[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:00.36,Default,,0000,0000,0000,,
Dialogue: 0,0:00:00.36,0:00:03.87,Default,,0000,0000,0000,,In this video I want to do a\Nbunch of examples of factoring
Dialogue: 0,0:00:03.87,0:00:06.59,Default,,0000,0000,0000,,a second degree polynomial,\Nwhich is
Dialogue: 0,0:00:06.59,0:00:08.87,Default,,0000,0000,0000,,often called a quadratic.
Dialogue: 0,0:00:08.87,0:00:12.65,Default,,0000,0000,0000,,Sometimes a quadratic\Npolynomial, or just a
Dialogue: 0,0:00:12.65,0:00:15.85,Default,,0000,0000,0000,,quadratic itself, or quadratic\Nexpression, but all it means
Dialogue: 0,0:00:15.85,0:00:18.30,Default,,0000,0000,0000,,is a second degree polynomial.
Dialogue: 0,0:00:18.30,0:00:22.44,Default,,0000,0000,0000,,So something that's going to\Nhave a variable raised to the
Dialogue: 0,0:00:22.44,0:00:23.08,Default,,0000,0000,0000,,second power.
Dialogue: 0,0:00:23.08,0:00:26.46,Default,,0000,0000,0000,,In this case, in all of the\Nexamples we'll do, it'll be x.
Dialogue: 0,0:00:26.46,0:00:30.74,Default,,0000,0000,0000,,So let's say I have the\Nquadratic expression, x
Dialogue: 0,0:00:30.74,0:00:35.28,Default,,0000,0000,0000,,squared plus 10x, plus 9.
Dialogue: 0,0:00:35.28,0:00:39.87,Default,,0000,0000,0000,,And I want to factor it into the\Nproduct of two binomials.
Dialogue: 0,0:00:39.87,0:00:41.52,Default,,0000,0000,0000,,How do we do that?
Dialogue: 0,0:00:41.52,0:00:44.44,Default,,0000,0000,0000,,Well, let's just think about\Nwhat happens if we were to
Dialogue: 0,0:00:44.44,0:00:51.69,Default,,0000,0000,0000,,take x plus a, and multiply\Nthat by x plus b.
Dialogue: 0,0:00:51.69,0:00:55.05,Default,,0000,0000,0000,,If we were to multiply these\Ntwo things, what happens?
Dialogue: 0,0:00:55.05,0:00:57.09,Default,,0000,0000,0000,,Well, we have a little bit\Nof experience doing this.
Dialogue: 0,0:00:57.09,0:01:03.17,Default,,0000,0000,0000,,This will be x times x, which is\Nx squared, plus x times b,
Dialogue: 0,0:01:03.17,0:01:12.69,Default,,0000,0000,0000,,which is bx, plus a times x,\Nplus a times b-- plus ab.
Dialogue: 0,0:01:12.69,0:01:15.80,Default,,0000,0000,0000,,Or if we want to add these two\Nin the middle right here,
Dialogue: 0,0:01:15.80,0:01:18.89,Default,,0000,0000,0000,,because they're both\Ncoefficients of x.
Dialogue: 0,0:01:18.89,0:01:22.10,Default,,0000,0000,0000,,We could right this as x squared\Nplus-- I can write it
Dialogue: 0,0:01:22.10,0:01:29.70,Default,,0000,0000,0000,,as b plus a, or a plus\Nb, x, plus ab.
Dialogue: 0,0:01:29.70,0:01:34.39,Default,,0000,0000,0000,,So in general, if we assume that\Nthis is the product of
Dialogue: 0,0:01:34.39,0:01:40.76,Default,,0000,0000,0000,,two binomials, we see that this\Nmiddle coefficient on the
Dialogue: 0,0:01:40.76,0:01:43.25,Default,,0000,0000,0000,,x term, or you could say the\Nfirst degree coefficient
Dialogue: 0,0:01:43.25,0:01:49.04,Default,,0000,0000,0000,,there, that's going to be\Nthe sum of our a and b.
Dialogue: 0,0:01:49.04,0:01:51.36,Default,,0000,0000,0000,,And then the constant term is\Ngoing to be the product
Dialogue: 0,0:01:51.36,0:01:52.52,Default,,0000,0000,0000,,of our a and b.
Dialogue: 0,0:01:52.52,0:01:57.29,Default,,0000,0000,0000,,Notice, this would map\Nto this, and this
Dialogue: 0,0:01:57.29,0:01:58.79,Default,,0000,0000,0000,,would map to this.
Dialogue: 0,0:01:58.79,0:02:02.60,Default,,0000,0000,0000,,And, of course, this is the\Nsame thing as this.
Dialogue: 0,0:02:02.60,0:02:05.59,Default,,0000,0000,0000,,So can we somehow pattern\Nmatch this to that?
Dialogue: 0,0:02:05.59,0:02:14.06,Default,,0000,0000,0000,,Is there some a and b where\Na plus b is equal to 10?
Dialogue: 0,0:02:14.06,0:02:22.07,Default,,0000,0000,0000,,And a times b is equal to 9?
Dialogue: 0,0:02:22.07,0:02:23.85,Default,,0000,0000,0000,,Well, let's just think about\Nit a little bit.
Dialogue: 0,0:02:23.85,0:02:25.47,Default,,0000,0000,0000,,What are the factors of 9?
Dialogue: 0,0:02:25.47,0:02:27.77,Default,,0000,0000,0000,,What are the things that a\Nand b could be equal to?
Dialogue: 0,0:02:27.77,0:02:29.17,Default,,0000,0000,0000,,And we're assuming that\Neverything is an integer.
Dialogue: 0,0:02:29.17,0:02:32.35,Default,,0000,0000,0000,,And normally when we're\Nfactoring, especially when
Dialogue: 0,0:02:32.35,0:02:33.95,Default,,0000,0000,0000,,we're beginning to factor,\Nwe're dealing
Dialogue: 0,0:02:33.95,0:02:35.58,Default,,0000,0000,0000,,with integer numbers.
Dialogue: 0,0:02:35.58,0:02:37.08,Default,,0000,0000,0000,,So what are the factors of 9?
Dialogue: 0,0:02:37.08,0:02:40.73,Default,,0000,0000,0000,,They're 1, 3, and 9.
Dialogue: 0,0:02:40.73,0:02:45.00,Default,,0000,0000,0000,,So this could be a 3 and a 3,\Nor it could be a 1 and a 9.
Dialogue: 0,0:02:45.00,0:02:48.63,Default,,0000,0000,0000,,Now, if it's a 3 and a 3, then\Nyou'll have 3 plus 3-- that
Dialogue: 0,0:02:48.63,0:02:49.84,Default,,0000,0000,0000,,doesn't equal 10.
Dialogue: 0,0:02:49.84,0:02:53.76,Default,,0000,0000,0000,,But if it's a 1 and a\N9, 1 times 9 is 9.
Dialogue: 0,0:02:53.76,0:02:56.67,Default,,0000,0000,0000,,1 plus 9 is 10.
Dialogue: 0,0:02:56.67,0:02:57.57,Default,,0000,0000,0000,,So it does work.
Dialogue: 0,0:02:57.57,0:03:04.19,Default,,0000,0000,0000,,So a could be equal to 1, and\Nb could be equal to 9.
Dialogue: 0,0:03:04.19,0:03:08.92,Default,,0000,0000,0000,,So we could factor this\Nas being x plus 1,
Dialogue: 0,0:03:08.92,0:03:12.97,Default,,0000,0000,0000,,times x plus 9.
Dialogue: 0,0:03:12.97,0:03:15.85,Default,,0000,0000,0000,,And if you multiply these two\Nout, using the skills we
Dialogue: 0,0:03:15.85,0:03:18.97,Default,,0000,0000,0000,,developed in the last few\Nvideos, you'll see that it is
Dialogue: 0,0:03:18.97,0:03:22.88,Default,,0000,0000,0000,,indeed x squared plus\N10x, plus 9.
Dialogue: 0,0:03:22.88,0:03:25.15,Default,,0000,0000,0000,,So when you see something like\Nthis, when the coefficient on
Dialogue: 0,0:03:25.15,0:03:28.07,Default,,0000,0000,0000,,the x squared term, or the\Nleading coefficient on this
Dialogue: 0,0:03:28.07,0:03:31.65,Default,,0000,0000,0000,,quadratic is a 1, you can just\Nsay, all right, what two
Dialogue: 0,0:03:31.65,0:03:34.53,Default,,0000,0000,0000,,numbers add up to this\Ncoefficient right here?
Dialogue: 0,0:03:34.53,0:03:37.72,Default,,0000,0000,0000,,
Dialogue: 0,0:03:37.72,0:03:39.87,Default,,0000,0000,0000,,And those same two numbers, when\Nyou take their product,
Dialogue: 0,0:03:39.87,0:03:41.66,Default,,0000,0000,0000,,have to be equal to 9.
Dialogue: 0,0:03:41.66,0:03:43.51,Default,,0000,0000,0000,,And of course, this has to\Nbe in standard form.
Dialogue: 0,0:03:43.51,0:03:46.00,Default,,0000,0000,0000,,Or if it's not in standard form,\Nyou should put it in
Dialogue: 0,0:03:46.00,0:03:48.50,Default,,0000,0000,0000,,that form, so that you can\Nalways say, OK, whatever's on
Dialogue: 0,0:03:48.50,0:03:51.53,Default,,0000,0000,0000,,the first degree coefficient,\Nmy a and b
Dialogue: 0,0:03:51.53,0:03:52.30,Default,,0000,0000,0000,,have to add to that.
Dialogue: 0,0:03:52.30,0:03:55.89,Default,,0000,0000,0000,,Whatever's my constant term, my\Na times b, the product has
Dialogue: 0,0:03:55.89,0:03:56.37,Default,,0000,0000,0000,,to be that.
Dialogue: 0,0:03:56.37,0:03:58.15,Default,,0000,0000,0000,,Let's do several\Nmore examples.
Dialogue: 0,0:03:58.15,0:04:00.51,Default,,0000,0000,0000,,I think the more examples\Nwe do the more
Dialogue: 0,0:04:00.51,0:04:02.63,Default,,0000,0000,0000,,sense this'll make.
Dialogue: 0,0:04:02.63,0:04:08.70,Default,,0000,0000,0000,,Let's say we had x squared\Nplus 10x, plus-- well, I
Dialogue: 0,0:04:08.70,0:04:11.10,Default,,0000,0000,0000,,already did 10x, let's do a\Ndifferent number-- x squared
Dialogue: 0,0:04:11.10,0:04:15.37,Default,,0000,0000,0000,,plus 15x, plus 50.
Dialogue: 0,0:04:15.37,0:04:17.47,Default,,0000,0000,0000,,And we want to factor this.
Dialogue: 0,0:04:17.47,0:04:20.34,Default,,0000,0000,0000,,Well, same drill.
Dialogue: 0,0:04:20.34,0:04:22.60,Default,,0000,0000,0000,,We have an x squared term.
Dialogue: 0,0:04:22.60,0:04:25.13,Default,,0000,0000,0000,,We have a first degree term.
Dialogue: 0,0:04:25.13,0:04:27.96,Default,,0000,0000,0000,,This right here should be\Nthe sum of two numbers.
Dialogue: 0,0:04:27.96,0:04:30.62,Default,,0000,0000,0000,,And then this term, the constant\Nterm right here,
Dialogue: 0,0:04:30.62,0:04:32.87,Default,,0000,0000,0000,,should be the product\Nof two numbers.
Dialogue: 0,0:04:32.87,0:04:35.64,Default,,0000,0000,0000,,So we need to think of two\Nnumbers that, when I multiply
Dialogue: 0,0:04:35.64,0:04:39.22,Default,,0000,0000,0000,,them I get 50, and when\NI add them, I get 15.
Dialogue: 0,0:04:39.22,0:04:41.91,Default,,0000,0000,0000,,And this is going to be a bit of\Nan art that you're going to
Dialogue: 0,0:04:41.91,0:04:44.53,Default,,0000,0000,0000,,develop, but the more practice\Nyou do, you're going to see
Dialogue: 0,0:04:44.53,0:04:45.73,Default,,0000,0000,0000,,that it'll start to\Ncome naturally.
Dialogue: 0,0:04:45.73,0:04:47.33,Default,,0000,0000,0000,,So what could a and b be?
Dialogue: 0,0:04:47.33,0:04:48.98,Default,,0000,0000,0000,,Let's think about the\Nfactors of 50.
Dialogue: 0,0:04:48.98,0:04:52.23,Default,,0000,0000,0000,,It could be 1 times 50.
Dialogue: 0,0:04:52.23,0:04:55.07,Default,,0000,0000,0000,,2 times 25.
Dialogue: 0,0:04:55.07,0:04:57.52,Default,,0000,0000,0000,,Let's see, 4 doesn't\Ngo into 50.
Dialogue: 0,0:04:57.52,0:05:02.45,Default,,0000,0000,0000,,It could be 5 times 10.
Dialogue: 0,0:05:02.45,0:05:03.58,Default,,0000,0000,0000,,I think that's all of them.
Dialogue: 0,0:05:03.58,0:05:05.92,Default,,0000,0000,0000,,Let's try out these numbers,\Nand see if any of
Dialogue: 0,0:05:05.92,0:05:07.31,Default,,0000,0000,0000,,these add up to 15.
Dialogue: 0,0:05:07.31,0:05:12.63,Default,,0000,0000,0000,,So 1 plus 50 does not\Nadd up to 15.
Dialogue: 0,0:05:12.63,0:05:16.16,Default,,0000,0000,0000,,2 plus 25 does not\Nadd up to 15.
Dialogue: 0,0:05:16.16,0:05:19.26,Default,,0000,0000,0000,,But 5 plus 10 does\Nadd up to 15.
Dialogue: 0,0:05:19.26,0:05:24.28,Default,,0000,0000,0000,,So this could be 5 plus 10, and\Nthis could be 5 times 10.
Dialogue: 0,0:05:24.28,0:05:28.76,Default,,0000,0000,0000,,So if we were to factor this,\Nthis would be equal to x plus
Dialogue: 0,0:05:28.76,0:05:32.63,Default,,0000,0000,0000,,5, times x plus 10.
Dialogue: 0,0:05:32.63,0:05:33.84,Default,,0000,0000,0000,,And multiply it out.
Dialogue: 0,0:05:33.84,0:05:36.71,Default,,0000,0000,0000,,I encourage you to multiply this\Nout, and see that this is
Dialogue: 0,0:05:36.71,0:05:39.66,Default,,0000,0000,0000,,indeed x squared plus\N15x, plus 10.
Dialogue: 0,0:05:39.66,0:05:43.32,Default,,0000,0000,0000,,In fact, let's do it. x\Ntimes x, x squared. x
Dialogue: 0,0:05:43.32,0:05:45.80,Default,,0000,0000,0000,,times 10, plus 10x.
Dialogue: 0,0:05:45.80,0:05:48.60,Default,,0000,0000,0000,,5 times x, plus 5x.
Dialogue: 0,0:05:48.60,0:05:51.62,Default,,0000,0000,0000,,5 times 10, plus 50.
Dialogue: 0,0:05:51.62,0:05:55.22,Default,,0000,0000,0000,,Notice, the 5 times\N10 gave us the 50.
Dialogue: 0,0:05:55.22,0:06:00.89,Default,,0000,0000,0000,,The 5x plus the 10x is giving\Nus the 15x in between.
Dialogue: 0,0:06:00.89,0:06:06.56,Default,,0000,0000,0000,,So it's x squared plus\N15x, plus 50.
Dialogue: 0,0:06:06.56,0:06:09.44,Default,,0000,0000,0000,,Let's up the stakes a little\Nbit, introduce some negative
Dialogue: 0,0:06:09.44,0:06:11.04,Default,,0000,0000,0000,,signs in here.
Dialogue: 0,0:06:11.04,0:06:18.89,Default,,0000,0000,0000,,Let's say I had x squared\Nminus 11x, plus 24.
Dialogue: 0,0:06:18.89,0:06:21.61,Default,,0000,0000,0000,,Now, it's the exact\Nsame principle.
Dialogue: 0,0:06:21.61,0:06:24.58,Default,,0000,0000,0000,,I need to think of two numbers,\Nthat when I add them,
Dialogue: 0,0:06:24.58,0:06:26.57,Default,,0000,0000,0000,,need to be equal\Nto negative 11.
Dialogue: 0,0:06:26.57,0:06:30.15,Default,,0000,0000,0000,,a plus b need to be equal\Nto negative 11.
Dialogue: 0,0:06:30.15,0:06:37.90,Default,,0000,0000,0000,,And a times b need to\Nbe equal to 24.
Dialogue: 0,0:06:37.90,0:06:41.29,Default,,0000,0000,0000,,Now, there's something for\Nyou to think about.
Dialogue: 0,0:06:41.29,0:06:43.96,Default,,0000,0000,0000,,When I multiply both of these\Nnumbers, I'm getting a
Dialogue: 0,0:06:43.96,0:06:45.08,Default,,0000,0000,0000,,positive number.
Dialogue: 0,0:06:45.08,0:06:46.96,Default,,0000,0000,0000,,I'm getting a 24.
Dialogue: 0,0:06:46.96,0:06:50.26,Default,,0000,0000,0000,,That means that both of these\Nneed to be positive, or both
Dialogue: 0,0:06:50.26,0:06:51.38,Default,,0000,0000,0000,,of these need to be negative.
Dialogue: 0,0:06:51.38,0:06:55.05,Default,,0000,0000,0000,,That's the only way I'm going to\Nget a positive number here.
Dialogue: 0,0:06:55.05,0:06:58.29,Default,,0000,0000,0000,,Now, if when I add them, I get\Na negative number, if these
Dialogue: 0,0:06:58.29,0:07:00.72,Default,,0000,0000,0000,,were positive, there's no way I\Ncan add two positive numbers
Dialogue: 0,0:07:00.72,0:07:03.28,Default,,0000,0000,0000,,and get a negative number, so\Nthe fact that their sum is
Dialogue: 0,0:07:03.28,0:07:05.75,Default,,0000,0000,0000,,negative, and the fact that\Ntheir product is positive,
Dialogue: 0,0:07:05.75,0:07:10.42,Default,,0000,0000,0000,,tells me that both a\Nand b are negative.
Dialogue: 0,0:07:10.42,0:07:13.20,Default,,0000,0000,0000,,a and b have to be negative.
Dialogue: 0,0:07:13.20,0:07:15.73,Default,,0000,0000,0000,,Remember, one can't be negative\Nand the other one
Dialogue: 0,0:07:15.73,0:07:18.71,Default,,0000,0000,0000,,can't be positive, because the\Nproduct would be negative.
Dialogue: 0,0:07:18.71,0:07:22.69,Default,,0000,0000,0000,,And they both can't be positive,\Nbecause when you add
Dialogue: 0,0:07:22.69,0:07:24.52,Default,,0000,0000,0000,,them it would get you\Na positive number.
Dialogue: 0,0:07:24.52,0:07:27.50,Default,,0000,0000,0000,,So let's just think about\Nwhat a and b can be.
Dialogue: 0,0:07:27.50,0:07:28.99,Default,,0000,0000,0000,,So two negative numbers.
Dialogue: 0,0:07:28.99,0:07:31.26,Default,,0000,0000,0000,,So let's think about\Nthe factors of 24.
Dialogue: 0,0:07:31.26,0:07:33.14,Default,,0000,0000,0000,,And we'll kind of have to think\Nof the negative factors.
Dialogue: 0,0:07:33.14,0:07:44.67,Default,,0000,0000,0000,,But let me see, it could be 1\Ntimes 24, 2 times 11, 3 times
Dialogue: 0,0:07:44.67,0:07:48.07,Default,,0000,0000,0000,,8, or 4 times 6.
Dialogue: 0,0:07:48.07,0:07:51.22,Default,,0000,0000,0000,,Now, which of these when I\Nmultiply these-- well,
Dialogue: 0,0:07:51.22,0:07:54.38,Default,,0000,0000,0000,,obviously when I multiply\N1 times 24, I get 24.
Dialogue: 0,0:07:54.38,0:07:58.91,Default,,0000,0000,0000,,When I get 2 times 11-- sorry,\Nthis is 2 times 12.
Dialogue: 0,0:07:58.91,0:07:59.79,Default,,0000,0000,0000,,I get 24.
Dialogue: 0,0:07:59.79,0:08:03.09,Default,,0000,0000,0000,,So we know that all these,\Nthe products are 24.
Dialogue: 0,0:08:03.09,0:08:07.47,Default,,0000,0000,0000,,But which two of these, which\Ntwo factors, when I add them,
Dialogue: 0,0:08:07.47,0:08:08.79,Default,,0000,0000,0000,,should I get 11?
Dialogue: 0,0:08:08.79,0:08:09.88,Default,,0000,0000,0000,,And then we could say,\Nlet's take the
Dialogue: 0,0:08:09.88,0:08:11.45,Default,,0000,0000,0000,,negative of both of those.
Dialogue: 0,0:08:11.45,0:08:15.47,Default,,0000,0000,0000,,So when you look at these,\N3 and 8 jump out.
Dialogue: 0,0:08:15.47,0:08:19.15,Default,,0000,0000,0000,,3 times 8 is equal to 24.
Dialogue: 0,0:08:19.15,0:08:22.81,Default,,0000,0000,0000,,3 plus 8 is equal to 11.
Dialogue: 0,0:08:22.81,0:08:24.68,Default,,0000,0000,0000,,But that doesn't quite\Nwork out, right?
Dialogue: 0,0:08:24.68,0:08:26.51,Default,,0000,0000,0000,,Because we have a negative\N11 here.
Dialogue: 0,0:08:26.51,0:08:29.69,Default,,0000,0000,0000,,But what if we did negative\N3 and negative 8?
Dialogue: 0,0:08:29.69,0:08:37.79,Default,,0000,0000,0000,,Negative 3 times negative 8\Nis equal to positive 24.
Dialogue: 0,0:08:37.79,0:08:43.58,Default,,0000,0000,0000,,Negative 3 plus negative 8\Nis equal to negative 11.
Dialogue: 0,0:08:43.58,0:08:46.60,Default,,0000,0000,0000,,So negative 3 and\Nnegative 8 work.
Dialogue: 0,0:08:46.60,0:08:53.84,Default,,0000,0000,0000,,So if we factor this, x squared\Nminus 11x, plus 24 is
Dialogue: 0,0:08:53.84,0:09:02.94,Default,,0000,0000,0000,,going to be equal to x minus\N3, times x minus 8.
Dialogue: 0,0:09:02.94,0:09:06.27,Default,,0000,0000,0000,,Let's do another\None like that.
Dialogue: 0,0:09:06.27,0:09:08.33,Default,,0000,0000,0000,,Actually, let's mix it\Nup a little bit.
Dialogue: 0,0:09:08.33,0:09:19.81,Default,,0000,0000,0000,,Let's say I had x squared\Nplus 5x, minus 14.
Dialogue: 0,0:09:19.81,0:09:21.77,Default,,0000,0000,0000,,So here we have a different\Nsituation.
Dialogue: 0,0:09:21.77,0:09:26.46,Default,,0000,0000,0000,,The product of my two numbers is\Nnegative, right? a times b
Dialogue: 0,0:09:26.46,0:09:28.19,Default,,0000,0000,0000,,is equal to negative 14.
Dialogue: 0,0:09:28.19,0:09:29.92,Default,,0000,0000,0000,,My product is negative.
Dialogue: 0,0:09:29.92,0:09:32.93,Default,,0000,0000,0000,,That tells me that one of them\Nis positive, and one of them
Dialogue: 0,0:09:32.93,0:09:33.91,Default,,0000,0000,0000,,is negative.
Dialogue: 0,0:09:33.91,0:09:39.20,Default,,0000,0000,0000,,And when I add the two, a plus\Nb, it'd be equal to 5.
Dialogue: 0,0:09:39.20,0:09:41.36,Default,,0000,0000,0000,,So let's think about\Nthe factors of 14.
Dialogue: 0,0:09:41.36,0:09:44.30,Default,,0000,0000,0000,,And what combinations of them,\Nwhen I add them, if one is
Dialogue: 0,0:09:44.30,0:09:46.56,Default,,0000,0000,0000,,positive and one is negative,\Nor I'm really kind of taking
Dialogue: 0,0:09:46.56,0:09:49.78,Default,,0000,0000,0000,,the difference of the\Ntwo, do I get 5?
Dialogue: 0,0:09:49.78,0:09:53.45,Default,,0000,0000,0000,,So if I take 1 and 14-- I'm just\Ngoing to try out things--
Dialogue: 0,0:09:53.45,0:10:01.82,Default,,0000,0000,0000,,1 and 14, negative 1 plus\N14 is negative 13.
Dialogue: 0,0:10:01.82,0:10:04.26,Default,,0000,0000,0000,,Negative 1 plus 14 is 13.
Dialogue: 0,0:10:04.26,0:10:07.43,Default,,0000,0000,0000,,So let me write all of the\Ncombinations that I could do.
Dialogue: 0,0:10:07.43,0:10:09.44,Default,,0000,0000,0000,,And eventually your brain\Nwill just zone in on it.
Dialogue: 0,0:10:09.44,0:10:16.49,Default,,0000,0000,0000,,So you've got negative 1\Nplus 14 is equal to 13.
Dialogue: 0,0:10:16.49,0:10:20.46,Default,,0000,0000,0000,,And 1 plus negative 14 is\Nequal to negative 13.
Dialogue: 0,0:10:20.46,0:10:21.38,Default,,0000,0000,0000,,So those don't work.
Dialogue: 0,0:10:21.38,0:10:22.95,Default,,0000,0000,0000,,That doesn't equal 5.
Dialogue: 0,0:10:22.95,0:10:24.86,Default,,0000,0000,0000,,Now what about 2 and 7?
Dialogue: 0,0:10:24.86,0:10:29.60,Default,,0000,0000,0000,,If I do negative 2-- let me do\Nthis in a different color-- if
Dialogue: 0,0:10:29.60,0:10:35.29,Default,,0000,0000,0000,,I do negative 2 plus 7,\Nthat is equal to 5.
Dialogue: 0,0:10:35.29,0:10:35.75,Default,,0000,0000,0000,,We're done!
Dialogue: 0,0:10:35.75,0:10:36.67,Default,,0000,0000,0000,,That worked!
Dialogue: 0,0:10:36.67,0:10:39.44,Default,,0000,0000,0000,,I mean, we could have tried 2\Nplus negative 7, but that'd be
Dialogue: 0,0:10:39.44,0:10:41.07,Default,,0000,0000,0000,,equal to negative 5, so that\Nwouldn't have worked.
Dialogue: 0,0:10:41.07,0:10:42.96,Default,,0000,0000,0000,,But negative 2 plus 7 works.
Dialogue: 0,0:10:42.96,0:10:46.59,Default,,0000,0000,0000,,And negative 2 times\N7 is negative 14.
Dialogue: 0,0:10:46.59,0:10:47.60,Default,,0000,0000,0000,,So there we have it.
Dialogue: 0,0:10:47.60,0:10:53.21,Default,,0000,0000,0000,,We know it's x minus\N2, times x plus 7.
Dialogue: 0,0:10:53.21,0:10:54.33,Default,,0000,0000,0000,,That's pretty neat.
Dialogue: 0,0:10:54.33,0:10:56.95,Default,,0000,0000,0000,,Negative 2 times 7\Nis negative 14.
Dialogue: 0,0:10:56.95,0:11:00.88,Default,,0000,0000,0000,,Negative 2 plus 7\Nis positive 5.
Dialogue: 0,0:11:00.88,0:11:03.76,Default,,0000,0000,0000,,
Dialogue: 0,0:11:03.76,0:11:07.68,Default,,0000,0000,0000,,Let's do several more of these,\Njust to really get well
Dialogue: 0,0:11:07.68,0:11:09.52,Default,,0000,0000,0000,,honed this skill.
Dialogue: 0,0:11:09.52,0:11:16.36,Default,,0000,0000,0000,,So let's say we have x squared\Nminus x, minus 56.
Dialogue: 0,0:11:16.36,0:11:19.57,Default,,0000,0000,0000,,So the product of the two\Nnumbers have to be minus 56,
Dialogue: 0,0:11:19.57,0:11:21.62,Default,,0000,0000,0000,,have to be negative 56.
Dialogue: 0,0:11:21.62,0:11:24.43,Default,,0000,0000,0000,,And their difference, because\None is going to be positive,
Dialogue: 0,0:11:24.43,0:11:26.28,Default,,0000,0000,0000,,and one is going to be\Nnegative, right?
Dialogue: 0,0:11:26.28,0:11:28.35,Default,,0000,0000,0000,,Their difference has\Nto be negative 1.
Dialogue: 0,0:11:28.35,0:11:30.13,Default,,0000,0000,0000,,And the numbers that immediately\Njump out in my
Dialogue: 0,0:11:30.13,0:11:31.90,Default,,0000,0000,0000,,brain-- and I don't know if they\Njump out in your brain,
Dialogue: 0,0:11:31.90,0:11:33.74,Default,,0000,0000,0000,,we just learned this in\Nthe times tables--
Dialogue: 0,0:11:33.74,0:11:36.48,Default,,0000,0000,0000,,56 is 8 times 7.
Dialogue: 0,0:11:36.48,0:11:37.48,Default,,0000,0000,0000,,I mean, there's other numbers.
Dialogue: 0,0:11:37.48,0:11:39.95,Default,,0000,0000,0000,,It's also 28 times 2.
Dialogue: 0,0:11:39.95,0:11:41.14,Default,,0000,0000,0000,,It's all sorts of things.
Dialogue: 0,0:11:41.14,0:11:44.30,Default,,0000,0000,0000,,But 8 times 7 really jumped\Nout into my brain, because
Dialogue: 0,0:11:44.30,0:11:45.47,Default,,0000,0000,0000,,they're very close\Nto each other.
Dialogue: 0,0:11:45.47,0:11:47.73,Default,,0000,0000,0000,,And we need numbers that are\Nvery close to each other.
Dialogue: 0,0:11:47.73,0:11:50.43,Default,,0000,0000,0000,,And one of these has to be\Npositive, and one of these has
Dialogue: 0,0:11:50.43,0:11:51.81,Default,,0000,0000,0000,,to be negative.
Dialogue: 0,0:11:51.81,0:11:55.25,Default,,0000,0000,0000,,Now, the fact that when their\Nsum is negative, tells me that
Dialogue: 0,0:11:55.25,0:11:58.46,Default,,0000,0000,0000,,the larger of these two should\Nprobably be negative.
Dialogue: 0,0:11:58.46,0:12:01.35,Default,,0000,0000,0000,,So if we take negative\N8 times 7, that's
Dialogue: 0,0:12:01.35,0:12:03.32,Default,,0000,0000,0000,,equal to negative 56.
Dialogue: 0,0:12:03.32,0:12:08.47,Default,,0000,0000,0000,,And then if we take negative\N8 plus 7, that is equal to
Dialogue: 0,0:12:08.47,0:12:12.10,Default,,0000,0000,0000,,negative 1, which is exactly the\Ncoefficient right there.
Dialogue: 0,0:12:12.10,0:12:16.49,Default,,0000,0000,0000,,So when I factor this, this\Nis going to be x minus 8,
Dialogue: 0,0:12:16.49,0:12:18.69,Default,,0000,0000,0000,,times x plus 7.
Dialogue: 0,0:12:18.69,0:12:21.70,Default,,0000,0000,0000,,This is often one of the hardest\Nconcepts people learn
Dialogue: 0,0:12:21.70,0:12:23.92,Default,,0000,0000,0000,,in algebra, because it\Nis a bit of an art.
Dialogue: 0,0:12:23.92,0:12:26.71,Default,,0000,0000,0000,,You have to look at all of the\Nfactors here, play with the
Dialogue: 0,0:12:26.71,0:12:29.71,Default,,0000,0000,0000,,positive and negative signs,\Nsee which of those factors
Dialogue: 0,0:12:29.71,0:12:31.90,Default,,0000,0000,0000,,when one is positive, one is\Nnegative, add up to the
Dialogue: 0,0:12:31.90,0:12:33.59,Default,,0000,0000,0000,,coefficient on the x term.
Dialogue: 0,0:12:33.59,0:12:35.86,Default,,0000,0000,0000,,But as you do more and more\Npractice, you'll see that
Dialogue: 0,0:12:35.86,0:12:39.28,Default,,0000,0000,0000,,it'll become a bit\Nof second nature.
Dialogue: 0,0:12:39.28,0:12:42.35,Default,,0000,0000,0000,,Now let's step up the stakes\Na little bit more.
Dialogue: 0,0:12:42.35,0:12:46.14,Default,,0000,0000,0000,,Let's say we had negative x\Nsquared-- everything we've
Dialogue: 0,0:12:46.14,0:12:49.04,Default,,0000,0000,0000,,done so far had a positive\Ncoefficient, a positive 1
Dialogue: 0,0:12:49.04,0:12:50.69,Default,,0000,0000,0000,,coefficient on the\Nx squared term.
Dialogue: 0,0:12:50.69,0:12:55.59,Default,,0000,0000,0000,,But let's say we had a\Nnegative x squared
Dialogue: 0,0:12:55.59,0:12:59.44,Default,,0000,0000,0000,,minus 5x, plus 24.
Dialogue: 0,0:12:59.44,0:13:00.91,Default,,0000,0000,0000,,How do we do this?
Dialogue: 0,0:13:00.91,0:13:03.42,Default,,0000,0000,0000,,Well, the easiest way I can\Nthink of doing it is factor
Dialogue: 0,0:13:03.42,0:13:05.67,Default,,0000,0000,0000,,out a negative 1, and then it\Nbecomes just like the problems
Dialogue: 0,0:13:05.67,0:13:07.26,Default,,0000,0000,0000,,we've been doing before.
Dialogue: 0,0:13:07.26,0:13:11.66,Default,,0000,0000,0000,,So this is the same thing as\Nnegative 1 times positive x
Dialogue: 0,0:13:11.66,0:13:15.99,Default,,0000,0000,0000,,squared, plus 5x, minus 24.
Dialogue: 0,0:13:15.99,0:13:16.30,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:13:16.30,0:13:18.08,Default,,0000,0000,0000,,I just factored a\Nnegative 1 out.
Dialogue: 0,0:13:18.08,0:13:20.18,Default,,0000,0000,0000,,You can multiply negative 1\Ntimes all of these, and you'll
Dialogue: 0,0:13:20.18,0:13:21.69,Default,,0000,0000,0000,,see it becomes this.
Dialogue: 0,0:13:21.69,0:13:23.56,Default,,0000,0000,0000,,Or you could factor the negative\N1 out and divide all
Dialogue: 0,0:13:23.56,0:13:25.00,Default,,0000,0000,0000,,of these by negative 1.
Dialogue: 0,0:13:25.00,0:13:26.76,Default,,0000,0000,0000,,And you get that right there.
Dialogue: 0,0:13:26.76,0:13:29.36,Default,,0000,0000,0000,,Now, same game as before.
Dialogue: 0,0:13:29.36,0:13:33.51,Default,,0000,0000,0000,,I need two numbers, that when\NI take their product I get
Dialogue: 0,0:13:33.51,0:13:34.69,Default,,0000,0000,0000,,negative 24.
Dialogue: 0,0:13:34.69,0:13:37.12,Default,,0000,0000,0000,,So one will be positive,\None will be negative.
Dialogue: 0,0:13:37.12,0:13:41.55,Default,,0000,0000,0000,,
Dialogue: 0,0:13:41.55,0:13:43.77,Default,,0000,0000,0000,,When I take their sum,\Nit's going to be 5.
Dialogue: 0,0:13:43.77,0:13:48.85,Default,,0000,0000,0000,,So let's think about\N24 is 1 and 24.
Dialogue: 0,0:13:48.85,0:13:55.75,Default,,0000,0000,0000,,Let's see, if this is negative 1\Nand 24, it'd be positive 23,
Dialogue: 0,0:13:55.75,0:13:57.54,Default,,0000,0000,0000,,if it was the other way around,\Nit'd be negative 23.
Dialogue: 0,0:13:57.54,0:13:58.50,Default,,0000,0000,0000,,Doesn't work.
Dialogue: 0,0:13:58.50,0:14:01.21,Default,,0000,0000,0000,,What about 2 and 12?
Dialogue: 0,0:14:01.21,0:14:04.53,Default,,0000,0000,0000,,Well, if this is negative--\Nremember, one of these has to
Dialogue: 0,0:14:04.53,0:14:05.04,Default,,0000,0000,0000,,be negative.
Dialogue: 0,0:14:05.04,0:14:07.57,Default,,0000,0000,0000,,If the 2 is negative, their\Nsum would be 10.
Dialogue: 0,0:14:07.57,0:14:09.75,Default,,0000,0000,0000,,If the 12 is negative, their\Nsum would be negative 10.
Dialogue: 0,0:14:09.75,0:14:11.30,Default,,0000,0000,0000,,Still doesn't work.
Dialogue: 0,0:14:11.30,0:14:13.29,Default,,0000,0000,0000,,3 and 8.
Dialogue: 0,0:14:13.29,0:14:16.85,Default,,0000,0000,0000,,If the 3 is negative,\Ntheir sum will be 5.
Dialogue: 0,0:14:16.85,0:14:17.91,Default,,0000,0000,0000,,So it works!
Dialogue: 0,0:14:17.91,0:14:24.62,Default,,0000,0000,0000,,So if we pick negative 3 and\N8, negative 3 and 8 work.
Dialogue: 0,0:14:24.62,0:14:26.63,Default,,0000,0000,0000,,Because negative\N3 plus 8 is 5.
Dialogue: 0,0:14:26.63,0:14:29.51,Default,,0000,0000,0000,,Negative 3 times 8\Nis negative 24.
Dialogue: 0,0:14:29.51,0:14:31.64,Default,,0000,0000,0000,,So this is going to be equal\Nto-- can't forget that
Dialogue: 0,0:14:31.64,0:14:35.23,Default,,0000,0000,0000,,negative 1 out front, and then\Nwe factor the inside.
Dialogue: 0,0:14:35.23,0:14:42.99,Default,,0000,0000,0000,,Negative 1 times x minus\N3, times x plus 8.
Dialogue: 0,0:14:42.99,0:14:44.80,Default,,0000,0000,0000,,And if you really wanted to,\Nyou could multiply the
Dialogue: 0,0:14:44.80,0:14:46.45,Default,,0000,0000,0000,,negative 1 times this,\Nyou would get 3
Dialogue: 0,0:14:46.45,0:14:47.56,Default,,0000,0000,0000,,minus x if you did.
Dialogue: 0,0:14:47.56,0:14:48.81,Default,,0000,0000,0000,,Or you don't have to.
Dialogue: 0,0:14:48.81,0:14:51.81,Default,,0000,0000,0000,,
Dialogue: 0,0:14:51.81,0:14:53.22,Default,,0000,0000,0000,,Let's do one more of these.
Dialogue: 0,0:14:53.22,0:14:56.06,Default,,0000,0000,0000,,The more practice, the\Nbetter, I think.
Dialogue: 0,0:14:56.06,0:15:01.63,Default,,0000,0000,0000,,All right, let's say I had\Nnegative x squared
Dialogue: 0,0:15:01.63,0:15:06.69,Default,,0000,0000,0000,,plus 18x, minus 72.
Dialogue: 0,0:15:06.69,0:15:09.32,Default,,0000,0000,0000,,So once again, I like to factor\Nout the negative 1.
Dialogue: 0,0:15:09.32,0:15:13.04,Default,,0000,0000,0000,,So this is equal to negative\N1 times x squared,
Dialogue: 0,0:15:13.04,0:15:16.91,Default,,0000,0000,0000,,minus 18x, plus 72.
Dialogue: 0,0:15:16.91,0:15:19.65,Default,,0000,0000,0000,,Now we just have to think of\Ntwo numbers, that when I
Dialogue: 0,0:15:19.65,0:15:22.33,Default,,0000,0000,0000,,multiply them I get\Npositive 72.
Dialogue: 0,0:15:22.33,0:15:24.48,Default,,0000,0000,0000,,So they have to be\Nthe same sign.
Dialogue: 0,0:15:24.48,0:15:26.93,Default,,0000,0000,0000,,And that makes it easier in our\Nhead, at least in my head.
Dialogue: 0,0:15:26.93,0:15:29.26,Default,,0000,0000,0000,,When I multiply them,\NI get positive 72.
Dialogue: 0,0:15:29.26,0:15:32.04,Default,,0000,0000,0000,,When I add them, I\Nget negative 18.
Dialogue: 0,0:15:32.04,0:15:34.41,Default,,0000,0000,0000,,So they're the same sign, and\Ntheir sum is a negative
Dialogue: 0,0:15:34.41,0:15:36.47,Default,,0000,0000,0000,,number, they both must\Nbe negative.
Dialogue: 0,0:15:36.47,0:15:41.47,Default,,0000,0000,0000,,
Dialogue: 0,0:15:41.47,0:15:43.79,Default,,0000,0000,0000,,And we could go through all\Nof the factors of 72.
Dialogue: 0,0:15:43.79,0:15:48.51,Default,,0000,0000,0000,,But the one that springs up,\Nmaybe you think of 8 times 9,
Dialogue: 0,0:15:48.51,0:15:53.27,Default,,0000,0000,0000,,but 8 times 9, or negative 8\Nminus 9, or negative 8 plus
Dialogue: 0,0:15:53.27,0:15:55.21,Default,,0000,0000,0000,,negative 9, doesn't work.
Dialogue: 0,0:15:55.21,0:15:58.06,Default,,0000,0000,0000,,That turns into 17.
Dialogue: 0,0:15:58.06,0:15:58.86,Default,,0000,0000,0000,,That was close.
Dialogue: 0,0:15:58.86,0:15:59.56,Default,,0000,0000,0000,,Let me show you that.
Dialogue: 0,0:15:59.56,0:16:04.40,Default,,0000,0000,0000,,Negative 9 plus negative 8, that\Nis equal to negative 17.
Dialogue: 0,0:16:04.40,0:16:06.14,Default,,0000,0000,0000,,Close, but no cigar.
Dialogue: 0,0:16:06.14,0:16:06.99,Default,,0000,0000,0000,,So what other ones are there?
Dialogue: 0,0:16:06.99,0:16:08.00,Default,,0000,0000,0000,,We have 6 and 12.
Dialogue: 0,0:16:08.00,0:16:09.70,Default,,0000,0000,0000,,That actually seems\Npretty good.
Dialogue: 0,0:16:09.70,0:16:13.71,Default,,0000,0000,0000,,If we have negative 6 plus\Nnegative 12, that is equal to
Dialogue: 0,0:16:13.71,0:16:15.36,Default,,0000,0000,0000,,negative 18.
Dialogue: 0,0:16:15.36,0:16:16.59,Default,,0000,0000,0000,,Notice, it's a bit of an art.
Dialogue: 0,0:16:16.59,0:16:18.81,Default,,0000,0000,0000,,You have to try the different\Nfactors here.
Dialogue: 0,0:16:18.81,0:16:22.36,Default,,0000,0000,0000,,So this will become negative\N1-- don't want to forget
Dialogue: 0,0:16:22.36,0:16:29.44,Default,,0000,0000,0000,,that-- times x minus 6,\Ntimes x minus 12.
Dialogue: 0,0:16:29.44,0:16:29.87,Default,,0000,0000,0000,,