Introduction

From Whole Polynomials to Their Quotients

The previous chapter worked entirely with polynomial equations — expressions built purely from sums of powers of $x$. But polynomials are just as often divided by one another, as in $\dfrac{f(x)}{g(x)}$, and such a quotient is called a rational fraction. These come up constantly once you start doing calculus: evaluating certain integrals, solving differential equations, and expanding or summing infinite power series are all far easier if the fraction is first broken apart into simpler pieces.

What This Chapter Covers

This chapter develops the technique of partial-fraction decomposition — rewriting a single rational fraction $\dfrac{f(x)}{g(x)}$ as a sum of simpler fractions, each with a lower-degree denominator. Three cases are treated in turn, depending on the structure of the denominator $g(x)$: when $g(x)$ factors into distinct (non-repeated) linear factors, when it has repeated linear factors, and when it contains irreducible quadratic factors that cannot be split further over the reals. Along the way, the chapter also distinguishes a proper fraction (numerator's degree less than denominator's) from an improper one, since the decomposition method is built for proper fractions and an improper one must first be reduced by division.