One of the methods of decomposing a fraction and finding the constants, $A, B, C,$ etc., is to "pick values of $x$". For example, to find $A$ and $B$ of $${{3x}\over(x-1)(x+2)} = {A\over(x-1)} + {B\over(x+2)}$$
$${{3x}} = {A(x+2)} + {B(x-1)} $$
one is supposed to pick values of $x$. To find $A$, one would pick $x=1$, and then $A=1$. To find $B$, one would pick $x=-2$, and then $B=2$.
I am having a hard time understanding the rationale behind this method. It looks like we are trying to pick values of $x$ that will cancel out the other constant. Are these the only values that could be picked, or could any values theoretically be picked, but they fail to cancel out the other constant?
For example, once I find out that $A=1$ by picking $x=1$, could I then say ${{3x}} - {(x+2)}= {B(x-1)}$ and then solve for $B$ by picking out any value of $x$? Or am I only allowed to pick $x=-2$? If so, why?
You can pick any value of $x$ you like, but if you don't pick those special values then you will end up with one equation in 2 variables $A$ and $B$. So then you'll have to pick two such values of $x$ and then solve a simultaneous system of two equations in two variables, as opposed to two separate equations in one variable. You would still get the same answer, because the partial fraction decomposition is unique. But it is more work.