Cubic Polynomial and Marden's Theorem
- Steve Phelps
With one real and two complex roots, the three roots can be represented as points in the complex plane, as can the two roots of the cubic's derivative. Marden's Theorem descirbes the geometrical relationship among all these roots. The points in the complex plane representing the three roots serve as the vertices of an isosceles triangle. Marden's Theorem says that the points representing the roots of the derivative of the cubic are the foci of the Steiner inellipse, the unique ellipse that is tangent to the triangle at the midpoints of its sides. In the case the triangle is equilateral, the Steiner inellipse is simply the triangle's incircle, its foci coincide with each other at the incenter, which lies on the real axis, and hence the derivative has duplicate real roots.