1 00:00:00,830 --> 00:00:05,897 As well known investigation at GCSE level called frogs. 2 00:00:06,470 --> 00:00:09,971 Many of you might well have attempted that particular 3 00:00:09,971 --> 00:00:13,472 investigation and want to start this particular section by 4 00:00:13,472 --> 00:00:16,584 having a look at that investigation, because it's 5 00:00:16,584 --> 00:00:20,863 going to show us why we might want to use brackets. 6 00:00:21,580 --> 00:00:26,416 Now, how does the investigation work? Well, what we have to do 7 00:00:26,416 --> 00:00:30,849 is try and interchange these coins at this side the pound 8 00:00:30,849 --> 00:00:36,708 coins. With the 10 P coins at this site and we have to do it 9 00:00:36,708 --> 00:00:41,284 by using one of two kinds of move, we can either slide into 10 00:00:41,284 --> 00:00:46,564 an empty space, or we can hop or jump over a coin of the opposite 11 00:00:46,564 --> 00:00:51,140 kind. So here a pound coin jumped over the 10 P coin. We 12 00:00:51,140 --> 00:00:53,604 can never go back. We must keep 13 00:00:53,604 --> 00:00:59,385 going. Forwards and so these 3 coins have with those rules to 14 00:00:59,385 --> 00:01:04,280 interchange with these 3 coins over here. So let's just see 15 00:01:04,280 --> 00:01:10,077 what happens. Let's see if we can keep count of the number of 16 00:01:10,077 --> 00:01:14,721 moves, not only keep count of the number of moves, but the 17 00:01:14,721 --> 00:01:18,591 number of different kinds of moves hops remember all slides. 18 00:01:18,591 --> 00:01:20,139 That was a slide. 19 00:01:20,890 --> 00:01:25,440 That's a hop, and the object is to do it in the minimum number 20 00:01:25,440 --> 00:01:29,990 of moves. Now when I do it, it will be the minimum number of 21 00:01:29,990 --> 00:01:34,215 moves. Don't worry about that, but we need to keep count of how 22 00:01:34,215 --> 00:01:37,790 many hops, how many slides and how many moves altogether so. 23 00:01:38,690 --> 00:01:45,645 Let's begin and will just keep a tally here of hops and slides. 24 00:01:45,645 --> 00:01:48,320 So we begin one slide. 25 00:01:49,510 --> 00:01:52,610 Over there one hop. 26 00:01:53,620 --> 00:01:59,748 One slide. And now with that we want one. 27 00:02:00,350 --> 00:02:02,640 Two hops 28 00:02:03,990 --> 00:02:06,369 There one slide. 29 00:02:06,900 --> 00:02:13,519 Now one. 2 three hops. 30 00:02:14,990 --> 00:02:17,545 A slide gets us that one home. 31 00:02:18,120 --> 00:02:24,676 And then two hops a slide gets us that one home. 32 00:02:25,220 --> 00:02:29,835 And then Hopkins that one home and then a slide gets that one 33 00:02:29,835 --> 00:02:36,068 home. So let's just talk these up 123456 slides. 34 00:02:37,030 --> 00:02:43,982 369 hops so altogether a grand total of 35 00:02:43,982 --> 00:02:48,890 15 moves. OK, when you were doing this investigation at 36 00:02:48,890 --> 00:02:52,730 school, or possibly you may not even have done it. What you 37 00:02:52,730 --> 00:02:57,530 would have done would be draw up a table so we want the number of 38 00:02:57,530 --> 00:03:00,330 coins. On 39 00:03:00,330 --> 00:03:08,870 each side. Hops? 40 00:03:09,930 --> 00:03:11,720 Slides. 41 00:03:12,740 --> 00:03:19,089 Moves Got our table there. So number of coins on 42 00:03:19,089 --> 00:03:24,703 each side. We could have one 2, three which is the case. We did 43 00:03:24,703 --> 00:03:25,906 four or five. 44 00:03:26,780 --> 00:03:32,212 Hops? Well, let's fill in this one for this table here. We had 45 00:03:32,212 --> 00:03:36,776 nine hops, six slides, and that gave us a total of 15 moves. Now 46 00:03:36,776 --> 00:03:38,732 we could do it for one. 47 00:03:39,290 --> 00:03:41,342 On each side, just do it 48 00:03:41,342 --> 00:03:43,360 quickly. One slide. 49 00:03:44,160 --> 00:03:49,010 One hop, one slide. So altogether two slides and one 50 00:03:49,010 --> 00:03:51,518 hop. And we can do it again. 51 00:03:52,210 --> 00:03:54,520 Quickly for these. 52 00:03:55,930 --> 00:03:56,850 A slide. 53 00:03:58,220 --> 00:03:59,050 A hop. 54 00:04:00,340 --> 00:04:04,942 A slide, so that's two slides all together in one hop, and now 55 00:04:04,942 --> 00:04:06,712 we have another two hops. 56 00:04:07,760 --> 00:04:10,120 A slide. A hop. 57 00:04:10,850 --> 00:04:13,796 And the slide. So altogether we 58 00:04:13,796 --> 00:04:19,440 had there. Four hops and for slides. I'm not going to go 59 00:04:19,440 --> 00:04:25,194 ahead and do it for 4 coins. It's quite easy to do and the 60 00:04:25,194 --> 00:04:30,126 results that we get will be 16 and 8 and 25 and. 61 00:04:30,910 --> 00:04:36,798 10 Giving us a total number of moves of 62 00:04:36,798 --> 00:04:38,962 three, 824 and 35. 63 00:04:40,050 --> 00:04:43,405 Now the object with most of these investigations is to try 64 00:04:43,405 --> 00:04:47,065 and arrive at a prediction. What would happen if we had any 65 00:04:47,065 --> 00:04:51,640 number of coins? Can we say what the result would be if we had 10 66 00:04:51,640 --> 00:04:55,910 coins on each side? If we had 50 coins on each side, 'cause this 67 00:04:55,910 --> 00:04:59,570 is the power that maths gives us the power to model it? 68 00:05:00,200 --> 00:05:05,324 In symbols and be able to use those symbols to make our 69 00:05:05,324 --> 00:05:09,594 predictions, so that's what we're looking at. Can we make 70 00:05:09,594 --> 00:05:14,728 that prediction? Well, let's have a little look. Want to 71 00:05:14,728 --> 00:05:19,438 concentrate here on this column and here on this column. 72 00:05:20,300 --> 00:05:24,116 I actually need a few more columns because I want to look 73 00:05:24,116 --> 00:05:28,886 at this set of numbers and I want to look at how this set of 74 00:05:28,886 --> 00:05:33,338 numbers is made up or created. So what I'm going to do is now 75 00:05:33,338 --> 00:05:37,154 we've got the data that we need. I'm going to turn over 76 00:05:37,154 --> 00:05:40,652 the sheet and I'm going to rewrite this table on a 77 00:05:40,652 --> 00:05:44,150 separate sheet and then we can analyze it without having all 78 00:05:44,150 --> 00:05:47,966 these bits and pieces around the edge. So we pick up the 79 00:05:47,966 --> 00:05:48,284 coins. 80 00:05:50,960 --> 00:05:52,890 Put them to one side. 81 00:05:53,720 --> 00:06:01,412 Turn over the sheet an begin again. So here we've got the 82 00:06:01,412 --> 00:06:09,104 coins and the number of coins on each side. 12345 and we 83 00:06:09,104 --> 00:06:16,796 have the number of hops that was one 4 nine 1625. Then 84 00:06:16,796 --> 00:06:20,642 we have the number of slides. 85 00:06:21,370 --> 00:06:25,122 And that was 2468 86 00:06:25,122 --> 00:06:32,122 ten. And then the total number of moves which we 87 00:06:32,122 --> 00:06:38,920 got from the total of the hops and the slides. So 88 00:06:38,920 --> 00:06:40,774 that's three 815-2435. 89 00:06:42,020 --> 00:06:46,205 Now what we said was we want it to be able to make a prediction 90 00:06:46,205 --> 00:06:50,111 no matter how many coins we've got on each side. We want it to 91 00:06:50,111 --> 00:06:54,017 be able to say how many moves it was going to take us to 92 00:06:54,017 --> 00:06:55,412 interchange those two sets of 93 00:06:55,412 --> 00:06:58,102 coins. Let's have a look at this 94 00:06:58,102 --> 00:07:04,290 set of numbers. And let's try and relate it to this set of 95 00:07:04,290 --> 00:07:08,538 numbers. Well, if we look at three an 15. 96 00:07:09,550 --> 00:07:14,116 15 is 3 times by 5. 97 00:07:15,250 --> 00:07:18,708 Just a lucky guess. Well, let's have a look at the next one. 98 00:07:19,850 --> 00:07:26,990 4 and 24 will 24 is 4 times by 6. 99 00:07:27,500 --> 00:07:34,612 An again 5 and 35 that's five times by 7, so eight is 8 100 00:07:34,612 --> 00:07:40,708 the same relationship with two? Yes, it is. That's two times by 101 00:07:40,708 --> 00:07:47,312 4. And even for this top one we've got one times by three. 102 00:07:48,250 --> 00:07:53,434 So let's look at these numbers here. These two columns. Here we 103 00:07:53,434 --> 00:07:56,026 have this number. The number of 104 00:07:56,026 --> 00:08:02,306 coins. The number of coins and here we seem to have two more 105 00:08:02,306 --> 00:08:08,576 than the number of coins, so if this was an what we seem to be 106 00:08:08,576 --> 00:08:11,502 saying is that here we've got N. 107 00:08:12,020 --> 00:08:16,520 Multiplied by, I'm not going to write anything in there at the 108 00:08:16,520 --> 00:08:21,395 moment, and plus two now it's the whole of this that we want 109 00:08:21,395 --> 00:08:26,270 to multiply by N everything, not just the N, not just the two, 110 00:08:26,270 --> 00:08:31,145 but all of it, because when we do one times by three, we 111 00:08:31,145 --> 00:08:36,395 multiply all of the three by one. When we do the two times by 112 00:08:36,395 --> 00:08:42,020 4, we multiply all of the four by the two, and so on. So this 113 00:08:42,020 --> 00:08:43,145 has to be. 114 00:08:43,460 --> 00:08:48,620 Help together in a bracket so that we can show we are 115 00:08:48,620 --> 00:08:52,490 multiplying everything inside the bracket by what is outside 116 00:08:52,490 --> 00:08:57,528 the bracket. So let's just leave that there for the moment. 117 00:08:58,470 --> 00:09:02,480 Let's have a look now at these sets of numbers. 118 00:09:03,250 --> 00:09:06,877 And what we do know is that if we add these sets of 119 00:09:06,877 --> 00:09:09,667 numbers together, we do get the total number of moves. 120 00:09:10,690 --> 00:09:13,822 Can we see any kind of 121 00:09:13,822 --> 00:09:18,684 relationship between? These numbers the number of coins and 122 00:09:18,684 --> 00:09:24,820 these two was a fairly obvious one. This is 1 squared. This is 123 00:09:24,820 --> 00:09:30,012 2 squared, 3 squared, 4 squared and five squared. In other 124 00:09:30,012 --> 00:09:34,732 words, it's this number multiplied by itself. So if we 125 00:09:34,732 --> 00:09:38,036 end we can see we've got end 126 00:09:38,036 --> 00:09:41,690 squared. If we look down here. 127 00:09:42,500 --> 00:09:49,172 Each one of these relates in the same way four is 2 * 2 six is 128 00:09:49,172 --> 00:09:55,844 2 times by three, 8 is 2 times by 410, two times by 5, so again 129 00:09:55,844 --> 00:10:02,933 2 times by 1, two times by two 2 times by 3, two times by 4, two 130 00:10:02,933 --> 00:10:09,605 times by 5. This is our N again, so we must have two times by N 131 00:10:09,605 --> 00:10:10,856 or two N. 132 00:10:11,810 --> 00:10:17,770 Now the number of moves can't change simply because we've 133 00:10:17,770 --> 00:10:20,154 written it down differently. 134 00:10:20,720 --> 00:10:25,881 We know the total number of moves is the sum of these two 135 00:10:25,881 --> 00:10:32,725 terms. That must be the same as that the number of moves 136 00:10:32,725 --> 00:10:37,450 simply can't change just because it written down differently. 137 00:10:38,200 --> 00:10:42,787 Doesn't mean that the value changes, and So what we must 138 00:10:42,787 --> 00:10:45,706 have is that the total number of 139 00:10:45,706 --> 00:10:49,595 moves. Is equal to both of those expressions, and so they must 140 00:10:49,595 --> 00:10:55,520 both be equal. So that shows us two things. First of all, it 141 00:10:55,520 --> 00:11:00,524 shows us why we need brackets to keep things together as entities 142 00:11:00,524 --> 00:11:01,775 as a whole. 143 00:11:02,320 --> 00:11:06,511 It also shows us how we can remove brackets because we 144 00:11:06,511 --> 00:11:11,464 multiply that with that to give us the end squared and we can 145 00:11:11,464 --> 00:11:15,274 multiply that with that to give us the two N. 146 00:11:16,140 --> 00:11:20,807 So bearing these things in mind, what we're going to have a look 147 00:11:20,807 --> 00:11:24,038 at is multiplying out brackets and removing brackets from 148 00:11:24,038 --> 00:11:28,705 expressions. Now we've seen why we need it now. We need to be 149 00:11:28,705 --> 00:11:33,372 able to work both ways across this equal sign. We need to be 150 00:11:33,372 --> 00:11:37,680 able to work from the brackets to the expression and from the 151 00:11:37,680 --> 00:11:41,988 expression to the brackets. Now we're only going to go one way 152 00:11:41,988 --> 00:11:45,578 in this particular clip. That's from the brackets to the 153 00:11:45,578 --> 00:11:51,064 multiplied out. Or expanded xpression coming back this way 154 00:11:51,064 --> 00:11:57,784 is factorizing and we look at that in another section. 155 00:11:57,784 --> 00:12:04,504 So let's take some examples 3 times by X +2. 156 00:12:05,530 --> 00:12:10,800 Everything inside has to be multiplied by what is outside, 157 00:12:10,800 --> 00:12:18,178 so we need 3 times by X and we need 3 times by two 158 00:12:18,178 --> 00:12:23,448 which is 6. The plus sign, because that's plus there. 159 00:12:24,020 --> 00:12:29,641 X times X minus Y everything inside must be multiplied by 160 00:12:29,641 --> 00:12:36,284 what is outside, so we've X times by XX squared X times by 161 00:12:36,284 --> 00:12:43,438 Y is X why we are taking the wire way so we've got X 162 00:12:43,438 --> 00:12:49,059 times by minus Y is going to give us minus XY. 163 00:12:50,450 --> 00:12:53,519 Minus 3A squared. 164 00:12:54,020 --> 00:12:59,135 And then in the bracket, 3 minus B. Now we're multiplying 165 00:12:59,135 --> 00:13:02,855 everything inside by what's outside. What's outsiders got 166 00:13:02,855 --> 00:13:08,900 this minus sign attached to it? So we gotta take care. So we 167 00:13:08,900 --> 00:13:14,015 have minus 3A squared times by 3 - 9 A squared. 168 00:13:14,780 --> 00:13:21,050 Minus 3A squared times minus B, so the minus minus together 169 00:13:21,050 --> 00:13:26,750 gives us a plus and we have 3A squared B. 170 00:13:27,950 --> 00:13:33,080 We can do this across a number of terms, so if we minus two X 171 00:13:33,080 --> 00:13:35,474 let's say X minus Y minus Z. 172 00:13:36,500 --> 00:13:42,730 So we've minus two X times by X. That's that times by that minus 173 00:13:42,730 --> 00:13:44,510 two X squared we've. 174 00:13:45,740 --> 00:13:52,856 Minus two X times Y minus Y. That gives us +2 XY 175 00:13:52,856 --> 00:13:59,972 and then minus two X times Y minus said plus 2X Zedd. 176 00:14:01,550 --> 00:14:05,150 OK, those are reasonably straightforward and you can find 177 00:14:05,150 --> 00:14:09,550 plenty of those in textbooks and other sources for you to 178 00:14:09,550 --> 00:14:14,750 practice. Let's now have a look at what happens if we want to 179 00:14:14,750 --> 00:14:17,950 multiply out expressions where there are two brackets 180 00:14:17,950 --> 00:14:19,150 multiplying each other. 181 00:14:19,170 --> 00:14:26,842 So let's take X +5 times by X 182 00:14:26,842 --> 00:14:30,678 plus 10. What now? 183 00:14:32,120 --> 00:14:37,468 Well, what I want just to go back to is a simple one like 184 00:14:37,468 --> 00:14:40,524 let's say 3 times by X plus 10. 185 00:14:41,100 --> 00:14:44,223 If you remember what we did, we said, well, 186 00:14:44,223 --> 00:14:45,611 we'll take the three. 187 00:14:46,830 --> 00:14:52,402 And multiply it with the X and will take the three an. We will 188 00:14:52,402 --> 00:14:54,392 multiply it by the 10. 189 00:14:55,920 --> 00:15:01,908 So that this was the entity we were multiplied by. Let's think 190 00:15:01,908 --> 00:15:09,892 of this X +5 as being the three, so for this one we would do 3 191 00:15:09,892 --> 00:15:16,878 times by X. So for this one will do X +5 times by X. 192 00:15:16,910 --> 00:15:23,630 So the X +5 is behaving exactly like the three. 193 00:15:25,240 --> 00:15:30,565 Now here we would do 3 times by 10 and rather than right it is 194 00:15:30,565 --> 00:15:34,115 30. I will write it as three times by 10. 195 00:15:34,640 --> 00:15:42,018 So this is taking the place of the three, so it's going to be 196 00:15:42,018 --> 00:15:44,653 X +5 times by 10. 197 00:15:45,720 --> 00:15:49,200 Here we would have a plus sign, so here we 198 00:15:49,200 --> 00:15:50,940 will have a plus sign. 199 00:15:52,040 --> 00:15:57,515 Now, the fact that I've got the X and the 10 at the end doesn't 200 00:15:57,515 --> 00:16:01,165 make any difference. I'm still multiplying them together so the 201 00:16:01,165 --> 00:16:04,815 same rules apply. I still have to multiply everything inside 202 00:16:04,815 --> 00:16:09,560 this bracket by what is outside this bracket. So now I need that 203 00:16:09,560 --> 00:16:14,305 with that, which gives me an X squared and I need that with 204 00:16:14,305 --> 00:16:19,050 that, which gives me 5X. I need that with that, which gives me 205 00:16:19,050 --> 00:16:22,700 10X and I need that with that, which gives me. 206 00:16:22,730 --> 00:16:25,938 50 and so we tidy up the middle 207 00:16:25,938 --> 00:16:29,534 bit. 5X plus 10X is 208 00:16:29,534 --> 00:16:32,638 15 X. And there we have. 209 00:16:33,210 --> 00:16:34,330 Our expression. 210 00:16:35,810 --> 00:16:37,010 OK. 211 00:16:38,100 --> 00:16:44,540 This. Here. Is an extra step we can manage without 212 00:16:44,540 --> 00:16:48,410 it provided we keep track of what we've done. 213 00:16:49,550 --> 00:16:55,106 Notice I've got an arrow from here going to each of these. 214 00:16:55,690 --> 00:16:59,331 Alternatively, if you've got an arrow going from that extra that 215 00:16:59,331 --> 00:17:01,648 X and an arrow going from that 216 00:17:01,648 --> 00:17:07,420 extra that 10. So every term is got sort of two links associated 217 00:17:07,420 --> 00:17:12,581 with it. So let's have a look at doing that without going into 218 00:17:12,581 --> 00:17:18,139 this step. Now we've seen how we can do it can, we shorten the 219 00:17:18,139 --> 00:17:19,727 process a little bit. 220 00:17:19,750 --> 00:17:24,976 So we've X +5 times by 221 00:17:24,976 --> 00:17:27,589 X plus 10. 222 00:17:28,920 --> 00:17:33,860 So we've got to multiply everything in this bracket by 223 00:17:33,860 --> 00:17:40,282 everything in this bracket. So remember we had to have X by X, 224 00:17:40,282 --> 00:17:47,198 so that's their X squared. Then we have to have X times by 10, 225 00:17:47,198 --> 00:17:54,114 so that's there 10X. Now we need to make the same links here, so 226 00:17:54,114 --> 00:17:56,090 that's X by 5. 227 00:17:57,920 --> 00:18:05,600 Giving us 5X and then 5 by 10, giving us 50 and so we end up 228 00:18:05,600 --> 00:18:09,920 with exactly the same expression as we had before. 229 00:18:09,920 --> 00:18:13,550 Gain, we've got two hours going into this bracket. Two 230 00:18:13,550 --> 00:18:17,180 arrows going into this bracket and then coming back. We've 231 00:18:17,180 --> 00:18:21,173 got the same from 10. There are two arrows linking back. 232 00:18:22,230 --> 00:18:26,509 We can use this process for brackets like this that leaders 233 00:18:26,509 --> 00:18:30,399 to quadratic expressions an for multiplying any pair of brackets 234 00:18:30,399 --> 00:18:35,067 together, and in fact the arrows help us to see that we've 235 00:18:35,067 --> 00:18:38,957 completed the multiplication 'cause they help us to see that 236 00:18:38,957 --> 00:18:43,236 if we've got two terms here, then there are two arrows 237 00:18:43,236 --> 00:18:47,904 linking this first term, one with each term there, and so on. 238 00:18:47,904 --> 00:18:52,183 So let's have another look at some examples of this working 239 00:18:52,183 --> 00:18:55,818 our way. Down some quite complicated ones, but beginning 240 00:18:55,818 --> 00:19:01,070 with some simple ones, very much like we've just had a look at. 241 00:19:01,070 --> 00:19:06,322 So first of all, we got X takeaway 7 times by X takeaway 242 00:19:06,322 --> 00:19:09,554 10 or X minus Seven times by X 243 00:19:09,554 --> 00:19:17,150 minus 10. Let's expand this as we've done before, so X times my 244 00:19:17,150 --> 00:19:20,390 X that gives us X squared. 245 00:19:20,970 --> 00:19:27,330 And X times by now this is minus 10, so it gives us minus 10X. 246 00:19:28,180 --> 00:19:35,304 Minus 7 times by X minus 7X and minus Seven times by minus 247 00:19:35,304 --> 00:19:42,360 10. Plus 70 and we can simplify that X 248 00:19:42,360 --> 00:19:46,010 squared minus 17X plus 70. 249 00:19:46,800 --> 00:19:53,919 Take X plus six times by X minus 6. 250 00:19:55,010 --> 00:20:00,110 Everything in this bracket must be multiplied by everything in 251 00:20:00,110 --> 00:20:06,740 this bracket, so we have X times by X. That's how X squared. 252 00:20:07,340 --> 00:20:12,954 X. Times by minus six gives us minus 6X. 253 00:20:14,530 --> 00:20:18,387 Plus six times by X their loss 254 00:20:18,387 --> 00:20:25,900 6X. And then six times by minus 6 - 36 and now again we need 255 00:20:25,900 --> 00:20:31,420 to simplify this. Looking at these ex is so with X squared. 256 00:20:31,420 --> 00:20:38,320 Now we have to take away 6X and add on 6X. The net effect is 257 00:20:38,320 --> 00:20:44,300 nothing so there are no excess minus 36, so we don't always get 258 00:20:44,300 --> 00:20:49,360 this middle term sometimes and this is an example notice it 259 00:20:49,360 --> 00:20:53,200 structure. Certain symmetry about it, a 6 and a six hour 260 00:20:53,200 --> 00:20:54,205 plus under minus. 261 00:20:54,900 --> 00:20:57,618 And the ex is will disappear. 262 00:20:59,240 --> 00:21:06,260 Take another one, 2X minus 3X plus one. This time it's 2X not 263 00:21:06,260 --> 00:21:12,200 just X, but again, everything in here must be multiplied by 264 00:21:12,200 --> 00:21:19,220 everything in there. So we have two X times by X. That gives 265 00:21:19,220 --> 00:21:21,380 us two X squared. 266 00:21:21,970 --> 00:21:28,346 2X times by one that gives us 2X. 267 00:21:28,940 --> 00:21:31,862 Minus three times by X that's 268 00:21:31,862 --> 00:21:37,480 minus 3X. And minus three times by one is minus 3. 269 00:21:38,210 --> 00:21:42,242 Here we've got some excess. We must simplify them. We've got to 270 00:21:42,242 --> 00:21:45,266 gather them together so we have two X squared. 271 00:21:45,960 --> 00:21:53,196 Plus 2X minus three X. We're adding on 2X taking away 3X. 272 00:21:53,196 --> 00:21:59,829 We still have their foreign X to take away, minus three. 273 00:22:00,720 --> 00:22:06,960 Now let's have a look at just two or three more of 274 00:22:06,960 --> 00:22:12,680 these. If we take 3X minus two and three X +2. 275 00:22:14,680 --> 00:22:18,502 Have we seen something like this before? A minus sign in a plus 276 00:22:18,502 --> 00:22:22,618 sign a two and a 2A3X and 3X. Not just an expert of 3X. 277 00:22:23,200 --> 00:22:24,586 Are we going to get the same 278 00:22:24,586 --> 00:22:28,419 sorts of things? Symmetry suggests that we might, so let's 279 00:22:28,419 --> 00:22:29,460 have a look. 280 00:22:30,250 --> 00:22:37,615 3X times by three X. That will give us nine X squared 3 * 3 281 00:22:37,615 --> 00:22:39,579 X times by X. 282 00:22:40,360 --> 00:22:48,040 3X times by +2 is plus 6X minus two times 283 00:22:48,040 --> 00:22:51,880 by three. X is minus 284 00:22:51,880 --> 00:22:59,350 6 X. And minus two times by +2 is minus four. Gotta 285 00:22:59,350 --> 00:23:02,050 tidy up these middle terms. 286 00:23:02,050 --> 00:23:08,130 Nynex squared. Plus 6X minus 6X no access, and we left with 287 00:23:08,130 --> 00:23:13,980 minus four on the end. So yes, it was another example of that 288 00:23:13,980 --> 00:23:18,480 kind of expression where the middle term is going to 289 00:23:18,480 --> 00:23:24,452 disappear. Let's Make it a little bit harder and 290 00:23:24,452 --> 00:23:28,106 let's have a look at something with three terms 291 00:23:28,106 --> 00:23:32,572 in this bracket and only two in this, so we make 292 00:23:32,572 --> 00:23:36,632 that One X +2, but let's put in X squared. 293 00:23:38,020 --> 00:23:42,620 Minus two X cubed +8. 294 00:23:44,340 --> 00:23:49,826 Got three terms here. We need to keep very careful track and make 295 00:23:49,826 --> 00:23:54,468 sure that we multiply everything in here by everything in there. 296 00:23:54,468 --> 00:23:55,734 So let's begin. 297 00:23:56,400 --> 00:24:03,570 X squared times by X? That's X cubed X squared 298 00:24:03,570 --> 00:24:07,155 times by two. That's two 299 00:24:07,155 --> 00:24:14,460 X squared. Minus 2X now minus two X cubed. Now times by 300 00:24:14,460 --> 00:24:18,212 X is minus 2 X till the 301 00:24:18,212 --> 00:24:25,678 4th. And minus two X cubed times by two is minus 4X cubed, 302 00:24:25,678 --> 00:24:29,670 and now we've got to do the 8. 303 00:24:30,340 --> 00:24:37,750 So we've got 8 times by X Plus 8X and we've got 8. 304 00:24:38,740 --> 00:24:42,639 Times by two. So that's plus 60. 305 00:24:43,140 --> 00:24:47,729 What we can see is that from each of these terms we've got 306 00:24:47,729 --> 00:24:48,788 two arrows coming. 307 00:24:49,610 --> 00:24:53,731 And that's what we must have. 'cause we got two terms here so 308 00:24:53,731 --> 00:24:58,169 we can be pretty happy that in fact, we've done what we set out 309 00:24:58,169 --> 00:25:01,656 to do. And now we need to simplify. And it's usually 310 00:25:01,656 --> 00:25:05,143 better to write down expressions with the highest power in X 311 00:25:05,143 --> 00:25:09,581 first. This here is the highest power in X. It's X to the power 312 00:25:09,581 --> 00:25:14,019 4. So we write that one down first, minus 2X to the power 4. 313 00:25:14,900 --> 00:25:20,195 What else we got? We got any ex cubes. Well yes we have. We've A 314 00:25:20,195 --> 00:25:24,784 plus X cubed an A minus 4X cubed. So that's going to give 315 00:25:24,784 --> 00:25:26,549 us minus three X cubed. 316 00:25:27,380 --> 00:25:33,451 Sometimes helps just to put a little tick on top of them, just 317 00:25:33,451 --> 00:25:38,121 to show that, yeah, we've actually dealt with those X 318 00:25:38,121 --> 00:25:43,258 squared's only one that's there plus two X squared. So we've 319 00:25:43,258 --> 00:25:49,329 dealt with that one plus 8X. We've dealt with that 1 + 16 320 00:25:49,329 --> 00:25:53,532 we've dealt with that one, so there our answer. 321 00:25:55,120 --> 00:26:02,322 Let's see if we can just take this a little bit further by 322 00:26:02,322 --> 00:26:05,092 multiplying together 2 quadratic expressions. 323 00:26:06,440 --> 00:26:10,521 Normally come up with something as difficult as this, or perhaps 324 00:26:10,521 --> 00:26:14,602 as tedious, but it's a good exercise to see if you've 325 00:26:14,602 --> 00:26:19,054 actually got the hang of what you're supposed to be doing. So 326 00:26:19,054 --> 00:26:23,135 we've got to multiply everything in this bracket by everything in 327 00:26:23,135 --> 00:26:27,587 that bracket. There are three terms here, so if we've done it 328 00:26:27,587 --> 00:26:31,668 right, we ought to have three arrows coming from each of 329 00:26:31,668 --> 00:26:35,749 these, associating with each of these. If you think about that 330 00:26:35,749 --> 00:26:37,975 ought to give us 9 terms. 331 00:26:38,030 --> 00:26:39,577 In the same way that this one. 332 00:26:40,130 --> 00:26:45,260 Three terms here, 2 terms that gave us six terms 333 00:26:45,260 --> 00:26:49,877 altogether. 123456 before we simplified it, so we're going 334 00:26:49,877 --> 00:26:52,442 to have nine terms here. 335 00:26:53,550 --> 00:26:55,866 Let's work our way through it. 336 00:26:56,570 --> 00:27:01,050 X squared times by X squared X to the 4th. 337 00:27:01,740 --> 00:27:07,644 X squared times by X Plus X cubed. 338 00:27:08,250 --> 00:27:15,700 X squared times by minus 6 - 6 X squared. 339 00:27:15,700 --> 00:27:21,433 Finished with that one on to this One X times by X squared 340 00:27:21,433 --> 00:27:22,756 plus X cubed. 341 00:27:23,720 --> 00:27:30,877 X times by X Plus X squared X times by minus 6 - 6 X. Now this 342 00:27:30,877 --> 00:27:35,929 one here minus two times by X squared getting a bit fraught 343 00:27:35,929 --> 00:27:40,139 with the brackets here with the arrows linking them, but 344 00:27:40,139 --> 00:27:45,612 nevertheless we can still use it to help us check. So minus two 345 00:27:45,612 --> 00:27:48,559 times by X squared minus two X 346 00:27:48,559 --> 00:27:55,374 squared. Minus 2 times by X so that gives us minus 2X and 347 00:27:55,374 --> 00:28:00,918 then finally minus two times by minus six gives us plus 12. 348 00:28:01,620 --> 00:28:05,748 This now needs tidying up, so we start with the highest powers 349 00:28:05,748 --> 00:28:09,876 1st and write down the highest power, then the next, then the 350 00:28:09,876 --> 00:28:15,380 next and so on down the line we take them off as we did here so 351 00:28:15,380 --> 00:28:19,508 we can show that we've coped with them. We've taken them into 352 00:28:19,508 --> 00:28:23,980 account, so our highest term is this one. Here X to the power 353 00:28:23,980 --> 00:28:28,452 four, and there are no others. Now we want the ex cubes. Well, 354 00:28:28,452 --> 00:28:32,236 there's one there, and there's one there. So that's plus 2X 355 00:28:32,236 --> 00:28:33,612 cubed, so we can. 356 00:28:33,640 --> 00:28:37,558 Take off those two X squared's. 357 00:28:38,360 --> 00:28:39,948 Minus six X squared. 358 00:28:40,570 --> 00:28:45,922 Plus, X squared minus two X squared, so we've got minus six 359 00:28:45,922 --> 00:28:50,828 X squared minus two X squared. That's minus eight X squared, 360 00:28:50,828 --> 00:28:55,734 and then we're adding on One X squared, so altogether that's 361 00:28:55,734 --> 00:29:00,640 minus Seven X squared, and we can take them off. We've 362 00:29:00,640 --> 00:29:01,978 accounted for them. 363 00:29:02,550 --> 00:29:07,902 Minus six X minus two X the X terms next altogether that's 364 00:29:07,902 --> 00:29:14,662 minus 8X. And then at the end we just got this number 12 that 365 00:29:14,662 --> 00:29:18,872 we have not accounted for, and so that's plus 12. 366 00:29:19,380 --> 00:29:21,580 So that's multiplying out 367 00:29:21,580 --> 00:29:26,730 brackets. Every pair of brackets that needs to be multiplied out, 368 00:29:26,730 --> 00:29:31,826 or indeed if it's three sets of brackets, you do 2 together, and 369 00:29:31,826 --> 00:29:35,354 then the third would. The resulting expression is done 370 00:29:35,354 --> 00:29:41,234 with the third one one. I want to have a look at now is just 371 00:29:41,234 --> 00:29:44,370 simply removing brackets. Sometimes we get collections of 372 00:29:44,370 --> 00:29:47,114 expressions nested in various sets of brackets. 373 00:29:47,120 --> 00:29:53,137 So let's have a look at the sorts of horrible, awful 374 00:29:53,137 --> 00:29:59,154 expressions that people write down in textbooks and say have a 375 00:29:59,154 --> 00:30:02,436 go at these and remove the 376 00:30:02,436 --> 00:30:07,200 brackets. So here we are, whole set of letters jumbled together 377 00:30:07,200 --> 00:30:11,737 with brackets in and what we're being asked to do is just remove 378 00:30:11,737 --> 00:30:15,576 these brackets and simplify. So the first thing is remove the 379 00:30:15,576 --> 00:30:20,113 brackets, get rid of them. So let's have a look. OK, now we're 380 00:30:20,113 --> 00:30:24,301 taking away B minus C, so we've got to take away everything 381 00:30:24,301 --> 00:30:28,838 that's in the brackets, and we've got to take away the B and 382 00:30:28,838 --> 00:30:33,375 then takeaway minus see which of course is the same as adding see 383 00:30:33,375 --> 00:30:40,390 on. Plus a no problem there plus B because we've got to add on 384 00:30:40,390 --> 00:30:45,230 everything that's in this bracket. So plus be plus minus C 385 00:30:45,230 --> 00:30:47,430 which is just minus C. 386 00:30:47,470 --> 00:30:50,930 Plus B and then again. We've gotta take away everything 387 00:30:50,930 --> 00:30:55,082 that's in this bracket. So we gotta take away, see an then 388 00:30:55,082 --> 00:30:58,888 takeaway? Minus A and that's the same as adding a on. 389 00:31:00,110 --> 00:31:05,234 Now we have to simplify, so we have to look for all the days, 390 00:31:05,234 --> 00:31:09,992 all the bees and all the Seas. So let's take the ace Pursuivant 391 00:31:09,992 --> 00:31:14,750 A&A&A and they're all, plus. So that's three a we can just take 392 00:31:14,750 --> 00:31:18,044 them off to show that we've dealt with them. 393 00:31:18,580 --> 00:31:22,252 Bees minus B Plus B plus 394 00:31:22,252 --> 00:31:28,612 B. Minus B Plus B know bees and then another be to add 395 00:31:28,612 --> 00:31:32,972 on plus be. So we've dealt with all the bees. 396 00:31:34,370 --> 00:31:42,070 Plus C minus C minus C Plus C minus C, no seas and the 397 00:31:42,070 --> 00:31:45,920 sea still to take away minus C. 398 00:31:45,920 --> 00:31:49,336 So that was fairly straightforward. No nesting of 399 00:31:49,336 --> 00:31:54,460 brackets, so let's have a look at one where we've actually got 400 00:31:54,460 --> 00:31:56,595 some brackets nested so minus. 401 00:31:56,610 --> 00:32:02,238 5X inside that bracket minus 11 Y. 402 00:32:03,010 --> 00:32:07,320 Minus 3X. Minus. 403 00:32:08,580 --> 00:32:15,300 5 Y minus three X minus six Y. 404 00:32:17,550 --> 00:32:25,470 Now that looks pretty awful. What's a brackets? 405 00:32:26,340 --> 00:32:29,520 And if we look, we've got a pair of brackets here covering 406 00:32:29,520 --> 00:32:33,830 everything. We've got an expression in a pair of 407 00:32:33,830 --> 00:32:37,730 brackets. There got another pair of brackets here and inside. 408 00:32:37,730 --> 00:32:41,630 We've got another pair of brackets. Where do we start? 409 00:32:41,630 --> 00:32:47,090 Looks a bit tricky, but what you do is you start right inside the 410 00:32:47,090 --> 00:32:52,160 brackets so you look inside each set of brackets and you say what 411 00:32:52,160 --> 00:32:58,010 do I need to do to get rid of one set of brackets and usually 412 00:32:58,010 --> 00:33:00,350 go right inside the most complex 413 00:33:00,350 --> 00:33:05,882 bit? And this bit here is by far the most complex bit. So let's 414 00:33:05,882 --> 00:33:09,428 begin by going right inside and looking at this. 415 00:33:10,160 --> 00:33:15,224 Really nested embedded set of brackets and get rid of them to 416 00:33:15,224 --> 00:33:22,770 begin with. So if we're going to do that, we write the rest 417 00:33:22,770 --> 00:33:25,520 down just as it was. 418 00:33:25,530 --> 00:33:30,138 We won't touch it at all. 419 00:33:31,420 --> 00:33:35,996 We're just going to look at this bit. We're taking away 420 00:33:35,996 --> 00:33:39,740 everything there, so we're taking away 3X and we're 421 00:33:39,740 --> 00:33:43,484 taking away minus six Y, so altogether. That's takeaway 422 00:33:43,484 --> 00:33:48,892 3X, and then the minus minus is plus six Y. And then that 423 00:33:48,892 --> 00:33:50,972 Square bracket and that one. 424 00:33:52,780 --> 00:33:58,228 It's beginning to look a little bit simpler. We've now got this 425 00:33:58,228 --> 00:34:03,222 big set of square brackets outside, but we've got that one 426 00:34:03,222 --> 00:34:09,124 there and that one there, and we can deal with both of these 427 00:34:09,124 --> 00:34:13,664 together. Be'cause, they're no different from what we've got up 428 00:34:13,664 --> 00:34:19,112 there, really. So equals minus 5X minus 11 Y and then minus 429 00:34:19,112 --> 00:34:22,744 minus is A plus 3X minus five Y 430 00:34:22,744 --> 00:34:30,452 minus. Minus 3X is A plus, 3X minus plus six Y, so we're going 431 00:34:30,452 --> 00:34:36,546 to have minus six Y and restore the Big Square bracket. 432 00:34:37,350 --> 00:34:40,875 I think we want to tidy this up. We've got lots of axes and wise 433 00:34:40,875 --> 00:34:43,695 floating about so really we've got to tighten this up, so let's 434 00:34:43,695 --> 00:34:49,790 have a look. And we'll do it simply inside the bracket. 435 00:34:50,500 --> 00:34:54,616 So we don't need to think at the moment about the effect of this 436 00:34:54,616 --> 00:34:57,850 minus sign with just looking at that bit there. That's inside 437 00:34:57,850 --> 00:35:04,315 the bracket. So we 5X3X and 3X, that's 11X. 438 00:35:05,630 --> 00:35:12,590 Minus 11 Y minus five Y minus six Y all together we 439 00:35:12,590 --> 00:35:19,550 are taking away 22Y and close the big bracket and now we've 440 00:35:19,550 --> 00:35:26,510 got to remove this final bracket so we have minus 11X plus 441 00:35:26,510 --> 00:35:29,410 minus takeaway minus 22 Y. 442 00:35:30,410 --> 00:35:36,242 So what? Looked quite an awful lump of algebra has now come 443 00:35:36,242 --> 00:35:41,102 down to be something that's really quite small. An quite 444 00:35:41,102 --> 00:35:46,448 manageable will just take one more 'cause it's helpful to get 445 00:35:46,448 --> 00:35:51,308 this focus on getting right inside those brackets and what 446 00:35:51,308 --> 00:35:57,140 you will notice, I think, is that we tend to use different 447 00:35:57,140 --> 00:36:00,542 sets of brackets, different types of brackets. 448 00:36:00,580 --> 00:36:06,531 To draw our attention tool, the different lumps of algebra that 449 00:36:06,531 --> 00:36:09,236 we've got to deal with. 450 00:36:09,760 --> 00:36:14,476 So here in this expression we've got three sets of brackets, the 451 00:36:14,476 --> 00:36:17,620 curly brackets, the square brackets, and the round 452 00:36:17,620 --> 00:36:22,336 brackets, and notice I've worked my way in, so I'm now right 453 00:36:22,336 --> 00:36:27,052 inside, right in the middle of this expression. So this is the 454 00:36:27,052 --> 00:36:31,768 one I'm going to go for first. Everything else for the moment 455 00:36:31,768 --> 00:36:36,484 is going to stay the same, not going to touch anything else, 456 00:36:36,484 --> 00:36:40,414 just this round bracket here. So I've got to multiply. 457 00:36:40,480 --> 00:36:48,096 By two so 2 * 10 A is 20 A and two times by minus B 458 00:36:48,096 --> 00:36:53,808 is minus 2B. Put my square bracket in. Put my curly bracket 459 00:36:53,808 --> 00:37:01,284 here. What now? I think it would be a good idea to 460 00:37:01,284 --> 00:37:08,356 try and tidy this up before we went any further, so we've got 461 00:37:08,356 --> 00:37:14,884 5A Minus Square bracket. I've got 6A plus 28, that's 26 a 462 00:37:14,884 --> 00:37:16,516 takeaway to be. 463 00:37:17,250 --> 00:37:20,022 Close up the brackets at the 464 00:37:20,022 --> 00:37:25,509 end. Equals 3B, Now I've gotta go for the square bracket. Now 465 00:37:25,509 --> 00:37:29,399 these are inside the curly brackets so it's the square 466 00:37:29,399 --> 00:37:34,067 brackets we tackle. I've made that look like a six. Let's make 467 00:37:34,067 --> 00:37:36,012 it more clearly. AB minus. 468 00:37:36,650 --> 00:37:42,354 5A. Take away, take away everything inside this 469 00:37:42,354 --> 00:37:47,546 bracket. So takeaway 26 a takeaway minus 2B so it's the 470 00:37:47,546 --> 00:37:53,210 same as adding on to be and keep the curly brackets. Now 471 00:37:53,210 --> 00:37:55,098 we can simplify this. 472 00:37:56,270 --> 00:38:00,490 3B minus. 473 00:38:01,540 --> 00:38:09,340 5A takeaway 26. So that's take away altogether or minus 21 A 474 00:38:09,340 --> 00:38:12,590 plus 2B. Close the bracket. 475 00:38:13,620 --> 00:38:18,900 And now come to the final step. Almost the final step. Anyway, 476 00:38:18,900 --> 00:38:23,740 we can get rid of these curly brackets 3B. Take away 477 00:38:23,740 --> 00:38:27,700 everything in the curly brackets. Minus minus gives us 478 00:38:27,700 --> 00:38:33,860 plus 21 A minus +2 BS. That's taking away to be and then. Now 479 00:38:33,860 --> 00:38:39,580 we just need to simplify this and what we've got is to be 480 00:38:39,580 --> 00:38:43,980 taken 3 be sorry 3B takeaway to beat is B. 481 00:38:44,610 --> 00:38:49,650 Plus 21 a so we were able to reduce that. 482 00:38:50,760 --> 00:38:55,248 Huge lump of algebra with three sets of brackets to a very 483 00:38:55,248 --> 00:38:58,988 simple expression. And again, notice where did we start right 484 00:38:58,988 --> 00:39:03,476 in the middle with the most nested set of brackets that we 485 00:39:03,476 --> 00:39:07,964 could find right in the middle. And we simplified that. And then 486 00:39:07,964 --> 00:39:09,834 we gradually worked our way 487 00:39:09,834 --> 00:39:13,832 outwards. Dealing with each set of brackets intern and 488 00:39:13,832 --> 00:39:18,140 simplifying as we went along so as to make the job easier 489 00:39:18,140 --> 00:39:22,089 for ourselves and be more sure that we've got the right 490 00:39:22,089 --> 00:39:23,525 answer at the end.