repeated-linear fractions

Partial Fractions

The Strange exceptions of partial fraction

(i.e. repeated-linear fractions is illustrated here. 

For a repeated linear fraction, i.e. (1 + x)2

the format will look something like:

              A              B               C

fraction = ------------ + ---------   +   ---------

            expression    (1 + x)         (1 + x)²

Example: 

Express as a partial fraction.

           x

    ---------------

    (1-x)(x + 1)²

      x             A         B         C

--------------- = ------- + ------- + --------

(1-x)(x + 1)²     (1 - x)   (x + 1)   (x + 1)²

multiply by the denominator.

x = A(x + 1)² + B(1 - x)(x + 1) + C(1 - x)

multiply out (in this case its only the 'A' term).

x = A(x +1)(x +1) + B(1 - x)(x + 1) +   C(1 - x)

x = Ax2 + 2Ax + A + Bx – Bx2 + B –Bx + C – Cx

solve for the coefficient:

A = 1/4,  B = 1/4, C = -1/2 

   x                1           1          1

--------------- = -------- - --------- - ---------

(1 - x)(x + 1)²   4(1 - x)    4(x + 1)   2(x + 1)²

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