Partial fraction decomposition

[Taylor_1989]

I have been having trouble of late with partial fraction decomposition. Not so much the maths, but the intuition behind it. What I mean by this, but a question in front of me, I now what procedure to follow to get the answer, but I don't get why you follow the said produced. I will give an example to show what parts I don't understand.

Partial fraction decomposition: 8x-42/x^2+3x-18

Factor the denominator: I understand it a quadratic: which gives me
8x-42/(x+6)(x-3)

Now this is the part I do not understand: Why do you use A and B, why do they appear? I really can't see the intuition behind this.
8x-42/x^2+3x-18 = A/x+6 + B/x-3 The rest from here I know how to do: find the LCM and the replace x with and number to = one equation to 0 to see find the value for the A or B.

I do hope I have put this in the right forum, as is my first time posting maths equation.

 

[Nessdude14]

Once you factor the denominator, you know that you can express the fraction as the sum of the two fractions:
$\Large \frac{(Some Number)}{x+6}+\frac{(Some Other Number)}{x-3}$
You don't know what the numerator on the fractions are, so you just give them temporary names until you can find the actual value that goes in place of them. "Some other number" doesn't look very nice in an equation, so instead they're typically called "A" and "B." You can give them any temporary name you want, but really it's just a placeholder for the real value.

 

[Taylor_1989]

I see now, thanks for the fast response, it has been bother me for sometime. Once again thanks big help.