[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:00.51,Default,,0000,0000,0000,,
Dialogue: 0,0:00:00.51,0:00:02.40,Default,,0000,0000,0000,,We're asked to factor\Nthis expression.
Dialogue: 0,0:00:02.40,0:00:05.46,Default,,0000,0000,0000,,And there's going to be simpler\Nways to factor it, but
Dialogue: 0,0:00:05.46,0:00:08.39,Default,,0000,0000,0000,,in this video I'm going to\Nfactor it by grouping.
Dialogue: 0,0:00:08.39,0:00:10.46,Default,,0000,0000,0000,,And when you factor by grouping,\Nwhat you need to do
Dialogue: 0,0:00:10.46,0:00:13.17,Default,,0000,0000,0000,,is think about two numbers\Nwhose products are
Dialogue: 0,0:00:13.17,0:00:14.05,Default,,0000,0000,0000,,going to be equal to.
Dialogue: 0,0:00:14.05,0:00:16.53,Default,,0000,0000,0000,,You have actually one\Ncoefficient right here, right?
Dialogue: 0,0:00:16.53,0:00:19.02,Default,,0000,0000,0000,,t squared is the same\Nthing as 1t squared.
Dialogue: 0,0:00:19.02,0:00:21.07,Default,,0000,0000,0000,,So we're looking for two\Nnumbers, let's call them a and
Dialogue: 0,0:00:21.07,0:00:23.51,Default,,0000,0000,0000,,b, a times b.
Dialogue: 0,0:00:23.51,0:00:25.75,Default,,0000,0000,0000,,the product of these two\Nnumbers needs to be the
Dialogue: 0,0:00:25.75,0:00:29.82,Default,,0000,0000,0000,,product of the coefficient on\Nthe t squared, which is 1, and
Dialogue: 0,0:00:29.82,0:00:32.09,Default,,0000,0000,0000,,the negative 15 right here.
Dialogue: 0,0:00:32.09,0:00:35.95,Default,,0000,0000,0000,,So a times b has to be equal\Nto 1 times negative 15, or
Dialogue: 0,0:00:35.95,0:00:38.51,Default,,0000,0000,0000,,just negative 15.
Dialogue: 0,0:00:38.51,0:00:43.77,Default,,0000,0000,0000,,And the sum of a and b, and a\Nplus b, needs to be equal to
Dialogue: 0,0:00:43.77,0:00:45.04,Default,,0000,0000,0000,,negative 2.
Dialogue: 0,0:00:45.04,0:00:49.61,Default,,0000,0000,0000,,
Dialogue: 0,0:00:49.61,0:00:51.94,Default,,0000,0000,0000,,And once we have these two\Nnumbers, I can show you how we
Dialogue: 0,0:00:51.94,0:00:54.24,Default,,0000,0000,0000,,can use those to factor\Nby grouping.
Dialogue: 0,0:00:54.24,0:00:56.62,Default,,0000,0000,0000,,And in other videos I've\Nactually broken down to why
Dialogue: 0,0:00:56.62,0:00:58.43,Default,,0000,0000,0000,,this technique works.
Dialogue: 0,0:00:58.43,0:01:01.37,Default,,0000,0000,0000,,Now, let's think of the\Ndifferent factors of negative
Dialogue: 0,0:01:01.37,0:01:04.29,Default,,0000,0000,0000,,15 when we take their product,\Nand if we take the sum, if we
Dialogue: 0,0:01:04.29,0:01:06.03,Default,,0000,0000,0000,,can somehow get to negative 2.
Dialogue: 0,0:01:06.03,0:01:10.59,Default,,0000,0000,0000,,So let's look at the different\Nfactors of negative 15.
Dialogue: 0,0:01:10.59,0:01:15.26,Default,,0000,0000,0000,,So we could do-- let me do it in\Nthis other, let me do it in
Dialogue: 0,0:01:15.26,0:01:19.25,Default,,0000,0000,0000,,pink-- see if 1 and negative 15,\Nthese are-- so everything
Dialogue: 0,0:01:19.25,0:01:22.57,Default,,0000,0000,0000,,I list here, their product is\Ngoing to be negative 15.
Dialogue: 0,0:01:22.57,0:01:24.22,Default,,0000,0000,0000,,But let's think about\Nwhat happens when
Dialogue: 0,0:01:24.22,0:01:25.79,Default,,0000,0000,0000,,you take their sum.
Dialogue: 0,0:01:25.79,0:01:28.79,Default,,0000,0000,0000,,So 1 and negative 15, the\Nsum is negative 14.
Dialogue: 0,0:01:28.79,0:01:33.12,Default,,0000,0000,0000,,And if you did negative 1 and\N15, you're just going to get
Dialogue: 0,0:01:33.12,0:01:33.93,Default,,0000,0000,0000,,the negative of that.
Dialogue: 0,0:01:33.93,0:01:34.73,Default,,0000,0000,0000,,You're going to get 14.
Dialogue: 0,0:01:34.73,0:01:36.24,Default,,0000,0000,0000,,It does not equal negative 2.
Dialogue: 0,0:01:36.24,0:01:42.09,Default,,0000,0000,0000,,So what happens if you take\N3 and negative 5?
Dialogue: 0,0:01:42.09,0:01:45.43,Default,,0000,0000,0000,,So their product is definitely\Nnegative 15, 3, plus negative
Dialogue: 0,0:01:45.43,0:01:48.15,Default,,0000,0000,0000,,5 is negative 2,\Nso that works.
Dialogue: 0,0:01:48.15,0:01:51.91,Default,,0000,0000,0000,,And if we tried negative 3 and\N5 first, we would have gotten
Dialogue: 0,0:01:51.91,0:01:52.78,Default,,0000,0000,0000,,that to be positive 2.
Dialogue: 0,0:01:52.78,0:01:54.47,Default,,0000,0000,0000,,It's just like, oh, we just have\Nto swap the signs and we
Dialogue: 0,0:01:54.47,0:01:55.51,Default,,0000,0000,0000,,would have gotten\Na negative 2.
Dialogue: 0,0:01:55.51,0:01:58.28,Default,,0000,0000,0000,,So these work, 3 and\Nnegative 5 work.
Dialogue: 0,0:01:58.28,0:02:02.43,Default,,0000,0000,0000,,3 times negative 5 is negative\N15, 3 plus negative 5 is
Dialogue: 0,0:02:02.43,0:02:03.42,Default,,0000,0000,0000,,negative 2.
Dialogue: 0,0:02:03.42,0:02:06.13,Default,,0000,0000,0000,,So what we want to do\Nhere is break this
Dialogue: 0,0:02:06.13,0:02:07.23,Default,,0000,0000,0000,,middle term up here.
Dialogue: 0,0:02:07.23,0:02:10.55,Default,,0000,0000,0000,,We know that 3 plus negative\N5 is equal to negative 2.
Dialogue: 0,0:02:10.55,0:02:14.01,Default,,0000,0000,0000,,So we can break up this middle\Nterm here as a sum of-- and
Dialogue: 0,0:02:14.01,0:02:15.37,Default,,0000,0000,0000,,I'll do it right here; I'll\Nactually do it in the same
Dialogue: 0,0:02:15.37,0:02:18.91,Default,,0000,0000,0000,,color-- so this thing here, we\Ncan rewrite as t squared.
Dialogue: 0,0:02:18.91,0:02:21.23,Default,,0000,0000,0000,,I'll put the minus\N15 out here.
Dialogue: 0,0:02:21.23,0:02:27.32,Default,,0000,0000,0000,,But the negative 2t we can\Nrewrite as the sum of 3t.
Dialogue: 0,0:02:27.32,0:02:34.96,Default,,0000,0000,0000,,We could write it here\Nas plus 3t, minus 5t.
Dialogue: 0,0:02:34.96,0:02:38.52,Default,,0000,0000,0000,,
Dialogue: 0,0:02:38.52,0:02:41.07,Default,,0000,0000,0000,,And when you're trying to figure\Nout which one to put
Dialogue: 0,0:02:41.07,0:02:44.54,Default,,0000,0000,0000,,first or second, you should look\Nat these other terms, and
Dialogue: 0,0:02:44.54,0:02:46.29,Default,,0000,0000,0000,,say which ones have\Ncommon factors?
Dialogue: 0,0:02:46.29,0:02:51.57,Default,,0000,0000,0000,,The 3 and 5 both have a common\Nfactor with 15, so it's not as
Dialogue: 0,0:02:51.57,0:02:53.46,Default,,0000,0000,0000,,obvious which one to put first,\Nso we're just going to
Dialogue: 0,0:02:53.46,0:02:53.89,Default,,0000,0000,0000,,go with this.
Dialogue: 0,0:02:53.89,0:02:58.25,Default,,0000,0000,0000,,3t minus 5t, I got that from\N3t minus 5t is equal to
Dialogue: 0,0:02:58.25,0:02:59.39,Default,,0000,0000,0000,,negative 2t.
Dialogue: 0,0:02:59.39,0:03:02.30,Default,,0000,0000,0000,,Positive 3 times negative 5\Nis equal to negative 15.
Dialogue: 0,0:03:02.30,0:03:03.69,Default,,0000,0000,0000,,That's where it came from.
Dialogue: 0,0:03:03.69,0:03:06.48,Default,,0000,0000,0000,,Now, we're ready to factor\Nby grouping.
Dialogue: 0,0:03:06.48,0:03:09.53,Default,,0000,0000,0000,,So let's take the first group,\Nlet's take these first two
Dialogue: 0,0:03:09.53,0:03:11.23,Default,,0000,0000,0000,,terms, right there.
Dialogue: 0,0:03:11.23,0:03:13.06,Default,,0000,0000,0000,,And what's the common\Nfactor there?
Dialogue: 0,0:03:13.06,0:03:14.51,Default,,0000,0000,0000,,Well, the common factor\Nthere is t.
Dialogue: 0,0:03:14.51,0:03:17.81,Default,,0000,0000,0000,,So if I factor a t out, that\Nbecomes t times-- t squared
Dialogue: 0,0:03:17.81,0:03:19.27,Default,,0000,0000,0000,,divided by t is t.
Dialogue: 0,0:03:19.27,0:03:22.53,Default,,0000,0000,0000,,3t divided by t is 3.
Dialogue: 0,0:03:22.53,0:03:25.12,Default,,0000,0000,0000,,So these first two terms\Nare the same thing as t
Dialogue: 0,0:03:25.12,0:03:27.09,Default,,0000,0000,0000,,times t plus 3.
Dialogue: 0,0:03:27.09,0:03:32.50,Default,,0000,0000,0000,,Now, let's look at the\Nsecond two terms.
Dialogue: 0,0:03:32.50,0:03:34.15,Default,,0000,0000,0000,,What's a common factor?
Dialogue: 0,0:03:34.15,0:03:36.68,Default,,0000,0000,0000,,Well, they're both divisible by\Nnegative 5, so let's factor
Dialogue: 0,0:03:36.68,0:03:37.93,Default,,0000,0000,0000,,out a negative 5.
Dialogue: 0,0:03:37.93,0:03:40.68,Default,,0000,0000,0000,,
Dialogue: 0,0:03:40.68,0:03:44.45,Default,,0000,0000,0000,,And negative 5t divided by\Nnegative 5, if you factor out
Dialogue: 0,0:03:44.45,0:03:46.40,Default,,0000,0000,0000,,the negative 5, you're just\Ngoing to have a t there.
Dialogue: 0,0:03:46.40,0:03:49.62,Default,,0000,0000,0000,,And then negative 15, if you\Nfactor out a negative 5, you
Dialogue: 0,0:03:49.62,0:03:51.70,Default,,0000,0000,0000,,divide negative 15 by negative\N5, you're just going to have a
Dialogue: 0,0:03:51.70,0:03:54.97,Default,,0000,0000,0000,,positive 3.
Dialogue: 0,0:03:54.97,0:03:59.29,Default,,0000,0000,0000,,And then notice, you now have\Ntwo terms here, two products,
Dialogue: 0,0:03:59.29,0:04:05.66,Default,,0000,0000,0000,,and they both have the common\Nfactor of t plus 3.
Dialogue: 0,0:04:05.66,0:04:10.20,Default,,0000,0000,0000,,So we can rewrite this right\Nhere as a product of t plus 3.
Dialogue: 0,0:04:10.20,0:04:12.56,Default,,0000,0000,0000,,We're undistributing\Nthe t plus 3.
Dialogue: 0,0:04:12.56,0:04:20.14,Default,,0000,0000,0000,,We're factoring out the t plus\N3. t plus 3 times t, right?
Dialogue: 0,0:04:20.14,0:04:23.92,Default,,0000,0000,0000,,Times t minus 5.
Dialogue: 0,0:04:23.92,0:04:26.31,Default,,0000,0000,0000,,And I want you to really make\Nsure you feel good that these
Dialogue: 0,0:04:26.31,0:04:28.03,Default,,0000,0000,0000,,are really the same thing.
Dialogue: 0,0:04:28.03,0:04:31.40,Default,,0000,0000,0000,,If you take t times t plus 3 and\Nfactor out the t plus 3,
Dialogue: 0,0:04:31.40,0:04:33.55,Default,,0000,0000,0000,,you're just left with that t.
Dialogue: 0,0:04:33.55,0:04:36.18,Default,,0000,0000,0000,,If you take negative 5 times t\Nplus 3 and you factor out the
Dialogue: 0,0:04:36.18,0:04:38.95,Default,,0000,0000,0000,,t plus 3, you're just left\Nwith that negative 5.
Dialogue: 0,0:04:38.95,0:04:41.26,Default,,0000,0000,0000,,But once you factor out the t\Nplus 3, and you're just left
Dialogue: 0,0:04:41.26,0:04:44.47,Default,,0000,0000,0000,,with the t minus 5, you\Nhave fully factored
Dialogue: 0,0:04:44.47,0:04:45.83,Default,,0000,0000,0000,,this expression here.
Dialogue: 0,0:04:45.83,0:04:47.58,Default,,0000,0000,0000,,And in the future we're going\Nto see easier ways of doing
Dialogue: 0,0:04:47.58,0:04:50.37,Default,,0000,0000,0000,,this, but factoring by grouping\Nis actually the
Dialogue: 0,0:04:50.37,0:04:55.13,Default,,0000,0000,0000,,easiest way to do it if you have\Na coefficient higher than
Dialogue: 0,0:04:55.13,0:04:56.39,Default,,0000,0000,0000,,1, or a non-one coefficient.
Dialogue: 0,0:04:56.39,0:04:58.83,Default,,0000,0000,0000,,It could also be a negative\Ncoefficient out front here.
Dialogue: 0,0:04:58.83,0:05:00.65,Default,,0000,0000,0000,,When you have 1 as your\Ncoefficient here, there's
Dialogue: 0,0:05:00.65,0:05:03.71,Default,,0000,0000,0000,,actually much easier ways to\Nfactor something like this,
Dialogue: 0,0:05:03.71,0:05:06.12,Default,,0000,0000,0000,,but it's really the same\Nthought process.
Dialogue: 0,0:05:06.12,0:05:06.80,Default,,0000,0000,0000,,