We now need to look at rational expressions. A rational
expression is nothing more than a fraction in which the numerator and/or the denominator are
polynomials. Here are some examples
of rational expressions.
The last one may look a little strange since it is more commonly
written 4x2 + 6x −10. However, it’s important to note that
polynomials can be thought of as rational expressions if we need to, although they rarely are.
There is an unspoken rule when dealing with rational expressions
that we now need to address. When
dealing with numbers we know that division by zero is not allowed. Well the
same is true for rational expressions. So, when
dealing with rational expressions we will always assume that whatever x is it won’t give
division by zero. We rarely write these restrictions down, but we will always need to keep them in mind.
For the first one listed we need to avoid x = 1. The second
rational expression is never zero in the denominator and so we don’t need to worry about any restrictions.
Note as well that the numerator of
the second rational expression will be zero. That is okay, we just need to
avoid division by
zero. For the third rational expression we will need to avoid m = 3 and m = -2.
The final rational
expression listed above will never be zero in the denominator so again we don’t
need to have any restrictions.
The first topic that we need to discuss here is reducing a rational
expression to lowest terms. A rational
expression has been reduced to lowest terms if all common factors from
the numerator and denominator have been canceled.
We already know how to do this with number fractions so let’s take a quick look at an
example.
With rational expression it works exactly the same way.
We do have to be careful with canceling however. There are some
common mistakes that students often
make with these problems. Recall that in order to cancel a factor it must
multiply the whole
numerator and the whole denominator. So, the x-3 above could cancel
since it multiplied the
whole numerator and the whole denominator. However, the x’s in the
reduced form can’t
cancel since the x in the numerator is not times the whole numerator.
To see why the x’s don’t cancel in the reduced form above
put a number in and see what happens. Let’s plug in x = 4.
Clearly the two aren’t the same number!
So, be careful with canceling. As a general rule of thumb remember
that you can’t cancel something if
it’s got a “+” or a “-” on one side of it. There is one exception to this rule
of thumb with “-” that
we’ll deal with in an example later on down the road.
Before moving on let’s briefly discuss the answer in the second part
of this example. Notice that we
moved the minus sign from the denominator to the front of the rational
expression in the final form. This can
always be done when we need to. Recall that the following are all equivalent.
In other words, a minus sign in front of a rational expression can
be moved onto the whole numerator or
whole denominator if it is convenient to do that. We do have to be careful with this however. Consider the following rational expression.
In this case the “-” on the x can’t be moved to the front of
the rational expression since it is only on the x. In order to move a minus sign to the front of a
rational expression it needs to be times the whole numerator or denominator. So, if we factor a minus out of
the numerator we could then move
it into the front of the rational expression as follows,
The moral here is that we need to be careful with moving minus
signs around in rational expressions.
We now need to move into adding, subtracting, multiplying and
dividing rational expressions.
Let’s start with multiplying and dividing rational expressions. The
general formulas are as follows,
Note the two different forms for denoting division. We will use
either as needed so make sure you
are familiar with both. Note as well that to do division of rational
expressions all that we need to do is
multiply the numerator by the reciprocal of the denominator (i.e. the
fraction with the numerator
and denominator switched).
Before doing a couple of examples there are a couple of special cases
of division that we should look
at. In the general case above both the numerator and the denominator of the
rational expression
where fractions, however, what if one of them isn’t a fraction. So let’s look
at the following
cases.
Students often make mistakes with these initially. To correctly deal
with these we will turn the numerator
(first case) or denominator (second case) into a fraction and then do the
general division on
them.
Be careful with these cases. It is easy to make a mistake with
these and incorrectly do the division.
Okay, it’s time to move on to addition and subtraction of rational
expressions. Here are the general
formulas.
As these have shown we’ve got to remember that in order to add or subtract
rational expression or fractions we
MUST have common denominators. If we don’t have common denominators then we need to first get common
denominators.
Let’s remember how do to do this with a quick number example.
In this case we need a common denominator and recall that it’s
usually best to use the least common denominator, often denoted
lcd. In this case the least common denominator is 12. So we need to get the denominators of these two fractions to a 12.
This is easy to do. In the first case we need to multiply the denominator by 2 to get 12 so we will
multiply the numerator and denominator
of the first fraction by 2. Remember that we’ve got to multiply both the
numerator and denominator
by the same number since we aren’t allowed to actually change the problem and this is equivalent to multiplying
the fraction by 1 since
. For the second term we’ll need to multiply the numerator and
denominator by a 3.
. For the second term we’ll need to multiply the numerator and
denominator by a 3.
Now, the process for rational expressions is identical. The main
difficulty is in finding the least common denominator. However, there is a really simple process for
finding the least common denominator for
rational expressions. Here is it.
- Factor all the denominators.
- Write down each
factor that appears at least once in any of the denominators. Do NOT write down the power that is on each
factor, only write down the factor
- Now, for each
factor written down in the previous step and write down the largest power that occurs in all the denominators
containing that factor.
- The product all the factors from the previous step is the least common denominator.
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