+145 There are numbers \(A\) and \(B\) for which \(\frac A{x-1}+\frac B{x+1}=\frac{x+2}{x^2-1}\) for every number \(x\not=\pm1\). Find \(A-B\)
+128475 We can use partial fractions, here
Note that x^2 -1 can be factored as ( x -1) ( x + 2)
So we have
( x + 2) A + B
___________ = ___ ___
(x -1) ( x + 1) x -1 x + 1
Multiply through by (x -1) ( x + 1)
x + 2 = A(x + 1) + B(x -1) simplify
x+2 = Ax + A + Bx - B
1x + 2 = (A + B)x + (A - B ) equate coefficients to form this system
A + B =1
A - B = 2
So A - B = 2
