Post #4010174

@bemmesr@mathstodon.xyz so what I would say if we did that is that first of all there is no unique solution for D(x) and E(x) - they could conceivably be all sorts of expressions. There will in general be solutions that aren't even rational functions. As with the example of the 3x + 5 situation above, a complicated expression in D can be cancelled out by a different complicated expression in E. So this way of looking at the problem is not *that* useful. But we can *work out* that one particularly nice solution among the many has the shape "E(x) is just a constant, and D(x) is an order 1 polynomial in x". So the partial fraction step of just writing in the constants of the different polynomial numerators can be seen as doing this all in one go: we know that there is a solution where the numerators *are* be written in this way, so we're going to do that. You might be wondering *how* we can work out that this nice shape of solution works - well, the answer to that is simply that, once you're done you can put your solution over a common denominator and see that it comes out to the same thing you started with, so that proves it! I think you still had another question, but let me pause here in case this isn't clear.