Why Use Ax + B in Partial Fraction Decomposition?

[shamieh]

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

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