## If f(x) divides g(x), then f(c) divides g(c) for all c

Let $R$ be a commutative ring with 1. If $p,q \in R[x]$ such that $p|q$, for which $r \in R$ is it true that $p(r)$ divides $q(r)$?

Suppose $p(x) = q(x)t(x)$. Via the evaluation homomorphism, we have $p(r) = q(r)t(r)$ for all $r \in R$. That is, $q(r)$ divides $p(r)$ for all $r$.