Web hosting rating - 474 ARITHMETIC 4.6.4 Theorem E. Every nth degree
474 ARITHMETIC 4.6.4 Theorem E. Every nth degree polynomial (1) with real coefficients, n 2 3, can be evaluated by the scheme = y=x+c, w y2; z = (unv + ~O)Y + PO (n even), z = u,y + PO (n odd); U(X) = (. . . ((z(w -al) + Pl)(W -a2) + P2). . .>(w -GL) + Pm; (20) for suitable real parameters c, ok and Pk, where m = [n/2] -1. In fact, it iS possible to select these parameters so that ,& = 0. Proof. Let us first examine the circumstances under which the Q S and p s can be chosen in (20), if c is fixed. Let p(x) = u(x-cc) = u,xn +a,-lxn-l +~~~+alx+uo. (21) We want to show that p(x) has the form pi(x)(x2–cw,)+& for some polynomial pi(z) and some constants a,, ,&. If we divide p(z) by x2 - cr,, we can see that the remainder pm is a constant only if the auxiliary polynomial q(x) = u2m+lxm + a2m–15m-1 + . . . + al, (22) formed from every odd-numbered coefficient of p(x), is a multiple of x -Q,. Conversely, if q(x) has x -CY, as a factor, then p(x) = pi(x)(x2 -a,) + pm, for some constant & that may be determined by division. Similarly, we want PI(X) to have the form ps(z)(x2 -a,_~) + ,&-I, and this is the same as saying that g(x)/(x -a,) is a multiple of x -CZ,-1; for if Q(X) is the polynomial corresponding to pi(x) as q(x) corresponds to p(x), we have Q(X) = 4(x)/(x -am). C on t inuing in the same way, we find that the parameters ai, pi, . . . , cy,, Pm will exist if and only if q(x) = U2mfl(X -al). . .(x -%n). (23) In other words, either q(x) is identically zero (and this can happen only when n is even), or else q(x) is an mth degree polynomial having all real roots. Now we have a surprising fact discovered by J. Eve [Numer. Math. 6 (1964), 17-211: If p(x) has at least n -1 complex roots whose real parts are all nonnega- tive, or all nonpositive, then the corresponding polynomial q(x) is identically zero or has all real roots. (See exercise 23.) Since U(X) = 0 if and only if p(x+c) = 0, we need merely choose the parameter c large enough that at least n -1 of the roots of U(X) = 0 have a real part 2 -c, and (20) will apply whenever u,-~ = u,-1 -ncu, # 0. We can also determine c so that these conditions are fulfilled and also that Pm = 0. First the n roots of U(X) = 0 are determined. If a + bi is a root having the largest or the smallest real part, and if b # 0, let c = –a and Qm= -b2; then x2 -oym is a factor of U(X -c). If the root with smallest or
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