  Let $f=a_0+ a_{1}x+\cdots+a_m x^m\in \Bbb{Z}[x]$ be a primitive polynomial.
Suppose that there exists a positive real number $\alpha$ such that $|a_m|
\alpha^m>|a_0|+|a_1|\alpha+\cdots+|a_{m-1}|\alpha^{m-1}$. We prove that if
there exist natural numbers $n$ and $d$ satisfying $n\geq \alpha+ d$ for which
either $|f(n)|/d$ is a prime, or $|f(n)|/d$ is a prime-power coprime to
$|f'(n)|$, then $f$ is irreducible in $\mathbb{Z}[x]$.
