SOLUTION: Find the partial fraction decomposition for each rational expression. (11-2x)/(x^2-8x+16)

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Question 611509: Find the partial fraction decomposition for each rational expression.
(11-2x)/(x^2-8x+16)
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Finding partial fractions essentially the process of "un-adding" fractions.

As you know, when fractions are added the denominators must be the same. And the matching denominator was the least common multiple of the denominators of the fractions to be added. So the denominator we have, x%5E-8x%2B16 represents the least common multiple of the fractions that we added to get %2811-2x%29%2F%28x%5E2-8x%2B16%29.

So we interested in finding what denominators could there have been to have x%5E2-8x%2B16 as the least common multiple. For this we need to see the factors of x%5E2-8x%2B16:
x%5E2-8x%2B16+=+%28x-4%29%28x-4%29

With these factors the fractions we could have that would have x%5E2-8x%2B16 as the least common multiple would be:
A%2F%28x-4%29+%2B+B%2F%28%28x-4%29%28x-4%29%29

Now we just have to figure out what "A" and what "B" would make these two fractions add up to %2811-2x%29%2F%28x%5E2-8x%2B16%29. To do this we first add them! Of course we need a common denominator:
%28A%2F%28x-4%29%29%28%28x-4%29%2F%28x-4%29%29+%2B+B%2F%28%28x-4%29%28x-4%29%29
%28Ax-4A%29%2F%28%28x-4%29%28x-4%29%29+%2B+B%2F%28%28x-4%29%28x-4%29%29
Now we add:
%28Ax-4A%2BB%29%2F%28%28x-4%29%28x-4%29%29
This fraction is supposed to be equal to %2811-2x%29%2F%28x%5E2-8x%2B16%29. The denominators are the same so the numerators must be the same. So:
Ax - 4A + B = 11 - 2x

From this we can, believe it or not, figure out the "A" and the "B". The x term on the left is Ax and on the right it is -2x. They have to match so A = -2.

The constant term on the left is -4A + B and on the right it is 11. These have to match also. So -4A + B = 11. Using the found value for A we get:
-4(-2) + B = 11
Solving for B...
8 + B = 11
B = 3

So
%2811-2x%29%2F%28x%5E2-8x%2B16%29+=+%28-2%29%2F%28x-4%29+%2B+3%2F%28%28x-4%29%28x-4%29%29