[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:00.63,Default,,0000,0000,0000,,
Dialogue: 0,0:00:00.63,0:00:03.16,Default,,0000,0000,0000,,We're asked to multiply\Nand to simplify.
Dialogue: 0,0:00:03.16,0:00:05.79,Default,,0000,0000,0000,,And we have x squared minus\Nthe principal square root
Dialogue: 0,0:00:05.79,0:00:11.42,Default,,0000,0000,0000,,of 6 times x squared plus the\Nprincipal square root of 2.
Dialogue: 0,0:00:11.42,0:00:13.43,Default,,0000,0000,0000,,And so we really just\Nhave two binomials,
Dialogue: 0,0:00:13.43,0:00:15.63,Default,,0000,0000,0000,,two two-term expressions\Nthat we want to multiply,
Dialogue: 0,0:00:15.63,0:00:17.29,Default,,0000,0000,0000,,and there's multiple\Nways to do this.
Dialogue: 0,0:00:17.29,0:00:19.10,Default,,0000,0000,0000,,I'll show you the\Nmore intuitive way,
Dialogue: 0,0:00:19.10,0:00:19.99,Default,,0000,0000,0000,,and then I'll show\Nyou the way it's
Dialogue: 0,0:00:19.99,0:00:21.42,Default,,0000,0000,0000,,taught in some\Nalgebra classes, which
Dialogue: 0,0:00:21.42,0:00:23.07,Default,,0000,0000,0000,,might be a little bit\Nfaster, but requires
Dialogue: 0,0:00:23.07,0:00:24.28,Default,,0000,0000,0000,,a little bit of memorization.
Dialogue: 0,0:00:24.28,0:00:26.32,Default,,0000,0000,0000,,So I'll show you the\Nintuitive way first.
Dialogue: 0,0:00:26.32,0:00:28.52,Default,,0000,0000,0000,,So if you have\Nanything-- so let's say
Dialogue: 0,0:00:28.52,0:00:31.40,Default,,0000,0000,0000,,I have a times x\Nplus y-- we know
Dialogue: 0,0:00:31.40,0:00:33.96,Default,,0000,0000,0000,,from the distributive\Nproperty that this
Dialogue: 0,0:00:33.96,0:00:39.23,Default,,0000,0000,0000,,is the same thing as ax plus ay.
Dialogue: 0,0:00:39.23,0:00:40.98,Default,,0000,0000,0000,,And so we can do the\Nsame thing over here.
Dialogue: 0,0:00:40.98,0:00:44.81,Default,,0000,0000,0000,,If you view a as x squared--\Nas this whole expression
Dialogue: 0,0:00:44.81,0:00:48.40,Default,,0000,0000,0000,,over here-- x squared minus\Nthe principal square root of 6,
Dialogue: 0,0:00:48.40,0:00:51.22,Default,,0000,0000,0000,,and you view x plus y\Nas this thing over here,
Dialogue: 0,0:00:51.22,0:00:52.01,Default,,0000,0000,0000,,you can distribute.
Dialogue: 0,0:00:52.01,0:00:55.65,Default,,0000,0000,0000,,
Dialogue: 0,0:00:55.65,0:00:58.58,Default,,0000,0000,0000,,We can distribute all\Nof this onto-- let
Dialogue: 0,0:00:58.58,0:01:02.88,Default,,0000,0000,0000,,me do it this way-- distribute\Nthis entire term onto this term
Dialogue: 0,0:01:02.88,0:01:04.58,Default,,0000,0000,0000,,and onto that term.
Dialogue: 0,0:01:04.58,0:01:05.67,Default,,0000,0000,0000,,So let's do that.
Dialogue: 0,0:01:05.67,0:01:09.47,Default,,0000,0000,0000,,So we get x squared minus\Nthe principal square root
Dialogue: 0,0:01:09.47,0:01:14.69,Default,,0000,0000,0000,,of 6 times this term-- I'll do\Nit in yellow-- times x squared.
Dialogue: 0,0:01:14.69,0:01:17.52,Default,,0000,0000,0000,,And then we have plus\Nthis thing again.
Dialogue: 0,0:01:17.52,0:01:18.64,Default,,0000,0000,0000,,We're just distributing it.
Dialogue: 0,0:01:18.64,0:01:19.59,Default,,0000,0000,0000,,It's just like they say.
Dialogue: 0,0:01:19.59,0:01:20.96,Default,,0000,0000,0000,,It's sometimes\Nnot that intuitive
Dialogue: 0,0:01:20.96,0:01:22.55,Default,,0000,0000,0000,,because this is\Na big expression,
Dialogue: 0,0:01:22.55,0:01:25.30,Default,,0000,0000,0000,,but you can treat it just like\Nyou would treat a variable over
Dialogue: 0,0:01:25.30,0:01:25.80,Default,,0000,0000,0000,,here.
Dialogue: 0,0:01:25.80,0:01:28.84,Default,,0000,0000,0000,,You're distributing it over\Nthis expression over here.
Dialogue: 0,0:01:28.84,0:01:33.73,Default,,0000,0000,0000,,And so then we have x squared\Nminus the principal square root
Dialogue: 0,0:01:33.73,0:01:37.22,Default,,0000,0000,0000,,of 6 times the principal\Nsquare root of 2.
Dialogue: 0,0:01:37.22,0:01:41.05,Default,,0000,0000,0000,,
Dialogue: 0,0:01:41.05,0:01:44.29,Default,,0000,0000,0000,,And now we can do the\Ndistributive property again,
Dialogue: 0,0:01:44.29,0:01:48.76,Default,,0000,0000,0000,,but what we'll do is we'll\Ndistribute this x squared
Dialogue: 0,0:01:48.76,0:01:51.81,Default,,0000,0000,0000,,onto each of these terms and\Ndistribute the square root of 2
Dialogue: 0,0:01:51.81,0:01:54.36,Default,,0000,0000,0000,,onto each of these terms.
Dialogue: 0,0:01:54.36,0:01:55.96,Default,,0000,0000,0000,,It's the exact\Nsame thing as here,
Dialogue: 0,0:01:55.96,0:01:58.23,Default,,0000,0000,0000,,it's just you could imagine\Nwriting it like this.
Dialogue: 0,0:01:58.23,0:02:04.77,Default,,0000,0000,0000,,x plus y times a is still\Ngoing to be ax plus ay.
Dialogue: 0,0:02:04.77,0:02:07.60,Default,,0000,0000,0000,,And just to see the pattern, how\Nthis is really the same thing
Dialogue: 0,0:02:07.60,0:02:09.14,Default,,0000,0000,0000,,as this up here,\Nwe're just switching
Dialogue: 0,0:02:09.14,0:02:10.62,Default,,0000,0000,0000,,the order of the multiplication.
Dialogue: 0,0:02:10.62,0:02:13.35,Default,,0000,0000,0000,,You can kind of view it as we're\Ndistributing from the right.
Dialogue: 0,0:02:13.35,0:02:16.59,Default,,0000,0000,0000,,And so if you do this, you\Nget x squared times x squared,
Dialogue: 0,0:02:16.59,0:02:21.06,Default,,0000,0000,0000,,which is x to the fourth,\Nthat's that times that, and then
Dialogue: 0,0:02:21.06,0:02:24.36,Default,,0000,0000,0000,,minus x squared times the\Nprincipal square root of 6.
Dialogue: 0,0:02:24.36,0:02:28.30,Default,,0000,0000,0000,,
Dialogue: 0,0:02:28.30,0:02:30.98,Default,,0000,0000,0000,,And then over here you\Nhave square root of 2 times
Dialogue: 0,0:02:30.98,0:02:36.57,Default,,0000,0000,0000,,x squared, so plus x squared\Ntimes the square root of 2.
Dialogue: 0,0:02:36.57,0:02:39.36,Default,,0000,0000,0000,,And then you have square root\Nof 2 times the square root of 6.
Dialogue: 0,0:02:39.36,0:02:41.44,Default,,0000,0000,0000,,And we have a negative\Nsign out here.
Dialogue: 0,0:02:41.44,0:02:43.11,Default,,0000,0000,0000,,Now if you take the\Nsquare root of 2--
Dialogue: 0,0:02:43.11,0:02:44.98,Default,,0000,0000,0000,,let me do this on the\Nside-- square root of 2
Dialogue: 0,0:02:44.98,0:02:48.68,Default,,0000,0000,0000,,times the square root of 6, we\Nknow from simplifying radicals
Dialogue: 0,0:02:48.68,0:02:51.78,Default,,0000,0000,0000,,that this is the exact same\Nthing as the square root of 2
Dialogue: 0,0:02:51.78,0:02:55.50,Default,,0000,0000,0000,,times 6, or the principal\Nsquare root of 12.
Dialogue: 0,0:02:55.50,0:02:57.46,Default,,0000,0000,0000,,So the square root of 2\Ntimes square root of 6,
Dialogue: 0,0:02:57.46,0:02:58.84,Default,,0000,0000,0000,,we have a negative\Nsign out here,
Dialogue: 0,0:02:58.84,0:03:02.29,Default,,0000,0000,0000,,it becomes minus the\Nsquare root of 12.
Dialogue: 0,0:03:02.29,0:03:05.17,Default,,0000,0000,0000,,And let's see if we can\Nsimplify this at all.
Dialogue: 0,0:03:05.17,0:03:05.67,Default,,0000,0000,0000,,Let's see.
Dialogue: 0,0:03:05.67,0:03:08.32,Default,,0000,0000,0000,,You have an x to\Nthe fourth term.
Dialogue: 0,0:03:08.32,0:03:10.73,Default,,0000,0000,0000,,And then here you\Nhave-- well depending
Dialogue: 0,0:03:10.73,0:03:12.73,Default,,0000,0000,0000,,on how you want to view\Nit, you could say, look,
Dialogue: 0,0:03:12.73,0:03:14.16,Default,,0000,0000,0000,,we have to second degree terms.
Dialogue: 0,0:03:14.16,0:03:15.86,Default,,0000,0000,0000,,We have something\Ntimes x squared,
Dialogue: 0,0:03:15.86,0:03:17.97,Default,,0000,0000,0000,,and we have something\Nelse times x squared.
Dialogue: 0,0:03:17.97,0:03:19.97,Default,,0000,0000,0000,,So if you want,\Nyou could simplify
Dialogue: 0,0:03:19.97,0:03:21.82,Default,,0000,0000,0000,,these two terms over here.
Dialogue: 0,0:03:21.82,0:03:25.44,Default,,0000,0000,0000,,So I have square\Nroot of 2 x squareds
Dialogue: 0,0:03:25.44,0:03:27.81,Default,,0000,0000,0000,,and then I'm going to subtract\Nfrom that square root of 6
Dialogue: 0,0:03:27.81,0:03:28.96,Default,,0000,0000,0000,,x squareds.
Dialogue: 0,0:03:28.96,0:03:32.49,Default,,0000,0000,0000,,So you could view this\Nas square root of 2
Dialogue: 0,0:03:32.49,0:03:35.51,Default,,0000,0000,0000,,minus the square root of 6, or\Nthe principal square root of 2
Dialogue: 0,0:03:35.51,0:03:40.26,Default,,0000,0000,0000,,minus the principal square\Nroot of 6, x squared.
Dialogue: 0,0:03:40.26,0:03:44.01,Default,,0000,0000,0000,,And then, if you want,\Nsquare root of 12,
Dialogue: 0,0:03:44.01,0:03:45.56,Default,,0000,0000,0000,,you might be able\Nto simplify that.
Dialogue: 0,0:03:45.56,0:03:48.50,Default,,0000,0000,0000,,12 is the same\Nthing as 3 times 4.
Dialogue: 0,0:03:48.50,0:03:52.05,Default,,0000,0000,0000,,So the square root of 12\Nis equal to square root
Dialogue: 0,0:03:52.05,0:03:54.71,Default,,0000,0000,0000,,of 3 times square root of 4.
Dialogue: 0,0:03:54.71,0:03:57.17,Default,,0000,0000,0000,,And the square root of 4, or\Nthe principal square root of 4
Dialogue: 0,0:03:57.17,0:03:58.76,Default,,0000,0000,0000,,I should say, is 2.
Dialogue: 0,0:03:58.76,0:04:00.51,Default,,0000,0000,0000,,So the square root of\N12 is the same thing
Dialogue: 0,0:04:00.51,0:04:02.62,Default,,0000,0000,0000,,as 2 square roots of 3.
Dialogue: 0,0:04:02.62,0:04:04.90,Default,,0000,0000,0000,,So instead of writing the\Nprincipal square root of 12,
Dialogue: 0,0:04:04.90,0:04:08.90,Default,,0000,0000,0000,,we could write minus 2 times\Nthe principal square root of 3.
Dialogue: 0,0:04:08.90,0:04:13.91,Default,,0000,0000,0000,,And then out here you have\Nan x to the fourth plus this.
Dialogue: 0,0:04:13.91,0:04:15.95,Default,,0000,0000,0000,,And you see, if you\Ndistributed this out,
Dialogue: 0,0:04:15.95,0:04:18.24,Default,,0000,0000,0000,,if you distribute this x\Nsquared, you get this term,
Dialogue: 0,0:04:18.24,0:04:19.100,Default,,0000,0000,0000,,negative x squared,\Nsquare root of 6,
Dialogue: 0,0:04:19.100,0:04:22.33,Default,,0000,0000,0000,,and if you distribute it onto\Nthis, you'd get that term.
Dialogue: 0,0:04:22.33,0:04:27.31,Default,,0000,0000,0000,,So you could debate which\Nof these two is more simple.
Dialogue: 0,0:04:27.31,0:04:29.08,Default,,0000,0000,0000,,Now I mentioned\Nthat this way I just
Dialogue: 0,0:04:29.08,0:04:30.58,Default,,0000,0000,0000,,did the distributive\Nproperty twice.
Dialogue: 0,0:04:30.58,0:04:31.82,Default,,0000,0000,0000,,Nothing new, nothing fancy.
Dialogue: 0,0:04:31.82,0:04:35.38,Default,,0000,0000,0000,,But in some classes, you will\Nsee something called FOIL.
Dialogue: 0,0:04:35.38,0:04:38.04,Default,,0000,0000,0000,,And I think we've done\Nthis in previous videos.
Dialogue: 0,0:04:38.04,0:04:39.54,Default,,0000,0000,0000,,FOIL.
Dialogue: 0,0:04:39.54,0:04:41.50,Default,,0000,0000,0000,,I'm not a big fan of\Nit because it's really
Dialogue: 0,0:04:41.50,0:04:44.00,Default,,0000,0000,0000,,a way to memorize a process\Nas opposed to understanding
Dialogue: 0,0:04:44.00,0:04:46.46,Default,,0000,0000,0000,,that this is really just from\Nthe common-sense distributive
Dialogue: 0,0:04:46.46,0:04:47.27,Default,,0000,0000,0000,,property.
Dialogue: 0,0:04:47.27,0:04:48.81,Default,,0000,0000,0000,,But all this is is\Na way to make sure
Dialogue: 0,0:04:48.81,0:04:50.48,Default,,0000,0000,0000,,that you're multiplying\Neverything times
Dialogue: 0,0:04:50.48,0:04:52.57,Default,,0000,0000,0000,,everything when\Nyou're multiplying
Dialogue: 0,0:04:52.57,0:04:55.29,Default,,0000,0000,0000,,two binomials times\Neach other like this.
Dialogue: 0,0:04:55.29,0:05:00.37,Default,,0000,0000,0000,,And FOIL just says, look,\Nfirst multiply the first term.
Dialogue: 0,0:05:00.37,0:05:04.16,Default,,0000,0000,0000,,So x squared times x\Nsquared is x to the fourth.
Dialogue: 0,0:05:04.16,0:05:06.67,Default,,0000,0000,0000,,Then multiply the outside.
Dialogue: 0,0:05:06.67,0:05:09.10,Default,,0000,0000,0000,,So then multiply--\NI'll do this in green--
Dialogue: 0,0:05:09.10,0:05:10.36,Default,,0000,0000,0000,,then multiply the outside.
Dialogue: 0,0:05:10.36,0:05:14.30,Default,,0000,0000,0000,,So the outside terms are x\Nsquared and square root of 2.
Dialogue: 0,0:05:14.30,0:05:16.01,Default,,0000,0000,0000,,And so x squared times\Nsquare root of 2--
Dialogue: 0,0:05:16.01,0:05:20.21,Default,,0000,0000,0000,,and they are positive-- so\Nplus square root of 2 times x
Dialogue: 0,0:05:20.21,0:05:21.04,Default,,0000,0000,0000,,squared.
Dialogue: 0,0:05:21.04,0:05:23.70,Default,,0000,0000,0000,,And then multiply the inside.
Dialogue: 0,0:05:23.70,0:05:25.16,Default,,0000,0000,0000,,And you can see\Nwhy I don't like it
Dialogue: 0,0:05:25.16,0:05:26.73,Default,,0000,0000,0000,,that much is because you\Nreally don't know you're doing.
Dialogue: 0,0:05:26.73,0:05:28.71,Default,,0000,0000,0000,,You're just applying\Nan algorithm.
Dialogue: 0,0:05:28.71,0:05:30.81,Default,,0000,0000,0000,,Then you'll multiply the inside.
Dialogue: 0,0:05:30.81,0:05:32.85,Default,,0000,0000,0000,,And so negative square\Nroot of 6 times x squared.
Dialogue: 0,0:05:32.85,0:05:36.02,Default,,0000,0000,0000,,
Dialogue: 0,0:05:36.02,0:05:40.30,Default,,0000,0000,0000,,And then you multiply\Nthe last terms.
Dialogue: 0,0:05:40.30,0:05:42.47,Default,,0000,0000,0000,,So negative square root of\N6 times square root of 2,
Dialogue: 0,0:05:42.47,0:05:44.49,Default,,0000,0000,0000,,that is-- and we\Nalready know that-- that
Dialogue: 0,0:05:44.49,0:05:49.02,Default,,0000,0000,0000,,is negative square root of\N12, which you can also then
Dialogue: 0,0:05:49.02,0:05:51.92,Default,,0000,0000,0000,,simplify to that expression\Nright over there.
Dialogue: 0,0:05:51.92,0:05:55.91,Default,,0000,0000,0000,,So it's fine to use\Nthis, although it's good,
Dialogue: 0,0:05:55.91,0:05:58.93,Default,,0000,0000,0000,,even if you do use this, to\Nknow where FOIL comes from.
Dialogue: 0,0:05:58.93,0:06:01.64,Default,,0000,0000,0000,,It really just comes from\Nusing the distributive property
Dialogue: 0,0:06:01.64,0:06:03.19,Default,,0000,0000,0000,,twice.