It's because when you changed it, you effectively isolated the numerator, which does not have any restrictions.
What I mean is, take this as an example (instead of your complicated one lol):
1/(x-1)(x-2)=A/(x-1)+B/(x-2)=[A(x-2)+B(x-1)]/[(x-1)(x-2)]
Now, eliminate the denominator (i.e...