# Separating a fraction

Discussion in 'Math' started by boks, Dec 1, 2008.

1. ### boks Thread Starter Active Member

Oct 10, 2008
218
0
I don't remember what this method is named in English, but I want to write the fraction

$\frac{1}{(s^2 + 1)(s-3)(s+2)}$

in the form

$\frac{A}{s^2 + 1} + \frac{B}{s-3} + \frac{C}{s+2}$

I multiply A with (s-3)(s+2), B with (s^2 + 1)(s+2) and C with (s^2 + 1)(s-3), to get

$As^2 - As - 6A + Bs^3 + 2Bs^2 + Bs + 2B + Cs^3 - 3Cs^2 + Cs - 3C = 1$

and the equations

(1) B + C = 0
(2) A + 2B - 3C = 0
(3) -A + B + C = 0
(4) -6A + 2B - 3C = 1

(1) and (3) gives A = 0
(2) then gives B = C = 0

This is obviously wrong. The correct answer is

A = 2s - 4
B = 2
C = -4

according to my book. There must be something I don't understand here...

2. ### mik3 Senior Member

Feb 4, 2008
4,846
63
You have to put As+D over (s^2+1) and not just A.

3. ### boks Thread Starter Active Member

Oct 10, 2008
218
0
OK, thanks. What is the method called? I'm trying to find something on the internet.

4. ### mik3 Senior Member

Feb 4, 2008
4,846
63
The general is partial fractions, I think this is called partial fractions with repeated and quadratic roots.

5. ### blazedaces Active Member

Jul 24, 2008
130
0
It's also called partial fraction expansion.
Good luck,
-blazed