MathDB

Let

P(z)=a_d z^d+\dots+ a_1z+a_0

be a polynomial with complex coefficients. The

reverse

P

is defined by

P^*(z)=\overline{a_0}z^d+\overline{a_1}z^{d-1}+\dots+\overline{a_d}

(a) Prove that

P^*(z)=z^d \overline{ P\left( \frac{1}{\overline{z}} \right) }

(b) Let

m

be a positive integer and let

q(z)

be a monic nonconstant polynomial with complex coefficients. Suppose that all roots of

q(z)

lie inside or on the unit circle. Prove that all roots of the polynomial

Q(z)=z^m q(z)+ q^*(z)

lie on the unit circle.

Problem Statement