WEBVTT 00:00:01.070 --> 00:00:04.850 In simplifying algebraic fractions, we occasionally need 00:00:04.850 --> 00:00:07.010 a process known as. 00:00:07.710 --> 00:00:14.950 Polynomial. Division. 00:00:17.110 --> 00:00:22.486 Before we do that, I want to take you back to something 00:00:22.486 --> 00:00:26.518 that you actually know very well indeed, and that's 00:00:26.518 --> 00:00:27.862 ordinary long division. 00:00:28.920 --> 00:00:33.886 You know how to do long division, but I want to go over 00:00:33.886 --> 00:00:38.600 it again. 'cause I want to point out certain things to you. 00:00:39.250 --> 00:00:43.246 'cause the things that are important about long division 00:00:43.246 --> 00:00:45.466 are also important in polynomial 00:00:45.466 --> 00:00:52.369 division so. Let's have a look at a long division. Some 00:00:52.369 --> 00:00:55.975 supposing I want to divide 25. 00:00:56.490 --> 00:01:02.502 Into Let's say 00:01:02.502 --> 00:01:04.708 2675. 00:01:05.960 --> 00:01:10.107 When I would have to do is look at 25 in tool 2. 00:01:10.840 --> 00:01:17.080 No way 25 into 26. It goes once and write the one there. 00:01:17.640 --> 00:01:20.965 Add multiply the one by the 25. 00:01:21.680 --> 00:01:25.688 And subtract and have one left. 00:01:28.170 --> 00:01:31.943 Hope you remember doing that. You were probably taught how to 00:01:31.943 --> 00:01:35.716 do that at primary school or the beginnings of Secondary School. 00:01:35.716 --> 00:01:40.175 Next step is to bring down the next number, so we bring down 00:01:40.175 --> 00:01:45.320 17. Well, we bring down Seven to make it 17 and now we say how 00:01:45.320 --> 00:01:47.721 many times does 25 going to 17. 00:01:48.340 --> 00:01:52.302 It doesn't go at all. It's not enough, so we have to record the 00:01:52.302 --> 00:01:54.283 fact that it doesn't go with a 00:01:54.283 --> 00:02:01.834 0. Next we bring down the five. So now we've got 175 and 00:02:01.834 --> 00:02:09.492 we say how many times does 25 go into that? And it goes 7 00:02:09.492 --> 00:02:17.150 and we can check that Seven 535, five down three to carry. 7 twos 00:02:17.150 --> 00:02:20.432 are 14 and three is 17. 00:02:20.440 --> 00:02:24.100 Tracked, we get nothing left. 00:02:24.660 --> 00:02:28.630 So this is our answer. We've nothing left there, no 00:02:28.630 --> 00:02:31.806 remainder, nothing left over. And there's our answer. 00:02:32.320 --> 00:02:39.041 2675 divides by 25 and the answer is 107. They just look at 00:02:39.041 --> 00:02:44.728 what we did. We did 25 into 26 because that went. 00:02:45.820 --> 00:02:52.244 We then recorded that once that it went there, multiplied, wrote 00:02:52.244 --> 00:02:54.580 the answer and subtracted. 00:02:55.260 --> 00:02:58.038 We brought down the next number. 00:02:58.610 --> 00:03:03.758 Asked how many times 25 went into it, it didn't go. We 00:03:03.758 --> 00:03:08.477 recorded that and brought down the next number. Then we said 00:03:08.477 --> 00:03:13.625 how many times does 25 going to that Seven we did the 00:03:13.625 --> 00:03:17.057 multiplication, wrote it down, subtracted, got nothing left 00:03:17.057 --> 00:03:18.344 so it finished. 00:03:19.600 --> 00:03:25.294 What we're going to do now is take that self same process and 00:03:25.294 --> 00:03:27.046 do it with algebra. 00:03:28.160 --> 00:03:31.220 So let us 00:03:31.220 --> 00:03:36.938 take. This 27 X cubed. 00:03:38.070 --> 00:03:41.550 +9 X squared. 00:03:42.460 --> 00:03:47.240 Minus 3X. Minus 10. 00:03:48.340 --> 00:03:49.560 All over. 00:03:50.960 --> 00:03:54.779 3X minus 2. 00:03:55.840 --> 00:04:01.378 We want to divide that into that. We want to know how many 00:04:01.378 --> 00:04:07.342 times that will fit into there, so we set it up exactly like a 00:04:07.342 --> 00:04:13.070 long division. Problem by dividing by this. This is what 00:04:13.070 --> 00:04:19.610 we're dividing into 27 X cubed plus nine X squared minus three 00:04:19.610 --> 00:04:21.245 X minus 10. 00:04:22.480 --> 00:04:26.704 So we ask ourselves, how many times does well? How many times 00:04:26.704 --> 00:04:30.576 does that go into that? But difficult what we ask ourselves 00:04:30.576 --> 00:04:34.448 is how many times does the excpet go into this bit? 00:04:37.220 --> 00:04:41.796 Just like we asked ourselves how many times the 25 went into the 00:04:41.796 --> 00:04:43.908 26, how many times does 3X? 00:04:44.590 --> 00:04:50.154 Go into 27 X cubed. The answer must be 9 X squared because 00:04:50.154 --> 00:04:56.146 Nynex squared times by three X gives us 27 X cubed and we need 00:04:56.146 --> 00:05:02.138 to record that. But we need to record it in the right place and 00:05:02.138 --> 00:05:06.418 because these are the X squared's we record that above 00:05:06.418 --> 00:05:07.702 the X squares. 00:05:08.950 --> 00:05:11.656 So now we do the multiplication. 00:05:12.180 --> 00:05:15.967 Nine X squared times 3X is 27 00:05:15.967 --> 00:05:23.281 X cubed. Nine X squared times minus two is minus 18 X squared. 00:05:24.600 --> 00:05:28.670 Just like we did for long division, we now do the 00:05:28.670 --> 00:05:34.368 Subtraction. 27 X cubed takeaway 27 X cubed none of 00:05:34.368 --> 00:05:39.670 them, because we arrange for it to be so Nynex squared 00:05:39.670 --> 00:05:44.972 takeaway minus 18 X squared gives us plus 27 X squared. 00:05:46.300 --> 00:05:52.660 Now we do what we did before we bring down the next one, so we 00:05:52.660 --> 00:05:54.780 bring down the minus 3X. 00:05:55.360 --> 00:06:00.880 How many times does 3X go into 27 X squared? 00:06:01.570 --> 00:06:09.132 Answer. It goes 9X times and we write that in the X Column. 00:06:09.132 --> 00:06:16.426 So now we have 9X times 3 X 27 X squared 9X times, Y 00:06:16.426 --> 00:06:19.031 minus 2 - 18 X. 00:06:19.700 --> 00:06:21.468 And we subtract again. 00:06:22.370 --> 00:06:27.854 27 X squared takeaway, 27 X squared, no X squared, but we 00:06:27.854 --> 00:06:33.795 arrange for it to be like that, minus three X minus minus 18X. 00:06:33.795 --> 00:06:38.822 Well, that's going to give us plus 15X altogether, and we 00:06:38.822 --> 00:06:41.107 bring down the minus 10. 00:06:42.940 --> 00:06:49.310 3X into 15X. This time it goes five times, so we can say plus 00:06:49.310 --> 00:06:53.860 five there. And again it's in the numbers. The constants 00:06:53.860 --> 00:06:55.680 column at the end. 00:06:56.330 --> 00:07:02.196 Five times by 15 times by three X gives us 15X. Write it down 00:07:02.196 --> 00:07:08.062 there five times by minus two gives us minus 10 and we can see 00:07:08.062 --> 00:07:12.671 that when we take these two away. Got exactly the same 00:07:12.671 --> 00:07:16.442 expression. 15X minus 10 takeaway. 50X minus 10 nothing 00:07:16.442 --> 00:07:21.051 left. So there's our answer, just as in the long division. 00:07:21.051 --> 00:07:22.727 The answer was there. 00:07:23.390 --> 00:07:29.006 It's there now so we can say that this expression is equal to 00:07:29.006 --> 00:07:31.166 9 X squared plus 9X. 00:07:31.730 --> 00:07:38.390 Plus 5. Let's take another one. 00:07:39.200 --> 00:07:42.847 So we'll take X to the 4th. 00:07:43.550 --> 00:07:46.469 Plus X cubed. 00:07:47.340 --> 00:07:54.340 Plus Seven X squared minus six X +8. 00:07:54.980 --> 00:08:02.420 Divided by all over X squared, +2 X 00:08:02.420 --> 00:08:07.808 +8. So this is what we're dividing by and this is what 00:08:07.808 --> 00:08:11.030 we're dividing into is not immediately obvious what the 00:08:11.030 --> 00:08:15.684 answer is going to be. Let's have a look X squared plus 2X 00:08:15.684 --> 00:08:16.758 plus 8IN tool. 00:08:17.410 --> 00:08:19.189 All of this. 00:08:22.610 --> 00:08:27.381 Our first question is how many times does X squared going to X 00:08:27.381 --> 00:08:32.519 to the 4th? We don't need to worry about the rest, we just do 00:08:32.519 --> 00:08:38.391 it on the first 2 bits in each one, just as the same as we did 00:08:38.391 --> 00:08:42.795 with the previous example. How many times X squared going to X 00:08:42.795 --> 00:08:47.566 to the four will it goes X squared times? So we write it 00:08:47.566 --> 00:08:51.603 there over the X squared's. Now we do the multiplication X 00:08:51.603 --> 00:08:55.273 squared times. My X squared is X to the 4th. 00:08:55.430 --> 00:09:03.158 X squared by two X is plus 2X cubed X squared by 8 is 00:09:03.158 --> 00:09:04.814 plus 8X squared. 00:09:07.450 --> 00:09:13.932 And now we do the Subtraction X. The four takeaway X to the 4th 00:09:13.932 --> 00:09:19.488 there Arnold, but we arranged it that way. X cubed takeaway 2X 00:09:19.488 --> 00:09:24.118 cubed minus X cubed. Seven X squared takeaway, 8X squared 00:09:24.118 --> 00:09:28.285 minus X squared and bring down the next term. 00:09:29.080 --> 00:09:34.064 Now we say how many times does X squared going to minus X cubed, 00:09:34.064 --> 00:09:39.404 and it must be minus X, and so we write it in the X Column. 00:09:39.970 --> 00:09:45.305 And above the line there, next the multiplication minus X times 00:09:45.305 --> 00:09:52.580 by X squared is minus X cubed minus X times 2X is minus two X 00:09:52.580 --> 00:09:57.430 squared and minus X times by 8 is minus 8X. 00:09:59.630 --> 00:10:04.706 Do the subtraction minus X cubed takeaway minus X cubed. No ex 00:10:04.706 --> 00:10:09.782 cubes minus X squared minus minus two X squared or the minus 00:10:09.782 --> 00:10:13.589 minus A plus, so that effectively that's minus X 00:10:13.589 --> 00:10:17.396 squared +2 X squared just gives us X squared. 00:10:17.480 --> 00:10:23.924 Minus six X minus minus 8X. Well, that's minus 6X Plus 8X 00:10:23.924 --> 00:10:29.294 gives us plus 2X and bring down the next one. 00:10:30.090 --> 00:10:35.550 X squared plus 2X plus a 12 X squared goes into X squared 00:10:35.550 --> 00:10:40.700 once. And so X squared plus 2X plus eight. And again we 00:10:40.700 --> 00:10:45.140 can see these two are the same when I take them away, 00:10:45.140 --> 00:10:49.210 I will have nothing left and so this is my answer. 00:10:50.570 --> 00:10:54.810 The result of doing that division is that. 00:10:55.600 --> 00:11:01.492 Well, the one that started us off on doing this was if 00:11:01.492 --> 00:11:02.474 you remember. 00:11:03.600 --> 00:11:09.144 X cubed minus one over X minus one. 00:11:10.180 --> 00:11:13.519 This looks a little bit different, doesn't it? Because 00:11:13.519 --> 00:11:17.971 whereas the space between the X Cube term and the constant term 00:11:17.971 --> 00:11:20.197 was filled with all the terms? 00:11:21.130 --> 00:11:22.048 This one isn't. 00:11:23.210 --> 00:11:24.638 How do we cope with the? 00:11:25.220 --> 00:11:28.899 Let's have a look. Remember, we know what the answer to this one 00:11:28.899 --> 00:11:35.360 is already. So what we must do is right in X cubed and then 00:11:35.360 --> 00:11:41.080 leave space for the X squared term, the X term and then the 00:11:41.080 --> 00:11:47.217 constant term. So what I asked myself is how many times does X 00:11:47.217 --> 00:11:53.055 go into XQ, and the answer goes in X squared. So I write the 00:11:53.055 --> 00:11:56.808 answer there where the X squared term would be. 00:11:57.560 --> 00:12:00.616 X squared times by X is X cubed. 00:12:01.340 --> 00:12:05.700 X squared times by minus one is minus X squared. 00:12:07.400 --> 00:12:12.390 And subtract X cubed takeaway X cubed no ex cubes. 00:12:12.990 --> 00:12:18.552 0 minus minus X squared is plus X squared. 00:12:19.250 --> 00:12:24.372 Bring down the next term. There is no next term to bring down. 00:12:24.372 --> 00:12:26.736 There's no X to bring down. 00:12:27.250 --> 00:12:33.200 So it's as though I got zero X. There was no point in writing 00:12:33.200 --> 00:12:39.575 it. If it's not there, so let's carry on X in two X squared that 00:12:39.575 --> 00:12:45.525 goes X times. So record the X there above where the X is would 00:12:45.525 --> 00:12:50.625 be. Let's do the multiplication X times by X. Is X squared. 00:12:51.600 --> 00:12:57.307 X times Y minus one is minus X. Do the subtraction X squared 00:12:57.307 --> 00:12:59.502 takeaway X squared is nothing. 00:13:00.530 --> 00:13:04.482 Nothing takeaway minus X. It's minus minus X. 00:13:04.482 --> 00:13:06.952 That gives us Plus X. 00:13:08.000 --> 00:13:12.186 Bring down the next term. We have got a term here to bring 00:13:12.186 --> 00:13:13.474 down it's minus one. 00:13:14.310 --> 00:13:18.908 How many times does X going to X? It goes once. 00:13:20.110 --> 00:13:24.868 Long times by XX. One times by minus one is minus one. Take 00:13:24.868 --> 00:13:29.626 them away and we've got nothing left there and so this is my 00:13:29.626 --> 00:13:33.652 answer X squared plus X plus one, and that's exactly the 00:13:33.652 --> 00:13:37.678 answer that we had before. So where you've got terms missing? 00:13:37.678 --> 00:13:42.070 You can still do the same division. You can still do the 00:13:42.070 --> 00:13:46.462 same process, but you just leave the gaps where the terms would 00:13:46.462 --> 00:13:50.854 be and you'll need the gaps because you're going to have to 00:13:50.854 --> 00:13:54.793 write something. Up here in what's going to be the answer.