Canceling Terms in Fractions

Once a fraction is in reduced form, you may begin trying to cancel terms. Only
when factors in the numerator match exactly with factors in the denominator can the
terms
be canceled. One of the most consistent ways to ensure that your cancellations are
appropriate is to factor the numerator and denominator into smaller pieces before
attempting to cancel.

Example 1: Simplify

Solution:

In Example 1, we can factor 2x from both the numerator and denominator. Thus, we can
cancel the 2x terms from the ex pression . A common mistake made by students is to
“cancel” pieces that are not factors.

In this bad example, the term   is a factor of the numerator, but it is not a factor of
the denominator. This cancellation is therefore invalid. (verify this by plugging in a
numerical
value for x .)

The Least Common Denominator

You should already be familiar with the standard technique for adding fractions
together. Generally, we find a common denominator between the terms in our
expression
. In order to ensure that the fraction we get as a result of addition (or
subtraction ) is in reduced form , we are interested in finding the least common
denominator (LCD) for our sum. To find the LCD, write the denominator of each
fraction as a product of its prime factors. The LCD will be the smallest number that is a
multiple
of each denominator.

Example 2: Find the LCD for

The LCD of this sum is (x + 1)(x + 2)(x + 3).

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