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. This problem
came out of an undergraduate research project conducted at
Randolph-Macon College in 2006. The ellipse associated with the
derivative has semimajor axis and semiminor axis . Further
investigation showed that these results could be generalized to
a broader class of polynomials. These results have been
submitted for publication in the Pi Mu Epsilon Journal. The
interested reader might also enjoy the a