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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 \(A-B\)

DanielCai  Jun 28, 2018
 #1
avatar+87309 
+2

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

 

 

cool cool cool

CPhill  Jun 28, 2018

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