Hints 189
- 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.
- 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.
- 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.
- 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. - 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.