WEBVTT 00:00:01.070 --> 00:00:03.842 When we come across fractions, one of the things that we have 00:00:03.842 --> 00:00:07.769 to do is to look at them and see if we can put them in a simple 00:00:07.769 --> 00:00:14.514 form. In fact, see if we can put them in a form that might be 00:00:14.514 --> 00:00:19.566 called the lowest terms. So for instance, if we have a fraction 00:00:19.566 --> 00:00:25.460 12 over 36, then what we want is a fraction in its lowest terms. 00:00:25.460 --> 00:00:31.775 Now 12 will divide into both 12 and 36, so we can divide 12 into 00:00:31.775 --> 00:00:37.669 12 one and 12 into 36 three. So that reduces to the fraction 1/3 00:00:37.669 --> 00:00:40.195 the two fractions are the same. 00:00:40.210 --> 00:00:44.000 They are equivalent fractions. Now that's OK with numbers, but 00:00:44.000 --> 00:00:49.306 we want to be able to do it with similar things, but with algebra 00:00:49.306 --> 00:00:53.096 in other similar expressions that have got letters in them. 00:00:53.710 --> 00:00:59.305 And where is here? We look for a number that was in fact a common 00:00:59.305 --> 00:01:03.408 factor. That's a number that will divide into both the top 00:01:03.408 --> 00:01:08.257 and the bottom. What we've now got to look for is a common 00:01:08.257 --> 00:01:11.987 factor. That's an expression that will divide into both the 00:01:11.987 --> 00:01:16.463 top and the bottom. So instead of talking about it, let's have 00:01:16.463 --> 00:01:20.939 a look at some examples. So supposing we've got 3X cubed all 00:01:20.939 --> 00:01:22.804 over X to the 5th. 00:01:23.340 --> 00:01:29.025 We've got to look at is what's the same on the top and on the 00:01:29.025 --> 00:01:33.703 bottom? Well, we've got a 3 here, but no numbers down here. 00:01:33.703 --> 00:01:38.548 We've got X is here and we've got X is here and what we can 00:01:38.548 --> 00:01:43.070 see is that we've got X cubed here and X to the fifth here. 00:01:43.710 --> 00:01:49.665 Now X cubed times by X squared gives us X to the 5th, so any 00:01:49.665 --> 00:01:55.620 fact we've got a common factor of X cubed on the top and on the 00:01:55.620 --> 00:02:00.781 bottom. So let's just write that down so we can see it more 00:02:00.781 --> 00:02:05.780 clearly. Thanks for the 5th is X cubed times by X squared? What 00:02:05.780 --> 00:02:11.100 we can see very clearly here is we've got a common factor of X 00:02:11.100 --> 00:02:16.040 cubed, so we can divide top and bottom by X cubed. That leaves 00:02:16.040 --> 00:02:20.980 us with three over X squared. Now we don't need to do this 00:02:20.980 --> 00:02:26.300 middle step every time we have to be able to do is to see 00:02:26.300 --> 00:02:28.200 exactly what that common factor 00:02:28.200 --> 00:02:33.984 is. So let's have a look at some examples and the examples will 00:02:33.984 --> 00:02:37.904 increase in terms of their complexity as we move through 00:02:37.904 --> 00:02:42.608 them in terms of the difficulty. So what's common here to both 00:02:42.608 --> 00:02:48.096 the top and the bottom? On the top? We can see we've got X 00:02:48.096 --> 00:02:53.192 cubed, and on the bottom X squared, and we know that if we 00:02:53.192 --> 00:02:59.072 multiply X squared by X, it will give us X cubed. So in effect we 00:02:59.072 --> 00:03:04.446 can divide. Top and bottom by X squared X squared in two X 00:03:04.446 --> 00:03:09.854 squared goals once and X squared into X cubed leaves us with X 00:03:09.854 --> 00:03:15.262 because X Times X squared gives us X cubed. Similarly, Y into Y 00:03:15.262 --> 00:03:20.670 goes once and Y into Y cubed. Well why times by Y squared 00:03:20.670 --> 00:03:26.910 gives us Y cubed. So if we divide in what we get there is Y 00:03:26.910 --> 00:03:31.070 squared. So on the top we've got X times by. 00:03:31.120 --> 00:03:36.112 Squared on the bottom. We've got one times by one so we don't 00:03:36.112 --> 00:03:40.336 really need to write the one there. And there's our answer. 00:03:41.740 --> 00:03:48.952 You can also involve numbers in this, so we have alot minus 00:03:48.952 --> 00:03:54.962 16 X squared Y squared over 4X cubed Y squared. 00:03:56.400 --> 00:04:03.600 Here minus 16 an 4 four goes into minus 16 - 4 00:04:03.600 --> 00:04:09.121 times. X squared on the top and X cubed underneath X squared 00:04:09.121 --> 00:04:14.204 goes into X squared. Once an X squared goes into X cubed X 00:04:14.204 --> 00:04:19.678 times and here Y squared is the same on both top and bottom, so 00:04:19.678 --> 00:04:24.370 Y squared into Y squared goes once and Y squared into Y 00:04:24.370 --> 00:04:25.934 squared ones as well. 00:04:26.530 --> 00:04:33.478 So here we minus 4 * 1 * 1 - 4 over. 00:04:34.290 --> 00:04:39.603 14 went into 4 one select remember one times X times 00:04:39.603 --> 00:04:44.433 one that's X. So we have minus four over X. 00:04:46.300 --> 00:04:50.704 Sometimes that common factor may not be obvious. We may have to 00:04:50.704 --> 00:04:56.209 work to see it. We may have to work so that it stands out over 00:04:56.209 --> 00:05:00.613 what we've got. So let's have a look at something like this. 00:05:01.400 --> 00:05:04.922 X squared minus two XY all 00:05:04.922 --> 00:05:08.742 over X. Well, is there a 00:05:08.742 --> 00:05:13.890 common factor? The best way to look at this is not so much. Is 00:05:13.890 --> 00:05:17.855 there a common factor on top and bottom, but is there a common 00:05:17.855 --> 00:05:19.380 factor here? Can we factorize 00:05:19.380 --> 00:05:25.622 this expression? Well, in each term that is at least an X as an 00:05:25.622 --> 00:05:32.278 X here and an X squared here. So we can take that X out as a 00:05:32.278 --> 00:05:36.854 common factor X times by X gives us the X squared. 00:05:37.660 --> 00:05:43.420 Taking X out of here, we're left with minus two Y so that X times 00:05:43.420 --> 00:05:48.412 Y minus two Y would give us minus two XY, and then that's 00:05:48.412 --> 00:05:54.172 all over X. Now we can cancel the X. We can divide the top a 00:05:54.172 --> 00:06:00.316 numerator by X and the bottom by X&X in 2X goes one and X in 2X 00:06:00.316 --> 00:06:04.924 goes once, and So what we're left with is one times that 00:06:04.924 --> 00:06:07.612 divided by one. So we just left. 00:06:07.670 --> 00:06:10.520 With X minus two Y. 00:06:11.470 --> 00:06:16.090 Have a look at another One X to the power 6. 00:06:16.890 --> 00:06:24.330 Minus Seven X to the fifth plus 4X to the 00:06:24.330 --> 00:06:31.256 8th. All over X squared. Again, what we want to do is look at 00:06:31.256 --> 00:06:36.008 this top line and see. Have we got a common factor. 00:06:36.720 --> 00:06:41.114 Well, yes, we have its X to the power five 'cause X the 00:06:41.114 --> 00:06:46.184 power five is included in X to the power 6 and X to the power 00:06:46.184 --> 00:06:51.254 8. So we can take that out as a common factor X to the power 00:06:51.254 --> 00:06:55.310 five times by X will give us X to the power 6. 00:06:57.070 --> 00:07:01.390 Minus Seven X to the power five. We've got the X to the power 5 00:07:01.390 --> 00:07:03.118 outside, so we want minus 7. 00:07:04.310 --> 00:07:07.470 Plus X to the power 8. 00:07:08.030 --> 00:07:13.294 We're taking out X the power 5th, so it's going to be 4X to 00:07:13.294 --> 00:07:17.430 the power three. Close the bracket and all over X squared. 00:07:17.430 --> 00:07:21.566 Now we can see what we're looking at is. Is there 00:07:21.566 --> 00:07:25.326 something common now between these two? And clearly we can 00:07:25.326 --> 00:07:30.590 divide X squared by X squared and we can divide X to the power 00:07:30.590 --> 00:07:32.094 5 by X squared. 00:07:32.760 --> 00:07:35.768 So we'll do that X in two X 00:07:35.768 --> 00:07:42.072 squared once. X squared into X to the power five is X cubed. 00:07:42.820 --> 00:07:50.380 And so we end up with X cubed times by X minus 7 + 00:07:50.380 --> 00:07:52.000 4 X cubed. 00:07:52.010 --> 00:07:54.710 We don't worry about the dividing by one. That's not 00:07:54.710 --> 00:07:55.790 going to change anything. 00:07:56.640 --> 00:08:00.160 Notice we don't multiply out the brackets. It's better to keep 00:08:00.160 --> 00:08:04.320 the brackets there. We may want it in that form to work with 00:08:04.320 --> 00:08:11.032 later. What if we 00:08:11.032 --> 00:08:14.840 have something 00:08:14.840 --> 00:08:23.072 like this? No obvious common factor, 00:08:23.072 --> 00:08:26.784 but this is a 00:08:26.784 --> 00:08:31.275 quadratic. And so because it's a quadratic expression, there is 00:08:31.275 --> 00:08:32.575 the possibility of factorizing 00:08:32.575 --> 00:08:36.930 it. If there's the possibility of Factorizing it, then what we 00:08:36.930 --> 00:08:41.760 might find is that one of those factors could be X plus one, in 00:08:41.760 --> 00:08:45.900 which case we could then divide top on bottom by that common 00:08:45.900 --> 00:08:52.070 factor. So leave the top as it is and let's concentrate 00:08:52.070 --> 00:08:57.394 on this bottom. We want to be able to factorize that. 00:08:57.394 --> 00:08:59.330 That means two brackets. 00:09:01.160 --> 00:09:03.240 An X in each bracket. 00:09:03.740 --> 00:09:06.350 That ensures is the X squared. 00:09:07.350 --> 00:09:08.538 We want to have. 00:09:09.250 --> 00:09:13.397 2 as a result of multiplying these two numbers together that 00:09:13.397 --> 00:09:17.544 go in here and here. So we'll have two and one. 00:09:18.340 --> 00:09:23.800 Now we need a plus sign to give us the Plus 3X. 00:09:24.400 --> 00:09:29.594 And what we can see is that X Plus one is a common factor, 00:09:29.594 --> 00:09:33.675 so I'm going to put the brackets around that one on 00:09:33.675 --> 00:09:38.498 top to show us. We've got a common factor and X plus one 00:09:38.498 --> 00:09:42.950 into there goes 1X plus one into. There goes one and so 00:09:42.950 --> 00:09:45.547 we're left with one over X +2. 00:09:47.650 --> 00:09:54.235 You can have one that is if you like the other way up, so let's 00:09:54.235 --> 00:10:00.820 have a look at that X minus 11 A plus 30. Sorry, A squared minus 00:10:00.820 --> 00:10:04.332 11 A plus 30 over a minus 5. 00:10:04.390 --> 00:10:09.034 Again, the A minus five. That's nice. That's OK. Let's keep that 00:10:09.034 --> 00:10:13.678 together with a bracket, but let's have a look at this. A 00:10:13.678 --> 00:10:19.096 squared minus eleven 8 + 30. Can we factorize it? Can we break it 00:10:19.096 --> 00:10:20.644 down into two factors? 00:10:21.400 --> 00:10:26.968 We've got a squared, so we need an A and and a. We now need 2 00:10:26.968 --> 00:10:30.796 numbers that are going to multiply together to give us 30, 00:10:30.796 --> 00:10:34.972 but add together to give us minus eleven. Well, six and five 00:10:34.972 --> 00:10:40.540 seem a good bet for the 30, and if we make a minus six and minus 00:10:40.540 --> 00:10:45.064 five, that ensures the plus 30 minus times by A minus. And it 00:10:45.064 --> 00:10:49.240 also ensures the minus 11 a 'cause will have minus 6A and 00:10:49.240 --> 00:10:50.284 minus 5A there. 00:10:51.410 --> 00:10:56.912 We've now got a common factor of a minus five, so we can divide 00:10:56.912 --> 00:11:02.807 the top by A minus five on the bottom by A minus five. So we 00:11:02.807 --> 00:11:07.130 end up with just a minus 6, and that's our answer. 00:11:08.060 --> 00:11:14.092 Now that's the proper way to do them. Some of you might be 00:11:14.092 --> 00:11:18.732 tempted. Might be tempted when you see something like this. 00:11:18.770 --> 00:11:25.298 Till forget what it is you're supposed to 00:11:25.298 --> 00:11:30.256 do. So you might suddenly think are lots of threes. I can get 00:11:30.256 --> 00:11:34.000 rid of some three, so let's cancel 3 here and a three there. 00:11:35.080 --> 00:11:40.630 One problem. We said we had to cancel common factors. 00:11:41.850 --> 00:11:47.454 3 doesn't appear here in this term, there's no factor of three 00:11:47.454 --> 00:11:52.124 in this term. We cannot do this kind of canceling. 00:11:52.670 --> 00:11:57.485 Let me just show you why not. If we have a look at a numerical 00:11:57.485 --> 00:12:05.000 example. Supposing we had, let's say 5 + 3 over 3 00:12:05.000 --> 00:12:10.852 + 1. Now if we do the computation without doing 00:12:10.852 --> 00:12:12.728 the canceling we get. 00:12:13.500 --> 00:12:16.098 8 over 4. 00:12:16.670 --> 00:12:21.080 And that's two, everybody is happy that that's the case. 00:12:21.660 --> 00:12:26.080 But if I do what I did here and suddenly go absolutely bananas 00:12:26.080 --> 00:12:29.820 and cancel the threes, then that's a one there and one 00:12:29.820 --> 00:12:35.516 there. So what I seem to have now is 5 + 1, which is 6 over 1 00:12:35.516 --> 00:12:37.924 + 1, which is 2 gives me 3. 00:12:38.890 --> 00:12:43.257 Two and three are not the same. Now we've agreed that 00:12:43.257 --> 00:12:46.830 tools the correct answer, where have we gone wrong? 00:12:48.040 --> 00:12:51.538 We cancel these threes divided by these threes, but three is 00:12:51.538 --> 00:12:55.354 not a common factor because it doesn't appear in the Five and 00:12:55.354 --> 00:12:57.262 it doesn't appear in the one. 00:12:58.140 --> 00:13:05.070 So we can't do this over here by the same reasons what we have to 00:13:05.070 --> 00:13:11.538 do is look and see if we can factorize what is on the top 00:13:11.538 --> 00:13:16.620 three X squared plus 10X plus three. Can we factorize it? 00:13:16.620 --> 00:13:22.626 Well, the X +3 is fine, let's keep it all together in a 00:13:22.626 --> 00:13:25.950 bracket. Brackets here. 00:13:26.760 --> 00:13:31.120 Three X squared will need a 3X and an X. 00:13:31.720 --> 00:13:36.494 Will also need a three under one to make up this number here, and 00:13:36.494 --> 00:13:41.268 we've got to get 10 out of it, so we're probably have to have 00:13:41.268 --> 00:13:45.360 the three there to give us Nynex across there under one there. 00:13:46.110 --> 00:13:50.842 Plus signs because these are all plus signs here and now. We can 00:13:50.842 --> 00:13:55.574 see we have got this common factor of X plus three, so we 00:13:55.574 --> 00:14:00.306 can divide the bottom by X +3 and the top by X +3. 00:14:00.830 --> 00:14:08.032 That will give us 3X Plus One and that is the correct answer. 00:14:08.630 --> 00:14:15.590 Sometimes we have to work again back a little bit harder, so 00:14:15.590 --> 00:14:18.490 let's take this example 6X 00:14:18.490 --> 00:14:26.247 cubed. Minus Seven X squared minus 5X all 00:14:26.247 --> 00:14:30.011 over 2X plus one. 00:14:30.560 --> 00:14:35.344 That doesn't give me much hope for is here, but there is a 00:14:35.344 --> 00:14:39.760 common factor of X in each term, so perhaps we can take 00:14:39.760 --> 00:14:44.176 that out as a common factor. To begin with. We might find 00:14:44.176 --> 00:14:48.592 we can factorize what's left, so taking that X out as a 00:14:48.592 --> 00:14:52.640 common factor is going to give us six X squared there. 00:14:53.780 --> 00:14:56.760 Minus Seven X there. 00:14:57.290 --> 00:15:01.886 And minus five there closed the bracket and we still to divide 00:15:01.886 --> 00:15:07.248 by the 2X plus one. Also we hope. So now let's have a look 00:15:07.248 --> 00:15:09.929 at this. 'cause this is now an 00:15:09.929 --> 00:15:12.170 ordinary quadratics. X. 00:15:12.670 --> 00:15:20.038 Six X squared? Well, let's have a guess at 3X and 2X. 00:15:20.038 --> 00:15:23.108 After all, there's 2X down 00:15:23.108 --> 00:15:27.981 there. What do we need now? We've got minus five to deal 00:15:27.981 --> 00:15:32.946 with. I want if I can have it 2X plus one. So let's just be 00:15:32.946 --> 00:15:36.918 guided by that. For the moment, let's make that 2X plus one. 00:15:37.560 --> 00:15:43.286 Minus 5 is what I need to multiply by the one to give me 00:15:43.286 --> 00:15:49.413 minus 5. Have I got it right? I need to check on that minus 7X 00:15:49.413 --> 00:15:53.976 will hear I plus 3X and here I've minus 10X and that does 00:15:53.976 --> 00:15:58.539 give me minus 7X. So that's right. Now I need to divide by 00:15:58.539 --> 00:16:02.400 the 2X plus one. Let's get it together in a bracket. 00:16:03.570 --> 00:16:08.619 I can divide top and bottom now by 2X plus one. 00:16:09.210 --> 00:16:11.742 Goes into itself once and once 00:16:11.742 --> 00:16:17.144 there. The answer is just what I'm left with here X times 3X 00:16:17.144 --> 00:16:21.980 minus five, and again I leave it in its factorized form, not try 00:16:21.980 --> 00:16:26.816 to multiply it out again 'cause I may need that form later on. 00:16:27.670 --> 00:16:32.980 Supposing we take something like this 00:16:32.980 --> 00:16:40.060 X cubed minus one over X minus one. 00:16:41.270 --> 00:16:45.072 What now? Doesn't seem to be a common factor 00:16:45.072 --> 00:16:46.702 in X cubed minus one. 00:16:48.430 --> 00:16:55.570 Difficult. But you may know a factorization 4X cubed 00:16:55.570 --> 00:17:02.920 minus one. It does in fact factorize as X minus 00:17:02.920 --> 00:17:05.860 one brackets X squared. 00:17:06.570 --> 00:17:10.138 Plus X plus one. 00:17:10.670 --> 00:17:14.970 And so, because we know that factorization, we can see 00:17:14.970 --> 00:17:18.410 straight away, we can simplify by dividing the 00:17:18.410 --> 00:17:24.000 bottom by X minus one, and that goes in once and the top 00:17:24.000 --> 00:17:29.160 by X minus one. And so we just left with the other 00:17:29.160 --> 00:17:32.170 factor X squared plus X plus one.