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.

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