Polinômios, equações algébricas e o estudo de suas raízes reais

Abstract

This work aims to help students and high school teachers to improve their math skills in complex numbers, polynomials and polynomial equations. Initially it analysed the historical context of complex numbers then were seen some important concepts such as the body of complex numbers, imaginary unit and complex plane. In addition, the properties and basic operations of the polynomials were presented, the Briot-Ruffini device, through which we can get the quotient and remainder of the division of a polynomial p(x) by a linear polynomial. Significant part of this work was devoted to the study of algebraic equations. In this perspective, were discussed some theorems and methods of resolution of equations such as the method of Gustavo, who helps us in the resolution of equations of the third and fourth degrees, the theorem of rational roots, among others. For both, it was essential to prove the Fundamental Theorem of Algebra, which says that all polynomial not constant with complex coeficients has at least one complex root. Furthermore, we show how we can analyze the number of real roots of a polynomial equation with real coeficients. In this sense, we will prove the Theorem of Descartes, which says that the number of positive roots of an equation does not exceed the number of signal changes following its non-zero coeficients. We prove the theorem of Bolzano, which investigates the number of real roots of an equation in a real interval and finally the theorem of Lagrange the establishes an upper limit on roots of an equation.