Extensions of Rolle’s theorem

Abstract

Repeatedly differentiating a polynomial with distinct real roots and then finding the roots of each derivative produces a sequence of real numbers. The classical Rolle’s theorem, typically studied in first-semester calculus, provides some constraints on the ordering of these roots. However, not all root sequences that are allowed by Rolle’s theorem occur for polynomials with all real roots. We use elementary methods to prove several Rolle’s-type theorems that further constrain the orderings of the roots of polynomials and their derivatives.