Expert: Scott A Wilson - 10/18/2009

Question
Express the following equation is simplest form:

36-x^2/x^2+8x+12 / x^2-6x/x-2

Answer
Taking that as written, * and / are always done first.
This gives 36 - x²/x² + 8x  + 12/x² - 6x/x - 2
= 36 - 1 + 8x + 12/x² - 6 - 2 = 27 + 8x + 12/x².


However, there might be some parenthesis that are suppose to be there.
It almost looks like it should be (36-x²)/x² + (8x+12)/x² - 6x/(x-2).

The first two fractions need to be multiplied by (x-2)/(x-2) and the last by x/x.
This gives (36-x²)(x-2)/(x²(x-2)) + (8x+12)(x-2)/(x²(x-2)) - 6x•x/(x(x-2)).

They can now be combined, since the denominators are all the same.
That gives [(36-x²)(x-2) + (8x+12)(x-2) - 6x²]/(x(x-2)).

Multiplying out the numerator gives -x³+2x²+36x-72 + 8x²-4x-24 - 6x = -x³ + 10x² + 26x - 96.

Using numerical analysis, I found the roots were -3.751541598, 2.218867994, and 11.5326736,
so the top factors into (x-3.751541598)(x-2.218867994)(x-11.5326736),
so none of the facors are x or x-2.

Thus, my final answer is (-x³ + 10x² + 26x - 96)/[x(x-2)].