# Reducing Rational Functions

The goal of this section is to learn how to reduce a
rational expression to “lowest terms.”

Of course, that means that we will have to understand what is meant by the
phrase

“lowest terms.” With that thought in mind, we begin with a discussion of the
greatest

common divisor of a pair of integers.

First, we define what we mean by “divisibility.”

Definition 1. Suppose that we have a pair
of integers a and b. We say that “ais a divisor of b,” or “a divides b” if and only if there is another integer k so that b = ak. Another way of saying the same thing is to say that a divides b if, upon dividing b by a, the remainder is zero |

Let’s look at an example.

**Example 2. **What are the divisors of
12?

Because 12 = 1×12, both 1 and 12 are divisors of 12. Because 12 = 2×6, both 2

and 6 are divisors of 12. Finally, because 12 = 3 × 4, both 3 and 4 are divisors
of 12.

If we list them in ascending order , the divisors of 12 are

1, 2, 3, 4, 6, and 12.

Let’s look at another example.

** Example 3. **What are the divisors of
18?

Because 18 = 1 × 18, both 1 and 18 are divisors of 18. Similarly, 18 = 2 × 9 and

18 = 3 × 6, so in ascending order, the divisors of 18 are

1, 2, 3, 6, 9, and 18.

The greatest common divisor of two or more integers is the
largest divisor the

integers share in common. An example should make this clear.

**Example 4**. What is the greatest
common divisor of 12 and 18?

In **Example 2** and **Example 3**, we saw the following.

Divisors of 12 : | |

Divisors of 18 : |

We’ve framed the divisors that 12 and 18 have in common.
They are 1, 2, 3, and 6. The

“greatest” of these “common” divisors is 6. Hence, we say that “the greatest
common

divisor of 12 and 18 is 6.”

Definition 5. The greatest common divisor
of two integers a and b is the largestdivisor they have in common. We will use the notation GCD(a, b) to represent the greatest common divisor of a and b. |

Thus, as we saw in **Example 4,** GCD(12, 18) = 6.

When the greatest common divisor of a pair of integers is one, we give that pair
a

special name.

Definition 6. Let a and b be integers. If
the greatest common divisor of a andb is one, that is, if GCD(a, b) = 1, then we say that a and b are relatively prime. |

For example:

• 9 and 12 are not relatively prime because GCD(9, 12) = 3.

• 10 and 15 are not relatively prime because GCD(10, 15) = 5.

• 8 and 21 are relatively prime because GCD(8, 21) = 1.

We can now define what is meant when we say that a rational number is reduced

to lowest terms.

Definition 7. A rational number in the
form p/q, where p and q are integers,is said to be reduced to lowest terms if and only if GCD(p, q) = 1. That is, p/q is reduced to lowest terms if the greatest common divisor of both numerator and denominator is 1. |

As we saw in** Example 4**, the greatest common divisor
of 12 and 18 is 6. Therefore,

the fraction 12/18 is not reduced to lowest terms. However, we can reduce 12/18
to

lowest terms by dividing both numerator and denominator by their greatest common

divisor. That is,

Note that GCD(2, 3) = 1, so 2/3 is reduced to lowest terms.

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.

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.

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, notaddends. To avoid cancellation mistakes, factor completely before you begin to cancel. |

Warning 9. Many of the ensuing
calculations are incorrect. They are examplesof common mistakes that are made when performing cancellation. Make sure that you read carefully and avoid just “scanning” these calculations . |

As a first example, consider the rational expression

which clearly equals 8/2, or 4. However, if you cancel in this situation, as in

you certainly do not get the same result. So, what
happened?

Note that in the numerator of **equation (10)**, the 2 and the 6 are
separated by a

plus sign . Thus, they are not factors; they are addends! You are not allowed to
cancel

addends, only factors.

Suppose, for comparison , that the rational expression had been

which clearly equals 12/2, or 6. In this case, the 2 and
the 6 in the numerator are

separated by a multiplication symbol , so they are factors and cancellation is
allowed,

as in

Now, before you dismiss these examples as trivial,
consider the following examples

which are identical in structure. First, consider

This cancellation is identical to that performed in **
equation (10)** and is not allowed.

In the numerator, note that x and (x+2) are separated by an addition symbol , so
they

are addends. You are not allowed to cancel addends!

Conversely, consider the following example.

In the numerator of this example, x and (x+2) are
separated by implied multiplication.

Hence, they are factors and cancellation is permissible.

Look again at **equation (10)**, where the correct answer should have been
8/2, or 4.

We mistakenly found the answer to be 6, because we cancelled addends. A
workaround

would be to first factor the numerator of **equation (10)**, then cancel, as
follows.

Note that we cancelled factors in this approach, which is
permissible, and got the

correct answer 4.

Warning 12. We are finished discussing
common cancellation mistakes andyou may not continue reading with confidence that all mathematics is correctly presented. |

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