www.mathcentre.ac.uk/.../Simplifying%20Algebraic%20Fractions.mp4

When we come across fractions,
one of the things that we have

to do is to look at them and see
if we can put them in a simple

form. In fact, see if we can put
them in a form that might be

called the lowest terms. So for
instance, if we have a fraction

12 over 36, then what we want is
a fraction in its lowest terms.

Now 12 will divide into both 12
and 36, so we can divide 12 into

12 one and 12 into 36 three. So
that reduces to the fraction 1/3

the two fractions are the same.

They are equivalent fractions.
Now that's OK with numbers, but

we want to be able to do it with
similar things, but with algebra

in other similar expressions
that have got letters in them.

And where is here? We look for a
number that was in fact a common

factor. That's a number that
will divide into both the top

and the bottom. What we've now
got to look for is a common

factor. That's an expression
that will divide into both the

top and the bottom. So instead
of talking about it, let's have

a look at some examples. So
supposing we've got 3X cubed all

over X to the 5th.

We've got to look at is what's
the same on the top and on the

bottom? Well, we've got a 3
here, but no numbers down here.

We've got X is here and we've
got X is here and what we can

see is that we've got X cubed
here and X to the fifth here.

Now X cubed times by X squared
gives us X to the 5th, so any

fact we've got a common factor
of X cubed on the top and on the

bottom. So let's just write that
down so we can see it more

clearly. Thanks for the 5th is X
cubed times by X squared? What

we can see very clearly here is
we've got a common factor of X

cubed, so we can divide top and
bottom by X cubed. That leaves

us with three over X squared.
Now we don't need to do this

middle step every time we have
to be able to do is to see

exactly what that common factor

is. So let's have a look at some
examples and the examples will

increase in terms of their
complexity as we move through

them in terms of the difficulty.
So what's common here to both

the top and the bottom? On the
top? We can see we've got X

cubed, and on the bottom X
squared, and we know that if we

multiply X squared by X, it will
give us X cubed. So in effect we

can divide. Top and bottom by X
squared X squared in two X

squared goals once and X squared
into X cubed leaves us with X

because X Times X squared gives
us X cubed. Similarly, Y into Y

goes once and Y into Y cubed.
Well why times by Y squared

gives us Y cubed. So if we
divide in what we get there is Y

squared. So on the top we've got
X times by.

Squared on the bottom. We've got
one times by one so we don't

really need to write the one
there. And there's our answer.

You can also involve numbers in
this, so we have alot minus

16 X squared Y squared over
4X cubed Y squared.

Here minus 16 an 4 four
goes into minus 16 - 4

times. X squared on the top and
X cubed underneath X squared

goes into X squared. Once an X
squared goes into X cubed X

times and here Y squared is the
same on both top and bottom, so

Y squared into Y squared goes
once and Y squared into Y

squared ones as well.

So here we minus 4 *
1 * 1 - 4 over.

14 went into 4 one select
remember one times X times

one that's X. So we have
minus four over X.

Sometimes that common factor may
not be obvious. We may have to

work to see it. We may have to
work so that it stands out over

what we've got. So let's have a
look at something like this.

X squared minus two XY all

over X. Well, is there a

common factor? The best way to
look at this is not so much. Is

there a common factor on top and
bottom, but is there a common

factor here? Can we factorize

this expression? Well, in each
term that is at least an X as an

X here and an X squared here. So
we can take that X out as a

common factor X times by X gives
us the X squared.

Taking X out of here, we're left
with minus two Y so that X times

Y minus two Y would give us
minus two XY, and then that's

all over X. Now we can cancel
the X. We can divide the top a

numerator by X and the bottom by
X&X in 2X goes one and X in 2X

goes once, and So what we're
left with is one times that

divided by one. So we just left.

With X minus two Y.

Have a look at another One X to
the power 6.

Minus Seven X to the
fifth plus 4X to the

8th. All over X squared. Again,
what we want to do is look at

this top line and see. Have we
got a common factor.

Well, yes, we have its X to
the power five 'cause X the

power five is included in X to
the power 6 and X to the power

8. So we can take that out as
a common factor X to the power

five times by X will give us X
to the power 6.

Minus Seven X to the power five.
We've got the X to the power 5

outside, so we want minus 7.

Plus X to the power 8.

We're taking out X the power
5th, so it's going to be 4X to

the power three. Close the
bracket and all over X squared.

Now we can see what we're
looking at is. Is there

something common now between
these two? And clearly we can

divide X squared by X squared
and we can divide X to the power

5 by X squared.

So we'll do that X in two X

squared once. X squared into X
to the power five is X cubed.

And so we end up with X
cubed times by X minus 7 +

4 X cubed.

We don't worry about the
dividing by one. That's not

going to change anything.

Notice we don't multiply out the
brackets. It's better to keep

the brackets there. We may want
it in that form to work with

later. What
if we

have something

like this?
No obvious common factor,

but this is a

quadratic. And so because it's a
quadratic expression, there is

the possibility of factorizing

it. If there's the possibility
of Factorizing it, then what we

might find is that one of those
factors could be X plus one, in

which case we could then divide
top on bottom by that common

factor. So leave the top as
it is and let's concentrate

on this bottom. We want to
be able to factorize that.

That means two brackets.

An X in each bracket.

That ensures is the X squared.

We want to have.

2 as a result of multiplying
these two numbers together that

go in here and here. So we'll
have two and one.

Now we need a plus sign to give
us the Plus 3X.

And what we can see is that X
Plus one is a common factor,

so I'm going to put the
brackets around that one on

top to show us. We've got a
common factor and X plus one

into there goes 1X plus one
into. There goes one and so

we're left with one over X +2.

You can have one that is if you
like the other way up, so let's

have a look at that X minus 11 A
plus 30. Sorry, A squared minus

11 A plus 30 over a minus 5.

Again, the A minus five. That's
nice. That's OK. Let's keep that

together with a bracket, but
let's have a look at this. A

squared minus eleven 8 + 30. Can
we factorize it? Can we break it

down into two factors?

We've got a squared, so we need
an A and and a. We now need 2

numbers that are going to
multiply together to give us 30,

but add together to give us
minus eleven. Well, six and five

seem a good bet for the 30, and
if we make a minus six and minus

five, that ensures the plus 30
minus times by A minus. And it

also ensures the minus 11 a
'cause will have minus 6A and

minus 5A there.

We've now got a common factor of
a minus five, so we can divide

the top by A minus five on the
bottom by A minus five. So we

end up with just a minus 6, and
that's our answer.

Now that's the proper way to do
them. Some of you might be

tempted. Might be tempted when
you see something like this.

Till forget what it
is you're supposed to

do. So you might suddenly think
are lots of threes. I can get

rid of some three, so let's
cancel 3 here and a three there.

One problem. We said we had
to cancel common factors.

3 doesn't appear here in this
term, there's no factor of three

in this term. We cannot do this
kind of canceling.

Let me just show you why not. If
we have a look at a numerical

example. Supposing we had, let's
say 5 + 3 over 3

+ 1. Now if we do
the computation without doing

the canceling we get.

8 over 4.

And that's two, everybody is
happy that that's the case.

But if I do what I did here and
suddenly go absolutely bananas

and cancel the threes, then
that's a one there and one

there. So what I seem to have
now is 5 + 1, which is 6 over 1

+ 1, which is 2 gives me 3.

Two and three are not the
same. Now we've agreed that

tools the correct answer,
where have we gone wrong?

We cancel these threes divided
by these threes, but three is

not a common factor because it
doesn't appear in the Five and

it doesn't appear in the one.

So we can't do this over here by
the same reasons what we have to

do is look and see if we can
factorize what is on the top

three X squared plus 10X plus
three. Can we factorize it?

Well, the X +3 is fine, let's
keep it all together in a

bracket. Brackets here.

Three X squared will need a 3X
and an X.

Will also need a three under one
to make up this number here, and

we've got to get 10 out of it,
so we're probably have to have

the three there to give us Nynex
across there under one there.

Plus signs because these are all
plus signs here and now. We can

see we have got this common
factor of X plus three, so we

can divide the bottom by X +3
and the top by X +3.

That will give us 3X Plus One
and that is the correct answer.

Sometimes we have to work again
back a little bit harder, so

let's take this example 6X

cubed. Minus Seven X
squared minus 5X all

over 2X plus one.

That doesn't give me much hope
for is here, but there is a

common factor of X in each
term, so perhaps we can take

that out as a common factor.
To begin with. We might find

we can factorize what's left,
so taking that X out as a

common factor is going to give
us six X squared there.

Minus Seven X there.

And minus five there closed the
bracket and we still to divide

by the 2X plus one. Also we
hope. So now let's have a look

at this. 'cause this is now an

ordinary quadratics. X.

Six X squared? Well, let's have
a guess at 3X and 2X.

After all, there's 2X down

there. What do we need now?
We've got minus five to deal

with. I want if I can have it 2X
plus one. So let's just be

guided by that. For the moment,
let's make that 2X plus one.

Minus 5 is what I need to
multiply by the one to give me

minus 5. Have I got it right? I
need to check on that minus 7X

will hear I plus 3X and here
I've minus 10X and that does

give me minus 7X. So that's
right. Now I need to divide by

the 2X plus one. Let's get it
together in a bracket.

I can divide top and bottom now
by 2X plus one.

Goes into itself once and once

there. The answer is just what
I'm left with here X times 3X

minus five, and again I leave it
in its factorized form, not try

to multiply it out again 'cause
I may need that form later on.

Supposing we take
something like this

X cubed minus one
over X minus one.

What now? Doesn't seem
to be a common factor

in X cubed minus one.

Difficult. But you may
know a factorization 4X cubed

minus one. It does in
fact factorize as X minus

one brackets X squared.

Plus X plus one.

And so, because we know that
factorization, we can see

straight away, we can
simplify by dividing the

bottom by X minus one, and
that goes in once and the top

by X minus one. And so we
just left with the other

factor X squared plus X plus
one.