+23246 To find which two rational expressions sum to (3x - 2) / (x2 - x - 12):
1) Factor the denominator: (3x - 2) / [ (x - 4)(x + 3) ]
2) This means that the two rational expressions are: A / (x - 4) and B / (x + 3)
3) Start with: A / (x - 4) + B / (x + 3)
write as one fraction: [ A(x + 3) + B(x - 4) ] / [ (x - 4)(x + 3) ]
4) Simplify this: = ( Ax + 3A + Bx - 4B ) / [ (x - 4)(x + 3) ]
5) When you compare this result with (3x - 2) / [ (x - 4)(x + 3) ]
notice that the numerators must be equal: ( Ax + 3A + Bx - 4B ) = (3x - 2)
6) Rearranging the numerator: [ Ax + Bx + 3A - 4B ] = (3x - 2)
7) This means that Ax + Bx = 3x and 3A - 4B = -2
---> Ax + Bx = 3x ---> A + B = 3
---> 3A - 4B = -2
8) Solving for A and B: A = 10/7 and B = 11/7
9) Which means that the two rational expressions are: ( 10/7 ) / (x - 4) and ( 11/7 ) / (x + 3)
