  The Gauss-Lucas theorem says that for any complex polynomial $P$, the roots
of the derivative $P'$ lie in the convex hull of the roots of $P$. In other
words, the roots of $P'$ lie inside the smallest convex subset of the complex
plane containing all the roots of $P$. This theorem is not hard to prove, but
is there an intuitive explanation? In fact there is, using physics -- or more
precisely, electrostatics in 2-dimensional space

Complex Variables