Before we can apply Horner's method we must know the first significant figure of the root to be found. In other words, we must "locate" the root. This can always be done by Sturm's theorem, but usually the following theorem is more convenient. If two real num bers a and b, when substituted for x in f(x), give to f(x) contrary signs, an odd number of roots of the equation f(x)=0 lies between a and b. Thus, to locate the roots of s' 3.r' 46x 71 0, substitute for x, in succession, the values 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. It is found that f(-5) and f(--4), f(-2) and f(-1). f(8) and f (9) are pairs of values of f(x) having opposite signs. As there are three roots in all, we conclude that there is just one root between each of the pairs of values 5 and 4, 2 and 1, 8 and 9. To reduce the number of trials in more difficult examples, there are theorems on the upper and lower limits of roots which may be applied.
Horner's method consists of successive transformations of an equation. Each transfor mation diminishes the root by a certain amount If the required root is 1.95,, then the root is diminished successively by 1, .9, .05, .005. Syn thetic division is employed. Suppose we desire to find, to three decimals, the root between 1 and 2 in the above example. It is convenient first to transform the equation so that the root becomes positive. We get + 3x' 46x + 71 = 0. The first significant figure in the root is 1. To diminish the roots by 1 we perform by synthetic division the following operation: 1 + 3-46+ 71 1 1+ 4-42 4-42 + 29 1 4- 5 1 6 The transformed equation, whose root under consideration now lies between 0 and 1, is s'+ 6?-37x + 29 = 0. This root being less than unity, x' and s' are less than x. Neglect ing x' and 6x', we obtain an approximate value for x from 37x + 29=0, viz., x = .7. As in the process of ordinary long division or in the extraction of roots, so here the digit ob tained by the first approximate division may be too large or too small and may need correction. An error of this sort will reveal itself later in the attempt to find the third digit of the root. Such correction is needed here. Actually = .9. Diminish the roots of the last trans formed cubic by .9, then find the third digit by the process just indicated for finding the sec ond digit, then diminish the roots again, and so on. The entire operation is as follows: The broken lines indicate the conclusion of the successive transformations. For advanced reading on the solution of numerical equations consult McClintock, E., in Am. Jour. of Maths.,) Vol. XVII, pp. 89-110; Carvallo, M. E., 'Resolution numerique complete d. Equations algebriques ou transcendantes> (Paris 1896) ; A. Xavier, 'Approximations numeriques) (Paris 1909).
Multiple Roots.Suppose that in f(x);) there are m multiple roots; that is, as roots are equal to each other. Then f r)nlio (x), and the first derivative isf'(x)(x r)nng(s) + m (x r) The fact that f(x) and f' (x) have the factor in common suggests the following rule for the discovery of multiple roots: Find the highest common factor of f(x) and f(x). If that factor is
(x Os, then r occurs as a root s + 1 times. If the highest common factor is cx ri)', then r occurs as a root s +1 times and r, occurs t + 1 times. If f 6x + 9, then f' (x) 24x' 40x + 6, and the H.C.F. is 2x-3. Hence is a double root.
Elimination. Take the equations, + F(x).7_-'ss-Faix + a, = 0, and let r, and r, be the roots of the second equation. The necessary and sufficient con ditions that the two equations shall have a root in common is that f(ri) or Ars) shall vanish; that is, that the product shall be zero. Multiply together f + + ru, f(rs)-=_-r" + + as, we get Ora' -1-a, (roe + r,'r,) + a, (r,' + r,') +Orin r,) + a,'.
Expressing the symmetric functions of r, and r, in terms of the coefficients of the second of the given equations, we get ri're=b2', ri-l-rs= Substituting these values, we have bea,14,-1- titbit-214, a,'b,a,a,b, + This expression, involving the coefficients of the two given equations, is called the eliminant or resultant. Its vanishing is the condition that these equations have a root in common. More generally, if from a equations with n 1 varia bles we eliminate the variables and obtain an equation R=.0, involving only the coefficients, the expression R is called the eliminant or re sultant of the given equations.
In the above example the elimination was performed with the aid of symmetric functions. Of other methods of elimination the best known are those of Euler, Bezout and Sylvester. We outline the last, known as Sylvester's Dialytic Method. To eliminate x between .f(x)=-T-_a0xn airs-'+. . . +an=0, . . . multiply the first successively by x°, st, and the second successively by x°, . . ., and we obtain in +a equations. The highest power of x is in + n-1. If f(x) 0 and F(x) =0 have a common root, it will satisfy all the m + n equations. If the differ ent powers of x, viz., x, . . . , be taken as m + n 1 unknown quantities, satisfying + n linear equations, a relation will exist be tween the coefficients. This condition of con sistency is the vanishing of the resultant. This resultant Sylvester expressed neatly in the form of a determinant. See DETERMINANTS.
Discriminants. It has been shown that a multiple root of is also a root of f' (x) = 0. But the condition that these two equations have a common root is expressed by the vanishing of the resultant.
The resultant of f(x)0 and f (x) =0 is called the discriminant of f(x) =0. It may be otherwise defined as the simplest function of the coefficients, or of the roots, whose vanishing signifies that the equation has equal roots.
To the references already given we add the following: 'Encyklopidie der mathematischen Wissenschaften,) Band I; Cajori, Florian, 'In troduction to the Modern Theory of Equations) (New York 1904) ; Netto, E., Worlesungen fiber Algebra) (Leipzig, Vol. I, 1896, Vol. II, 1900); Serret, J. A., Tours d'Algebre Supe rieure) (Paris, 2 vols.) ; Todhunter, 'Theory of Equations) (London 1880) ; Weber, H., 'Lehr buch der Algebra) (Braunschweig, Vol. I, 1898, Vol. II, 1896) ; (Encyklopidie der elem. Algebra and Analysis) (Leipzig 1903).