It’s Professor Dave; let’s work with rational expressions.
We’ve learned a lot of algebra by now, and we haven’t even started with all the graphing,
which is a bit overdue at this point.
But before we start with graphs, let’s just do a little bit of bookkeeping regarding rational
expressions.
In algebra, we will frequently need to simplify, add, subtract, multiply, and divide these
expressions, so let’s make sure we have plenty of practice doing that so there’s
no trouble later.
First, we want to be able to simplify rational expressions.
To do this, we must completely factor any polynomials, and get rid of any common factors
in the numerator and denominator.
For example, if we have X cubed plus X squared over X plus one, we can see that in the numerator
we could factor an X squared out of each term.
That would leave us with X squared times the quantity X plus one.
The reason this is important to do is that now we can cancel out both of the X plus one
terms, leaving us with simply X squared.
Let’s try another one.
X squared minus one over X squared plus two X plus one.
In the numerator, it might not seem like we can factor this, but we have to realize that
when there is a second order polynomial with no X term, we can pretend there is zero X.
This will allow us to factor like we normally do, because there are indeed factors of negative
one that add up to zero, and those are one and negative one.
So this expression can be written as the product of X plus one and X minus one.
Then on the bottom, this can be factored to get X plus one quantity squared, which we
can express as X plus one times X plus one.
We cancel out an X plus one from the top and bottom, and we are left with X minus one over
X plus one, which is as far as we can go.
This kind of simplification is important, because it will help us do arithmetic with
these kinds of expressions.
Let’s say we want to multiply X plus two over X minus one, times X squared plus four
X minus five divided by X plus three.
If we factor this trinomial, we get X minus one times X plus five, allowing us to cancel
out some terms, and we get this as the product.
Division works very similarly.
If we have X plus three over X minus four divided by X squared plus X minus six over
X squared minus three X minus four, we have to realize that dividing by a fraction is
the same thing as multiplying by its reciprocal, so all we have to do is flip this second fraction
over, and multiply.
Of course we first have to factor, so up top we get X minus four times X plus one, and
on the bottom we get X plus three times X minus two.
A number of terms will therefore cancel, and we are left with this as our answer.
So multiplying and dividing are no sweat.
It’s actually addition and subtraction that are a little trickier, because to do this
we must have a common denominator.
Say we have the following expression.
We can see that the denominators are different.
Well when we first learned about adding fractions, how did we add one third and one fourth together?
We had to find the least common multiple.
Many times, that will just be the product of the two denominators.
If we want both of these to be over twelve, we just multiply one third by four over four,
giving us four twelfths, and we multiply one fourth by three over three, giving us three
twelfths.
Then we are free to add, and we get seven twelfths.
Here it’s exactly the same idea, just with algebraic expressions.
We multiply the first fraction by the denominator of the second fraction over itself.
We multiply the second fraction by the denominator of the first fraction over itself.
That gives us a common denominator, and we are safe to add.
Now with the numerators, we do have to FOIL both of these expressions, because that will
give us a bunch of like terms that we can them combine.
Once we’ve done that, we see that the resulting expression can’t be factored, and so we
just leave it like this.
Subtracting works exactly the same way, we just have to make sure that when we combine
the numerators, we are careful with our signs, and distribute this minus sign over all the
terms in the second numerator.
If the denominators are more complex terms, we may have to factor them first, and once
we have them factored, we can more easily see what the lowest common denominator will
be, it will just be the product of all the terms in both denominators.
Sometimes we will need to simplify complex rational expressions, which means there are
fractions within the fraction.
Here is one plus (one over X) divided by one minus (one over X).
At first glance, there doesn’t seem to be much we can do, but there is if we are clever.
Let’s take the numerator first, how can we turn this into one term?
Well, let’s express one as one over one.
Now we have two fractions to add if we can get a common denominator.
That will be easy, just turn one over one into X over X.
Now we can add them and get X plus one over X.
Now let’s do the same with the denominator.
We get X over X and then subtract to get X minus one over X.
Now remember, dividing by a fraction is like multiplying by the reciprocal, so let’s
take this whole denominator, flip it over, and bring it over here.
Now we can cancel out the X’s, and we get X plus one over X minus one, which is definitely
more aesthetically pleasing than what we started with.
The last thing we want to mention is that for whatever reason, we hate having radicals
in the denominators of fractions.
We always get rid of these when we see them.
If we have something like one over root two, we always multiply this fraction by the root
in the denominator over itself, so root two over root two.
That gives us root two in the numerator, and for the denominator, root two times root two
is the same thing as two to the one half times two to the one half, and one half plus one
half is one, so two to the one, or simply two.
So a radical expression times itself just gets rid of the radical, leaving us with root
two over two.
If things are a little trickier and we have one over one minus root two, then instead
of simply multiplying by root two over root two, we need to multiply by the conjugate
of this expression, because we need to get rid of the radical, so we will multiply by
one plus root two over one plus root two.
The numerator simply becomes this term, and then on the bottom we can FOIL this out.
One times one is one, then we get plus root two and minus root two, and lastly negative
root two times root two, which will simply be negative two.
The reason we do this is so that the terms with radicals cancel, and we are left with
one minus two, or negative one.
Dividing by negative one is the same as multiplying by negative one, so we can distribute the
negative sign across the numerator, and we are left with negative one minus root two.
That covers most of the ways we will need to manipulate rational expressions, so let’s
check comprehension.