The previous post was a warmup for this post. It gave an example of the theorem that if *p* is a polynomial, the roots of its derivative *p*′ lie inside the convex hull of the roots of *p*. If *p* is a cubic polynomial, we can say much more.

Suppose *p*(*z*) is a polynomial with three distinct roots, not all on a line. Then the roots of *p* are the vertices of a triangle *T* in the complex plane. We know that the roots of *p*′ lie inside *T*, but **Marden’s theorem** says *where* they lie inside *T*.

Let *E* be the unique ellipse inscribed inside *T* that touches *T* at the midpoint of each side. Then Marden’s theorem says that the roots of *p*′ lie on the foci of *E*.

The ellipse *E* is sometimes called the **Steiner inellipse**.

For an example, let

*p*(*z*) = *z* (*z* – 3) (*z* – 2 – 4*i*)

The roots of the derivative of *p* are 1.4495 + 0.3100*i* and 1.8838 + 2.3566*i*.

Here’s a plot of the roots of *p* (blue dots), the roots of *p*′ (orange ×), and the Steiner ellipse. I used a parametric equation for the Steiner ellipse from here.

Might want to add that p’ is the derivative of p to this blog too. Had to click the other one to see what it was, thought it might have been some other polynomial.