If $z_0$ is a solution for $z^{13}-13z^7+7z^3-3z+1=0$, does also $\bar z_0$ is a solution?

Question

If $z_0$ is a solution for $z^{13}-13z^7+7z^3-3z+1=0$, does also $\bar z_0$ is a solution for the equation?

I dont realy have an idea, i would like a hint.

Answer 1

Just take conjugate on both sides. Use the facts that conjugate of $a+b$ is the sum of the conjugates of $a$ and $b$ and the conjugate of $z^{n}$ is the $n-th$ power of the conjugate of $z$.