Return to Video

## www.mathcentre.ac.uk/.../polynomial%20division.mp4

• 0:01 - 0:05
In simplifying algebraic
fractions, we occasionally need
• 0:05 - 0:07
a process known as.
• 0:08 - 0:15
Polynomial.
Division.
• 0:17 - 0:22
Before we do that, I want to
take you back to something
• 0:22 - 0:27
that you actually know very
well indeed, and that's
• 0:27 - 0:28
ordinary long division.
• 0:29 - 0:34
You know how to do long
division, but I want to go over
• 0:34 - 0:39
it again. 'cause I want to point
out certain things to you.
• 0:39 - 0:43
'cause the things that are
important about long division
• 0:43 - 0:45
are also important in polynomial
• 0:45 - 0:52
division so. Let's have a look
at a long division. Some
• 0:52 - 0:56
supposing I want to divide 25.
• 0:56 - 1:03
Into Let's
say
• 1:03 - 1:05
2675.
• 1:06 - 1:10
When I would have to do is look
at 25 in tool 2.
• 1:11 - 1:17
No way 25 into 26. It goes once
and write the one there.
• 1:18 - 1:21
Add multiply the one by the 25.
• 1:22 - 1:26
And subtract and
have one left.
• 1:28 - 1:32
Hope you remember doing that.
You were probably taught how to
• 1:32 - 1:36
do that at primary school or the
beginnings of Secondary School.
• 1:36 - 1:40
Next step is to bring down the
next number, so we bring down
• 1:40 - 1:45
17. Well, we bring down Seven to
make it 17 and now we say how
• 1:45 - 1:48
many times does 25 going to 17.
• 1:48 - 1:52
It doesn't go at all. It's not
enough, so we have to record the
• 1:52 - 1:54
fact that it doesn't go with a
• 1:54 - 2:02
0. Next we bring down the
five. So now we've got 175 and
• 2:02 - 2:09
we say how many times does 25
go into that? And it goes 7
• 2:09 - 2:17
and we can check that Seven 535,
five down three to carry. 7 twos
• 2:17 - 2:20
are 14 and three is 17.
• 2:20 - 2:24
Tracked, we get nothing left.
• 2:25 - 2:29
So this is our answer. We've
nothing left there, no
• 2:29 - 2:32
remainder, nothing left over.
And there's our answer.
• 2:32 - 2:39
2675 divides by 25 and the
answer is 107. They just look at
• 2:39 - 2:45
what we did. We did 25 into
26 because that went.
• 2:46 - 2:52
We then recorded that once that
it went there, multiplied, wrote
• 2:52 - 2:55
the answer and subtracted.
• 2:55 - 2:58
We brought down the next number.
• 2:59 - 3:04
Asked how many times 25 went
into it, it didn't go. We
• 3:04 - 3:08
recorded that and brought down
the next number. Then we said
• 3:08 - 3:14
how many times does 25 going
to that Seven we did the
• 3:14 - 3:17
multiplication, wrote it down,
subtracted, got nothing left
• 3:17 - 3:18
so it finished.
• 3:20 - 3:25
What we're going to do now is
take that self same process and
• 3:25 - 3:27
do it with algebra.
• 3:28 - 3:31
So let us
• 3:31 - 3:37
take. This
27 X cubed.
• 3:38 - 3:42
+9 X squared.
• 3:42 - 3:47
Minus 3X. Minus
10.
• 3:48 - 3:50
All over.
• 3:51 - 3:55
3X minus 2.
• 3:56 - 4:01
We want to divide that into
that. We want to know how many
• 4:01 - 4:07
times that will fit into there,
so we set it up exactly like a
• 4:07 - 4:13
long division. Problem by
dividing by this. This is what
• 4:13 - 4:20
we're dividing into 27 X cubed
plus nine X squared minus three
• 4:20 - 4:21
X minus 10.
• 4:22 - 4:27
So we ask ourselves, how many
times does well? How many times
• 4:27 - 4:31
does that go into that? But
difficult what we ask ourselves
• 4:31 - 4:34
is how many times does the
excpet go into this bit?
• 4:37 - 4:42
Just like we asked ourselves how
many times the 25 went into the
• 4:42 - 4:44
26, how many times does 3X?
• 4:45 - 4:50
Go into 27 X cubed. The answer
must be 9 X squared because
• 4:50 - 4:56
Nynex squared times by three X
gives us 27 X cubed and we need
• 4:56 - 5:02
to record that. But we need to
record it in the right place and
• 5:02 - 5:06
because these are the X
squared's we record that above
• 5:06 - 5:08
the X squares.
• 5:09 - 5:12
So now we do the multiplication.
• 5:12 - 5:16
Nine X squared times 3X is 27
• 5:16 - 5:23
X cubed. Nine X squared times
minus two is minus 18 X squared.
• 5:25 - 5:29
Just like we did for long
division, we now do the
• 5:29 - 5:34
Subtraction. 27 X cubed
takeaway 27 X cubed none of
• 5:34 - 5:40
them, because we arrange for
it to be so Nynex squared
• 5:40 - 5:45
takeaway minus 18 X squared
gives us plus 27 X squared.
• 5:46 - 5:53
Now we do what we did before we
bring down the next one, so we
• 5:53 - 5:55
bring down the minus 3X.
• 5:55 - 6:01
How many times does 3X go into
27 X squared?
• 6:02 - 6:09
Answer. It goes 9X times and
we write that in the X Column.
• 6:09 - 6:16
So now we have 9X times 3
X 27 X squared 9X times, Y
• 6:16 - 6:19
minus 2 - 18 X.
• 6:20 - 6:21
And we subtract again.
• 6:22 - 6:28
27 X squared takeaway, 27 X
squared, no X squared, but we
• 6:28 - 6:34
arrange for it to be like that,
minus three X minus minus 18X.
• 6:34 - 6:39
Well, that's going to give us
plus 15X altogether, and we
• 6:39 - 6:41
bring down the minus 10.
• 6:43 - 6:49
3X into 15X. This time it goes
five times, so we can say plus
• 6:49 - 6:54
five there. And again it's in
the numbers. The constants
• 6:54 - 6:56
column at the end.
• 6:56 - 7:02
Five times by 15 times by three
X gives us 15X. Write it down
• 7:02 - 7:08
there five times by minus two
gives us minus 10 and we can see
• 7:08 - 7:13
that when we take these two
away. Got exactly the same
• 7:13 - 7:16
expression. 15X minus 10
takeaway. 50X minus 10 nothing
• 7:16 - 7:21
left. So there's our answer,
just as in the long division.
• 7:21 - 7:23
The answer was there.
• 7:23 - 7:29
It's there now so we can say
that this expression is equal to
• 7:29 - 7:31
9 X squared plus 9X.
• 7:32 - 7:38
Plus 5. Let's
take another one.
• 7:39 - 7:43
So we'll take X to the 4th.
• 7:44 - 7:46
Plus X cubed.
• 7:47 - 7:54
Plus Seven X squared
minus six X +8.
• 7:55 - 8:02
Divided by all over
X squared, +2 X
• 8:02 - 8:08
+8. So this is what we're
dividing by and this is what
• 8:08 - 8:11
we're dividing into is not
immediately obvious what the
• 8:11 - 8:16
answer is going to be. Let's
have a look X squared plus 2X
• 8:16 - 8:17
plus 8IN tool.
• 8:17 - 8:19
All of this.
• 8:23 - 8:27
Our first question is how many
times does X squared going to X
• 8:27 - 8:33
to the 4th? We don't need to
worry about the rest, we just do
• 8:33 - 8:38
it on the first 2 bits in each
one, just as the same as we did
• 8:38 - 8:43
with the previous example. How
many times X squared going to X
• 8:43 - 8:48
to the four will it goes X
squared times? So we write it
• 8:48 - 8:52
there over the X squared's. Now
we do the multiplication X
• 8:52 - 8:55
squared times. My X squared is X
to the 4th.
• 8:55 - 9:03
X squared by two X is plus
2X cubed X squared by 8 is
• 9:03 - 9:05
plus 8X squared.
• 9:07 - 9:14
And now we do the Subtraction X.
The four takeaway X to the 4th
• 9:14 - 9:19
there Arnold, but we arranged it
that way. X cubed takeaway 2X
• 9:19 - 9:24
cubed minus X cubed. Seven X
squared takeaway, 8X squared
• 9:24 - 9:28
minus X squared and bring down
the next term.
• 9:29 - 9:34
Now we say how many times does X
squared going to minus X cubed,
• 9:34 - 9:39
and it must be minus X, and so
we write it in the X Column.
• 9:40 - 9:45
And above the line there, next
the multiplication minus X times
• 9:45 - 9:53
by X squared is minus X cubed
minus X times 2X is minus two X
• 9:53 - 9:57
squared and minus X times by 8
is minus 8X.
• 10:00 - 10:05
Do the subtraction minus X cubed
takeaway minus X cubed. No ex
• 10:05 - 10:10
cubes minus X squared minus
minus two X squared or the minus
• 10:10 - 10:14
minus A plus, so that
effectively that's minus X
• 10:14 - 10:17
squared +2 X squared just gives
us X squared.
• 10:17 - 10:24
Minus six X minus minus 8X.
Well, that's minus 6X Plus 8X
• 10:24 - 10:29
gives us plus 2X and bring down
the next one.
• 10:30 - 10:36
X squared plus 2X plus a 12 X
squared goes into X squared
• 10:36 - 10:41
once. And so X squared plus
2X plus eight. And again we
• 10:41 - 10:45
can see these two are the
same when I take them away,
• 10:45 - 10:49
I will have nothing left
and so this is my answer.
• 10:51 - 10:55
The result of doing that
division is that.
• 10:56 - 11:01
Well, the one that started
us off on doing this was if
• 11:01 - 11:02
you remember.
• 11:04 - 11:09
X cubed minus one over
X minus one.
• 11:10 - 11:14
This looks a little bit
different, doesn't it? Because
• 11:14 - 11:18
whereas the space between the X
Cube term and the constant term
• 11:18 - 11:20
was filled with all the terms?
• 11:21 - 11:22
This one isn't.
• 11:23 - 11:25
How do we cope with the?
• 11:25 - 11:29
Let's have a look. Remember, we
know what the answer to this one
• 11:29 - 11:35
is already. So what we must do
is right in X cubed and then
• 11:35 - 11:41
leave space for the X squared
term, the X term and then the
• 11:41 - 11:47
constant term. So what I asked
myself is how many times does X
• 11:47 - 11:53
go into XQ, and the answer goes
in X squared. So I write the
• 11:53 - 11:57
answer there where the X squared
term would be.
• 11:58 - 12:01
X squared times by X is X cubed.
• 12:01 - 12:06
X squared times by minus one is
minus X squared.
• 12:07 - 12:12
And subtract X cubed takeaway X
cubed no ex cubes.
• 12:13 - 12:19
0 minus minus X squared is
plus X squared.
• 12:19 - 12:24
Bring down the next term. There
is no next term to bring down.
• 12:24 - 12:27
There's no X to bring down.
• 12:27 - 12:33
So it's as though I got zero X.
There was no point in writing
• 12:33 - 12:40
it. If it's not there, so let's
carry on X in two X squared that
• 12:40 - 12:46
goes X times. So record the X
there above where the X is would
• 12:46 - 12:51
be. Let's do the multiplication
X times by X. Is X squared.
• 12:52 - 12:57
X times Y minus one is minus X.
Do the subtraction X squared
• 12:57 - 13:00
takeaway X squared is nothing.
• 13:01 - 13:04
Nothing takeaway minus
X. It's minus minus X.
• 13:04 - 13:07
That gives us Plus X.
• 13:08 - 13:12
Bring down the next term. We
have got a term here to bring
• 13:12 - 13:13
down it's minus one.
• 13:14 - 13:19
How many times does X going to
X? It goes once.
• 13:20 - 13:25
Long times by XX. One times by
minus one is minus one. Take
• 13:25 - 13:30
them away and we've got nothing
left there and so this is my
• 13:30 - 13:34
answer X squared plus X plus
one, and that's exactly the
• 13:34 - 13:38
answer that we had before. So
where you've got terms missing?
• 13:38 - 13:42
You can still do the same
division. You can still do the
• 13:42 - 13:46
same process, but you just leave
the gaps where the terms would
• 13:46 - 13:51
be and you'll need the gaps
because you're going to have to
• 13:51 - 13:55
write something. Up here in
what's going to be the answer.
Title:
www.mathcentre.ac.uk/.../polynomial%20division.mp4
Video Language:
English

• Uploaded
mathcentre