### Video Transcript

In this lesson, what weโll learn to
do is how to multiply and divide rational expressions. So, the objectives weโll be looking
at are how to multiply rational expressions, expressing amounts in its simplest
form, so that might mean that weโre going to factor the numerator and denominator,
if necessary. Then, weโre also gonna look at
dividing rational expressions, expressing an answer in its simplest form. And weโll do this with expressions
with two terms and then finally finish up with one with three terms.

So before we start with looking at
some examples, we firstly think, โWell, what is a rational expression?โ Well, with rational expressions,
there are a few definitions we can look at. The first of them being the
quotient of two polynomials, which we can think of in a more simple way as a
fraction whose numerator and denominator are polynomials, like the example we have
here: ๐ฅ plus four over ๐ฅ squared plus five ๐ฅ plus four. Finally, another way we could think
of it is the ratio of two polynomials.

Okay, so now, we should be pretty
clear about what a rational expression is. But in this lesson, weโre gonna
look at multiplying and dividing rational expressions. Well, if weโre gonna multiply and
divide rational expressions, then weโre going to use exactly same rules we would for
any fractions. So, for example, if we have ๐ over
๐ multiplied by ๐ over ๐, this is equal to ๐๐ over ๐๐. And if we divided fractions, what
weโd have is ๐ over ๐ divided by ๐ over ๐ is equal to ๐ over ๐ multiplied by
the reciprocal of the second fraction, so ๐ over ๐, which would give us ๐๐ over
๐๐. However, as we are dealing with
expressions, what we will be using is a number of different skills that we already
know, such as factoring, to help us carry out these operations.

So, now weโve looked at what a
rational expression is and weโve reminded ourselves how we multiply and divide
fractions. Weโre gonna have a look at first
question.

Simplify ๐ฅ plus two over ๐ฅ plus
three multiplied by two ๐ฅ plus six over ๐ฅ plus one.

So with a problem like this, we
could just multiply the numerators and multiply the denominators. However, what we always do when we
get a problem like this is look at them first to see, are there any section, so
numerators or denominators, that can be factored? Well, on inspection, we can see
that the numerator from the right-hand side can in fact be factored. So, what weโre gonna do is factor
this first. Well, what we can see is that with
our numerator on the right-hand side, we can take out two as a factor. When we do, weโll have two
multiplied by ๐ฅ plus three. So therefore, what weโve got now in
our expression is ๐ฅ plus two over ๐ฅ plus three multiplied by two multiplied by ๐ฅ
plus three over ๐ฅ plus one.

So, in these types of questions,
what weโre always looking to do is to divide through by any common factors. So, these common factors will
usually be apparent once weโve done any factoring. And this is the case in this
question cause, as you can see, on the left-hand side, the denominator, weโve got ๐ฅ
plus three. And now, on the numerator on the
right-hand side, we also have the factor ๐ฅ plus three. So, what we now can do is divide
through by ๐ฅ plus three. So now, using the standard rules
for multiplying fractions, what weโre gonna have is ๐ฅ plus two multiplied by two
for our numerator, so that gives us two, multiplied by ๐ฅ plus two. And then for our denominator, weโve
got one multiplied by ๐ฅ plus one, which just gives us ๐ฅ plus one.

Now, we could leave it in this
form. But what weโre gonna do is one more
step before we give our answer. And that is to distribute across
our parentheses. And then if we distribute across
our parentheses, we have two multiplied by ๐ฅ and two multiplied by two. So, therefore, we can say that if
we simplify ๐ฅ plus two over ๐ฅ plus three multiplied by two ๐ฅ plus six over ๐ฅ
plus one, the result is two ๐ฅ plus four over ๐ฅ plus one.

Okay, so in this question, what we
looked at was multiplying two rational expressions. Now, weโre gonna take a look at an
example where weโre going to divide two rational expressions.

Simplify two ๐ฅ over ๐ฅ plus three
divided by two ๐ฅ over five ๐ฅ plus 15.

So to solve this problem, the first
thing we need to do is remind ourselves of how we divide fractions. So, if we have ๐ over ๐ divided
by ๐ over ๐, this is equal to ๐ over ๐ multiplied by ๐ over ๐, so the
reciprocal of the second fraction, which is equal to ๐๐ over ๐๐. You might also remember the skill
using the phrase โkeep it, change it, flip it.โ Keep the first fraction, change the
sign to multiply, and flip the second fraction. So, in our question, what this
means is that our expression is gonna become two ๐ฅ over ๐ฅ plus three multiplied by
five ๐ฅ plus 15 over two ๐ฅ. And thatโs because thatโs
reciprocal of two ๐ฅ over five ๐ฅ plus 15.

So, the next thing we always look
to do in this type of problem is take a look at numerators and denominators and see
if any of them could be factored. And in fact, yes, in this problem
we can see that the right-hand side numerator can be factored. And if we factor five ๐ฅ plus 15,
well, what we can do is take out five as a factor. So, when we do that, weโll have
five multiplied by ๐ฅ plus three. So now our expression is two ๐ฅ
over ๐ฅ plus three multiplied by five multiplied by ๐ฅ plus three over two ๐ฅ. Now, the purpose of this factoring
is to see if we can uncover any common factors in the numerator and denominator. And here we have because we have ๐ฅ
plus three on the left-hand side is the denominator. And we have ๐ฅ plus three on the
right-hand side in our numerator. So, what this means is that we can
divide through by that common factor, as we have done here.

But also, if we take a look at this
question, we can also see another common factor. And that common factor is two ๐ฅ
because on the numerator on the left-hand side, we have two ๐ฅ and on the
denominator on the right-hand side we have two ๐ฅ. So, we divide through by this as
well. So now, what weโre left with is one
over one multiplied by five multiplied by one over one which neatly leaves us with a
result of five.

Okay, so weโve now looked at
multiplying and dividing our rational expressions. Weโll now then look at another
example of dividing rational expressions. And in this example, our numerators
and denominators are already going to be in factored form.

Simplify 24 over 12 multiplied by
๐ plus five divided by ๐ plus three over ๐ plus five multiplied by ๐ plus
three.

So, the first thing we do is remind
ourselves of how we divide fractions. And to do that, what we have is ๐
over ๐ divided by ๐ over ๐ is equal to ๐ over ๐ multiplied by ๐ over ๐, so
the reciprocal of the second fraction, which is equal to ๐๐ over ๐๐. So, what weโve got here is two
rational expressions, and weโre gonna divide them. So, using our rule for dividing
fractions, what weโre gonna get is 24 over 12 multiplied by ๐ plus five multiplied
by ๐ plus five multiplied by ๐ plus three over ๐ plus three.

So, in a problem like this, before
we carry out any multiplying of numerators and denominators, what I like to do is
see if we got any common factors that we can divide through by. But we can see that this is the
case because weโve got ๐ plus five is a common factor in the left-hand denominator
and the right-hand numerator. And then if we look at the
right-hand expression, weโve got a common factor of ๐ plus three on the numerator
and dominator. So first of all, we can divide
through by ๐ plus five, which cancel these out, and then we can divide through by
๐ plus three. So now, what we can do is work out
the result. But what weโre left with is 24 over
12 multiplied by one multiplied by one multiplied by one over one. Well, the right-hand expression is
just one. So, weโve got 24 over 12 multiplied
by one, which is just equal to two.

Okay, great. So, weโve now looked at an example
which was already in fully factored form. Now, what weโre gonna do is look at
how weโre going to multiply two rational expressions. But this time, weโre gonna have to
do some factoring of a quadratic.

Simplify ๐ฅ plus four over ๐ฅ minus
one multiplied by ๐ฅ plus one over ๐ฅ squared plus three ๐ฅ minus four.

So, whenever weโre trying to solve
this type of problem, before we look at multiplying numerators and multiplying
denominators, what we want to do is see if any of the terms that weโve got can be
factored. Well, in fact, we can see yes,
thereโs one that can be because the denominator from the right-hand side is a
quadratic. Well, we think it can be
factored. But letโs check. Because even though itโs a
quadratic, this doesnโt automatically mean it can be factored. So, weโve got ๐ฅ squared plus three
๐ฅ minus four.

Now, to remind ourselves what we do
if we want to factor quadratic, what weโre looking for is two numbers whose product
would give us, in this case, the negative four and whose sum would give us positive
three. Well, the two values we need would
be four and negative one. Thatโs because four multiplied by
negative one is negative four and four add negative one is positive three. So therefore, we know that this can
be factored. So, ๐ฅ squared plus three ๐ฅ minus
four becomes ๐ฅ plus four multiplied by ๐ฅ minus one.

So now, what we can do is
substitute this back into our expression. So when we do that, what we have is
๐ฅ plus four over ๐ฅ minus one multiplied by ๐ฅ plus one over ๐ฅ plus four
multiplied by ๐ฅ minus one. Now the reason why we do this while
weโre factoring a thing that can be factored is to search for any common factors on
the numerators and denominators. And now, once weโve done this, we
can see that weโve got one because weโve got ๐ฅ plus four in the numerator of the
left-hand side and ๐ฅ plus four in the denominator of the right-hand term. So now, what we can do is divide
through by this factor, which weโve done. So now, what weโve got is one over
๐ฅ minus one multiplied by ๐ฅ plus one over one multiplied by ๐ฅ minus one.

So, now, what we need to do is
multiply the numerators and multiply the denominators. So, when we do that, weโre left
with our fully simplified answer, which is ๐ฅ plus one over ๐ฅ minus one
squared. And we got ๐ฅ minus one squared in
the denominator because we had ๐ฅ minus one multiplied by ๐ฅ minus one.

Okay, so for our last problem, what
weโre gonna take a look at is a more complex example. This one will have three terms, and
it will also include multiplication and division.

Simplify ๐ฅ plus five over ๐ฅ plus
six divided by ๐ฅ squared plus 10๐ฅ plus 25 over ๐ฅ squared plus six ๐ฅ multiplied
by six ๐ฅ over ๐ฅ squared plus 11๐ฅ plus 30.

So what we can see is that in this
problem, weโve got three expressions or terms, and they are three rational
expressions. So, what weโre going to do is
divide and multiply them. And what weโre gonna start with is
the bit inside the parentheses. Because if we think about our order
of operations, PEMDAS, then we know that parentheses come first. So therefore, what weโre gonna do
is take a look at ๐ฅ squared plus 10๐ฅ plus 25 over ๐ฅ squared plus six ๐ฅ
multiplied by six ๐ฅ over ๐ฅ squared plus 11๐ฅ plus 30.

Well, the first thing we do with
anything like this is we take a look at the numerators and dominators to see if any
of them could be factored. And here, weโve identified
three. So now, what we want to do is
factor each of them. But if we take a look at the
numerator on the left-hand side, this is a quadratic, and this will factor to ๐ฅ
plus five multiplied by ๐ฅ plus five or ๐ฅ plus five squared. And the reason we know that this is
the case is because weโre looking for two values that whose product is positive 25
and whose sum is positive 10. And we know that five multiplied by
five is 25 and five add five is 10. Okay, great. So thatโs the numerator on the
left-hand side factored.

Now, if we take a look at the
denominator on the left-hand side, we can see that a common factor here is ๐ฅ. So, we can take ๐ฅ outside the
parentheses. So, what weโve got is ๐ฅ multiplied
by ๐ฅ plus six. So then, if we take a look at the
numerator of the right-hand term, this is six ๐ฅ because it canโt be factored. And then what weโll do is take a
look at the dominator. Well, once again, itโs a
quadratic. So, we want the product of the two
numbers to be 30 and the sum to be positive 11. So therefore, what weโre gonna have
is the factored form is ๐ฅ plus six multiplied by ๐ฅ plus five. Thatโs cause six multiplied by five
is 30, six add five is 11. So now, what we have is our
expressions in fully factored form.

Okay, great. So now, whatโs the next step? Well, the next step is to see if
thereโre any common factors in the numerators and denominators. Well, we can see that ๐ฅ plus five
is a common factor cause itโs in the numerator of the left-hand term and the
denominator of the right-hand term. So now what we do is we take a look
to see if there are any other factors. And we can see, well, yes, thereโs
an ๐ฅ on the right-hand numerator and an ๐ฅ on the left-hand denominator. So, we can divide through by
๐ฅ. So now, what we can do is simplify
what weโre left behind with. And what that will give us is six
multiplied by ๐ฅ plus five over ๐ฅ plus six squared or ๐ฅ plus six multiplied by ๐ฅ
plus six. Now, this certainly could have been
written as ๐ฅ plus six squared on the denominator. But Iโve just kept it in this
format so that itโs gonna be easier to cancel in a minute.

So now, we deal with the
parenthesis. What weโre gonna do is go back to
the original calculation. So now what weโve got is ๐ฅ plus
five over ๐ฅ plus six divided by six multiplied by ๐ฅ plus five over ๐ฅ plus six
squared. So now to carry out this
calculation, we just need to remind ourselves about how we divide a fraction. So thatโs ๐ over ๐ divided by ๐
over ๐ is equal to ๐ over ๐ multiplied by ๐ over ๐, which is the same as ๐๐
over ๐๐. So what we do is we multiply by the
reciprocal of the second fraction. So what we do now is we apply this
rule. And we get ๐ฅ plus five over ๐ฅ
plus six multiplied by ๐ฅ plus six squared over six multiplied by ๐ฅ plus five.

So now as we did before, we
identify any common factors and we divide through by these. So we can see weโve got ๐ฅ plus
five and ๐ฅ plus six as common factors. So first of all, we can divide
through by ๐ฅ plus five and then through by ๐ฅ plus six. So now, all we need to do is
multiply what weโre left with, so multiply our numerators and multiply our
denominators. And when we do this, what weโre
left with is ๐ฅ plus six over six. And thatโs our fully simplified
form because neither the numerator nor the denominator can be simplified any
further.

Okay, great. So weโve looked at a range of
problems. And theyโve included multiplying
and dividing rational expressions; theyโve included factoring. And weโve even looked at an example
that included both multiplying and dividing because it had three expressions. So now, what weโre gonna do is take
a look at the key points of the lesson. So, our first key point was to have
a look at what a rational expression was. And a rational expression is a
quotient of two polynomials or, more simply put, is a fraction whose numerator and
denominator are polynomials.

We also reminded ourselves how we
actually multiply and divide fractions because this is the same when weโre dealing
with rational expressions. So, for example, ๐ over ๐
multiplied by ๐ over ๐ is equal to ๐๐ over ๐๐ and ๐ over ๐ divided by ๐
over ๐ is equal to ๐ over ๐ multiplied by ๐ over ๐. So, we multiply by the reciprocal
of the second fraction, which is equal to ๐๐ over ๐๐.

And finally, to remind us the
method that we used, if weโre multiplying two rational expressions, first of all, we
factor where possible. So, we look at numerator and
dominator to see if thereโre any terms that can be factored. Then, we identify any common
factors and divide through by these common factors and then simplify whatโs
left. So, we multiply the numerators and
denominators. The only difference with this when
weโre doing division is that we, first of all, carry out like we said before the
rule for division, which means that we flip the second fraction or find the
reciprocal and then we carry out our multiplication, given these four steps.