How Do You Factor Transfer Functions in S-Plane Analysis?

[FrankJ777]

Hi all. I'm trying to relearn S-plane analysis and filter transfer functions. What I'm having problems with is simple algebra i think, because it's been a while. Right now I've been reading http://www.ee.up.ac.za/main/_media/en/undergrad/subjects/eli220/polezero.pdf. What I'm confused about, among other things, is factoring the transfer function H(s).
In their example:

H(s) = [itex]\frac{2s + 1}{(s^2 + 5S + 6)}[/itex]

factored

H(s) = [itex]\frac{1}{2}[/itex] [itex]\frac{s+1/2}{(s+3)(s+2)}[/itex]

I don't understand how they get that. The way I see it, because:
2s+1 = 2(s+1/2)
and
[itex]s^{2}[/itex]+5s+6 = (s+3)(s+2)
so

H(s) = [itex]\frac{2s + 1}{(s^2 + 5S + 6)}[/itex] = [itex]\frac{2}{1}[/itex] [itex]\frac{s+1/2}{(s+3)(s+2)}[/itex]

Also It seems that:

[itex]\frac{2s + 1}{(s^2 + 5S + 6)}[/itex] ≠ [itex]\frac{1}{2}[/itex] [itex]\frac{s+1/2}{(s+3)(s+2)}[/itex]

So I guess what I don't understand is why it seems that instead of factoring out 2 from the numerator, we are dividing the numerator by 2 and multiplying the denominator by 2.

Can someone please set me straight. I know I did this a few years ago and it seemed to make more sense then, than it does now.

Thanks a lot.

 

[NascentOxygen]

Hi FrankJ777. The explanation is quite simple: you are right and they are wrong.

 

[FrankJ777]

Really? I hope that's the case. Too bad I wasted several study hours trying to get it.

Thanks