Chapter 7 Rational Functions - College of the Redwoods

Section 7.2 Reducing Rational Functions 621
When it is difficult to ascertain the greatest common divisor, we’ll find it more
efficient to proceed as follows:
• Prime factor both numerator and denominator.
• Cancel common factors.
Thus, to reduce 12/18 to lowest terms, first express both numerator and denominator
as a product of prime numbers, then cancel common primes.
12
18 = 2 · 2 · 3
2 · 3 · 3 = 2 · 2 · 3
2 · 3 · 3 = 2 3
When you cancel a 2, you’re actually dividing both numerator and denominator by 2.
When you cancel a 3, you’re actually dividing both numerator and denominator by 3.
Note that doing both (dividing by 2 and then dividing by 3) is equivalent to dividing
both numerator and denominator by 6.
We will favor this latter technique, precisely because it is identical to the technique
we will use to reduce rational functions to lowest terms. However, this “cancellation”
technique has some pitfalls, so let’s take a moment to discuss some common cancellation
mistakes.
Cancellation
You can spark some pretty heated debate amongst mathematics educators by innocently
mentioning the word “cancellation.” There seem to be two diametrically opposed camps,
those who don’t mind when their students use the technique of cancellation, and on
the other side, those that refuse to even use the term “cancellation” in their classes.
Both sides of the argument have merit. As we showed in equation (8), we can
reduce 12/18 quite efficiently by simply canceling common factors. On the other hand,
instructors from the second camp prefer to use the phrase “factor out a 1” instead of
the phrase “cancel,” encouraging their students to reduce 12/18 as follows.
12
18 = 2 · 2 · 3
2 · 3 · 3 = 2 3 · 2 · 3
2 · 3 = 2 3 · 1 = 2 3
This is a perfectly valid technique and one that, quite honestly, avoids the quicksand
of “cancellation mistakes.” Instructors who grow weary of watching their students
“cancel” when they shouldn’t are quite likely to promote this latter technique.
However, if we can help our students avoid “cancellation mistakes,” we prefer to
allow our students to cancel common factors (as we did in equation (8)) when reducing
fractions such as 12/18 to lowest terms. So, with these thoughts in mind, let’s discuss
some of the most common cancellation mistakes.
Let’s begin with a most important piece of advice.
How to Avoid Cancellation Mistakes. You may only cancel factors, not
addends. To avoid cancellation mistakes, factor completely before you begin to
cancel.
(8)
Version: Fall 2007