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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 \(x\neq\pm1\). Find \(A-B\).

 Apr 12, 2020
 #1
avatar+21004 
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The common denominator is  (x + 1)(x - 1)

 

Multiply each term by this and the equation becomes:   A(x + 1) + B(x - 1)  =   x + 2

Multiplying out:                                                                    Ax + A + Bx - B  =  x + 2

Factoring:                                                                         (A + B)x + (A - B)  =  x + 2

 

Setting the X-terms to each other:        (A + B)x  =  x   --->   A + B  =  1

Setting the constants to each other:                                       A - B  =  2

 

Solving these two equations:  A  =  1.5   and   B  = - 0.5  

 Apr 12, 2020

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