Gauss-Lukas theorem:

For any non-constant polinomial P with complex coefficients, the zeroes of P' belong to the convex hull of the zeroes of P.

Proof is by differentiating P in its product form (or its logarithm) and showing any zero of P' is a convex combination of the zeroes of P.

https://en.m.wikipedia.org/wiki/Gauss%E2%80%93Lucas_theorem