Uma introdução aos polinômios simétricos e aplicações

Abstract

This paper aims to reduce the difficulties that many students have in solving nonlinear systems. To this end, we have a valuable tool, namely, symmetric polynomials. The discussion of this topic in high school has generated great interest among students, since the ideas that will be presented here have been applied in special classes that take the entrance exam for the Military Institute of Engineering (IME) or the Technological Institute of Aeronautics (ITA), and in training for the Mathematics Olympiads. The successful experience with these classes motivated us to take the discussion of symmetric polynomials to conventional classes in Basic Education, particularly in high school. Thus, in Chapter 1 we introduce the concepts and some definitions and present some problems that gradually suggest the importance of the theory. We especially highlight the use of elementary symmetric polynomials and how they are associated with Girard's relations and Newton's sums. In Chapter 2, we present the Van Der Waerden algorithm that shows how to write any symmetric polynomial in terms of elementary symmetric polynomials. Next, we highlight the study of the discriminant for polynomial equations of the 2nd and 3rd degrees. Finally, in Chapter 3, we include the resolution of several problems that serve as support for solving other possible problem situations involving nonlinear systems.