A sufficient condition for checking the robust stability of a polytope of polynomial matrices is proposed. A simple feasibility test performed in a convex set of linear matrix inequalities defined at the vertices of the polytope yields sufficient conditions for the robust stability of the entire domain. Both continuous-time (left half-plane) and discrete-time stability (unit disk) are investigated. Numerical comparisons with quadratic stability results and with another method recently appeared in the literature show that the conditions proposed provide, in general, less conservative results.