Geometry of Cubic Polynomials

Abstract

SummaryImagine a sphere with its equator inscribed in an equilateral triangle. This Saturn-like figure will help us understand from where Cardano's formula for finding the roots of a cubic polynomial p(z) comes. It will also help us find a new proof of Marden's theorem, the surprising result that the roots of the derivative p′(z) are the foci of the ellipse inscribed in and tangent to the midpoints of the triangle determined by the roots of the polynomial.