Fatoração de expressões polinomiais e decomposição em frações parciais

Abstract

This dissertation investigates the factorization of polynomials and the decomposition into partial fractions within the context of Basic and Higher Education, articulating algebraic foundations with proposals aimed at teaching practice. The study begins with the extension of number systems, emphasizing the introduction of complex numbers as a natural extension of R, preserving its algebraic properties while enlarging the set in which solutions exist. It is shown that the solution of the quadratic equation ax^2+bx+c = 0, with real coefficients, maintains the same structure expressed by Bhaskara’s formula, the essential difference lying in the numerical set in which the discriminant is interpreted. In C, regardless of the sign of Δ = b^2 − 4ac, the square root of Δ always exists, ensuring the existence of solutions. It is further emphasized that algebraic properties valid in R, such as special products and completing the square, remain valid in C, helping to clarify frequent difficulties both in the factorization process and in solving polynomial equations. Classical methods are presented, including Euclidean Division for polynomials, the Briot-Ruffini algorithm, and the Rational Root Theorem, highlighting that the factorization of a polynomial depends on the coefficient ring. The case of x^2 + 1 in R[x] and C[x] is discussed, showing that in C[x] every polynomial can be factored into linear factors, according to the Fundamental Theorem of Algebra. This fact structurally simplifies the partial fraction decomposition of rational functions with real coefficients when analyzed in the complex setting. As an educational product, a Workbook of Commented Activities is presented, in which these structural observations are explored didactically through examples and proposed activities. The material is intended for high school students, early undergraduate students, and mathematics teachers, contributing to a more structured understanding of Algebra and its articulation with Calculus.