  Let $p(z)$ be a monic polynomial of degree $n$, with complex coefficients,
and let $q(z)$ be its monic factor. We prove an asymptotically sharp inequality
of the form $\|q\|_{E} \le C^n \|p\|_E$, where $\|\cdot\|_E$ denotes the sup
norm on a compact set $E$ in the plane. The best constant $C_E$ in this
inequality is found by potential theoretic methods. We also consider
applications of the general result to the cases of a disk and a segment.
