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 $p$.

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