0:00:00.760,0:00:03.400
In this expression, we're[br]dividing this third degree

0:00:03.400,0:00:06.300
polynomial by this[br]first degree polynomial.

0:00:06.300,0:00:08.360
And we could simplify[br]this by using

0:00:08.360,0:00:10.229
traditional algebraic[br]long division.

0:00:10.229,0:00:12.020
But what we're going[br]to cover in this video

0:00:12.020,0:00:13.436
is a slightly[br]different technique,

0:00:13.436,0:00:15.990
and we call it[br]synthetic division.

0:00:15.990,0:00:17.640
And synthetic division[br]is going to seem

0:00:17.640,0:00:20.470
like a little bit of voodoo[br]in the context of this video.

0:00:20.470,0:00:21.886
In the next few[br]videos we're going

0:00:21.886,0:00:24.450
to think about why it actually[br]makes sense, why you actually

0:00:24.450,0:00:28.690
get the same result as[br]traditional algebraic long

0:00:28.690,0:00:29.540
division.

0:00:29.540,0:00:32.830
My personal tastes are not[br]to like synthetic division

0:00:32.830,0:00:35.240
because it is very,[br]very, very algorithmic.

0:00:35.240,0:00:38.150
I prefer to do traditional[br]algebraic long division.

0:00:38.150,0:00:40.670
But I think you'll see that[br]this has some advantages.

0:00:40.670,0:00:41.950
It can be faster.

0:00:41.950,0:00:44.900
And it also uses a lot[br]less space on your paper.

0:00:44.900,0:00:47.380
So let's actually perform[br]this synthetic division.

0:00:47.380,0:00:49.750
Let's actually simplify[br]this expression.

0:00:49.750,0:00:52.640
Before we start, there's[br]two important things

0:00:52.640,0:00:53.450
to keep in mind.

0:00:53.450,0:00:55.400
We're doing, kind of,[br]the most basic form

0:00:55.400,0:00:56.730
of synthetic division.

0:00:56.730,0:00:59.940
And to do this most basic[br]algorithm, this most basic

0:00:59.940,0:01:01.650
process, you have[br]to look for two

0:01:01.650,0:01:04.370
things in this[br]bottom expression.

0:01:04.370,0:01:09.970
The first is that it has to[br]be a polynomial of degree 1.

0:01:09.970,0:01:11.262
So you have just an x here.

0:01:11.262,0:01:13.220
You don't have an x[br]squared, an x to the third,

0:01:13.220,0:01:15.220
an x to the fourth or[br]anything like that.

0:01:15.220,0:01:19.470
The other thing is, is that[br]the coefficient here is a 1.

0:01:19.470,0:01:21.910
There are ways to do it if[br]the coefficient was different,

0:01:21.910,0:01:23.493
but then our synthetic[br]division, we'll

0:01:23.493,0:01:26.329
have to add a little bit of[br]bells and whistles to it.

0:01:26.329,0:01:27.870
So in general, what[br]I'm going to show

0:01:27.870,0:01:30.170
you now will work if[br]you have something

0:01:30.170,0:01:33.580
of the form x plus or[br]minus something else.

0:01:33.580,0:01:35.230
So with that said,[br]let's actually

0:01:35.230,0:01:38.114
perform the synthetic division.

0:01:38.114,0:01:39.530
So the first thing[br]I'm going to do

0:01:39.530,0:01:42.260
is write all the coefficients[br]for this polynomial

0:01:42.260,0:01:43.800
that's in the numerator.

0:01:43.800,0:01:45.230
So let's write all of them.

0:01:45.230,0:01:47.210
So we have a 3.

0:01:47.210,0:01:50.780
We have a 4, that's[br]a positive 4.

0:01:50.780,0:01:54.460
We have a negative 2.

0:01:54.460,0:01:55.900
And a negative 1.

0:01:59.670,0:02:02.380
And you'll see different people[br]draw different types of signs

0:02:02.380,0:02:04.220
here depending on how they're[br]doing synthetic division.

0:02:04.220,0:02:05.800
But this is the[br]most traditional.

0:02:05.800,0:02:07.350
And you want to leave[br]some space right here

0:02:07.350,0:02:08.473
for another row of numbers.

0:02:08.473,0:02:11.000
So that's why I've gone[br]all the way down here.

0:02:11.000,0:02:13.130
And then we look[br]at the denominator.

0:02:13.130,0:02:15.200
And in particular, we're[br]going to look whatever

0:02:15.200,0:02:17.340
x plus or minus is,[br]right over here.

0:02:17.340,0:02:20.570
So we'll look at, right over[br]here, we have a positive 4.

0:02:20.570,0:02:24.540
Instead of writing a positive 4,[br]we write the negative of that.

0:02:24.540,0:02:30.090
So we write the negative,[br]which would be negative 4.

0:02:33.470,0:02:35.250
And now we are[br]all set up, and we

0:02:35.250,0:02:38.660
are ready to perform[br]our synthetic division.

0:02:38.660,0:02:40.150
And it's going to[br]seem like voodoo.

0:02:40.150,0:02:43.350
In future videos, we'll[br]explain why this works.

0:02:43.350,0:02:45.700
So first, this first[br]coefficient, we literally just

0:02:45.700,0:02:47.130
bring it straight down.

0:02:47.130,0:02:48.990
And so you put the 3 there.

0:02:48.990,0:02:53.200
Then you multiply what you[br]have here times the negative 4.

0:02:53.200,0:02:55.820
So you multiply it[br]times the negative 4.

0:02:55.820,0:02:59.840
3 times negative[br]4 is negative 12.

0:02:59.840,0:03:02.820
Then you add the 4[br]to the negative 12.

0:03:02.820,0:03:06.960
4 plus negative[br]12 is negative 8.

0:03:06.960,0:03:10.500
Then you multiply negative[br]8 times the negative 4.

0:03:10.500,0:03:12.480
I think you see the pattern.

0:03:12.480,0:03:17.610
Negative 8 times negative[br]4 is positive 32.

0:03:17.610,0:03:21.180
Now we add negative[br]2 plus positive 32.

0:03:21.180,0:03:24.400
That gives us positive 30.

0:03:24.400,0:03:28.600
Then you multiply the positive[br]30 times the negative 4.

0:03:28.600,0:03:34.200
And that gives you negative 120.

0:03:34.200,0:03:38.220
And then you add the negative[br]1 plus the negative 120.

0:03:38.220,0:03:43.272
And you end up with[br]a negative 121.

0:03:43.272,0:03:44.980
Now the last thing[br]you do is say, well, I

0:03:44.980,0:03:45.970
have one term here.

0:03:45.970,0:03:47.924
And in this plain,[br]vanilla, simple version

0:03:47.924,0:03:50.090
of synthetic division, we're[br]only dealing, actually,

0:03:50.090,0:03:51.820
when you have x plus[br]or minus something.

0:03:51.820,0:03:53.760
So you're only going[br]to have one term there.

0:03:53.760,0:03:57.760
So you separate out one term[br]from the right, just like that.

0:03:57.760,0:03:59.770
And we essentially[br]have our answer,

0:03:59.770,0:04:02.250
even though it[br]seems like voodoo.

0:04:02.250,0:04:07.230
So to simplify this, you get,[br]and you could have a drum roll

0:04:07.230,0:04:11.040
right over here,[br]this right over here,

0:04:11.040,0:04:13.750
it's going to be[br]a constant term.

0:04:13.750,0:04:15.460
You could think of it[br]as a degree 0 term.

0:04:15.460,0:04:16.769
This is going to be an x term.

0:04:16.769,0:04:18.928
And this is going to[br]be an x squared term.

0:04:18.928,0:04:20.594
You can kind of just[br]build up from here,

0:04:20.594,0:04:22.520
saying this first one is[br]going to be a constant.

0:04:22.520,0:04:24.810
Then this is going to be an[br]x term, then an x squared.

0:04:24.810,0:04:26.851
If we had more you'd have[br]an x to the third, an x

0:04:26.851,0:04:28.740
to the fourth, so[br]on and so forth.

0:04:28.740,0:04:41.750
So this is going to be equal[br]to 3x squared minus 8x plus 30.

0:04:44.420,0:04:46.060
And this right over[br]here you can view

0:04:46.060,0:04:53.910
as the remainder, so minus[br]121 over the x plus 4.

0:04:53.910,0:04:55.740
This didn't divide perfectly.

0:04:55.740,0:05:00.397
So over the x plus 4.

0:05:00.397,0:05:02.730
Another way you could have[br]done it, you could have said,

0:05:02.730,0:05:03.760
this is the remainder.

0:05:03.760,0:05:07.850
So I'm going to have a[br]negative 121 over x plus 4.

0:05:07.850,0:05:13.222
And this is going to be plus[br]30 minus 8x plus 3x squared.

0:05:13.222,0:05:14.680
So hopefully that[br]makes some sense.

0:05:14.680,0:05:16.510
I'll do another example[br]in the next video.

0:05:16.510,0:05:19.795
And then we'll think about[br]why this actually works.