Partial Fractions

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Partial Fractions

• Lesson 8.5

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Partial Fraction Decomposition

• Consider adding two algebraic fractions
• Partial fraction decomposition reverses the
  process

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Partial Fraction Decomposition

• Motivation for this process
• The separate terms are easier to integrate

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The Process

• Given
• Where polynomial P(x) has degree lt n
• P(r) ? 0
• Then f(x) can be decomposed with this cascading
  form

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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)

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Strategy

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

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Strategy

1. Quadratic factors For each factor of the form
   , the partial fraction
   decomposition must include the following sum of n
   fractions.

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A Variation

• Suppose rational function has distinct linear
  factors
• Then we know

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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

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What If

• Single irreducible quadratic factor
• But P(x) degree lt 2m
• Then cascading form is

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Gotta Try It

• Given
• Then

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Gotta Try It

• Now equate corresponding coefficients on each
  side
• Solve for A, B, C, and D

?
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Even More Exciting

• When but
• P(x) and D(x) are polynomials with no common
  factors
• D(x) ? 0
• Example

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Combine the Methods

• Consider where
• P(x), D(x) have no common factors
• D(x) ? 0
• Express as cascading functions of

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Try It This Time

• Given
• Now manipulate the expression to determine A, B,
  and C

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Partial Fractions for Integration

• Use these principles for the following integrals

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Why Are We Doing This?

• Remember, the whole idea is tomake the rational
  function easier to integrate

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Assignment

• Lesson 8.5
• Page 559
• Exercises 1 45 EOO