Unknown

Hints 189

31. Show that every zero z of the complex polynomial

f(%) = %” + an-l%“-’ +. * * + aI% + 00

satisfies -b 5 Re z < a, where a and b are the unique positive roots

of the equations

z”+(Rea,-r)~“-~ - la,-211”--2 - IQ,-312--3 - - * * - Ia1 Ix - laoI = 0

x” - (Re Q,-$z”-~ - 1*,4~2”-2- IO”-3lt”-3-.. .- lqlz- laoI = 0.

32. Suppose that p(t) h as n distinct real zeros exceeding 1. Show that

a(t) = o2 + l)PWP’W + tL.Pw2 + (P’W21

has at least 2n - 1 distinct real zeros.

33. Let p(z) be a polynomial of degree n with only real zeros and real
    coefficients. Show that

(n - %w12 - np(z)p”(z)^2 0.

34. Show that there exist infinitely many manic polynomial equations
    over Z of degree n such that n - 1 of the roots occur within a specified
    interval, however small.


35. Let m be a positive integer and define the real polynomials f(z) and
    s(x) by
    (1+ k??), = f(z) + ig(x).
    Prove that, for arbitrary real numbers a and b, not both zero,

d(x) + b(x)

has only real zeros.

Hints

Chapter 5

2.10. How many sign changes can there be altogether in the polynomial at
t and at -t?

2.11. Multiply by t - 1 and apply Exercise 10.

3.5. (b) Note that the polynomial in (a) is positive when t > r.