Partial Fractions in Euclidean Domains

Abstract

When we teach the technique of using partial fractions to integrate a rational function, we usually first tell the students that a rational function which is proper (i.e., in which the degree of the top is less than the degree of the bottom) can be broken up into partial fractions, and we show the students the algebraic tricks involved in solving for the constants. We then mention that to integrate any rational function whatever, first do the division to get a polynomial plus a proper rational function, and then apply the previous theory. Thus, we produce the two statements below. In each of these, F[x] stands for the set of polynomials over the reals.