[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.76,0:00:03.40,Default,,0000,0000,0000,,In this expression, we're\Ndividing this third degree
Dialogue: 0,0:00:03.40,0:00:06.30,Default,,0000,0000,0000,,polynomial by this\Nfirst degree polynomial.
Dialogue: 0,0:00:06.30,0:00:08.36,Default,,0000,0000,0000,,And we could simplify\Nthis by using
Dialogue: 0,0:00:08.36,0:00:10.23,Default,,0000,0000,0000,,traditional algebraic\Nlong division.
Dialogue: 0,0:00:10.23,0:00:12.02,Default,,0000,0000,0000,,But what we're going\Nto cover in this video
Dialogue: 0,0:00:12.02,0:00:13.44,Default,,0000,0000,0000,,is a slightly\Ndifferent technique,
Dialogue: 0,0:00:13.44,0:00:15.99,Default,,0000,0000,0000,,and we call it\Nsynthetic division.
Dialogue: 0,0:00:15.99,0:00:17.64,Default,,0000,0000,0000,,And synthetic division\Nis going to seem
Dialogue: 0,0:00:17.64,0:00:20.47,Default,,0000,0000,0000,,like a little bit of voodoo\Nin the context of this video.
Dialogue: 0,0:00:20.47,0:00:21.89,Default,,0000,0000,0000,,In the next few\Nvideos we're going
Dialogue: 0,0:00:21.89,0:00:24.45,Default,,0000,0000,0000,,to think about why it actually\Nmakes sense, why you actually
Dialogue: 0,0:00:24.45,0:00:28.69,Default,,0000,0000,0000,,get the same result as\Ntraditional algebraic long
Dialogue: 0,0:00:28.69,0:00:29.54,Default,,0000,0000,0000,,division.
Dialogue: 0,0:00:29.54,0:00:32.83,Default,,0000,0000,0000,,My personal tastes are not\Nto like synthetic division
Dialogue: 0,0:00:32.83,0:00:35.24,Default,,0000,0000,0000,,because it is very,\Nvery, very algorithmic.
Dialogue: 0,0:00:35.24,0:00:38.15,Default,,0000,0000,0000,,I prefer to do traditional\Nalgebraic long division.
Dialogue: 0,0:00:38.15,0:00:40.67,Default,,0000,0000,0000,,But I think you'll see that\Nthis has some advantages.
Dialogue: 0,0:00:40.67,0:00:41.95,Default,,0000,0000,0000,,It can be faster.
Dialogue: 0,0:00:41.95,0:00:44.90,Default,,0000,0000,0000,,And it also uses a lot\Nless space on your paper.
Dialogue: 0,0:00:44.90,0:00:47.38,Default,,0000,0000,0000,,So let's actually perform\Nthis synthetic division.
Dialogue: 0,0:00:47.38,0:00:49.75,Default,,0000,0000,0000,,Let's actually simplify\Nthis expression.
Dialogue: 0,0:00:49.75,0:00:52.64,Default,,0000,0000,0000,,Before we start, there's\Ntwo important things
Dialogue: 0,0:00:52.64,0:00:53.45,Default,,0000,0000,0000,,to keep in mind.
Dialogue: 0,0:00:53.45,0:00:55.40,Default,,0000,0000,0000,,We're doing, kind of,\Nthe most basic form
Dialogue: 0,0:00:55.40,0:00:56.73,Default,,0000,0000,0000,,of synthetic division.
Dialogue: 0,0:00:56.73,0:00:59.94,Default,,0000,0000,0000,,And to do this most basic\Nalgorithm, this most basic
Dialogue: 0,0:00:59.94,0:01:01.65,Default,,0000,0000,0000,,process, you have\Nto look for two
Dialogue: 0,0:01:01.65,0:01:04.37,Default,,0000,0000,0000,,things in this\Nbottom expression.
Dialogue: 0,0:01:04.37,0:01:09.97,Default,,0000,0000,0000,,The first is that it has to\Nbe a polynomial of degree 1.
Dialogue: 0,0:01:09.97,0:01:11.26,Default,,0000,0000,0000,,So you have just an x here.
Dialogue: 0,0:01:11.26,0:01:13.22,Default,,0000,0000,0000,,You don't have an x\Nsquared, an x to the third,
Dialogue: 0,0:01:13.22,0:01:15.22,Default,,0000,0000,0000,,an x to the fourth or\Nanything like that.
Dialogue: 0,0:01:15.22,0:01:19.47,Default,,0000,0000,0000,,The other thing is, is that\Nthe coefficient here is a 1.
Dialogue: 0,0:01:19.47,0:01:21.91,Default,,0000,0000,0000,,There are ways to do it if\Nthe coefficient was different,
Dialogue: 0,0:01:21.91,0:01:23.49,Default,,0000,0000,0000,,but then our synthetic\Ndivision, we'll
Dialogue: 0,0:01:23.49,0:01:26.33,Default,,0000,0000,0000,,have to add a little bit of\Nbells and whistles to it.
Dialogue: 0,0:01:26.33,0:01:27.87,Default,,0000,0000,0000,,So in general, what\NI'm going to show
Dialogue: 0,0:01:27.87,0:01:30.17,Default,,0000,0000,0000,,you now will work if\Nyou have something
Dialogue: 0,0:01:30.17,0:01:33.58,Default,,0000,0000,0000,,of the form x plus or\Nminus something else.
Dialogue: 0,0:01:33.58,0:01:35.23,Default,,0000,0000,0000,,So with that said,\Nlet's actually
Dialogue: 0,0:01:35.23,0:01:38.11,Default,,0000,0000,0000,,perform the synthetic division.
Dialogue: 0,0:01:38.11,0:01:39.53,Default,,0000,0000,0000,,So the first thing\NI'm going to do
Dialogue: 0,0:01:39.53,0:01:42.26,Default,,0000,0000,0000,,is write all the coefficients\Nfor this polynomial
Dialogue: 0,0:01:42.26,0:01:43.80,Default,,0000,0000,0000,,that's in the numerator.
Dialogue: 0,0:01:43.80,0:01:45.23,Default,,0000,0000,0000,,So let's write all of them.
Dialogue: 0,0:01:45.23,0:01:47.21,Default,,0000,0000,0000,,So we have a 3.
Dialogue: 0,0:01:47.21,0:01:50.78,Default,,0000,0000,0000,,We have a 4, that's\Na positive 4.
Dialogue: 0,0:01:50.78,0:01:54.46,Default,,0000,0000,0000,,We have a negative 2.
Dialogue: 0,0:01:54.46,0:01:55.90,Default,,0000,0000,0000,,And a negative 1.
Dialogue: 0,0:01:59.67,0:02:02.38,Default,,0000,0000,0000,,And you'll see different people\Ndraw different types of signs
Dialogue: 0,0:02:02.38,0:02:04.22,Default,,0000,0000,0000,,here depending on how they're\Ndoing synthetic division.
Dialogue: 0,0:02:04.22,0:02:05.80,Default,,0000,0000,0000,,But this is the\Nmost traditional.
Dialogue: 0,0:02:05.80,0:02:07.35,Default,,0000,0000,0000,,And you want to leave\Nsome space right here
Dialogue: 0,0:02:07.35,0:02:08.47,Default,,0000,0000,0000,,for another row of numbers.
Dialogue: 0,0:02:08.47,0:02:11.00,Default,,0000,0000,0000,,So that's why I've gone\Nall the way down here.
Dialogue: 0,0:02:11.00,0:02:13.13,Default,,0000,0000,0000,,And then we look\Nat the denominator.
Dialogue: 0,0:02:13.13,0:02:15.20,Default,,0000,0000,0000,,And in particular, we're\Ngoing to look whatever
Dialogue: 0,0:02:15.20,0:02:17.34,Default,,0000,0000,0000,,x plus or minus is,\Nright over here.
Dialogue: 0,0:02:17.34,0:02:20.57,Default,,0000,0000,0000,,So we'll look at, right over\Nhere, we have a positive 4.
Dialogue: 0,0:02:20.57,0:02:24.54,Default,,0000,0000,0000,,Instead of writing a positive 4,\Nwe write the negative of that.
Dialogue: 0,0:02:24.54,0:02:30.09,Default,,0000,0000,0000,,So we write the negative,\Nwhich would be negative 4.
Dialogue: 0,0:02:33.47,0:02:35.25,Default,,0000,0000,0000,,And now we are\Nall set up, and we
Dialogue: 0,0:02:35.25,0:02:38.66,Default,,0000,0000,0000,,are ready to perform\Nour synthetic division.
Dialogue: 0,0:02:38.66,0:02:40.15,Default,,0000,0000,0000,,And it's going to\Nseem like voodoo.
Dialogue: 0,0:02:40.15,0:02:43.35,Default,,0000,0000,0000,,In future videos, we'll\Nexplain why this works.
Dialogue: 0,0:02:43.35,0:02:45.70,Default,,0000,0000,0000,,So first, this first\Ncoefficient, we literally just
Dialogue: 0,0:02:45.70,0:02:47.13,Default,,0000,0000,0000,,bring it straight down.
Dialogue: 0,0:02:47.13,0:02:48.99,Default,,0000,0000,0000,,And so you put the 3 there.
Dialogue: 0,0:02:48.99,0:02:53.20,Default,,0000,0000,0000,,Then you multiply what you\Nhave here times the negative 4.
Dialogue: 0,0:02:53.20,0:02:55.82,Default,,0000,0000,0000,,So you multiply it\Ntimes the negative 4.
Dialogue: 0,0:02:55.82,0:02:59.84,Default,,0000,0000,0000,,3 times negative\N4 is negative 12.
Dialogue: 0,0:02:59.84,0:03:02.82,Default,,0000,0000,0000,,Then you add the 4\Nto the negative 12.
Dialogue: 0,0:03:02.82,0:03:06.96,Default,,0000,0000,0000,,4 plus negative\N12 is negative 8.
Dialogue: 0,0:03:06.96,0:03:10.50,Default,,0000,0000,0000,,Then you multiply negative\N8 times the negative 4.
Dialogue: 0,0:03:10.50,0:03:12.48,Default,,0000,0000,0000,,I think you see the pattern.
Dialogue: 0,0:03:12.48,0:03:17.61,Default,,0000,0000,0000,,Negative 8 times negative\N4 is positive 32.
Dialogue: 0,0:03:17.61,0:03:21.18,Default,,0000,0000,0000,,Now we add negative\N2 plus positive 32.
Dialogue: 0,0:03:21.18,0:03:24.40,Default,,0000,0000,0000,,That gives us positive 30.
Dialogue: 0,0:03:24.40,0:03:28.60,Default,,0000,0000,0000,,Then you multiply the positive\N30 times the negative 4.
Dialogue: 0,0:03:28.60,0:03:34.20,Default,,0000,0000,0000,,And that gives you negative 120.
Dialogue: 0,0:03:34.20,0:03:38.22,Default,,0000,0000,0000,,And then you add the negative\N1 plus the negative 120.
Dialogue: 0,0:03:38.22,0:03:43.27,Default,,0000,0000,0000,,And you end up with\Na negative 121.
Dialogue: 0,0:03:43.27,0:03:44.98,Default,,0000,0000,0000,,Now the last thing\Nyou do is say, well, I
Dialogue: 0,0:03:44.98,0:03:45.97,Default,,0000,0000,0000,,have one term here.
Dialogue: 0,0:03:45.97,0:03:47.92,Default,,0000,0000,0000,,And in this plain,\Nvanilla, simple version
Dialogue: 0,0:03:47.92,0:03:50.09,Default,,0000,0000,0000,,of synthetic division, we're\Nonly dealing, actually,
Dialogue: 0,0:03:50.09,0:03:51.82,Default,,0000,0000,0000,,when you have x plus\Nor minus something.
Dialogue: 0,0:03:51.82,0:03:53.76,Default,,0000,0000,0000,,So you're only going\Nto have one term there.
Dialogue: 0,0:03:53.76,0:03:57.76,Default,,0000,0000,0000,,So you separate out one term\Nfrom the right, just like that.
Dialogue: 0,0:03:57.76,0:03:59.77,Default,,0000,0000,0000,,And we essentially\Nhave our answer,
Dialogue: 0,0:03:59.77,0:04:02.25,Default,,0000,0000,0000,,even though it\Nseems like voodoo.
Dialogue: 0,0:04:02.25,0:04:07.23,Default,,0000,0000,0000,,So to simplify this, you get,\Nand you could have a drum roll
Dialogue: 0,0:04:07.23,0:04:11.04,Default,,0000,0000,0000,,right over here,\Nthis right over here,
Dialogue: 0,0:04:11.04,0:04:13.75,Default,,0000,0000,0000,,it's going to be\Na constant term.
Dialogue: 0,0:04:13.75,0:04:15.46,Default,,0000,0000,0000,,You could think of it\Nas a degree 0 term.
Dialogue: 0,0:04:15.46,0:04:16.77,Default,,0000,0000,0000,,This is going to be an x term.
Dialogue: 0,0:04:16.77,0:04:18.93,Default,,0000,0000,0000,,And this is going to\Nbe an x squared term.
Dialogue: 0,0:04:18.93,0:04:20.59,Default,,0000,0000,0000,,You can kind of just\Nbuild up from here,
Dialogue: 0,0:04:20.59,0:04:22.52,Default,,0000,0000,0000,,saying this first one is\Ngoing to be a constant.
Dialogue: 0,0:04:22.52,0:04:24.81,Default,,0000,0000,0000,,Then this is going to be an\Nx term, then an x squared.
Dialogue: 0,0:04:24.81,0:04:26.85,Default,,0000,0000,0000,,If we had more you'd have\Nan x to the third, an x
Dialogue: 0,0:04:26.85,0:04:28.74,Default,,0000,0000,0000,,to the fourth, so\Non and so forth.
Dialogue: 0,0:04:28.74,0:04:41.75,Default,,0000,0000,0000,,So this is going to be equal\Nto 3x squared minus 8x plus 30.
Dialogue: 0,0:04:44.42,0:04:46.06,Default,,0000,0000,0000,,And this right over\Nhere you can view
Dialogue: 0,0:04:46.06,0:04:53.91,Default,,0000,0000,0000,,as the remainder, so minus\N121 over the x plus 4.
Dialogue: 0,0:04:53.91,0:04:55.74,Default,,0000,0000,0000,,This didn't divide perfectly.
Dialogue: 0,0:04:55.74,0:05:00.40,Default,,0000,0000,0000,,So over the x plus 4.
Dialogue: 0,0:05:00.40,0:05:02.73,Default,,0000,0000,0000,,Another way you could have\Ndone it, you could have said,
Dialogue: 0,0:05:02.73,0:05:03.76,Default,,0000,0000,0000,,this is the remainder.
Dialogue: 0,0:05:03.76,0:05:07.85,Default,,0000,0000,0000,,So I'm going to have a\Nnegative 121 over x plus 4.
Dialogue: 0,0:05:07.85,0:05:13.22,Default,,0000,0000,0000,,And this is going to be plus\N30 minus 8x plus 3x squared.
Dialogue: 0,0:05:13.22,0:05:14.68,Default,,0000,0000,0000,,So hopefully that\Nmakes some sense.
Dialogue: 0,0:05:14.68,0:05:16.51,Default,,0000,0000,0000,,I'll do another example\Nin the next video.
Dialogue: 0,0:05:16.51,0:05:19.80,Default,,0000,0000,0000,,And then we'll think about\Nwhy this actually works.