# Polynomial Division question

If $f(x)$ and $g(x)$ are polynomials over $\mathbb{Z}$ and $f(n)|g(n)$ for all $n\in \mathbb{Z}$ then does it follow that $f$ divides $g$ in $\mathbb{Z}[x]?$

I'm pretty sure the answer is yes and that the proof should be easy (in fact I think it should be true with far weaker assumptions), but can't seem to get too far. If it helps, note in particular that $f$ has no integral roots.

-
Nope, but for silly reasons. Take $f(x) = 2, g(x) = x^2 + x$.
If $f$ is monic, the proof is straightforward: using the division algorithm we may assume WLOG that $g$ has degree strictly less than $f$ and then $\frac{g}{f}$ eventually tends to zero, so must itself be zero.