All of the coefficients of $P(x)$ are divisible by $p$ 
Find a counterexample to the following: If $P(x)$ is a polynomial with integer coefficients and $p$ a prime, then if $p \mid P(x)$ for all $x$ then all of the coefficients of $P(x)$ are divisible by $p$.

I thought about proving this by a proof by contradiction but wasn't sure how to do that. How should we prove the statement?

I think this is false.
Consider the polynomial $x^p-x$, by Fermat's little theorem $x^p\equiv x \bmod p$ so $x^p
-x\equiv 0 \bmod p$
