Question 182
Let $n$ and $k$ be positive integers with $k \lt n$. Let $P(x)$ be a polynomial of degree $n$ with real coefficients, nonzero constant term, and no repeated roots. Suppose that for any real numbers $a_0, a_1, \dots, a_k$ such that the polynomial $a_kx^k + \dots + a_1x + a_0$ divides $P(x)$, the product $a_0a_1\dots a_k$ is zero. Prove that $P(x)$ has a nonreal root.