Objectives Decompose where Q has only distinct linear

Objectives: • Decompose where Q has only distinct linear factors. • Decompose where Q has repeated linear factors. • Decompose where Q has a nonrepeated prime quadratic factor. • Decompose where Q has a prime, repeated quadratic factor. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

The Partial Fraction Decomposition of a Rational Expression with Distinct Linear Factors in the Denominator Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2

Steps in Partial Fraction Decomposition Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3

Example: Partial Fraction Decomposition with Distinct Linear Factors Find the partial fraction decomposition of Step 1 Set up the partial fraction decomposition with the unknown constants. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4

Example: Partial Fraction Decomposition with Distinct Linear Factors (continued) Find the partial fraction decomposition of Step 2 Multiply both sides of the resulting equation by the least common denominator. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5

Example: Partial Fraction Decomposition with Distinct Linear Factors (continued) Find the partial fraction decomposition of Step 3 Simplify the right side of the equation. Step 4 Write both sides in descending powers, equate coefficients of like powers of x, and equate constant terms. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6

Example: Partial Fraction Decomposition with Distinct Linear Factors (continued) Find the partial fraction decomposition of Step 5 Solve the resulting system for A and B. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7

Example: Partial Fraction Decomposition with Distinct Linear Factors (continued) Find the partial fraction decomposition of Step 6 Substitute values of A and B and write the partial fraction decomposition. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8

The Partial Fraction Decomposition of Repeated Linear Factors Copyright © 2014, 2010, 2007 Pearson Education, Inc. Has 9

Example: Partial Fraction Decomposition with Repeated Linear Factors Find the partial fraction decomposition of Step 1 Set up the partial fraction decomposition with the unknown constants. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10

Example: Partial Fraction Decomposition with Repeated Linear Factors (continued) Find the partial fraction decomposition of Step 2 Multiply both sides of the resulting equation by the least common denominator. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11

Example: Partial Fraction Decomposition with Repeated Linear Factors (continued) Find the partial fraction decomposition of Step 3 Simplify the right side of the equation. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12

Example: Partial Fraction Decomposition with Repeated Linear Factors (continued) Find the partial fraction decomposition of Step 4 Write both sides in descending powers, equate coefficients of like powers of x, and equate constant terms. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13

Example: Partial Fraction Decomposition with Repeated Linear Factors (continued) Find the partial fraction decomposition of Step 5 Solve the resulting system for A and B. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14

Example: Partial Fraction Decomposition with Repeated Linear Factors (continued) Find the partial fraction decomposition of Step 6 Substitute values of A and B and write the partial fraction decomposition. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15

The Partial Fraction Decomposition of a Rational Expression with Prime, Nonrepeated Quadratic Factors in the Denominator Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16

Example: Partial Fraction Decomposition Find the partial fraction decomposition of Step 1 Set up the partial fraction decomposition with the unknown constants. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17

Example: Partial Fraction Decomposition (continued) Find the partial fraction decomposition of Step 2 Multiply both sides of the resulting equation by the least common denominator. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 18

Example: Partial Fraction Decomposition (continued) Find the partial fraction decomposition of Step 3 Simplify the right side of the equation. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 19

Example: Partial Fraction Decomposition (continued) Find the partial fraction decomposition of Step 4 Write both sides in descending powers, equate coefficients of like powers of x, and equate constant terms. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 20

Example: Partial Fraction Decomposition (continued) Find the partial fraction decomposition of Step 5 Solve the resulting system for A and B. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 21

Example: Partial Fraction Decomposition (continued) Find the partial fraction decomposition of Step 5 (cont) Solve the resulting system for A and B. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 22

Example: Partial Fraction Decomposition (continued) Find the partial fraction decomposition of Step 6 Substitute values of A and B and write the partial fraction decomposition. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 23

The Partial Fraction Decomposition of a Rational Expression with a Prime, Repeated Quadratic Factor in the Denominator Copyright © 2014, 2010, 2007 Pearson Education, Inc. 24

Example: Partial Fraction Decomposition with a Repeated Quadratic Factor Find the partial fraction decomposition of Step 1 Set up the partial fraction decomposition with the unknown constants. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 25

Example: Partial Fraction Decomposition with a Repeated Quadratic Factor (continued) Find the partial fraction decomposition of Step 2 Multiply both sides of the resulting equation by the least common denominator. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 26

Example: Partial Fraction Decomposition with a Repeated Quadratic Factor (continued) Find the partial fraction decomposition of Step 3 Simplify the right side of the equation. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 27

Example: Partial Fraction Decomposition with a Repeated Quadratic Factor (continued) Find the partial fraction decomposition of Step 4 Write both sides in descending powers, equate coefficients of like powers of x, and equate constant terms. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 28

Example: Partial Fraction Decomposition with a Repeated Quadratic Factor (continued) Find the partial fraction decomposition of Step 5 Solve the resulting system for A and B. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 29

Example: Partial Fraction Decomposition with a Repeated Quadratic Factor (continued) Find the partial fraction decomposition of Step 6 Substitute values of A and B and write the partial fraction decomposition. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 30