A Divisor problem for polynomials

We characterize all monic polynomials $f(x)\in \mathbb{Z}[x]$ that have the property that $$f(p)\mid f(p^p),\text{ for all sufficiently large primes }p\ge N(f)\mathrm{.}$$ 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 $p$.