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# Simplifying Fractions

An indispensable aspect of any work with algebraic fractions is simplification of the result. We will demonstrate techniques for simplification of fractions throughout these notes where the need arises, but in this specific document, we’ll set the stage by working through a number of examples, focussing primarily on the strategies for simplification themselves. Despite the differences in appearance of the examples to follow, there really is just one approach available for simplifying fractions:

(i) factor the numerator and denominator as completely as possible, and then

(ii) cancel any factors the numerator and denominator have in common.

If you take care to always do step (i) as a separate explicit step of work, you will rarely go wrong in simplifying fractions. What follows in this document is a series of quite a few worked out examples. They tend to be quite repetitive because there is just the one simple strategy just described, which must be applied to each one. Don’t just read through the rest of this document quickly to notice how easily we’ve worked out the solutions (and perhaps to notice how boring this sort of stuff can become). Instead, after studying the first few examples carefully, use the remaining examples as practice problems: cover up the solutions and try to work them out on your own. When you are done in each case, compare your method and final result with the one we’ve given.

Example 1:

Simplify .

solution:

The numerator and denominator of this fraction are already nearly fully factored. Using brackets to make these factors explicit, we get

as the final result.

Example 2:

Simplify .

solution:

Writing the numerator and denominator with the factors explicitly separated gives

Example 3:

Simplify .

solution:

The denominator is factored, but the numerator is not factored. Using the stepwise approach for factoring illustrated earlier in these notes, we get

Thus

as the final result. There is no need to multiply this last form out to remove the brackets, unless there had been some specific instruction to do so.

Note that the first step of factoring is absolutely essential here in order to get a correct result – a simpler mathematical expression which is mathematically equivalent to the original expression. A common error is to just focus on one of the terms in the numerator:

This “simplified” expression is not equivalent to the original fraction, and so an error has been made. You can easily demonstrate this by doing a test calculation:

but

ince for this one value of x, the expression 20x – 45x 3 does not give the same value that the original fraction gives, we know that 20x – 45x 3 cannot be a correct simplification of the original fraction. The error came from the fact that we cancelled the 3x of the denominator into just one of the terms in the numerator. Instead, a valid cancellation must be against factors of the entire numerator, as was done in the original solution of this example.