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Extensions of Rolle's theorem

Laura J. Batts, Megan E. Moran and Courtney K. Taylor

Vol. 15 (2022), No. 4, 641–648
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.

Keywords
Rolle's theorem, polynomial root sequences
Mathematical Subject Classification
Primary: 26A06
Milestones
Received: 5 August 2021
Revised: 30 December 2021
Accepted: 31 December 2021
Published: 7 January 2023

Communicated by Michael Dorff
Authors
Laura J. Batts
Department of Mathematics
Anderson University
Anderson, IN
United States
Megan E. Moran
Department of Mathematics
Anderson University
Anderson, IN
United States
Courtney K. Taylor
Department of Mathematics
Anderson University
Anderson, IN
United States