A polynomial $p \in \Bbb C[z]$ has the property that $p(x)$ is real for all $x$ real. Then prove that $p$ has real coefficients.

Question

I was thinking that since $p(x)$ is real if and only if $x$ is real, then $p$ is linear. And since real polynomials are equal to the products of real linear and quadratic polynomials, $p$ must be real.