
06331444
j
2014e.00582
Northshield, Sam
Geometry of cubic polynomials.
Math. Mag. 86, No. 2, 136143 (2013).
2013
Mathematical Association of America (MAA), Washington, DC
EN
G75
Marden's theorem
cubic polynomials
ellipses
Cardano's formula
doi:10.4169/math.mag.86.2.136
Summary: Imagine a sphere with its equator inscribed in an equilateral triangle. This Saturnlike 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 $p^{\prime}(z)$ are the foci of the ellipse inscribed in and tangent to the midpoints of the triangle determined by the roots of the polynomial.