  Let $\mathcal{A}$ and $\mathcal{B}$ be sets of polynomials of degree $n$ over
a finite field. We show, that if $\mathcal{A}$ and $\mathcal{B}$ are large
enough, then $A+B$ has an irreducible divisor of large degree for some
$A\in\mathcal{A}$ and $B\in \mathcal{B}$.
