SOLUTION: Decomposing fractions into partial fraction x-2 /(x+2)^2(x-4)

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Decomposing fractions into partial fraction x-2 /(x+2)^2(x-4)      Log On


   

Question 1180207: Decomposing fractions into partial fraction
x-2 /(x+2)^2(x-4)
Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


By basic rules for decomposing fractions, the decomposition is of the form



Multiply by the least common denominator:

x-2=A%28x%2B2%29%28x-4%29%2BB%28x-4%29%2BC%28x%2B2%29%5E2

x-2=A%28x%5E2-2x-8%29%2BB%28x-4%29%2BC%28x%5E2%2B4x%2B4%29

x-2=%28A%2BC%29x%5E2%2B%28-2A%2BB%2B4C%29x%2B%28-8A-4B%2B4C%29

Equate the coefficients on the two sides of the equation:

(1) A%2BC=0
(2) -2A%2BB%2B4C=1
(3) -8A-4B%2B4C=-2

Solve (1) to get C=-A and substitute in (2) and (3).

(4) -6A%2BB=1
(5) -12A-4B=-2

Eliminate A...

12A-2B=-2
-12A-4B=-2
-6B=-4
B=2%2F3

-6A%2B2%2F3=1
-6A=1%2F3
A=-1%2F18

-1%2F18%2BC=0
C=1%2F18

ANSWER:


or



-------------------------------------------------------------------------

If you have more problems like this to do, you can check your answers on wolframalpha.com. Just go to the website and enter e.g. "decompose (x-2)/((x+2)^2(x-4))".