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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 Then f(x) can be decomposed with this cascading form
Where polynomial P(x) has degree < 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 obtain Degree 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 Factor the denominator into factors of the form where is irreducible For each factor the partial fraction must include the following sum of m fractions
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Strategy 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 Then cascading form is
But P(x) degree < 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 Example
P(x) and D(x) are polynomials with no common factors D(x) ≠ 0 Example
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Combine the Methods Consider where Express as cascading functions of
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 to make the rational function easier to integrate
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Assignment Lesson 8.5 Page 559 Exercises 1 – 29 EOO
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