How can I prove that $\displaystyle z_0$ and $\displaystyle \overline{z_{0}}$ is a root of the polynomial

$\displaystyle a_{n}z^n + a_{(n-1)} z^{(n-1)} +....+a_{1} z + a_{0} = 0$

Using $\displaystyle (\bar{z})^k = \overline{z^k}$

I have been able to prove $\displaystyle (\bar{z})^k = \overline{z^k}$ but I dont get how to use that to show that $\displaystyle z_0$ and $\displaystyle \overline{z_{0}}$ are roots of the polynomial.

Thanks in advance for your help!