Question 56
Let $P(z) = a_d z^d + a_{d-1} z^{d-1} + \dots + a_1 z + a_0$ be a polynomial with complex coefficients, the **reverse** of this polynomial is defined as
$$
P^*(z) = \overline{a_0}z^d + \overline{a_1}z^{d-1} + \dots + \overline{a_d}
$$
a. Prove that
$$
P^*(z) = z^d \overline{P\left(\frac{1}{z}\right)}
$$
b. Let $m$ be a positive integer. Let $q(z)$ be a non-constant monic polynomial which all of its roots lie on the perimeter or the interior of the unit circle. Prove that all the roots of the polynomial
$$
Q(z) = z^m q(z) + q^*(z)
$$
lie on the **perimeter** of the unit circle.
$$
P^*(z) = \overline{a_0}z^d + \overline{a_1}z^{d-1} + \dots + \overline{a_d}
$$
a. Prove that
$$
P^*(z) = z^d \overline{P\left(\frac{1}{z}\right)}
$$
b. Let $m$ be a positive integer. Let $q(z)$ be a non-constant monic polynomial which all of its roots lie on the perimeter or the interior of the unit circle. Prove that all the roots of the polynomial
$$
Q(z) = z^m q(z) + q^*(z)
$$
lie on the **perimeter** of the unit circle.