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Geometry of cubic polynomials. (English)
Math. Mag. 86, No. 2, 136-143 (2013).
Summary: Imagine 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 $p^{\prime}(z)$ are the foci of the ellipse inscribed in and tangent to the midpoints of the triangle determined by the roots of the polynomial.
Classification: G75