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Find the product CD of the integers C and D for which (C)/(x-3) + (D)/(x+8) = (4x-23)/(x^2 + 5x - 24) for all real values of x except -8 and 3. 

Guest Feb 27, 2018
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Common denominator for the first two terms is

  (x-3)(x+8) = x^2+5x-24

 

so (cross multiplying the first two terms) results in

{c(x+8)  + d(x-3) }/(x^2+5x-24)  =  (4x-23) / (x^2+5x-24)    

   Both left and right sides have the same denominator ....so the numerator is equal

c(x+8) + d(x-3) = 4x- 23

cx + 8c + dx - d3  = 4x- 23

 

so cx + dx = 4x   (1)  and  8c-d3 = -23   (2)

    c+d = 4

         d = 4-c     substitute this into (2)

8c -3(4-c) = -23

8c-12+3c = -23

11c= -11

c = -1               d=4-c = 4-(-1) = 5       The product of  c x d  = (-1) x 5 = -5        

ElectricPavlov  Feb 27, 2018

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