decompose 5x+1/x^2-x-12 into partial functions?
I think you mean partial fractions. That is what this method was called when I went to school. First factor the denominator.
(5x+1)/(x^2-x-12) = (5x+1)/[(x-4)(x+3)]
This can be written as
A/(x-4) + B/(x+3) where A and B are certain constants. Here is how you can solve for A and B.
Require that
(5x+1)/[(x-4)(x+3)] = A/(x-4) + B/(x+3)
= [A(x+3) +B(x-4)]/[(x-4)(x+3)]
= (A+B)x +3A-4B
This will be true if A+B = 5 and 3A-4B = 1
4A+4B = 20
Solve for A and B
3A+3B = 15
3A-4B = 1
7B = 14
B = 2 A = 3
So the answer is
3/(x-4) + 2/(x+3)