Teorema de Rolle e a regra de Descartes

Tese mest. , Matemática para o Ensino, 2007, Universidade do AlgarveFrom algebra we know that a polynomial of degree n with real coe±cients has at most n real zeros. But this result does not give any information about the number of positive zeros of such a polynomial. This question take us to Descarte's Rule of Signs, which give an upper bound for the number of positive zeros of a real polynomial. In this work we study Descarte's Rule of Signs following the work of P¶olya and SzegÄo in [8, Parte 5, cap.1]. We successively study the zeros and sign variations of a function, the sign changes of a sequence and present an algebraic proof of Descarte's Rule of Signs. We show some applications of Descartes's Rule of Signs and Rolle's Theorem to the resolution of some problems from algebra and analysis. Using Rolle's Theorem we prove analytically the Rule of Signs and use this method of proof to extend Descarte's Rule of Signs to di®erent systems of functions, ending with a necessary and su±cient condition for a system of functions to satisfy Descarte's Rule of Signs