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12. Write the partial fraction decomposition for the rational expression. 3x2 - 2x- 1 x(x+ 1)2(x² + 4)2

12. Write the partial fraction decomposition for the rational expression. 3x2 - 2x- 1 x(x+ 1)2(x² + 4)2

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.2: Exponents And Radicals
Problem 91E
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Case 4

6. What are the forms (constant, linear, or quadratic) of the numerators in each fraction above? 7. How does the form of the numerator compare with its denominator? 8. Explain the differences in the form of the numerators for terms with irreducible quadratics (case 3) and repeating linear terms (case 2). 9. Evaluate the integral. 2x2-x+4 x3+ 4x xp Case 4. Q(x) Contains Repeated Irreducible Quadratic Factors Something like (x2 +2)2 occurs in the factorization of Q(x). In this case we need to write a fraction for each power of x2 + 2. For example, Cx + D (x² + 2)²* 8x3 +13x Ax + B (x2 + 2)2 %3D 11. How is the partial fraction decomposition in case 4 similar to case 2 and case 3? 12. Write the partial fraction decomposition for the rational expression. 3x2 - 2x-1 | x(x + 1)²(x² + 4)2 13. Evaluate the integral. EX-Z+xーI xp x(x² + 1)2 14. Is partial fraction decomposition necessary for integrating all rational functions? How are the rational functions in problems 1, 2, and 3 different than the others? 15. Using compete sentences, summarize the process of using partial fractions for integrating a rational function. Specify the algebraic tools needed throughout the process.
Transcribed Image Text:6. What are the forms (constant, linear, or quadratic) of the numerators in each fraction above? 7. How does the form of the numerator compare with its denominator? 8. Explain the differences in the form of the numerators for terms with irreducible quadratics (case 3) and repeating linear terms (case 2). 9. Evaluate the integral. 2x2-x+4 x3+ 4x xp Case 4. Q(x) Contains Repeated Irreducible Quadratic Factors Something like (x2 +2)2 occurs in the factorization of Q(x). In this case we need to write a fraction for each power of x2 + 2. For example, Cx + D (x² + 2)²* 8x3 +13x Ax + B (x2 + 2)2 %3D 11. How is the partial fraction decomposition in case 4 similar to case 2 and case 3? 12. Write the partial fraction decomposition for the rational expression. 3x2 - 2x-1 | x(x + 1)²(x² + 4)2 13. Evaluate the integral. EX-Z+xーI xp x(x² + 1)2 14. Is partial fraction decomposition necessary for integrating all rational functions? How are the rational functions in problems 1, 2, and 3 different than the others? 15. Using compete sentences, summarize the process of using partial fractions for integrating a rational function. Specify the algebraic tools needed throughout the process.
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