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Another irreducibility criterion

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 $α$ such that $|a_m| α^m>|a_0|+|a_1|α+\cdots+|a_{m-1}|α^{m-1}$. We prove that if there exist natural numbers $n$ and $d$ satisfying $n\geq α+ d$ 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 $\mathbb{Z}[x]$.