Prove $f(z) =f(z ̄)$ if $f(z) = 0$

Question

Prove that $f$ of $z $ where $z$ is a complex number is equal to $f$ of $z ̄$ (which is the conjugate of $z$) if $f(z)=0$. $f(x) =x^2+8x+16$

Edit: $f(z ̄)= z^2+8+16$ with a bar over everything.

Answer 1

Hint: what is $\overline{x^2 + 8 x + 16}$?

Answer 2

This can be partially extended. If $f(z) = \sum_{j=0}^m a_j z^j$ is a polynomial of real coefficients, then $f(z)$ is symmetric about the real line, a.k.a. $f(z) = f(\overline z)$.