WEBVTT 00:00:01.000 --> 00:00:08.420 A binomial expression is the sum or difference of two 00:00:08.420 --> 00:00:14.356 terms. So for example 2X plus three Y. 00:00:15.000 --> 00:00:17.226 Is an example of a binomial 00:00:17.226 --> 00:00:22.313 expression. Because it's the sum of the term 2X and the term 3 Y 00:00:22.313 --> 00:00:24.560 is the sum of these two terms. 00:00:25.470 --> 00:00:30.293 Some of the terms could be just numbers, so for example X plus 00:00:30.293 --> 00:00:36.224 one. Is the sum of the term X and the term one, so that's two 00:00:36.224 --> 00:00:37.668 is a binomial expression. 00:00:39.070 --> 00:00:45.890 A-B is the difference of the two terms A&B, so 00:00:45.890 --> 00:00:49.300 that too is a binomial 00:00:49.300 --> 00:00:54.010 expression. Now in your previous work you have seen many binomial 00:00:54.010 --> 00:00:57.343 expressions and you have raised them to different powers. So you 00:00:57.343 --> 00:00:59.161 have squared them, cube them and 00:00:59.161 --> 00:01:03.884 so on. You probably already be very familiar with working with 00:01:03.884 --> 00:01:07.844 the binomial expression like X Plus One and squaring it. 00:01:08.520 --> 00:01:13.968 And you have done that by remembering that when we want to 00:01:13.968 --> 00:01:17.600 square a bracket when multiplying the bracket by 00:01:17.600 --> 00:01:25.065 itself. So X Plus One squared is X Plus One multiplied by X plus 00:01:25.065 --> 00:01:29.740 one. And we remove the brackets by multiplying all the terms in 00:01:29.740 --> 00:01:33.460 the first bracket by all the terms in the SEC bracket, so 00:01:33.460 --> 00:01:35.630 they'll be an X multiplied by X. 00:01:35.630 --> 00:01:36.898 Which is X squared. 00:01:38.360 --> 00:01:42.848 X multiplied by one which is just X. 00:01:44.470 --> 00:01:47.002 1 multiplied by X, which is 00:01:47.002 --> 00:01:51.896 another X. And 1 * 1, which is just one. 00:01:52.850 --> 00:01:58.670 So to tide you all that up X Plus One squared is equal to X 00:01:58.670 --> 00:02:04.530 squared. As an X plus another X which is 2 X. 00:02:04.850 --> 00:02:07.616 Plus the one at the end. 00:02:07.620 --> 00:02:11.796 Note in particular that we have two X here and that came 00:02:11.796 --> 00:02:16.320 from this X here and another X there. I'll come back to that 00:02:16.320 --> 00:02:19.452 point later on and will see why that's important. 00:02:20.640 --> 00:02:24.396 Now suppose we want to raise a binomial expression to our power 00:02:24.396 --> 00:02:27.839 that's higher than two. So suppose we want to cube it, 00:02:27.839 --> 00:02:31.282 raise it to the power four or five or even 32. 00:02:31.820 --> 00:02:34.780 The process of removing the brackets by multiplying term by 00:02:34.780 --> 00:02:38.628 term over and over again is very very cumbersome. I mean, if we 00:02:38.628 --> 00:02:42.180 wanted to workout X plus one to the Seven, you wouldn't really 00:02:42.180 --> 00:02:45.140 want to multiply a pair of brackets by itself several 00:02:45.140 --> 00:02:48.988 times. So what we want is a better way. Better way of doing 00:02:48.988 --> 00:02:54.069 that. And one way of doing it is by means of a triangle of 00:02:54.069 --> 00:02:55.947 numbers, which is called Pascals Triangle. 00:02:57.010 --> 00:03:01.220 Pascal was a 17th century French mathematician and he derived 00:03:01.220 --> 00:03:05.430 this triangle of numbers that will repeat for ourselves now, 00:03:05.430 --> 00:03:10.482 and this is how we form the triangle. We start by writing 00:03:10.482 --> 00:03:12.166 down the number one. 00:03:14.260 --> 00:03:20.126 Then we form a new row and on this nuro we have a one. 00:03:20.630 --> 00:03:21.908 And another one. 00:03:23.730 --> 00:03:27.726 We're going to build up a triangle like this and each nuro 00:03:27.726 --> 00:03:29.724 that we write down will start 00:03:29.724 --> 00:03:32.440 with a one. And will end with a 00:03:32.440 --> 00:03:36.520 one. So my third row is going to begin with the 00:03:36.520 --> 00:03:38.320 one and end with a one. 00:03:39.490 --> 00:03:41.443 And in a few minutes, we'll write a number 00:03:41.443 --> 00:03:42.528 in there in the gap. 00:03:43.810 --> 00:03:47.845 The next row will begin with a one and end with a one and will 00:03:47.845 --> 00:03:49.728 write a number in there and a 00:03:49.728 --> 00:03:53.940 number in there. And in this way we can build a triangle of 00:03:53.940 --> 00:03:56.928 numbers and we can build it as big as we want to. 00:03:58.030 --> 00:04:00.039 How do we find this number in 00:04:00.039 --> 00:04:03.770 here? Well, the number that goes in here we find by 00:04:03.770 --> 00:04:05.160 looking on the row above. 00:04:06.460 --> 00:04:10.233 And looking above to the left and above to the right. 00:04:11.320 --> 00:04:15.701 And adding what we find, there's a one here. There's a one there. 00:04:15.701 --> 00:04:20.419 We add them one and one gives 2 and we write the result in 00:04:20.419 --> 00:04:21.767 there. So there's two. 00:04:22.310 --> 00:04:27.662 On the 3rd row has come by adding that one and that 00:04:27.662 --> 00:04:28.554 one together. 00:04:30.220 --> 00:04:34.518 Let's look at the next row down the number that's going to go in 00:04:34.518 --> 00:04:38.284 here. Is found by looking on the previous row. 00:04:39.910 --> 00:04:43.745 And we look above left which gives us the one we look above 00:04:43.745 --> 00:04:47.285 to the right, which gives us two, and we add the numbers 00:04:47.285 --> 00:04:51.415 that we find, so we're adding a one and two which is 3 and 00:04:51.415 --> 00:04:52.890 we write that in there. 00:04:54.210 --> 00:04:55.740 What about the number here? 00:04:57.920 --> 00:05:02.041 Well again previous row above to the left is 2 above to the 00:05:02.041 --> 00:05:06.796 right is one. We add what we find 2 plus one is 3 and that 00:05:06.796 --> 00:05:07.747 goes in there. 00:05:08.840 --> 00:05:12.090 And we can carry on building this triangle as big as we want 00:05:12.090 --> 00:05:13.840 to. Let's just do one more row. 00:05:14.370 --> 00:05:16.030 We start the row. 00:05:16.820 --> 00:05:20.945 With a one and we finished with a one and we put some numbers in 00:05:20.945 --> 00:05:22.870 here and in here and in here. 00:05:24.270 --> 00:05:26.166 The number that's going to go in here. 00:05:27.260 --> 00:05:31.498 Is found from the previous row by adding the one and the three. 00:05:31.498 --> 00:05:32.802 So 1 + 3. 00:05:33.340 --> 00:05:35.628 Is 4 let me write that in there. 00:05:37.050 --> 00:05:38.891 The number that's going to go in 00:05:38.891 --> 00:05:42.320 here. Well, we look in the previous row above left and 00:05:42.320 --> 00:05:46.491 above right. 3 + 3 is 6 and we write that in there. 00:05:48.500 --> 00:05:53.260 And finally 3 Plus One is 4 when we write that in there. So 00:05:53.260 --> 00:05:54.280 that's another row. 00:05:54.920 --> 00:05:59.370 And what you should do now is practice generating additional 00:05:59.370 --> 00:06:02.930 rows for yourself, and altogether this triangle of 00:06:02.930 --> 00:06:04.710 numbers is called pascals. 00:06:05.810 --> 00:06:08.950 Pascal's triangle. 00:06:09.060 --> 00:06:15.790 OK. Now we're going to use this 00:06:15.790 --> 00:06:19.690 triangle to expand binomial expressions and will see that it 00:06:19.690 --> 00:06:22.420 can make life very easy for us. 00:06:22.440 --> 00:06:28.662 We'll start by going back to 00:06:28.662 --> 00:06:31.773 the expression A+B. 00:06:31.880 --> 00:06:33.428 To the power 2. 00:06:34.150 --> 00:06:37.450 So we have binomial binomial expression here, a I'd be and 00:06:37.450 --> 00:06:39.550 we're raising it to the power 2. 00:06:40.130 --> 00:06:47.046 Let's do it the old way. First of all by multiplying A&B by 00:06:47.046 --> 00:06:50.880 itself. Because we're squaring 00:06:50.880 --> 00:06:55.036 A&B. Let's write down what will get. 00:06:56.570 --> 00:07:00.474 A multiplied by a gives us a squared. 00:07:02.630 --> 00:07:10.260 A multiplied by B will give us a Times B or just a B. 00:07:10.370 --> 00:07:17.298 Be multiplied by a. Gives us a BA. 00:07:18.220 --> 00:07:23.181 And finally, be multiplied by B. Give us a B squared. 00:07:25.000 --> 00:07:29.788 And if we just tidy it, what we found, there's a squared. 00:07:30.480 --> 00:07:31.539 There's an AB. 00:07:32.080 --> 00:07:37.190 And because BA is the same as a bee, there's another a be here. 00:07:37.190 --> 00:07:40.110 So altogether there's two lots of a B. 00:07:40.130 --> 00:07:45.898 And finally, AB squared at the end. 00:07:46.400 --> 00:07:49.580 Now that's the sort of expansion. This sort of removing 00:07:49.580 --> 00:07:52.760 brackets that you've seen many times before. You were already 00:07:52.760 --> 00:07:56.576 very familiar with, but what I want to do is make some 00:07:56.576 --> 00:07:57.848 observations about this result. 00:07:58.890 --> 00:08:04.259 When we expanded A+B to the power two, what we find is that 00:08:04.259 --> 00:08:08.389 as we successively move through these terms that we've written 00:08:08.389 --> 00:08:13.758 down the power of a decreases, it starts off here with an A 00:08:13.758 --> 00:08:17.908 squared. The highest power being two corresponding to the power 00:08:17.908 --> 00:08:20.644 in the original binomial expression, and then every 00:08:20.644 --> 00:08:24.748 subsequent term that power drops. So it was 8 to the power 00:08:24.748 --> 00:08:29.149 2. There's A to the power one in here, although we don't 00:08:29.149 --> 00:08:33.296 normally right the one in and then know as at all, so the 00:08:33.296 --> 00:08:36.805 powers of a decrease as we move from left to right. 00:08:38.400 --> 00:08:39.510 What about bees? 00:08:40.110 --> 00:08:43.883 There's no bees in here. There's a beta. The one in 00:08:43.883 --> 00:08:46.970 there, although we just normally right B&AB to the 00:08:46.970 --> 00:08:51.429 power two there. So as we move from left to right, the powers 00:08:51.429 --> 00:08:55.202 of B increase until we reach the highest power B squared, 00:08:55.202 --> 00:08:58.632 and the squared corresponds to the two in the original 00:08:58.632 --> 00:08:58.975 problem. 00:09:00.550 --> 00:09:03.994 What else can we observe if we look at the coefficients of 00:09:03.994 --> 00:09:07.438 these terms now the coefficients are the numbers in front of each 00:09:07.438 --> 00:09:10.595 of these terms. Well, there's a one in here, although we 00:09:10.595 --> 00:09:13.752 wouldn't normally write it in, there's a two there, and there's 00:09:13.752 --> 00:09:16.622 a one inference of the B squared, although we wouldn't 00:09:16.622 --> 00:09:17.770 normally write it in. 00:09:19.210 --> 00:09:21.890 So the coefficients are 1, two and one. 00:09:23.050 --> 00:09:27.131 Now let me remind you again about pascals triangle. Have a 00:09:27.131 --> 00:09:32.696 copy of the triangle here so we can refer to it. If we look at 00:09:32.696 --> 00:09:37.148 pascals triangle here will see that one 2 one is the numbers 00:09:37.148 --> 00:09:41.600 that's in the 3rd row of pascals triangle. 121 other numbers that 00:09:41.600 --> 00:09:45.310 occur in the expansion of A+B to the power 2. 00:09:46.550 --> 00:09:51.126 There's something else I want to point out that this 2A. B in 00:09:51.126 --> 00:09:56.686 here. Came from a term here 1A B on one BA in there and together 00:09:56.686 --> 00:10:01.362 the one plus the one gave the two in exactly the same way as 00:10:01.362 --> 00:10:05.704 the two in pascals triangle came from adding the one and the one 00:10:05.704 --> 00:10:07.040 in the previous row. 00:10:08.140 --> 00:10:12.472 So Pascal's triangle will give us an easy way of evaluating a 00:10:12.472 --> 00:10:16.804 binomial expression when we want to raise it to an even higher 00:10:16.804 --> 00:10:21.858 power. Let me look at what happens if we want a plus B to 00:10:21.858 --> 00:10:26.190 the power three and will see that we can do this almost 00:10:26.190 --> 00:10:31.039 straight away. What we note is that the highest power now is 3. 00:10:31.610 --> 00:10:34.330 So we start with an A to the power 3. 00:10:35.460 --> 00:10:41.616 Each successive term that power of a will reduce, so they'll be 00:10:41.616 --> 00:10:44.181 a term in a squared. 00:10:45.120 --> 00:10:46.668 That'll be a term in A. 00:10:47.460 --> 00:10:50.001 And then they'll be a term without any Asian at all. 00:10:50.820 --> 00:10:55.929 So as we move from left to right, the powers of a decrease. 00:10:58.470 --> 00:11:01.850 Similarly, as we move from left to right, we want the powers of 00:11:01.850 --> 00:11:05.490 be to increase just as they did here. There will be no bees in 00:11:05.490 --> 00:11:09.566 the first term. ABB to the power one or just B. 00:11:10.300 --> 00:11:11.420 In the second term. 00:11:12.150 --> 00:11:14.110 B to the power two in the next 00:11:14.110 --> 00:11:18.555 term. And then finally there will be a B to the power 00:11:18.555 --> 00:11:22.575 three and we stop it be to the power three that highest 00:11:22.575 --> 00:11:25.255 power corresponding to the power in the original 00:11:25.255 --> 00:11:25.925 binomial expression. 00:11:27.920 --> 00:11:31.160 We need some coefficients. That's the numbers in front of 00:11:31.160 --> 00:11:32.456 each of these terms. 00:11:33.350 --> 00:11:36.750 And the numbers come from the relevant row in pascals 00:11:36.750 --> 00:11:40.830 triangle, and we want the row that begins 1, three and the 00:11:40.830 --> 00:11:44.570 reason why we want the row beginning 1. Three is because 00:11:44.570 --> 00:11:48.310 three is the power in the original expression. So I go 00:11:48.310 --> 00:11:52.730 back to my pascals triangle and I look for the robe beginning 1 00:11:52.730 --> 00:11:54.090 three, which is 1331. 00:11:55.120 --> 00:11:58.135 So these numbers are the coefficients that I need. 00:11:58.710 --> 00:12:01.440 In this expansion I want one. 00:12:02.390 --> 00:12:10.028 331 And just to tidy that up a 00:12:10.028 --> 00:12:13.152 little bit 1A Cube would normally just be written as a 00:12:13.152 --> 00:12:16.610 cubed. 3A squared 00:12:16.610 --> 00:12:19.816 B. 3A B 00:12:19.816 --> 00:12:25.464 squared. And finally 1B cubed which would normally write as 00:12:25.464 --> 00:12:27.012 just be cubed. 00:12:27.570 --> 00:12:33.576 Now, I hope you'll agree that using pascals triangle to expand 00:12:33.576 --> 00:12:36.306 A+B to the power 3. 00:12:36.970 --> 00:12:43.170 Is much simpler than multiplying A+B Times A+B times A+B? 00:12:43.670 --> 00:12:47.254 What I want to do for just before we go on is just actually 00:12:47.254 --> 00:12:50.326 go and do it the long way, just to point something out. 00:12:50.330 --> 00:12:56.028 Let's go back to a plus B. 00:12:56.050 --> 00:13:01.367 To the power three and work it out the long way by noting 00:13:01.367 --> 00:13:07.093 that we can work this out as a plus B multiplied by a plus 00:13:07.093 --> 00:13:08.320 B or squared. 00:13:09.890 --> 00:13:14.906 We've already expanded A+B to the power two, so let's write 00:13:14.906 --> 00:13:22.470 that down. Well remember A+B to the power two we've already seen 00:13:22.470 --> 00:13:24.429 is A squared. 00:13:24.430 --> 00:13:29.410 2-AB And B squared. 00:13:31.710 --> 00:13:34.854 Now to expand this, everything in the first 00:13:34.854 --> 00:13:37.605 bracket must multiply everything in the SEC 00:13:37.605 --> 00:13:41.142 bracket, so we've been a multiplied by a squared 00:13:41.142 --> 00:13:42.714 which is a cubed. 00:13:45.330 --> 00:13:47.580 A multiplied by two AB. 00:13:48.370 --> 00:13:49.438 Which is 2. 00:13:50.000 --> 00:13:52.799 A squared B. 00:13:53.080 --> 00:13:56.530 A multiplied by 00:13:56.530 --> 00:14:02.470 B squared. Which is a B squared. 00:14:04.970 --> 00:14:08.936 We be multiplied by a squared. 00:14:08.940 --> 00:14:10.638 She's BA squared. 00:14:12.390 --> 00:14:18.594 We be multiplied by two AB which is 2A B squared. 00:14:18.690 --> 00:14:24.760 And finally, be multiplied by AB squared is AB cubed. 00:14:25.340 --> 00:14:33.152 To tidy this up as a cubed and then notice there's a 00:14:33.152 --> 00:14:36.407 squared B terms in here. 00:14:37.520 --> 00:14:41.888 And there's also an A squared B turn there, one of them, so 00:14:41.888 --> 00:14:46.592 we've into there and a one there too, and the one gives you three 00:14:46.592 --> 00:14:48.272 lots of A squared fee. 00:14:50.670 --> 00:14:52.658 There's an AB squared. 00:14:53.370 --> 00:14:57.267 Here, and there's more AB squared's there. There's one 00:14:57.267 --> 00:15:02.030 there, two of them there so altogether will have three lots 00:15:02.030 --> 00:15:03.329 of AB squared. 00:15:04.570 --> 00:15:07.790 And finally, the last term at the end B cubed. 00:15:08.600 --> 00:15:12.212 That's working out the expansion the long way. Why have I done 00:15:12.212 --> 00:15:15.824 that? Well, I've only done that just to point out something to 00:15:15.824 --> 00:15:20.640 you and I want to point out that the three in here in the three A 00:15:20.640 --> 00:15:22.747 squared B came from adding a 2 00:15:22.747 --> 00:15:28.384 here. And a one in there 2 plus the one gave you the three. 00:15:29.180 --> 00:15:33.188 Similarly, this three here came from a one lot of AB squared 00:15:33.188 --> 00:15:37.864 there and two lots of AB squared there. So the one plus the two 00:15:37.864 --> 00:15:41.872 gave you the three, and that mirrors exactly what we had when 00:15:41.872 --> 00:15:45.212 we generated the triangle, because the three here came back 00:15:45.212 --> 00:15:49.888 from adding the one in the two in the row above and the three 00:15:49.888 --> 00:15:53.562 here came from adding two and one in the row above. 00:15:53.620 --> 00:16:00.094 Let's have a look at another example and see if we can just 00:16:00.094 --> 00:16:04.576 write the answer down straightaway. Suppose we want to 00:16:04.576 --> 00:16:07.564 expand A+B or raised to the 00:16:07.564 --> 00:16:12.828 power 4. Well, this is straightforward to do. We know 00:16:12.828 --> 00:16:18.171 that when we expand this, our highest power of a will be 4 00:16:18.171 --> 00:16:21.459 because that's the power in the original expression. 00:16:22.650 --> 00:16:26.800 And thereafter every subsequent term will have a power reduced 00:16:26.800 --> 00:16:31.365 by one each time. So there will be an A cubed. 00:16:32.280 --> 00:16:36.680 And a squared and A and then, no worries at all. 00:16:38.090 --> 00:16:41.306 As we move from left to right, the powers of B will 00:16:41.306 --> 00:16:44.254 increase. There will be none at all in the first term. 00:16:45.470 --> 00:16:47.398 And they'll be a big to the one. 00:16:47.398 --> 00:16:50.010 Or just be. A bit of the two. 00:16:51.250 --> 00:16:55.904 Beta three will be cubed and finally the last term will be to 00:16:55.904 --> 00:16:59.842 the four and again the highest power corresponding to the power 00:16:59.842 --> 00:17:01.632 four in the original expression. 00:17:02.850 --> 00:17:06.600 And all we need now are the coefficients. The coefficients 00:17:06.600 --> 00:17:10.725 come from the appropriate role in the triangle and this time 00:17:10.725 --> 00:17:15.600 because we're looking at power four, we want to look at the Roo 00:17:15.600 --> 00:17:21.567 beginning 14. The row beginning 1 four is 14641. 00:17:22.220 --> 00:17:23.680 Those are the coefficients that 00:17:23.680 --> 00:17:30.580 will need. 14641 00:17:31.980 --> 00:17:37.740 And just to tidy it up, we wouldn't normally right the one 00:17:37.740 --> 00:17:45.420 in there and the one in there so A&B to the four is 8 to 4 00:17:45.420 --> 00:17:48.300 four A cubed B that's that. 00:17:48.850 --> 00:17:52.138 6A squared, B squared. 00:17:52.140 --> 00:17:55.572 4A B 00:17:55.572 --> 00:18:02.406 cubed. And finally, be to the power 4. 00:18:03.820 --> 00:18:07.672 OK, so I hope you'll agree that using pascals triangle to get 00:18:07.672 --> 00:18:10.561 this expansion was much simpler than multiplying this bracket 00:18:10.561 --> 00:18:14.734 over and over by itself. Lots and lots of times that way is 00:18:14.734 --> 00:18:18.907 also prone to error, so if you can get used to using pascals 00:18:18.907 --> 00:18:24.070 triangle. We can use the same technique even when we have 00:18:24.070 --> 00:18:27.101 slightly more complicated expressions. Let's do another 00:18:27.101 --> 00:18:34.740 example. Suppose we want to expand 2X plus Y all to 00:18:34.740 --> 00:18:36.705 the power 3. 00:18:37.840 --> 00:18:40.678 So it's more complicated this time because I just haven't got 00:18:40.678 --> 00:18:43.774 a single term here, but I've actually got a 2X in there. 00:18:45.070 --> 00:18:48.070 The principle is exactly the same. 00:18:49.510 --> 00:18:53.358 What will do is will write this term down first. The whole of 00:18:53.358 --> 00:18:58.158 2X. And just like before, it will be raised to the highest 00:18:58.158 --> 00:19:02.214 possible power which is 3 and that corresponds to the three in 00:19:02.214 --> 00:19:03.228 the original problem. 00:19:06.140 --> 00:19:11.138 Every subsequent term will have a 2X in it, but as we go from 00:19:11.138 --> 00:19:15.779 left to right, the power of 2X will decrease, so the next term 00:19:15.779 --> 00:19:17.921 will have a 2X or squared. 00:19:19.250 --> 00:19:23.675 The next term will have a 2X to the power one or just 2X, and 00:19:23.675 --> 00:19:26.920 then there won't be any at all in the last term. 00:19:29.950 --> 00:19:34.304 Powers of Y will increase as we move from the left to the right, 00:19:34.304 --> 00:19:36.481 so there won't be any in the 00:19:36.481 --> 00:19:38.778 first term. Then they'll be Y. 00:19:39.600 --> 00:19:43.353 Then they'll be Y squared and finally Y cubed. 00:19:45.830 --> 00:19:49.360 And then we remember the coefficients. Where do we get 00:19:49.360 --> 00:19:50.419 the coefficients from? 00:19:50.990 --> 00:19:54.708 Well, because we're looking at power three, we go to pascals 00:19:54.708 --> 00:19:57.074 triangle and we look for the row 00:19:57.074 --> 00:20:01.330 beginning 13. You might even remember those numbers now. 00:20:01.330 --> 00:20:06.660 We've seen it so many times. The numbers are 1331. Those are the 00:20:06.660 --> 00:20:08.300 coefficients we require, 1331. 00:20:09.010 --> 00:20:13.498 So I want one of those three of those three of 00:20:13.498 --> 00:20:15.130 those, one of those. 00:20:16.790 --> 00:20:20.552 And there's just a bit more tidying up to do to 00:20:20.552 --> 00:20:21.578 finish it off. 00:20:22.760 --> 00:20:26.480 Here we've got 2 to the Power 3, two cubed that's eight. 00:20:27.230 --> 00:20:33.022 X cubed And the one just is, one could 00:20:33.022 --> 00:20:36.220 just stay there 1. Multiply by all that is not going to do 00:20:36.220 --> 00:20:37.450 anything else, just 8X cubed. 00:20:38.390 --> 00:20:42.849 What about this term? There's a 2 squared, which is 4, and it's 00:20:42.849 --> 00:20:47.651 got to be multiplied by three. So 4 threes are 12, so we have 00:20:47.651 --> 00:20:53.541 12. What about powers of X? Well, there be an X squared. 00:20:53.640 --> 00:20:54.830 Why? 00:20:56.220 --> 00:21:00.828 In this term, we've just got 2X to the power one. That's 00:21:00.828 --> 00:21:05.436 just 2X, so this is just three times 2X, which is 6X, 00:21:05.436 --> 00:21:07.356 and there's a Y squared. 00:21:08.750 --> 00:21:12.544 And finally, there's just the Y cubed at the end. One Y cubed is 00:21:12.544 --> 00:21:16.960 just Y cubed. So there we've expanded the binomial expression 00:21:16.960 --> 00:21:22.560 2X plus Y to the power three in just a couple of lines using 00:21:22.560 --> 00:21:26.996 pascals triangle. Let's look at another one. Suppose this time 00:21:26.996 --> 00:21:31.160 we want one plus P different letter just for a change one 00:21:31.160 --> 00:21:33.936 plus P or raised to the power 4. 00:21:34.490 --> 00:21:38.942 In lots of ways, this is going to be a bit simpler. 00:21:39.630 --> 00:21:43.110 Because as we move through the terms from left to right, we 00:21:43.110 --> 00:21:44.850 want powers of the first term, 00:21:44.850 --> 00:21:49.202 which is one. It won't want to the Power 4 one to the Power 3, 00:21:49.202 --> 00:21:52.892 one to the power two and so on, but want to any power is still 00:21:52.892 --> 00:21:54.368 one that's going to make life 00:21:54.368 --> 00:21:58.143 easier for ourselves. So 1 to the power four is just one. 00:21:58.880 --> 00:22:00.065 And then thereafter they'll be 00:22:00.065 --> 00:22:02.590 just one. All the way through. 00:22:04.270 --> 00:22:07.678 We want the powers of P to increase. We don't want any 00:22:07.678 --> 00:22:09.098 peace in the first term. 00:22:09.870 --> 00:22:11.460 We want to be there. 00:22:12.170 --> 00:22:17.154 P squared there the next time will have a P cubed in and the 00:22:17.154 --> 00:22:21.070 last term will have a Peter. The four in these ones. 00:22:21.730 --> 00:22:25.630 Are the powers of the first term one, so 1 to the 4th, one to 00:22:25.630 --> 00:22:29.010 three, 1 to the two, 1 to the one which is just one? 00:22:29.860 --> 00:22:31.348 And no ones there at all. 00:22:33.450 --> 00:22:36.290 And finally, we want some coefficients and the 00:22:36.290 --> 00:22:39.485 coefficients come from pascals triangle. This time the row 00:22:39.485 --> 00:22:42.325 beginning 1, four. Because of this powerful here. 00:22:43.060 --> 00:22:48.709 So the numbers we want our 14641. 00:22:49.800 --> 00:22:51.120 1. 00:22:52.040 --> 00:22:56.320 4. 6. 00:22:57.450 --> 00:23:02.955 4. One, let's just tidy it 00:23:02.955 --> 00:23:05.046 up as one. 00:23:06.400 --> 00:23:09.865 4 * 1 is just four P. 00:23:09.940 --> 00:23:13.915 6 * 1 is 66 00:23:13.915 --> 00:23:20.935 P squared. 4 * 1 is 4 P cubed. 00:23:20.940 --> 00:23:25.518 And last of all, one times Peter the four is just Peter the four. 00:23:26.290 --> 00:23:30.580 Again, another example of a binomial expression raised to a 00:23:30.580 --> 00:23:35.299 power, and we can almost write the answer straight down using 00:23:35.299 --> 00:23:38.731 the triangle instead of multiplying those brackets out 00:23:38.731 --> 00:23:40.447 over and over again. 00:23:40.460 --> 00:23:46.928 Now, sometimes either or both of the terms in the binomial 00:23:46.928 --> 00:23:49.280 expression might be negative. 00:23:49.790 --> 00:23:54.197 So let's have a look at an example where one of the terms 00:23:54.197 --> 00:23:56.231 is negative. So suppose we want 00:23:56.231 --> 00:23:56.909 to expand. 00:23:57.710 --> 00:24:04.032 3A. Minus 2B, so I've got a term 00:24:04.032 --> 00:24:07.398 that's negative now, minus 2B, and let's suppose we want this 00:24:07.398 --> 00:24:08.622 to the power 5. 00:24:09.610 --> 00:24:15.010 3A minus 2B all raised to the power 5. 00:24:15.620 --> 00:24:17.895 This is going to be a bit more complicated this time, so let's 00:24:17.895 --> 00:24:19.120 see how we get on with it. 00:24:19.760 --> 00:24:25.120 As before. We want to take our first term. 00:24:25.660 --> 00:24:28.720 And raise it to the highest power, the highest 00:24:28.720 --> 00:24:29.740 power being 5. 00:24:30.940 --> 00:24:34.314 So our first term will be 3A. 00:24:34.320 --> 00:24:36.348 All raised to the power 5. 00:24:36.870 --> 00:24:40.699 The next term will have a 3A 00:24:40.699 --> 00:24:45.122 in it. And this time it will be raised to the power 4. 00:24:48.910 --> 00:24:53.995 There be another term with a 3A in. It'll be 3A to the power 3. 00:24:55.220 --> 00:24:58.250 Then 3A to the power 2. 00:24:59.920 --> 00:25:04.288 Then 3A to the power one, and then they'll be a final term 00:25:04.288 --> 00:25:06.640 that doesn't have 3A in it at 00:25:06.640 --> 00:25:10.588 all. That deals with this first term. 00:25:12.810 --> 00:25:14.896 Let's deal with the minus 2B now. 00:25:16.830 --> 00:25:21.094 In the first term here, there won't be any minus two BS at 00:25:21.094 --> 00:25:24.374 all, but there after the powers of this term will 00:25:24.374 --> 00:25:28.310 increase as we move from left to right exactly as before. So 00:25:28.310 --> 00:25:32.574 when we get to the second term here will need a minus two 00:25:32.574 --> 00:25:32.902 fee. 00:25:35.640 --> 00:25:39.982 When we get to the next term will leave minus 2B and we're 00:25:39.982 --> 00:25:41.318 going to square it. 00:25:43.180 --> 00:25:47.107 Minus 2B raised to the power 3. 00:25:48.630 --> 00:25:51.906 Minus two be raised to the power 00:25:51.906 --> 00:25:58.305 4. And the last term will be minus two be raised to the power 00:25:58.305 --> 00:26:01.776 5. The power five corresponding to the highest 00:26:01.776 --> 00:26:03.466 power in the original problem. 00:26:06.570 --> 00:26:11.370 We also need our coefficients. The numbers in front of each of 00:26:11.370 --> 00:26:12.570 these six terms. 00:26:13.420 --> 00:26:18.108 The coefficients come from the row beginning 15. 00:26:18.640 --> 00:26:21.124 Because the problem has a power five in it. 00:26:22.330 --> 00:26:26.590 The coefficients are one 510-1051. 00:26:27.880 --> 00:26:32.344 One 510-1051 so we want one of those. 00:26:33.810 --> 00:26:36.039 Five of those. 00:26:37.180 --> 00:26:40.052 Ten of 00:26:40.052 --> 00:26:43.522 those. Ten of 00:26:43.522 --> 00:26:47.367 those. Five of those, and finally one of those you can see 00:26:47.367 --> 00:26:50.965 now why I left a lot of space when I was writing all this 00:26:50.965 --> 00:26:53.792 down. There's a lot of things to tidy up in here. 00:26:55.060 --> 00:26:58.852 Just to tidy all this up, we need to remember that when 00:26:58.852 --> 00:27:02.328 we raise a negative number to say the power two, the 00:27:02.328 --> 00:27:06.120 results going to be positive when we raise it to an even 00:27:06.120 --> 00:27:09.596 even power, the result would be positive. So this term is 00:27:09.596 --> 00:27:13.388 going to be positive and the minus 2B to the power four 00:27:13.388 --> 00:27:14.652 will also become positive. 00:27:15.820 --> 00:27:19.825 When we raise it to an odd power like 3 or the five, the result 00:27:19.825 --> 00:27:23.296 is going to be negative. So our answer is going to have some 00:27:23.296 --> 00:27:24.364 positive and some negative 00:27:24.364 --> 00:27:27.620 numbers in it. Let's tidy it all up. 00:27:28.720 --> 00:27:31.030 Go to Calculator for this, 'cause I'm going to raise some 00:27:31.030 --> 00:27:32.290 of these numbers to some powers. 00:27:33.050 --> 00:27:36.098 First of all I want to raise 3 to the power 5. 00:27:38.480 --> 00:27:41.924 3 to the power five is 00:27:41.924 --> 00:27:45.088 243. So I have 243. 00:27:45.930 --> 00:27:48.640 A to the power 5. 00:27:48.640 --> 00:27:51.250 And it's all multiplied by one which isn't going to 00:27:51.250 --> 00:27:51.772 change anything. 00:27:53.220 --> 00:27:56.160 Now here we've got a negative number because this is minus 2 00:27:56.160 --> 00:27:59.345 be raised to the power one is going to be negative, so this 00:27:59.345 --> 00:28:01.060 term is going to have a minus 00:28:01.060 --> 00:28:05.466 sign at the front. We've got 3 to the power 4. 00:28:06.670 --> 00:28:11.742 Well, I know 3 squared is 9 and 9, nine 481, so 3 to the power 00:28:11.742 --> 00:28:14.690 four is 81. Five 210 00:28:15.430 --> 00:28:21.018 So I'm going to multiply 81 by 10, which is 810th. 00:28:21.970 --> 00:28:25.652 There will be 8 to the power 00:28:25.652 --> 00:28:29.228 4. And a single be. 00:28:29.280 --> 00:28:30.560 So that's my next term. 00:28:31.280 --> 00:28:35.720 Now what have we got left? There's 3 to the power three 00:28:35.720 --> 00:28:38.310 which is 3 cubed, which is 27. 00:28:39.230 --> 00:28:42.310 Multiplied by two squared, which is 4. 00:28:45.370 --> 00:28:46.910 All multiplied by 10. 00:28:47.950 --> 00:28:50.929 Which is 1080. 00:28:50.930 --> 00:28:54.350 8 to the 00:28:54.350 --> 00:28:59.760 power 3. B to the power 2. 00:29:01.440 --> 00:29:04.128 And here we have two cubed which 00:29:04.128 --> 00:29:07.320 is 8. 3 squared which is 9. 00:29:08.300 --> 00:29:14.495 9 eight 472 * 10 is 720. 00:29:14.500 --> 00:29:18.378 There will be an A squared from 00:29:18.378 --> 00:29:23.980 this term. And not be a B cubed from the last time. 00:29:25.260 --> 00:29:27.108 What about here? 00:29:27.990 --> 00:29:29.817 Well, we've 2 to the power 4. 00:29:30.490 --> 00:29:31.630 Which is 16. 00:29:32.200 --> 00:29:34.648 5 three is a 15 here. 00:29:35.330 --> 00:29:42.142 And 15 * 16 is 240. It'll be positive because here we been 00:29:42.142 --> 00:29:48.954 negative number to an even power 248 to the power one or just 00:29:48.954 --> 00:29:52.750 a. B to the power 4. 00:29:55.040 --> 00:29:59.429 And finally. There will be one more term and that will be minus 00:29:59.429 --> 00:30:01.501 2 to the Power 5, which is going 00:30:01.501 --> 00:30:07.280 to be negative. 32 B to the power 5. 00:30:07.400 --> 00:30:11.666 And that's the expansion of this rather complicated expression, 00:30:11.666 --> 00:30:16.406 which had both positive and negative quantities in it. And 00:30:16.406 --> 00:30:20.198 again, we've used pascals triangle to do that. 00:30:21.260 --> 00:30:27.299 We can use exactly the same method even if there are 00:30:27.299 --> 00:30:33.338 fractions involved, so let's have a look at an example where 00:30:33.338 --> 00:30:37.730 there's some fractions. Suppose we want to expand. 00:30:37.730 --> 00:30:44.113 This time 1 + 2 over X, so I've deliberately put a fraction 00:30:44.113 --> 00:30:47.550 in there all to the power 3. 00:30:48.790 --> 00:30:50.210 Let's see what happens. 00:30:51.140 --> 00:30:57.604 1 + 2 over X to the power 00:30:57.604 --> 00:31:03.334 3. Well. We start with one raised to the highest 00:31:03.334 --> 00:31:05.718 power which is 1 to the power 3. 00:31:06.460 --> 00:31:07.888 Which is still 1. 00:31:08.750 --> 00:31:12.230 And once at, any power will still be one's remove all the 00:31:12.230 --> 00:31:13.390 way through the calculation. 00:31:14.870 --> 00:31:21.383 Will have two over X raised first of all to the power one. 00:31:21.480 --> 00:31:27.199 Two over X to the power 2. 00:31:27.750 --> 00:31:32.468 And two over X to the power three and we stop there. When we 00:31:32.468 --> 00:31:33.816 reached the highest power. 00:31:34.470 --> 00:31:36.927 Which corresponds to the power in the original problem. 00:31:38.980 --> 00:31:43.534 We need the coefficients of each of these terms from pascals 00:31:43.534 --> 00:31:47.260 triangle and the row in the triangle beginning 13. 00:31:48.090 --> 00:31:54.679 Those numbers are 1331, so there's one of these three of 00:31:54.679 --> 00:31:56.600 those. Three of those. 00:31:57.220 --> 00:31:59.040 I'm one of those. 00:31:59.800 --> 00:32:03.128 And all we need to do now is tidy at what we've got. 00:32:04.180 --> 00:32:05.488 So there's once. 00:32:07.030 --> 00:32:10.982 Two over X to the power one is just two over X. We're 00:32:10.982 --> 00:32:14.630 going to multiply it by three, so 3 twos are six will 00:32:14.630 --> 00:32:16.150 have 6 divided by X. 00:32:17.980 --> 00:32:23.328 Here there's a 2 squared, which is 4. Multiply it by three so we 00:32:23.328 --> 00:32:28.294 have 12 divided by X to the power 2 divided by X squared. 00:32:29.490 --> 00:32:31.566 And finally, there's 2 to the 00:32:31.566 --> 00:32:33.929 power 3. Which is 8. 00:32:34.840 --> 00:32:40.864 And this time it's divided by X to the power 34X cubed. 00:32:41.470 --> 00:32:45.122 So that's a simple example which illustrates how we can apply 00:32:45.122 --> 00:32:48.442 exactly the same technique even when the refraction is involved. 00:32:49.330 --> 00:32:57.258 Now, that's not quite the end of the 00:32:57.258 --> 00:33:02.782 story. The problem is, supposing I were to ask you to expand a 00:33:02.782 --> 00:33:05.842 binomial expression to a very large power, suppose I wanted 00:33:05.842 --> 00:33:10.738 one plus X to the power 32 or one plus X to the power 127. You 00:33:10.738 --> 00:33:14.410 have an awful lot of rows of pascals triangle to generate if 00:33:14.410 --> 00:33:16.552 you wanted to do it this way. 00:33:17.250 --> 00:33:20.450 Fortunately, there's an alternative way, and it involves 00:33:20.450 --> 00:33:22.450 a theorem called the binomial 00:33:22.450 --> 00:33:26.660 theorem. So let's just have a look at what the binomial 00:33:26.660 --> 00:33:27.320 theorem says. 00:33:28.540 --> 00:33:36.000 The binomial theorem allows us to develop an expansion of 00:33:36.000 --> 00:33:42.714 the binomial expression A+B raised to the power N. 00:33:44.220 --> 00:33:48.510 And it allows us to get an expansion in terms of 00:33:48.510 --> 00:33:52.020 decreasing powers of a, exactly as we've seen before. 00:33:53.310 --> 00:33:56.790 And increasing powers of B exactly as we've seen before. 00:33:56.790 --> 00:34:02.010 And it I'm going to quote the theorem for the case when N is a 00:34:02.010 --> 00:34:03.054 positive whole number. 00:34:04.270 --> 00:34:08.806 This theorem will actually work when is negative and when it's a 00:34:08.806 --> 00:34:11.452 fraction, but only under exceptional circumstances, which 00:34:11.452 --> 00:34:16.366 we're not going to discuss here. So in all these examples, N will 00:34:16.366 --> 00:34:18.256 be a positive whole number. 00:34:19.430 --> 00:34:25.742 Now what the theorem 00:34:25.742 --> 00:34:28.898 says is 00:34:28.898 --> 00:34:34.693 this. A+B to the power N is given by the following expansion 00:34:34.693 --> 00:34:36.608 A to the power N. 00:34:37.230 --> 00:34:40.035 Now that looks familiar, doesn't it? Because as in all the 00:34:40.035 --> 00:34:42.585 examples we've seen before, we've taken the first term and 00:34:42.585 --> 00:34:45.390 raised it to the highest power. The power in the original 00:34:45.390 --> 00:34:46.920 question 8 to the power N. 00:34:47.670 --> 00:34:51.730 Then there's a next term, and the next term will have an A to 00:34:51.730 --> 00:34:53.180 the power N minus one. 00:34:53.920 --> 00:34:57.892 And a B in it. That's exactly as we've seen before, because we're 00:34:57.892 --> 00:35:00.732 starting to see the terms involving be appear and the 00:35:00.732 --> 00:35:02.720 powers event at the powers of a 00:35:02.720 --> 00:35:07.018 a decreasing. We want a coefficient in here and the 00:35:07.018 --> 00:35:10.267 binomial theorem tells us that the coefficient is NTH. 00:35:12.140 --> 00:35:15.740 The next term. 00:35:15.740 --> 00:35:19.337 As an A to the power N minus two in it. 00:35:20.540 --> 00:35:23.873 Along with the line we had before of decreasing the powers 00:35:23.873 --> 00:35:27.509 and increasing the power of be will give us a B squared. 00:35:28.130 --> 00:35:31.847 And the binomial theorem tells us the coefficient to 00:35:31.847 --> 00:35:35.977 right in here and the coefficient this time is NN 00:35:35.977 --> 00:35:38.042 minus one over 2 factorial. 00:35:39.350 --> 00:35:43.860 In case you don't know what this notation means, 2 factorial 00:35:43.860 --> 00:35:45.500 means 2 * 1. 00:35:46.430 --> 00:35:49.438 That's called 2 factorial. 00:35:50.370 --> 00:35:57.720 And this series goes on and on and on. The next term will be 00:35:57.720 --> 00:36:04.020 NN minus one and minus two over 3 factorial, and there's a 00:36:04.020 --> 00:36:09.795 pattern developing here. You see, here we had an N&NN minus 00:36:09.795 --> 00:36:13.048 one. And minus one and minus 2. 00:36:13.550 --> 00:36:19.160 With a 3 factorial at the bottom where we had a two factor at the 00:36:19.160 --> 00:36:22.900 bottom before 3 factorial means 3 * 2 * 1. 00:36:23.870 --> 00:36:28.880 The power of a will be 1 less again, which this time will be A 00:36:28.880 --> 00:36:30.550 to the N minus three. 00:36:31.470 --> 00:36:35.264 And we want to power of bee which is B to the power 3. 00:36:36.920 --> 00:36:40.496 So all the way through this theorem you'll see the powers of 00:36:40.496 --> 00:36:45.472 a are decreasing. And the powers of B are increasing. Now this 00:36:45.472 --> 00:36:50.992 series goes on and on and on until we reach the term B to the 00:36:50.992 --> 00:36:55.776 power N. When it stops. So this is a finite series. It stops 00:36:55.776 --> 00:36:57.984 after a finite number of terms. 00:36:58.910 --> 00:37:02.331 Now, the theorems often quoted in this form, but it's also 00:37:02.331 --> 00:37:03.886 often quoted in a slightly 00:37:03.886 --> 00:37:08.346 simpler form. And it's quoted in the form for which a is the 00:37:08.346 --> 00:37:09.776 simple value of just one. 00:37:10.410 --> 00:37:16.258 And B is X. Now when a is one, all of these A to the power ends 00:37:16.258 --> 00:37:22.106 or A to the N minus one A to the N minus two. Each one of those 00:37:22.106 --> 00:37:26.234 terms will just simplify to the number one, so the whole thing 00:37:26.234 --> 00:37:29.674 looks simpler. So let's write down the binomial theorem again 00:37:29.674 --> 00:37:33.458 for the special case when a is one and these X. 00:37:33.500 --> 00:37:40.380 This time will get one plus X raised to the 00:37:40.380 --> 00:37:43.388 power N. Is equal to. 00:37:44.040 --> 00:37:44.880 1. 00:37:45.980 --> 00:37:50.070 Plus N. X. 00:37:51.870 --> 00:37:56.154 Plus NN minus one over 2 factorial X squared, and you can 00:37:56.154 --> 00:37:59.724 see what's happening. This second term X is starting to 00:37:59.724 --> 00:38:03.651 appear and and its powers increasing as we move from left 00:38:03.651 --> 00:38:08.292 to right. So even X&X squared the next time will have an X 00:38:08.292 --> 00:38:13.290 cubed in it, one to any power is still one, so I don't actually 00:38:13.290 --> 00:38:15.075 need to write it down. 00:38:15.710 --> 00:38:22.118 The next term will be NN minus one and 00:38:22.118 --> 00:38:27.102 minus two over 3 factorial X cubed. 00:38:28.330 --> 00:38:33.650 The next term will be NN minus one and minus 2 N minus three 00:38:33.650 --> 00:38:39.350 over 4 factorial X to the four, and this will go on and on until 00:38:39.350 --> 00:38:43.910 eventually you'll get to the stage where you get to the last 00:38:43.910 --> 00:38:49.230 term raised to the highest power you'll get to X to the power N, 00:38:49.230 --> 00:38:51.130 and the series will stop. 00:38:52.120 --> 00:38:55.288 So this is a slightly simpler form of the theorem, and it's 00:38:55.288 --> 00:38:56.608 often quoted in this form. 00:38:57.260 --> 00:39:02.562 Now let's use it to examine some binomial expressions that you're 00:39:02.562 --> 00:39:04.490 already very familiar with. 00:39:04.490 --> 00:39:08.988 Let's suppose we want to expand one plus X or raised to the 00:39:08.988 --> 00:39:13.140 power two. Now I've written down the theorem again so we can 00:39:13.140 --> 00:39:18.330 refer to it and this is printed in the notes. If you want to use 00:39:18.330 --> 00:39:20.060 the one in the notes. 00:39:21.300 --> 00:39:27.045 So we've one plus X to the power N. In our problem, we've got one 00:39:27.045 --> 00:39:33.556 plus X to the power two, so all we have to do is let NB two in 00:39:33.556 --> 00:39:35.471 all of this formula through 00:39:35.471 --> 00:39:40.460 here. So let's see what we get or from the theorem. 00:39:40.980 --> 00:39:44.040 The first thing will write down is just the one. 00:39:45.680 --> 00:39:52.904 Then we want NX, but N IS two, so will just put plus 2X. 00:39:52.910 --> 00:39:56.628 And then the next term we want is going to be a term 00:39:56.628 --> 00:39:59.774 involving X to the power two, but that's the highest power 00:39:59.774 --> 00:40:03.492 we want because we've got a power to in here. We want to 00:40:03.492 --> 00:40:07.210 stop when we get to X to the power two, so we're actually 00:40:07.210 --> 00:40:10.928 already at the end with the next term, and we just want an 00:40:10.928 --> 00:40:13.216 X to the power two on its own. 00:40:14.720 --> 00:40:18.392 1 + 2 X plus X squared and that's the expansion that 00:40:18.392 --> 00:40:21.452 you're already very familiar with, and you'll notice in it 00:40:21.452 --> 00:40:25.430 that the powers of X increase as we move through from left to 00:40:25.430 --> 00:40:29.102 right, and there's powers of one in there, but we don't see 00:40:29.102 --> 00:40:32.468 them, and the one 2 one other numbers in pascals triangle. 00:40:33.970 --> 00:40:40.262 Let's look at the theorem for the case when is 3, let's expand 00:40:40.262 --> 00:40:43.650 one plus X to the power 3. 00:40:44.810 --> 00:40:47.461 I'm going to use the theorem again, but this time we're 00:40:47.461 --> 00:40:48.666 going to let NB 3. 00:40:51.020 --> 00:40:54.608 So we want 1 + 3 00:40:54.608 --> 00:41:01.220 X. And then we want 3. 00:41:01.820 --> 00:41:07.604 3 - 1 three minus one is 2. 00:41:08.480 --> 00:41:11.250 All divided by 2 factorial. 00:41:12.130 --> 00:41:14.758 Than an X squared. 00:41:16.150 --> 00:41:19.776 And then the next term will be a term involving X cubed, which is 00:41:19.776 --> 00:41:22.884 the term that we stop with because we're only working 2X to 00:41:22.884 --> 00:41:26.510 the power three here. So the last term will be just a plus X 00:41:26.510 --> 00:41:31.760 cubed. We can tie this up to 1 + 3 X. 00:41:32.630 --> 00:41:37.432 2 factorial is 2 * 1, which is just two little cancel with the 00:41:37.432 --> 00:41:42.234 two at the top, so will be left with just three X squared, and 00:41:42.234 --> 00:41:43.606 finally an X cubed. 00:41:43.650 --> 00:41:47.268 And again, that's something that you're already very familiar 00:41:47.268 --> 00:41:50.484 with. You'll notice the coefficients, the 1331 other 00:41:50.484 --> 00:41:55.308 numbers we've seen many times in pascals triangle the powers of X 00:41:55.308 --> 00:42:00.615 increase. As we move from the left to the right, and this is a 00:42:00.615 --> 00:42:04.635 finite series, it stops when we get to the term involving X 00:42:04.635 --> 00:42:07.315 cubed corresponding to this highest power over there. 00:42:07.600 --> 00:42:14.812 Now. That suppose we want to look at 00:42:14.812 --> 00:42:18.618 it and more complicated problem. Suppose we want to workout one 00:42:18.618 --> 00:42:21.386 plus X to the power 32. Now you 00:42:21.386 --> 00:42:25.077 would never. Use pascals triangle to attempt this problem 00:42:25.077 --> 00:42:28.938 because you'd have to generate so many rows of the triangle, 00:42:28.938 --> 00:42:31.044 but we can use the binomial 00:42:31.044 --> 00:42:35.032 theorem. What I'm going to do is I'm going to write down 00:42:35.032 --> 00:42:37.752 the first three terms of the series using the binomial 00:42:37.752 --> 00:42:41.288 theorem, and I'm going to use it with N being equal to 32. 00:42:43.290 --> 00:42:50.360 So we're putting any 32 in. The theorem will get 1 + 32 X. 00:42:50.360 --> 00:42:53.390 That's the one plus the NX. 00:42:54.400 --> 00:42:57.406 We want an which is 32. 00:42:58.390 --> 00:43:01.134 And minus one which will be 31. 00:43:01.730 --> 00:43:04.038 All over 2 factorial. 00:43:05.380 --> 00:43:06.490 X squared 00:43:07.500 --> 00:43:11.920 And we know that this series will go on and on until we 00:43:11.920 --> 00:43:16.000 reached the term, the last term being X to the power 32. 00:43:16.660 --> 00:43:20.911 But I only want to look at the first three terms here in 00:43:20.911 --> 00:43:24.508 this problem, so the first three terms are just going to 00:43:24.508 --> 00:43:29.413 be 1 + 32 X and we want to simplify this. We've got 32 * 00:43:29.413 --> 00:43:31.375 31 and then divided by two. 00:43:34.940 --> 00:43:38.219 Which is 496. 00:43:38.310 --> 00:43:43.000 And I just put some dots there to show that this series goes on 00:43:43.000 --> 00:43:46.685 a lot further than the terms that I've just written down 00:43:46.685 --> 00:43:53.306 there. I'm going to have a look at a couple more examples with 00:43:53.306 --> 00:43:58.289 some ingenuity. We can use the theorem in a slightly different 00:43:58.289 --> 00:44:02.819 form. Suppose we want to expand this binomial expression this 00:44:02.819 --> 00:44:09.161 time, I'm going to look at one plus Y divided by 3. All raised 00:44:09.161 --> 00:44:14.144 to the power 10 and suppose that I'm interested. I'm interested 00:44:14.144 --> 00:44:15.956 in generating the first. 00:44:17.000 --> 00:44:23.369 Four terms. Let's see how we can do that. Well, we've got our 00:44:23.369 --> 00:44:28.517 theorem. I've written it down again here for us in terms of 00:44:28.517 --> 00:44:35.381 one plus X to the power N. We can use it in this problem if we 00:44:35.381 --> 00:44:36.668 replace every X. 00:44:37.200 --> 00:44:40.088 In the theorem with a Y over 3. 00:44:41.060 --> 00:44:45.344 So everywhere there's an X in the theorem, I'm going to write 00:44:45.344 --> 00:44:49.628 Y divided by three and then the pattern will match exactly what 00:44:49.628 --> 00:44:54.269 we have in the theorem ends going to be 10 in this problem. 00:44:54.270 --> 00:44:59.665 So let's see what we get will have one plus Y over three 00:44:59.665 --> 00:45:02.570 raised to the power 10 is equal 00:45:02.570 --> 00:45:06.198 to. Well, we start with a one 00:45:06.198 --> 00:45:10.060 as always. Then we 00:45:10.060 --> 00:45:14.036 want NX. And 00:45:14.036 --> 00:45:18.730 it's 10. And we said that instead of X, but replacing the 00:45:18.730 --> 00:45:20.446 X with a Y over 3. 00:45:20.960 --> 00:45:24.184 So we have a Y over three there. 00:45:25.470 --> 00:45:30.390 What's the next term we want NN minus one over 2 factorial? 00:45:31.110 --> 00:45:32.520 Which is 10. 00:45:34.150 --> 00:45:38.214 10 - 1 is 9 over 2 factorial. 00:45:39.610 --> 00:45:45.115 And then we'd want an X squared. So in this case we want X being 00:45:45.115 --> 00:45:46.216 why over 3? 00:45:46.790 --> 00:45:49.920 All square 00:45:51.690 --> 00:45:55.335 I want to generate one more term 'cause I said I want to look for 00:45:55.335 --> 00:45:58.737 four terms, so the next term is going to be an which was 10. 00:46:00.030 --> 00:46:02.256 N minus one which is 9. 00:46:02.800 --> 00:46:06.880 And minus two, which is 8 and this time over 3 factorial. 00:46:07.450 --> 00:46:13.443 So I'm here NN minus one and minus two over 3 factorial and 00:46:13.443 --> 00:46:20.358 we want X cubed X is Y over three, so we want why over 3 00:46:20.358 --> 00:46:26.076 cubed. And the series goes on and on. Let's just tidy up what 00:46:26.076 --> 00:46:27.828 we've got. There's one. 00:46:29.070 --> 00:46:32.658 That'll be 10, why over 3? 00:46:32.660 --> 00:46:37.070 What if we got in here? Well, there's a 3 square at the bottom 00:46:37.070 --> 00:46:41.795 which is 9, and there's a 9 at the top, so the three squared in 00:46:41.795 --> 00:46:44.000 here is going to cancel with the 00:46:44.000 --> 00:46:46.610 nine there. 2 factorial 00:46:47.210 --> 00:46:48.209 Is just two. 00:46:49.430 --> 00:46:55.418 And choosing to 10 is 5, so will have five 5 squared. 00:46:56.210 --> 00:47:02.306 And then this is a bit more complicated. We've got a 3 00:47:02.306 --> 00:47:05.354 factorial which is 3 * 2. 00:47:05.360 --> 00:47:10.940 And three cubed. At the bottom there, which is 3 * 3 * 3. Some 00:47:10.940 --> 00:47:16.148 of this will cancel down. 3 * 3 will cancel, with the nine in 00:47:16.148 --> 00:47:21.316 here. The two will cancel their with the eight will have four 00:47:21.316 --> 00:47:26.160 and let's see what we're left with at the top will have 10 * 00:47:26.160 --> 00:47:27.544 4, which is 40. 00:47:28.140 --> 00:47:34.404 And at the bottom will have 3 * 3, which is 9. 00:47:34.960 --> 00:47:37.888 And they'll be a Y cubed. 00:47:37.890 --> 00:47:44.460 So altogether we've 1 + 10 Y over 3 five Y squared, 40 over 9 00:47:44.460 --> 00:47:50.154 Y cubed, and those are the first four terms of a series which 00:47:50.154 --> 00:47:54.972 will actually continue until you get to a term involving the 00:47:54.972 --> 00:48:00.666 highest power, which will be a Y over 3 to the power 10. 00:48:00.900 --> 00:48:04.428 So you can still use the theorem in slightly different form if 00:48:04.428 --> 00:48:08.250 you use a bit of ingenuity. Want to look at one final example 00:48:08.250 --> 00:48:14.949 before we finish? And this time I want to look at the example 3 00:48:14.949 --> 00:48:16.386 - 5 Z. 00:48:16.390 --> 00:48:20.242 To the power 40 again, it's an example where you wouldn't want 00:48:20.242 --> 00:48:24.094 to use pascals triangle because the power for teens too high and 00:48:24.094 --> 00:48:27.304 you have too many rose to generate in your triangle. 00:48:27.880 --> 00:48:31.286 I'm going to use the original form of the theorem, the One I 00:48:31.286 --> 00:48:33.382 have here in terms of A+B to the 00:48:33.382 --> 00:48:36.910 power N. A will be 3. 00:48:38.630 --> 00:48:42.722 Now B is a negative number be will be minus five said. 00:48:44.770 --> 00:48:46.330 Ends going to be 14. 00:48:46.920 --> 00:48:50.270 But we can still use the theorem. Let's see what 00:48:50.270 --> 00:48:53.620 happens and in this problem I'm just going to generate 00:48:53.620 --> 00:48:54.960 the first three terms. 00:48:56.260 --> 00:49:02.512 OK so A is 3 and we want to raise the three. 00:49:02.520 --> 00:49:04.992 To the highest power which is 00:49:04.992 --> 00:49:09.780 14. So my first term is 3 to the power 14. 00:49:12.070 --> 00:49:14.638 My second term is this one. 00:49:15.400 --> 00:49:16.480 Begins with an N. 00:49:17.860 --> 00:49:21.046 The power in the original expression, which was 14. 00:49:21.940 --> 00:49:26.700 Multiplied by A to the power N 00:49:26.700 --> 00:49:29.420 minus one AS 3. 00:49:30.620 --> 00:49:35.772 And we want to raise it to the power N minus 114 - 1 is 30. 00:49:36.430 --> 00:49:37.840 And we want to be. 00:49:39.500 --> 00:49:41.680 B is minus five set. 00:49:42.320 --> 00:49:47.060 So our second term looking ahead is going to be negative because 00:49:47.060 --> 00:49:49.430 of that minus five in there. 00:49:50.640 --> 00:49:55.704 My third term and I'll stop after the third term, his N, 00:49:55.704 --> 00:49:56.970 which is 14. 00:49:57.600 --> 00:50:03.400 And minus one which is 13 all over 2 factorial. 00:50:04.700 --> 00:50:10.172 A to the power N minus two will be 3 to the power N minus two 00:50:10.172 --> 00:50:12.908 will be 14 - 2 which is 12. 00:50:13.560 --> 00:50:16.560 And finally AB, which was minus five said. 00:50:18.310 --> 00:50:20.240 Raised to the power 2. 00:50:20.850 --> 00:50:25.162 And we know this goes on and on until we reach term instead to 00:50:25.162 --> 00:50:28.550 the power 14. But we've only written down the first three 00:50:28.550 --> 00:50:32.246 terms there. Perhaps we should just tidy it up a little bit. 00:50:32.250 --> 00:50:35.122 There's a 3 to the power 14 at 00:50:35.122 --> 00:50:42.582 the beginning. There's a minus five here, minus 5. Four 00:50:42.582 --> 00:50:45.578 teens are minus 70. 00:50:45.580 --> 00:50:48.844 As a 3 to the power 13, let me just leave it like that for the 00:50:48.844 --> 00:50:51.450 time being. And then they'll be as Ed. 00:50:54.130 --> 00:50:56.818 Over here there's a 3 to the 00:50:56.818 --> 00:51:00.312 power 12. That's this term in 00:51:00.312 --> 00:51:05.510 here. And I'm reaching for my Calculator again because this is 00:51:05.510 --> 00:51:08.930 a bit more complicated. We have got a 14. 00:51:09.530 --> 00:51:11.249 Multiplied by 13. 00:51:12.450 --> 00:51:15.834 When multiplied by 5 squared, which is 25. 00:51:17.880 --> 00:51:23.520 And divided by the two factorial that's divided by two, this will 00:51:23.520 --> 00:51:25.870 be multiplied by two 275. 00:51:26.450 --> 00:51:31.480 And this expression will be positive because we've got a 00:51:31.480 --> 00:51:32.989 minus 5 squared. 00:51:34.170 --> 00:51:38.680 And we need to remember to include zed squared in there. 00:51:39.210 --> 00:51:42.402 OK, we observe as before that the powers of zed are increasing 00:51:42.402 --> 00:51:46.126 as we move from the left to the right. Now we could leave it 00:51:46.126 --> 00:51:49.850 like that. I'm just going to tidy it up and write it in a 00:51:49.850 --> 00:51:52.776 slightly different form because this is often the way you see 00:51:52.776 --> 00:51:56.234 answers in the back of textbooks or people ask you to give an 00:51:56.234 --> 00:51:59.692 answer in a particular form and the form I'm going to write it 00:51:59.692 --> 00:52:03.682 in is one obtained by taking out a factor of 3 to the power 40. 00:52:04.740 --> 00:52:08.520 If I take her three to 14 out from the first term, I'll be 00:52:08.520 --> 00:52:09.600 just left with one. 00:52:10.610 --> 00:52:13.970 Now 3 to the 13. In the second 00:52:13.970 --> 00:52:18.650 term. But if I multiply top and bottom by three, I'll have a 3 00:52:18.650 --> 00:52:20.890 to the 14th at the top, which I 00:52:20.890 --> 00:52:25.150 can take out. But have multiplied the bottom by 00:52:25.150 --> 00:52:29.550 three as well, which will leave me with minus 70 Zedd 00:52:29.550 --> 00:52:30.750 divided by three. 00:52:32.880 --> 00:52:34.888 Here with a 3 to the power 12. 00:52:35.420 --> 00:52:37.589 And I want to take out a 3 to 00:52:37.589 --> 00:52:42.910 14. If I multiply the top and bottom by three squared or nine, 00:52:42.910 --> 00:52:45.710 I affectively get a 3 to 14 in 00:52:45.710 --> 00:52:50.878 this term. So I'm multiplying top and bottom by 9, taking 00:52:50.878 --> 00:52:56.234 the three to the 14 out, and that will leave me here with 00:52:56.234 --> 00:52:57.882 two 275 over 9. 00:52:58.980 --> 00:53:04.008 Said squad And this series continues. As I said before, 00:53:04.008 --> 00:53:08.640 until you get to a term involving zed to the power 14, 00:53:08.640 --> 00:53:12.500 but those are the first three terms of the series.