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Complex Fractions: Technicalities

I started with:

• Simplify the following expression:

...and ended up with:

To get this "x not equal to zero" restriction, I had to consider all of the denominators, both of the entire fraction and of the sub-fractions. The two sub-fractions were the 1/x and the 2/x2 in the numerator and denominator:   Copyright © Elizabeth Stapel 2003-2011 All Rights Reserved

and

Each of these expressions is undefined if x = 0.

Then I also have to consider the denominator of the whole complex fraction. Recall that this was rearranged to be:

A fraction is zero when its numerator is zero, so the above fraction would make the complex fraction undefined when 3x2 + 2 = 0. However, this is never equal to zero, so I got no further restrictions on my answer.

Note: If you're going to have to find these restrictions on your answers, you may want to stick with the "flip-n-multiply" method of simplification, instead of the "multiply through by the common denominator" method, since you'll need to convert the complex fraction's denominator into a fraction anyway. If you don't have to find these restrictions, be grateful, and don't worry about these notes on the rest of the examples in this lesson.