Problem

Source: Iran 3rd round 2014-Algebra exam-P3

Tags: algebra, polynomial, complex analysis, algebra unsolved



Let $p,q\in \mathbb{R}[x]$ such that $p(z)q(\overline{z})$ is always a real number for every complex number $z$. Prove that $p(x)=kq(x)$ for some constant $k \in \mathbb{R}$ or $q(x)=0$. Proposed by Mohammad Ahmadi