2007/08 AHSMC Part 1 Problem 8

Suppose there exists such a polynomial $p$. Since $a^n - b^n$ is divisible by $a - b$ for all positive integers $a$, $b$ and $n$ with $a \ne b$, $13 - 9$ must divide $p(13) - p(9)$. However, $4$ does not divide $7$, and we have a contradiction. The answer is (A). A parity argument also works.