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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 13, 2023
 #1
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Note that   x^2 - x - 2   can  be factored as  (x - 2) (x +1)

 

Multiply throuh by this  factored form  and we  get

 

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

 

x - 7 = Ax + A  + Bx - 2B           

 

1x - 7= (A + B)x + (A - 2B)      equate coefficients and we  get this  system

 

1 =  A + B

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

 

8 = 3B

B = 3/8

 

And

1 =  A + 3/8

A = 5/8

 

(A, B)  = ( 5/8, 3/8 )

 

cool cool cool

 Aug 13, 2023

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