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Partial Fractions

Partial Fractions. Lesson 8.5. Partial Fraction Decomposition. Consider adding two algebraic fractions Partial fraction decomposition reverses the process. Partial Fraction Decomposition. Motivation for this process The separate terms are easier to integrate. The Process. Given

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Partial Fractions

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  1. Partial Fractions Lesson 8.5

  2. Partial Fraction Decomposition • Consider adding two algebraic fractions • Partial fraction decomposition reverses the process

  3. Partial Fraction Decomposition • Motivation for this process • The separate terms are easier to integrate

  4. The Process • Given • Where polynomial P(x) has degree < n • P(r) ≠ 0 • Then f(x) can be decomposed with this cascading form

  5. Strategy Given N(x)/D(x) • If degree of N(x) greater than degree of D(x) divide the denominator into the numerator to obtainDegree of N1(x) will be less than that of D(x) • Now proceed with following steps for N1(x)/D(x)

  6. Strategy • Factor the denominator into factors of the formwhere is irreducible • For each factor the partial fraction must include the following sum of m fractions

  7. Strategy • Quadratic factors: For each factor of the form , the partial fraction decomposition must include the following sum of n fractions.

  8. A Variation • Suppose rational function has distinct linear factors • Then we know

  9. A Variation • Now multiply through by the denominator to clear them from the equation • Let x = 1 and x = -1 • Solve for A and B

  10. What If • Single irreducible quadratic factor • But P(x) degree < 2m • Then cascading form is

  11. Gotta Try It • Given • Then

  12. Gotta Try It • Now equate corresponding coefficients on each side • Solve for A, B, C, and D ?

  13. Even More Exciting • When but • P(x) and D(x) are polynomials with no common factors • D(x) ≠ 0 • Example

  14. Combine the Methods • Consider where • P(x), D(x) have no common factors • D(x) ≠ 0 • Express as cascading functions of

  15. Try It This Time • Given • Now manipulate the expression to determine A, B, and C

  16. Partial Fractions for Integration • Use these principles for the following integrals

  17. Why Are We Doing This? • Remember, the whole idea is tomake the rational function easier to integrate

  18. Assignment • Lesson 8.5 • Page 559 • Exercises 1 – 29 EOO

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