id: 06331444
dt: j
an: 2014e.00582
au: Northshield, Sam
ti: Geometry of cubic polynomials.
so: Math. Mag. 86, No. 2, 136-143 (2013).
py: 2013
pu: Mathematical Association of America (MAA), Washington, DC
la: EN
cc: G75
ut: Marden’s theorem; cubic polynomials; ellipses; Cardano’s formula
ci:
li: doi:10.4169/math.mag.86.2.136
ab: 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.
rv: