Question 2
Let $n \gt k \ge 1$ be integers.
Let $P(x) \in \mathbb{R}[x]$ be a polynomial of degree $n$
with no repeated roots and $P(0) \neq 0$.
Suppose that for any real numbers $a_0$, $\dots$, $a_k$
such that the polynomial $a_k x^k + \dots + a_1 x + a_0$ divides $P(x)$,
the product $a_0 a_1 \dots a_k$ is zero.
Prove that $P(x)$ has a nonreal root.
Proposed by Carl Schildkraut.
Manual essay; 1 editorial point, not official contest scoring. Automatic answer fields are intentionally empty.
Let $P(x) \in \mathbb{R}[x]$ be a polynomial of degree $n$
with no repeated roots and $P(0) \neq 0$.
Suppose that for any real numbers $a_0$, $\dots$, $a_k$
such that the polynomial $a_k x^k + \dots + a_1 x + a_0$ divides $P(x)$,
the product $a_0 a_1 \dots a_k$ is zero.
Prove that $P(x)$ has a nonreal root.
Proposed by Carl Schildkraut.
Manual essay; 1 editorial point, not official contest scoring. Automatic answer fields are intentionally empty.