Writing $\frac{1}{x+5}$ as $\frac{A}{x}+\frac{B}{5}$ Is this possible? (in this case specifically, but also in general for any fraction with addition/subtraction in the denominator).
I thought this would be easy, but upon trying it and thinking some more


*

*theres 2 unknowns and one equation, so we probably can't identify a unique $A$ and $B$?


For example, I feel like the simplest manipulation is to multiple both sides by $5$ and move the $A$ term over, which gives
$$
B=\frac{5}{x+5} - \frac{5A}{x} =_{A=1} \frac{-25}{x(x+5)}
$$
which is not the same as if we assumed $A=2$

Note that 
$\frac{A}{x}+\frac{B}{5}=\frac{5A+Bx}{5x}=\frac{1}{x+5}\implies 5Ax+25A+Bx^2+5Bx=5x$$
then for $B=0$
$$5Ax+25A=5x \implies A=\frac{x}{x+5}$$
and for $A=0$
$$Bx^2+5Bx=5x\implies B=\frac{5}{x+5}$$
which are trivial solutions and then it is not possible obtain the equality for $A,B$ constants.
