1 00:00:01,070 --> 00:00:03,842 When we come across fractions, one of the things that we have 2 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 3 00:00:07,769 --> 00:00:14,514 form. In fact, see if we can put them in a form that might be 4 00:00:14,514 --> 00:00:19,566 called the lowest terms. So for instance, if we have a fraction 5 00:00:19,566 --> 00:00:25,460 12 over 36, then what we want is a fraction in its lowest terms. 6 00:00:25,460 --> 00:00:31,775 Now 12 will divide into both 12 and 36, so we can divide 12 into 7 00:00:31,775 --> 00:00:37,669 12 one and 12 into 36 three. So that reduces to the fraction 1/3 8 00:00:37,669 --> 00:00:40,195 the two fractions are the same. 9 00:00:40,210 --> 00:00:44,000 They are equivalent fractions. Now that's OK with numbers, but 10 00:00:44,000 --> 00:00:49,306 we want to be able to do it with similar things, but with algebra 11 00:00:49,306 --> 00:00:53,096 in other similar expressions that have got letters in them. 12 00:00:53,710 --> 00:00:59,305 And where is here? We look for a number that was in fact a common 13 00:00:59,305 --> 00:01:03,408 factor. That's a number that will divide into both the top 14 00:01:03,408 --> 00:01:08,257 and the bottom. What we've now got to look for is a common 15 00:01:08,257 --> 00:01:11,987 factor. That's an expression that will divide into both the 16 00:01:11,987 --> 00:01:16,463 top and the bottom. So instead of talking about it, let's have 17 00:01:16,463 --> 00:01:20,939 a look at some examples. So supposing we've got 3X cubed all 18 00:01:20,939 --> 00:01:22,804 over X to the 5th. 19 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 20 00:01:29,025 --> 00:01:33,703 bottom? Well, we've got a 3 here, but no numbers down here. 21 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 22 00:01:38,548 --> 00:01:43,070 see is that we've got X cubed here and X to the fifth here. 23 00:01:43,710 --> 00:01:49,665 Now X cubed times by X squared gives us X to the 5th, so any 24 00:01:49,665 --> 00:01:55,620 fact we've got a common factor of X cubed on the top and on the 25 00:01:55,620 --> 00:02:00,781 bottom. So let's just write that down so we can see it more 26 00:02:00,781 --> 00:02:05,780 clearly. Thanks for the 5th is X cubed times by X squared? What 27 00:02:05,780 --> 00:02:11,100 we can see very clearly here is we've got a common factor of X 28 00:02:11,100 --> 00:02:16,040 cubed, so we can divide top and bottom by X cubed. That leaves 29 00:02:16,040 --> 00:02:20,980 us with three over X squared. Now we don't need to do this 30 00:02:20,980 --> 00:02:26,300 middle step every time we have to be able to do is to see 31 00:02:26,300 --> 00:02:28,200 exactly what that common factor 32 00:02:28,200 --> 00:02:33,984 is. So let's have a look at some examples and the examples will 33 00:02:33,984 --> 00:02:37,904 increase in terms of their complexity as we move through 34 00:02:37,904 --> 00:02:42,608 them in terms of the difficulty. So what's common here to both 35 00:02:42,608 --> 00:02:48,096 the top and the bottom? On the top? We can see we've got X 36 00:02:48,096 --> 00:02:53,192 cubed, and on the bottom X squared, and we know that if we 37 00:02:53,192 --> 00:02:59,072 multiply X squared by X, it will give us X cubed. So in effect we 38 00:02:59,072 --> 00:03:04,446 can divide. Top and bottom by X squared X squared in two X 39 00:03:04,446 --> 00:03:09,854 squared goals once and X squared into X cubed leaves us with X 40 00:03:09,854 --> 00:03:15,262 because X Times X squared gives us X cubed. Similarly, Y into Y 41 00:03:15,262 --> 00:03:20,670 goes once and Y into Y cubed. Well why times by Y squared 42 00:03:20,670 --> 00:03:26,910 gives us Y cubed. So if we divide in what we get there is Y 43 00:03:26,910 --> 00:03:31,070 squared. So on the top we've got X times by. 44 00:03:31,120 --> 00:03:36,112 Squared on the bottom. We've got one times by one so we don't 45 00:03:36,112 --> 00:03:40,336 really need to write the one there. And there's our answer. 46 00:03:41,740 --> 00:03:48,952 You can also involve numbers in this, so we have alot minus 47 00:03:48,952 --> 00:03:54,962 16 X squared Y squared over 4X cubed Y squared. 48 00:03:56,400 --> 00:04:03,600 Here minus 16 an 4 four goes into minus 16 - 4 49 00:04:03,600 --> 00:04:09,121 times. X squared on the top and X cubed underneath X squared 50 00:04:09,121 --> 00:04:14,204 goes into X squared. Once an X squared goes into X cubed X 51 00:04:14,204 --> 00:04:19,678 times and here Y squared is the same on both top and bottom, so 52 00:04:19,678 --> 00:04:24,370 Y squared into Y squared goes once and Y squared into Y 53 00:04:24,370 --> 00:04:25,934 squared ones as well. 54 00:04:26,530 --> 00:04:33,478 So here we minus 4 * 1 * 1 - 4 over. 55 00:04:34,290 --> 00:04:39,603 14 went into 4 one select remember one times X times 56 00:04:39,603 --> 00:04:44,433 one that's X. So we have minus four over X. 57 00:04:46,300 --> 00:04:50,704 Sometimes that common factor may not be obvious. We may have to 58 00:04:50,704 --> 00:04:56,209 work to see it. We may have to work so that it stands out over 59 00:04:56,209 --> 00:05:00,613 what we've got. So let's have a look at something like this. 60 00:05:01,400 --> 00:05:04,922 X squared minus two XY all 61 00:05:04,922 --> 00:05:08,742 over X. Well, is there a 62 00:05:08,742 --> 00:05:13,890 common factor? The best way to look at this is not so much. Is 63 00:05:13,890 --> 00:05:17,855 there a common factor on top and bottom, but is there a common 64 00:05:17,855 --> 00:05:19,380 factor here? Can we factorize 65 00:05:19,380 --> 00:05:25,622 this expression? Well, in each term that is at least an X as an 66 00:05:25,622 --> 00:05:32,278 X here and an X squared here. So we can take that X out as a 67 00:05:32,278 --> 00:05:36,854 common factor X times by X gives us the X squared. 68 00:05:37,660 --> 00:05:43,420 Taking X out of here, we're left with minus two Y so that X times 69 00:05:43,420 --> 00:05:48,412 Y minus two Y would give us minus two XY, and then that's 70 00:05:48,412 --> 00:05:54,172 all over X. Now we can cancel the X. We can divide the top a 71 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 72 00:06:00,316 --> 00:06:04,924 goes once, and So what we're left with is one times that 73 00:06:04,924 --> 00:06:07,612 divided by one. So we just left. 74 00:06:07,670 --> 00:06:10,520 With X minus two Y. 75 00:06:11,470 --> 00:06:16,090 Have a look at another One X to the power 6. 76 00:06:16,890 --> 00:06:24,330 Minus Seven X to the fifth plus 4X to the 77 00:06:24,330 --> 00:06:31,256 8th. All over X squared. Again, what we want to do is look at 78 00:06:31,256 --> 00:06:36,008 this top line and see. Have we got a common factor. 79 00:06:36,720 --> 00:06:41,114 Well, yes, we have its X to the power five 'cause X the 80 00:06:41,114 --> 00:06:46,184 power five is included in X to the power 6 and X to the power 81 00:06:46,184 --> 00:06:51,254 8. So we can take that out as a common factor X to the power 82 00:06:51,254 --> 00:06:55,310 five times by X will give us X to the power 6. 83 00:06:57,070 --> 00:07:01,390 Minus Seven X to the power five. We've got the X to the power 5 84 00:07:01,390 --> 00:07:03,118 outside, so we want minus 7. 85 00:07:04,310 --> 00:07:07,470 Plus X to the power 8. 86 00:07:08,030 --> 00:07:13,294 We're taking out X the power 5th, so it's going to be 4X to 87 00:07:13,294 --> 00:07:17,430 the power three. Close the bracket and all over X squared. 88 00:07:17,430 --> 00:07:21,566 Now we can see what we're looking at is. Is there 89 00:07:21,566 --> 00:07:25,326 something common now between these two? And clearly we can 90 00:07:25,326 --> 00:07:30,590 divide X squared by X squared and we can divide X to the power 91 00:07:30,590 --> 00:07:32,094 5 by X squared. 92 00:07:32,760 --> 00:07:35,768 So we'll do that X in two X 93 00:07:35,768 --> 00:07:42,072 squared once. X squared into X to the power five is X cubed. 94 00:07:42,820 --> 00:07:50,380 And so we end up with X cubed times by X minus 7 + 95 00:07:50,380 --> 00:07:52,000 4 X cubed. 96 00:07:52,010 --> 00:07:54,710 We don't worry about the dividing by one. That's not 97 00:07:54,710 --> 00:07:55,790 going to change anything. 98 00:07:56,640 --> 00:08:00,160 Notice we don't multiply out the brackets. It's better to keep 99 00:08:00,160 --> 00:08:04,320 the brackets there. We may want it in that form to work with 100 00:08:04,320 --> 00:08:11,032 later. What if we 101 00:08:11,032 --> 00:08:14,840 have something 102 00:08:14,840 --> 00:08:23,072 like this? No obvious common factor, 103 00:08:23,072 --> 00:08:26,784 but this is a 104 00:08:26,784 --> 00:08:31,275 quadratic. And so because it's a quadratic expression, there is 105 00:08:31,275 --> 00:08:32,575 the possibility of factorizing 106 00:08:32,575 --> 00:08:36,930 it. If there's the possibility of Factorizing it, then what we 107 00:08:36,930 --> 00:08:41,760 might find is that one of those factors could be X plus one, in 108 00:08:41,760 --> 00:08:45,900 which case we could then divide top on bottom by that common 109 00:08:45,900 --> 00:08:52,070 factor. So leave the top as it is and let's concentrate 110 00:08:52,070 --> 00:08:57,394 on this bottom. We want to be able to factorize that. 111 00:08:57,394 --> 00:08:59,330 That means two brackets. 112 00:09:01,160 --> 00:09:03,240 An X in each bracket. 113 00:09:03,740 --> 00:09:06,350 That ensures is the X squared. 114 00:09:07,350 --> 00:09:08,538 We want to have. 115 00:09:09,250 --> 00:09:13,397 2 as a result of multiplying these two numbers together that 116 00:09:13,397 --> 00:09:17,544 go in here and here. So we'll have two and one. 117 00:09:18,340 --> 00:09:23,800 Now we need a plus sign to give us the Plus 3X. 118 00:09:24,400 --> 00:09:29,594 And what we can see is that X Plus one is a common factor, 119 00:09:29,594 --> 00:09:33,675 so I'm going to put the brackets around that one on 120 00:09:33,675 --> 00:09:38,498 top to show us. We've got a common factor and X plus one 121 00:09:38,498 --> 00:09:42,950 into there goes 1X plus one into. There goes one and so 122 00:09:42,950 --> 00:09:45,547 we're left with one over X +2. 123 00:09:47,650 --> 00:09:54,235 You can have one that is if you like the other way up, so let's 124 00:09:54,235 --> 00:10:00,820 have a look at that X minus 11 A plus 30. Sorry, A squared minus 125 00:10:00,820 --> 00:10:04,332 11 A plus 30 over a minus 5. 126 00:10:04,390 --> 00:10:09,034 Again, the A minus five. That's nice. That's OK. Let's keep that 127 00:10:09,034 --> 00:10:13,678 together with a bracket, but let's have a look at this. A 128 00:10:13,678 --> 00:10:19,096 squared minus eleven 8 + 30. Can we factorize it? Can we break it 129 00:10:19,096 --> 00:10:20,644 down into two factors? 130 00:10:21,400 --> 00:10:26,968 We've got a squared, so we need an A and and a. We now need 2 131 00:10:26,968 --> 00:10:30,796 numbers that are going to multiply together to give us 30, 132 00:10:30,796 --> 00:10:34,972 but add together to give us minus eleven. Well, six and five 133 00:10:34,972 --> 00:10:40,540 seem a good bet for the 30, and if we make a minus six and minus 134 00:10:40,540 --> 00:10:45,064 five, that ensures the plus 30 minus times by A minus. And it 135 00:10:45,064 --> 00:10:49,240 also ensures the minus 11 a 'cause will have minus 6A and 136 00:10:49,240 --> 00:10:50,284 minus 5A there. 137 00:10:51,410 --> 00:10:56,912 We've now got a common factor of a minus five, so we can divide 138 00:10:56,912 --> 00:11:02,807 the top by A minus five on the bottom by A minus five. So we 139 00:11:02,807 --> 00:11:07,130 end up with just a minus 6, and that's our answer. 140 00:11:08,060 --> 00:11:14,092 Now that's the proper way to do them. Some of you might be 141 00:11:14,092 --> 00:11:18,732 tempted. Might be tempted when you see something like this. 142 00:11:18,770 --> 00:11:25,298 Till forget what it is you're supposed to 143 00:11:25,298 --> 00:11:30,256 do. So you might suddenly think are lots of threes. I can get 144 00:11:30,256 --> 00:11:34,000 rid of some three, so let's cancel 3 here and a three there. 145 00:11:35,080 --> 00:11:40,630 One problem. We said we had to cancel common factors. 146 00:11:41,850 --> 00:11:47,454 3 doesn't appear here in this term, there's no factor of three 147 00:11:47,454 --> 00:11:52,124 in this term. We cannot do this kind of canceling. 148 00:11:52,670 --> 00:11:57,485 Let me just show you why not. If we have a look at a numerical 149 00:11:57,485 --> 00:12:05,000 example. Supposing we had, let's say 5 + 3 over 3 150 00:12:05,000 --> 00:12:10,852 + 1. Now if we do the computation without doing 151 00:12:10,852 --> 00:12:12,728 the canceling we get. 152 00:12:13,500 --> 00:12:16,098 8 over 4. 153 00:12:16,670 --> 00:12:21,080 And that's two, everybody is happy that that's the case. 154 00:12:21,660 --> 00:12:26,080 But if I do what I did here and suddenly go absolutely bananas 155 00:12:26,080 --> 00:12:29,820 and cancel the threes, then that's a one there and one 156 00:12:29,820 --> 00:12:35,516 there. So what I seem to have now is 5 + 1, which is 6 over 1 157 00:12:35,516 --> 00:12:37,924 + 1, which is 2 gives me 3. 158 00:12:38,890 --> 00:12:43,257 Two and three are not the same. Now we've agreed that 159 00:12:43,257 --> 00:12:46,830 tools the correct answer, where have we gone wrong? 160 00:12:48,040 --> 00:12:51,538 We cancel these threes divided by these threes, but three is 161 00:12:51,538 --> 00:12:55,354 not a common factor because it doesn't appear in the Five and 162 00:12:55,354 --> 00:12:57,262 it doesn't appear in the one. 163 00:12:58,140 --> 00:13:05,070 So we can't do this over here by the same reasons what we have to 164 00:13:05,070 --> 00:13:11,538 do is look and see if we can factorize what is on the top 165 00:13:11,538 --> 00:13:16,620 three X squared plus 10X plus three. Can we factorize it? 166 00:13:16,620 --> 00:13:22,626 Well, the X +3 is fine, let's keep it all together in a 167 00:13:22,626 --> 00:13:25,950 bracket. Brackets here. 168 00:13:26,760 --> 00:13:31,120 Three X squared will need a 3X and an X. 169 00:13:31,720 --> 00:13:36,494 Will also need a three under one to make up this number here, and 170 00:13:36,494 --> 00:13:41,268 we've got to get 10 out of it, so we're probably have to have 171 00:13:41,268 --> 00:13:45,360 the three there to give us Nynex across there under one there. 172 00:13:46,110 --> 00:13:50,842 Plus signs because these are all plus signs here and now. We can 173 00:13:50,842 --> 00:13:55,574 see we have got this common factor of X plus three, so we 174 00:13:55,574 --> 00:14:00,306 can divide the bottom by X +3 and the top by X +3. 175 00:14:00,830 --> 00:14:08,032 That will give us 3X Plus One and that is the correct answer. 176 00:14:08,630 --> 00:14:15,590 Sometimes we have to work again back a little bit harder, so 177 00:14:15,590 --> 00:14:18,490 let's take this example 6X 178 00:14:18,490 --> 00:14:26,247 cubed. Minus Seven X squared minus 5X all 179 00:14:26,247 --> 00:14:30,011 over 2X plus one. 180 00:14:30,560 --> 00:14:35,344 That doesn't give me much hope for is here, but there is a 181 00:14:35,344 --> 00:14:39,760 common factor of X in each term, so perhaps we can take 182 00:14:39,760 --> 00:14:44,176 that out as a common factor. To begin with. We might find 183 00:14:44,176 --> 00:14:48,592 we can factorize what's left, so taking that X out as a 184 00:14:48,592 --> 00:14:52,640 common factor is going to give us six X squared there. 185 00:14:53,780 --> 00:14:56,760 Minus Seven X there. 186 00:14:57,290 --> 00:15:01,886 And minus five there closed the bracket and we still to divide 187 00:15:01,886 --> 00:15:07,248 by the 2X plus one. Also we hope. So now let's have a look 188 00:15:07,248 --> 00:15:09,929 at this. 'cause this is now an 189 00:15:09,929 --> 00:15:12,170 ordinary quadratics. X. 190 00:15:12,670 --> 00:15:20,038 Six X squared? Well, let's have a guess at 3X and 2X. 191 00:15:20,038 --> 00:15:23,108 After all, there's 2X down 192 00:15:23,108 --> 00:15:27,981 there. What do we need now? We've got minus five to deal 193 00:15:27,981 --> 00:15:32,946 with. I want if I can have it 2X plus one. So let's just be 194 00:15:32,946 --> 00:15:36,918 guided by that. For the moment, let's make that 2X plus one. 195 00:15:37,560 --> 00:15:43,286 Minus 5 is what I need to multiply by the one to give me 196 00:15:43,286 --> 00:15:49,413 minus 5. Have I got it right? I need to check on that minus 7X 197 00:15:49,413 --> 00:15:53,976 will hear I plus 3X and here I've minus 10X and that does 198 00:15:53,976 --> 00:15:58,539 give me minus 7X. So that's right. Now I need to divide by 199 00:15:58,539 --> 00:16:02,400 the 2X plus one. Let's get it together in a bracket. 200 00:16:03,570 --> 00:16:08,619 I can divide top and bottom now by 2X plus one. 201 00:16:09,210 --> 00:16:11,742 Goes into itself once and once 202 00:16:11,742 --> 00:16:17,144 there. The answer is just what I'm left with here X times 3X 203 00:16:17,144 --> 00:16:21,980 minus five, and again I leave it in its factorized form, not try 204 00:16:21,980 --> 00:16:26,816 to multiply it out again 'cause I may need that form later on. 205 00:16:27,670 --> 00:16:32,980 Supposing we take something like this 206 00:16:32,980 --> 00:16:40,060 X cubed minus one over X minus one. 207 00:16:41,270 --> 00:16:45,072 What now? Doesn't seem to be a common factor 208 00:16:45,072 --> 00:16:46,702 in X cubed minus one. 209 00:16:48,430 --> 00:16:55,570 Difficult. But you may know a factorization 4X cubed 210 00:16:55,570 --> 00:17:02,920 minus one. It does in fact factorize as X minus 211 00:17:02,920 --> 00:17:05,860 one brackets X squared. 212 00:17:06,570 --> 00:17:10,138 Plus X plus one. 213 00:17:10,670 --> 00:17:14,970 And so, because we know that factorization, we can see 214 00:17:14,970 --> 00:17:18,410 straight away, we can simplify by dividing the 215 00:17:18,410 --> 00:17:24,000 bottom by X minus one, and that goes in once and the top 216 00:17:24,000 --> 00:17:29,160 by X minus one. And so we just left with the other 217 00:17:29,160 --> 00:17:32,170 factor X squared plus X plus one.