Originally Posted by

**Oipiah** I can't figure out where to start on this one.

Let $\displaystyle p(z) = a_nz^n+\ldots+a_1z+a_0$ be a polynomial of degree n with complex coefficients and suppose there exist $\displaystyle \{z_1,\ldots,z_n\}$ distinct complex numbers each having $\displaystyle \text{Im}(z_j)>0$ such that $\displaystyle p(z_j)=0$ for $\displaystyle j\in\{1,\ldots,n\}.$ Set $\displaystyle q(z)=\text{Re}(a_n)z^n+\ldots+\text{Re}(a_1)z+ \text{Re}(a_0).$ Prove that if $\displaystyle q(w) = 0$ then w must be real.

(Here, $\displaystyle \text{Im}(z_j)$ denotes the imaginary part of $\displaystyle z_j$ and $\displaystyle \text{Re}(a_j)$ denotes the real part of $\displaystyle a_j$.)