SOLUTION: add or subtract as indicated. x/x+4-36/x^2-x-20

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Question 86680: add or subtract as indicated. x/x+4-36/x^2-x-20
Answer by bucky(2100) About Me  (Show Source):
You can put this solution on YOUR website!
Given:
.
%28x%2F%28x%2B4%29%29-%2836%2F%28x%5E2-x-20%29%29
.
Subtract
.
Notice first that x%5E2-x-20 factors into %28x+%2B4%29%2A%28x+-5%29. Substitute the factored
form into the problem, and the problem becomes:
.
%28x%2F%28x%2B4%29%29-%2836%2F%28%28x%2B4%29%2A%28x-5%29%29%29%29
.
Next, create a common denominator in the two terms by multiplying the first term by
%28x-5%29%2F%28x-5%29 which is equivalent to multiplying by 1 because the numerator of this
multiplier is the same as the denominator. This multiplication makes the problem
become:
.
%28x%2F%28x%2B4%29%29%2A%28%28x-5%29%2F%28x-5%29%29-%2836%2F%28%28x%2B4%29%2A%28x-5%29%29%29%29
.
and this simplifies to:
.
%28%28x%2A%28x-5%29%29%2F%28%28x%2B4%29%2A%28x-5%29%29%29-%2836%2F%28%28x%2B4%29%2A%28x-5%29%29%29%29
.
The numerator of the first term multiplies out to x%5E2-5x. Substituting this into
the polynomial makes it become:
.
%28%28x%5E2+-+5x%29%2F%28%28x%2B4%29%2A%28x-5%29%29%29-%2836%2F%28%28x%2B4%29%2A%28x-5%29%29%29%29
.
Now since the two terms have the same denominator, they can be combined by just subtracting
the two numerators and putting that result over the common denominator. This results
in:
.
%28%28x%5E2+-+5x+-36%29%2F%28%28x%2B4%29%2A%28x-5%29%29%29%29
.
Now notice that the numerator x%5E2+-+5x+-+36 factors into %28x+-+9%29%2A%28x+%2B+4%29.
Substitute this for the numerator and the result is:
.
%28%28%28x-9%29%2A%28x%2B4%29%29%29%2F%28%28x%2B4%29%2A%28x-5%29%29%29%29
.
Cancel the like terms in the numerator and denominator:
.
%28%28%28x-9%29%2A%28cross%28x%2B4%29%29%29%29%2F%28%28cross%28x%2B4%29%29%2A%28x-5%29%29%29%29
.
and this reduces to the form:
.
%28x-9%29%2F%28x-5%29
.
This is the answer you are looking for. Hope this helps you to understand the problem
and how to work it through to an answer.
.