0:00:01.070,0:00:03.842
When we come across fractions,[br]one of the things that we have

0:00:03.842,0:00:07.769
to do is to look at them and see[br]if we can put them in a simple

0:00:07.769,0:00:14.514
form. In fact, see if we can put[br]them in a form that might be

0:00:14.514,0:00:19.566
called the lowest terms. So for[br]instance, if we have a fraction

0:00:19.566,0:00:25.460
12 over 36, then what we want is[br]a fraction in its lowest terms.

0:00:25.460,0:00:31.775
Now 12 will divide into both 12[br]and 36, so we can divide 12 into

0:00:31.775,0:00:37.669
12 one and 12 into 36 three. So[br]that reduces to the fraction 1/3

0:00:37.669,0:00:40.195
the two fractions are the same.

0:00:40.210,0:00:44.000
They are equivalent fractions.[br]Now that's OK with numbers, but

0:00:44.000,0:00:49.306
we want to be able to do it with[br]similar things, but with algebra

0:00:49.306,0:00:53.096
in other similar expressions[br]that have got letters in them.

0:00:53.710,0:00:59.305
And where is here? We look for a[br]number that was in fact a common

0:00:59.305,0:01:03.408
factor. That's a number that[br]will divide into both the top

0:01:03.408,0:01:08.257
and the bottom. What we've now[br]got to look for is a common

0:01:08.257,0:01:11.987
factor. That's an expression[br]that will divide into both the

0:01:11.987,0:01:16.463
top and the bottom. So instead[br]of talking about it, let's have

0:01:16.463,0:01:20.939
a look at some examples. So[br]supposing we've got 3X cubed all

0:01:20.939,0:01:22.804
over X to the 5th.

0:01:23.340,0:01:29.025
We've got to look at is what's[br]the same on the top and on the

0:01:29.025,0:01:33.703
bottom? Well, we've got a 3[br]here, but no numbers down here.

0:01:33.703,0:01:38.548
We've got X is here and we've[br]got X is here and what we can

0:01:38.548,0:01:43.070
see is that we've got X cubed[br]here and X to the fifth here.

0:01:43.710,0:01:49.665
Now X cubed times by X squared[br]gives us X to the 5th, so any

0:01:49.665,0:01:55.620
fact we've got a common factor[br]of X cubed on the top and on the

0:01:55.620,0:02:00.781
bottom. So let's just write that[br]down so we can see it more

0:02:00.781,0:02:05.780
clearly. Thanks for the 5th is X[br]cubed times by X squared? What

0:02:05.780,0:02:11.100
we can see very clearly here is[br]we've got a common factor of X

0:02:11.100,0:02:16.040
cubed, so we can divide top and[br]bottom by X cubed. That leaves

0:02:16.040,0:02:20.980
us with three over X squared.[br]Now we don't need to do this

0:02:20.980,0:02:26.300
middle step every time we have[br]to be able to do is to see

0:02:26.300,0:02:28.200
exactly what that common factor

0:02:28.200,0:02:33.984
is. So let's have a look at some[br]examples and the examples will

0:02:33.984,0:02:37.904
increase in terms of their[br]complexity as we move through

0:02:37.904,0:02:42.608
them in terms of the difficulty.[br]So what's common here to both

0:02:42.608,0:02:48.096
the top and the bottom? On the[br]top? We can see we've got X

0:02:48.096,0:02:53.192
cubed, and on the bottom X[br]squared, and we know that if we

0:02:53.192,0:02:59.072
multiply X squared by X, it will[br]give us X cubed. So in effect we

0:02:59.072,0:03:04.446
can divide. Top and bottom by X[br]squared X squared in two X

0:03:04.446,0:03:09.854
squared goals once and X squared[br]into X cubed leaves us with X

0:03:09.854,0:03:15.262
because X Times X squared gives[br]us X cubed. Similarly, Y into Y

0:03:15.262,0:03:20.670
goes once and Y into Y cubed.[br]Well why times by Y squared

0:03:20.670,0:03:26.910
gives us Y cubed. So if we[br]divide in what we get there is Y

0:03:26.910,0:03:31.070
squared. So on the top we've got[br]X times by.

0:03:31.120,0:03:36.112
Squared on the bottom. We've got[br]one times by one so we don't

0:03:36.112,0:03:40.336
really need to write the one[br]there. And there's our answer.

0:03:41.740,0:03:48.952
You can also involve numbers in[br]this, so we have alot minus

0:03:48.952,0:03:54.962
16 X squared Y squared over[br]4X cubed Y squared.

0:03:56.400,0:04:03.600
Here minus 16 an 4 four[br]goes into minus 16 - 4

0:04:03.600,0:04:09.121
times. X squared on the top and[br]X cubed underneath X squared

0:04:09.121,0:04:14.204
goes into X squared. Once an X[br]squared goes into X cubed X

0:04:14.204,0:04:19.678
times and here Y squared is the[br]same on both top and bottom, so

0:04:19.678,0:04:24.370
Y squared into Y squared goes[br]once and Y squared into Y

0:04:24.370,0:04:25.934
squared ones as well.

0:04:26.530,0:04:33.478
So here we minus 4 *[br]1 * 1 - 4 over.

0:04:34.290,0:04:39.603
14 went into 4 one select[br]remember one times X times

0:04:39.603,0:04:44.433
one that's X. So we have[br]minus four over X.

0:04:46.300,0:04:50.704
Sometimes that common factor may[br]not be obvious. We may have to

0:04:50.704,0:04:56.209
work to see it. We may have to[br]work so that it stands out over

0:04:56.209,0:05:00.613
what we've got. So let's have a[br]look at something like this.

0:05:01.400,0:05:04.922
X squared minus two XY all

0:05:04.922,0:05:08.742
over X. Well, is there a

0:05:08.742,0:05:13.890
common factor? The best way to[br]look at this is not so much. Is

0:05:13.890,0:05:17.855
there a common factor on top and[br]bottom, but is there a common

0:05:17.855,0:05:19.380
factor here? Can we factorize

0:05:19.380,0:05:25.622
this expression? Well, in each[br]term that is at least an X as an

0:05:25.622,0:05:32.278
X here and an X squared here. So[br]we can take that X out as a

0:05:32.278,0:05:36.854
common factor X times by X gives[br]us the X squared.

0:05:37.660,0:05:43.420
Taking X out of here, we're left[br]with minus two Y so that X times

0:05:43.420,0:05:48.412
Y minus two Y would give us[br]minus two XY, and then that's

0:05:48.412,0:05:54.172
all over X. Now we can cancel[br]the X. We can divide the top a

0:05:54.172,0:06:00.316
numerator by X and the bottom by[br]X&X in 2X goes one and X in 2X

0:06:00.316,0:06:04.924
goes once, and So what we're[br]left with is one times that

0:06:04.924,0:06:07.612
divided by one. So we just left.

0:06:07.670,0:06:10.520
With X minus two Y.

0:06:11.470,0:06:16.090
Have a look at another One X to[br]the power 6.

0:06:16.890,0:06:24.330
Minus Seven X to the[br]fifth plus 4X to the

0:06:24.330,0:06:31.256
8th. All over X squared. Again,[br]what we want to do is look at

0:06:31.256,0:06:36.008
this top line and see. Have we[br]got a common factor.

0:06:36.720,0:06:41.114
Well, yes, we have its X to[br]the power five 'cause X the

0:06:41.114,0:06:46.184
power five is included in X to[br]the power 6 and X to the power

0:06:46.184,0:06:51.254
8. So we can take that out as[br]a common factor X to the power

0:06:51.254,0:06:55.310
five times by X will give us X[br]to the power 6.

0:06:57.070,0:07:01.390
Minus Seven X to the power five.[br]We've got the X to the power 5

0:07:01.390,0:07:03.118
outside, so we want minus 7.

0:07:04.310,0:07:07.470
Plus X to the power 8.

0:07:08.030,0:07:13.294
We're taking out X the power[br]5th, so it's going to be 4X to

0:07:13.294,0:07:17.430
the power three. Close the[br]bracket and all over X squared.

0:07:17.430,0:07:21.566
Now we can see what we're[br]looking at is. Is there

0:07:21.566,0:07:25.326
something common now between[br]these two? And clearly we can

0:07:25.326,0:07:30.590
divide X squared by X squared[br]and we can divide X to the power

0:07:30.590,0:07:32.094
5 by X squared.

0:07:32.760,0:07:35.768
So we'll do that X in two X

0:07:35.768,0:07:42.072
squared once. X squared into X[br]to the power five is X cubed.

0:07:42.820,0:07:50.380
And so we end up with X[br]cubed times by X minus 7 +

0:07:50.380,0:07:52.000
4 X cubed.

0:07:52.010,0:07:54.710
We don't worry about the[br]dividing by one. That's not

0:07:54.710,0:07:55.790
going to change anything.

0:07:56.640,0:08:00.160
Notice we don't multiply out the[br]brackets. It's better to keep

0:08:00.160,0:08:04.320
the brackets there. We may want[br]it in that form to work with

0:08:04.320,0:08:11.032
later. What[br]if we

0:08:11.032,0:08:14.840
have something

0:08:14.840,0:08:23.072
like this?[br]No obvious common factor,

0:08:23.072,0:08:26.784
but this is a

0:08:26.784,0:08:31.275
quadratic. And so because it's a[br]quadratic expression, there is

0:08:31.275,0:08:32.575
the possibility of factorizing

0:08:32.575,0:08:36.930
it. If there's the possibility[br]of Factorizing it, then what we

0:08:36.930,0:08:41.760
might find is that one of those[br]factors could be X plus one, in

0:08:41.760,0:08:45.900
which case we could then divide[br]top on bottom by that common

0:08:45.900,0:08:52.070
factor. So leave the top as[br]it is and let's concentrate

0:08:52.070,0:08:57.394
on this bottom. We want to[br]be able to factorize that.

0:08:57.394,0:08:59.330
That means two brackets.

0:09:01.160,0:09:03.240
An X in each bracket.

0:09:03.740,0:09:06.350
That ensures is the X squared.

0:09:07.350,0:09:08.538
We want to have.

0:09:09.250,0:09:13.397
2 as a result of multiplying[br]these two numbers together that

0:09:13.397,0:09:17.544
go in here and here. So we'll[br]have two and one.

0:09:18.340,0:09:23.800
Now we need a plus sign to give[br]us the Plus 3X.

0:09:24.400,0:09:29.594
And what we can see is that X[br]Plus one is a common factor,

0:09:29.594,0:09:33.675
so I'm going to put the[br]brackets around that one on

0:09:33.675,0:09:38.498
top to show us. We've got a[br]common factor and X plus one

0:09:38.498,0:09:42.950
into there goes 1X plus one[br]into. There goes one and so

0:09:42.950,0:09:45.547
we're left with one over X +2.

0:09:47.650,0:09:54.235
You can have one that is if you[br]like the other way up, so let's

0:09:54.235,0:10:00.820
have a look at that X minus 11 A[br]plus 30. Sorry, A squared minus

0:10:00.820,0:10:04.332
11 A plus 30 over a minus 5.

0:10:04.390,0:10:09.034
Again, the A minus five. That's[br]nice. That's OK. Let's keep that

0:10:09.034,0:10:13.678
together with a bracket, but[br]let's have a look at this. A

0:10:13.678,0:10:19.096
squared minus eleven 8 + 30. Can[br]we factorize it? Can we break it

0:10:19.096,0:10:20.644
down into two factors?

0:10:21.400,0:10:26.968
We've got a squared, so we need[br]an A and and a. We now need 2

0:10:26.968,0:10:30.796
numbers that are going to[br]multiply together to give us 30,

0:10:30.796,0:10:34.972
but add together to give us[br]minus eleven. Well, six and five

0:10:34.972,0:10:40.540
seem a good bet for the 30, and[br]if we make a minus six and minus

0:10:40.540,0:10:45.064
five, that ensures the plus 30[br]minus times by A minus. And it

0:10:45.064,0:10:49.240
also ensures the minus 11 a[br]'cause will have minus 6A and

0:10:49.240,0:10:50.284
minus 5A there.

0:10:51.410,0:10:56.912
We've now got a common factor of[br]a minus five, so we can divide

0:10:56.912,0:11:02.807
the top by A minus five on the[br]bottom by A minus five. So we

0:11:02.807,0:11:07.130
end up with just a minus 6, and[br]that's our answer.

0:11:08.060,0:11:14.092
Now that's the proper way to do[br]them. Some of you might be

0:11:14.092,0:11:18.732
tempted. Might be tempted when[br]you see something like this.

0:11:18.770,0:11:25.298
Till forget what it[br]is you're supposed to

0:11:25.298,0:11:30.256
do. So you might suddenly think[br]are lots of threes. I can get

0:11:30.256,0:11:34.000
rid of some three, so let's[br]cancel 3 here and a three there.

0:11:35.080,0:11:40.630
One problem. We said we had[br]to cancel common factors.

0:11:41.850,0:11:47.454
3 doesn't appear here in this[br]term, there's no factor of three

0:11:47.454,0:11:52.124
in this term. We cannot do this[br]kind of canceling.

0:11:52.670,0:11:57.485
Let me just show you why not. If[br]we have a look at a numerical

0:11:57.485,0:12:05.000
example. Supposing we had, let's[br]say 5 + 3 over 3

0:12:05.000,0:12:10.852
+ 1. Now if we do[br]the computation without doing

0:12:10.852,0:12:12.728
the canceling we get.

0:12:13.500,0:12:16.098
8 over 4.

0:12:16.670,0:12:21.080
And that's two, everybody is[br]happy that that's the case.

0:12:21.660,0:12:26.080
But if I do what I did here and[br]suddenly go absolutely bananas

0:12:26.080,0:12:29.820
and cancel the threes, then[br]that's a one there and one

0:12:29.820,0:12:35.516
there. So what I seem to have[br]now is 5 + 1, which is 6 over 1

0:12:35.516,0:12:37.924
+ 1, which is 2 gives me 3.

0:12:38.890,0:12:43.257
Two and three are not the[br]same. Now we've agreed that

0:12:43.257,0:12:46.830
tools the correct answer,[br]where have we gone wrong?

0:12:48.040,0:12:51.538
We cancel these threes divided[br]by these threes, but three is

0:12:51.538,0:12:55.354
not a common factor because it[br]doesn't appear in the Five and

0:12:55.354,0:12:57.262
it doesn't appear in the one.

0:12:58.140,0:13:05.070
So we can't do this over here by[br]the same reasons what we have to

0:13:05.070,0:13:11.538
do is look and see if we can[br]factorize what is on the top

0:13:11.538,0:13:16.620
three X squared plus 10X plus[br]three. Can we factorize it?

0:13:16.620,0:13:22.626
Well, the X +3 is fine, let's[br]keep it all together in a

0:13:22.626,0:13:25.950
bracket. Brackets here.

0:13:26.760,0:13:31.120
Three X squared will need a 3X[br]and an X.

0:13:31.720,0:13:36.494
Will also need a three under one[br]to make up this number here, and

0:13:36.494,0:13:41.268
we've got to get 10 out of it,[br]so we're probably have to have

0:13:41.268,0:13:45.360
the three there to give us Nynex[br]across there under one there.

0:13:46.110,0:13:50.842
Plus signs because these are all[br]plus signs here and now. We can

0:13:50.842,0:13:55.574
see we have got this common[br]factor of X plus three, so we

0:13:55.574,0:14:00.306
can divide the bottom by X +3[br]and the top by X +3.

0:14:00.830,0:14:08.032
That will give us 3X Plus One[br]and that is the correct answer.

0:14:08.630,0:14:15.590
Sometimes we have to work again[br]back a little bit harder, so

0:14:15.590,0:14:18.490
let's take this example 6X

0:14:18.490,0:14:26.247
cubed. Minus Seven X[br]squared minus 5X all

0:14:26.247,0:14:30.011
over 2X plus one.

0:14:30.560,0:14:35.344
That doesn't give me much hope[br]for is here, but there is a

0:14:35.344,0:14:39.760
common factor of X in each[br]term, so perhaps we can take

0:14:39.760,0:14:44.176
that out as a common factor.[br]To begin with. We might find

0:14:44.176,0:14:48.592
we can factorize what's left,[br]so taking that X out as a

0:14:48.592,0:14:52.640
common factor is going to give[br]us six X squared there.

0:14:53.780,0:14:56.760
Minus Seven X there.

0:14:57.290,0:15:01.886
And minus five there closed the[br]bracket and we still to divide

0:15:01.886,0:15:07.248
by the 2X plus one. Also we[br]hope. So now let's have a look

0:15:07.248,0:15:09.929
at this. 'cause this is now an

0:15:09.929,0:15:12.170
ordinary quadratics. X.

0:15:12.670,0:15:20.038
Six X squared? Well, let's have[br]a guess at 3X and 2X.

0:15:20.038,0:15:23.108
After all, there's 2X down

0:15:23.108,0:15:27.981
there. What do we need now?[br]We've got minus five to deal

0:15:27.981,0:15:32.946
with. I want if I can have it 2X[br]plus one. So let's just be

0:15:32.946,0:15:36.918
guided by that. For the moment,[br]let's make that 2X plus one.

0:15:37.560,0:15:43.286
Minus 5 is what I need to[br]multiply by the one to give me

0:15:43.286,0:15:49.413
minus 5. Have I got it right? I[br]need to check on that minus 7X

0:15:49.413,0:15:53.976
will hear I plus 3X and here[br]I've minus 10X and that does

0:15:53.976,0:15:58.539
give me minus 7X. So that's[br]right. Now I need to divide by

0:15:58.539,0:16:02.400
the 2X plus one. Let's get it[br]together in a bracket.

0:16:03.570,0:16:08.619
I can divide top and bottom now[br]by 2X plus one.

0:16:09.210,0:16:11.742
Goes into itself once and once

0:16:11.742,0:16:17.144
there. The answer is just what[br]I'm left with here X times 3X

0:16:17.144,0:16:21.980
minus five, and again I leave it[br]in its factorized form, not try

0:16:21.980,0:16:26.816
to multiply it out again 'cause[br]I may need that form later on.

0:16:27.670,0:16:32.980
Supposing we take[br]something like this

0:16:32.980,0:16:40.060
X cubed minus one[br]over X minus one.

0:16:41.270,0:16:45.072
What now? Doesn't seem[br]to be a common factor

0:16:45.072,0:16:46.702
in X cubed minus one.

0:16:48.430,0:16:55.570
Difficult. But you may[br]know a factorization 4X cubed

0:16:55.570,0:17:02.920
minus one. It does in[br]fact factorize as X minus

0:17:02.920,0:17:05.860
one brackets X squared.

0:17:06.570,0:17:10.138
Plus X plus one.

0:17:10.670,0:17:14.970
And so, because we know that[br]factorization, we can see

0:17:14.970,0:17:18.410
straight away, we can[br]simplify by dividing the

0:17:18.410,0:17:24.000
bottom by X minus one, and[br]that goes in once and the top

0:17:24.000,0:17:29.160
by X minus one. And so we[br]just left with the other

0:17:29.160,0:17:32.170
factor X squared plus X plus[br]one.