
In simplifying algebraic
fractions, we occasionally need

a process known as.

Polynomial.
Division.

Before we do that, I want to
take you back to something

that you actually know very
well indeed, and that's

ordinary long division.

You know how to do long
division, but I want to go over

it again. 'cause I want to point
out certain things to you.

'cause the things that are
important about long division

are also important in polynomial

division so. Let's have a look
at a long division. Some

supposing I want to divide 25.

Into Let's
say

2675.

When I would have to do is look
at 25 in tool 2.

No way 25 into 26. It goes once
and write the one there.

Add multiply the one by the 25.

And subtract and
have one left.

Hope you remember doing that.
You were probably taught how to

do that at primary school or the
beginnings of Secondary School.

Next step is to bring down the
next number, so we bring down

17. Well, we bring down Seven to
make it 17 and now we say how

many times does 25 going to 17.

It doesn't go at all. It's not
enough, so we have to record the

fact that it doesn't go with a

0. Next we bring down the
five. So now we've got 175 and

we say how many times does 25
go into that? And it goes 7

and we can check that Seven 535,
five down three to carry. 7 twos

are 14 and three is 17.

Tracked, we get nothing left.

So this is our answer. We've
nothing left there, no

remainder, nothing left over.
And there's our answer.

2675 divides by 25 and the
answer is 107. They just look at

what we did. We did 25 into
26 because that went.

We then recorded that once that
it went there, multiplied, wrote

the answer and subtracted.

We brought down the next number.

Asked how many times 25 went
into it, it didn't go. We

recorded that and brought down
the next number. Then we said

how many times does 25 going
to that Seven we did the

multiplication, wrote it down,
subtracted, got nothing left

so it finished.

What we're going to do now is
take that self same process and

do it with algebra.

So let us

take. This
27 X cubed.

+9 X squared.

Minus 3X. Minus
10.

All over.

3X minus 2.

We want to divide that into
that. We want to know how many

times that will fit into there,
so we set it up exactly like a

long division. Problem by
dividing by this. This is what

we're dividing into 27 X cubed
plus nine X squared minus three

X minus 10.

So we ask ourselves, how many
times does well? How many times

does that go into that? But
difficult what we ask ourselves

is how many times does the
excpet go into this bit?

Just like we asked ourselves how
many times the 25 went into the

26, how many times does 3X?

Go into 27 X cubed. The answer
must be 9 X squared because

Nynex squared times by three X
gives us 27 X cubed and we need

to record that. But we need to
record it in the right place and

because these are the X
squared's we record that above

the X squares.

So now we do the multiplication.

Nine X squared times 3X is 27

X cubed. Nine X squared times
minus two is minus 18 X squared.

Just like we did for long
division, we now do the

Subtraction. 27 X cubed
takeaway 27 X cubed none of

them, because we arrange for
it to be so Nynex squared

takeaway minus 18 X squared
gives us plus 27 X squared.

Now we do what we did before we
bring down the next one, so we

bring down the minus 3X.

How many times does 3X go into
27 X squared?

Answer. It goes 9X times and
we write that in the X Column.

So now we have 9X times 3
X 27 X squared 9X times, Y

minus 2  18 X.

And we subtract again.

27 X squared takeaway, 27 X
squared, no X squared, but we

arrange for it to be like that,
minus three X minus minus 18X.

Well, that's going to give us
plus 15X altogether, and we

bring down the minus 10.

3X into 15X. This time it goes
five times, so we can say plus

five there. And again it's in
the numbers. The constants

column at the end.

Five times by 15 times by three
X gives us 15X. Write it down

there five times by minus two
gives us minus 10 and we can see

that when we take these two
away. Got exactly the same

expression. 15X minus 10
takeaway. 50X minus 10 nothing

left. So there's our answer,
just as in the long division.

The answer was there.

It's there now so we can say
that this expression is equal to

9 X squared plus 9X.

Plus 5. Let's
take another one.

So we'll take X to the 4th.

Plus X cubed.

Plus Seven X squared
minus six X +8.

Divided by all over
X squared, +2 X

+8. So this is what we're
dividing by and this is what

we're dividing into is not
immediately obvious what the

answer is going to be. Let's
have a look X squared plus 2X

plus 8IN tool.

All of this.

Our first question is how many
times does X squared going to X

to the 4th? We don't need to
worry about the rest, we just do

it on the first 2 bits in each
one, just as the same as we did

with the previous example. How
many times X squared going to X

to the four will it goes X
squared times? So we write it

there over the X squared's. Now
we do the multiplication X

squared times. My X squared is X
to the 4th.

X squared by two X is plus
2X cubed X squared by 8 is

plus 8X squared.

And now we do the Subtraction X.
The four takeaway X to the 4th

there Arnold, but we arranged it
that way. X cubed takeaway 2X

cubed minus X cubed. Seven X
squared takeaway, 8X squared

minus X squared and bring down
the next term.

Now we say how many times does X
squared going to minus X cubed,

and it must be minus X, and so
we write it in the X Column.

And above the line there, next
the multiplication minus X times

by X squared is minus X cubed
minus X times 2X is minus two X

squared and minus X times by 8
is minus 8X.

Do the subtraction minus X cubed
takeaway minus X cubed. No ex

cubes minus X squared minus
minus two X squared or the minus

minus A plus, so that
effectively that's minus X

squared +2 X squared just gives
us X squared.

Minus six X minus minus 8X.
Well, that's minus 6X Plus 8X

gives us plus 2X and bring down
the next one.

X squared plus 2X plus a 12 X
squared goes into X squared

once. And so X squared plus
2X plus eight. And again we

can see these two are the
same when I take them away,

I will have nothing left
and so this is my answer.

The result of doing that
division is that.

Well, the one that started
us off on doing this was if

you remember.

X cubed minus one over
X minus one.

This looks a little bit
different, doesn't it? Because

whereas the space between the X
Cube term and the constant term

was filled with all the terms?

This one isn't.

How do we cope with the?

Let's have a look. Remember, we
know what the answer to this one

is already. So what we must do
is right in X cubed and then

leave space for the X squared
term, the X term and then the

constant term. So what I asked
myself is how many times does X

go into XQ, and the answer goes
in X squared. So I write the

answer there where the X squared
term would be.

X squared times by X is X cubed.

X squared times by minus one is
minus X squared.

And subtract X cubed takeaway X
cubed no ex cubes.

0 minus minus X squared is
plus X squared.

Bring down the next term. There
is no next term to bring down.

There's no X to bring down.

So it's as though I got zero X.
There was no point in writing

it. If it's not there, so let's
carry on X in two X squared that

goes X times. So record the X
there above where the X is would

be. Let's do the multiplication
X times by X. Is X squared.

X times Y minus one is minus X.
Do the subtraction X squared

takeaway X squared is nothing.

Nothing takeaway minus
X. It's minus minus X.

That gives us Plus X.

Bring down the next term. We
have got a term here to bring

down it's minus one.

How many times does X going to
X? It goes once.

Long times by XX. One times by
minus one is minus one. Take

them away and we've got nothing
left there and so this is my

answer X squared plus X plus
one, and that's exactly the

answer that we had before. So
where you've got terms missing?

You can still do the same
division. You can still do the

same process, but you just leave
the gaps where the terms would

be and you'll need the gaps
because you're going to have to

write something. Up here in
what's going to be the answer.