INTRODUCTORY EXERCISES

(2) Decompose each of the following fractions into partial fractions.

(a) x2 + 1
    -------
    x3 + x2

(b)      x2
    -----------
    x2 - 3x + 2

(a) x2 + 1      x2 + 1        A      B    C
    -------  =  ------  =  ------ + -- + --  , where A, B, C are constants.
    x3 + x2     x2(x+1)     (x+1)    x    x2

    Finding the common denominator on the left we get:

    x2 + 1      Ax2 + Bx(x+1) + C(x+1)
    ------  =  -----------------------
    x2(x+1)     x2(x+1)

    ==> x2 + 1  =  Ax2 + Bx2 + Bx + Cx + C

    Therefore we get:
           C = 1
       B + C = 0   ==>   B = -C  ==>  B = -1
       A + B = 1   ==>   A = 1 - B  ==>  A = 1 - (-1) = 2

    Hence we have:

    x2 + 1        2      -1    1
    -------  =  ------ + -- + --
    x3 + x2      (x+1)    x    x2



(b)      x2               1             A       B
    -----------  =  x2----------  =  x2--- + x2---
    x2 - 3x + 2       (x-2)(x-1)       x-2     x-1

    Finding a common denominator on the left we get:
        x2         x2(A(x-1) + B(x-2))
    ----------  =  ------------------
    (x-2)(x-1)         (x-2)(x-1)

    Therefore we get that:

    1 = Ax - A + Bx - 2B

    A + B = 0    ==> A = -B
    -A - 2B = 1  ==> -(-B) - 2B = 1  ==> -B = 1  ==>  B = -1

    Therefore A = -(-1) = 1

    Hence we see that:

         x2             1        -1         x2      x2
    -----------  =  x2----- + x2-----  =  ----- - -----
    x2 - 3x + 2       (x-2)     (x-1)     (x-2)   (x-1)