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Find constants A and B such that
{x + 7}/{x^2 - x - 2} = {A}/{x - 2} + {B}/{x + 1}
for all x such that x is not equal to -1 and x is not equal to 2. Give your answer as the ordered pair (A,B).

 Jun 26, 2018
 #1
avatar+98182 
+3

(x + 7 ) / ( x^2 - x - 2)   =   A  / (x - 2)   +  B / ( x + 1)

 

We can use partial fractions, here

 

We can write

 

( x + 7)  /[( ( x + 1)( x - 2)]  =  A / ( x - 2)  + B / ( x + 1)

 

 

Multiply through by  (x + 1) ( x -2)

 

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

 

x + 7   = Ax   +  A   +   Bx - 2B

 

x + 7  =  ( A + B) x  +  ( A - 2B)

 

Equating coefficients, we have

 

A + B   = 1

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

 

-3B =  6

B  = -2

 

And  we can find  A  as  A + B  = 1 ⇒    A - 2  = 1   ⇒  A  = 3

 

So  (A, B)   = ( 3, - 2)

 

 

cool cool cool

 Jun 26, 2018
edited by CPhill  Jun 26, 2018

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