ENGR 2422 Engineering Mathematics 2
Three Examples of
Partial Fractions

        Express   f (x)   in partial fractions:

  1. 4 / (x^2 – 4)
    a / (x+2) + b / (x – 2)]
    Both denominators are linear non-repeated factors.   The cover-up rule may be used:
    [evaluation of a and b]
    Therefore

    1 / (x – 2) – 1 / (x+2)

  2. (3 x^2 – 1) / (x^3 – x)
    a / x + b / (x+1) + c / (x–1)
    All three denominators are linear non-repeated factors.   The cover-up rule may be used:
    [evaluation of a, b and c]
    Therefore

    1 / x + 1 / (x+1) + 1 / (x–1)

  3. 1 / (x^3 + x)
    a / x + (bx + c) / (x^2 + 1)
    Note that the polynomial in the numerator of a partial fraction must be of order one less than that of the denominator of that partial fraction.   Linear denominators need constant numerators, while quadratic denominators require linear numerators.
    Only the first denominator is a linear non-repeated factor.   The cover-up rule may be used to find a:
    [evaluation of a]
    One of the standard methods must be used to find b and c.
    Clearing the denominators:
    1 = a(x^2 + 1) + (bx + c)x   for all x
    Matching coefficients of   x2:     0   =   1 + b     implies b = – 1
    Matching coefficients of   x1:     0   =   c     implies c = 0
    [The coefficients of   x0   already match, because we found a = 1 by the cover-up rule.]
    Therefore

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

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    Created 2006 02 15 and most recently modified 2007 02 13 by Dr. G.H. George