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Find constants A and B such that $\frac{x - 7}{x^2 - x - 2} = \frac{A}{x - 2} + \frac{B}{x + 1}$ for all x such that  $x \neq -1$ and $x \neq 2$. Give your answer as the ordered pair (A,B).

 Apr 12, 2021
 #1
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Note that  x^2  - x  -2   can be factored  as   ( X -2) ( x + 1)....multiply through  by  this and we  get that

 

x - 7  =  A( x + 1)  + B( x - 2)

 

1x -  7  =  (A + B)x  + ( A - 2B)

 

Which  implies  that

 

A + B  = 1   ⇒    -A - B  =  -1    (1)

A - 2B  =  - 7           (2)

 

Add (1) and (2)   and we  get  that

 

-3B  =  -8

B  =  8/3

 

And  

A + 8/3  =  1

A  =  -5/3

 

(A , B)  =  (  -5/3, 8/3)

 

 

cool cool cool

 Apr 12, 2021

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