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Enter (A,B,C) in order below if A, B, and C are the coefficients of the partial fractions expansion of (2x^2 + 3x - 5)/(x(x^2 - 1)) = A/x + B/(x - 1) + C/(x + 1).

 Feb 16, 2022
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Rewrite   A/x  +  B/(x-1)  +  C(x + 1)  with the common denominator  x(x - 1)(x + 1):

 

   A / x · (x + 1)(x - 1) / [(x + 1)(x - 1)]  =  A(x + 1)(x - 1) / [x(x + 1)(x - 1)]

   +     B / (x - 1) · x(x + 1) / [x(x + 1)]  =  B(x)(x + 1) / [x(x + 1)(x - 1)]

   +     C / (x + 1) · x(x - 1) / [x(x - 1)]  =  C(x)(x - 1) / [x(x + 1)(x - 1)] 

 

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

 

=  [ Ax2 - A + Bx2 + Bx + Cx2 - Cx ] / [x(x + 1)(x - 1)] 

 

=  [ (A + C)x2 + (B - C)x - A ] / [x(x + 1)(x - 1)] 

 

Comparing the original numerator to this numerator:

 

2x2 + 3x - 5  =  (A + C)x2 + (B - C)x - A

 

So:  A + C  =  2

        B - C  =  3

             -A  =  -5

 

Solve these equations for A, B, and C to get the answer.

 Feb 17, 2022

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