Why Use Ax + B in Partial Fraction Decomposition?

[shamieh]

When I'm evaluating a problem like
\(\displaystyle
\int \frac{2x^2 + 8x + 9}{(x^2 + 2x + 5)(x + 2)} = \frac{Ax + B}{x^2 + 2x + 5} + \frac{C}{x+2}\)

I understand how to get the C part, that's simple. But what is a Good trick to know that I need to have \(\displaystyle Ax + B\) over the \(\displaystyle x^2 + 2x + 5\) denominator? Is there a way I can remember it easier? Because sometimes I will mistakenly put \(\displaystyle Ax + Bx\) or \(\displaystyle (A + B)\)/denominator + \(\displaystyle C\)/denominator

Thanks for your time

 

[MarkFL]

If you have an irreducible factor in the denominator, then in the partial fraction decomposition, you want to put a polynomial one degree less than this factor in the numerator.

So, if for example, you have an irreducible quadratic factor in the denominator, you want to put a general linear polynomial over this factor in the decomposition. The general linear polynomial is of the form $Ax+B$.

 

[shamieh]

Awesome, thanks Mark.