4.6.4 EVALUATION OF (Graphic web design) POLYNOMIALS 499 23. [HM30] (J.
4.6.4 EVALUATION OF POLYNOMIALS 499 23. [HM30] (J. Eve.) Let f(z) = a,zn + an-lzn- + … + a0 be a polynomial of degree n with real coefficients, having at least n -1 roots with a nonnegative real part. Let g(z) = a,zn + un-2Zn–2 + . . + hm0dzZ nmod2 7 h(z) = a,-lz–l + &-3zn-3 + f ~(,-l)~~d2Z(n-1)mod2. Assume that h(z) is not identically zero. a) Show that g(z) has at least n -2 imaginary roots (i.e., roots whose real part is zero), and h(z) has at least n -3 imaginary roots. [Hint: Consider the number of times the path f(z) circles the origin as z goes around the path shown in Fig. 15, for a sufficiently large radius R.] b) Prove that the squares of the roots of g(z) = 0 and h(z) = 0 are all real. - R Fig. 15. Proof of Eve s theorem. b 24. [h .Z4] Find values of c and &k, Pk satisfying the conditions of Theorem E, for the polynomial U(Z) = (Z + 7)(x* + 6~ + 10)(~ $4~ j-5)(2 + 1). Choose these values so that PZ = 0. Give two different solutions to this problem! 25. [M.Z0] When the construction in the proof of Theorem M is applied to the (ineffi- cient) polynomial chain Xl = 0.1 +x0, x2 = —x0 -x0, x3 = Xl + Xl, x4 = a2 x X3, x5 = x0 –x0, X6 = cl6 -x5, x7 = ff7 +x63, X8 = x7 x X7, x9 = Xl x x4, x10 = Q8 -x9, 111 = x3 -x10, how can PI, fiz, . . . , /3s be expressed in terms of CQ, , cvs? b 26. [M.21] (a) Give the polynomial chain corresponding to Horner s rule for evaluating polynomials of degree n = 3. (b) Using the construction that appears in the text s proof of Theorem A, express ~1, ~2, 1c3, and the result polynomial U(X) in terms of ,&, Pz, ps, p4, and 5. (c) Show that the result set obtained in (b), as PI, &, /33, and p4 independently assume all real values, omits certain vectors in the result set of (a). 27. [Mz?] Let R be a set that inciudes all (n+l)-tuples (q,, . . , ~,qo) of real numbers such that qn # 0; prove that R dpes not have at most n degrees of freedom. 28. [HMz?~] Show that if fo(crl,. . , as), . . . , fs(o~, . . , cu,) are multivariate polyno- mials with integer coefficients, then there is a nonzero polynomial g(zo, . . , z~) with integer coefficients such that g(f ( . , fs(crl,. = all real al, 0 CXI, , as), . . , as)) 0 for . ..) as. (Hence any polynomial chain with s parameters has at most s degrees of freedom.) [Hint: Use the theorems about algebraic dependence that are found, for example, in B. L. van der Waerden s Modern Algebra, tr. by Fred Blum (New York: Ungar, 1949), Section 64.1