 03- z bo h. hi ho hy 62 &0 61 62 0, take z = f{0i) . Multiply the elements of the first four columns by jSi"*, /3/, /Si^, j8i, respectively, and add the products to the last column. All of the elements of the new last column are zero. Hence /(/3i) and /(jSa) are the roots of (9). Since the equation is of the form 60V + ( )2 + F = 0, 164 THEORY OF EQUATIONS ICn. xii where F is given by (4), we have Hence the Sylvester determinant F is the resultant R(J, g). Moreover, the equation in z is the eliminant of g{x) =Q, Z =f{x), and hence gives explicitly the equation obtained from g(x) =0 by apply- ing the transformation z = f{x) of Tschirnhausen (Ch. VII, § 13). 10. t Theorem. Necessary and sufident conditions that f(x) and g(x) shall have a common divisor of degree d, hut none of higher degree, are R = 0, Ri = 0, . . . , Rd-i = 0, Rd 9^ 0, where R is the determinant (5), and Rk is the determinant derived from R by deleting the last k rows of a's, the last k rows of b\s, and the last 2 k columns. For example, if 7n = n = 4, Oo Oi Qo 03 04 (10) Ri = Go Ol 02 az 04 Oo fli 02 03 bo 61 h ^3 b, bo bi 62 63 64 bo bi &2 63 To prove the theorem for the case d = 1, set /i = pix'"-- + • • • + pm-h gi = gix"-2 -1- The conditions for an identity of the form (11) fgi-gfi^cx + c' are flog. — &0P1 + (7n-l aiqi + 00(72 bipi — 60P2 = 0, = 0, amqn-2 + Om-lQ'n-l — hnPm-2 — hn-\Pm-\ = C, — bnPm-1 = C'. § 10] RESULTANTS AND DISCRIMINANTS 165 Omitting the last equation, we liave m -\- n — 2 linear equations for the same number of unknowns qi, —pi- The determinant of the coefficients equals Ri with the rows and columns interchanged. Hence if i^i ?^ we may choose c = Ri and find values not all zero of the unknowns satis- fying all of the above equations except the last, and then choose c' so that the last holds. Let R = 0. Then / and g have a common linear factor, but no common factor of degree > 1 since the right member of (11) is of degree unity. But li R = Ri = 0, we may take c = and find values not all zero of Qi, Pi satisfying all but the last of the above equations. The resulting value of c' is zero by (11), with c = 0, since/ and g have a common factor X — r. Then ^ -.or - ~^U - 0. X — r X — r Since not all of the m — 1 linear factors of the first fraction are factors of /i (of degree m — 2), at least one is a factor of the second fraction. Hence \i R = Rx — (i, J and g have a common factor of degree > 1. To prove the theorem for d = 2, we employ functions J^. and g2 of de- grees m — 3 and n — 3, respectively. Of the conditions for the identity (12) Jgo - gfo = ca:2 + ^'x + c", we omit the two in which c' and c" occur and see that the determinant of the coefficients of the remaining equations is Ro. Then if R ^ Ri = 0, R. ^ 0, we may take c = Ro and satisfy all of the conditions for (12). Thus / and g have no common factor of degree > 2. EXERCISES t 1. By performing on (10) exactly the same operations as used in § 8 to reduce a determinant of order 6 to one of order 3, show that (0063) (0064) + {aih) {aihi) + (0063) Ri = (00^2) (00^3) + {axbij {(lobi) + (a 163) (aobi) (ao^o) (0063) Note that if 04 = 64 = 0, the present work reduces to the former. 166 THEORY OF EQUATIONS [Ch. XII 2. In the notation of Ex. 4, p. 162, the preceding Ri with its first and third rows interchanged becomes Di: Ry = -A. dn (/l2 dn c/21 c/22 d-iz C^31 dsz dzz 3. For7/i = n, T>k dn di „-k d,,-. dn-k n-k 4. Hence, if m = n, f and g have a common divisor of degree d, but none of degree > d, if and only if /> = 0, A =0, ... , Dd-i = 0, Dj 9^ 0. 5. Give a direct proof of Ex. 4 by multiplying the ith function in Ex. 4, p. 162, by a variable ?/,- and summing for i = 1, . . . , ^ Thus g-laoUi+iaox -}- ai)y2+ • • • +(aox'-i+ • • • )ytl - f'\b(iiji+{bo.t + hi)y2+ • • •] = 5ix"-i + 52X"--+ • • • +5n, where 5i = dnVi + • • • + d^yt, . . . , 5„= dmyi + • • • + dmyi. The determinant of the coefficients of 7/1, ... , yt in 5i, . . . , St is Dn~t. If Z) = 0, take i = n; then we can choose ?/i, . . . , i/„ not all zero so that 5i = 0, , . . , dn = 0. Then gfi — fgi =0 for functions /i and gi of degree w — 1, so that / and g have a linear divisor. If also Di = 0, take t = 71 — 1; then we can make 5i = 0, . . . , 5n-i = 0. Hence gfo — fg2 = 5n for functions Ji and ^2 of degree n — 2. Since/ and g have a common divisor, the constant 5n is zero, and hence they have a common divisor of degree ^ 2. But if Di 7^ 0, we can make gh - f92 = 5«-lX + 5„, Sn-l9^ 0, so that the only common divisor is linear. MISCELLANEOUS EXERCISES 167 MISCELLANEOUS EXERCISES 1. Find a necessary and sufficient condition that the roots a, /3, 7 of x^ + px'^ + gx + r = shall be in geometrical progression. 2. For the same equation find 2q:^/3^. [Repjace x by 1/x.] 3. Find the equation with the roots a^ + /3-, a^ + 7^, /S^ + 7^. 4. Find the equation with the roots a~ + /3'^ — 7^, a- + 7" — /3", etc. 5. Find the equation with the roots a- + «/3 + 0', etc. 6. Solve the equation in Ex. 1 by forming and solving the quadratic equation with the roots (a + co/3 + 0)^7)^ and {a + co'/3 + wyY, where co- + w + 1 = 0. (Lagrange.) 7. Solve x^ — 28 X + 48 = 0, given that two roots differ by 2. 8. Find a necessary and sufficient condition that /(x) = x^ -\- px^ + 5.C' + r.c + s = shall have one root the negative of another root. When this condition is satisfied, what are the quadratic factors of /(x)? 9. Solve J{x) = x^ — Q x^ -\- 13 .r^ — 14 .r + 6 = 0, given that two roots a and j8 are such that 2 a + /3 = 5. Hint: f(x) and /(o — 2 x) have a common factor. 10. Diminish the roots oi x'* -{- qx- -\- rx -\- s = (s ^ 0) by such a number that the roots of the transformed equation shall be of the form a, in /a, h, m/h, and show how the latter equation may be solved. 11. Solve x" - 2 x2 - 16 .r + 1 = by the method of Ex. 10. 12. By use of the equation whose roots are the] squares of the roots of x^ + a;' — x2 + 2a; — 3 = and Descartes' rule, show that the latter equation has four imaginary roots. 13. Similarly, x^ + .r^ + 8 .c + 6 = has imaginary roots. 14. If all of the roots of x" + ax"~^+ 6x"~^ + • • • =0 are real, a2-26>0, 62_2ac + 2d>0, c^ - 2 6d + 2ae - 2/> 0, . . . . Hint: Form the equation in ?/ = x^. 15. Solve x^ + px + 5 = by eliminating x between it and x- -\- vx -{■ w = y by the greatest common divisor process, and choosing v and w so that in the result- ing cubic equation for y the coefficients of y and y- are zero. The next to the last step of the elimination gives x as a rational function of y. (Tschirnhausen, Acta Erudit., Lipsiae, H, 1683, p. 204.) 16. Find the preceding y-cnhio, as follows. Multiply x"^ + vx -\- w = y hy x and replace x^ hy —px — q; then multiply the resulting quadratic equation in x by X and replace x^ by its value. The determinant of the coefficients of x^, x, 1 must vanish. 17. Eliminate y between y^ = v, x = ry -\- sy"^, and get x' — 3 rsvx — (r^v + s^v~) = 0. Take s = 1 and choose r and v so that this equation shall be identical with x' + px -\- q = 0, and hence solve the latter. (Euler, 1764.) 168 THEORY OF EQUATIONS 18. Eliminate y between 1/ = v, x = f -{- ey + y- and get 1 e f — X e f — X V J — X V ev = 0. This cubic equation in x may be identified with the general cubic equation by choice of e, /, V. Hence solve the latter. 19. Determine r, s and v so that the resultant of 7/3 = V, shall be identical with x^ + px -\- q = 0. (Bezout, 1762.) 20. Show that the reduction of a cubic equation in x to the form y^ = y by the substitution _ r-\- sy ""' 1 + 2/ is not essentially different from the method of Ex. 18. [Multiply the numerator and denominator of .c by 1 — ?/ + ?/-.] 21. If the discriminant of a cubic equation is positive, the number of positive roots equals the number of variations of signs of the coefficients. 22. Descartes' rule gives the exact nmiiber of positive roots only when all the coefl&cients are of Uke sign or when each Pi being = 0. Without using that rule, show that the latter equation has one and only one positive root r. Hints: There is a positive root r by Ch. I, § 12 (a = 0, 6 = 00 ). Call F{x) the quotient of the sum of the positive terms by x', and call — A7'(x) that of the negative terms. Then A' (x) is a sum of powers of 1/x with positive coefficients. If x>r, P{x)>P{r), .V(.c) < .Y(r), /(.r) > 0; If X < r, P(x) < F(/-), .V(x) > .V(r), /(x) < 0. (Lagrange.) 23. If /(x)= /i(x) + • • • +/a(.c), where each /j(.c) is like the /in Ex. 22, and if ii is the greatest of the single positive roots of /i = 0, . . . , fk = 0, then R is an upper Umit to the positive roots of / = 0. 24. Any cubic or quartic equation in x can be transformed into a reciprocal equation by a substitution x = ry + s. 25. Admitting that an equation f(x) = x"-\- • • • =0 with real coefficients has n roots, show algebraically that there is a real root between a and b if /(a) and /(6) have opposite signs. Note that a pair of conjugate imaginary roots c ± di are the roots of (x — c)^ + (/'- = and that this quadratic function ia MISCELLANEOUS EXERCISES 169 positive if x is real. Hence if Xi, . . . , Xr are the real roots and {a) and 4>{b) have opposite signs. Thus a — Xi and b — Xi have opposite signs for at least one real root x,. (Lagrange.) 26. If s, is the sum of the jth powers of the roots of an equation of degree n and if m is any integer, the equation is 1 X" X" Sm+n+1 Sm-{-n . X . Sm+2 Sm+1 Sm+2 n—1 S„j-|-2 71—2 Sm+n Sm+n—l = 0. Hint: Use the second set of Newton's identities. (Jacobi.) 27. li a < b < c . . . < I, and a, 0, . . . , \ are positive, X X — a + X — b + re — c + + X — I -\-t = has a real root between a and &, one between b and c, . . . , one between k and i, and if t is negative one greater than I, but if t is positive one less than a. 28. Verify that the equation in Ex. 27 has no imaginary root by substituting r + si and r — si in turn for x, and subtracting the results. 29. In the problem of three astronomical bodies occurs the equation r'+iS- m)/-" + (3 - 2 ,.y - ixr'' - 2 ^.r - ^ = 0, where < m < 1. Why is there a single positive real root? As ju approaches zero, two complex roots and the real root approach zero. 30. Discuss the equation obtained from the preceding by changing the signs of the coefficients of r^ and r. 31. By Newton's identities. 1 Pl S3 = - Pi 1 2p, = -Pi' + Sp P2 Pi 3p3 1 ... Pi Pi 1 ... 2p2 Sk= - P2 Pi 1 ... 3p3 Ps P2 Pi ... 4p4 Pk- 1 Pk- -2 Pk- i . . . Pi kpk where all but the last term in the main diagonal is 1, and all terms above the diagonal are zero except those in the last column. If A; > n, we must set py = 0{j > n). 170 THEORY OF EQUATIONS 32. By Newton's identities, 1 si 3!p3 = -| si 2 S2 Si Si S3 Jc\p,= - 1 . . Si Si 2 . . S2 S2 Si 3 . . S3 Sk- 1 Sa-2 Sk-3 . . . k Sk iik = n. But if A; = 7i, Sk Sk-l Sk+l Sk Sk-2 Sk-l Sk-n Sk-n+l = 0. Sk+n Sk+n-1 Sk+n-2 . . . Sk 33. Let Si = ai^ + • • • + aj. Let ar, . . . , «„- be the roots of yn ^ p^yn-1 + . . . + p^^ = 0. Set y = a:/ and multiply the result by ay^'^- " , where k = 2 7i. Sum f or j = 1 , . Thus Hence ,n, Sk + PlSk-2 + P^Sk-i + + PnSk-2n = 0. Sk Sk-2 Sk+l Sk-l Sk-i Sk-3 Sk-2 n Sk-2n+l 0. Sk+n Sk+n-2 Sk+n-i • • ■ Sk-n 34. Obtain a vanisliing determinant similar to that in Ex. 33 but having the subscripts of the s's in each row decreased by 3. 10 35. Si Si S2 S3 = S3 Si So Si S2 So Si S2 S3 Si S2 S3 Si 1 Vi P2 Ps So Si + Pi So S2 + PiSi + poSo So Si + PlSo S2 + PiSi + P2S0 S3 + P1S2 + V-iSi + P3S0 Si S2 + PiSi S3 + P1S2 + P2S1 Si + P1S3 + P2S2 + P3S1 1 Vi Pi V3 On (71 - l)ih (n - 2)]h n (n — l)7;i (n — 2)p2 (n - 3)p3 Pi 2p2 3p3 4 2^4 MISCELLANEOUS EXERCISES 171 36. If n = 3, the last determinant may be obtained from the Sylvester resultant R oi x^ -\- pi.r^ + Ihx + ]h and its derivative by multiplying the elements of the first row of jR by —3 and adding tlie products to the elements of the third row. 37. Express the determinant of order 4 in the Sj (analogous to the first one in Ex. 35) as a determinant of order 6 in the p's. For n = 4, identify the latter with the resultant of x^ + jhx^ + P-^^~ + Ih^^ + Pi and its derivative. 38. Let Sk be the sum of the kih powers of the roots .Ci, . . . , a;„ of a given equation. The coeflScients of the equation having as its roots the 5 n(n — 1) squares of the differences of the x's can be found from Si, S2, . . . , where Sp is the sum of the pth powers of the roots of the latter equation. Expand by the binomial theorem (X - Xi)2P + (X - XoyP + . . . + (x - XnY^, set X = Xi, . . . , X = Xn in turn, add and divide by 2. Thus (2p)(2p-l) Op = ns2p — 2 ps2p-iSi -\ ~-^ S2P-2S2 , 2p(2p-l) ■ . . (p + 1) , - " ' ^~- 1-2 ... p '^"• (Lagrange.) 39. In particular, *Si = ns2 — sr, S2 = WS4 — 4 S1S3 + 3 S2^ S3 = nse — 6 S1S5 + 15 S2S4 — 10 S3''. Hence give the equation whose roots are the squares of the differences of the roots of a given cubic equation. Deduce the discriminant of the latter. 40. The equation whose roots are the n(?i — 1) differences Xj — Xk of the roots of /(x) = may be obtained hy eliminating x between the latter and f{x -\- y) = Q and deleting the factor y" (arising from y = xj — xj = 0) from the eliminant. The equation free of this factor may be obtained by eliminating x between /(x) = and l/(x + 2/)-/(.r)|/y=/'(x)+r(.r)^+ • • • +/"(.r) ^ ^^^""' _ ^^ = 0. This eliminant involves only even powers of y, so that if we set y"^ = z \nq obtain an equation in z having as its roots the squares of the differences of the roots of /(x) = 0. (Lagrange.) 4L Compute by Ex. 40 the ^-equation when/(.i-) = x^ + px + g. 42. Except for h — Q, the equation a — X b b f-x 0, has a real root exceeding a and /, and one less than a and /. [Substitute a and / for X in turn]. 172 THEORY OF EQUATIONS 43. Let the equation in Ex. 42 have distinct real roots a, /3, where a > fi. Then there are three real roots of* a — X b c D{x) - f-x g 9 h — X = 0. Hint: The results of substituting a and /3 for x in D(x) are [c V^^f + g V^^]', - [c Vf-i5- g Va - 0]\ where the product of the radicals in each is +6. Hence if neither a nor /3 is a root, there is a root > a, one < /3, and one between a and /3. If a is a root, there is a root < /3 and hence thi-ee real roots. 44. li a = fi in Ex. 43, then a = / is a root of D{x) = and there are two further real roots. 45. aa' + bb' + cc' ae' + bj' + eg' b' a' -\-be r c ea'+jb' + gc' ee'+Jf' + gg' b' c' r g' + bg + ce = af c' a' g' e' a'b' e'f + ag + cf c' b' g' r Combine the first and third, second and fifth, fourth and sixth: a b a' b' + a c a' c' + b c ^ b' c' e J e' r e g e' g' J g r g' a b 2 a c 2 b c = + + , e J e g f g 46. Hence, in particular, a^ + 62 -(- c2 ae + bf -{- eg ae + bf-\- eg e- + p -f g- 47. Hence if a, b, c and e, f, g are the direction cosines of two lines in space, and if e is the angle between them, so that cos d — ae -\- bf -\- eg, then sin'- d equals the above sum of three squares. 48. For the determinant in Ex. 43, o2 + ^2 + c2 — .1-2 ab -\' bf + eg ae -\- bg + eh D{x) 'D{-x)= ab + bf + eg b'- + P + (72 - .1-2 be + Jg + gh ac-\-bg + eh be -\- fg + gh c- + ^2 _|_ /,2 _ ^2 = -.T« + x\a' +P + h' + 2b- + 2e- + 2 g-) - x'-iD, + D, + Ds) + D2(0), * This theorem is important in many branches of pure and applied mathematics. Besides this proof and that in Ex. 48, other more advanced proof.s, including that by Borchardt, are given in Salmon's Modern Higher Algebra, pp. 48-56. MISCELLANEOUS EXERCISES 173 where D3 is the first determinant in Ex. 46 for e = b and Di and D2 are analogous minors of elements in the main diagonal of the present determinant of order 3 with X = 0. Hence the coefficient of —x- is a sum of squares (Ex. 49). Since the function of degree 6 is not zero for a negative value of x^, D{x) = has no purely imaginary root. If it had an imaginary root r + si, then D{x -\- r) = would have a purely imaginary root si. But D{x + r) is of the form in Ex. 43 with a, f, h replaced by a — r, f — r, h — r. Hence D{x) = has only real roots. The method is applicable to such determinants of order n. (Sylvester.) 49. In Ex. 48, Di -\- Do + D3 equals (af - br- + {ah - c-y- + (fh - g-'-y + 2 {ag - bcY + 2 (of - bgY + 2 {bh - cg)\ 50. Without using its solution by radicals, prove that .f* + bx^ + cx" + dx + e has a factor x- — s.r + p, where s is a root of a sextic equation, and that p is a rational function of s and the coefficients. Hints: There are six functions like s = Xi-{- Xt, next, C = SX1X2 = s{Xz + .1-4) + J) -\- XzXi, —d= Sx-iX2X3 = SX3X4 + (.I's + Xi)p. Replace Xs + .r4 by —b — s and solve for p the resulting linear equations in X3X4 and p. The case b -\- 2 s = may be avoided by starting with another pair of roots. 51. Prove Ex. 50 by di\dding the quartic by the quadratic function and requir- ing that the linear remainder shall be zero identically, 52. Prove Ex. 50 by use of (3) and (8) in Ch. IV. 53. x^ + bx^ + cx^ + dx^ -{- ex'^ -{- fx -^ g has a factor x^ — sx + P, where s is a root of an equation of degree 15, and p is a rational function of s and the coefficients. Hints: Write 0-1 = X3 + X4 + Xs + a.-6, t' «i'+ «/+ «A,-' "i-* + «/ + ak* 1 "i "i" 1 aj af 1 ayt "A;" = X i r, each product involves two equal as and hence Dk = 0. If A; = r, the only term not zero is that involving the r distinct as, so that Dr 7^ 0. (L. Baur, Malh. Annalen, vols. 50, 52.) MISCELLANEOUS EXERCISES 175 60. The n roots are all real and distinct if and only if D2, . . positive. (Weber, Algebra, 2d ed., I, p. 322.) 61. If each cv is real and if the numbers , Z>„ are all Co, Co Ci . Cn Ci C2 . Cn+\ Cn Cn+l- C2n+2 + C2 ^2n — are positive, all of the roots of Co + CiX + CiX^ + • are imaginary, and all but one of the roots of Co + cix + CoX- + • • • + C2„+ia;2"+i = are imaginary. (Van Vleck, Annals of Math., 4 (1903), p. 191.) C2 1 C2 t+i C2t+1 C2t+2 62. The results in Ex. 61 follow if the Cti and are all positive. (Kellogg, Annals of Math., 9 (1907), p. 97.) 63. If the terms with negative coefficients in an equation of degree n are — ax""", — /3x"~*, — 7X"~<', . . . , no positive root exceeds the sum of the two largest of the numbers v a, V /3, V 7, ... . (Lagrange.) 64. In Ex. 63, no positive root exceeds the greatest of the numbers y/ka, Vkl3, . . . , where k is the number of the negative coefficients —a, (Cauchy.) 65.* Define W as in Ch. IX, § 8, and let /(a) ^ 0, f{b) ^ 0. If /(x) = has imaginary roots, Va — Vb cannot give the exact number of real roots in every interval [a, b] ; but, if f{x) = has no imaginary roots, Va — Vb gives the exact number of real roots in every interval [a, b]. Hint: Use (14), Ch. IX. 66. Budan's Theorem gives the exact number of real roots of f(x) = in [a, b] if /(rt) 9^ 0,/(6) 9^ 0, iirovided that, for r = 0, 1, . . . , n — 2, real roots of /('■^(x) — separate those of f^'"^'^Kx) — in that interval from each other and from a and 6. The term "separate" here excludes the case of coincidence. Hint: At a root oi f^'"^^^(x) = 0, the functions f^^'Kx) and f^'"^^Kx) must be of opposite sign. 67. Descartes' Rule gives the exact number of real roots only when Budan's Rule is exact for every positive interval [a, b]. Thus it is exact for an equation having only real roots. 68. We define as generalized Sturm's functions for an interval [a, b] a sequence of polynomials /(x),/i(x), . . . , /^(x), with the following properties: * The author is indebted to Professor D. R. Curtiss for Exs. 65-72. 176 THEORY OF EQUATIONS (a) No two consecutive functions vanish simultaneously at any point of [a, b]; (6) fr{x) does not vanish in [a, b]; (c) When, for 1 = i ^ r — 1, fi{x) vanishes for a value of Xi in [a, b], fi-i{xi) and /t+i(a;i) have opposite signs; (d) When/(.r) vanishes for a value .ri in [a, b], fi{xi) has the same sign as/'(xi). Prove that the number of real roots of /(.r) = in [a, b] is equal to the difference between the numbers of variations of signs in such a sequence for x — a and for X = b. Prove the corresponding statement for an interval [c, d] within [a, b]. 69. Prove that generalized Sturm's functions for any interval [a, b], where a and b are both positive or both negative and f{x) = has no multiple roots, may be obtained as foHows: Take/i(.c) =/'(x). Arrange /(.r) and/i(a;) in ascending powers of x, and divide the former by the latter (using negative powers of x in the quotient, if necessary) ; let the last remainder of degree equal to that of fix) be designated by r^ix); then /2(.c) = —ro{x) -r- x'^. Define /i(.r) similarly by division of fi-2(.x) by fi-i(x), both being arranged according to ascending powers of x; the last remainder of degree ecjual to that of /i-2(-r) is divided by —x- and the quotient taken as fi{x). Show that the sequence thus obtained is valid for [—GO, go], provided no one of the functions vanishes for x = 0. 70. Prove that generalized Stunn's functions for any interval [a, b], where a and b are both positive or both negative and /(.r) = has no multiple roots, may be obtained by the greatest common di\asor process for/(.c) = oox" +OiX"~^+ • • • +a„ and/i(x), with the signs of the remainders changed (as in Sturm's method), if we take Mx) = 4>(x) s a,x"-' + 2aoa;"-2 + . • • + jiUn (x < 0), but Mx) = - 0. Hint: .r/'(.r) + 0. Hint : If as ^ 0, discuss whether variations of signs for x very near zero and negative = variations of signs for x veiy near zero and positive. ANSWERS Page 2. 1. 1.6,4.4. 2. No real. 4. 1.2,-1.8,-3.4. Page 7. 2. 2.1. 3. (-0.845, 4.921), (-3.155, 11.079); between -4 and -5. 4. 1.1, —1.3. 5. Between and 1, and —1, 2.5 and 3, —2.5 and —3. 9. 120 {x^ + x), 120 x^ - 42. Page 9. I. 3. 2. 2, —2. 3. —1. 4. Double roots 1, 3. Page 10. 3. Use Ex. 3, p. 9, abscissas —1, 3. 4. Use Ex. 2, p. 9. Page II. I. One. 2. Three. 3. Three. 4. 1, 1, -2. 5. One. Page 16. 7. 0.3,1.5,-1.8. 8. 1.2. 9. 1.3,1.7,-3.0. 15. 1,2,3,-6. Page 23. 3. TV(19^-9), ^'~J+^?"^^ i(6 + V5-3r + 2V5t). 4. Commutative and associative laws of addition and multiplication. Distributive law. Page 24. 1. ±(3 + 4i). 2. ±(5 + 6t). 3. ±(3-2^■). 4. ±[c + d + (c - d)z]. 5. ±(c - di). Page 26 (middle). 2. — 3, — 3w, — 3a;^; I, coi, co-i. 3. cos^ + zsin A (A = 40°, 160°, 280°). 177 178 THEORY OF EQUATIONS Page 26 (bottom). 1. -1, cosA + isiiiil (A = 36°, 108°, 252°, 324°). Page 30. 2. 5, -1 ± V-3. 3. 1 ± i, 1 ±V2. 4. a:^ - 7 a;2 +19 a: - 13 = 0. 5. r* _^ (1 _^ i^^2 -|- 1 = 0. 7. ±1,2 ± V3. 8, 9. V3, 2 ± /. 10. a;3 - I x2 - I a; + I - 0. Page 32. 2. -5, H5±V^). 3- -6, ±V^. 4. -2, l±i. 5. i f (-2±V-3). Page 34. I. Three. 2. Two. 3. Two. Page 35. I. -4, 2zhV3; 3, 3, -6. 2. Page 37. 3. 1.3569, 1.6920, -3.0489- Page 37. 1. Sce3, p. 35. 2. -1.2017, 1.3300, -3.1284. 3. 1.24698, -1.80194, -0.44504. 4. 1.1642, -1.7728, -3.3914. Page 39. 2. -1, -2,2, 3. 3. 1, -1, 4± V6. Page 43. 2. l±v^2, -1±V^. 3. 4, -2, -l±i. 4. See Ex. 3, p. 39. Page 53. I. z = —I — 2i, ooz, o}^z. Page 56. I. x' - 8^2 + 16 = 0. 2. 1, 3. 3. 4, 1 -V^. 5. 2+\/3, x2 + 2x + 2 = 0. 6. 1,2. 7. -3,1,5. 8. 4, f, -|. 9. 1, 3, 5. 10. 2, -6, 18. II. 5, 2, -1, -4. 12. 1, 1, 1, 3. 13- Ps = ViJh' 16. if — V2y - 12 = 0. Page 58. 3. 6, 4. 2. 5. 3. ANSWERS 179 Page 6i. I. 1, 3, 6. 2. 2, -1, -4, 5. 3. -12, -35. 4. 2, 2, - 3. Page 62. 1. 3, 1, O, J. 2. X, 2> 3- 3' 6- 4' 4> 4> 2- Page 65. I. g- — 2 pr + 2 s. 2. p-g — 2 g^ — p^- -)_ 4 s. 3. p'^ — 4 p-g + 2 g- + 4 pr — 4 s. 4. ?/^ - (p^ - 2 g)2/2 -f (52 _ 2 pr)y _ ^2 = 0. 5. 2/3 — qy"^ + pr?/ — r^ = 0. 6. rif -\- 2 qif -\- 4: py + 8 = 0. 7. ^i2 - 2 ^2. 8. ^lE's - 3 E3. 9. ^1^2. 10. Ei^ - S E1E2 -\- 3 E3. II. Ei'-SEiEi. 12. ^1^3 - 4 ^4 if n > 3, £Ji^3 if n = 3. Page 71. 2. SaSbScSd —USaSbSc+d + 2IlSaS6+c+d +2Sa+6S,.+d — 6 Sa+ft+c+rf, if «, &, C, d aVQ distinct; but if all are equal, j\ {Sj — 6 Sa%a + 8 SaSsa + 3 S2a^ — 6 S4 a). 3. See Exs. 1, 2, 12, 13, page 65. Page 76. 3. ?/ -7qi/-^ 14 gV _ 7 qSy = c. 4. e'" - 2 e^'" (?« = 0, . . . , 4). Page 77. .. P'-^P'l + 5pr + l\ ,.2p'-2q. 8. -p' + 24r. r — pq I r 1 A' I 3 p'^if — 4 p^r — 4 g^ — 2 pqr — 9 r^ ^' (r - pqf 10. 27 r^ — 9 pgr + 2 g^ = 0. 12. y = q-\- r/x. 13. re = , , .f^ - 2 + 2y Page 83. I. 1, -I (1 ±\/^), i (7 ±V45). 2. 1, a;2 + i (l ± V5)a: +1=0. 3. ±1, a;2 ± a: + 1 = 0. 5. 2^ _^ 2^ _ 2 ^ _ 1 = 0. 6. 2^ + 2^ - 4 2^ - 3 22 _j_ 3 2; -1- 1 = 0. 8. 2 cos 2 7r/7, etc. 180 THEORY OF EQUATIONS Page 88. 2. g = 2, r + r8 -I- ri2 _|_ ^s^ g^^., 2^-{-z^-4:Z-{-l=0. Page 98. I. One, between -2 and -3. 2. One, between 1 and 2. Page 99. I. Ex. 2, p. 37. 2. (0, 1), (-1.1, -1). 3. a;= -i/inEx. 1, p.37. 4. (0, 1), (-2, -1). 5. (1, 2), (-7, -6). 6. (0, 1), (3, 4). Page 103. !• 2, —2. 4. 1, 1, two imaginary. Page 105. I. (-2, -1), (0, 1), (1, 2). 2. (-4, -3), (-2, -1), (1, 2). Page 113. I. Page 117. 2, 3. Exs. 1, 3, page 119. Page 119. 1. -1.7728656. 2. y =-xm Ex. 3, p. 35. 3. Single, -2.46954. 4, 5. Exs. 2, 3, p. 37. 6. Two negative and 2.121 + , 2.123 + . 7. 3.45592, 21.43067. 8. 2.24004099. Page 121. 2. Darwin's quartic: -12.4433 ± 19.7596 ^. 3. -0.59308, -2.04727, 1.32048 ± 2.0039 ^•. 4. 0.35098, 12.75644, 32.0602, 34.8322. Pages 123-24. 1. 4.0644364, -0.89196520, 0.82752156. 2. §1, -1.04727 ± 1.13594 i. 5. -2.46955. Page 128. I. a; = 5, y = Q. 2. a; = 2, y = 1. 3. x = a, y = 0. ANSWERS 181 Page 130. 1. X = —8, y = —7, 2 = 26. 2. a; = 3, y = —5, 2 = 2. Page 134. 2. X = Q, y = Z, z = 12. 3. a: = 5, ?/ = 4, 2 = 3. ^ (fe - b)(c - k) ^ k(b-k)(c-k)(k-\-h + c) '^' ^ {a-h){c-a)' ^' ^ a{h-a){c-a){a + h + c)' Page 137. 1. —a'>J)iCidi + a-ihicdi + a-ihzCicU — a-ih^ddi — a-zhiCids + a-ihiCzdi. 2. +, +. Page 148. 2. Consistent: 2/ = —8/7 — 2x, 2 = 5/7 (common line). 3. Inconsistent, case (j8). 4. Inconsistent (two parallel planes). 5. Consistent (single plane). Page 149. I. a: : ?/ : 2 = — 4 : 1 : 1. 2. x : |/ : 2 = — 10 : 8 : 7. 3. Two unknowns arbitrary. 4. x : y : z : w = Qt : ?> : 12 : \. 11 19 10 17 S-z=-^x-—y, w=-—x- — y. 6. y = -8/7 -2a;, 2 = 5/7. 7. Inconsistent (determinant 4th order 9^ 0). Page 167. I. ph = q^. 2. 3 r- — 3 pqr + cf". 3. Eliminate a: by ?/ = S2 — x^. 5. Eliminate a; by p- — g + px = ?/. 7. 2, 4, — 6. 8. yqr — p'S — r- = 0, X" -\- rjp, x^ •\- px ■\- ps/r. 9. 1, 3, 1 ±1. 12. 2^ + 224 + 5 23 + 3 22-22-9 = 0. 13. 2^ + 15 2- + 52 2 - 36 = 0. 89. See Ex. 17, p. 78. INDEX (The numbers refer to pages.) Abscissa, 1 Absolute value, 24 Amplitude, 24 Argument, 24 Axes, 1 Bend point, 3, 9, 11 Bezout on elimination, 160 — determinant, 162 Binomial equation, 84 Borchardt's theorem, 174 Budan's theorem, 103, 175 Cardan's formula?, 32 Cauchy: symmetric fimctions, 78 Columns of determinant, 128 ahke, 130, 138 Common divisor, 8, 95, 164 Complementary minors, 141 Complex number, 21, 47 , geometrical representation, 24 Conjugate imaginary, 22, 29 Continued fraction, 125 Continuity, 12, 51 Coordinates, 1 Cube roots, 26, 35 of unity, 23, 24 Cubic equation, 10, 11, 16, 17, 31, 40, 75, 80, 90, 99, 119, 122, 167-8 Decagon, 88 De Moivre's quintic, 76 — theorem, 25 Derivative, 5, 69, 93, 96, 103, 106, 110, 158 Descartes on the quartic, 42 — rule of signs, 105, 168, 175 Determinant, 127, 134, 154-175 Diagonal term, 134 Discriminant, 33, 41, 157 — of cubic, 33, 78, 99, 158-9, 162 — of quartic, 41, 45, 100 Divisor, 58, 164 Double root, 8 (see Discriminant) Duphcation of cube, 90 Element of determinant, 128 EUminant, 150 Ehmination, 152-160 Equal roots, 8, 11, 33, 34, 45 Equation of squared differences, 78, 125, 171 Euler on elimination, 159 quartic, 44 Expansion of determinant, 130, 133, 138 Factor theorem, 8 — of determinant, 131 Ferrari on the quartic, 38 Fold, 8 Foui'ier on roots, 106 Fundamental theorem of algebra, 47 Geometrical (see Complex) — construction, 16, 87, 90, 92 Griiffe on finding roots, 121 Graph, 2, 14 Graphical solution, 2, 15-17, 110 Homogeneous equations, 148 Horner's method, 115 Imaginary number, 21 — roots, 28, 102, 107, 120, 124, 174-5 Inflexion point, 9, 44 — tangent, 10 Integral root, 59 Intercihange in determinants, 132-4, 137-8 Interpolation, 111 183 184 THEORY OF EQUATIONS Interval, 97 Invariant, 42 Irrational, 30 Irreduc'ible case, 34, 35 Irreducibility of resultants, 152 Isolated, 95 Lagrange on the quartic, 40 — , solution of equations, 125 Laplace's development, 141-3 Linear equations, 127-9, 144-9 Lower limit to roots, 58 Minor, 129 Modulus, 24, 25 Moivre (see De) Multiple root, 8 (see Discriminant) Newton's formula;, 70, 169, 170 — method of solution, 109-115 for integral roots, 59 Ordinate, 1 Pentagon, 88 Periods, 85 Plotting, 2 Polar coordinates, 19 Polynomial, 3, 6 — , sign of, 14 Primitive root, 28 Product of determinants, 143 Quadratic equation, 15 Quartic, 16, 38, 42, 80, 99, 120, 173 Quintic, 76, 80, 83 Radian, 27 Rank of determinant, 145 Rational root, 61 — integral function, 3 Reciprocal equation, 81 Reduced cubic, 10, 31 — quartic, 42 Regula falsi, 111 Regular polygon, 27, 87-90 Relations between roots and coefficients, 32, 39, 55, Ch. VII, 169, 170 Relatively prime, 28 Remainder theorem, 8 ■• Resolvent cubic, 38, 39 Resultant, 150 RoUe's theorem, 93 Roots, theorems on, 13, 15, 30, 34, 39, 45, 47, 55, 59, 61, 64, 69, 76, 90, 93, 96, 100, 103, 105, 112, 174-6 — of unity, 27 — , nth, 26 Rows of determinant, 128 ahke, 138 2-polynomial, 63 2-function, 68, 70 Slope, 3, 6 Solution of numerical equations, 109 Solvable by radicals, 75, 84 Square root, 23 Sturm's functions, 98 , generalized, 176 — theorem, 95-102 Sum of roots, 39, 56 powers of roots, 69, 72, 169 Sylvester on eUmination, 154 Symmetric function, 63, 121 — in all roots but one, 76 Synthetic division, 116 Taylor's theorem, 6, 113 Transformed equation, 10, 115, 119 Trigonometric form, 24 — solution of cubic, 34, 36 Trinomial equation, 11 Triple root, 8 Trisection of angle, 90 Tschirnhausen, 79, 164 Upper hmit to roots, 57, 175 Variation in signs, 95 Vector, 18 Vieta on the cubic, 31 Waring's formula, 72 o RETURN CIRCULATION DEPARTMENT 202 Main Library LOAN PERIOD 1 HOME USE 2 3 4 5 6 ALL BOOKS MAY BE RECALLED AFTER 7 DAYS 1 -month loans may be renewed by calling 642-3405 Y 6-month loans may be recharged by bringmg books to Circulation Desk Renewals and recharges may be made 4 days prior to due dote DUE AS STAMPED BELOW EEC. ClUfX U 77 iUL 3teK i 1 1 FORM NO. DD 6, UNIVERSITY OF CALIFORNIA, BERKELEY BERKELEY, CA 94720 ® 1 (^UV CDbl34fl51S ivi61?94 1)5 THE UNIVERSITY OF CALIFORNIA LIBRARY