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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 $(A,B)$.

 Aug 18, 2022
 #1
avatar+124596 
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Note that  x^2 - x - 2 can be factored as ( x -2) (x + 1)

 

Multiply through by this factorization and we  have

 

x -7 =  A(x + 1) + B( x -2)       simplify

 

x - 7 =  (A + B)x  + ( A - 2B)     equating terms we  have this system

 

A + B  = 1

 

A -2B =  -7     

 

subtract the second equation  from the first and we  have

 

3B = 8

B  = 8/3

 

and

A + B = 1

A + 8/3  = 1

A = 1 - 8/3

A = - 5/3

 

 

cool cool cool

 Aug 18, 2022
 #2
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Thanks!

 Aug 19, 2022

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