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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. 

 

Or if you prefer 
https://gyazo.com/cdbf05411074209828b51f33c00774d2

 Dec 24, 2018
 #1
avatar+98005 
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   x + 7                       A                B            

________   =        ______   +  ____

x^2 - x - 2               x - 2           x + 1

 

This is a partial fraction decomposition

 

The denominator on the left can be factored as  (x - 2) (x + 1)

Multiply through by this denominator and we get

 

1x + 7  =   A(x + 1)   +   B(x - 2)

1x + 7  =  (A + B) x  + A - 2B

 

Equate coefiicients and we get the system

 

A + B  =  1

A - 2B = 7        subtract the  second from the first

 

3B = - 6

B = - 2

 

So A =   A + -2 = 1   ⇒   A =  3

 

cool cool cool

 Dec 24, 2018

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