Problem

Source: Turkey TST 2022 P6 Day 2

Tags: algebra, polynomial, Integer Polynomial, number theory



For a polynomial $P(x)$ with integer coefficients and a prime $p$, if there is no $n \in \mathbb{Z}$ such that $p|P(n)$, we say that polynomial $P$ excludes $p$. Is there a polynomial with integer coefficients such that having degree of 5, excluding exactly one prime and not having a rational root?