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In this talk I will outline some new results on the location in the complex plane of the critical points (and the roots of the higher derivatives) of a class of real polynomials with one real root. These investigations, while easily accessible to undergraduate and even advanced high school mathematics students, reveal a beautiful structure that is easily explored with Mathematica, both graphically and algebraically. In particular, the roots of the higher derivatives of the polynomial a, where a, a, and a is a positive integer, will either be real or will lie on a fixed ellipse as the real root a varies. I will outline not only the mathematical results themselves in the context of known results (such as the Lucas-Gauss theorem), but also the manner in which Mathematica aided in the discovery and the proof of these results. The broader purpose is to demonstrate by example how Mathematica can be an integral component for student research projects in pure mathematics.
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