  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 $p$.
