7.2 Reducing Rational Functions

622 Chapter 7 Rational Functions
Warning 9. Many of the ensuing calculations are incorrect. They are examples
of 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
2 + 6
2 ,
which clearly equals 8/2, or 4. However, if you cancel in this situation, as in
2 + 6
2
you certainly do not get the same result. So, what happened
= 2 + 6
2 , (10)
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
2 · 6
2 ,
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
2 · 6
2 = 2 · 6 = 6. (11)
2
Now, before you dismiss these examples as trivial, consider the following examples
which are identical in structure. First, consider
x + (x + 2)
x
=
x + (x + 2)
x
= x + 2.
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.
x(x + 2)
x
=
x(x + 2)
x
= x + 2
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.
Version: Fall 2007