Gauss-Lucas theorem
= Gauss-Lucas theorem
{c}
{wiki=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> give
$$
0=\frac{Q'(\beta)}{Q(\beta)}=\sum_k\frac1{\beta-\alpha_k}.
$$
After taking <complex conjugates>, set $w_k=|\beta-\alpha_k|^{-2}>0$ to obtain $\beta=\sum_kw_k\alpha_k/\sum_kw_k$. 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.