how to prove z0 is a root of the polynomial

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!

Re: how to prove z0 is a root of the polynomial

Hey flametag3.

Is this asking to show that if z0 is a root then z0_bar is a root or is it asking that if the condition (z_bar)^k = (z^k)_bar, then that particular z is a root for when that is true?

Re: how to prove z0 is a root of the polynomial

Hey Chiro,

thanks for your reply. Its asking to show that if z0 is a root then z0_bar is a root as well.

Re: how to prove z0 is a root of the polynomial

You'll have to show that if z gives 0 then z_bar also gives 0, and one to show that given what you have is to show that the equation with z_bar is zero if the one with z is zero and you will need to apply this term by term in your polynomial to show that both equal zero and thus both are roots of the polynomial if they have a specific f(z) form (note f(z) doesn't change but z = z or z = z_bar both need to give f(z) = 0 to both be roots).