Aspectos teóricos e computacionais em polinômios

Abstract

In this work we study ways to find polynomial roots computationally and their algebraic properties. We use two excellent tools for that: the Newton-Raphson Method and the Symmetric Polynomials theory. In the first part, we demonstrate the local convergence of the Newton-Raphson Method for real functions. We also perform a study to verify the existence and location of the roots of a polynomial through the Descartes’ Rule of

Signs and Sturm Sequences. In particular, we make a detailed study of the Newton- Raphson Method to find the nth root of a positive number. Next, we study symmetric

polynomials: a very useful tool for solving algebraic problems of factorization, non-linear equation systems, linear recurrence equations and some algebraic equations. At the end, a list of solved exercises is presented, aiming at fixing these subjects.