The existence of partial fraction decompositions I'm sure you are all familiar with partial fraction decomposition, but I seem to be having trouble understanding the way it works. If we have a fraction f(x)/[g(x)h(x)], it seems only logical that it can be "split up" into A/g(x) + B/h(x) for some A and B, because there must be some A and B that make this work, right? But I don't see why this wouldn't work when we have f(x)/[g(x)]^2. Couldn't we just split it like A/g(x) + B/g(x) for some A and B? I am missing something, but I don't know what. Could someone please enlighten me as to why my reasoning is incorrect? Thank you.

HINT  $\rm\displaystyle\qquad \frac{1}{g\:h}\ =\ \frac{a}g\ +\ \frac{b}h\ \iff\ 1\ =\ a\ h\: +\: b\ g\ \iff\ 1\: =\: \gcd(g,h)$ 
