Descartes' Rule of Signs by an Easy Induction

TL;DR

This paper presents an inductive proof of Descartes' rule of signs, demonstrating how multiplying a polynomial by (c - x) affects the sign changes in its coefficients.

Contribution

It introduces an easy inductive method to prove Descartes' rule of signs, providing a new perspective on the classical theorem.

Findings

Proves the effect of multiplying by (c - x) on sign changes in polynomial coefficients

Derives Descartes' rule of signs from the inductive proof

Offers a simplified approach to understanding polynomial sign variations

Abstract

If c is a positive number, Descartes' rule of signs implies that multiplying a polynomial f(x) by c - x introduces an odd number of changes of sign in the coefficients. We turn this around, proving this fact about sign changes inductively and deriving Descartes' rule from it.