if a polynomial $P(x)\in \mathbb {Z} [x]$ is a power of $n$ for all integer values of x, then $P(x)$ is the $n$th power of a polynomial?

Question

How can I use Hilbert's irreducibility theorem to prove that if a polynomial $P(x)\in \mathbb {Z} [x]$ is a power of $n$ for all integer values of x, then $P(x)$ is the power of $n$ of a polynomial in $ \mathbb {Z} [x]$ ?