Conjugate of Real Polynomial is Polynomial in Conjugate

Theorem
Let $P \left({z}\right)$ be a polynomial in a complex number $z$.

Let the coefficients of $P$ all be real.

Then:
 * $\overline {P \left({z}\right)} = P \left({\overline z}\right)$

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

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

Then: