A Divisor problem for polynomials

We characterize all monic polynomials $f(x) \in \mathbb{Z}[x]$ that have the property that $$f(p) \mid f(p^{p}),~\text{for all sufficiently large primes }p \geq N(f).$$ We also give necessary conditions and a sufficient condition for monic polynomials $f(x) \in \mathbb{Z}[x]$ to satisfy $f(p) \mid f(p^{p})$ for all primes $p$.