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Partial-Fraction Decomposition: Examples (page 3 of 3)

Sections: General techniques, How to handle repeated and irreducible factors, Examples


If the denominator of your rational expression has repeated unfactorable quadratics, then you use linear-factor numerators and follow the pattern that we used for repeated linear factors in the denominator; that is, you'll use fractions with increasing powers of the repeated factors in the denominator.

  • Set up, but do not solve, the decomposition equality for the following:
    • [ x^4 + 3x - 2 ] / [ (x^2 + 1)^3 (x - 4)^2 ]

    Since x2 + 1 is not factorable, I'll have to use numerators with linear factors. Then the decomposition set-up looks like this:

      A/(x - 4) + B/(x - 4)^2 + (Cx + D)/(x^2 + 1) + (Ex + F)/(x^2 + 1)^2 + (Gx + H)/(x^2 + 1)^3

Thankfully, I don't have to try to solve this one.


One additional note: Partial-fraction decomposition only works for "proper" fractions. That is, if the denominator's degree is not larger than the numerator's degree (so you have, in effect, an "improper" polynomial fraction), then you first have to use long division to get the "mixed number" form of the rational expression. Then decompose the remaining fractional part.

  • Decompose the following:   Copyright Elizabeth Stapel 2006-2011 All Rights Reserved
    • (x^5 - 2x^4 + x^3 + x + 5) / (x^3 - 2x^2 + x - 2)

    The numerator is of degree 5; the denominator is of degree 3. So first I have to do the long division:

      long division

    The long division rearranges the rational expression to give me:

      x^2 + (2x^2 + x + 5) / (x^3 - 2x^2 + x - 2)

    Now I can decompose the fractional part. The denominator factors as (x2 + 1)(x 2).

      x^2 + (2x^2 + x + 5) / [ (x^2 + 1)(x - 2) ]

    The x2 + 1 is irreducible, so the decomposition will be of the form:

      (2x^2 + x + 5)/(x^3 - 2x^2 + x - 2) = A/(x - 2) + (Bx + C)/(x^2 + 1)

    Multiplying out and solving, I get:

      2x2 + x + 5 = A(x2 + 1) + (Bx + C)(x 2)
      x = 2: 8 + 2 + 5 = A(5) + (2B + C)(0), 15 = 5A, and A = 3
      x = 0: 0 + 0 + 5 = 3(1) + (0 + C)(0 2),
                5 = 3 2C, 2 = 2C, and C = 1
      x = 1: 2 + 1 + 5 = 3(1 + 1) + (1B 1)(1 2),
                8 = 6 + (B  1)(1) = 6 B + 1,

                8 = 7  B, 1 = B,
      and B = 1

    Then the complete expansion is:

      x^2 + 3/(x - 2) + (-x - 1)/(x^2 + 1) = x^2 + 3/(x - 2) - (x + 1)/(x^2 + 1)

The preferred placement of the "minus" signs, either "inside" the fraction or "in front", may vary from text to text. Just don't leave a "minus" sign hanging loose underneath.

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Cite this article as:

Stapel, Elizabeth. "Partial-Fraction Decomposition: Examples." Purplemath. Available from
    http://www.purplemath.com/modules/partfrac3.htm. Accessed
 

 



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