Real Roots of Polynomials with Real Coefficients

Abstract

Polynomial equations are used throughout mathematics. When solving polynomials many questions arise such as: Are there any real roots? If so, how many? Where are they located? Are these roots positive or negative? Depending on the problem being solved sometimes a rough estimate for the interval where a root is located is enough.

There are many methods that can be used to answer these questions. We will focus on Descartes' Rule of Signs, the Budan-Fourier theorem and Sturm's theorem. Descartes' Rule of Signs traditionally is used to determine the possible number of positive real roots of a polynomial. This method can be modified to also find the possible negative roots for a polynomial. The Budan-Fourier theorem takes advantage of the derivatives of a polynomial to determine the number of possible number of roots. While Sturm's theorem uses a blend of derivatives and the Euclidean Algorithm to determine the exact number of roots.

In some cases, an interval where a root of the polynomial exists is not enough. Two methods, Horner and Newton's methods, to numerically approximate roots up to a given precision are also discussed. We will also give a real world application that uses Sturm's theorem to solve a problem.