Complex Fractions: Techinicalities
Purplemath
In the lesson, I'd been given this exercise:
- Simplify the following expression:
![[ 1/(x - 1) + x + 3 ] / [ x - 3 + 1/(x + 4) ]](rational/compfr03.gif)
...and, after simplification, I'd ended up with this:
How did I come up with the restrictions on the domain of this expression? How did I find the restrictions on x?
The first two domain restrictions were easy. The sub-fraction in the complex fraction's numerator will be undefined if x − 1 = 0, and the sub-fraction in the denominator will be undefined if x + 4 = 0.
The other restriction is harder. I had to consider the denominator of the simplified fractional expression. This would be zero when the numerator (of the denominator) was zero. Remember that the denominator in the simplified form factored as (x − 1)(x2 + x − 11).
I'd already handled the (x − 1) factor, so now I need to find the zeroes of the remaining quadratic, (x2 + x − 11). Applying the Quadratic Formula, I get:
![x = [-1 ± 3sqrt(5)] / 2](rational/compfr22.gif)
This gives me the remaining restrictions.
By the way, if I'd simplified by converting the addition in the numerator of the original fraction to a common denominator, and converted similarly for the denominator, my work would have looked like this:
![[(x^2 + 2x - 2)/(x - 1)] / [(x^2 + x -11)/(x + 4)]](rational/compfr21.gif)
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