+145 There are numbers A and B for which Ax−1+Bx+1=x+2x2−1 for every number x≠±1. Find A−B
+130475 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
