(4) Decompose the following fraction into partial fractions.
3x - 13
------------
6x2 - x - 12
SOLUTION:
3x - 13 3x - 13 A B
------------ = ------------ = ---- + ----
6x2 - x - 12 (2x-3)(3x+4) 2x-3 3x+4
Find the common denominator to get:
3x - 13 A(3x+4) + B(2x-3)
------------ = -----------------
6x2 - x - 12 (2x-3)(3x+4)
Therefore,
3x - 13 = 3Ax + 4A + 2Bx -3B
= (3A + 2B)x + (4A - 3B)
3A + 2B = 3 ==> A = 1 - 2B/3
4A - 3B = -13 ==> 4(1 - 2B/3) - 3B = -13 ==> 4 - 8B/3 - 3B = -13
==> -17B/3 = -17
==> B = 3
A = 1 - 2(3)/3 = -1
The answer is:
3x - 13 -1 3
------------ = ---- + ----
6x2 - x - 12 2x-3 3x+4