  Let $\mathbb{F}_q$ be the finite field with $q$ elements, where $q$ is a
prime power and $n$ be a positive integer. In this paper, we explore the
factorization of $f(x^{n})$ over $\mathbb{F}_q$, where $f(x)$ is an irreducible
polynomial over $\mathbb F_q$. Our main results provide generalizations of
recent works on the factorization of binomials $x^n-1$. As an application, we
provide an explicit formula for the number of irreducible factors of $f(x^n)$
under some generic conditions on $f$ and $n$.
