Roots of the Derivatives of a Certain Real Polynomial in the Complex Plane

With and real, is a real polynomial with three roots: and . This Demonstration shows the roots of , , …, . Each of these derivatives has at most two nonreal roots (depending on the value of the real root ). As varies over the reals, these nonreal roots trace out ellipses in the complex plane. The nonreal roots of the first derivative lie on a circle, while those of the higher derivatives lie on successively narrower ellipses. Set the value of , and slide along the real line to see the roots of the derivatives move in real time.