9 - TOPICS IN ALGEBRA

Abstract

In arithmetic, an improper fraction rational number is reduced to the sum of an integer and a proper fraction. There is an analogous reduction for a rational function if the degree of its numerator is greater than or equal to the degree of its denominator. In each case, the top-heavy rational function on the left is reduced to a polynomial plus a bottom-heavy rational function with the same denominator as the given rational function. This chapter discusses the basic property of polynomials, which is the division algorithm. According to the remainder theorem, if f(x) is a polynomial and a is a real number, then f(a) is the remainder when f(x) is divided by x - a. Thus f(x) = (x-a) q(x) + f(a). A zero of a function f(x) is a real number r such that f(r) = 0.