Binômio de Newton com expoente negativo e fracionário

Abstract

The discussion in this work revolves around the binomial expansion. We are not interested in exploring the expansion only for positive integer exponents, as is in the usual scope of a Secondary School, where with the aid of counting techniques, students learn a practical device for computations. Such content is usually introduced without any demonstration, since the demonstration for natural exponents, attributed to Pascal, requires knowledge at the highest level of teaching. When we look for a purely algebraic demonstration that is valid also for negative and rational exponents and that can be understood by students, we nd a demonstration proposed by Euler, which we present at the end of the text. For, as of its origins, the binomial expansion was not exclusively for natural exponents nor for any other previously xed numerical set, we believe that an approach that reconciles the presentation of Euler's demonstration with an appropriate way of approaching the subject in a Secondary School could be presented so to allow the teaching and application of the binomial expansion for exponents in a set of values (rationals) relatively larger than the current.