The Geometry of Polynomial Roots

It's well-known that any polynomial with complex coefficients has all of its roots in the complex plane. Moreover, if one counts multiplicity of roots, the function $f(x)$ has $\text{deg}(f)$ number of roots. This means that each polynomial yields a set of points in the plane. What is less-commonly known is the location of these points with respect to those of its derivative. In this project, I show three things:

• Part 1: The roots of the derivative of $f$ all lie within the convex hull of the roots of $f$
• Part 2: Playground (interactive implementation)
• Part 3: The roots of a cubic $f$ form the focii of a special ellipse