Example 1 Reduce
the following rational expression to lowest terms.
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!
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.
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 (x +1) ’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.
Example 2 Perform the
indicated operation and reduce the answer to lowest terms.
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.
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.
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.
Example 3 Perform the
indicated operation.
Solution
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,
lcd : 6x5
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.
In this case there are only two factors and they both occur to the
first power and so the least common denominator is.
lcd : (z +1)(z + 2)
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 z + 2
and so that’s what we multiply the numerator and denominator by. In the second term we’re “missing” a z +1 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.
Sumber
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