Conjugate of Real Polynomial is Polynomial in Conjugate

Theorem
Let $\map P z$ be a polynomial in a complex number $z$.

Let the coefficients of $P$ all be real.

Then:
 * $\overline {\map P z} = \map P {\overline z}$

where $\overline z$ denotes the complex conjugate of $z$.

Proof
Let $\map P z$ be expressed as:
 * $a_n z^n + a_{n - 1} z^{n - 1} + \cdots + a_1 z + a_0$

Then: