ADVANCED EXERCISES

Decompose each fraction into partial fractions:

(4)  4x2 + 5x - 9
     ------------
      x3 -12x + 9

Let 4x2 + 5x - 9      A         Bx + C
    ------------  =  ---- + ------------
    x3 - 12x + 9      x-3    x2 + 3x - 3

We then get that:
    4x2 + 5x - 9 = A(x2 + 3x - 3) + (Bx + C)(x-3)
                            = Ax2 + 3Ax - 3A + Bx2 - 3Bx + Cx - 3C
                            = (A + B)x2 + (3A - 3B + C)x + (-3A - 3C)

This implies :

          A + B = 4   ==>  A = 4 - B
    3A - 3B + C = 5   ==>  3(4-B) - 3B + C = 5  ==>  C = 6B - 7
       -3A - 3C = -9  ==>  -3(4-B) - 3(6B-7) = -9  ==>  -12 + 3B - 18B + 21 = -9
						   ==>  -15B = -18
						   ==>  B = 6/5

	  A = 4 - 6/5 = 14/5
          C = 6(6/5) - 7 = 1/5

The answer is:

    4x2 + 5x - 9     14/5    6x/5 + 1/5
    ------------  =  ---- + ------------
     x3 - 12x + 9     x-3    x2 + 3x - 3


		      14         6x + 1
                  = ------ + ---------------
                    5(x-3)   5(x2 + 3x - 3)