POLINÔMIOS E SINAIS: UMA PROVA ALGÉBRICA DA REGRA DE DESCARTES PARA PROFESSORES E ESTUDANTES

ABSTRACT This article presents an algebraic and inductive proof of Descartes’ Rule of Signs, dispensing with the use of Analysis tools. This article presents an algebraic and inductive proof of Descartes’ Rule of Signs, avoiding the use of Analysis tools such as Rolle’s Theorem or the Intermediate Value Theorem. The theorem establishes an upper bound on the number of positive (or negative) real roots of a real polynomial based on the number of sign variations of its coefficients. The novelty of this proof lies in its purely algebraic and constructive nature, structured around elementary lemmas on signs and induction on the number of positive roots, which gives it educational value for secondary and lower secondary education. We discuss the history of the theorem, various existing proofs, and indicate how the topic can be explored pedagogically in the classroom. Keywords: Algebraic proof, Descartes’ Theorem, lemmas, induction, didactic value, High School.