Partial Fraction Decomposition - Multiple Answers-Question

Question

Now I do understand how partial fraction decomposition works and why you can do it, but there is one case that I don´t understand. And that is, the following: $$\frac{A_{1}}{(x-x_{1})} + \frac{A_{2}}{(x-x_{1})^2}$$ up until $$\frac{A_{n}}{(x-x_{1})^n}$$ I know we use this when rewriting the denominator in linear and irreducible quadratic terms, and I accept that it works (which can of course be seen in examples) but could you give me a particular reason or deeper understanding of this (no need to explain the decomposition)? Also when having this: $$\frac{Ax + B}{(ax^2 + bx + c)}$$ I understand that the nominator must be smaller, as the term must be smaller than one. But why does it have to to be linear when the denom. is quad. and quad. when the denom. is cubed and on and on. Yes, that works, but the particular reason is not that obvious to me, especially with the first question. Thank you!.

Answer 1

The number of unknowns must be equal to the degree of the denominator-polynomial.

The reason is that the numerator-polynomial can have degree $n-1$, if the denominator-polynomial has degree $n$, so $n$ terms.

Since the representation uniquely determines the numerator, we need $n$ conditions and therefore $n$ unknowns to ensure a unique solution.