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 
. 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.
Let’s take a look at a couple of examples.
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Example 1 Reduce
the following rational expression to lowest terms.
(a) [Solution]
(b) [Solution]
(c) [Solution]
Solution
When reducing a rational expression to lowest terms the
first thing that we will do is factor both the numerator and denominator as
much as possible. That should always
be the first step in these problems.
Also, the factoring in this section, and all successive
section for that matter, will be done without explanation. It will be assumed that you are capable of
doing and/or checking the factoring on your own. In other words, make sure that you can
factor!
(a)
We’ll first factor things out as completely as
possible. Remember that we can’t
cancel anything at this point in time since every term has a “+” or a “-” on
one side of it! We’ve got to factor
first!
At this point we can see that we’ve got a common factor in
both the numerator and the denominator and so we can cancel the x-4 from both. Doing this gives,
This is also all the farther that we can go. Nothing else will cancel and so we have
reduced this expression to lowest terms.
[Return to Problems]
(b)
Again, the first thing that we’ll do here is factor the
numerator and denominator.
At first glance it looks there is nothing that will
cancel. Notice however that there is a
term in the denominator that is almost the same as a term in the numerator
except all the signs are the opposite.
We can use the following fact on the second term in the
denominator.
This is commonly referred to as factoring a minus sign out because that is exactly what we’ve
done. There are two forms here that
cover both possibilities that we are liable to run into. In our case however we need the first form.
Because of some notation issues let’s just work with the
denominator for a while.
Notice the steps used here. In the first step we factored out the minus
sign, but we are still multiplying the terms and so we put in an added set of
brackets to make sure that we didn’t forget that. In the second step we acknowledged that a
minus sign in front is the same as multiplication by “-1”. Once we did that we didn’t really need the
extra set of brackets anymore so we dropped them in the third step. Next, we recalled that we change the order
of a multiplication if we need to so we flipped the x and the “-1”. Finally,
we dropped the “-1” and just went back to a negative sign in the front.
Typically when we factor out minus signs we skip all the
intermediate steps and go straight to the final step.
Let’s now get back to the problem. The rational expression becomes,
At this point we can see that we do have a common factor
and so we can cancel the x-5.
[Return to Problems]
(c)
In this case the denominator is already factored for us to
make our life easier. All we need to
do is factor the numerator.
Now we reach the point of this part of the example. There are 5 x’s in the numerator and 3 in the denominator so when we cancel
there will be 2 left in the numerator.
Likewise, there are 2 ’s in the numerator and 8 in the
denominator so when we cancel there will be 6 left in the denominator. Here is the rational expression reduced to
lowest terms.
[Return to Problems]
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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.
Now let’s take a look at a couple of examples.
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Example 2 Perform
the indicated operation and reduce the answer to lowest terms.
(a) [Solution]
(b) [Solution]
(c) [Solution]
Solution
Notice that with this problem we have started to move away
from x as the main variable in the
examples. Do not get so used to seeing
x’s that you always expect
them. The problems will work the same
way regardless of the letter we use for the variable so don’t get excited
about the different letters here.
(a)
Okay, this is a multiplication. The first thing that we should always do in
the multiplication is to factor everything in sight as much as possible.
Now, recall that we can cancel things across a
multiplication as follows.
Note that this ONLY works for multiplication and NOT for
division!
In this case we do have multiplication so cancel as much
as we can and then do the multiplication to get the answer.
[Return to Problems]
(b)
With division problems it is very easy to mistakenly
cancel something that shouldn’t be canceled and so the first thing we do here
(before factoring!!!!) is do the division.
Once we’ve done the division we have a multiplication problem and we
factor as much as possible, cancel everything that can be canceled and
finally do the multiplication.
So, let’s get started on this problem.
Now, notice that there will be a lot of canceling
here. Also notice that if we factor a
minus sign out of the denominator of the second rational expression. Let’s do some of the canceling and then do
the multiplication.
Remember that when we cancel all the terms out of a
numerator or denominator there is actually a “1” left over! Now, we didn’t finish the canceling to make
a point. Recall that at the start of
this discussion we said that as a rule of thumb we can only cancel terms if
there isn’t a “+” or a “-” on either side of it with one exception for the
“-”. We are now at that
exception. If there is a “-” if front
of the whole numerator or denominator, as we’ve got here, then we can still
cancel the term. In this case the “-”
acts as a “-1” that is multiplied by the whole denominator and so is a factor
instead of an addition or subtraction.
Here is the final answer for this part.
In this case all the terms canceled out and we were left
with a number. This doesn’t happen all
that often, but as this example has shown it clearly can happen every once in
a while so don’t get excited about it when it does happen.
[Return to Problems]
(c)
This is one of the special cases for division. So, as with the previous part, we will
first do the division and then we will factor and cancel as much as we can.
Here is the work for this part.
[Return to Problems]
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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.
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.
Let’s work some examples.
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Example 3 Perform
the indicated operation.
(a) [Solution]
(b) [Solution]
(c) [Solution]
(d) [Solution]
(e) [Solution]
Solution
(a)
For this problem there are coefficients on each term in
the denominator so we’ll first need the least common denominator for the
coefficients. This is 6. Now, x
(by itself with a power of 1) is the only factor that occurs in any of the
denominators. So, the least common
denominator for this part is x with
the largest power that occurs on all the x’s
in the problem, which is 5. So, the
least common denominator for this set of rational expression is
So, we simply need to multiply each term by an appropriate
quantity to get this in the denominator and then do the addition and
subtraction. Let’s do that.
[Return to Problems]
(b)
In this case there are only two factors and they both
occur to the first power and so the least common denominator is.
Now, in determining what to multiply each part by simply
compare the current denominator to the least common denominator and multiply
top and bottom by whatever is “missing”.
In the first term we’re “missing” a and so that’s what we multiply the numerator
and denominator by. In the second term
we’re “missing” a and so that’s what we’ll multiply in that
term.
Here is the work for this problem.
The final step is to do any multiplication in the
numerator and simplify that up as much as possible.
Be careful with minus signs and parenthesis when doing the
subtraction.
[Return to Problems]
(c)
Let’s first factor the denominators and determine the
least common denominator.
So, there are two factors in the denominators a y-1 and a y+2. So we will write both
of those down and then take the highest power for each. That means a 2 for the y-1 and a 1 for the y+2. Here is the least common denominator for
this rational expression.
Now determine what’s missing in the denominator for each
term, multiply the numerator and denominator by that and then finally do the
subtraction and addition.
Okay now let’s multiply the numerator out and
simplify. In the last term recall that
we need to do the multiplication prior to distributing the 3 through the
parenthesis. Here is the
simplification work for this part.
[Return to Problems]
(d)
Again, factor the denominators and get the least common
denominator.
The least common denominator is,
Notice that the first rational expression already contains
this in its denominator, but that is okay.
In fact, because of that the work will be slightly easier in this
case. Here is the subtraction for this
problem.
Notice that we can actually go one step further here. If we factor a minus out of the numerator we
can do some canceling.
Sometimes this kind of canceling will happen after the
addition/subtraction so be on the lookout for it.
[Return to Problems]
(e)
The point of this problem is that “1” sitting out behind
everything. That isn’t really the
problem that it appears to be. Let’s
first rewrite things a little here.
In this way we see that we really have three fractions
here. One of them simply has a
denominator of one. The least common
denominator for this part is,
Here is the addition and subtraction for this problem.
Notice the set of parenthesis we added onto the second
numerator as we did the subtraction.
We are subtracting off the whole numerator and so we need the
parenthesis there to make sure we don’t make any mistakes with the subtraction.
Here is the simplification for this rational expression.
[Return to Problems]
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