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What You Should Learn Recognize partial fraction decompositions of rational expressions. Find partial fraction decompositions of rational expressions.

Introduction We will learn to write a rational expression as the sum of two or more simpler rational expressions. For example, the rational expression can be written as the sum of two fractions with first-degree denominators. That is,

Introduction Each fraction on the right side of the equation is a partial fraction, and together they make up the partial fraction decomposition of the left side.

Introduction

Partial Fraction Decomposition Algebraic techniques for determining the constants in the numerators of partial fractions are demonstrated in the examples that follow. Note that the techniques vary slightly, depending on the type of factors of the denominator: linear or quadratic, distinct or repeated.

Example 1 – Distinct Linear Factors Write the partial fraction decomposition of .

Partial Fraction Decomposition

Example 2 Write the partial fraction decomposition of

Example 3 Write the partial fraction decomposition of

Example 4 Write the partial fraction decomposition of

Partial Fraction Decomposition Keep in mind that for improper rational expressions such as you must first divide before applying partial fraction decomposition.

Example 6 Write the partial fraction decomposition of