  In this paper, we provide the degree distribution of irreducible factors of
the composed polynomial $f(L(x))$ over $\mathbb F_q$, where $f(x)\in \mathbb
F_q[x]$ is irreducible and $L(x)\in \mathbb F_q[x]$ is a linearized polynomial.
We further provide some applications of our main result, including lower bounds
for the number of irreducible factors of $f(L(x))$, constructions of high
degree irreducible polynomials and the explicit factorization of $f(x^q-x)$
under certain conditions on $f(x)$.
