factor $1/(x\pm y)$ as $f(x)g(y)$ Is it possible to rewrite
$$\frac{1}{x\pm y}$$
as multiplication of an expression containing only $x$ and another containing only $y$ ($x$ and $y$ are real independent variables), i.e.
$$\frac{1}{x\pm y}\stackrel{?}{=}f(x)g(y)$$
If it is possible, what are $f(x)$ and $g(y)$?

Sure it's possible. Take $A=\frac{a}{a+b}$ and take $B=0$. Then it should be trivial to show that $$\frac Aa + \frac Bb = \frac{\frac{a}{a+b}}{a} + \frac0b = \frac{1}{a+b}.$$
In fact, taking any value of $B$, you can set $A=a\cdot\left(\frac{1}{a+b} - \frac Bb\right)$ and get the equality
$$\frac{A}{a} + \frac Bb = \frac1{a+b}.$$

However, if you demand that $a,b, A, B$ are all integers, then the answer is no. A simple counterexample can be found by taking $a=b=1$. Then, $\frac{1}{a+b}=\frac12$, while $\frac Aa + \frac Bb = \frac A1 + \frac B1 = A + B$, and since $A$ and $B$ are integers, this means $\frac Aa + \frac Bb$ must also be an integer.
Since $\frac12$ is not an integer, it's clear that $\frac Aa + \frac Bb$ cannot be equal to $\frac1{a+b}$.
