[Script Info]
Title:
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:01.07,0:00:04.85,Default,,0000,0000,0000,,In simplifying algebraic\Nfractions, we occasionally need
Dialogue: 0,0:00:04.85,0:00:07.01,Default,,0000,0000,0000,,a process known as.
Dialogue: 0,0:00:07.71,0:00:14.95,Default,,0000,0000,0000,,Polynomial.\NDivision.
Dialogue: 0,0:00:17.11,0:00:22.49,Default,,0000,0000,0000,,Before we do that, I want to\Ntake you back to something
Dialogue: 0,0:00:22.49,0:00:26.52,Default,,0000,0000,0000,,that you actually know very\Nwell indeed, and that's
Dialogue: 0,0:00:26.52,0:00:27.86,Default,,0000,0000,0000,,ordinary long division.
Dialogue: 0,0:00:28.92,0:00:33.89,Default,,0000,0000,0000,,You know how to do long\Ndivision, but I want to go over
Dialogue: 0,0:00:33.89,0:00:38.60,Default,,0000,0000,0000,,it again. 'cause I want to point\Nout certain things to you.
Dialogue: 0,0:00:39.25,0:00:43.25,Default,,0000,0000,0000,,'cause the things that are\Nimportant about long division
Dialogue: 0,0:00:43.25,0:00:45.47,Default,,0000,0000,0000,,are also important in polynomial
Dialogue: 0,0:00:45.47,0:00:52.37,Default,,0000,0000,0000,,division so. Let's have a look\Nat a long division. Some
Dialogue: 0,0:00:52.37,0:00:55.98,Default,,0000,0000,0000,,supposing I want to divide 25.
Dialogue: 0,0:00:56.49,0:01:02.50,Default,,0000,0000,0000,,Into Let's\Nsay
Dialogue: 0,0:01:02.50,0:01:04.71,Default,,0000,0000,0000,,2675.
Dialogue: 0,0:01:05.96,0:01:10.11,Default,,0000,0000,0000,,When I would have to do is look\Nat 25 in tool 2.
Dialogue: 0,0:01:10.84,0:01:17.08,Default,,0000,0000,0000,,No way 25 into 26. It goes once\Nand write the one there.
Dialogue: 0,0:01:17.64,0:01:20.96,Default,,0000,0000,0000,,Add multiply the one by the 25.
Dialogue: 0,0:01:21.68,0:01:25.69,Default,,0000,0000,0000,,And subtract and\Nhave one left.
Dialogue: 0,0:01:28.17,0:01:31.94,Default,,0000,0000,0000,,Hope you remember doing that.\NYou were probably taught how to
Dialogue: 0,0:01:31.94,0:01:35.72,Default,,0000,0000,0000,,do that at primary school or the\Nbeginnings of Secondary School.
Dialogue: 0,0:01:35.72,0:01:40.18,Default,,0000,0000,0000,,Next step is to bring down the\Nnext number, so we bring down
Dialogue: 0,0:01:40.18,0:01:45.32,Default,,0000,0000,0000,,17. Well, we bring down Seven to\Nmake it 17 and now we say how
Dialogue: 0,0:01:45.32,0:01:47.72,Default,,0000,0000,0000,,many times does 25 going to 17.
Dialogue: 0,0:01:48.34,0:01:52.30,Default,,0000,0000,0000,,It doesn't go at all. It's not\Nenough, so we have to record the
Dialogue: 0,0:01:52.30,0:01:54.28,Default,,0000,0000,0000,,fact that it doesn't go with a
Dialogue: 0,0:01:54.28,0:02:01.83,Default,,0000,0000,0000,,0. Next we bring down the\Nfive. So now we've got 175 and
Dialogue: 0,0:02:01.83,0:02:09.49,Default,,0000,0000,0000,,we say how many times does 25\Ngo into that? And it goes 7
Dialogue: 0,0:02:09.49,0:02:17.15,Default,,0000,0000,0000,,and we can check that Seven 535,\Nfive down three to carry. 7 twos
Dialogue: 0,0:02:17.15,0:02:20.43,Default,,0000,0000,0000,,are 14 and three is 17.
Dialogue: 0,0:02:20.44,0:02:24.10,Default,,0000,0000,0000,,Tracked, we get nothing left.
Dialogue: 0,0:02:24.66,0:02:28.63,Default,,0000,0000,0000,,So this is our answer. We've\Nnothing left there, no
Dialogue: 0,0:02:28.63,0:02:31.81,Default,,0000,0000,0000,,remainder, nothing left over.\NAnd there's our answer.
Dialogue: 0,0:02:32.32,0:02:39.04,Default,,0000,0000,0000,,2675 divides by 25 and the\Nanswer is 107. They just look at
Dialogue: 0,0:02:39.04,0:02:44.73,Default,,0000,0000,0000,,what we did. We did 25 into\N26 because that went.
Dialogue: 0,0:02:45.82,0:02:52.24,Default,,0000,0000,0000,,We then recorded that once that\Nit went there, multiplied, wrote
Dialogue: 0,0:02:52.24,0:02:54.58,Default,,0000,0000,0000,,the answer and subtracted.
Dialogue: 0,0:02:55.26,0:02:58.04,Default,,0000,0000,0000,,We brought down the next number.
Dialogue: 0,0:02:58.61,0:03:03.76,Default,,0000,0000,0000,,Asked how many times 25 went\Ninto it, it didn't go. We
Dialogue: 0,0:03:03.76,0:03:08.48,Default,,0000,0000,0000,,recorded that and brought down\Nthe next number. Then we said
Dialogue: 0,0:03:08.48,0:03:13.62,Default,,0000,0000,0000,,how many times does 25 going\Nto that Seven we did the
Dialogue: 0,0:03:13.62,0:03:17.06,Default,,0000,0000,0000,,multiplication, wrote it down,\Nsubtracted, got nothing left
Dialogue: 0,0:03:17.06,0:03:18.34,Default,,0000,0000,0000,,so it finished.
Dialogue: 0,0:03:19.60,0:03:25.29,Default,,0000,0000,0000,,What we're going to do now is\Ntake that self same process and
Dialogue: 0,0:03:25.29,0:03:27.05,Default,,0000,0000,0000,,do it with algebra.
Dialogue: 0,0:03:28.16,0:03:31.22,Default,,0000,0000,0000,,So let us
Dialogue: 0,0:03:31.22,0:03:36.94,Default,,0000,0000,0000,,take. This\N27 X cubed.
Dialogue: 0,0:03:38.07,0:03:41.55,Default,,0000,0000,0000,,+9 X squared.
Dialogue: 0,0:03:42.46,0:03:47.24,Default,,0000,0000,0000,,Minus 3X. Minus\N10.
Dialogue: 0,0:03:48.34,0:03:49.56,Default,,0000,0000,0000,,All over.
Dialogue: 0,0:03:50.96,0:03:54.78,Default,,0000,0000,0000,,3X minus 2.
Dialogue: 0,0:03:55.84,0:04:01.38,Default,,0000,0000,0000,,We want to divide that into\Nthat. We want to know how many
Dialogue: 0,0:04:01.38,0:04:07.34,Default,,0000,0000,0000,,times that will fit into there,\Nso we set it up exactly like a
Dialogue: 0,0:04:07.34,0:04:13.07,Default,,0000,0000,0000,,long division. Problem by\Ndividing by this. This is what
Dialogue: 0,0:04:13.07,0:04:19.61,Default,,0000,0000,0000,,we're dividing into 27 X cubed\Nplus nine X squared minus three
Dialogue: 0,0:04:19.61,0:04:21.24,Default,,0000,0000,0000,,X minus 10.
Dialogue: 0,0:04:22.48,0:04:26.70,Default,,0000,0000,0000,,So we ask ourselves, how many\Ntimes does well? How many times
Dialogue: 0,0:04:26.70,0:04:30.58,Default,,0000,0000,0000,,does that go into that? But\Ndifficult what we ask ourselves
Dialogue: 0,0:04:30.58,0:04:34.45,Default,,0000,0000,0000,,is how many times does the\Nexcpet go into this bit?
Dialogue: 0,0:04:37.22,0:04:41.80,Default,,0000,0000,0000,,Just like we asked ourselves how\Nmany times the 25 went into the
Dialogue: 0,0:04:41.80,0:04:43.91,Default,,0000,0000,0000,,26, how many times does 3X?
Dialogue: 0,0:04:44.59,0:04:50.15,Default,,0000,0000,0000,,Go into 27 X cubed. The answer\Nmust be 9 X squared because
Dialogue: 0,0:04:50.15,0:04:56.15,Default,,0000,0000,0000,,Nynex squared times by three X\Ngives us 27 X cubed and we need
Dialogue: 0,0:04:56.15,0:05:02.14,Default,,0000,0000,0000,,to record that. But we need to\Nrecord it in the right place and
Dialogue: 0,0:05:02.14,0:05:06.42,Default,,0000,0000,0000,,because these are the X\Nsquared's we record that above
Dialogue: 0,0:05:06.42,0:05:07.70,Default,,0000,0000,0000,,the X squares.
Dialogue: 0,0:05:08.95,0:05:11.66,Default,,0000,0000,0000,,So now we do the multiplication.
Dialogue: 0,0:05:12.18,0:05:15.97,Default,,0000,0000,0000,,Nine X squared times 3X is 27
Dialogue: 0,0:05:15.97,0:05:23.28,Default,,0000,0000,0000,,X cubed. Nine X squared times\Nminus two is minus 18 X squared.
Dialogue: 0,0:05:24.60,0:05:28.67,Default,,0000,0000,0000,,Just like we did for long\Ndivision, we now do the
Dialogue: 0,0:05:28.67,0:05:34.37,Default,,0000,0000,0000,,Subtraction. 27 X cubed\Ntakeaway 27 X cubed none of
Dialogue: 0,0:05:34.37,0:05:39.67,Default,,0000,0000,0000,,them, because we arrange for\Nit to be so Nynex squared
Dialogue: 0,0:05:39.67,0:05:44.97,Default,,0000,0000,0000,,takeaway minus 18 X squared\Ngives us plus 27 X squared.
Dialogue: 0,0:05:46.30,0:05:52.66,Default,,0000,0000,0000,,Now we do what we did before we\Nbring down the next one, so we
Dialogue: 0,0:05:52.66,0:05:54.78,Default,,0000,0000,0000,,bring down the minus 3X.
Dialogue: 0,0:05:55.36,0:06:00.88,Default,,0000,0000,0000,,How many times does 3X go into\N27 X squared?
Dialogue: 0,0:06:01.57,0:06:09.13,Default,,0000,0000,0000,,Answer. It goes 9X times and\Nwe write that in the X Column.
Dialogue: 0,0:06:09.13,0:06:16.43,Default,,0000,0000,0000,,So now we have 9X times 3\NX 27 X squared 9X times, Y
Dialogue: 0,0:06:16.43,0:06:19.03,Default,,0000,0000,0000,,minus 2 - 18 X.
Dialogue: 0,0:06:19.70,0:06:21.47,Default,,0000,0000,0000,,And we subtract again.
Dialogue: 0,0:06:22.37,0:06:27.85,Default,,0000,0000,0000,,27 X squared takeaway, 27 X\Nsquared, no X squared, but we
Dialogue: 0,0:06:27.85,0:06:33.80,Default,,0000,0000,0000,,arrange for it to be like that,\Nminus three X minus minus 18X.
Dialogue: 0,0:06:33.80,0:06:38.82,Default,,0000,0000,0000,,Well, that's going to give us\Nplus 15X altogether, and we
Dialogue: 0,0:06:38.82,0:06:41.11,Default,,0000,0000,0000,,bring down the minus 10.
Dialogue: 0,0:06:42.94,0:06:49.31,Default,,0000,0000,0000,,3X into 15X. This time it goes\Nfive times, so we can say plus
Dialogue: 0,0:06:49.31,0:06:53.86,Default,,0000,0000,0000,,five there. And again it's in\Nthe numbers. The constants
Dialogue: 0,0:06:53.86,0:06:55.68,Default,,0000,0000,0000,,column at the end.
Dialogue: 0,0:06:56.33,0:07:02.20,Default,,0000,0000,0000,,Five times by 15 times by three\NX gives us 15X. Write it down
Dialogue: 0,0:07:02.20,0:07:08.06,Default,,0000,0000,0000,,there five times by minus two\Ngives us minus 10 and we can see
Dialogue: 0,0:07:08.06,0:07:12.67,Default,,0000,0000,0000,,that when we take these two\Naway. Got exactly the same
Dialogue: 0,0:07:12.67,0:07:16.44,Default,,0000,0000,0000,,expression. 15X minus 10\Ntakeaway. 50X minus 10 nothing
Dialogue: 0,0:07:16.44,0:07:21.05,Default,,0000,0000,0000,,left. So there's our answer,\Njust as in the long division.
Dialogue: 0,0:07:21.05,0:07:22.73,Default,,0000,0000,0000,,The answer was there.
Dialogue: 0,0:07:23.39,0:07:29.01,Default,,0000,0000,0000,,It's there now so we can say\Nthat this expression is equal to
Dialogue: 0,0:07:29.01,0:07:31.17,Default,,0000,0000,0000,,9 X squared plus 9X.
Dialogue: 0,0:07:31.73,0:07:38.39,Default,,0000,0000,0000,,Plus 5. Let's\Ntake another one.
Dialogue: 0,0:07:39.20,0:07:42.85,Default,,0000,0000,0000,,So we'll take X to the 4th.
Dialogue: 0,0:07:43.55,0:07:46.47,Default,,0000,0000,0000,,Plus X cubed.
Dialogue: 0,0:07:47.34,0:07:54.34,Default,,0000,0000,0000,,Plus Seven X squared\Nminus six X +8.
Dialogue: 0,0:07:54.98,0:08:02.42,Default,,0000,0000,0000,,Divided by all over\NX squared, +2 X
Dialogue: 0,0:08:02.42,0:08:07.81,Default,,0000,0000,0000,,+8. So this is what we're\Ndividing by and this is what
Dialogue: 0,0:08:07.81,0:08:11.03,Default,,0000,0000,0000,,we're dividing into is not\Nimmediately obvious what the
Dialogue: 0,0:08:11.03,0:08:15.68,Default,,0000,0000,0000,,answer is going to be. Let's\Nhave a look X squared plus 2X
Dialogue: 0,0:08:15.68,0:08:16.76,Default,,0000,0000,0000,,plus 8IN tool.
Dialogue: 0,0:08:17.41,0:08:19.19,Default,,0000,0000,0000,,All of this.
Dialogue: 0,0:08:22.61,0:08:27.38,Default,,0000,0000,0000,,Our first question is how many\Ntimes does X squared going to X
Dialogue: 0,0:08:27.38,0:08:32.52,Default,,0000,0000,0000,,to the 4th? We don't need to\Nworry about the rest, we just do
Dialogue: 0,0:08:32.52,0:08:38.39,Default,,0000,0000,0000,,it on the first 2 bits in each\None, just as the same as we did
Dialogue: 0,0:08:38.39,0:08:42.80,Default,,0000,0000,0000,,with the previous example. How\Nmany times X squared going to X
Dialogue: 0,0:08:42.80,0:08:47.57,Default,,0000,0000,0000,,to the four will it goes X\Nsquared times? So we write it
Dialogue: 0,0:08:47.57,0:08:51.60,Default,,0000,0000,0000,,there over the X squared's. Now\Nwe do the multiplication X
Dialogue: 0,0:08:51.60,0:08:55.27,Default,,0000,0000,0000,,squared times. My X squared is X\Nto the 4th.
Dialogue: 0,0:08:55.43,0:09:03.16,Default,,0000,0000,0000,,X squared by two X is plus\N2X cubed X squared by 8 is
Dialogue: 0,0:09:03.16,0:09:04.81,Default,,0000,0000,0000,,plus 8X squared.
Dialogue: 0,0:09:07.45,0:09:13.93,Default,,0000,0000,0000,,And now we do the Subtraction X.\NThe four takeaway X to the 4th
Dialogue: 0,0:09:13.93,0:09:19.49,Default,,0000,0000,0000,,there Arnold, but we arranged it\Nthat way. X cubed takeaway 2X
Dialogue: 0,0:09:19.49,0:09:24.12,Default,,0000,0000,0000,,cubed minus X cubed. Seven X\Nsquared takeaway, 8X squared
Dialogue: 0,0:09:24.12,0:09:28.28,Default,,0000,0000,0000,,minus X squared and bring down\Nthe next term.
Dialogue: 0,0:09:29.08,0:09:34.06,Default,,0000,0000,0000,,Now we say how many times does X\Nsquared going to minus X cubed,
Dialogue: 0,0:09:34.06,0:09:39.40,Default,,0000,0000,0000,,and it must be minus X, and so\Nwe write it in the X Column.
Dialogue: 0,0:09:39.97,0:09:45.30,Default,,0000,0000,0000,,And above the line there, next\Nthe multiplication minus X times
Dialogue: 0,0:09:45.30,0:09:52.58,Default,,0000,0000,0000,,by X squared is minus X cubed\Nminus X times 2X is minus two X
Dialogue: 0,0:09:52.58,0:09:57.43,Default,,0000,0000,0000,,squared and minus X times by 8\Nis minus 8X.
Dialogue: 0,0:09:59.63,0:10:04.71,Default,,0000,0000,0000,,Do the subtraction minus X cubed\Ntakeaway minus X cubed. No ex
Dialogue: 0,0:10:04.71,0:10:09.78,Default,,0000,0000,0000,,cubes minus X squared minus\Nminus two X squared or the minus
Dialogue: 0,0:10:09.78,0:10:13.59,Default,,0000,0000,0000,,minus A plus, so that\Neffectively that's minus X
Dialogue: 0,0:10:13.59,0:10:17.40,Default,,0000,0000,0000,,squared +2 X squared just gives\Nus X squared.
Dialogue: 0,0:10:17.48,0:10:23.92,Default,,0000,0000,0000,,Minus six X minus minus 8X.\NWell, that's minus 6X Plus 8X
Dialogue: 0,0:10:23.92,0:10:29.29,Default,,0000,0000,0000,,gives us plus 2X and bring down\Nthe next one.
Dialogue: 0,0:10:30.09,0:10:35.55,Default,,0000,0000,0000,,X squared plus 2X plus a 12 X\Nsquared goes into X squared
Dialogue: 0,0:10:35.55,0:10:40.70,Default,,0000,0000,0000,,once. And so X squared plus\N2X plus eight. And again we
Dialogue: 0,0:10:40.70,0:10:45.14,Default,,0000,0000,0000,,can see these two are the\Nsame when I take them away,
Dialogue: 0,0:10:45.14,0:10:49.21,Default,,0000,0000,0000,,I will have nothing left\Nand so this is my answer.
Dialogue: 0,0:10:50.57,0:10:54.81,Default,,0000,0000,0000,,The result of doing that\Ndivision is that.
Dialogue: 0,0:10:55.60,0:11:01.49,Default,,0000,0000,0000,,Well, the one that started\Nus off on doing this was if
Dialogue: 0,0:11:01.49,0:11:02.47,Default,,0000,0000,0000,,you remember.
Dialogue: 0,0:11:03.60,0:11:09.14,Default,,0000,0000,0000,,X cubed minus one over\NX minus one.
Dialogue: 0,0:11:10.18,0:11:13.52,Default,,0000,0000,0000,,This looks a little bit\Ndifferent, doesn't it? Because
Dialogue: 0,0:11:13.52,0:11:17.97,Default,,0000,0000,0000,,whereas the space between the X\NCube term and the constant term
Dialogue: 0,0:11:17.97,0:11:20.20,Default,,0000,0000,0000,,was filled with all the terms?
Dialogue: 0,0:11:21.13,0:11:22.05,Default,,0000,0000,0000,,This one isn't.
Dialogue: 0,0:11:23.21,0:11:24.64,Default,,0000,0000,0000,,How do we cope with the?
Dialogue: 0,0:11:25.22,0:11:28.90,Default,,0000,0000,0000,,Let's have a look. Remember, we\Nknow what the answer to this one
Dialogue: 0,0:11:28.90,0:11:35.36,Default,,0000,0000,0000,,is already. So what we must do\Nis right in X cubed and then
Dialogue: 0,0:11:35.36,0:11:41.08,Default,,0000,0000,0000,,leave space for the X squared\Nterm, the X term and then the
Dialogue: 0,0:11:41.08,0:11:47.22,Default,,0000,0000,0000,,constant term. So what I asked\Nmyself is how many times does X
Dialogue: 0,0:11:47.22,0:11:53.06,Default,,0000,0000,0000,,go into XQ, and the answer goes\Nin X squared. So I write the
Dialogue: 0,0:11:53.06,0:11:56.81,Default,,0000,0000,0000,,answer there where the X squared\Nterm would be.
Dialogue: 0,0:11:57.56,0:12:00.62,Default,,0000,0000,0000,,X squared times by X is X cubed.
Dialogue: 0,0:12:01.34,0:12:05.70,Default,,0000,0000,0000,,X squared times by minus one is\Nminus X squared.
Dialogue: 0,0:12:07.40,0:12:12.39,Default,,0000,0000,0000,,And subtract X cubed takeaway X\Ncubed no ex cubes.
Dialogue: 0,0:12:12.99,0:12:18.55,Default,,0000,0000,0000,,0 minus minus X squared is\Nplus X squared.
Dialogue: 0,0:12:19.25,0:12:24.37,Default,,0000,0000,0000,,Bring down the next term. There\Nis no next term to bring down.
Dialogue: 0,0:12:24.37,0:12:26.74,Default,,0000,0000,0000,,There's no X to bring down.
Dialogue: 0,0:12:27.25,0:12:33.20,Default,,0000,0000,0000,,So it's as though I got zero X.\NThere was no point in writing
Dialogue: 0,0:12:33.20,0:12:39.58,Default,,0000,0000,0000,,it. If it's not there, so let's\Ncarry on X in two X squared that
Dialogue: 0,0:12:39.58,0:12:45.52,Default,,0000,0000,0000,,goes X times. So record the X\Nthere above where the X is would
Dialogue: 0,0:12:45.52,0:12:50.62,Default,,0000,0000,0000,,be. Let's do the multiplication\NX times by X. Is X squared.
Dialogue: 0,0:12:51.60,0:12:57.31,Default,,0000,0000,0000,,X times Y minus one is minus X.\NDo the subtraction X squared
Dialogue: 0,0:12:57.31,0:12:59.50,Default,,0000,0000,0000,,takeaway X squared is nothing.
Dialogue: 0,0:13:00.53,0:13:04.48,Default,,0000,0000,0000,,Nothing takeaway minus\NX. It's minus minus X.
Dialogue: 0,0:13:04.48,0:13:06.95,Default,,0000,0000,0000,,That gives us Plus X.
Dialogue: 0,0:13:08.00,0:13:12.19,Default,,0000,0000,0000,,Bring down the next term. We\Nhave got a term here to bring
Dialogue: 0,0:13:12.19,0:13:13.47,Default,,0000,0000,0000,,down it's minus one.
Dialogue: 0,0:13:14.31,0:13:18.91,Default,,0000,0000,0000,,How many times does X going to\NX? It goes once.
Dialogue: 0,0:13:20.11,0:13:24.87,Default,,0000,0000,0000,,Long times by XX. One times by\Nminus one is minus one. Take
Dialogue: 0,0:13:24.87,0:13:29.63,Default,,0000,0000,0000,,them away and we've got nothing\Nleft there and so this is my
Dialogue: 0,0:13:29.63,0:13:33.65,Default,,0000,0000,0000,,answer X squared plus X plus\None, and that's exactly the
Dialogue: 0,0:13:33.65,0:13:37.68,Default,,0000,0000,0000,,answer that we had before. So\Nwhere you've got terms missing?
Dialogue: 0,0:13:37.68,0:13:42.07,Default,,0000,0000,0000,,You can still do the same\Ndivision. You can still do the
Dialogue: 0,0:13:42.07,0:13:46.46,Default,,0000,0000,0000,,same process, but you just leave\Nthe gaps where the terms would
Dialogue: 0,0:13:46.46,0:13:50.85,Default,,0000,0000,0000,,be and you'll need the gaps\Nbecause you're going to have to
Dialogue: 0,0:13:50.85,0:13:54.79,Default,,0000,0000,0000,,write something. Up here in\Nwhat's going to be the answer.