Investigando raízes reais de polinômios: uma abordagem com o teorema de Bolzano.

Abstract

The search for the real roots of a polynomial is a classic problem in Mathematics and of fundamental importance for solving various problems in High School. Thus, this article brings together three theorems that facilitate this search, highlighting Bolzano's Theorem, the Theorem of Rational Roots, and Descartes' Rule of Signs, through which it is possible to establish an upper limit for the number of positive or negative real roots, considering their multiplicities, simply by analyzing the number of sign variations in the sequence of their real coefficients.  This article demonstrates that the combined use of these theorems can constitute a didactic tool for high school students in investigating the real roots of a polynomial with real coefficients. Among the results presented, Bolzano's Theorem stands out, which, under appropriate conditions, guarantees the existence of at least one real root in a given interval. In order to facilitate the understanding of the theorems, the definitions of polynomial, numerical value, and real root are presented, followed by some examples and applications.  Furthermore, the relationships between the real roots of a polynomial and the x-coordinates of the points where the graph intersects the x-axis are discussed. Thus, the article presents a didactic approach, using examples of equation solving and their respective geometric interpretations, with the aim of assisting students in investigating the real roots of a polynomial.