WEBVTT

00:00:01.070 --> 00:00:04.850
In simplifying algebraic
fractions, we occasionally need

00:00:04.850 --> 00:00:07.010
a process known as.

00:00:07.710 --> 00:00:14.950
Polynomial.
Division.

00:00:17.110 --> 00:00:22.486
Before we do that, I want to
take you back to something

00:00:22.486 --> 00:00:26.518
that you actually know very
well indeed, and that's

00:00:26.518 --> 00:00:27.862
ordinary long division.

00:00:28.920 --> 00:00:33.886
You know how to do long
division, but I want to go over

00:00:33.886 --> 00:00:38.600
it again. 'cause I want to point
out certain things to you.

00:00:39.250 --> 00:00:43.246
'cause the things that are
important about long division

00:00:43.246 --> 00:00:45.466
are also important in polynomial

00:00:45.466 --> 00:00:52.369
division so. Let's have a look
at a long division. Some

00:00:52.369 --> 00:00:55.975
supposing I want to divide 25.

00:00:56.490 --> 00:01:02.502
Into Let's
say

00:01:02.502 --> 00:01:04.708
2675.

00:01:05.960 --> 00:01:10.107
When I would have to do is look
at 25 in tool 2.

00:01:10.840 --> 00:01:17.080
No way 25 into 26. It goes once
and write the one there.

00:01:17.640 --> 00:01:20.965
Add multiply the one by the 25.

00:01:21.680 --> 00:01:25.688
And subtract and
have one left.

00:01:28.170 --> 00:01:31.943
Hope you remember doing that.
You were probably taught how to

00:01:31.943 --> 00:01:35.716
do that at primary school or the
beginnings of Secondary School.

00:01:35.716 --> 00:01:40.175
Next step is to bring down the
next number, so we bring down

00:01:40.175 --> 00:01:45.320
17. Well, we bring down Seven to
make it 17 and now we say how

00:01:45.320 --> 00:01:47.721
many times does 25 going to 17.

00:01:48.340 --> 00:01:52.302
It doesn't go at all. It's not
enough, so we have to record the

00:01:52.302 --> 00:01:54.283
fact that it doesn't go with a

00:01:54.283 --> 00:02:01.834
0. Next we bring down the
five. So now we've got 175 and

00:02:01.834 --> 00:02:09.492
we say how many times does 25
go into that? And it goes 7

00:02:09.492 --> 00:02:17.150
and we can check that Seven 535,
five down three to carry. 7 twos

00:02:17.150 --> 00:02:20.432
are 14 and three is 17.

00:02:20.440 --> 00:02:24.100
Tracked, we get nothing left.

00:02:24.660 --> 00:02:28.630
So this is our answer. We've
nothing left there, no

00:02:28.630 --> 00:02:31.806
remainder, nothing left over.
And there's our answer.

00:02:32.320 --> 00:02:39.041
2675 divides by 25 and the
answer is 107. They just look at

00:02:39.041 --> 00:02:44.728
what we did. We did 25 into
26 because that went.

00:02:45.820 --> 00:02:52.244
We then recorded that once that
it went there, multiplied, wrote

00:02:52.244 --> 00:02:54.580
the answer and subtracted.

00:02:55.260 --> 00:02:58.038
We brought down the next number.

00:02:58.610 --> 00:03:03.758
Asked how many times 25 went
into it, it didn't go. We

00:03:03.758 --> 00:03:08.477
recorded that and brought down
the next number. Then we said

00:03:08.477 --> 00:03:13.625
how many times does 25 going
to that Seven we did the

00:03:13.625 --> 00:03:17.057
multiplication, wrote it down,
subtracted, got nothing left

00:03:17.057 --> 00:03:18.344
so it finished.

00:03:19.600 --> 00:03:25.294
What we're going to do now is
take that self same process and

00:03:25.294 --> 00:03:27.046
do it with algebra.

00:03:28.160 --> 00:03:31.220
So let us

00:03:31.220 --> 00:03:36.938
take. This
27 X cubed.

00:03:38.070 --> 00:03:41.550
+9 X squared.

00:03:42.460 --> 00:03:47.240
Minus 3X. Minus
10.

00:03:48.340 --> 00:03:49.560
All over.

00:03:50.960 --> 00:03:54.779
3X minus 2.

00:03:55.840 --> 00:04:01.378
We want to divide that into
that. We want to know how many

00:04:01.378 --> 00:04:07.342
times that will fit into there,
so we set it up exactly like a

00:04:07.342 --> 00:04:13.070
long division. Problem by
dividing by this. This is what

00:04:13.070 --> 00:04:19.610
we're dividing into 27 X cubed
plus nine X squared minus three

00:04:19.610 --> 00:04:21.245
X minus 10.

00:04:22.480 --> 00:04:26.704
So we ask ourselves, how many
times does well? How many times

00:04:26.704 --> 00:04:30.576
does that go into that? But
difficult what we ask ourselves

00:04:30.576 --> 00:04:34.448
is how many times does the
excpet go into this bit?

00:04:37.220 --> 00:04:41.796
Just like we asked ourselves how
many times the 25 went into the

00:04:41.796 --> 00:04:43.908
26, how many times does 3X?

00:04:44.590 --> 00:04:50.154
Go into 27 X cubed. The answer
must be 9 X squared because

00:04:50.154 --> 00:04:56.146
Nynex squared times by three X
gives us 27 X cubed and we need

00:04:56.146 --> 00:05:02.138
to record that. But we need to
record it in the right place and

00:05:02.138 --> 00:05:06.418
because these are the X
squared's we record that above

00:05:06.418 --> 00:05:07.702
the X squares.

00:05:08.950 --> 00:05:11.656
So now we do the multiplication.

00:05:12.180 --> 00:05:15.967
Nine X squared times 3X is 27

00:05:15.967 --> 00:05:23.281
X cubed. Nine X squared times
minus two is minus 18 X squared.

00:05:24.600 --> 00:05:28.670
Just like we did for long
division, we now do the

00:05:28.670 --> 00:05:34.368
Subtraction. 27 X cubed
takeaway 27 X cubed none of

00:05:34.368 --> 00:05:39.670
them, because we arrange for
it to be so Nynex squared

00:05:39.670 --> 00:05:44.972
takeaway minus 18 X squared
gives us plus 27 X squared.

00:05:46.300 --> 00:05:52.660
Now we do what we did before we
bring down the next one, so we

00:05:52.660 --> 00:05:54.780
bring down the minus 3X.

00:05:55.360 --> 00:06:00.880
How many times does 3X go into
27 X squared?

00:06:01.570 --> 00:06:09.132
Answer. It goes 9X times and
we write that in the X Column.

00:06:09.132 --> 00:06:16.426
So now we have 9X times 3
X 27 X squared 9X times, Y

00:06:16.426 --> 00:06:19.031
minus 2 - 18 X.

00:06:19.700 --> 00:06:21.468
And we subtract again.

00:06:22.370 --> 00:06:27.854
27 X squared takeaway, 27 X
squared, no X squared, but we

00:06:27.854 --> 00:06:33.795
arrange for it to be like that,
minus three X minus minus 18X.

00:06:33.795 --> 00:06:38.822
Well, that's going to give us
plus 15X altogether, and we

00:06:38.822 --> 00:06:41.107
bring down the minus 10.

00:06:42.940 --> 00:06:49.310
3X into 15X. This time it goes
five times, so we can say plus

00:06:49.310 --> 00:06:53.860
five there. And again it's in
the numbers. The constants

00:06:53.860 --> 00:06:55.680
column at the end.

00:06:56.330 --> 00:07:02.196
Five times by 15 times by three
X gives us 15X. Write it down

00:07:02.196 --> 00:07:08.062
there five times by minus two
gives us minus 10 and we can see

00:07:08.062 --> 00:07:12.671
that when we take these two
away. Got exactly the same

00:07:12.671 --> 00:07:16.442
expression. 15X minus 10
takeaway. 50X minus 10 nothing

00:07:16.442 --> 00:07:21.051
left. So there's our answer,
just as in the long division.

00:07:21.051 --> 00:07:22.727
The answer was there.

00:07:23.390 --> 00:07:29.006
It's there now so we can say
that this expression is equal to

00:07:29.006 --> 00:07:31.166
9 X squared plus 9X.

00:07:31.730 --> 00:07:38.390
Plus 5. Let's
take another one.

00:07:39.200 --> 00:07:42.847
So we'll take X to the 4th.

00:07:43.550 --> 00:07:46.469
Plus X cubed.

00:07:47.340 --> 00:07:54.340
Plus Seven X squared
minus six X +8.

00:07:54.980 --> 00:08:02.420
Divided by all over
X squared, +2 X

00:08:02.420 --> 00:08:07.808
+8. So this is what we're
dividing by and this is what

00:08:07.808 --> 00:08:11.030
we're dividing into is not
immediately obvious what the

00:08:11.030 --> 00:08:15.684
answer is going to be. Let's
have a look X squared plus 2X

00:08:15.684 --> 00:08:16.758
plus 8IN tool.

00:08:17.410 --> 00:08:19.189
All of this.

00:08:22.610 --> 00:08:27.381
Our first question is how many
times does X squared going to X

00:08:27.381 --> 00:08:32.519
to the 4th? We don't need to
worry about the rest, we just do

00:08:32.519 --> 00:08:38.391
it on the first 2 bits in each
one, just as the same as we did

00:08:38.391 --> 00:08:42.795
with the previous example. How
many times X squared going to X

00:08:42.795 --> 00:08:47.566
to the four will it goes X
squared times? So we write it

00:08:47.566 --> 00:08:51.603
there over the X squared's. Now
we do the multiplication X

00:08:51.603 --> 00:08:55.273
squared times. My X squared is X
to the 4th.

00:08:55.430 --> 00:09:03.158
X squared by two X is plus
2X cubed X squared by 8 is

00:09:03.158 --> 00:09:04.814
plus 8X squared.

00:09:07.450 --> 00:09:13.932
And now we do the Subtraction X.
The four takeaway X to the 4th

00:09:13.932 --> 00:09:19.488
there Arnold, but we arranged it
that way. X cubed takeaway 2X

00:09:19.488 --> 00:09:24.118
cubed minus X cubed. Seven X
squared takeaway, 8X squared

00:09:24.118 --> 00:09:28.285
minus X squared and bring down
the next term.

00:09:29.080 --> 00:09:34.064
Now we say how many times does X
squared going to minus X cubed,

00:09:34.064 --> 00:09:39.404
and it must be minus X, and so
we write it in the X Column.

00:09:39.970 --> 00:09:45.305
And above the line there, next
the multiplication minus X times

00:09:45.305 --> 00:09:52.580
by X squared is minus X cubed
minus X times 2X is minus two X

00:09:52.580 --> 00:09:57.430
squared and minus X times by 8
is minus 8X.

00:09:59.630 --> 00:10:04.706
Do the subtraction minus X cubed
takeaway minus X cubed. No ex

00:10:04.706 --> 00:10:09.782
cubes minus X squared minus
minus two X squared or the minus

00:10:09.782 --> 00:10:13.589
minus A plus, so that
effectively that's minus X

00:10:13.589 --> 00:10:17.396
squared +2 X squared just gives
us X squared.

00:10:17.480 --> 00:10:23.924
Minus six X minus minus 8X.
Well, that's minus 6X Plus 8X

00:10:23.924 --> 00:10:29.294
gives us plus 2X and bring down
the next one.

00:10:30.090 --> 00:10:35.550
X squared plus 2X plus a 12 X
squared goes into X squared

00:10:35.550 --> 00:10:40.700
once. And so X squared plus
2X plus eight. And again we

00:10:40.700 --> 00:10:45.140
can see these two are the
same when I take them away,

00:10:45.140 --> 00:10:49.210
I will have nothing left
and so this is my answer.

00:10:50.570 --> 00:10:54.810
The result of doing that
division is that.

00:10:55.600 --> 00:11:01.492
Well, the one that started
us off on doing this was if

00:11:01.492 --> 00:11:02.474
you remember.

00:11:03.600 --> 00:11:09.144
X cubed minus one over
X minus one.

00:11:10.180 --> 00:11:13.519
This looks a little bit
different, doesn't it? Because

00:11:13.519 --> 00:11:17.971
whereas the space between the X
Cube term and the constant term

00:11:17.971 --> 00:11:20.197
was filled with all the terms?

00:11:21.130 --> 00:11:22.048
This one isn't.

00:11:23.210 --> 00:11:24.638
How do we cope with the?

00:11:25.220 --> 00:11:28.899
Let's have a look. Remember, we
know what the answer to this one

00:11:28.899 --> 00:11:35.360
is already. So what we must do
is right in X cubed and then

00:11:35.360 --> 00:11:41.080
leave space for the X squared
term, the X term and then the

00:11:41.080 --> 00:11:47.217
constant term. So what I asked
myself is how many times does X

00:11:47.217 --> 00:11:53.055
go into XQ, and the answer goes
in X squared. So I write the

00:11:53.055 --> 00:11:56.808
answer there where the X squared
term would be.

00:11:57.560 --> 00:12:00.616
X squared times by X is X cubed.

00:12:01.340 --> 00:12:05.700
X squared times by minus one is
minus X squared.

00:12:07.400 --> 00:12:12.390
And subtract X cubed takeaway X
cubed no ex cubes.

00:12:12.990 --> 00:12:18.552
0 minus minus X squared is
plus X squared.

00:12:19.250 --> 00:12:24.372
Bring down the next term. There
is no next term to bring down.

00:12:24.372 --> 00:12:26.736
There's no X to bring down.

00:12:27.250 --> 00:12:33.200
So it's as though I got zero X.
There was no point in writing

00:12:33.200 --> 00:12:39.575
it. If it's not there, so let's
carry on X in two X squared that

00:12:39.575 --> 00:12:45.525
goes X times. So record the X
there above where the X is would

00:12:45.525 --> 00:12:50.625
be. Let's do the multiplication
X times by X. Is X squared.

00:12:51.600 --> 00:12:57.307
X times Y minus one is minus X.
Do the subtraction X squared

00:12:57.307 --> 00:12:59.502
takeaway X squared is nothing.

00:13:00.530 --> 00:13:04.482
Nothing takeaway minus
X. It's minus minus X.

00:13:04.482 --> 00:13:06.952
That gives us Plus X.

00:13:08.000 --> 00:13:12.186
Bring down the next term. We
have got a term here to bring

00:13:12.186 --> 00:13:13.474
down it's minus one.

00:13:14.310 --> 00:13:18.908
How many times does X going to
X? It goes once.

00:13:20.110 --> 00:13:24.868
Long times by XX. One times by
minus one is minus one. Take

00:13:24.868 --> 00:13:29.626
them away and we've got nothing
left there and so this is my

00:13:29.626 --> 00:13:33.652
answer X squared plus X plus
one, and that's exactly the

00:13:33.652 --> 00:13:37.678
answer that we had before. So
where you've got terms missing?

00:13:37.678 --> 00:13:42.070
You can still do the same
division. You can still do the

00:13:42.070 --> 00:13:46.462
same process, but you just leave
the gaps where the terms would

00:13:46.462 --> 00:13:50.854
be and you'll need the gaps
because you're going to have to

00:13:50.854 --> 00:13:54.793
write something. Up here in
what's going to be the answer.