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Find constants a and b such that (2x + 17)/(x^2 - x - 2) = a/(x - 2) + b/(x + 1) for all x such that x≠-1 and x≠2.

 Jun 28, 2021
 #1
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X^2   - x  - 2   = (x -2)  ( x + 1)

 

Multiply through by   ( x -2) ( x + 1)    and we have  that

 

2x + 17 =   a( x + 1)  +  b ( x + 2)

 

2x + 17  =  (a + b)x  + (a   + 2b)         which implies that

 

a +  b = 2       ⇒    - a   - b   =  -2        (1)

a + 2b = 17          (2)

 

Add (1)  and (2)

 

b  =  15

a  + b = 2   ⇒    a + 15  = 2  ⇒   a  =   2 -15   ⇒  a  =   -13

 

 

cool cool cool

 Jun 28, 2021

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