The set of all such polynomials of degree less than or equal to $n$, denoted by $\mathbb{Z}_n[x]$ is clearly countable via the obvious identification $\displaystyle f:\mathbb{Z}_n[x]\to\mathbb{Z}^{n+1}:\sum_{k=0}^{n}\alpha_k x^k\mapsto (\alpha_0,\cdots,\alpha_k)$. Note then that $\displaystyle \mathbb{Z}[x]=\bigcup_{n\in\mathbb{N}}\mathbb{Z}_n[x]$ and so..