[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.55,0:00:03.96,Default,,0000,0000,0000,,When we have two
Dialogue: 0,0:00:03.96,0:00:11.30,Default,,0000,0000,0000,,brackets. X +2 times by X\N+3 and we know how to multiply
Dialogue: 0,0:00:11.30,0:00:13.29,Default,,0000,0000,0000,,these two brackets out.
Dialogue: 0,0:00:13.97,0:00:21.82,Default,,0000,0000,0000,,We have X kinds by X that\Ngives us X squared. We have X
Dialogue: 0,0:00:21.82,0:00:25.19,Default,,0000,0000,0000,,times by three, gives us 3X.
Dialogue: 0,0:00:26.32,0:00:32.94,Default,,0000,0000,0000,,2 times by X gives us 2X\Nand then two times by three
Dialogue: 0,0:00:32.94,0:00:34.46,Default,,0000,0000,0000,,gives us 6.
Dialogue: 0,0:00:35.63,0:00:42.92,Default,,0000,0000,0000,,And we can simplify these two\Nterms. 3X plus 2X gives us 5X.
Dialogue: 0,0:00:44.18,0:00:49.16,Default,,0000,0000,0000,,This is an example of a\Nquadratic expression or
Dialogue: 0,0:00:49.16,0:00:54.13,Default,,0000,0000,0000,,quadratic function. It's gotta\Ntermine ex squared, which it
Dialogue: 0,0:00:54.13,0:00:57.45,Default,,0000,0000,0000,,must have to be a quadratic
Dialogue: 0,0:00:57.45,0:01:04.19,Default,,0000,0000,0000,,expression. It's got a term in\NX which it might or might not
Dialogue: 0,0:01:04.19,0:01:09.52,Default,,0000,0000,0000,,have, and it's got a constant\Nterm and there are no other
Dialogue: 0,0:01:09.52,0:01:12.62,Default,,0000,0000,0000,,possibilities, so our most\Ngeneral quadratic expression
Dialogue: 0,0:01:12.62,0:01:15.73,Default,,0000,0000,0000,,would be AX squared plus BX plus
Dialogue: 0,0:01:15.73,0:01:21.91,Default,,0000,0000,0000,,C. What we're going to have a\Nlook at is how we factorise
Dialogue: 0,0:01:21.91,0:01:27.04,Default,,0000,0000,0000,,expressions like this in others.\NHow we go back from this kind of
Dialogue: 0,0:01:27.04,0:01:33.21,Default,,0000,0000,0000,,expression. To this now, why\Nmight we want to do that? Well,
Dialogue: 0,0:01:33.21,0:01:39.06,Default,,0000,0000,0000,,let's just take this X squared\Nplus 5X plus six. And let's say
Dialogue: 0,0:01:39.06,0:01:44.46,Default,,0000,0000,0000,,it's not just an expression, but\Nit's an equation and it says
Dialogue: 0,0:01:44.46,0:01:47.61,Default,,0000,0000,0000,,equals 0. What are the values of
Dialogue: 0,0:01:47.61,0:01:54.04,Default,,0000,0000,0000,,X? That will make it equal to\N0 that are answers to that
Dialogue: 0,0:01:54.04,0:01:59.80,Default,,0000,0000,0000,,equation. One of the things we\Ncan do is to rewrite this form.
Dialogue: 0,0:02:00.32,0:02:08.20,Default,,0000,0000,0000,,By this so we can say\NX +2 times by X +3
Dialogue: 0,0:02:08.20,0:02:14.01,Default,,0000,0000,0000,,equals 0. When we have\Ntwo numbers that multiply
Dialogue: 0,0:02:14.01,0:02:20.97,Default,,0000,0000,0000,,together to give zero and one of\Nthe things that must be true is
Dialogue: 0,0:02:20.97,0:02:27.43,Default,,0000,0000,0000,,that one of them zero or the\Nother one zero, or they're both
Dialogue: 0,0:02:27.43,0:02:35.38,Default,,0000,0000,0000,,0. So in this case, X +2 equals\N0 or X +3 equals 0, and so
Dialogue: 0,0:02:35.38,0:02:40.85,Default,,0000,0000,0000,,X would be minus two, or X would\Nbe minus three.
Dialogue: 0,0:02:41.46,0:02:47.35,Default,,0000,0000,0000,,So being able to factorize\Nactually helps us to solve a new
Dialogue: 0,0:02:47.35,0:02:48.82,Default,,0000,0000,0000,,kind of equation.
Dialogue: 0,0:02:49.36,0:02:56.02,Default,,0000,0000,0000,,So we're going to be having a\Nlook in this video that how you
Dialogue: 0,0:02:56.02,0:03:00.78,Default,,0000,0000,0000,,factorise this kind of function.\NThis kind of expression a
Dialogue: 0,0:03:00.78,0:03:06.92,Default,,0000,0000,0000,,quadratic expression. Now I'm\Ngoing to start by going back to
Dialogue: 0,0:03:06.92,0:03:09.82,Default,,0000,0000,0000,,this little piece of work again.
Dialogue: 0,0:03:09.83,0:03:13.74,Default,,0000,0000,0000,,So let's write it down
Dialogue: 0,0:03:13.74,0:03:20.43,Default,,0000,0000,0000,,X. Plus 3\N* 5 X +2
Dialogue: 0,0:03:20.43,0:03:24.59,Default,,0000,0000,0000,,and again. Will\Nmultiply out the
Dialogue: 0,0:03:24.59,0:03:28.38,Default,,0000,0000,0000,,brackets X times by X\Nis X squared.
Dialogue: 0,0:03:29.55,0:03:36.67,Default,,0000,0000,0000,,X times by two is 2 X\N3 times by X is 3X.
Dialogue: 0,0:03:37.24,0:03:43.88,Default,,0000,0000,0000,,3 times by two is 6.\NThis simplifies to X squared
Dialogue: 0,0:03:43.88,0:03:46.30,Default,,0000,0000,0000,,plus 5X or 6.
Dialogue: 0,0:03:47.49,0:03:52.21,Default,,0000,0000,0000,,So we've gone one way. What\Nhappens if we want to go back
Dialogue: 0,0:03:52.21,0:03:53.30,Default,,0000,0000,0000,,the other way?
Dialogue: 0,0:03:54.07,0:04:00.62,Default,,0000,0000,0000,,Let's have a look where this six\Ncame from. We know it came from
Dialogue: 0,0:04:00.62,0:04:02.49,Default,,0000,0000,0000,,3 times by two.
Dialogue: 0,0:04:03.73,0:04:11.09,Default,,0000,0000,0000,,Where did this five come from?\NWhere it came from 2 + 3?
Dialogue: 0,0:04:12.14,0:04:18.54,Default,,0000,0000,0000,,So if we were to reverse this\Nprocess, we be looking for two
Dialogue: 0,0:04:18.54,0:04:24.44,Default,,0000,0000,0000,,numbers that multiply together\Nto give us six an 2 numbers that
Dialogue: 0,0:04:24.44,0:04:27.39,Default,,0000,0000,0000,,added together to give us 5.
Dialogue: 0,0:04:28.03,0:04:34.66,Default,,0000,0000,0000,,The obvious ones that go in\Nthere are three 2.
Dialogue: 0,0:04:36.55,0:04:38.64,Default,,0000,0000,0000,,So if we began.
Dialogue: 0,0:04:39.21,0:04:40.38,Default,,0000,0000,0000,,With this
Dialogue: 0,0:04:42.03,0:04:43.28,Default,,0000,0000,0000,,We would.
Dialogue: 0,0:04:44.31,0:04:49.13,Default,,0000,0000,0000,,Be looking to\Nbreak that 5X down
Dialogue: 0,0:04:49.13,0:04:55.33,Default,,0000,0000,0000,,as X squared plus\N3X plus 2X plus 6.
Dialogue: 0,0:04:56.86,0:05:02.71,Default,,0000,0000,0000,,Then we could look at these two\Nand see if there was a common
Dialogue: 0,0:05:02.71,0:05:08.56,Default,,0000,0000,0000,,factor and there is X leaving us\Nwith X Plus three. Then we would
Dialogue: 0,0:05:08.56,0:05:13.100,Default,,0000,0000,0000,,look at these two. Is there a\Ncommon factor and there is 2.
Dialogue: 0,0:05:14.53,0:05:17.95,Default,,0000,0000,0000,,Leaving us again with
Dialogue: 0,0:05:17.95,0:05:25.17,Default,,0000,0000,0000,,X +3. And then we've\Ngot this common factor of X plus
Dialogue: 0,0:05:25.17,0:05:31.96,Default,,0000,0000,0000,,three in each of these two lumps\Nof algebra, so we can take out
Dialogue: 0,0:05:31.96,0:05:38.75,Default,,0000,0000,0000,,that X +3, and we've got the\Nother factor left X times by X
Dialogue: 0,0:05:38.75,0:05:45.54,Default,,0000,0000,0000,,plus three and two times by X\N+3, and so we've arrived at that
Dialogue: 0,0:05:45.54,0:05:49.42,Default,,0000,0000,0000,,factorization. Those brackets\Nthat we started off with.
Dialogue: 0,0:05:50.56,0:05:57.26,Default,,0000,0000,0000,,Now. That's what we've done and\Nwhat we need to do is to be able
Dialogue: 0,0:05:57.26,0:06:00.62,Default,,0000,0000,0000,,to repeat this process of\Nlooking for numbers that
Dialogue: 0,0:06:00.62,0:06:04.36,Default,,0000,0000,0000,,multiply together to give the\Nconstant term and numbers that
Dialogue: 0,0:06:04.36,0:06:08.85,Default,,0000,0000,0000,,will add together to give the\Nexterm. So let's have a little
Dialogue: 0,0:06:08.85,0:06:10.72,Default,,0000,0000,0000,,bit of practice at that.
Dialogue: 0,0:06:11.82,0:06:18.94,Default,,0000,0000,0000,,Let's look at X squared\Nminus 7X plus 12.
Dialogue: 0,0:06:19.93,0:06:24.23,Default,,0000,0000,0000,,So we want two numbers that will\Nmultiply together to give us.
Dialogue: 0,0:06:24.82,0:06:32.32,Default,,0000,0000,0000,,12 Times together to give\Nus 12 and will add together
Dialogue: 0,0:06:32.32,0:06:35.26,Default,,0000,0000,0000,,to give us minus 7.
Dialogue: 0,0:06:36.35,0:06:42.65,Default,,0000,0000,0000,,Minus four times minus three is\N12 and minus four plus minus
Dialogue: 0,0:06:42.65,0:06:48.95,Default,,0000,0000,0000,,three gives us minus Seven, so\Nlet's just write those in minus
Dialogue: 0,0:06:48.95,0:06:55.25,Default,,0000,0000,0000,,four times, minus three gives us\Nplus 12 and minus four plus
Dialogue: 0,0:06:55.25,0:07:01.55,Default,,0000,0000,0000,,minus three gives us minus\NSeven, so that's given us a way
Dialogue: 0,0:07:01.55,0:07:07.32,Default,,0000,0000,0000,,of breaking down this minus\NSeven XX squared minus 4X Minus
Dialogue: 0,0:07:07.32,0:07:12.97,Default,,0000,0000,0000,,3X. Plus 12 so now we look at\Nthese two at the front.
Dialogue: 0,0:07:13.73,0:07:21.05,Default,,0000,0000,0000,,Take out X as a common factor\Nthat gives us X minus four, and
Dialogue: 0,0:07:21.05,0:07:24.19,Default,,0000,0000,0000,,now we look at these two.
Dialogue: 0,0:07:24.98,0:07:30.16,Default,,0000,0000,0000,,Well, I want to make sure I get\Nthe same factor X minus 4.
Dialogue: 0,0:07:31.47,0:07:36.34,Default,,0000,0000,0000,,Clearly, in these two terms,\Nthat is a factor of three. But
Dialogue: 0,0:07:36.34,0:07:42.03,Default,,0000,0000,0000,,here I've got minus three, so I\Nthink I'm going to have to make
Dialogue: 0,0:07:42.03,0:07:46.49,Default,,0000,0000,0000,,the factor, not three, but minus\Nthree. So that's minus three
Dialogue: 0,0:07:46.49,0:07:50.96,Default,,0000,0000,0000,,times X, so that insures when I\Nmultiply these two together
Dialogue: 0,0:07:50.96,0:07:56.64,Default,,0000,0000,0000,,minus three times by ex. I do\Nget minus 3X, but now I need
Dialogue: 0,0:07:56.64,0:08:01.51,Default,,0000,0000,0000,,minus three times by something\Nthat's got to give me plus 12.
Dialogue: 0,0:08:01.57,0:08:07.76,Default,,0000,0000,0000,,So that will have to be minus\Nfour and close the bracket. Now
Dialogue: 0,0:08:07.76,0:08:13.95,Default,,0000,0000,0000,,again I've got two lumps of\Nalgebra, and in each one that is
Dialogue: 0,0:08:13.95,0:08:19.66,Default,,0000,0000,0000,,the same factor. This common\Nfactor of X minus 4X minus four,
Dialogue: 0,0:08:19.66,0:08:25.85,Default,,0000,0000,0000,,so I'll take that as my common\Nfactor X minus four. Then I've
Dialogue: 0,0:08:25.85,0:08:32.03,Default,,0000,0000,0000,,got X minus four times by X&X\Nminus four times by minus three.
Dialogue: 0,0:08:32.37,0:08:35.72,Default,,0000,0000,0000,,That's my factorization of
Dialogue: 0,0:08:35.72,0:08:39.11,Default,,0000,0000,0000,,that. Let's take another
Dialogue: 0,0:08:39.11,0:08:41.100,Default,,0000,0000,0000,,one. X
Dialogue: 0,0:08:41.100,0:08:48.25,Default,,0000,0000,0000,,squared Minus 5X\Nminus 14. So now
Dialogue: 0,0:08:48.25,0:08:52.45,Default,,0000,0000,0000,,looking for two\Nnumbers to multiply
Dialogue: 0,0:08:52.45,0:08:58.74,Default,,0000,0000,0000,,together to give minus\N14 and add together to
Dialogue: 0,0:08:58.74,0:09:00.84,Default,,0000,0000,0000,,give minus 5.
Dialogue: 0,0:09:01.86,0:09:08.06,Default,,0000,0000,0000,,Fairly obvious factors of 14 R.\NSeven and two. So can we play
Dialogue: 0,0:09:08.06,0:09:14.26,Default,,0000,0000,0000,,with Seven and two? Well, if we\Nmade it minus Seven and plus
Dialogue: 0,0:09:14.26,0:09:20.46,Default,,0000,0000,0000,,two, then minus Seven times my\Nplus two would give us minus 14
Dialogue: 0,0:09:20.46,0:09:26.19,Default,,0000,0000,0000,,and minus Seven, plus the two\Nwould give us minus five, so
Dialogue: 0,0:09:26.19,0:09:31.91,Default,,0000,0000,0000,,minus Seven and two look like\Nthe two numbers that we need.
Dialogue: 0,0:09:32.53,0:09:39.09,Default,,0000,0000,0000,,So let's breakdown this minus\N5X as minus 7X Plus 2X.
Dialogue: 0,0:09:39.09,0:09:45.64,Default,,0000,0000,0000,,And let's not forget the\Nminus 14 that we had again.
Dialogue: 0,0:09:45.64,0:09:51.60,Default,,0000,0000,0000,,Let's look at these two. The\Nfront two terms. Common
Dialogue: 0,0:09:51.60,0:09:57.56,Default,,0000,0000,0000,,factor. Yes, it's X. Take\Nthat out X minus 7.
Dialogue: 0,0:09:58.75,0:10:02.70,Default,,0000,0000,0000,,And here a common factor of +2.
Dialogue: 0,0:10:03.21,0:10:07.06,Default,,0000,0000,0000,,Let's take that out and X
Dialogue: 0,0:10:07.06,0:10:14.80,Default,,0000,0000,0000,,minus 7. Two lumps of algebra\Nthat one and that one and each
Dialogue: 0,0:10:14.80,0:10:20.75,Default,,0000,0000,0000,,one's got the same. Factoring\Nthis X minus Seven, so we'll
Dialogue: 0,0:10:20.75,0:10:23.100,Default,,0000,0000,0000,,take that as our common factor.
Dialogue: 0,0:10:24.15,0:10:31.59,Default,,0000,0000,0000,,So X minus Seven is\Nmultiplying X and it's auto
Dialogue: 0,0:10:31.59,0:10:38.15,Default,,0000,0000,0000,,multiplying +2. So again, there\Nwe've arrived at a factorization
Dialogue: 0,0:10:38.15,0:10:45.70,Default,,0000,0000,0000,,of this X squared minus 5X\Nminus 14 factorizes as X minus
Dialogue: 0,0:10:45.70,0:10:47.59,Default,,0000,0000,0000,,Seven X +2.
Dialogue: 0,0:10:48.54,0:10:54.68,Default,,0000,0000,0000,,Type X\Nsquared minus
Dialogue: 0,0:10:54.68,0:10:59.28,Default,,0000,0000,0000,,9X plus\N20.
Dialogue: 0,0:11:00.93,0:11:04.47,Default,,0000,0000,0000,,Now, from what we've got\Nalready, it might be that some
Dialogue: 0,0:11:04.47,0:11:08.34,Default,,0000,0000,0000,,of you watching this might\Nthink, well, do I need to do
Dialogue: 0,0:11:08.34,0:11:12.52,Default,,0000,0000,0000,,that every time? The answer is\Nno. Sometimes you may be able to
Dialogue: 0,0:11:12.52,0:11:15.42,Default,,0000,0000,0000,,do these by inspection, which\Nmeans looking at it.
Dialogue: 0,0:11:15.96,0:11:20.42,Default,,0000,0000,0000,,And doing the working out in\Nyour head rather than on the
Dialogue: 0,0:11:20.42,0:11:24.89,Default,,0000,0000,0000,,paper. So to do it by\Ninspection. What we might do is
Dialogue: 0,0:11:24.89,0:11:28.98,Default,,0000,0000,0000,,right down the pair of brackets\Nto begin with. Recognize X
Dialogue: 0,0:11:28.98,0:11:33.82,Default,,0000,0000,0000,,squared means we're going to\Nhave to have an X and then X.
Dialogue: 0,0:11:34.69,0:11:41.23,Default,,0000,0000,0000,,Recognize 20 as being four times\Nby 5 and of course 4 + 5 would
Dialogue: 0,0:11:41.23,0:11:47.33,Default,,0000,0000,0000,,give me 9, but I want minus\Nnine, so perhaps what I need is
Dialogue: 0,0:11:47.33,0:11:51.69,Default,,0000,0000,0000,,minus four and minus five,\Nbecause minus four times by
Dialogue: 0,0:11:51.69,0:11:56.93,Default,,0000,0000,0000,,minus five is going to give me\Nplus 20 and minus 4X.
Dialogue: 0,0:11:57.51,0:12:04.61,Default,,0000,0000,0000,,Minus 5X is going to give me\Nminus 9X, so I've done that one
Dialogue: 0,0:12:04.61,0:12:11.10,Default,,0000,0000,0000,,by inspection. But I could have\Ndone it in exactly the same way
Dialogue: 0,0:12:11.10,0:12:16.91,Default,,0000,0000,0000,,as I did the other two. Let's\Ntake X squared minus nine X
Dialogue: 0,0:12:16.91,0:12:23.86,Default,,0000,0000,0000,,minus 22. And again, let's try\Nthis one out by inspection. So
Dialogue: 0,0:12:23.86,0:12:25.48,Default,,0000,0000,0000,,pair of brackets.
Dialogue: 0,0:12:26.19,0:12:29.94,Default,,0000,0000,0000,,X&X in front of each bracket.
Dialogue: 0,0:12:31.31,0:12:36.69,Default,,0000,0000,0000,,Let's have a look at minus 22\Ntwo numbers to multiply together
Dialogue: 0,0:12:36.69,0:12:42.06,Default,,0000,0000,0000,,to give minus 22 will likely\Ncandidates are minus 11 and two.
Dialogue: 0,0:12:42.74,0:12:49.67,Default,,0000,0000,0000,,Or minus 2 and 11, but at the\Nend of the day I need minus 9X
Dialogue: 0,0:12:49.67,0:12:54.86,Default,,0000,0000,0000,,and that kind of suggests that\Nperhaps we've got to have the
Dialogue: 0,0:12:54.86,0:13:00.49,Default,,0000,0000,0000,,bigger of 11 and two as being\Nnegative and the smaller one as
Dialogue: 0,0:13:00.49,0:13:07.20,Default,,0000,0000,0000,,being positive. Let's just check\Nminus 11 times +2 gives me minus
Dialogue: 0,0:13:07.20,0:13:14.45,Default,,0000,0000,0000,,22, and then I have minus 11X\Nand 2X, which gives me minus 9X.
Dialogue: 0,0:13:14.45,0:13:18.08,Default,,0000,0000,0000,,So again, we've done that one by
Dialogue: 0,0:13:18.08,0:13:23.19,Default,,0000,0000,0000,,inspection. Again, you don't\Nhave to do it by inspection. You
Dialogue: 0,0:13:23.19,0:13:25.10,Default,,0000,0000,0000,,can use the previous method.
Dialogue: 0,0:13:26.58,0:13:30.12,Default,,0000,0000,0000,,If we have quadratic expressions\Nwhich don't have a unit
Dialogue: 0,0:13:30.12,0:13:33.66,Default,,0000,0000,0000,,coefficient, now this is one\Nthat has a unit coefficient,
Dialogue: 0,0:13:33.66,0:13:35.78,Default,,0000,0000,0000,,'cause this is One X squared.
Dialogue: 0,0:13:37.22,0:13:42.22,Default,,0000,0000,0000,,It could be 2 X squared. It\Ncould be 6 X squared, could be
Dialogue: 0,0:13:42.22,0:13:46.50,Default,,0000,0000,0000,,11 X squared, could be anything\Ntimes by X squared. That would
Dialogue: 0,0:13:46.50,0:13:47.93,Default,,0000,0000,0000,,be harder to do.
Dialogue: 0,0:13:48.45,0:13:54.58,Default,,0000,0000,0000,,So let's have a look at how we\Nmight tackle some of those. So
Dialogue: 0,0:13:54.58,0:13:56.77,Default,,0000,0000,0000,,we take three X squared.
Dialogue: 0,0:13:57.54,0:14:01.80,Default,,0000,0000,0000,,Plus 5X minus\N2.
Dialogue: 0,0:14:03.34,0:14:08.02,Default,,0000,0000,0000,,I'm going to use a method that's\Nvery similar to the first method
Dialogue: 0,0:14:08.02,0:14:12.34,Default,,0000,0000,0000,,that we saw. I'm going to look\Nfor two numbers that multiply
Dialogue: 0,0:14:12.34,0:14:16.30,Default,,0000,0000,0000,,together to give, well, let's\Nleave that unsaid for them in
Dialogue: 0,0:14:16.30,0:14:20.98,Default,,0000,0000,0000,,it, but these two numbers are\Ngoing to add together to give I.
Dialogue: 0,0:14:20.98,0:14:25.66,Default,,0000,0000,0000,,They're going to add together to\Ngive this +5 the Exterm, so that
Dialogue: 0,0:14:25.66,0:14:29.26,Default,,0000,0000,0000,,hasn't changed. We're looking\Nfor two numbers that will add
Dialogue: 0,0:14:29.26,0:14:32.14,Default,,0000,0000,0000,,together to give us the\Ncoefficient of X.
Dialogue: 0,0:14:33.15,0:14:37.70,Default,,0000,0000,0000,,What do the two numbers have to\Nmultiply together to give us?
Dialogue: 0,0:14:37.70,0:14:42.25,Default,,0000,0000,0000,,Well, they have to multiply\Ntogether to give us 3 times by
Dialogue: 0,0:14:42.25,0:14:46.79,Default,,0000,0000,0000,,minus two, so we don't just take\Nthe constant term, we multiply
Dialogue: 0,0:14:46.79,0:14:51.72,Default,,0000,0000,0000,,it by the coefficient of the X\Nsquared and three times by minus
Dialogue: 0,0:14:51.72,0:14:57.03,Default,,0000,0000,0000,,two is minus 6, and I'll just\Nwrite that here at the side that
Dialogue: 0,0:14:57.03,0:15:01.58,Default,,0000,0000,0000,,the minus six came from the\Nthree times by the minus two.
Dialogue: 0,0:15:01.58,0:15:05.36,Default,,0000,0000,0000,,And if you think about it,\Nthat's actually consistent with
Dialogue: 0,0:15:05.36,0:15:07.26,Default,,0000,0000,0000,,what we were doing before.
Dialogue: 0,0:15:07.32,0:15:11.18,Default,,0000,0000,0000,,Because in the previous\Nexamples, this number in front
Dialogue: 0,0:15:11.18,0:15:14.18,Default,,0000,0000,0000,,of the X squared had been one.
Dialogue: 0,0:15:14.83,0:15:17.23,Default,,0000,0000,0000,,And so one times by minus two
Dialogue: 0,0:15:17.23,0:15:22.39,Default,,0000,0000,0000,,would be. The constant term, so\Nwe are looking now for two
Dialogue: 0,0:15:22.39,0:15:26.61,Default,,0000,0000,0000,,numbers that multiply together\Nto give us minus 6 and add
Dialogue: 0,0:15:26.61,0:15:28.53,Default,,0000,0000,0000,,together to give us 5.
Dialogue: 0,0:15:29.25,0:15:32.56,Default,,0000,0000,0000,,Well. 3 times by two.
Dialogue: 0,0:15:33.13,0:15:36.37,Default,,0000,0000,0000,,Well, three times by two would\Ngive us plus 6.
Dialogue: 0,0:15:37.45,0:15:41.73,Default,,0000,0000,0000,,Minus three times by minus two\Nwould also give us plus six, so
Dialogue: 0,0:15:41.73,0:15:45.55,Default,,0000,0000,0000,,that's not good. Six and one.
Dialogue: 0,0:15:46.86,0:15:51.48,Default,,0000,0000,0000,,Well, if we could have 6\Ntimes by minus one, that
Dialogue: 0,0:15:51.48,0:15:56.52,Default,,0000,0000,0000,,would give us minus six and\Nsix AD minus one would give
Dialogue: 0,0:15:56.52,0:16:01.14,Default,,0000,0000,0000,,us 5. So this looks like the\Ncombination that we want.
Dialogue: 0,0:16:02.20,0:16:04.35,Default,,0000,0000,0000,,So we take three X squared.
Dialogue: 0,0:16:05.17,0:16:09.15,Default,,0000,0000,0000,,Plus 6X minus X
Dialogue: 0,0:16:09.15,0:16:15.78,Default,,0000,0000,0000,,minus 2. Let's have a\Nlook for a common factor here.
Dialogue: 0,0:16:15.78,0:16:21.70,Default,,0000,0000,0000,,Well, there's a three X squared\Nand a 6X, so there's obviously a
Dialogue: 0,0:16:21.70,0:16:28.52,Default,,0000,0000,0000,,tree is a factor, and also an X.\NSo let's take out three X leaves
Dialogue: 0,0:16:28.52,0:16:29.89,Default,,0000,0000,0000,,me X +2.
Dialogue: 0,0:16:30.82,0:16:36.36,Default,,0000,0000,0000,,3X times my X gives us the three\NX squared 3X times by two gives
Dialogue: 0,0:16:36.36,0:16:41.15,Default,,0000,0000,0000,,us 6X and now want to common\Nfactor for these two terms minus
Dialogue: 0,0:16:41.15,0:16:45.21,Default,,0000,0000,0000,,X minus two. We don't seem to\Nshare anything in common.
Dialogue: 0,0:16:45.76,0:16:52.31,Default,,0000,0000,0000,,I've got a common factor and\Nit's minus 1 - 1 times minus one
Dialogue: 0,0:16:52.31,0:16:58.86,Default,,0000,0000,0000,,times by something has to give\Nme minus X, so that must be X
Dialogue: 0,0:16:58.86,0:17:04.95,Default,,0000,0000,0000,,and minus one times by something\Nhas to give me minus two. Well,
Dialogue: 0,0:17:04.95,0:17:07.29,Default,,0000,0000,0000,,that's got to be +2.
Dialogue: 0,0:17:08.30,0:17:14.75,Default,,0000,0000,0000,,So now I've got these two lumps\Nof algebra again, this one and
Dialogue: 0,0:17:14.75,0:17:21.20,Default,,0000,0000,0000,,this one, and each lump has the\Nsame factor in it. This common
Dialogue: 0,0:17:21.20,0:17:28.64,Default,,0000,0000,0000,,factor of X +2, so I'll take\Nthat one out X +2 and I've got
Dialogue: 0,0:17:28.64,0:17:33.60,Default,,0000,0000,0000,,X +2. Multiplying 3X and X +2,\Nmultiplying minus one.
Dialogue: 0,0:17:33.74,0:17:40.41,Default,,0000,0000,0000,,And so there's the factorization\Nof the expression that we began
Dialogue: 0,0:17:40.41,0:17:47.56,Default,,0000,0000,0000,,with. Let's take another\None, two X squared.
Dialogue: 0,0:17:48.17,0:17:53.51,Default,,0000,0000,0000,,Plus 5X\Nminus 7.
Dialogue: 0,0:17:55.22,0:18:00.13,Default,,0000,0000,0000,,So we're looking for two numbers\Nthat will multiply together to
Dialogue: 0,0:18:00.13,0:18:05.48,Default,,0000,0000,0000,,give us 2 times by minus Seven,\Nso they must multiply together
Dialogue: 0,0:18:05.48,0:18:11.72,Default,,0000,0000,0000,,to give us minus 14. Just write\Ndown again at the side that the
Dialogue: 0,0:18:11.72,0:18:15.74,Default,,0000,0000,0000,,minus 14 comes from minus Seven\Ntimes by two.
Dialogue: 0,0:18:16.38,0:18:21.74,Default,,0000,0000,0000,,And then these two numbers,\Nwhatever they are, I've got to
Dialogue: 0,0:18:21.74,0:18:26.61,Default,,0000,0000,0000,,add together to give us the\Ncoefficient of X +5.
Dialogue: 0,0:18:27.81,0:18:32.25,Default,,0000,0000,0000,,So what are these two numbers?\NWell, Seven and two seem
Dialogue: 0,0:18:32.25,0:18:36.70,Default,,0000,0000,0000,,reasonable factors of 14, and\Nthey are factors of 14 which
Dialogue: 0,0:18:36.70,0:18:41.95,Default,,0000,0000,0000,,have a difference. If you like a\Nfive, so they seem good options
Dialogue: 0,0:18:41.95,0:18:44.37,Default,,0000,0000,0000,,7 and 2. Seven and two.
Dialogue: 0,0:18:45.22,0:18:49.59,Default,,0000,0000,0000,,But we've got to make a balance\Nhere. We need +5 and we need
Dialogue: 0,0:18:49.59,0:18:53.96,Default,,0000,0000,0000,,minus 4T. So one of these is got\Nto be negative, and it looks
Dialogue: 0,0:18:53.96,0:18:58.01,Default,,0000,0000,0000,,like it's going to have to be\Nnegative two in order that 7
Dialogue: 0,0:18:58.01,0:18:59.88,Default,,0000,0000,0000,,plus negative two should give us
Dialogue: 0,0:18:59.88,0:19:05.22,Default,,0000,0000,0000,,the five there. So now we\Ncan write this down as two
Dialogue: 0,0:19:05.22,0:19:06.06,Default,,0000,0000,0000,,X squared.
Dialogue: 0,0:19:07.15,0:19:14.47,Default,,0000,0000,0000,,Breaking up that plus 5X as\Nplus 7X minus 2X and then
Dialogue: 0,0:19:14.47,0:19:17.52,Default,,0000,0000,0000,,minus Seven at the end.
Dialogue: 0,0:19:18.76,0:19:23.42,Default,,0000,0000,0000,,What have we got here as a\Ncommon factor? Well, we've got
Dialogue: 0,0:19:23.42,0:19:29.24,Default,,0000,0000,0000,,an X in each term, so we can\Ntake that out, giving us 2X plus
Dialogue: 0,0:19:29.24,0:19:34.28,Default,,0000,0000,0000,,Seven. And here again, what have\Nwe got for a common factor? Or
Dialogue: 0,0:19:34.28,0:19:39.32,Default,,0000,0000,0000,,the only thing that's in common\Nis one and there's a minus sign
Dialogue: 0,0:19:39.32,0:19:46.31,Default,,0000,0000,0000,,with each one, so it's minus 1 *\N2 X plus 7 - 1 times by two. X
Dialogue: 0,0:19:46.31,0:19:49.02,Default,,0000,0000,0000,,gives us minus two X minus one.
Dialogue: 0,0:19:49.09,0:19:54.44,Default,,0000,0000,0000,,Plus Seven gives us minus Seven\Nclose the bracket.
Dialogue: 0,0:19:55.12,0:20:00.86,Default,,0000,0000,0000,,Two lumps of algebra. Again,\Nthis one, and this one. In each
Dialogue: 0,0:20:00.86,0:20:06.59,Default,,0000,0000,0000,,one. There's this common factor\Nof 2X plus Seven, so we take
Dialogue: 0,0:20:06.59,0:20:11.85,Default,,0000,0000,0000,,that out 2X plus 7 and that's\Nmultiplying X and it's
Dialogue: 0,0:20:11.85,0:20:17.11,Default,,0000,0000,0000,,multiplying minus one, and so we\Nhave got this factorization of
Dialogue: 0,0:20:17.11,0:20:19.50,Default,,0000,0000,0000,,the expression that we began
Dialogue: 0,0:20:19.50,0:20:22.61,Default,,0000,0000,0000,,with. Take
Dialogue: 0,0:20:22.61,0:20:29.12,Default,,0000,0000,0000,,another example.\NSix X squared.
Dialogue: 0,0:20:29.79,0:20:34.43,Default,,0000,0000,0000,,Minus 5X minus four. Now what's\Ndifferent here is that this is
Dialogue: 0,0:20:34.43,0:20:40.24,Default,,0000,0000,0000,,not a prime number. We've had a\Ntwo and we've had a 3, but this
Dialogue: 0,0:20:40.24,0:20:41.40,Default,,0000,0000,0000,,is a 6.
Dialogue: 0,0:20:42.60,0:20:47.06,Default,,0000,0000,0000,,You might have been able to do\Nthe other two by inspection, but
Dialogue: 0,0:20:47.06,0:20:51.18,Default,,0000,0000,0000,,this one is more difficult to do\Nby inspection, and really, we
Dialogue: 0,0:20:51.18,0:20:55.29,Default,,0000,0000,0000,,perhaps are going to have to\Ndepend upon the method we just
Dialogue: 0,0:20:55.29,0:20:58.72,Default,,0000,0000,0000,,learned, so we're looking again\Nfor two numbers that will
Dialogue: 0,0:20:58.72,0:21:02.84,Default,,0000,0000,0000,,multiply together to give us 6\Ntimes by minus four. In other
Dialogue: 0,0:21:02.84,0:21:03.87,Default,,0000,0000,0000,,words, minus 24.
Dialogue: 0,0:21:04.44,0:21:09.100,Default,,0000,0000,0000,,OK, I'll just write that down so\Nwe can see where it's come from.
Dialogue: 0,0:21:09.100,0:21:15.16,Default,,0000,0000,0000,,Minus 24 is 6 times by minus\Nfour and we want these two
Dialogue: 0,0:21:15.16,0:21:19.13,Default,,0000,0000,0000,,numbers. Whatever they are.\NThey've also got to add together
Dialogue: 0,0:21:19.13,0:21:24.29,Default,,0000,0000,0000,,to give us the coefficient of X,\Nso they must add together to
Dialogue: 0,0:21:24.29,0:21:25.88,Default,,0000,0000,0000,,give us minus 5.
Dialogue: 0,0:21:26.51,0:21:31.89,Default,,0000,0000,0000,,So 2 numbers that might multiply\Nto give us 24, eight, and three
Dialogue: 0,0:21:31.89,0:21:36.45,Default,,0000,0000,0000,,good options, and eight and\Nthree do have a difference of
Dialogue: 0,0:21:36.45,0:21:41.41,Default,,0000,0000,0000,,five, so they look options we\Ncan use. Now let's juggle the
Dialogue: 0,0:21:41.41,0:21:46.38,Default,,0000,0000,0000,,signs we need to have minus\Nfive, so that would suggest that
Dialogue: 0,0:21:46.38,0:21:52.18,Default,,0000,0000,0000,,the 8's got to be the negative\None. So let's have minus 8 times
Dialogue: 0,0:21:52.18,0:21:56.73,Default,,0000,0000,0000,,by three. That will give us\Nminus 24 and minus 8.
Dialogue: 0,0:21:56.79,0:22:03.60,Default,,0000,0000,0000,,Plus three that will give us\Nminus five, so we've got six X
Dialogue: 0,0:22:03.60,0:22:10.93,Default,,0000,0000,0000,,squared. Minus 8X plus\N3X breaking down that
Dialogue: 0,0:22:10.93,0:22:14.62,Default,,0000,0000,0000,,5X. Minus 4.
Dialogue: 0,0:22:15.16,0:22:17.30,Default,,0000,0000,0000,,Common factor here.
Dialogue: 0,0:22:18.81,0:22:22.94,Default,,0000,0000,0000,,Well, the six and the three\Nshare a common factor of three.
Dialogue: 0,0:22:22.94,0:22:27.75,Default,,0000,0000,0000,,And of course we've X squared\Nand X common factor of X, so we
Dialogue: 0,0:22:27.75,0:22:29.47,Default,,0000,0000,0000,,can take out three X.
Dialogue: 0,0:22:30.05,0:22:34.31,Default,,0000,0000,0000,,And that will leave us 2X\Nplus one.
Dialogue: 0,0:22:35.71,0:22:40.90,Default,,0000,0000,0000,,These two terms, what do we got\Nfor a common factor? Well, they
Dialogue: 0,0:22:40.90,0:22:46.08,Default,,0000,0000,0000,,clearly share a common factor of\Nfour and also a minus sign. So
Dialogue: 0,0:22:46.08,0:22:48.48,Default,,0000,0000,0000,,we take minus four times by.
Dialogue: 0,0:22:49.16,0:22:54.11,Default,,0000,0000,0000,,Now, minus four times by\Nsomething has to give us minus
Dialogue: 0,0:22:54.11,0:22:59.96,Default,,0000,0000,0000,,8X, so that's going to be 2X and\Nthen minus four times by
Dialogue: 0,0:22:59.96,0:23:05.81,Default,,0000,0000,0000,,something has to give us minus\Nfour, so that's got to be plus
Dialogue: 0,0:23:05.81,0:23:10.76,Default,,0000,0000,0000,,one again to lump sum algebra\Nsharing. This common factor of
Dialogue: 0,0:23:10.76,0:23:13.91,Default,,0000,0000,0000,,2X plus one. So we'll take that
Dialogue: 0,0:23:13.91,0:23:20.42,Default,,0000,0000,0000,,out. And then we have two\NX plus one multiplying 3X.
Dialogue: 0,0:23:20.67,0:23:23.98,Default,,0000,0000,0000,,And two X plus one multiplying
Dialogue: 0,0:23:23.98,0:23:31.45,Default,,0000,0000,0000,,minus 4. Will take\N1 final example of
Dialogue: 0,0:23:31.45,0:23:38.41,Default,,0000,0000,0000,,this kind. So\NI've got 15 X
Dialogue: 0,0:23:38.41,0:23:42.07,Default,,0000,0000,0000,,squared. Minus three X
Dialogue: 0,0:23:42.07,0:23:47.60,Default,,0000,0000,0000,,minus 80. Now in all the\Nothers, the thing that we
Dialogue: 0,0:23:47.60,0:23:50.73,Default,,0000,0000,0000,,haven't checked at the\Nbeginning, and perhaps we should
Dialogue: 0,0:23:50.73,0:23:53.87,Default,,0000,0000,0000,,have done is do the\Ncoefficients. The numbers that
Dialogue: 0,0:23:53.87,0:23:55.26,Default,,0000,0000,0000,,multiply the X squared.
Dialogue: 0,0:23:56.65,0:24:00.29,Default,,0000,0000,0000,,That multiply the X\Nand the constant term
Dialogue: 0,0:24:00.29,0:24:02.11,Default,,0000,0000,0000,,share a common factor.
Dialogue: 0,0:24:03.48,0:24:08.09,Default,,0000,0000,0000,,And in this case they do as a\Ncommon factor of 3.
Dialogue: 0,0:24:09.42,0:24:13.02,Default,,0000,0000,0000,,And where there is a common\Nfactor, we need to take it
Dialogue: 0,0:24:13.02,0:24:16.02,Default,,0000,0000,0000,,out to begin with, so will\Ntake the three out.
Dialogue: 0,0:24:17.25,0:24:22.09,Default,,0000,0000,0000,,3 times by something as to give\Nus 15 X squared so three times
Dialogue: 0,0:24:22.09,0:24:24.52,Default,,0000,0000,0000,,by 5 X squared will do that.
Dialogue: 0,0:24:25.53,0:24:30.74,Default,,0000,0000,0000,,3 times by something has to give\Nus minus 3X so three times by
Dialogue: 0,0:24:30.74,0:24:32.60,Default,,0000,0000,0000,,minus X will do that.
Dialogue: 0,0:24:33.59,0:24:39.39,Default,,0000,0000,0000,,3 times by something has to give\Nus minus 18 and so minus six
Dialogue: 0,0:24:39.39,0:24:40.63,Default,,0000,0000,0000,,will do that.
Dialogue: 0,0:24:41.13,0:24:45.48,Default,,0000,0000,0000,,Now we're looking at Factorizing\Nthis xpression Here in the
Dialogue: 0,0:24:45.48,0:24:49.83,Default,,0000,0000,0000,,bracket and we're looking for\Ntwo numbers that were multiplied
Dialogue: 0,0:24:49.83,0:24:55.48,Default,,0000,0000,0000,,together to give us five times\Nby minus six, which is minus 30.
Dialogue: 0,0:24:55.48,0:25:00.70,Default,,0000,0000,0000,,And again, I'll just write down\Nwhere that came from. Minus 30
Dialogue: 0,0:25:00.70,0:25:06.36,Default,,0000,0000,0000,,was five times by minus six, and\NI'm looking for two numbers that
Dialogue: 0,0:25:06.36,0:25:11.14,Default,,0000,0000,0000,,will add together to give Maine.\NNow here the number that's
Dialogue: 0,0:25:11.14,0:25:12.88,Default,,0000,0000,0000,,multiplying the X is.
Dialogue: 0,0:25:12.95,0:25:18.68,Default,,0000,0000,0000,,Minus one. So I want two\Nnumbers to multiply together to
Dialogue: 0,0:25:18.68,0:25:23.39,Default,,0000,0000,0000,,give me minus 30 and add\Ntogether to give me minus one,
Dialogue: 0,0:25:23.39,0:25:27.32,Default,,0000,0000,0000,,well, five and six seem like\Nobvious choices 'cause they've
Dialogue: 0,0:25:27.32,0:25:32.43,Default,,0000,0000,0000,,got a difference of one and they\Nmultiply together to give 30. So
Dialogue: 0,0:25:32.43,0:25:37.15,Default,,0000,0000,0000,,how can I juggle the signs with\Nthe five and the six?
Dialogue: 0,0:25:38.60,0:25:44.60,Default,,0000,0000,0000,,Well, 5 + 6 has to give me minus\None. It looks like the six is
Dialogue: 0,0:25:44.60,0:25:49.85,Default,,0000,0000,0000,,going to have to carry the minus\Nsign, so 5 plus minus six does
Dialogue: 0,0:25:49.85,0:25:55.10,Default,,0000,0000,0000,,give me minus one and five times\Nby minus six does give me minus
Dialogue: 0,0:25:55.10,0:25:57.72,Default,,0000,0000,0000,,30, so I'm going to have three.
Dialogue: 0,0:25:58.53,0:26:05.94,Default,,0000,0000,0000,,Brackets five X squared plus 5X\Nminus six X, so we broken down
Dialogue: 0,0:26:05.94,0:26:13.92,Default,,0000,0000,0000,,this minus X into 5X, minus 6X\Nand then the final term on the
Dialogue: 0,0:26:13.92,0:26:17.34,Default,,0000,0000,0000,,end minus six and close the
Dialogue: 0,0:26:17.34,0:26:23.24,Default,,0000,0000,0000,,bracket. Keep the three outside.\NLet's look at the front two
Dialogue: 0,0:26:23.24,0:26:28.14,Default,,0000,0000,0000,,terms here. There's a common\Nfactor of 5X. Let's take that
Dialogue: 0,0:26:28.14,0:26:29.47,Default,,0000,0000,0000,,out. Five X.
Dialogue: 0,0:26:30.52,0:26:38.02,Default,,0000,0000,0000,,X plus One 5X times by X gives\Nme the five X squared 5X times
Dialogue: 0,0:26:38.02,0:26:41.02,Default,,0000,0000,0000,,by one gives me the 5X.
Dialogue: 0,0:26:42.01,0:26:46.63,Default,,0000,0000,0000,,Common factor here is minus six.\NEach of these terms shares A6
Dialogue: 0,0:26:46.63,0:26:51.64,Default,,0000,0000,0000,,and the minus sign, so will take\Nout the factor minus six, and
Dialogue: 0,0:26:51.64,0:26:57.41,Default,,0000,0000,0000,,then we need minus six times by\Nhas to give us minus six X, so
Dialogue: 0,0:26:57.41,0:27:02.03,Default,,0000,0000,0000,,that's times by X minus six\Ntimes by something has to give
Dialogue: 0,0:27:02.03,0:27:07.04,Default,,0000,0000,0000,,us minus six, so that's minus\Nsix times by one, and then I
Dialogue: 0,0:27:07.04,0:27:12.04,Default,,0000,0000,0000,,need to make sure I close the\Nwhole bracket with that big one
Dialogue: 0,0:27:12.04,0:27:16.58,Default,,0000,0000,0000,,there. Two lumps of algebra,\Neach sharing this common factor
Dialogue: 0,0:27:16.58,0:27:19.26,Default,,0000,0000,0000,,of X plus one. Let's take that
Dialogue: 0,0:27:19.26,0:27:26.30,Default,,0000,0000,0000,,out three. Bracket X plus\None times by the X
Dialogue: 0,0:27:26.30,0:27:29.18,Default,,0000,0000,0000,,Plus One Times 5X.
Dialogue: 0,0:27:29.19,0:27:36.40,Default,,0000,0000,0000,,The X plus one also times\Nminus six, and so we've
Dialogue: 0,0:27:36.40,0:27:38.36,Default,,0000,0000,0000,,completed that factorization.
Dialogue: 0,0:27:39.79,0:27:43.69,Default,,0000,0000,0000,,OK, we've looked at a series of
Dialogue: 0,0:27:43.69,0:27:46.88,Default,,0000,0000,0000,,examples. An we've developed.
Dialogue: 0,0:27:47.44,0:27:52.35,Default,,0000,0000,0000,,A way of handling these that\Nenables us to factorize these
Dialogue: 0,0:27:52.35,0:27:56.92,Default,,0000,0000,0000,,quadratics. Hasn't really been\Nanything special about them, but
Dialogue: 0,0:27:56.92,0:28:02.40,Default,,0000,0000,0000,,I want to do now is have a look\Nat three particular special
Dialogue: 0,0:28:02.40,0:28:09.11,Default,,0000,0000,0000,,cases. Let's begin with the\Nfirst special case. By having a
Dialogue: 0,0:28:09.11,0:28:12.37,Default,,0000,0000,0000,,look at X squared minus 9.
Dialogue: 0,0:28:13.52,0:28:16.47,Default,,0000,0000,0000,,Now it's obviously different\Nabout this one.
Dialogue: 0,0:28:17.10,0:28:21.39,Default,,0000,0000,0000,,From the previous examples is\Nthat there's no external, just
Dialogue: 0,0:28:21.39,0:28:23.54,Default,,0000,0000,0000,,says X squared minus 9.
Dialogue: 0,0:28:24.09,0:28:30.32,Default,,0000,0000,0000,,So let's do it in the way that\Nwe would do we look for two
Dialogue: 0,0:28:30.32,0:28:34.88,Default,,0000,0000,0000,,numbers that would multiply\Ntogether to give us minus 9 and
Dialogue: 0,0:28:34.88,0:28:40.28,Default,,0000,0000,0000,,add together to give us the X\Ncoefficient, but there is no X
Dialogue: 0,0:28:40.28,0:28:44.01,Default,,0000,0000,0000,,coefficient. That means the\Ncoefficient has to be 0.
Dialogue: 0,0:28:44.53,0:28:47.93,Default,,0000,0000,0000,,0 times by XOX is.
Dialogue: 0,0:28:48.93,0:28:53.56,Default,,0000,0000,0000,,So I've got to find 2 numbers\Nthat multiply together to give
Dialogue: 0,0:28:53.56,0:28:56.26,Default,,0000,0000,0000,,minus 9 and add together to give
Dialogue: 0,0:28:56.26,0:29:03.33,Default,,0000,0000,0000,,0. Obviously they've got to be\Nthe same size but different
Dialogue: 0,0:29:03.33,0:29:10.00,Default,,0000,0000,0000,,sign, so minus three and\Nthree fit the bill perfectly,
Dialogue: 0,0:29:10.00,0:29:16.67,Default,,0000,0000,0000,,so we have X squared\Nminus 3X plus 3X minus
Dialogue: 0,0:29:16.67,0:29:22.80,Default,,0000,0000,0000,,9. Look at the front two terms.\NThere's a common factor of X.
Dialogue: 0,0:29:23.32,0:29:30.43,Default,,0000,0000,0000,,Leaving me with X minus three.\NThe back two terms as a common
Dialogue: 0,0:29:30.43,0:29:36.100,Default,,0000,0000,0000,,factor of 3 leaving X minus\Nthree and now two lumps of
Dialogue: 0,0:29:36.100,0:29:43.56,Default,,0000,0000,0000,,algebra each sharing this common\Nfactor of X minus three X minus
Dialogue: 0,0:29:43.56,0:29:47.94,Default,,0000,0000,0000,,three, multiplies the X and\Nmultiplies the three.
Dialogue: 0,0:29:48.47,0:29:52.25,Default,,0000,0000,0000,,Well, we compare this
Dialogue: 0,0:29:52.25,0:29:58.42,Default,,0000,0000,0000,,with this. Not only is\Nthis X squared that we get X
Dialogue: 0,0:29:58.42,0:30:00.54,Default,,0000,0000,0000,,times by X, but this 9.
Dialogue: 0,0:30:01.50,0:30:04.18,Default,,0000,0000,0000,,Forgetting the minus sign for a\Nmoment is 3 squared.
Dialogue: 0,0:30:05.18,0:30:06.63,Default,,0000,0000,0000,,3 times by three.
Dialogue: 0,0:30:07.77,0:30:15.33,Default,,0000,0000,0000,,So in fact this expression could\Nbe rewritten as X squared minus
Dialogue: 0,0:30:15.33,0:30:20.72,Default,,0000,0000,0000,,3 squared. In other words, it's\Nthe difference of two squares.
Dialogue: 0,0:30:21.42,0:30:23.70,Default,,0000,0000,0000,,So let's do this again.
Dialogue: 0,0:30:24.59,0:30:29.69,Default,,0000,0000,0000,,But more generally, In other\Nwords, instead of minus nine,
Dialogue: 0,0:30:29.69,0:30:35.81,Default,,0000,0000,0000,,which is minus 3 squared, let's\Nwrite minus a squared. So we
Dialogue: 0,0:30:35.81,0:30:38.87,Default,,0000,0000,0000,,have X squared minus a squared.
Dialogue: 0,0:30:39.53,0:30:46.35,Default,,0000,0000,0000,,We want to factorize it, so\Nwe're looking for two numbers
Dialogue: 0,0:30:46.35,0:30:53.17,Default,,0000,0000,0000,,that multiply together to give\Nminus a squared and add together
Dialogue: 0,0:30:53.17,0:31:00.61,Default,,0000,0000,0000,,to give 0. Because there are\Nno access, so minus A and
Dialogue: 0,0:31:00.61,0:31:08.05,Default,,0000,0000,0000,,a fit the bill. So we're\Ngoing to have X squared minus
Dialogue: 0,0:31:08.05,0:31:10.53,Default,,0000,0000,0000,,8X Plus 8X minus.
Dialogue: 0,0:31:10.57,0:31:17.83,Default,,0000,0000,0000,,A squared. Common factor\Nof X here X minus
Dialogue: 0,0:31:17.83,0:31:25.16,Default,,0000,0000,0000,,a. Under common factor\Nof a here, X minus a
Dialogue: 0,0:31:25.16,0:31:32.39,Default,,0000,0000,0000,,again 2 lumps of algebra, each\None sharing this common factor
Dialogue: 0,0:31:32.39,0:31:35.02,Default,,0000,0000,0000,,of X minus a.
Dialogue: 0,0:31:35.03,0:31:42.28,Default,,0000,0000,0000,,X minus a multiplies\NX, an multiplies a.
Dialogue: 0,0:31:42.93,0:31:46.77,Default,,0000,0000,0000,,So we now have.
Dialogue: 0,0:31:47.39,0:31:53.71,Default,,0000,0000,0000,,What is, in effect a standard\Nresult we have what's called the
Dialogue: 0,0:31:53.71,0:31:56.35,Default,,0000,0000,0000,,difference of two squares and
Dialogue: 0,0:31:56.35,0:32:02.81,Default,,0000,0000,0000,,its factorization. So let's just\Nwrite that down again. X squared
Dialogue: 0,0:32:02.81,0:32:09.88,Default,,0000,0000,0000,,minus a squared is always equal\Nto X minus a X plus
Dialogue: 0,0:32:09.88,0:32:15.22,Default,,0000,0000,0000,,A. So that if we can identify\Nthis number that appears, here
Dialogue: 0,0:32:15.22,0:32:16.64,Default,,0000,0000,0000,,is a square number.
Dialogue: 0,0:32:17.60,0:32:20.70,Default,,0000,0000,0000,,We can use this\Nfactorization immediately.
Dialogue: 0,0:32:21.92,0:32:28.33,Default,,0000,0000,0000,,So what if we had something say\Nlike X squared minus 25?
Dialogue: 0,0:32:29.10,0:32:33.24,Default,,0000,0000,0000,,Well, we recognize 25 as\Nbeing 5 squared, so
Dialogue: 0,0:32:33.24,0:32:38.76,Default,,0000,0000,0000,,immediately we can write this\Ndown as X minus five X +5.
Dialogue: 0,0:32:39.80,0:32:46.22,Default,,0000,0000,0000,,What if we had something like\N2 X squared minus 32?
Dialogue: 0,0:32:47.55,0:32:51.88,Default,,0000,0000,0000,,Doesn't really look like that,\Ndoes it? But there is a common
Dialogue: 0,0:32:51.88,0:32:56.21,Default,,0000,0000,0000,,factor of 2, so as we've said\Nbefore, take the common factor
Dialogue: 0,0:32:56.21,0:32:57.66,Default,,0000,0000,0000,,out to begin with.
Dialogue: 0,0:32:58.37,0:33:03.75,Default,,0000,0000,0000,,Leaving us with X\Nsquared minus 16 and
Dialogue: 0,0:33:03.75,0:33:09.79,Default,,0000,0000,0000,,of course 16 is 4\Nsquared, so this is
Dialogue: 0,0:33:09.79,0:33:13.15,Default,,0000,0000,0000,,2X minus four X +4.
Dialogue: 0,0:33:16.14,0:33:21.86,Default,,0000,0000,0000,,Files and we had nine X\Nsquared minus 16.
Dialogue: 0,0:33:23.95,0:33:25.08,Default,,0000,0000,0000,,What about this one?
Dialogue: 0,0:33:25.70,0:33:32.78,Default,,0000,0000,0000,,Again, look at this term here\N9 X squared. It is a
Dialogue: 0,0:33:32.78,0:33:37.50,Default,,0000,0000,0000,,complete square. It's 3X times\Nby three X.
Dialogue: 0,0:33:38.01,0:33:42.47,Default,,0000,0000,0000,,So instead of just working with\Nan X, why can't we just work
Dialogue: 0,0:33:42.47,0:33:43.50,Default,,0000,0000,0000,,with a 3X?
Dialogue: 0,0:33:44.49,0:33:50.52,Default,,0000,0000,0000,,And of course, that's what we\Nare going to do. This must be 3X
Dialogue: 0,0:33:50.52,0:33:56.99,Default,,0000,0000,0000,,and 3X and the 16 is 4 squared,\Nso 3X minus four, 3X plus four.
Dialogue: 0,0:33:56.99,0:34:01.73,Default,,0000,0000,0000,,And again, this is still the\Ndifference of two squares, so
Dialogue: 0,0:34:01.73,0:34:06.90,Default,,0000,0000,0000,,that's one. The first special\Ncase, and we really do have to
Dialogue: 0,0:34:06.90,0:34:11.64,Default,,0000,0000,0000,,learn that one. And remember it\N'cause it's a very, very
Dialogue: 0,0:34:11.64,0:34:15.28,Default,,0000,0000,0000,,important factorization. Let's\Nhave a look now.
Dialogue: 0,0:34:15.78,0:34:18.71,Default,,0000,0000,0000,,Another factorization
Dialogue: 0,0:34:18.71,0:34:25.32,Default,,0000,0000,0000,,special case. Having\Njust on the difference of two
Dialogue: 0,0:34:25.32,0:34:29.22,Default,,0000,0000,0000,,squares looking at this\Nquadratic expression, we've got
Dialogue: 0,0:34:29.22,0:34:35.56,Default,,0000,0000,0000,,a square front X squared and the\Nsquare at the end 5 squared.
Dialogue: 0,0:34:37.14,0:34:39.20,Default,,0000,0000,0000,,But we've got 10X in the middle.
Dialogue: 0,0:34:39.89,0:34:44.90,Default,,0000,0000,0000,,OK, we know how to handle it, so\Nlet's not worry too much. We
Dialogue: 0,0:34:44.90,0:34:48.84,Default,,0000,0000,0000,,want two numbers that multiply\Ntogether to give 25 and two
Dialogue: 0,0:34:48.84,0:34:51.70,Default,,0000,0000,0000,,numbers that add together to\Ngive us 10.
Dialogue: 0,0:34:52.89,0:34:56.34,Default,,0000,0000,0000,,The obvious choice for that is 5
Dialogue: 0,0:34:56.34,0:35:02.96,Default,,0000,0000,0000,,and five. Five times by 5 is 20\Nfive 5 + 5 is 10.
Dialogue: 0,0:35:03.56,0:35:11.11,Default,,0000,0000,0000,,So we break that middle term\Ndown X squared plus 5X Plus
Dialogue: 0,0:35:11.11,0:35:12.100,Default,,0000,0000,0000,,5X plus 25.
Dialogue: 0,0:35:15.11,0:35:20.92,Default,,0000,0000,0000,,Look at the front. Two terms are\Ncommon factor of X, leaving us
Dialogue: 0,0:35:20.92,0:35:22.26,Default,,0000,0000,0000,,with X +5.
Dialogue: 0,0:35:23.06,0:35:29.73,Default,,0000,0000,0000,,Look at the back to terms are\Ncommon factor of +5 leaving us
Dialogue: 0,0:35:29.73,0:35:31.27,Default,,0000,0000,0000,,with X +5.
Dialogue: 0,0:35:32.21,0:35:37.68,Default,,0000,0000,0000,,Two lumps of algebra sharing a\Ncommon factor of X +5.
Dialogue: 0,0:35:38.23,0:35:44.13,Default,,0000,0000,0000,,X +5\Nmultiplies, X&X,
Dialogue: 0,0:35:44.13,0:35:47.08,Default,,0000,0000,0000,,+5, multiplies
Dialogue: 0,0:35:47.08,0:35:54.80,Default,,0000,0000,0000,,+5. These two\Nare the same, so this is
Dialogue: 0,0:35:54.80,0:35:57.44,Default,,0000,0000,0000,,X +5 all squared.
Dialogue: 0,0:35:58.46,0:36:05.39,Default,,0000,0000,0000,,In other words, what we've got\Nhere X squared plus 10X plus 25
Dialogue: 0,0:36:05.39,0:36:12.32,Default,,0000,0000,0000,,is a complete square X +5 all\Nsquared. We call that a complete
Dialogue: 0,0:36:12.32,0:36:16.18,Default,,0000,0000,0000,,square. Take\Nanother
Dialogue: 0,0:36:16.18,0:36:21.08,Default,,0000,0000,0000,,example, X\Nsquared minus
Dialogue: 0,0:36:21.08,0:36:24.76,Default,,0000,0000,0000,,8X plus 16.
Dialogue: 0,0:36:25.98,0:36:30.71,Default,,0000,0000,0000,,We recognize this is a square\Nnumber X squared and this is a
Dialogue: 0,0:36:30.71,0:36:33.97,Default,,0000,0000,0000,,square number. 16 is 4 squared.
Dialogue: 0,0:36:35.19,0:36:36.38,Default,,0000,0000,0000,,And of course.
Dialogue: 0,0:36:37.52,0:36:42.20,Default,,0000,0000,0000,,Minus 4 plus minus four would\Ngive us 8.
Dialogue: 0,0:36:42.83,0:36:48.58,Default,,0000,0000,0000,,As indeed minus four times, my\Nminus four would give us plus
Dialogue: 0,0:36:48.58,0:36:51.46,Default,,0000,0000,0000,,16. So we immediately.
Dialogue: 0,0:36:52.21,0:36:58.02,Default,,0000,0000,0000,,Recognizing this as this\Ncomplete square X minus
Dialogue: 0,0:36:58.02,0:37:00.20,Default,,0000,0000,0000,,four all squared.
Dialogue: 0,0:37:02.90,0:37:09.95,Default,,0000,0000,0000,,One more. 25\NX squared minus
Dialogue: 0,0:37:09.95,0:37:13.34,Default,,0000,0000,0000,,20 X +4.
Dialogue: 0,0:37:16.70,0:37:22.95,Default,,0000,0000,0000,,25 X squared is a complete\Nsquare. It's a square number.
Dialogue: 0,0:37:22.95,0:37:27.49,Default,,0000,0000,0000,,It's the result of multiplying\N5X by itself.
Dialogue: 0,0:37:28.15,0:37:34.72,Default,,0000,0000,0000,,4 is a square\Nnumber, it's 2
Dialogue: 0,0:37:34.72,0:37:37.53,Default,,0000,0000,0000,,times by two.
Dialogue: 0,0:37:38.76,0:37:41.81,Default,,0000,0000,0000,,Minus 20
Dialogue: 0,0:37:41.81,0:37:47.58,Default,,0000,0000,0000,,X. Well, if I say\Nthis is a complete square,
Dialogue: 0,0:37:47.58,0:37:52.04,Default,,0000,0000,0000,,then I've got to have minus\Ntwo in their minus two times,
Dialogue: 0,0:37:52.04,0:37:57.62,Default,,0000,0000,0000,,Y minus two would give me 4 -\N2 times by the five. X would
Dialogue: 0,0:37:57.62,0:38:02.09,Default,,0000,0000,0000,,give me minus 10X and I'm\Ngoing to have two of them
Dialogue: 0,0:38:02.09,0:38:05.81,Default,,0000,0000,0000,,minus 20X. So again I\Nrecognize this as a complete
Dialogue: 0,0:38:05.81,0:38:06.18,Default,,0000,0000,0000,,square.
Dialogue: 0,0:38:07.23,0:38:12.08,Default,,0000,0000,0000,,Might have found that a little\Nbit confusing and a little bit
Dialogue: 0,0:38:12.08,0:38:16.79,Default,,0000,0000,0000,,quick. That doesn't matter\Nbe'cause. If you don't recognize
Dialogue: 0,0:38:16.79,0:38:21.39,Default,,0000,0000,0000,,it, you can still use the\Nprevious method on it.
Dialogue: 0,0:38:22.39,0:38:28.75,Default,,0000,0000,0000,,Let's just check that 25 X\Nsquared minus 20X plus four. So
Dialogue: 0,0:38:28.75,0:38:35.11,Default,,0000,0000,0000,,if we didn't recognize this as a\Ncomplete square, we would be
Dialogue: 0,0:38:35.11,0:38:40.41,Default,,0000,0000,0000,,looking for two numbers that\Nwould multiply together to give
Dialogue: 0,0:38:40.41,0:38:47.83,Default,,0000,0000,0000,,us 100. The 100 is 4 times\Nby 25. Just write this at the
Dialogue: 0,0:38:47.83,0:38:53.13,Default,,0000,0000,0000,,side four times by 25 and two\Nnumbers that would.
Dialogue: 0,0:38:53.15,0:38:58.13,Default,,0000,0000,0000,,Add together to give us minus\N20. The obvious choices are 10
Dialogue: 0,0:38:58.13,0:39:03.52,Default,,0000,0000,0000,,and 10, well minus 10 and minus\N10 because we want minus 10
Dialogue: 0,0:39:03.52,0:39:09.34,Default,,0000,0000,0000,,times by minus 10 to give plus\N100 and minus 10 plus minus 10
Dialogue: 0,0:39:09.34,0:39:11.41,Default,,0000,0000,0000,,to give us minus 20.
Dialogue: 0,0:39:12.38,0:39:16.92,Default,,0000,0000,0000,,So we take this expression\N25 X squared.
Dialogue: 0,0:39:18.05,0:39:23.81,Default,,0000,0000,0000,,And we breakdown the minus 20X\Nminus 10X minus 10X.
Dialogue: 0,0:39:24.56,0:39:27.05,Default,,0000,0000,0000,,And we have the last terms plus
Dialogue: 0,0:39:27.05,0:39:33.10,Default,,0000,0000,0000,,4. We look at the front two\Nterms for a common factor and
Dialogue: 0,0:39:33.10,0:39:34.74,Default,,0000,0000,0000,,clearly there is 5X.
Dialogue: 0,0:39:34.75,0:39:40.43,Default,,0000,0000,0000,,Gives me 5X minus two. Now I\Nlook for a common factor in the
Dialogue: 0,0:39:40.43,0:39:46.52,Default,,0000,0000,0000,,last two terms there is a common\Nfactor of 2 here, because 2 * 5
Dialogue: 0,0:39:46.52,0:39:53.02,Default,,0000,0000,0000,,gives us 10 and 2 * 2 gives us\N4. But there is this minus sign,
Dialogue: 0,0:39:53.02,0:39:57.49,Default,,0000,0000,0000,,so perhaps we better take minus\Ntwo is our common factor.
Dialogue: 0,0:39:58.00,0:40:02.57,Default,,0000,0000,0000,,Which will give us minus two\Ntimes by something has to give
Dialogue: 0,0:40:02.57,0:40:07.52,Default,,0000,0000,0000,,us minus 10X. That's going to be\N5X and minus two times by
Dialogue: 0,0:40:07.52,0:40:12.48,Default,,0000,0000,0000,,something has to give us plus\Nfour which is going to have to
Dialogue: 0,0:40:12.48,0:40:13.62,Default,,0000,0000,0000,,be minus 2.
Dialogue: 0,0:40:14.19,0:40:19.95,Default,,0000,0000,0000,,Two lumps of algebra. The common\Nfactor is 5X minus 2.
Dialogue: 0,0:40:20.70,0:40:27.87,Default,,0000,0000,0000,,5X minus two\Nis multiplying 5X.
Dialogue: 0,0:40:27.87,0:40:31.64,Default,,0000,0000,0000,,And it's also multiplying minus
Dialogue: 0,0:40:31.64,0:40:38.12,Default,,0000,0000,0000,,2. So we have got this again\Nas a complete square, but not
Dialogue: 0,0:40:38.12,0:40:41.46,Default,,0000,0000,0000,,by inspection, but using our\Nstandard method.
Dialogue: 0,0:40:43.13,0:40:45.84,Default,,0000,0000,0000,,So that deals with the 2nd
Dialogue: 0,0:40:45.84,0:40:51.66,Default,,0000,0000,0000,,special case. Let's now have a\Nlook at the 3rd and final
Dialogue: 0,0:40:51.66,0:40:56.51,Default,,0000,0000,0000,,special case, and this is when\Nwe don't have a constant term
Dialogue: 0,0:40:56.51,0:41:02.16,Default,,0000,0000,0000,,and we've got something like 3 X\Nsquared minus 8X. What do we do
Dialogue: 0,0:41:02.16,0:41:07.42,Default,,0000,0000,0000,,with that? Well, clearly there\Nis a common factor of X, so we
Dialogue: 0,0:41:07.42,0:41:12.26,Default,,0000,0000,0000,,must take that out to begin\Nwith. So we take out X.
Dialogue: 0,0:41:14.63,0:41:20.17,Default,,0000,0000,0000,,We've got three X squared, so X\Ntimes by three X must give us
Dialogue: 0,0:41:20.17,0:41:21.76,Default,,0000,0000,0000,,the three X squared.
Dialogue: 0,0:41:22.36,0:41:27.68,Default,,0000,0000,0000,,And then X times by something to\Ngive us the minus 8X. Well it's
Dialogue: 0,0:41:27.68,0:41:33.00,Default,,0000,0000,0000,,got to be minus 8, and that's\Nall that we need to do. We've
Dialogue: 0,0:41:33.00,0:41:37.94,Default,,0000,0000,0000,,broken it down into two brackets\NX times by three X minus 8.
Dialogue: 0,0:41:38.83,0:41:42.58,Default,,0000,0000,0000,,What if say we had 10
Dialogue: 0,0:41:42.58,0:41:46.11,Default,,0000,0000,0000,,X squared? Plus
Dialogue: 0,0:41:46.11,0:41:50.32,Default,,0000,0000,0000,,5X. Again, we look\Nfor a common factor.
Dialogue: 0,0:41:51.40,0:41:54.72,Default,,0000,0000,0000,,Obviously there's an ex as a\Ncommon factor again, but
Dialogue: 0,0:41:54.72,0:41:57.71,Default,,0000,0000,0000,,there's more this time\Nbecause there's ten and five
Dialogue: 0,0:41:57.71,0:42:02.02,Default,,0000,0000,0000,,which share a common factor\Nof 5, so we need to pull out
Dialogue: 0,0:42:02.02,0:42:04.02,Default,,0000,0000,0000,,the whole of that as 5X.
Dialogue: 0,0:42:06.05,0:42:11.80,Default,,0000,0000,0000,,Five times by two will give us\Nthe 10, so that's 5X times by
Dialogue: 0,0:42:11.80,0:42:17.15,Default,,0000,0000,0000,,two. X will give us the 10 X\Nsquared plus 5X times by
Dialogue: 0,0:42:17.15,0:42:21.26,Default,,0000,0000,0000,,something to give us 5X that\Nmust just be 1.
Dialogue: 0,0:42:22.29,0:42:27.22,Default,,0000,0000,0000,,So, so long as we remember\Nto inspect the quadratic
Dialogue: 0,0:42:27.22,0:42:31.16,Default,,0000,0000,0000,,expression 1st and check\Nfor common factors, this
Dialogue: 0,0:42:31.16,0:42:34.62,Default,,0000,0000,0000,,particular one shouldn't\Ncause us any difficulties.