College Algebra Chapter 5 Systems of Equations and Inequalities Section 5.3 Partial Fraction Decomposition

Concepts 1. Set up a Partial Fraction Decomposition 2. Decompose , where is a Product of Linear Factors 3. Decompose , where has Irreducible Quadratic Factors

Set up a Partial Fraction Decomposition Previously, we have added or subtracted rational expressions. For example,

Set up a Partial Fraction Decomposition Now we will learn the technique of partial fraction decomposition to write a rational expression as the sum of simpler fractions. In this case, we start with the rational expression and decompose it into two simpler fractions.

Example: Find constants A and B such that Multiply both sides by the common denominator

Example continued: Simplify and combine like terms. Two polynomials are equal if and only if the coefficients on like terms are equal.

Example continued: Solve the system of linear equations.

Set up a Partial Fraction Decomposition How to begin a partial fraction decomposition? Make sure the degree of the numerator is less than the degree of the denominator.   Factor the denominator. The denominator will factor into linear factors (ax + b) and quadratic factors (ax2 + bx + c) that are irreducible over the integers.

Set up a Partial Fraction Decomposition There will be one (or more) of four types:

Example 1: Set up the form for the partial fraction decomposition for the given rational expressions. (distinct linear factors)

Example 2: Set up the form for the partial fraction decomposition for the given rational expressions. (repeated linear factors)

Example 3: Set up the form for the partial fraction decomposition for the given rational expressions. (repeated linear factors)

Example 4: Set up the form for the partial fraction decomposition for the given rational expressions. (linear/quadratic factors)

Example 5: Set up the form for the partial fraction decomposition for the given rational expressions. (repeated linear/quadratic factors)

Concepts 1. Set up a Partial Fraction Decomposition 2. Decompose , where is a Product of Linear Factors 3. Decompose , where has Irreducible Quadratic Factors

Example 6: Find the partial fraction decomposition.

Example 6 continued:

Example 6 continued:

Example 7: Find the partial fraction decomposition.

Example 7 continued:

Concepts 1. Set up a Partial Fraction Decomposition 2. Decompose , where is a Product of Linear Factors 3. Decompose , where has Irreducible Quadratic Factors

Example 8: Find the partial fraction decomposition.

Example 8 continued:

Example 8 continued:

Example 9: Find the partial fraction decomposition.

Example 9 continued: