Binomial theorem for negative/rational exponents

We have the binomial theorem:

which makes perfect sense to me for nonnegative integers.

Firstly, I don't understand how to extend this to the case of negative integer exponents, since I thought it would simply be:

This is fine, since the denominator still makes sense using , and we just have 1 over a polynomial, which also makes sense except for when p(x)=0.

However, what I've read talks about an infinite series:

But I don't understand why this is needed if we can just have 1 over a polynomial.

I aso don't get the case for rational exponents, but want to understand this first.