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Many fascinating facts are known relating the roots of a polynomial to the roots of its derivatives. And whereas many of these facts are easily accessible to an undergraduate audience (or even a high school audience if the fundamental theorem of algebra is granted), it is rare that they are discussed at this level. One reason for this omission is the absence, until relatively recently, of an easy way to visualize the complex roots of a polynomial. With Mathematica, such visualizations are a simple matter. One is easily led to explore this rich arena, and to contemplate conjectures such as the elusive Sendov's conjecture, which has evaded proof for over forty years.
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