There’s a well-known theorem in complex analysis that says that if *p* is a polynomial, then the zeros of its derivative *p*′ lie inside the convex hull of the zeros of *p*. The convex hull of a set of points is the smallest convex set containing those points.

This post gives a brief illustration of the theorem. I created a polynomial with roots at 0, *i*, 2 + *i*, 3-*i*, and 1.5+0.5*i*. The convex hull of these points is the quadrilateral with corners at the first four roots; the fifth root is inside the convex hull of the other roots.

The roots are plotted with blue dots. The roots of the derivative are plotted with orange ×’s.

In the special case of cubic polynomials, we can say a lot more about where the roots of the derivative lie. That is the topic of the next post.

## More complex analysis posts

Check out this cute proof using electrostatics:

https://johncarlosbaez.wordpress.com/2021/05/24/electrostatics-and-the-gauss-lucas-theorem/