SOLUTION: Write the partial decomposition of the rational expression: (-x^2+2x+4)/(x(x+2)^2) = A/x + B/(x+2) + C/(x+2)^2 Thank you!

Algebra ->  Rational-functions -> SOLUTION: Write the partial decomposition of the rational expression: (-x^2+2x+4)/(x(x+2)^2) = A/x + B/(x+2) + C/(x+2)^2 Thank you!      Log On


   

Question 1031246: Write the partial decomposition of the rational expression:
(-x^2+2x+4)/(x(x+2)^2) = A/x + B/(x+2) + C/(x+2)^2
Thank you!
Found 2 solutions by stanbon, Edwin McCravy:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
Write the partial decomposition of the rational expression:
(-x^2+2x+4)/(x(x+2)^2) = A/x + B/(x+2) + C/(x+2)^2
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Equate the numerators since the denominators are the same.
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-x^2 + 2x + 4 = A(x+2)^2 + Bx(x+2) + Cx
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-x^2 + 2x + 4 = Ax^2 + 4Ax + 4A + Bx^2 + 2Bx + Cx
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-x^2 + 2x + 4 = (A+B)x^2 + (4A + 2B + C)x + 4A
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Equate corresponding coefficients::
A+B = -1
4A + 2B + C = 2
4A = 4
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A = 1
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So, B = -2
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And C = 2-4+4 = 2
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Ans: 1/x - 2/(x+2) + 2/(x+2)^2
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Cheers,
Stan H.

Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!


Multiply both sides by the LCD  x(x+2)²

-x%5E2%2B2x%2B4+=+A%28x%2B2%29%5E2+%2B+Bx%28x%2B2%29+%2B+Cx

Substitute x=0

-0%5E2%2B2%280%29%2B4+=+A%280%2B2%29%5E2+%2B+B%280%29%280%2B2%29+%2B+C%280%29

4+=+A%282%5E2%29

4+=+4A

1=A

Substitute x=-2

-%28-2%29%5E2%2B2%28-2%29%2B4+=+A%28-2%2B2%29%5E2+%2B+Bx%28-2%2B2%29+%2B+C%28-2%29

-4+=+-2C

2=C

Substitute x=1, A=1, C=2

-1%5E2%2B2%281%29%2B4+=+1%281%2B2%29%5E2+%2B+B%281%29%281%2B2%29+%2B+2%281%29

5+=+9+%2B+B%283%29+%2B+2

5=11%2B3B%7B%7B%7B%0D%0A%0D%0A%7B%7B%7B-6=3B

-2=B 

So:



becomes:



Edwin