Factoring polynomials: common factor area model

WEBVTT
Kind: captions
Language: en

00:00:00.518 --> 00:00:02.084
- [Voiceover] The
rectangle below has an area

00:00:02.084 --> 00:00:05.218
of 12x to the fourth plus 6x to the third

00:00:05.218 --> 00:00:08.951
plus 15x squared square meters.

00:00:08.951 --> 00:00:10.319
And we can see the area right over here

00:00:10.319 --> 00:00:13.084
and they broke it up; this
green area is 12x to the fourth,

00:00:13.084 --> 00:00:15.185
this purple area is 6x to the third,

00:00:15.185 --> 00:00:16.818
this blue area is 15x squared.

00:00:16.818 --> 00:00:19.651
You add 'em all together,
you get this entire rectangle

00:00:19.651 --> 00:00:21.818
which would be the combined
area is 12x to the fourth

00:00:21.818 --> 00:00:24.718
plus 6x to the third plus 15x squared.

00:00:24.718 --> 00:00:26.851
The length of the rectangle in meters,

00:00:26.851 --> 00:00:28.384
so this is the length right over here

00:00:28.384 --> 00:00:29.617
that we're talking about, so we're talking

00:00:29.617 --> 00:00:31.950
about this distance.

00:00:31.950 --> 00:00:34.984
The length of the rectangle
in meters is equal to

00:00:34.984 --> 00:00:39.818
the greatest common monomial
factor of 12x to the fourth,

00:00:39.818 --> 00:00:41.784
6x to the third,

00:00:41.784 --> 00:00:45.218
and 15x squared.

00:00:45.218 --> 00:00:48.351
What is the length and
width of the rectangle?

00:00:48.351 --> 00:00:50.398
I encourage you to pause the video and try

00:00:50.398 --> 00:00:52.184
to work through it on your own.

00:00:52.184 --> 00:00:54.444
Well the key realization
here is that the length

00:00:54.444 --> 00:00:57.551
times the width, the
length times the width

00:00:57.551 --> 00:01:00.652
is going to be equal to this area.

00:01:00.652 --> 00:01:03.884
And if the length is the
greatest common monomial factor

00:01:03.884 --> 00:01:06.151
of these terms, of 12x to the fourth,

00:01:06.151 --> 00:01:08.418
6x third, and 15x squared,

00:01:08.418 --> 00:01:10.684
well then we can factor that
out and then what we have

00:01:10.684 --> 00:01:14.251
left over is going to be the width.

00:01:14.251 --> 00:01:16.118
So let's figure out what
is the greatest common

00:01:16.118 --> 00:01:17.194
monomial factor

00:01:17.194 --> 00:01:21.504
of these three terms.

00:01:21.504 --> 00:01:23.218
And the first thing we
could look at is let's look

00:01:23.218 --> 00:01:24.584
at the coefficients.

00:01:24.584 --> 00:01:26.384
Let's figure out what's
the greatest common factor

00:01:26.384 --> 00:01:30.585
of 12, six, and 15.

00:01:30.585 --> 00:01:32.218
And there is a couple
of ways you could do it.

00:01:32.218 --> 00:01:35.218
You could do it by looking
at a prime factorization,

00:01:35.218 --> 00:01:36.984
you could say alright,

00:01:36.984 --> 00:01:41.017
well 12 is two times six

00:01:41.017 --> 00:01:43.818
which is two times three,

00:01:43.818 --> 00:01:45.785
that's the prime factorization of 12.

00:01:45.785 --> 00:01:49.451
Prime factorization of six
is just two times three.

00:01:49.451 --> 00:01:51.941
Prime factorization of 15

00:01:51.941 --> 00:01:56.451
is three times five.

00:01:56.451 --> 00:01:58.751
And so the greatest common factor,

00:01:58.751 --> 00:02:03.118
the largest factor that's
divisible into all of them,

00:02:03.118 --> 00:02:07.684
so let's see, we can
throw a three in there.

00:02:07.684 --> 00:02:11.018
Three is divisible into all of them.

00:02:11.018 --> 00:02:13.684
And that's it 'cause we
can't say a three and a two.

00:02:13.684 --> 00:02:16.751
A three and a two would be
divisible into 12 and six,

00:02:16.751 --> 00:02:18.952
but there's no two
that's divisible into 15.

00:02:18.952 --> 00:02:21.151
We can't say a three and a five
'cause five isn't divisible

00:02:21.151 --> 00:02:24.884
into 12 or six; so the
greatest common factor

00:02:24.884 --> 00:02:26.218
is going to be three.

00:02:26.218 --> 00:02:27.818
Another way we could have done
this is we could have said

00:02:27.818 --> 00:02:31.751
where are the non-prime factors
of each of these numbers.

00:02:31.751 --> 00:02:34.084
12 you could have said, OK I can get 12

00:02:34.084 --> 00:02:37.551
by saying one times 12

00:02:37.551 --> 00:02:41.018
or two times six

00:02:41.018 --> 00:02:43.985
or three times four.

00:02:43.985 --> 00:02:45.985
Six you could have said,

00:02:45.985 --> 00:02:49.884
let's see, that could be one
times six or two times three.

00:02:49.884 --> 00:02:51.651
So those are the factors of six.

00:02:51.651 --> 00:02:55.518
And then 15 you could have
said well one times 15

00:02:55.518 --> 00:02:58.517
or three times five.

00:02:58.517 --> 00:03:01.218
And so you say the greatest common factor?

00:03:01.218 --> 00:03:04.985
Well three is the largest
number that I've listed here

00:03:04.985 --> 00:03:07.784
that is common to all
three of these factors.

00:03:07.784 --> 00:03:10.284
So once again, the greatest common factor

00:03:10.284 --> 00:03:12.851
of 12, six, and 15 is three.

00:03:12.851 --> 00:03:15.051
So when we're looking at the
greatest common monomial factor

00:03:15.051 --> 00:03:18.184
the coefficient is going to be three.

00:03:18.184 --> 00:03:20.218
And then we look at these powers of X.

00:03:20.218 --> 00:03:22.571
We have X to the fourth.

00:03:23.571 --> 00:03:25.318
I'm using a different color.

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We have X to the fourth,

00:03:27.858 --> 00:03:30.478
X to the third, and X squared.

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Well what's the largest power of X

00:03:31.850 --> 00:03:33.484
that's divisible into all of those?

00:03:33.484 --> 00:03:34.585
Well it's going to be X squared.

00:03:34.585 --> 00:03:36.117
X squared is divisible
into X to the fourth

00:03:36.117 --> 00:03:38.950
and X to the third and of
course X squared itself.

00:03:38.950 --> 00:03:42.351
So the greatest common
monomial factor is 3x squared.

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This length right over here,

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this is 3x squared.

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So if this is 3x squared,
we can then figure out

00:03:49.851 --> 00:03:52.368
what the width is.

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If we were to divide 12x to
the fourth by 3x squared,

00:03:57.218 --> 00:03:58.018
what do we get?

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Well 12 divided by three is four

00:04:01.452 --> 00:04:03.619
and X to the fourth divided by X squared

00:04:03.619 --> 00:04:05.485
is X squared.

00:04:05.485 --> 00:04:08.051
Notice 3x squared times 4x squared

00:04:08.051 --> 00:04:10.218
is 12x to the fourth.

00:04:10.218 --> 00:04:12.217
And then we move over
to this purple section.

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If we take 6x to the third
divided by 3x squared,

00:04:15.385 --> 00:04:18.884
six divided by three is two.

00:04:18.884 --> 00:04:21.151
And then X to the third
divided by X squared

00:04:21.151 --> 00:04:23.617
is just going to be X.

00:04:23.617 --> 00:04:27.718
And then last but not least, we have

00:04:27.718 --> 00:04:30.985
15 divided by three is going to be five.

00:04:30.985 --> 00:04:32.652
X squared divided by
X squared is just one,

00:04:32.652 --> 00:04:33.950
so it's just gonna be five.

00:04:33.950 --> 00:04:36.418
So the width is going to be 4x squared,

00:04:36.418 --> 00:04:39.251
plus 2x, plus five.

00:04:39.251 --> 00:04:42.218
So once again, the length,
we figure that out,

00:04:42.218 --> 00:04:45.217
is the greatest common
monomial factor of these terms.

00:04:45.217 --> 00:04:47.684
It's 3x squared.

00:04:47.684 --> 00:04:50.750
And the width is 4x squared

00:04:50.750 --> 00:04:53.584
plus 2x, plus five.

00:04:53.584 --> 00:04:56.512
And one way to think about
it is we just factored

00:04:56.512 --> 00:04:57.785
this expression over here.

00:04:57.785 --> 00:05:01.195
We could write that.

00:05:01.195 --> 00:05:03.485
Actually I wanna see the original thing.

00:05:03.485 --> 00:05:06.951
We could write that 3x squared

00:05:06.951 --> 00:05:09.445
times 4x squared,

00:05:09.445 --> 00:05:12.555
plus 2x, plus five

00:05:12.555 --> 00:05:14.818
which is the entire width,
well that's going to be

00:05:14.818 --> 00:05:16.284
equal to the area.

00:05:16.284 --> 00:05:18.385
That's going to be equal
to our original expression:

00:05:18.385 --> 00:05:20.484
12x to the fourth power

00:05:20.484 --> 00:05:22.751
plus 6x to the third

00:05:22.751 --> 00:05:26.132
plus 15x squared.


Factoring quadratics as (x+a)(x+b)

WEBVTT
Kind: captions
Language: en

00:00:00.298 --> 00:00:02.024
- [Voiceover] So we have a
quadratic expression here.

00:00:02.024 --> 00:00:04.402
X squared minus three x minus 10.

00:00:04.402 --> 00:00:05.946
And what I'd like to do in this video

00:00:05.946 --> 00:00:10.070
is I'd like to factor it as
the product of two binomials.

00:00:10.070 --> 00:00:11.939
Or to put it another way,

00:00:11.939 --> 00:00:16.409
I want to write it as
the product x plus a,

00:00:16.409 --> 00:00:19.009
that's one binomial, times x plus b,

00:00:19.009 --> 00:00:23.012
where we need to figure out
what a and b are going to be.

00:00:23.012 --> 00:00:24.190
So I encourage you to pause the video

00:00:24.190 --> 00:00:26.526
and see if you can figure
out what a and b need to be.

00:00:26.526 --> 00:00:28.456
Can we rewrite this
expression as the product

00:00:28.456 --> 00:00:32.032
of two binomials where
we know what a and b are?

00:00:32.032 --> 00:00:34.592
So let's work through this together now.

00:00:34.592 --> 00:00:37.456
I'll highlight a and
b in different colors.

00:00:37.456 --> 00:00:41.885
I'll put a in yellow and
I'll put b in magenta.

00:00:41.885 --> 00:00:44.384
So one way to think about
it is let's just multiply

00:00:44.384 --> 00:00:47.594
these two binomials using a and b,

00:00:47.594 --> 00:00:49.423
and we've done this in previous videos.

00:00:49.423 --> 00:00:51.373
You might want to review
multiplying binomials

00:00:51.373 --> 00:00:53.730
if any of this looks strange to you.

00:00:53.730 --> 00:00:55.274
But if you were to multiply what we have

00:00:55.274 --> 00:00:59.378
on the right-hand side
out it would be equal to,

00:00:59.378 --> 00:01:01.247
you're going to have the x times the x

00:01:01.247 --> 00:01:03.137
which is going to be x squared.

00:01:03.137 --> 00:01:06.591
Then you're going to
have the a times the x,

00:01:06.591 --> 00:01:07.891
which is ax.

00:01:07.891 --> 00:01:09.679
And then you're going to
have the b times the x,

00:01:09.679 --> 00:01:10.553
which is bx.

00:01:10.553 --> 00:01:12.056
Actually just let me, I'm
not gonna skip any steps

00:01:12.056 --> 00:01:13.539
here just to see it this time.

00:01:13.539 --> 00:01:15.287
This is all review, or should be review.

00:01:15.287 --> 00:01:18.395
So then we have, so we did
x times x to get x squared.

00:01:18.395 --> 00:01:23.395
Then we have a times x
to get ax, to get a x.

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And then we're gonna have x times b,

00:01:26.298 --> 00:01:28.513
so we're multiplying each
term times every other term.

00:01:28.513 --> 00:01:31.418
So then we have x times b to get bx.

00:01:31.418 --> 00:01:36.418
So plus bx. b x.

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And then finally we have
plus the a times the b,

00:01:41.455 --> 00:01:46.455
which is of course going to be ab.

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And now we can simplify this,

00:01:48.058 --> 00:01:49.724
and you might have been
able to go straight to this

00:01:49.724 --> 00:01:52.203
if you are familiar with
multiplying binomials.

00:01:52.203 --> 00:01:54.194
This would be x squared plus,

00:01:54.194 --> 00:01:55.799
we can add these two coefficients

00:01:55.799 --> 00:01:57.261
because they're both on
the first degree terms,

00:01:57.261 --> 00:01:58.420
they're both multiplied by x.

00:01:58.420 --> 00:02:01.122
If I have ax's and I add bx's to that

00:02:01.122 --> 00:02:04.941
I'm going to have a plus b x's.

00:02:04.941 --> 00:02:06.363
So let me write that down.

00:02:06.363 --> 00:02:11.363
a plus b x's, and then
finally I have the plus,

00:02:14.206 --> 00:02:17.192
I'll do that blue color,
finally I have it.

00:02:17.192 --> 00:02:22.089
I have plus ab. Plus ab.

00:02:22.089 --> 00:02:25.908
And now we can use this to think about

00:02:25.908 --> 00:02:27.534
what a and b need to be.

00:02:27.534 --> 00:02:29.667
If we do a little bit of pattern matching,

00:02:29.667 --> 00:02:31.333
we see we have an x squared there,

00:02:31.333 --> 00:02:33.162
we have an x squared there.

00:02:33.162 --> 00:02:36.798
We have something times x,

00:02:36.798 --> 00:02:38.830
in this case it's a
negative three times x.

00:02:38.830 --> 00:02:41.654
And here we have something times x.

00:02:41.654 --> 00:02:42.853
So one way to think about it is

00:02:42.853 --> 00:02:46.273
that a plus b needs to be
equal to negative three.

00:02:46.273 --> 00:02:49.993
They need to add up to
be this coefficient.

00:02:49.993 --> 00:02:51.923
So let me write that down.

00:02:51.923 --> 00:02:56.923
So we have a plus b needs to
be equal to negative three.

00:03:02.874 --> 00:03:04.337
And we're not done yet.

00:03:04.337 --> 00:03:06.836
We finally look at this last
term, we have a times b.

00:03:06.836 --> 00:03:10.006
Well a times b needs to
be equal to negative 10.

00:03:10.006 --> 00:03:12.287
So let's write that down.

00:03:12.423 --> 00:03:17.423
So we have a times b needs
to be equal to negative 10.

00:03:20.388 --> 00:03:22.480
And in general, whenever
you're factoring something,

00:03:22.480 --> 00:03:27.295
a quadratic expression that has
a one on second degree term,

00:03:27.295 --> 00:03:29.104
so it has a one coefficient
on the x squared,

00:03:29.104 --> 00:03:30.648
you don't even see it but
it's implicitly there.

00:03:30.648 --> 00:03:32.496
You could write this as one x squared.

00:03:32.496 --> 00:03:34.447
A way to factor it is to
come up with two numbers

00:03:34.447 --> 00:03:37.941
that add up to the coefficient
on the first degree term,

00:03:37.941 --> 00:03:40.298
so two numbers that add
up to negative three.

00:03:40.298 --> 00:03:42.574
And if I multiply those same two numbers,

00:03:42.574 --> 00:03:44.910
I'm going to get negative 10.

00:03:44.910 --> 00:03:46.617
So two numbers that add
up to negative three,

00:03:46.617 --> 00:03:48.019
to add up to the coefficient here.

00:03:48.019 --> 00:03:50.477
And now when I multiply it,
I get the constant term.

00:03:50.477 --> 00:03:52.529
I get this right over here.

00:03:52.529 --> 00:03:54.683
Two numbers when I
multiply I get negative 10.

00:03:54.683 --> 00:03:57.100
Well what could those numbers be?

00:03:57.100 --> 00:04:00.270
Well since when you multiply
them we get a negative number,

00:04:00.270 --> 00:04:03.033
we know that they're going
to have different signs.

00:04:03.033 --> 00:04:05.573
And so let's see how we
could think about it.

00:04:05.573 --> 00:04:07.503
And since when we add them
we get a negative number,

00:04:07.503 --> 00:04:10.835
we know that the negative
number must be the larger one.

00:04:10.835 --> 00:04:13.781
So if I were to just factor 10,

00:04:13.781 --> 00:04:18.781
10 you could do that as one times ten,

00:04:22.314 --> 00:04:24.874
or two times five.

00:04:24.874 --> 00:04:26.479
And two and five are interesting

00:04:26.479 --> 00:04:28.226
because if one of them are negative,

00:04:28.226 --> 00:04:29.364
their difference is three.

00:04:29.364 --> 00:04:30.908
So if one is negative...

00:04:30.908 --> 00:04:32.960
So let's see if we're
talking about negative 10,

00:04:34.590 --> 00:04:38.384
you could say negative two times five.

00:04:38.384 --> 00:04:40.274
And when you multiply them
you do get negative 10.

00:04:40.274 --> 00:04:41.289
But if you add these two,

00:04:41.289 --> 00:04:42.773
you're going to get positive three.

00:04:42.773 --> 00:04:46.552
But what if you went positive
two times negative five.

00:04:46.552 --> 00:04:48.665
Now this is interesting because still

00:04:48.665 --> 00:04:51.042
when you multiply them
you get negative 10.

00:04:51.042 --> 00:04:54.800
And when you add them,
two plus negative five

00:04:54.800 --> 00:04:56.487
is going to be negative three.

00:04:56.487 --> 00:05:00.834
So we have just figured
out our two numbers.

00:05:00.834 --> 00:05:04.857
We could say that a is two or
we could say that b is two,

00:05:04.857 --> 00:05:08.616
but I'll just say that a is equal to two

00:05:08.616 --> 00:05:13.616
and b is equal to negative five.

00:05:14.061 --> 00:05:15.483
And so our original expression,

00:05:15.483 --> 00:05:16.763
we can rewrite as,

00:05:16.763 --> 00:05:20.603
so we can rewrite x squared
minus three x minus 10.

00:05:20.603 --> 00:05:22.878
We can say that that
is going to be equal to

00:05:22.878 --> 00:05:27.878
x plus two times x, instead
of saying plus negative five

00:05:36.491 --> 00:05:38.543
which we could say, we could just say,

00:05:38.543 --> 00:05:39.315
actually let me write that down.

00:05:39.315 --> 00:05:42.545
I could write just plus
negative five right over there

00:05:42.545 --> 00:05:43.399
because that's our b.

00:05:43.399 --> 00:05:48.399
I could just write x minus
five, and we're done.

00:05:49.189 --> 00:05:51.383
We've just factored it as
a product of two binomials.

00:05:51.383 --> 00:05:53.151
Now, I did it fairly involved mainly

00:05:53.151 --> 00:05:54.918
so you see where all this came from.

00:05:54.918 --> 00:05:57.011
But in the future whenever you
see a quadratic expression,

00:05:58.041 --> 00:05:59.649
and you have a one coefficient

00:05:59.649 --> 00:06:01.803
on the second degree term right over here,

00:06:01.803 --> 00:06:03.936
you could say alright well I
need to figure out two numbers

00:06:03.936 --> 00:06:06.435
that add up to the coefficient
on the first degree term,

00:06:06.435 --> 00:06:08.508
on the x term, and those same two numbers

00:06:08.508 --> 00:06:11.271
when I multiply them need to
be equal to this constant term,

00:06:11.271 --> 00:06:12.571
need to be equal to negative 10.

00:06:12.571 --> 00:06:14.257
You say okay well let's see,

00:06:14.257 --> 00:06:15.618
they're gonna be different signs

00:06:15.618 --> 00:06:17.731
because when I multiply them
I get a negative number.

00:06:17.731 --> 00:06:19.621
The negative one is
gonna be the larger one,

00:06:19.621 --> 00:06:21.917
since when I add them I
got a negative number.

00:06:21.917 --> 00:06:24.700
So let's see, let's say five
and two seem interesting.

00:06:24.700 --> 00:06:26.833
Well negative five and positive two,

00:06:26.833 --> 00:06:28.865
when you add them you're
gonna get negative three,

00:06:28.865 --> 00:06:31.208
when you multiply them
you get negative 10.


Strategy of the Civil War

WEBVTT
Kind: captions
Language: en

00:00:00.283 --> 00:00:02.709
- [Voiceover] All right Kim,
so we left off in I guess,

00:00:02.709 --> 00:00:07.040
early-mid 1861, you have
Lincoln gets inagurated,

00:00:07.040 --> 00:00:11.660
Fort Sumter which is kind
of the first real conflict

00:00:11.660 --> 00:00:14.852
of the war, if not the first major battle.

00:00:14.852 --> 00:00:18.081
Lincoln forms his volunteer army, and then

00:00:18.081 --> 00:00:20.552
the rest of the southern
states secede, four more

00:00:20.552 --> 00:00:22.213
states secede.

00:00:22.213 --> 00:00:23.106
- [Kim] Right.

00:00:23.106 --> 00:00:25.289
- [Voiceover] And then what
was the first major conflict?

00:00:25.289 --> 00:00:28.145
- [Kim] So the first
major conflict comes after

00:00:28.145 --> 00:00:29.248
a number of months.

00:00:29.248 --> 00:00:31.906
There are a couple of little
skirmishes here and there,

00:00:31.906 --> 00:00:34.547
but nothing super large

00:00:34.547 --> 00:00:38.153
until about 60,000 troops

00:00:38.153 --> 00:00:41.845
meet outside of Manassas, Virginia,

00:00:41.845 --> 00:00:44.608
at a place called Bull Run.

00:00:44.608 --> 00:00:47.626
An interesting fact, I think, is that

00:00:47.626 --> 00:00:52.131
Union armies and
Confederate armies actually

00:00:52.131 --> 00:00:54.569
named battles different
things, if you've ever been

00:00:54.569 --> 00:00:55.706
confused about this.

00:00:55.706 --> 00:00:58.283
The Union armies tended
to name battles after

00:00:58.283 --> 00:01:01.847
bodies of water, whereas
the Confederate armies

00:01:01.847 --> 00:01:04.425
tended to name them by nearby towns.

00:01:04.425 --> 00:01:06.990
So if you've ever heard
the Battle of Manassas

00:01:06.990 --> 00:01:09.602
and the Battle of Bull Run,
they're the same thing,

00:01:09.602 --> 00:01:12.342
it's just the Union officers
are talking about this creek,

00:01:12.342 --> 00:01:15.801
Bull Run, whereas the
Confederates are talking about

00:01:15.801 --> 00:01:17.392
the town nearby.

00:01:17.392 --> 00:01:18.622
- [Voiceover] I see, and the 60,000 troops

00:01:18.622 --> 00:01:19.690
between the two of them.

00:01:19.690 --> 00:01:20.666
- [Kim] Right.

00:01:20.666 --> 00:01:25.666
So they meet, and this is very
close to Washington, D.C.,

00:01:25.669 --> 00:01:29.268
so much so that people go
out and they bring picnics

00:01:29.268 --> 00:01:31.084
to watch this battle.

00:01:31.084 --> 00:01:32.199
- [Voiceover] They think it's
going to be entertaining.

00:01:32.199 --> 00:01:33.267
- [Kim] Yeah, they think
it's going to be like

00:01:33.267 --> 00:01:34.927
a football game.

00:01:34.927 --> 00:01:37.608
And it is not like a football game.

00:01:37.608 --> 00:01:42.509
It is a gigantic battle,
800 people die that day,

00:01:42.509 --> 00:01:46.048
which doesn't sound like a lot to us, but

00:01:46.048 --> 00:01:50.882
it was the most deadly battle
ever in American history

00:01:50.882 --> 00:01:52.943
up until that point.

00:01:52.943 --> 00:01:57.043
So it's a Confederate victory,
which is very surprising

00:01:57.043 --> 00:02:01.095
to the Union, because they
think that they have such

00:02:01.095 --> 00:02:04.299
superior forces that this
is really going to be

00:02:04.299 --> 00:02:06.214
a very short war.

00:02:06.214 --> 00:02:10.161
And this is a quick rebellion, in 90 days

00:02:10.161 --> 00:02:12.959
we're going to be able to, you know,

00:02:12.959 --> 00:02:15.037
suppress this rebellion and that'll be it.

00:02:15.037 --> 00:02:17.661
But Bull Run is really the first sign that

00:02:17.661 --> 00:02:20.319
this is going to be a major war.

00:02:20.319 --> 00:02:22.931
It's not going to be quick
and it is going to be

00:02:22.931 --> 00:02:23.918
very deadly.

00:02:23.918 --> 00:02:24.812
- [Voiceover] This was July of...

00:02:24.812 --> 00:02:26.855
- [Both] 1861.

00:02:26.855 --> 00:02:30.315
- [Voiceover] Okay, so now
it's clear to both sides,

00:02:30.315 --> 00:02:32.091
especially, I guess you
could say the North,

00:02:32.091 --> 00:02:34.134
that this is not going to be a short war.

00:02:34.134 --> 00:02:36.038
So they need to prepare.

00:02:36.038 --> 00:02:38.406
How are they approaching this?

00:02:38.406 --> 00:02:41.541
- [Kim] Well, so both
sides have some advantages

00:02:41.541 --> 00:02:43.236
and disadvantages.

00:02:43.236 --> 00:02:45.697
For the South, they have some of the same

00:02:45.697 --> 00:02:49.145
advantages that the United
States would have had

00:02:49.145 --> 00:02:52.845
during the war for independence.

00:02:52.845 --> 00:02:56.330
They have home court
advantage, we could say,

00:02:56.330 --> 00:02:59.825
which is that they know
the territory very well

00:02:59.825 --> 00:03:04.155
and also there's a real
incentive for people to

00:03:04.155 --> 00:03:06.349
protect their homes, right.

00:03:06.349 --> 00:03:08.845
You're gonna care more
about a war that's happening

00:03:08.845 --> 00:03:11.666
on your property than a
war that's gonna happen

00:03:11.666 --> 00:03:13.744
very far away.

00:03:13.744 --> 00:03:17.946
The other advantage that they have is

00:03:17.946 --> 00:03:21.673
just really, really terrific
military leadership.

00:03:21.673 --> 00:03:25.364
So they have Robert E. Lee,

00:03:25.364 --> 00:03:28.034
who is

00:03:28.034 --> 00:03:32.759
widely considered the
greatest general of his era.

00:03:32.759 --> 00:03:34.999
He's truly a military genius.

00:03:34.999 --> 00:03:38.354
He, in fact, was offered a
commission in the Union army

00:03:38.354 --> 00:03:42.534
but when Virginia seceded,
he went with Virginia.

00:03:42.534 --> 00:03:45.413
He preferred his home state.

00:03:45.413 --> 00:03:49.592
So he is a terrific general.

00:03:49.592 --> 00:03:54.433
The Union is gonna really
struggle to come up with

00:03:54.433 --> 00:03:57.474
the kind of military
leadership that the South has.

00:03:57.474 --> 00:03:59.225
- [Voiceover] Who is in
charge of the Union or the

00:03:59.225 --> 00:03:59.887
Northern armies?

00:03:59.887 --> 00:04:00.862
You said, the United States Army.

00:04:00.862 --> 00:04:02.371
- [Kim] The United States Army.

00:04:02.371 --> 00:04:06.005
The first general that
Lincoln puts in charge is

00:04:06.005 --> 00:04:07.709
George B. McClellan.

00:04:10.036 --> 00:04:12.474
This is problematic for a lot of reasons.

00:04:12.474 --> 00:04:15.736
One is that George
McClellan is a Democrat,

00:04:15.736 --> 00:04:18.731
so he doesn't agree
politically with Lincoln.

00:04:18.731 --> 00:04:21.192
I think he would have
preferred peace, in fact

00:04:21.192 --> 00:04:24.698
in 1864 he runs against
Lincoln for President

00:04:24.698 --> 00:04:28.575
on a platform of letting
the South go, basically.

00:04:30.619 --> 00:04:35.164
And so Lincoln is struggling

00:04:35.164 --> 00:04:37.272
to match the South when it comes to

00:04:37.272 --> 00:04:39.345
military leadership, but he does have

00:04:39.345 --> 00:04:41.432
other advantages.

00:04:41.432 --> 00:04:46.383
For one thing, there are four
times as many free people

00:04:46.383 --> 00:04:49.181
in the North as there are in the South.

00:04:49.181 --> 00:04:51.403
- [Voiceover] And you made
the point, free people.

00:04:51.403 --> 00:04:52.819
- [Kim] Right.

00:04:52.819 --> 00:04:53.876
- [Voiceover] Because the
South, as you mentioned,

00:04:53.876 --> 00:04:58.275
it has a majority of the
population was not free.

00:04:58.275 --> 00:05:00.725
- [Kim] I wouldn't say a
majority of the population,

00:05:00.725 --> 00:05:03.604
in many states,
- [Voiceover] In Deep South.

00:05:03.604 --> 00:05:05.299
- [Kim] In the Deep South states, right.

00:05:05.299 --> 00:05:08.758
But so there are only
about 9,000,000 people

00:05:08.758 --> 00:05:11.833
living in the South, and
of those 9,000,000 people

00:05:11.833 --> 00:05:14.899
3,500,000 to 4,000,000
of them are enslaved.

00:05:14.899 --> 00:05:16.316
So they're not going to be fighting

00:05:16.316 --> 00:05:20.312
to continue the institution of slavery.

00:05:20.312 --> 00:05:24.606
By contrast, the North
has 22,000,000 people

00:05:25.888 --> 00:05:30.684
and it also has a
terrific industrial base.

00:05:30.684 --> 00:05:34.241
One of the major cultural differences

00:05:34.241 --> 00:05:35.994
between the North and South that leads to

00:05:35.994 --> 00:05:38.257
the Civil War is that
the South is primarily

00:05:38.257 --> 00:05:41.636
agrarian, and the North
becomes very industrial.

00:05:41.636 --> 00:05:44.073
But industry is really helpful in a war.

00:05:44.073 --> 00:05:47.278
They've got miles and
miles of railroad tracks

00:05:47.278 --> 00:05:49.181
which means that they can move supplies

00:05:49.181 --> 00:05:52.711
very quickly, and they also
have hundreds and hundreds

00:05:52.711 --> 00:05:57.110
of factories that make it easy
for them to make munitions.

00:05:57.110 --> 00:05:58.074
- [Voiceover] This is the middle of the

00:05:58.074 --> 00:05:59.676
Industrial Revolution
- [Kim] Right.

00:05:59.676 --> 00:06:02.950
- [Voiceover] and an
industrial base matters a lot.

00:06:02.950 --> 00:06:06.618
And so what's, given the North's

00:06:06.618 --> 00:06:09.950
advantages and the South's advantages,

00:06:09.950 --> 00:06:12.179
what's their strategies, how do they try

00:06:12.179 --> 00:06:13.978
to play to their strengths?

00:06:13.978 --> 00:06:16.939
- [Kim] Right, so the South, they

00:06:16.939 --> 00:06:20.525
are basically trying to outlast the North.

00:06:22.012 --> 00:06:26.121
They know that they have this territory,

00:06:26.121 --> 00:06:28.675
and if the North wants them to come back

00:06:28.675 --> 00:06:30.080
into the Union, they're going to have to

00:06:30.080 --> 00:06:32.030
conquer this territory.

00:06:32.030 --> 00:06:36.291
And even though it's hard to kind of tell,

00:06:36.291 --> 00:06:38.578
the territory of the
South is actually larger

00:06:38.578 --> 00:06:40.633
than Western Europe.
- [Voiceover] Wow.

00:06:40.633 --> 00:06:44.615
- [Kim] So in a way, the
North has a bigger job

00:06:44.615 --> 00:06:47.459
to conquer the South than the Allies did

00:06:47.459 --> 00:06:50.315
in World War II, to conquer Europe.

00:06:50.315 --> 00:06:55.016
So they know that the
North is gonna have to

00:06:55.016 --> 00:06:58.058
fight a war to conquer them, whereas

00:06:58.058 --> 00:07:02.920
the South just needs to
win the war of waiting.

00:07:02.920 --> 00:07:06.207
- [Voiceover] Of attrition.

00:07:06.207 --> 00:07:08.813
- [Kim] Yeah, they're
hoping that the North

00:07:08.813 --> 00:07:10.833
will get tired of fighting.

00:07:10.833 --> 00:07:12.760
- [Voiceover] Fighting
in another person's land,

00:07:12.760 --> 00:07:14.977
you're not defending your own land.

00:07:14.977 --> 00:07:16.892
- [Kim] Right, and they
know that there are

00:07:16.892 --> 00:07:20.212
plenty of whites in the North who don't

00:07:20.212 --> 00:07:21.292
care about slavery.

00:07:21.292 --> 00:07:23.196
It's not in their..

00:07:23.196 --> 00:07:24.217
- [Voiceover] They're
indifferent, what do they care.

00:07:24.217 --> 00:07:25.379
- [Kim] Yeah, what do they care, in fact

00:07:25.379 --> 00:07:29.152
some people are afraid that
if the slaves are freed

00:07:29.152 --> 00:07:31.334
in the South, they're
all gonna come up North

00:07:31.334 --> 00:07:34.178
and they're going to compete for labor

00:07:34.178 --> 00:07:35.316
with poor white people.

00:07:35.316 --> 00:07:37.382
So there are plenty of whites in the North

00:07:37.382 --> 00:07:40.517
who have no interest in the slaves

00:07:40.517 --> 00:07:42.758
in the South being free, even if that's

00:07:42.758 --> 00:07:45.672
not an early war aim of the North.

00:07:45.672 --> 00:07:48.376
So the South is hoping that maybe they can

00:07:48.376 --> 00:07:51.348
win a couple of really
big battles that show

00:07:51.348 --> 00:07:53.275
this isn't gonna be a big war.

00:07:53.275 --> 00:07:54.947
- [Voiceover] Or it'd be
so painful for the North

00:07:54.947 --> 00:07:57.977
to try to conquer the South, so to speak.

00:07:57.977 --> 00:07:59.788
- [Kim] And they're
also trying to show that

00:07:59.788 --> 00:08:02.388
they're serious, to an
international audience,

00:08:02.388 --> 00:08:05.975
particularly England, because
the South is producing

00:08:05.975 --> 00:08:09.806
3/4 of the world's supply
of cotton at this point,

00:08:09.806 --> 00:08:13.980
and England is an
industrial nation which is

00:08:15.805 --> 00:08:19.580
built in many cases around
textile manufacturing.

00:08:19.580 --> 00:08:22.517
So they're hoping that if they show that

00:08:22.517 --> 00:08:24.804
they are serious about
their own nationhood

00:08:24.804 --> 00:08:27.637
that they're going to win
this war against the North

00:08:27.637 --> 00:08:29.842
that England will
intercede on their behalf

00:08:29.842 --> 00:08:31.618
to protect their supply of cotton.

00:08:31.618 --> 00:08:32.780
- [Voiceover] So this would
be an appeal to England

00:08:32.780 --> 00:08:34.242
on purely economic grounds.

00:08:34.242 --> 00:08:35.334
- [Kim] Right.

00:08:35.334 --> 00:08:35.983
- [Voiceover] Fascinating.

00:08:35.983 --> 00:08:39.258
Because England, they didn't have slavery.

00:08:39.258 --> 00:08:40.279
- [Kim] No.

00:08:40.279 --> 00:08:42.554
- [Voiceover] But purely
economically, at least,

00:08:42.554 --> 00:08:43.541
appeal to them.

00:08:43.541 --> 00:08:46.780
- [Kim] So on the other
hand, the North's strategy

00:08:46.780 --> 00:08:50.333
is what they call "The Anaconda Plan".

00:08:50.333 --> 00:08:53.154
And the idea of the Anaconda Plan is that

00:08:53.154 --> 00:08:56.497
they are going to squeeze
the South, economically.

00:08:56.497 --> 00:08:57.856
What they want to do,

00:08:57.856 --> 00:08:58.679
- [Voiceover] Like an anaconda.

00:08:58.679 --> 00:09:00.154
- [Kim] Like an anaconda, right.

00:09:00.154 --> 00:09:04.437
So they want to blockade
the Atlantic ocean

00:09:04.437 --> 00:09:06.933
because they don't want
the South to be able

00:09:06.933 --> 00:09:09.755
to sell their cotton to
get money, and they also

00:09:09.755 --> 00:09:11.971
don't want the South to be able to buy

00:09:11.971 --> 00:09:13.841
the kinds of things that
they're going to need

00:09:13.841 --> 00:09:16.871
to make a war happen.

00:09:16.871 --> 00:09:20.761
They also want to control
the Mississippi River

00:09:20.761 --> 00:09:24.892
cuz that's the real main artery of

00:09:24.892 --> 00:09:26.635
commerce in the West.

00:09:26.635 --> 00:09:28.979
Anyone who is gonna be
shipping their grain

00:09:28.979 --> 00:09:31.035
or their cotton is gonna be shipping it

00:09:31.035 --> 00:09:33.914
down the Mississippi to
the port of New Orleans.

00:09:33.914 --> 00:09:38.115
So the Union hopes that
if they can basically

00:09:38.115 --> 00:09:40.159
surround the South, and make sure nothing

00:09:40.159 --> 00:09:42.133
gets in or out, then
eventually they're just

00:09:42.133 --> 00:09:43.258
gonna starve to death.

00:09:43.258 --> 00:09:44.373
- [Voiceover] This also
goes through the industrial

00:09:44.373 --> 00:09:47.032
bays, it can also produce
more ships and etc.

00:09:47.032 --> 00:09:49.167
- [Kim] Right, and it takes
them a while to do that,

00:09:49.167 --> 00:09:52.732
in fact at the start of
the war, the Union only has

00:09:52.732 --> 00:09:55.298
90 ships.

00:09:55.298 --> 00:09:57.689
I've heard it compared
to "Five leaky boats".

00:09:57.689 --> 00:10:00.672
Right, we're not a naval
power at this point

00:10:00.672 --> 00:10:03.088
and so it's gonna take them a while to

00:10:03.088 --> 00:10:04.874
build up the kind of naval power they need

00:10:04.874 --> 00:10:09.252
to do that, cuz this is
3500 miles of coastline

00:10:09.252 --> 00:10:11.435
that they're gonna need to patrol.

00:10:11.435 --> 00:10:13.070
- [Voiceover] I'm just
looking at this map, not

00:10:13.070 --> 00:10:14.405
getting too much into
details, it looks like

00:10:14.405 --> 00:10:16.752
a lot of the battles
are concentrated right

00:10:16.752 --> 00:10:19.014
in this Virginia/Maryland area, and then

00:10:19.014 --> 00:10:20.850
there's more, it's a
little bit more sparse

00:10:20.850 --> 00:10:23.647
but you have a few that
are in the Deep South

00:10:23.647 --> 00:10:26.885
and along this Mississippi corridor.

00:10:26.885 --> 00:10:28.732
- [Kim] There are two major theaters

00:10:28.732 --> 00:10:29.370
of the war.

00:10:29.370 --> 00:10:30.926
We'd say the Eastern Theater, and this is

00:10:30.926 --> 00:10:34.849
that 100-mile corridor between

00:10:34.849 --> 00:10:38.204
Washington and Richmond,
where a huge amount

00:10:38.204 --> 00:10:40.166
of the fighting takes place.

00:10:40.166 --> 00:10:43.370
It's important to
remember that the capital

00:10:43.370 --> 00:10:45.008
of the Confederacy and the capital of the

00:10:45.008 --> 00:10:47.584
United States are only 100 miles apart.

00:10:47.584 --> 00:10:49.315
- [Voiceover] This capital
is, you can't see it

00:10:49.315 --> 00:10:51.289
on this map but it's
someplace in the middle

00:10:51.289 --> 00:10:54.062
of Virginia, and then D.C.,
literally, as you mentioned

00:10:54.062 --> 00:10:55.246
you said it was 100 miles apart?

00:10:55.246 --> 00:10:57.453
- [Kim] Yep.

00:10:57.453 --> 00:10:59.154
- [Voiceover] Fascinating.


Start of the Civil War

WEBVTT
Kind: captions
Language: en

00:00:00.109 --> 00:00:01.491
- [Voiceover] So Kim, we've been talking

00:00:01.491 --> 00:00:03.176
about the run up to the Civil War.

00:00:03.176 --> 00:00:05.388
We talked about the Compromise of 1850,

00:00:05.388 --> 00:00:07.681
which angered a lot of anti-slavery

00:00:07.681 --> 00:00:09.801
and abolitionist folks in the North.

00:00:09.801 --> 00:00:13.252
As we get to the election of 1860,

00:00:13.252 --> 00:00:15.518
you have Abraham Lincoln

00:00:15.518 --> 00:00:16.531
getting elected.

00:00:16.531 --> 00:00:18.331
And a lot of folks view that as a bit

00:00:18.331 --> 00:00:21.558
of the final catalyst for the Civil War.

00:00:21.558 --> 00:00:22.493
What's happening?

00:00:22.493 --> 00:00:23.672
And is that accurate?

00:00:23.672 --> 00:00:25.665
- [Voiceover] So, Lincoln is elected

00:00:25.665 --> 00:00:28.254
as a Republican Party president.

00:00:28.254 --> 00:00:31.860
This is the first Republican
Party president, ever.

00:00:31.860 --> 00:00:35.280
And the real basis of the Republican Party

00:00:35.280 --> 00:00:37.701
is an anti-slavery platform.

00:00:37.701 --> 00:00:40.360
They really don't want slavery

00:00:40.360 --> 00:00:43.296
to extend into the Western territories

00:00:43.296 --> 00:00:46.950
that have been acquired
through the Mexican War.

00:00:46.950 --> 00:00:50.635
So, they have been
making both an economic,

00:00:50.635 --> 00:00:54.906
and to some extent, moral
argument against slavery.

00:00:54.906 --> 00:00:56.881
So, when Lincoln becomes president,

00:00:56.881 --> 00:00:59.922
the states of the South,
particularly the Deep South,

00:00:59.922 --> 00:01:04.438
or this Cotton Belt area,
whose entire economic system

00:01:04.438 --> 00:01:08.433
relies on slavery, they think
that they are under attack.

00:01:08.433 --> 00:01:12.303
That Lincoln is going
to be coming for slavery

00:01:12.303 --> 00:01:14.979
as soon as he gets a chance, as president.

00:01:14.979 --> 00:01:16.267
- [Voiceover] And we're talking about

00:01:16.267 --> 00:01:17.243
these states down here.

00:01:17.243 --> 00:01:19.393
This is Texas, Louisiana, Mississippi,

00:01:19.393 --> 00:01:21.981
Alabama, Georgia, South
Carolina, and Florida.

00:01:21.981 --> 00:01:24.333
- [Voiceover] Right; so these are the real

00:01:24.333 --> 00:01:29.124
cotton states, where slavery
is highly entrenched.

00:01:29.124 --> 00:01:32.159
There, more than 50% of the population

00:01:32.159 --> 00:01:35.528
is enslaved working on cotton plantations.

00:01:35.528 --> 00:01:37.970
And it's making the elite people,

00:01:37.970 --> 00:01:40.529
elite whites in the South, very wealthy.

00:01:40.529 --> 00:01:44.424
Cotton is just the
backbone of their economy.

00:01:44.424 --> 00:01:45.562
- [Voiceover] So what do they do?

00:01:45.562 --> 00:01:46.434
Lincoln gets elected.

00:01:46.434 --> 00:01:48.458
This is November of 1860?

00:01:48.458 --> 00:01:49.688
- [Voiceover] That's right.

00:01:49.688 --> 00:01:51.245
- [Voiceover] So what do they do about it?

00:01:51.245 --> 00:01:52.379
They're afraid.

00:01:52.379 --> 00:01:53.734
- [Voiceover] They're afraid that Lincoln

00:01:53.734 --> 00:01:56.164
is going to do something about slavery.

00:01:56.164 --> 00:02:01.121
So, over the course of this winter period,

00:02:01.121 --> 00:02:05.050
this is in a period before we moved

00:02:05.050 --> 00:02:07.284
the inauguration up to January,

00:02:07.284 --> 00:02:09.304
so it used to be that presidents

00:02:09.304 --> 00:02:11.600
would be elected in November

00:02:11.600 --> 00:02:13.950
and not take office until March.

00:02:13.950 --> 00:02:15.052
- [Voiceover] Yeah, we have here,

00:02:15.052 --> 00:02:16.352
this Lincoln gets elected in November,

00:02:16.352 --> 00:02:17.553
but then he doesn't get inaugurated

00:02:17.553 --> 00:02:18.649
until March, over here.

00:02:18.649 --> 00:02:19.877
- [Voiceover] So there's this long,

00:02:19.877 --> 00:02:21.900
lame duck period where everyone

00:02:21.900 --> 00:02:23.870
knows that a new political party

00:02:23.870 --> 00:02:25.819
is going to be in power, a new president

00:02:25.819 --> 00:02:28.770
is in power, but he's not in office yet.

00:02:28.770 --> 00:02:30.290
- [Voiceover] And so
you have James Buchanan

00:02:30.290 --> 00:02:32.986
sitting around; he's
still the president, but.

00:02:32.986 --> 00:02:36.177
- [Voiceover] But, yeah,
his days are numbered,

00:02:36.177 --> 00:02:38.419
and his power is pretty limited.

00:02:38.419 --> 00:02:40.241
So, over the course of what they call this

00:02:40.241 --> 00:02:43.983
Secession Winter, the seven states

00:02:43.983 --> 00:02:46.363
of the Deep South get together

00:02:46.363 --> 00:02:50.565
and they secede from the
Union, one after another.

00:02:50.565 --> 00:02:52.825
And this includes South Carolina,

00:02:52.825 --> 00:02:55.646
Mississippi, Florida, Alabama,

00:02:55.646 --> 00:02:58.070
Georgia, Louisiana, and Texas.

00:02:58.070 --> 00:02:59.540
They secede from the Union,

00:02:59.540 --> 00:03:00.918
and they form what they call

00:03:00.918 --> 00:03:02.816
The Confederate States of America,

00:03:02.816 --> 00:03:05.725
which is, basically, almost exactly

00:03:05.725 --> 00:03:07.765
the same as The United States of America.

00:03:07.765 --> 00:03:10.246
Their constitution is based very

00:03:10.246 --> 00:03:12.301
closely on the US Constitution.

00:03:12.301 --> 00:03:16.422
But it guarantees the
existence of slavery.

00:03:16.422 --> 00:03:17.996
It explicitly says that slavery

00:03:17.996 --> 00:03:20.270
is allowed and protected forever.

00:03:20.270 --> 00:03:24.390
And they elect Jefferson
Davis as their president.

00:03:24.390 --> 00:03:26.902
- [Voiceover] So when they seceded,

00:03:26.902 --> 00:03:28.512
for them it was clearly about slavery.

00:03:28.512 --> 00:03:30.417
- [Voiceover] Right;
everything is about slavery.

00:03:30.417 --> 00:03:33.107
They are concerned that Lincoln

00:03:33.107 --> 00:03:35.063
is going to interfere with slavery.

00:03:35.063 --> 00:03:37.315
They are afraid that because slavery

00:03:37.315 --> 00:03:39.598
is being outlawed many other places

00:03:39.598 --> 00:03:43.279
in the world, particularly
in the Western Hemisphere,

00:03:43.279 --> 00:03:46.630
that, one way or another,
slavery's days are numbered.

00:03:46.630 --> 00:03:49.308
And if they're going to
protect their livelihood

00:03:49.308 --> 00:03:52.070
as slave owners and as cotton planters,

00:03:52.070 --> 00:03:54.087
they're going to have
to form their own nation

00:03:54.087 --> 00:03:56.304
to make sure that it's protected.

00:03:56.304 --> 00:03:57.999
- [Voiceover] And James Buchanan

00:03:57.999 --> 00:04:00.197
is officially president when
all of this has happened.

00:04:00.197 --> 00:04:02.561
You have seven states
of the United States,

00:04:02.561 --> 00:04:05.286
I guess they're not so
united anymore, leaving.

00:04:05.286 --> 00:04:07.499
Is he just powerless to do anything?

00:04:07.499 --> 00:04:09.702
- [Voiceover] Well, he tries a few things.

00:04:10.655 --> 00:04:14.296
He was a very ineffective
president to begin with.

00:04:14.296 --> 00:04:16.115
It's hard to be an effective president

00:04:16.115 --> 00:04:19.383
when Congress is so divided over issues.

00:04:19.383 --> 00:04:21.819
One thing that Congress actually does

00:04:21.819 --> 00:04:24.432
before Lincoln is in office, and before

00:04:24.432 --> 00:04:26.270
these states officially secede,

00:04:26.270 --> 00:04:29.736
is they try what's called
the Crittenden Plan,

00:04:29.736 --> 00:04:32.896
proposed by John Crittenden of Kentucky,

00:04:32.896 --> 00:04:35.690
saying we will officially protect

00:04:35.690 --> 00:04:37.884
slavery in the Constitution.

00:04:37.884 --> 00:04:39.904
We will say that you can't outlaw

00:04:39.904 --> 00:04:42.647
slavery in the South, and we'll even

00:04:42.647 --> 00:04:44.207
extend this Missouri Compromise

00:04:44.207 --> 00:04:46.111
line, which was sort of the official line

00:04:46.111 --> 00:04:47.348
between North and South,

00:04:47.348 --> 00:04:50.036
between free states and slave states,

00:04:50.036 --> 00:04:51.351
all the way to the Pacific.

00:04:51.351 --> 00:04:53.489
So, just so you know, Southern states,

00:04:53.489 --> 00:04:55.562
we'll make sure that we
won't get rid of slavery.

00:04:55.562 --> 00:04:57.086
- [Voiceover] So this
Crittenden Compromise,

00:04:57.086 --> 00:04:58.425
this was kind of a last ditch effort.

00:04:58.425 --> 00:05:00.152
Everyone's started to see
the writing on the wall.

00:05:00.152 --> 00:05:01.784
These seven states, especially,

00:05:01.784 --> 00:05:04.088
were very loose-in-the-socket.

00:05:04.088 --> 00:05:05.414
- [Voiceover] And this was a last ditch

00:05:05.414 --> 00:05:09.065
effort to keep them in the Union, perhaps.

00:05:09.065 --> 00:05:10.753
- [Voiceover] Yeah,
and I don't want to say

00:05:10.753 --> 00:05:13.672
it's too little too
late, but for the South,

00:05:13.672 --> 00:05:16.720
they have seen the writing on the wall.

00:05:16.720 --> 00:05:18.356
They have seen that this is going

00:05:18.356 --> 00:05:21.014
to be their only opportunity to secede.

00:05:21.014 --> 00:05:22.449
- [Voiceover] Lincoln got elected, and his

00:05:22.449 --> 00:05:24.539
whole party is based
on being anti-slavery.

00:05:24.539 --> 00:05:25.561
- [Voiceover] Right.

00:05:25.561 --> 00:05:28.129
So, they want to get out
while the getting is good.

00:05:28.129 --> 00:05:29.406
So that they can make sure

00:05:29.406 --> 00:05:31.666
that slavery remains in their states.

00:05:31.666 --> 00:05:34.244
- [Voiceover] All right, so
during this lame duck period,

00:05:34.244 --> 00:05:37.449
the seven states, these Deep
South states, they secede.

00:05:37.449 --> 00:05:39.495
Then Lincoln gets inaugurated.

00:05:39.495 --> 00:05:41.032
He is now president.

00:05:41.032 --> 00:05:43.668
And we're not really in the Civil War yet.

00:05:43.668 --> 00:05:45.328
- [Voiceover] No, in fact, Lincoln's

00:05:45.328 --> 00:05:48.567
inaugural address is very conciliatory.

00:05:50.237 --> 00:05:53.598
We think of Lincoln as
being a really great orator.

00:05:53.598 --> 00:05:57.929
And he certainly was, but
his first inaugural address,

00:05:57.929 --> 00:06:00.499
if you read it, is very much

00:06:00.499 --> 00:06:02.962
a plea to the South,

00:06:02.962 --> 00:06:07.053
saying hey, really, I'm not planning

00:06:07.053 --> 00:06:08.989
on outlawing slavery.

00:06:08.989 --> 00:06:11.506
So, the anti-slavery platform that Lincoln

00:06:11.506 --> 00:06:14.099
ascribes to is specifically about

00:06:14.099 --> 00:06:16.746
not extending slavery to the West.

00:06:16.746 --> 00:06:19.483
So he's saying I'm not in favor

00:06:19.483 --> 00:06:22.097
of getting rid of slavery where it is,

00:06:22.097 --> 00:06:25.034
so there's no reason
for you all to secede.

00:06:25.034 --> 00:06:28.769
Come back, everything
will be situation normal.

00:06:28.769 --> 00:06:29.972
- [Voiceover] And they don't.

00:06:29.972 --> 00:06:32.666
- [Voiceover] No, as I said,

00:06:32.666 --> 00:06:35.770
they've already seen

00:06:35.770 --> 00:06:37.488
that this is their opportunity

00:06:37.488 --> 00:06:40.367
to make sure that slavery continues,

00:06:40.367 --> 00:06:43.026
by creating their own nation.

00:06:43.026 --> 00:06:45.999
So, in the South, there are

00:06:45.999 --> 00:06:48.463
a bunch of arsenals

00:06:48.463 --> 00:06:51.689
and forts that belong
to the United States;

00:06:51.689 --> 00:06:56.433
and most of these are taken
over by the Confederacy

00:06:56.433 --> 00:06:58.403
when it becomes its own nation.

00:06:58.403 --> 00:06:59.998
- [Voiceover] This is a
picture of one right here.

00:06:59.998 --> 00:07:01.839
This is Sumter.

00:07:01.839 --> 00:07:03.073
- [Voiceover] Right; so Fort Sumter

00:07:03.073 --> 00:07:05.669
is right in Charleston Harbor.

00:07:05.669 --> 00:07:08.750
And this is a Union fort,
or a United States fort,

00:07:08.750 --> 00:07:10.828
that's holding out, basically.

00:07:10.828 --> 00:07:13.271
They're running out of supplies,

00:07:13.271 --> 00:07:15.749
they have tried to have supplies brought

00:07:15.749 --> 00:07:18.952
in to them before, which
have been repelled.

00:07:20.101 --> 00:07:21.422
- [Voiceover] They're holding out.

00:07:21.422 --> 00:07:23.960
They're well in Confederate territory,

00:07:23.960 --> 00:07:28.647
but they're still controlled
by United States soldiers.

00:07:28.647 --> 00:07:31.216
- [Voiceover] Right; so, they do not

00:07:31.216 --> 00:07:32.900
want to surrender this fort.

00:07:32.900 --> 00:07:37.420
Lincoln lets the Confederates, the Rebels,

00:07:37.420 --> 00:07:40.929
know that he wants to resupply this fort.

00:07:43.038 --> 00:07:47.276
The Confederates instead
fire on Fort Sumter.

00:07:47.276 --> 00:07:50.861
They start lobbing artillery at it.

00:07:50.861 --> 00:07:55.435
Over the course of a day, they force

00:07:55.435 --> 00:07:59.913
the Union forces in Fort
Sumter to surrender.

00:07:59.913 --> 00:08:02.049
- [Voiceover] I guess this was the real

00:08:02.049 --> 00:08:04.299
matchstick for the war.

00:08:04.299 --> 00:08:05.854
But this wasn't the first tension.

00:08:05.854 --> 00:08:07.414
- [Voiceover] No; obviously this had been

00:08:07.414 --> 00:08:09.850
going on for some time.

00:08:09.850 --> 00:08:11.548
- [Voiceover] Even in Buchanan's

00:08:11.548 --> 00:08:13.882
lame duck period, there's
probably a little bit of tension.

00:08:13.882 --> 00:08:17.982
- [Voiceover] Yeah, if
you want to be expansive,

00:08:17.982 --> 00:08:20.425
you could say that this
tension is almost built into

00:08:20.425 --> 00:08:22.120
the Constitution when they don't--

00:08:22.120 --> 00:08:23.607
- [Voiceover] (laughs)
- [Voiceover] (laughs)

00:08:23.607 --> 00:08:24.307
secede.

00:08:24.307 --> 00:08:25.428
- [Voiceover] You got that right.

00:08:25.428 --> 00:08:26.678
But especially, even post-secession

00:08:26.678 --> 00:08:28.096
of these first seven states,

00:08:28.096 --> 00:08:30.016
there are already some tensions,

00:08:30.016 --> 00:08:32.410
especially if they're
taking over these forts.

00:08:32.410 --> 00:08:34.320
Former United States soldiers are now

00:08:34.320 --> 00:08:35.796
thinking about cutting off supplies

00:08:35.796 --> 00:08:39.009
to current United States soldiers.

00:08:39.009 --> 00:08:40.248
And then Fort Sumter, it sounds

00:08:40.248 --> 00:08:42.445
like this was definitely the straw

00:08:42.445 --> 00:08:44.275
that breaks the camel's back, so to speak.

00:08:44.275 --> 00:08:45.392
- [Voiceover] Yeah,
this is the tinder box.

00:08:45.392 --> 00:08:47.032
And I think it's maybe intended

00:08:47.032 --> 00:08:49.025
to be a tinder box on both sides.

00:08:49.025 --> 00:08:51.272
Because Lincoln wants to be sure

00:08:51.272 --> 00:08:53.142
that if there's going to be a war,

00:08:53.142 --> 00:08:55.516
the North isn't going
to fire the first shot.

00:08:55.516 --> 00:08:56.505
They want to make sure

00:08:56.505 --> 00:08:58.110
that this is the South's decision.

00:08:58.110 --> 00:09:00.890
It can be blamed on
them if it needs to be.

00:09:00.890 --> 00:09:02.190
- [Voiceover] This is a pattern you see

00:09:02.190 --> 00:09:04.906
throughout history, is
that no one at least

00:09:04.906 --> 00:09:07.716
wants to, officially, be the
person to fire the first shot.

00:09:07.716 --> 00:09:09.070
They often look for a good reason

00:09:09.070 --> 00:09:10.278
to fire the first shot because

00:09:10.278 --> 00:09:11.588
they want to get into war,

00:09:11.588 --> 00:09:13.747
but everyone wants

00:09:13.747 --> 00:09:14.818
to have the moral high ground.

00:09:14.818 --> 00:09:17.620
- [Voiceover] Right; and in the South,

00:09:17.620 --> 00:09:20.401
they are looking to make sure

00:09:20.401 --> 00:09:23.386
that this is kind of a
morale building moment.

00:09:23.386 --> 00:09:25.824
When they fire on Fort Sumter,

00:09:25.824 --> 00:09:28.984
they're firing on a federal fort, right?

00:09:28.984 --> 00:09:30.069
- [Voiceover] Mm-hmm.

00:09:30.069 --> 00:09:31.237
- [Voiceover] In any circumstances,

00:09:31.237 --> 00:09:32.765
that's going to bring on war.

00:09:32.765 --> 00:09:34.706
And they're hoping that if they can

00:09:34.706 --> 00:09:37.763
kind of get this fire started,

00:09:37.763 --> 00:09:39.992
then these four other slaveholding

00:09:39.992 --> 00:09:41.525
states, or actually eight other

00:09:41.525 --> 00:09:43.445
slaveholding states, in the South

00:09:43.445 --> 00:09:46.034
are going to join the effort.

00:09:46.034 --> 00:09:48.315
And that's pretty much
exactly what happens.

00:09:48.315 --> 00:09:50.805
So, after they fire on Fort Sumter,

00:09:50.805 --> 00:09:54.521
the fort is surrendered
to the Confederacy.

00:09:54.521 --> 00:09:57.962
Lincoln says, okay, you want
to start a war, we got a war.

00:09:57.962 --> 00:10:02.549
He calls for 75,000 troops,

00:10:02.549 --> 00:10:05.494
volunteers to put down the insurrection.

00:10:05.494 --> 00:10:08.889
He calls them for a 90-day service period,

00:10:08.889 --> 00:10:10.166
which tells you how long

00:10:10.166 --> 00:10:11.959
they thought this was gonna last.

00:10:11.959 --> 00:10:15.154
After Lincoln has asked for this army,

00:10:15.154 --> 00:10:19.236
four more slaveholding
states in the South secede.

00:10:19.236 --> 00:10:22.402
And that's Virginia, the
most important of these,

00:10:22.402 --> 00:10:25.307
it's gonna be the real
battleground of the Civil War.

00:10:25.307 --> 00:10:28.282
- [Voiceover] Today, we'd
consider that West Virginia

00:10:28.282 --> 00:10:30.515
and Virginia, but that
was Virginia back then.

00:10:30.515 --> 00:10:33.128
- [Voiceover] Right; Virginia, Arkansas,

00:10:33.128 --> 00:10:35.179
Tennessee, and North Carolina

00:10:35.179 --> 00:10:37.875
also join the Confederate
States of America.

00:10:37.875 --> 00:10:40.620
And the war is on.

00:10:40.620 --> 00:10:42.035
- [Voiceover] Fascinating.


Factoring polynomials: common binomial factor

WEBVTT
Kind: captions
Language: en

00:00:00.261 --> 00:00:02.808
- Whereas to factor the
polynomial below as the product

00:00:02.808 --> 00:00:04.324
of two binomials

00:00:04.324 --> 00:00:07.552
and we have n times n minus one plus

00:00:07.552 --> 00:00:10.023
3 times n minus one.

00:00:10.023 --> 00:00:11.441
So I encourage you to pause this video

00:00:11.441 --> 00:00:13.891
and see if you can figure this out.

00:00:13.891 --> 00:00:17.466
Well, the key is to realizing
that both of these terms

00:00:17.466 --> 00:00:19.684
have n minus one as a factor.

00:00:19.684 --> 00:00:21.089
Let me just rewrite the whole thing

00:00:21.089 --> 00:00:22.854
so we can work on it down here.

00:00:22.854 --> 00:00:26.116
So this is n times n minus one

00:00:26.116 --> 00:00:30.191
plus 3 times n minus one.

00:00:30.191 --> 00:00:33.249
And notice both of them
have an n minus one,

00:00:33.249 --> 00:00:36.054
have an n minus one as a factor.

00:00:36.054 --> 00:00:38.028
So what we can do is factor out

00:00:38.028 --> 00:00:39.224
the n minus one

00:00:39.224 --> 00:00:41.708
or you can view it as undistributing

00:00:41.708 --> 00:00:44.514
the n minus one and if we do that,

00:00:44.514 --> 00:00:46.596
we're gonna factor out the n minus one

00:00:46.596 --> 00:00:49.023
and what are we going to have left over?

00:00:49.023 --> 00:00:50.753
Well if you take out the n minus one here

00:00:50.753 --> 00:00:52.123
if you undistribute it out,

00:00:52.123 --> 00:00:55.770
you're just going to be left with

00:00:55.770 --> 00:00:57.185
that n.

00:00:57.185 --> 00:00:58.392
So you're going to have an n there

00:00:58.392 --> 00:01:01.155
and then for this second term,

00:01:01.155 --> 00:01:02.955
you factor this n minus one out.

00:01:02.955 --> 00:01:04.452
You're just going to be left with this

00:01:04.452 --> 00:01:07.088
positive three plus three.

00:01:07.088 --> 00:01:08.853
And just like that we are done.

00:01:08.853 --> 00:01:11.453
We have factored the polynomial below

00:01:11.453 --> 00:01:14.925
as a product of two binomials.

00:01:14.925 --> 00:01:17.328
So this is the same thing as

00:01:17.328 --> 00:01:22.328
n minus one times n plus three.

00:01:22.355 --> 00:01:23.690
And once again you can check this.

00:01:23.690 --> 00:01:25.292
You can take this n minus one

00:01:25.292 --> 00:01:26.186
and distribute it.

00:01:26.186 --> 00:01:30.354
n minus one times n is
this term right over here

00:01:30.354 --> 00:01:31.655
n times n minus one

00:01:31.655 --> 00:01:34.186
and then n minus one times three

00:01:34.186 --> 00:01:36.252
is this term right over here.

00:01:36.252 --> 00:01:37.551
n minus one times three

00:01:37.551 --> 00:01:40.705
or three times n minus one.


Monomial factorization

WEBVTT
Kind: captions
Language: en

00:00:00.411 --> 00:00:02.686
- "Theodore and Claire
were each asked to factor

00:00:02.686 --> 00:00:06.006
"the term 24 x to the
fifth as the product of

00:00:06.006 --> 00:00:07.608
"two monomials.

00:00:07.608 --> 00:00:09.466
"Their responses are shown below."

00:00:09.466 --> 00:00:11.950
So Theodore factored 24 x to the fifth

00:00:11.950 --> 00:00:13.947
as being equal to eight x third, times

00:00:13.947 --> 00:00:16.408
three x squared, and Claire factored

00:00:16.408 --> 00:00:19.229
24 x to the fifth as being equal to four x

00:00:19.229 --> 00:00:21.481
times six x to the fourth.

00:00:21.481 --> 00:00:22.921
And then they ask us,
"Which of the students

00:00:22.921 --> 00:00:26.624
"factored 24 x to the fifth correctly?"

00:00:26.624 --> 00:00:28.319
So I encourage you,
pause the video and see

00:00:28.319 --> 00:00:29.747
if you can figure this out.

00:00:29.747 --> 00:00:32.707
Which of them factored it correctly?

00:00:32.707 --> 00:00:34.959
All right, now let's
first look at Theodore.

00:00:34.959 --> 00:00:37.687
So he factored it into
these two monomials,

00:00:37.687 --> 00:00:39.986
eight x to the third and three x squared.

00:00:39.986 --> 00:00:41.483
Well, let's just see,
if we were to multiply

00:00:41.483 --> 00:00:44.397
these two things, do we
get 24 x to the fifth?

00:00:44.397 --> 00:00:48.101
So if you multiply eight times three,

00:00:48.101 --> 00:00:50.538
you do indeed get 24.

00:00:50.538 --> 00:00:52.048
And then all you have to do is multiply

00:00:52.048 --> 00:00:56.412
the x terms, or the powers of x.

00:00:56.412 --> 00:00:59.024
You have x to the third times x squared,

00:00:59.024 --> 00:01:02.056
that indeed does equal x to the fifth.

00:01:02.056 --> 00:01:04.666
So Theodore did factor
it correctly, this is

00:01:04.666 --> 00:01:07.011
one factorization, I guess you would say,

00:01:07.011 --> 00:01:08.613
of 24 x to the fifth.

00:01:08.613 --> 00:01:10.273
Now let's look at Claire.

00:01:10.273 --> 00:01:13.384
So Claire, if we were
taking just the coefficient,

00:01:13.384 --> 00:01:17.808
four times six is indeed equal to 24.

00:01:17.808 --> 00:01:19.886
And then if we were to
look at the powers of x

00:01:19.886 --> 00:01:22.265
we have x to the first power here, times

00:01:22.265 --> 00:01:24.471
x to the fourth power,
which is going to be

00:01:24.471 --> 00:01:25.841
x to the fifth power.

00:01:25.841 --> 00:01:28.732
So Claire also factored it correctly.

00:01:28.732 --> 00:01:30.508
And this just goes to
show you that there's

00:01:30.508 --> 00:01:33.201
more than one possible factorization of

00:01:33.201 --> 00:01:36.405
a monomial like 24 x to the fifth.

00:01:36.405 --> 00:01:37.404
I could come up with another one.

00:01:37.404 --> 00:01:39.307
I could write something like

00:01:39.307 --> 00:01:42.790
24 x to the fifth, I could say that

00:01:42.790 --> 00:01:46.147
that is 12 x,

00:01:48.368 --> 00:01:51.687
to the third, times, what
would have to be left?

00:01:51.687 --> 00:01:55.750
12 times two is 24, so two x squared.

00:01:55.750 --> 00:01:57.643
That's another possible factorization.

00:01:57.643 --> 00:02:00.104
So clearly there's more than one way

00:02:00.104 --> 00:02:04.161
to factor this monomial
into two other monomials.


Factoring binomials: common factor

WEBVTT
Kind: captions
Language: en

00:00:00.466 --> 00:00:02.707
- [Voiceover] So we're told
to factor the polynomial below

00:00:02.707 --> 00:00:07.467
by its greatest common monomial factor.

00:00:07.467 --> 00:00:08.303
So what does that mean?

00:00:08.303 --> 00:00:10.926
So we have these two terms,
and I want to figure out

00:00:10.926 --> 00:00:14.142
their greatest common monomial factor,

00:00:14.142 --> 00:00:15.662
and then I want to express this

00:00:15.662 --> 00:00:18.576
with that greatest common
monomial factor factored out.

00:00:18.576 --> 00:00:20.167
So how can we tackle it?

00:00:20.167 --> 00:00:21.119
Well, one way to start

00:00:21.119 --> 00:00:24.229
is I can look at just the constant terms,

00:00:24.229 --> 00:00:26.766
or not the constants, the
coefficients, I should say.

00:00:26.766 --> 00:00:29.030
So I have the 8 and the 12.

00:00:29.030 --> 00:00:29.472
Then I could say,

00:00:29.472 --> 00:00:32.849
"Well, what is just the greatest
common factor of 8 and 12?"

00:00:32.849 --> 00:00:36.809
The gcf of 8 and 12.

00:00:36.809 --> 00:00:39.131
And there are a lot of
common factors of 8 and 12.

00:00:39.131 --> 00:00:42.788
They're both divisible by 1,
they're both divisible by 2,

00:00:42.788 --> 00:00:44.365
they're both divisible by 4.

00:00:44.365 --> 00:00:48.592
But the greatest of their
common factors is going to be 4.

00:00:48.592 --> 00:00:50.705
So that is equal to 4.

00:00:50.705 --> 00:00:53.445
So, let me just leave that there.

00:00:53.445 --> 00:00:54.778
And then we could think
about, "What is...?"

00:00:54.778 --> 00:00:56.496
Well, let me actually
write it right over here.

00:00:56.496 --> 00:00:57.854
I'll put a 4 here.

00:00:57.854 --> 00:01:00.872
And now we can move on to the powers of x.

00:01:00.872 --> 00:01:03.739
We have an x squared and we have an x.

00:01:03.739 --> 00:01:04.274
And we could say,

00:01:04.274 --> 00:01:07.838
"What is the largest power
of x that is divisible

00:01:07.838 --> 00:01:10.671
into both x squared and x?"

00:01:10.671 --> 00:01:12.574
Well, that's just going to be x.

00:01:12.574 --> 00:01:14.838
x squared is clearly divisible by x,

00:01:14.838 --> 00:01:17.658
and x is clearly divisible by x,

00:01:17.658 --> 00:01:22.512
but x isn't going to have a
larger power of x as a factor.

00:01:22.512 --> 00:01:25.971
So this is the greatest
common monomial factor

00:01:25.971 --> 00:01:27.759
of x squared and x.

00:01:27.759 --> 00:01:30.673
Now we do the same thing for the ys.

00:01:30.673 --> 00:01:33.598
So, we have a y and a y squared.

00:01:33.598 --> 00:01:34.678
If we think in the same terms

00:01:34.678 --> 00:01:37.557
the largest power of y that's
divisible into both of these

00:01:37.557 --> 00:01:40.959
is going to be just y to
the first power, or y.

00:01:40.959 --> 00:01:45.010
And so 4xy is the greatest
common monomial factor.

00:01:45.010 --> 00:01:47.576
And to see that we could
express each of these terms

00:01:47.576 --> 00:01:51.291
as a product of 4xy and something else.

00:01:51.291 --> 00:01:53.751
So this first term right over here,

00:01:53.751 --> 00:01:55.052
so let me pick a color,

00:01:55.052 --> 00:02:00.052
so this term right over
here we could write as 4xy.

00:02:00.775 --> 00:02:02.714
That one's actually,
that color's hard to see,

00:02:02.714 --> 00:02:04.908
let me pick a darker color.

00:02:04.908 --> 00:02:09.908
We could write this right
over here as 4xy times what?

00:02:12.105 --> 00:02:15.287
And I encourage you to pause
the video and think about that.

00:02:15.287 --> 00:02:19.211
Let's see, 4 times what
is going to get us to 8?

00:02:19.211 --> 00:02:21.927
Well, 4 times 2 is going to get us to 8.

00:02:21.927 --> 00:02:25.410
x times what is going
to get us to x squared?

00:02:25.410 --> 00:02:28.427
Well, x times x is going
to get us to x squared.

00:02:28.427 --> 00:02:31.795
And then y times what
is going to get us to y?

00:02:31.795 --> 00:02:33.548
Well, it's just going to be y.

00:02:33.548 --> 00:02:36.391
So 4xy times 2x is
actually going to give us

00:02:36.391 --> 00:02:37.031
this first term.

00:02:37.031 --> 00:02:39.363
So actually just let me rewrite
a little bit differently.

00:02:39.363 --> 00:02:43.420
So it's 4xy times 2x is this first term,

00:02:43.420 --> 00:02:45.497
and you can verify that.

00:02:45.497 --> 00:02:49.642
4 times 2 is going to be equal to 8,

00:02:49.642 --> 00:02:52.045
x times x is equal to x squared,

00:02:52.045 --> 00:02:54.147
and then you just have the y.

00:02:54.147 --> 00:02:56.887
Now let's do the same
thing with the second term.

00:02:56.887 --> 00:02:58.280
And I just want to do this to show you

00:02:58.280 --> 00:03:01.937
that this is their largest
common monomial factor.

00:03:01.937 --> 00:03:03.458
So the second term,

00:03:03.458 --> 00:03:05.545
and I'll do this in a
slightly different color,

00:03:05.545 --> 00:03:07.235
do it in blue.

00:03:07.235 --> 00:03:11.657
I want to write this as the product of 4xy

00:03:11.657 --> 00:03:15.478
and another monomial.

00:03:15.478 --> 00:03:17.509
So 4 times what is 12?

00:03:17.509 --> 00:03:19.934
Well, 4 times 3 is 12.

00:03:19.934 --> 00:03:21.514
x times what is x?

00:03:21.514 --> 00:03:22.617
Well, it's just going to be 1,

00:03:22.617 --> 00:03:24.649
so we don't have to write a times 1 here.

00:03:24.649 --> 00:03:26.808
And then y times what is y squared?

00:03:26.808 --> 00:03:29.815
It's going to be y times y is y squared.

00:03:29.815 --> 00:03:30.558
And you can verify.

00:03:30.558 --> 00:03:34.412
If you multiply these two you're
going to get 12 xy-squared.

00:03:34.412 --> 00:03:36.792
4 times 3 is 12, you get your x,

00:03:36.792 --> 00:03:39.392
and then y times y is y squared.

00:03:39.392 --> 00:03:42.771
So, so far I've written
this exact same expression

00:03:42.771 --> 00:03:45.254
but I've taken each of those
terms and I've factored them

00:03:45.254 --> 00:03:49.433
into their greatest common monomial factor

00:03:49.433 --> 00:03:51.408
and then whatever is left over.

00:03:51.408 --> 00:03:55.052
And now I can factor the 4xy out.

00:03:55.052 --> 00:03:56.934
I can actually factor it out.

00:03:56.934 --> 00:03:58.907
So this is going to be equal to,

00:03:58.907 --> 00:04:02.866
if I factor the 4xys out,

00:04:02.866 --> 00:04:05.907
you can kind of say I
undistribute the 4xy.

00:04:05.907 --> 00:04:07.347
I factor it out.

00:04:07.347 --> 00:04:12.347
This is going to be equal
to 4xy times 2x plus --

00:04:20.906 --> 00:04:25.330
when I factor 4xy from here
I get the 3y left over.

00:04:25.330 --> 00:04:29.230
Plus 3y, and we're done.

00:04:29.230 --> 00:04:30.066
And you can verify it.

00:04:30.066 --> 00:04:31.645
If you were to go the other way,

00:04:31.645 --> 00:04:33.223
if you were to distribute this 4xy

00:04:33.223 --> 00:04:36.405
and multiply it times 2x,
you would get 8 x-squared y.

00:04:36.405 --> 00:04:39.167
And then when you distribute
the 4xy onto the 3y

00:04:39.167 --> 00:04:41.431
you get the 12xy-squared.

00:04:41.431 --> 00:04:42.453
And so we're done.

00:04:42.453 --> 00:04:44.530
This right over here is our answer.

00:04:44.530 --> 00:04:48.931
The answer is going to be 4xy,

00:04:48.931 --> 00:04:51.462
which is the greatest
common monomial factor,

00:04:51.462 --> 00:04:56.462
times 2x plus 3y.


Worked example: finding the missing monomial factor

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:02.150
- [Voiceover] So we have
negative 30x to the fifth

00:00:02.150 --> 00:00:06.090
is equal to negative 10x
to the third, times F.

00:00:06.090 --> 00:00:08.020
And I encourage you to pause this video

00:00:08.020 --> 00:00:11.250
and see if you can figure
out what F is going to be.

00:00:11.250 --> 00:00:12.650
Well, the way that we can tackle it,

00:00:12.650 --> 00:00:15.587
we could just isolate the F
on the right-hand side here

00:00:15.587 --> 00:00:19.490
if we divide by negative 10x to the third.

00:00:19.490 --> 00:00:21.150
So we might say, well, we want to divide

00:00:21.150 --> 00:00:23.590
this side by negative 10x to the third,

00:00:23.590 --> 00:00:25.390
but if we want the equality to be true,

00:00:25.390 --> 00:00:27.720
if we want the left side
to stay being the same

00:00:27.720 --> 00:00:29.800
as the right side,

00:00:29.800 --> 00:00:30.782
whatever we do with the right side

00:00:30.782 --> 00:00:32.550
we have to do to the left side as well.

00:00:32.550 --> 00:00:33.750
So we have to divide the left side

00:00:33.750 --> 00:00:37.520
by negative 10x to the third.

00:00:37.520 --> 00:00:40.120
And then what does that leave us with?

00:00:40.120 --> 00:00:42.150
Well, on the right-hand side, up top

00:00:42.150 --> 00:00:44.050
we're multiplying by
negative 10x to the third,

00:00:44.050 --> 00:00:46.520
and then we're dividing by
negative 10x to the third,

00:00:46.520 --> 00:00:47.790
well, multiplying by something

00:00:47.790 --> 00:00:49.150
and then dividing by that same thing

00:00:49.150 --> 00:00:51.250
is the same thing as
just multiplying by one,

00:00:51.250 --> 00:00:53.550
or one way to think about
it, they just cancel out.

00:00:53.550 --> 00:00:56.210
So we are just going to be left with an F.

00:00:56.210 --> 00:00:59.520
We're going to be left an
F on the right-hand side.

00:00:59.520 --> 00:01:01.650
And there's the whole point,
we wanted to solve for F.

00:01:01.650 --> 00:01:04.360
And on the left-hand side,

00:01:04.360 --> 00:01:06.050
we can first look at the coefficients.

00:01:06.050 --> 00:01:10.590
We could say negative 30 divided by,

00:01:10.590 --> 00:01:13.430
negative 30 divided by negative 10

00:01:13.430 --> 00:01:16.020
is positive three.

00:01:16.020 --> 00:01:17.820
So that's going to be three.

00:01:17.820 --> 00:01:22.320
And then x to the fifth power

00:01:22.320 --> 00:01:25.490
divided by x to the third power,

00:01:25.490 --> 00:01:28.640
well that's going to be x-squared,

00:01:29.720 --> 00:01:31.020
x-squared.

00:01:31.020 --> 00:01:32.820
You could either think of
it in terms of our exponent

00:01:32.820 --> 00:01:36.190
properties, we would
subtract these two exponents,

00:01:36.190 --> 00:01:38.820
x to the five minus
three, which is x-squared

00:01:38.820 --> 00:01:40.550
or you could say up on top that's

00:01:40.550 --> 00:01:43.550
x times x times x times x times x.

00:01:43.550 --> 00:01:45.320
Did I say that right? Five x's.

00:01:45.320 --> 00:01:46.820
You could do it as x,

00:01:46.820 --> 00:01:48.890
let me do to that, on top,

00:01:48.890 --> 00:01:52.400
you have x to the fifth which is this.

00:01:53.550 --> 00:01:54.910
I also like to remind myself

00:01:54.910 --> 00:01:56.850
why the exponent properties even work.

00:01:56.850 --> 00:01:58.650
And then on the denominator,

00:01:58.650 --> 00:02:00.930
on the denominator you have

00:02:00.930 --> 00:02:03.420
x times x times x

00:02:03.420 --> 00:02:05.420
and these three x's are going to cancel

00:02:05.420 --> 00:02:07.182
and you're just going to
be left with x times x

00:02:07.182 --> 00:02:09.282
which is just x-squared.

00:02:09.282 --> 00:02:11.720
So you get F is equal to three x-squared

00:02:11.720 --> 00:02:13.120
so you could write,

00:02:13.120 --> 00:02:17.490
we could write that
negative 30x to the fifth

00:02:17.490 --> 00:02:19.750
is equal to,

00:02:19.750 --> 00:02:22.060
is equal to the

00:02:22.060 --> 00:02:25.050
negative 10x to the third

00:02:25.050 --> 00:02:27.620
times F and now we know that F

00:02:27.620 --> 00:02:29.790
is three x-squared.

00:02:29.790 --> 00:02:32.150
Three x-squared.

00:02:32.150 --> 00:02:34.520
And so another way to
describe what's going on

00:02:34.520 --> 00:02:37.230
in this equation, we could say that

00:02:37.230 --> 00:02:41.350
negative 30x to the fifth
is divisible by either one

00:02:41.350 --> 00:02:42.320
of these factors.

00:02:42.320 --> 00:02:44.720
That negative 30x to the
fifth is divisible by

00:02:44.720 --> 00:02:46.520
negative 10x to the third

00:02:46.520 --> 00:02:48.790
or we could say negative 30x to the fifth

00:02:48.790 --> 00:02:50.882
is divisible by three x-squared

00:02:50.882 --> 00:02:52.650
or we could say that three x-squared

00:02:52.650 --> 00:02:55.850
is a factor of negative 30x to the fifth.

00:02:55.850 --> 00:02:57.790
And the way that we can
make these claims about

00:02:57.790 --> 00:03:01.030
factor and divisibility
is we're dealing with

00:03:01.030 --> 00:03:04.150
non-fractional coefficients
right over here

00:03:04.150 --> 00:03:07.290
and we're also dealing with
non-fractional exponents

00:03:07.290 --> 00:03:08.320
right over here

00:03:08.320 --> 00:03:10.180
so that's why we're saying,
hey, these are factors,

00:03:10.180 --> 00:03:12.483
this yellow thing and this magenta thing,

00:03:12.483 --> 00:03:14.020
factors of this blue thing

00:03:14.020 --> 00:03:16.840
or this blue thing is divisible
by either one of these.


Worked example: finding missing monomial side in area model

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:02.590
- [Voiceover] So we have a
rectangle right over here.

00:00:02.590 --> 00:00:04.550
Let's say that we know that its area is

00:00:04.550 --> 00:00:08.220
42x times y to the third.

00:00:08.220 --> 00:00:11.190
So that is the area of the rectangle.

00:00:11.190 --> 00:00:13.490
And we also know that the
height right over here

00:00:13.490 --> 00:00:17.590
is 14xy and what we
want to do in this video

00:00:17.590 --> 00:00:19.890
is figure out what the
length is going to be.

00:00:19.890 --> 00:00:22.610
And as you can imagine, it's going to be

00:00:22.610 --> 00:00:23.990
in algebraic terms.

00:00:23.990 --> 00:00:24.950
So I encourage you to pause the video

00:00:24.950 --> 00:00:26.850
and figure out what is
the length going to be

00:00:26.850 --> 00:00:30.520
if this height is 14xy and the area is

00:00:30.520 --> 00:00:33.500
42xy to the third.

00:00:34.790 --> 00:00:36.250
Well, how do you figure out area?

00:00:36.250 --> 00:00:38.760
You take your length, and
I'll just use L for length.

00:00:38.760 --> 00:00:40.390
I'll put L in parentheses.

00:00:40.390 --> 00:00:42.920
So you take your length
and you multiply it

00:00:42.920 --> 00:00:44.750
times your height.

00:00:44.750 --> 00:00:46.890
So let's multiply it times

00:00:46.890 --> 00:00:50.000
14xy

00:00:50.000 --> 00:00:52.320
and that's going to give you your area.

00:00:52.320 --> 00:00:54.290
So that's going to give us our area

00:00:54.290 --> 00:00:59.080
of 42xy to the third power.

00:00:59.080 --> 00:01:01.520
So how do we solve for our length?

00:01:01.520 --> 00:01:04.590
Well, we can just divide
both sides by 14xy.

00:01:04.590 --> 00:01:05.850
So let's do that.

00:01:05.850 --> 00:01:07.550
So, let's divide the left-hand side

00:01:07.550 --> 00:01:09.182
and the right-hand side

00:01:09.182 --> 00:01:13.120
by 14xy,

00:01:13.120 --> 00:01:15.650
14xy.

00:01:15.650 --> 00:01:17.720
Now on the left-hand
side, I'm multiplying by

00:01:17.720 --> 00:01:20.220
14xy and dividing by 14xy,

00:01:20.220 --> 00:01:22.950
that's the same thing as just
multiplying or dividing by one

00:01:22.950 --> 00:01:24.390
so that cancels out.

00:01:24.390 --> 00:01:27.290
So I'm just left with L or our length

00:01:27.290 --> 00:01:31.050
which is the whole point of
dividing both sides by 14xy.

00:01:31.050 --> 00:01:32.620
And on the right-hand side, I can look

00:01:32.620 --> 00:01:34.220
at the coefficients first.

00:01:34.220 --> 00:01:38.182
I could say 42 divided by 14,

00:01:38.182 --> 00:01:40.270
and that's going to be three.

00:01:41.320 --> 00:01:42.650
Three, and then I could say

00:01:42.650 --> 00:01:45.920
well, x divided by x that's
just going to be one,

00:01:45.920 --> 00:01:49.470
and then I have y to
the third divided by y.

00:01:49.470 --> 00:01:53.150
y to the third divided
by y, that is going to be

00:01:53.150 --> 00:01:57.660
y to the third divided by
y is going to be y-squared.

00:01:57.660 --> 00:01:58.550
And then we're done.

00:01:58.550 --> 00:02:01.650
Our length is three y-squared.

00:02:01.650 --> 00:02:04.590
So our length is equal to

00:02:04.590 --> 00:02:07.132
three y-squared.


Intro to factors & divisibility

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.651
- [Voiceover] You're probably
familiar with the general term

00:00:02.651 --> 00:00:03.818
factor.

00:00:03.818 --> 00:00:08.217
So if I were to say: What
are the factors of 12,

00:00:08.217 --> 00:00:10.285
you could say: Well what
are the whole numbers

00:00:10.285 --> 00:00:14.318
that I can multiply by another
whole number to get 12?

00:00:14.318 --> 00:00:17.286
So for examples, you
could say things like,

00:00:17.286 --> 00:00:22.051
well I could multiply
one times 12 to get 12.

00:00:22.051 --> 00:00:24.384
So you could say that
one is a factor of 12.

00:00:24.384 --> 00:00:26.484
You could even say that
12 is a factor of 12.

00:00:26.484 --> 00:00:29.384
You could say two times
six is equal to 12,

00:00:29.384 --> 00:00:31.984
so you could say that
two is a factor of 12

00:00:31.984 --> 00:00:34.452
and that six is also a factor of 12.

00:00:34.452 --> 00:00:37.584
And of course three times four is also 12,

00:00:37.584 --> 00:00:40.618
so both three and four are factors of 12.

00:00:40.618 --> 00:00:42.118
So if you said well what
are the factors of 12

00:00:42.118 --> 00:00:44.097
and you've seen this
before, well you could say:

00:00:44.097 --> 00:00:47.917
one, two, three, four, six, and 12,

00:00:47.917 --> 00:00:50.117
those are all factors of 12.

00:00:50.117 --> 00:00:51.884
And you could also phase
it the other way around,

00:00:51.884 --> 00:00:53.085
so let me just give an example.

00:00:53.085 --> 00:00:56.217
So if I were to pick on
three, I could say that three

00:00:56.217 --> 00:01:01.217
is a factor of 12.

00:01:01.484 --> 00:01:05.284
Or to phrase it slightly
differently, I could say that 12

00:01:05.284 --> 00:01:09.618
is divisible,

00:01:09.618 --> 00:01:12.684
12 is divisible by three.

00:01:12.684 --> 00:01:15.784
Now what I wanna do in this
video is extend this idea

00:01:15.784 --> 00:01:20.684
of being a factor or divisibility
into the algebraic world.

00:01:20.684 --> 00:01:23.017
So for example,

00:01:23.017 --> 00:01:24.971
if I were to take

00:01:24.971 --> 00:01:28.671
3xy.

00:01:28.671 --> 00:01:32.185
So this is a monomial with
an integer coefficient.

00:01:32.185 --> 00:01:34.518
Three is an integer right over here.

00:01:34.518 --> 00:01:37.085
And if I were to multiply
it with another monomial

00:01:37.085 --> 00:01:39.184
with an integer coefficient,

00:01:39.184 --> 00:01:42.385
I don't know, let's say times negative two

00:01:42.385 --> 00:01:46.418
X squared, Y to the third power,

00:01:46.418 --> 00:01:48.718
what is this going to be equal to?

00:01:48.718 --> 00:01:50.784
Well this would be equal to,

00:01:50.784 --> 00:01:54.951
if we multiply the coefficients,
three times negative two

00:01:54.951 --> 00:01:58.884
is going to be negative six.

00:01:58.884 --> 00:02:03.884
X times X squared is X to the third power.

00:02:03.984 --> 00:02:06.484
And then Y times Y to the third

00:02:06.484 --> 00:02:09.651
is Y to the fourth power.

00:02:09.651 --> 00:02:12.951
And so what we could say is,
if we wanted to say factors

00:02:12.951 --> 00:02:15.851
of negative six X to the
third, Y to the fourth,

00:02:15.851 --> 00:02:18.817
we could say that 3xy is a factor of this

00:02:18.817 --> 00:02:20.551
just as an example; so
let me write that down.

00:02:20.551 --> 00:02:25.001
We could write that 3xy

00:02:25.001 --> 00:02:27.220
is a factor of,

00:02:27.220 --> 00:02:32.220
is a factor of...

00:02:33.500 --> 00:02:38.084
of negative six X to the
third power, Y to the fourth,

00:02:38.084 --> 00:02:39.951
or we could phrase that
the other way around.

00:02:39.951 --> 00:02:43.384
We could say that negative
six X to the third,

00:02:43.384 --> 00:02:47.184
Y to the fourth, is divisible by,

00:02:47.184 --> 00:02:52.054
is divisible

00:02:52.054 --> 00:02:54.244
by 3xy.

00:02:54.244 --> 00:02:56.418
So hopefully you're seeing the parallels.

00:02:56.418 --> 00:02:59.450
If I'm taking these two monomials
with integer coefficients

00:02:59.450 --> 00:03:02.218
and I multiply 'em and I get
this other, in this case,

00:03:02.218 --> 00:03:05.285
this other monomial, I could
say that either one of these

00:03:05.285 --> 00:03:07.618
and there's actually
other factors of this,

00:03:07.618 --> 00:03:09.551
but I could say either
one of these is a factor

00:03:09.551 --> 00:03:12.214
of this monomial, or we
could say that negative six

00:03:12.214 --> 00:03:14.518
X to the third, Y to the four is divisible

00:03:14.518 --> 00:03:17.185
by one of its factors.

00:03:17.185 --> 00:03:21.351
And we could even extend this
to binomials or polynomials.

00:03:21.351 --> 00:03:24.748
For example, if I were to take,

00:03:24.748 --> 00:03:26.318
if I were to take,

00:03:26.318 --> 00:03:28.168
let me scroll down a little bit,

00:03:28.168 --> 00:03:32.085
whoops, if I were to take,

00:03:32.085 --> 00:03:36.384
let me say X plus three

00:03:36.384 --> 00:03:39.685
and I wanted to multiply it times

00:03:39.685 --> 00:03:43.298
X plus seven,

00:03:43.298 --> 00:03:45.818
we know that this is going to be equal to,

00:03:45.818 --> 00:03:47.417
if I were to write it as a trinomial,

00:03:47.417 --> 00:03:50.551
it's gonna be X times X, so X squared,

00:03:50.551 --> 00:03:53.117
and then it's gonna be
three X plus seven X,

00:03:53.117 --> 00:03:55.917
so plus 10x; and if any
of this looks familiar,

00:03:55.917 --> 00:03:57.617
we have a lot of videos
where we go in detail

00:03:57.617 --> 00:03:59.985
of multiplying binomials like this.

00:03:59.985 --> 00:04:02.751
And then three times seven is 21.

00:04:02.751 --> 00:04:05.255
Plus 21.

00:04:05.255 --> 00:04:08.917
So because I multiplied these
two, in this case binomials,

00:04:08.917 --> 00:04:11.484
or we could consider
themselves to be polynomials,

00:04:11.484 --> 00:04:14.684
polynomials or binomials
with integer coefficients.

00:04:14.684 --> 00:04:17.951
Notice the coefficients
here, they're one, one.

00:04:17.951 --> 00:04:20.018
The constants here, they're all integers.

00:04:20.018 --> 00:04:23.084
Because I'm dealing with all
integers here, we could say

00:04:23.084 --> 00:04:26.918
that either one of these
binomials is a factor

00:04:26.918 --> 00:04:28.618
of this trinomial,

00:04:28.618 --> 00:04:30.785
or we could say this
trinomial is divisible

00:04:30.785 --> 00:04:32.318
by either one of these.

00:04:32.318 --> 00:04:33.351
So let me write that down.

00:04:33.351 --> 00:04:35.618
So I could say, I'll just
pick on X plus seven.

00:04:35.618 --> 00:04:38.584
We could say that X plus seven

00:04:38.584 --> 00:04:40.848
is a factor,

00:04:40.848 --> 00:04:44.398
is a factor

00:04:44.398 --> 00:04:48.187
of X squared

00:04:48.187 --> 00:04:51.757
plus 10x plus 21;

00:04:51.757 --> 00:04:56.058
or we could say that X squared plus 10x

00:04:56.058 --> 00:04:59.845
plus 21 is divisible by,

00:04:59.845 --> 00:05:04.845
is divisible by

00:05:04.935 --> 00:05:07.285
I could say X plus three
or I could say X plus seven

00:05:07.285 --> 00:05:09.318
is divisible by,

00:05:09.318 --> 00:05:12.085
X plus seven.

00:05:12.085 --> 00:05:14.284
And the key is, is that
both of these binomials,

00:05:14.284 --> 00:05:16.151
or even if we were
dealing with polynomials,

00:05:16.151 --> 00:05:18.951
we are dealing with
things that have integer,

00:05:18.951 --> 00:05:23.734
we're dealing with things that
have integer coefficients.


Special products of the form (x+a)(x-a)

WEBVTT
Kind: captions
Language: en

00:00:00.527 --> 00:00:02.663
- [Voiceover] Let's see
if we can figure out what

00:00:02.663 --> 00:00:06.842
x plus three times

00:00:06.842 --> 00:00:09.106
x minus three is, and I encourage you

00:00:09.106 --> 00:00:10.290
to pause the video and see if you can

00:00:10.290 --> 00:00:12.380
work this out.

00:00:12.380 --> 00:00:13.842
Well, one way to tackle
it is the way that we've

00:00:13.842 --> 00:00:16.025
always tackled it when
we multiply binomials,

00:00:16.025 --> 00:00:18.730
is just apply the
distributive property twice.

00:00:18.730 --> 00:00:21.760
So first we can take this
entire yellow x plus three

00:00:21.760 --> 00:00:25.602
and multiply times each
of these two terms.

00:00:25.602 --> 00:00:28.203
So first we can multiply it times this x.

00:00:28.203 --> 00:00:32.972
So that's going to be
x times x plus three.

00:00:32.972 --> 00:00:36.158
And then we are going to

00:00:36.158 --> 00:00:39.455
multiply it times, we can
say, this negative three.

00:00:39.455 --> 00:00:42.636
So we could write minus three times,

00:00:42.636 --> 00:00:44.575
now that's going to be multiplied by

00:00:44.575 --> 00:00:47.446
x plus three again.

00:00:47.446 --> 00:00:49.632
And then we apply the
distributive property

00:00:49.632 --> 00:00:50.968
one more time.

00:00:50.968 --> 00:00:52.849
Where we take this magenta x and we

00:00:52.849 --> 00:00:55.569
distribute it across this x plus three

00:00:55.569 --> 00:00:58.045
so x times x is x squared,

00:00:58.045 --> 00:01:01.516
x times three is three x, and then

00:01:01.516 --> 00:01:03.826
we do it on this side.

00:01:03.826 --> 00:01:07.957
Negative three times x is negative three x

00:01:07.957 --> 00:01:11.253
and negative three times three

00:01:11.253 --> 00:01:13.586
is negative nine.

00:01:13.586 --> 00:01:15.235
And what does this simplify to?

00:01:15.235 --> 00:01:17.185
Well, we're gonna get x squared,

00:01:17.185 --> 00:01:19.008
and we have three x and minus three x

00:01:19.008 --> 00:01:20.889
so these two characters cancel out,

00:01:20.889 --> 00:01:24.592
and we are just left with x squared

00:01:24.592 --> 00:01:26.472
minus nine.

00:01:26.472 --> 00:01:27.773
And you might see a little pattern here,

00:01:27.773 --> 00:01:30.745
notice I added three and
then I subtracted three

00:01:30.745 --> 00:01:35.284
and I got this, I got the x squared

00:01:35.284 --> 00:01:37.826
and then if you take three and multiply it

00:01:37.826 --> 00:01:40.310
by negative three, you are going to get

00:01:40.310 --> 00:01:41.703
a negative nine.

00:01:41.703 --> 00:01:43.247
And notice, the middle terms canceled out.

00:01:43.247 --> 00:01:44.374
And one thing you might ask is,

00:01:44.374 --> 00:01:46.406
well, will that always be the case, if we

00:01:46.406 --> 00:01:48.971
add a number and we subtract that same

00:01:48.971 --> 00:01:50.410
number like that?

00:01:50.410 --> 00:01:51.292
And we could try it out.

00:01:51.292 --> 00:01:53.672
Let's talk in general terms.

00:01:53.672 --> 00:01:55.959
So if we, instead of doing x plus three

00:01:55.959 --> 00:01:58.270
times x minus three, we could write

00:01:58.270 --> 00:02:00.417
this same thing as, instead of three,

00:02:00.417 --> 00:02:04.771
let's just say you have x plus a

00:02:04.771 --> 00:02:06.602
times x minus a.

00:02:09.092 --> 00:02:10.926
And I encourage you to pause this video

00:02:10.926 --> 00:02:12.772
and work it all out, just assume

00:02:12.772 --> 00:02:14.525
a is some number, like three

00:02:14.525 --> 00:02:16.835
or some other number, and apply the

00:02:16.835 --> 00:02:20.144
distributive property
twice and see what you get.

00:02:20.144 --> 00:02:21.432
Well, let's work through it.

00:02:21.432 --> 00:02:25.194
So first we can distribute
this yellow x plus a

00:02:25.194 --> 00:02:27.608
onto the x and the negative a.

00:02:27.608 --> 00:02:29.466
So x plus a times x,

00:02:29.466 --> 00:02:32.252
or we could say x times x plus a,

00:02:32.252 --> 00:02:36.861
that's going to be x

00:02:36.861 --> 00:02:40.761
times x plus a,

00:02:40.761 --> 00:02:43.420
and then we're going to have

00:02:43.420 --> 00:02:47.185
minus a, or this negative
a, times x plus a.

00:02:47.185 --> 00:02:50.617
So minus, and then we're gonna have

00:02:50.617 --> 00:02:53.782
this minus a times x, plus a.

00:02:59.870 --> 00:03:02.493
Notice, all I did is I distributed

00:03:02.493 --> 00:03:05.511
this yellow, I distributed this big chunk

00:03:05.511 --> 00:03:08.078
of this expression, I just distributed it

00:03:08.078 --> 00:03:10.969
onto the x and onto this negative a.

00:03:10.969 --> 00:03:12.233
I'm multiplying it times the x

00:03:12.233 --> 00:03:14.591
and I'm multiplying it by the negative a.

00:03:14.591 --> 00:03:17.306
And now we can apply the
distributive property again.

00:03:17.306 --> 00:03:20.593
X times x is x squared,

00:03:20.593 --> 00:03:24.890
x times a is ax,

00:03:24.890 --> 00:03:28.267
and then we get

00:03:28.267 --> 00:03:32.192
negative a times x is negative ax,

00:03:32.192 --> 00:03:35.522
and then negative a times a, is negative

00:03:35.522 --> 00:03:37.044
a squared.

00:03:37.044 --> 00:03:39.609
And notice, regardless of my choice of a,

00:03:39.609 --> 00:03:42.163
I'm going to have ax and then minus ax.

00:03:42.163 --> 00:03:44.229
So this is always going to cancel out.

00:03:44.229 --> 00:03:46.865
It didn't just work for
the case when a was three.

00:03:46.865 --> 00:03:49.292
For any a, if I have a times x and then

00:03:49.292 --> 00:03:53.483
I subtract a times x, that's
just going to cancel out.

00:03:53.483 --> 00:03:56.130
So this is just going to cancel out,

00:03:56.130 --> 00:03:57.871
and what are we going to be left with?

00:03:57.871 --> 00:04:01.365
We are going to be left with x squared

00:04:01.365 --> 00:04:03.530
minus a squared.

00:04:06.430 --> 00:04:07.966
And you can view this as a special case.

00:04:07.966 --> 00:04:10.066
When you have something, x plus something,

00:04:10.066 --> 00:04:12.656
times x minus that same something,

00:04:12.656 --> 00:04:14.676
it's going to be x squared minus that

00:04:14.676 --> 00:04:16.347
something squared.

00:04:16.347 --> 00:04:20.236
And this is a good one to know in general.

00:04:20.236 --> 00:04:22.625
And we could use it to quickly figure out

00:04:22.625 --> 00:04:23.971
the products of other binomials that

00:04:23.971 --> 00:04:25.922
fit this pattern here.

00:04:25.922 --> 00:04:28.546
So if I were to say, quick, what is

00:04:28.546 --> 00:04:33.383
x plus 10, times x minus 10?

00:04:34.456 --> 00:04:35.948
Well, you could say, all right

00:04:35.948 --> 00:04:37.992
this fits the pattern, it's x plus a

00:04:37.992 --> 00:04:39.743
times x minus a,

00:04:39.743 --> 00:04:42.704
so it's going to be x
squared minus a squared.

00:04:42.704 --> 00:04:46.201
If a is 10, a squared is

00:04:46.201 --> 00:04:48.450
going to be 100.

00:04:48.450 --> 00:04:49.703
So you can do it really quick once you

00:04:49.703 --> 00:04:52.075
recognize the pattern.