# Math Help - Fun with Partial Fractions.. Kind of

1. ## Fun with Partial Fractions.. Kind of

Question on Textbook.

My teacher gave us some PDF's of a text... i'm wondering why this situation is the case...

$\frac{s^2+2s+3}{(s+1)(s^2+2s+2)} = A\frac{1}{s+1} + B\frac{s+1}{(s+1)+1} + C\frac{1}{(s+1)+1}$

it then states "where (s+1)(s^2+2s+2) = (s+1)((s+1)^2+1)"

I understand how the denominator can break down.. but why is the denominator of the B and C fractions NOT squared???? This is confusing me a lot... They then go on to multipy the fractions out very oddly...

$
\frac{[(s+1)((s+1)^2+1)](s^2+2s+3)}{(s+1)(s^2+2s+2)} =
$

$
A\frac{(s+1)((s+1)^2+1)}{s+1} + B\frac{(s+1)(s+1)((s+1)^2+1)}{(s+1)+1} + C\frac{(s+1)((s+1)^2+1)}{(s+1)+1}
$

Can someone clarify what is going on here by chance? I would think the denominators for B and C should be (s+1)^2+1 and that the numerators would only get multiplied so that the demoniators equaled that of the equation on the left....

Any help is much appreciated!

EDIT: I was so stumped at that part, i never looked further in the example... looking further in the problem, i think it was simply a typo. However, i still would like to ask why the coefficient for B is (s+1) instead of simply s (since only the s is squared before you break up the poly)..?

2. I'm having trouble with these last two terms.

$\frac{s^2+2s+3}{(s+1)(s^2+2s+2)} = A\frac{1}{s+1} + B\frac{s+1}{(s+1)+1} + C\frac{1}{(s+1)+1}$

$s^2+2s+2$ does NOT factor into $((s+1)+1)^2= (s+2)^2$

Is the orignal problem.... $\frac{s^2+2s+3}{(s+1)(s^2+4s+4)}$ ?