The Journal of Online Mathematics and Its Applications, Volume 8 (2008)
The Most Marvelous Theorem in Mathematics, Dan Kalman

Repeated Roots

In the restricted version of Marden's Theorem, it is assumed that the p(z) is a third degree polynomial with complex coefficients whose roots z1, z2, and z3 are non-collinear points in the complex plane. In particular, this implies that the roots are distinct. More generally, when dealing with complex numbers, any polynomial can be expressed in the form

where each mj is an integer greater than or equal to 1. Whenever mj is strictly greater than 1, we say that the corresponding zj is a repeated root. In this case, mj is called the multiplicity of the root zj.

Using the product rule, it is easy to see that if (z − zj)mj is a factor of p then (z − zj)mj − 1 is a factor of p. This shows that a repeated root of p is also a root of p, with multiplicity reduced by 1.

In considering the locations of the roots of p′ relative to the roots of p, we can observe at the start that any repeated roots of p will also appear as roots of p′. We can even count these roots. In p the multiplicity of root zj will be mj − 1. Overall, this accounts for all but n − 1 of the roots of p′, where n is the number of distinct roots of p. The remaining roots of p, those that are not also roots of p, are the roots of the logarithmic derivative (log f(z))′.

My version of Marden's Theorem concerns a polynomial with three distinct and unrepeated roots zj. Marden's more general version of the theorem particularizes to a polynomial with three distinct roots, but with multiplicities mj that may exceed one. In this case, there are exactly two roots of p that are not also roots of p, and they are again foci of an ellipse inscribed in the triangle with vertices at the roots of p. However, in this more general setting, the inscribed ellipse is not tangent to the sides of the triangle at their midpoints. Rather, the points of tangency are determined by proportions of multiplicities. Specifically, on side zj zk, the inscribed ellipse is tangent at a point w, where the ratio of the lengths of segments zj w and w zk is mj / mk.