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