The Joukowsky Map Reveals the Cubic Equation

Two canonical polynomials generate all cubics, via linear transformations of the polynomial map and the parameter: the cubic power function, with coincident critical points, and the third Chebyshev polynomial of the first kind, with two distinct critical points. Computing the roots of any cubic boils down to inverting these fundamental maps. In the more general case of distinct critical points, we show that the roots admit a startlingly simple expression in terms of a Joukowsky map and its inverse. Marden's theorem comes as a straightforward consequence, because the roots are the images, under a Joukowsky map, of the vertices of an equilateral triangle.