Partial Fraction Decomposition for Inverse Laplace Transform

[tandoorichicken]

The whole problem reads:
Find the inverse Laplace transform of
$$\frac{s-2}{s^3+2s^2+2s}$$

I think once I get the expression simplified I can do the rest by myself. I started to separate this expression out by partial fractions and got as far as this:
$$\frac{s-2}{s^3+2s^2+2s}=\frac{s-2}{(s^2+2s+2)s}=\frac{As+B}{s^2+2s+2} +\frac{C}{s}$$

$$(As+B)s+C(s^2+2s+2)=s-2$$
From this expression I got C=-1, which I checked was correct by using my calculator, but I still don't know how to find A, or B. Can anyone help with this please?

Any ideas or hints on how to do the inverse would be also be appreciated.

 

[Corneo]

Note that

$$s^2+2s+2 = (s+1)^2+1$$

Opps major brain laspe. That should be fixed

 

[whozum]

Note that

$$s^2+2s+2 = (s+1)^2$$

Nope, it equals [itex](s+1)^2 + 1.[/tex]<br /> <br /> edit: tex[/itex]

 

[Corneo]

Or perhaps that wasn't too helpful. Here is an trick I learned when doing these partial fractions problems.

Consider what you have
$$(As+B)s+C(s^2+2s+2)=s-2$$

Multiply out and gather the terms.
$$(A+C)s^2 + (2C+B)s + 2C = s-2$$

Is there a $s^2$ term on the right hand side of the equation? What does this tell you about $A+C$? What about $2C$, what should that be equal to? And $2C+B$?

 

[tandoorichicken]

Thanks, Corneo.