TY: THES
T1 - Teorema de Rolle e a regra de Descartes
A1 - Vaz, Dalila Maria Palma Afonso
N2 - From 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.
UR - https://sapientia.ualg.pt/handle/10400.1/843
Y1 - 2007
PB - No publisher defined