Gauss-Lucas theorem

Every root of the derivative of a nonconstant complex polynomial lies in the convex hull of its roots. If a critical point $\beta$ is itself a root, this is immediate. Otherwise factorization and the product rule giveAfter taking complex conjugates, set $w_k=|\beta -{\alpha }_k{|}^{-2}>0$ to obtain $\beta ={\sum }_k{w}_k{\alpha }_k/{\sum }_k{w}_k$. This is a convex combination, proving the theorem, with roots counted according to their multiplicities. The same formula shows that roots strictly inside an open half-plane or a convex set that is open keep all critical points there. For a degree-one polynomial the derivative has no zeros and the assertion is vacuous.