A divisor problem for polynomials

We characterize all monic polynomials f(x) ∈ Z[x] that have the property that f(p) | f(p), for all sufficiently large primes p ≥ N(f). We also give necessary conditions and a sufficient condition for monic polynomials f(x) ∈ Z[x] to satisfy f(p) | f(p) for all primes p.