1. Copyright © 2011 Pearson Education, Inc. Slide 7.8-1 7.8 Partial Fractions Partial Fraction Decomposition of Step 1If is not a proper fraction (a fraction with the numerator of lower degree than the denominator), divide f (x) by g(x). For example, Then apply the following steps to the remainder, which is a proper fraction. Step 2Factor g(x) completely into factors of the form (ax + b) m or (cx 2 + dx + e) n, where cx 2 + dx + e is irreducible and m and n are integers.

2. Copyright © 2011 Pearson Education, Inc. Slide 7.8-2 7.8 Partial Fractions Partial Fraction Decomposition of (continued) Step 3 (a)For each distinct linear factor (ax + b), the decomposition must include the term (b)For each repeated linear factor (ax + b) m, the decomposition must include the terms

3. Copyright © 2011 Pearson Education, Inc. Slide 7.8-3 Partial Fraction Decomposition of (continued) Step 4(a)For each distinct quadratic factor (cx 2 + dx + e), the decomposition must include the term (b)For each repeated factor (cx 2 + dx + e) n, the decomposition must include the terms Step 5Use algebraic techniques to solve for the constants in the numerators of the decomposition. 7.8 Partial Fractions

4. Copyright © 2011 Pearson Education, Inc. Slide 7.8-4 ExampleFind the partial fraction decomposition. SolutionWrite the fraction as a proper fraction using long division. 7.8 Finding a Partial Fraction Decomposition

5. Copyright © 2011 Pearson Education, Inc. Slide 7.8-5 Now work with the remainder fraction. Solve for the constants A, B, and C by multiplying both sides of the equation by x(x + 2)(x – 2), getting Substituting 0 in for x gives –2 = –4A, so Similarly, choosing x = –2 gives –12 = 8B, so Choosing x = 2 gives 8 = 8C, so C = 1. 7.8 Finding a Partial Fraction Decomposition

6. Copyright © 2011 Pearson Education, Inc. Slide 7.8-6 The remainder rational expression can be written as the following sum of partial fractions: The given rational expression can be written as Check the work by combining the terms on the right. 7.8 Finding a Partial Fraction Decomposition

7. Copyright © 2011 Pearson Education, Inc. Slide 7.8-7 ExampleFind the partial fraction decomposition. SolutionThis is a proper fraction and the denominator is already factored. We write the decomposition as follows: 7.8 Repeated Linear Factors

8. Copyright © 2011 Pearson Education, Inc. Slide 7.8-8 Multiplying both sides of the equation by (x – 1) 3 : Substitute 1 for x leads to C = 2, so Since any number can be substituted for x, choose x = –1, and the equation becomes 7.8 Repeated Linear Factors

9. Copyright © 2011 Pearson Education, Inc. Slide 7.8-9 Substituting 0 in for x in gives Now solve the two equations with the unknowns A and B to get A = 0 and B = 2. The partial fraction decomposition is 7.8 Repeated Linear Factors

10. Copyright © 2011 Pearson Education, Inc. Slide 7.8-10 ExampleFind the partial fraction decomposition. SolutionThe partial fraction decomposition is Multiply both sides by (x + 1)(x 2 + 2) to get 7.8 Distinct Linear and Quadratic Factors

11. Copyright © 2011 Pearson Education, Inc. Slide 7.8-11 First, substitute –1 in for x to get Replace A with –1 and substitute any value for x, say x = 0, in to get Solving now for B, we get B = 2, and our result is 7.8 Distinct Linear and Quadratic Factors

12. Copyright © 2011 Pearson Education, Inc. Slide 7.8-12 7.8 Techniques for Decomposition into Partial Fractions Method 1For Linear Factors Step 1Multiply each side of the resulting rational equation by the common denominator. Step 2Substitute the zero of each factor into the resulting equation. For repeated linear factors, substitute as many other numbers as is necessary to find all the constants in the numerators. The number of substitutions required will equal the number of constants A, B,...

13. Copyright © 2011 Pearson Education, Inc. Slide 7.8-13 7.8 Techniques for Decomposition into Partial Fractions Method 2For Quadratic Factors Step 1Multiply each side of the resulting rational equation by the common denominator. Step 2Collect terms on the right side of the equation. Step 3Equate the coefficients of like terms to get a system of equations. Step 4Solve the system to find the constants in the numerators.