# Simple algebra question - separating fractions

How does the following come about? I'm completely lost. Can anyone help me fill in the steps in between?

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

I figured that $$\frac{s+2}{s(s+1)} = \frac{s}{s(s+1)} + \frac{2}{s(s+1)} = \frac{1}{s+1} + (\text{not sure})$$

• Thanks @lab bhattacharjee ! I should've known to to that since this is one of the last steps of a differential equations problem. Thanks again! – user12279 Nov 19 '13 at 4:46

$$\frac{s+2}{s(s+1)}=\frac As+\frac B{s+1}$$
$$\implies s+2=A(s+1)+Bs$$ $$\implies s+2=s(A+B)+A$$
$$\text{Now, Compare the constants & the coefficients of s}$$
Well, you can note that $$s+2=(s+1)+1=(s+1)+(s+1-s)=2(s+1)-s,$$ so that $$\frac{s+2}{s(s+1)}=\frac{2(s+1)-s}{s(s+1)}=\frac{2(s+1)}{s(s+1)}-\frac s{s(s+1)}=\frac2s-\frac1{s+1}.$$