Polinômios Palindrômicos: Método Alternativo para a Redução de Grau

Abstract

This work presents a study on self-inverting polynomials, especially palindromic polynomials. For that, we present classic results referring to the classical theory of polynomials, as well as the study of the behavior of its roots in the Cartesian plane, definitions and properties about the auto-invertible and palindromic polynomials. We present an alternative method for reducing the degree of palindromic polynomials P, in the case of this being of degree 2n (n belonging to naturals) using the Viète relation and the characteristics inherent to its roots, we obtain a polynomial Q of degree n, which we will call polynomial reduced from P. In the case of P being of odd degree this is also possible, since -1 is a root of all palindromic polynomial of odd degree. Since the reduced polynomial Q associated with P contains all its roots, it is possible to determine them by solving n quadratic equations or from the coefficients of Q we can infer about the nature of its roots and consequently about the nature of the roots of P and their location in relation to the unit circle.