Multiplying binomials by polynomials challenge

WEBVTT
Kind: captions
Language: en

00:00:00.259 --> 00:00:02.442
- [Voiceover] So we've got
the expression two X plus four

00:00:02.442 --> 00:00:05.971
times five X minus nine
is equal to AX squared

00:00:05.971 --> 00:00:08.525
plus BX minus 36.

00:00:08.525 --> 00:00:10.429
And what we want to figure out is what are

00:00:10.429 --> 00:00:12.909
A and B going to be?

00:00:12.909 --> 00:00:14.139
And I encourage you to pause the video

00:00:14.139 --> 00:00:15.527
and try to figure it out.

00:00:16.206 --> 00:00:18.249
Well there's a coupe of
ways of trying to tackle it

00:00:18.249 --> 00:00:19.805
and the most straightforward would be

00:00:19.805 --> 00:00:21.639
just let's multiply these two binomials

00:00:21.639 --> 00:00:23.519
on the left-hand side
and let's see if we can

00:00:23.519 --> 00:00:26.375
match up the terms and
match up the coefficients.

00:00:26.375 --> 00:00:28.140
So let's multiply this left-hand side.

00:00:28.140 --> 00:00:30.788
There's a couple of ways to
think about tackling this.

00:00:30.788 --> 00:00:32.459
I like to think about it as applying

00:00:32.459 --> 00:00:34.317
the distributive property twice.

00:00:34.317 --> 00:00:36.476
So this expression on the left-hand side

00:00:36.476 --> 00:00:38.798
we can rewrite it, or one
way to think about it is

00:00:38.798 --> 00:00:41.399
we distribute the entire two X plus four

00:00:41.399 --> 00:00:43.303
onto the five X minus nine.

00:00:43.303 --> 00:00:45.103
So this is the same
thing as two X plus four

00:00:45.103 --> 00:00:48.517
times five X, plus two X plus four

00:00:48.517 --> 00:00:51.698
times negative nine.

00:00:51.698 --> 00:00:52.997
Or we could write it like this.

00:00:52.997 --> 00:00:53.910
It is

00:00:55.215 --> 00:00:56.344
five X

00:00:57.277 --> 00:01:00.635
five X times two X plus four.

00:01:01.314 --> 00:01:03.705
Two X plus four.

00:01:03.705 --> 00:01:05.238
And then we could either view this as a

00:01:05.238 --> 00:01:07.833
plus negative nine or just a minus nine.

00:01:08.767 --> 00:01:11.089
Minus nine times, once again,

00:01:11.089 --> 00:01:12.459
two X plus four.

00:01:12.459 --> 00:01:14.734
Two X plus four.

00:01:14.734 --> 00:01:16.128
And all we've done is we've distributed

00:01:16.128 --> 00:01:19.146
this two X plus four onto
the five X minus nine.

00:01:19.146 --> 00:01:20.980
Well now when we write it like this,

00:01:20.980 --> 00:01:23.186
when we look at the five X
times the two X plus four,

00:01:23.186 --> 00:01:25.369
we can distribute the five X.

00:01:25.369 --> 00:01:28.875
We can distribute the five
X onto the two X plus four.

00:01:28.875 --> 00:01:31.104
So what's five X times two X?

00:01:31.104 --> 00:01:34.378
Well that's going to be 10 X squared,

00:01:34.378 --> 00:01:38.395
five X times four is plus 20 X.

00:01:38.395 --> 00:01:40.298
Plus 20 X.

00:01:40.298 --> 00:01:42.597
And then we have,

00:01:42.597 --> 00:01:45.477
and then we have negative nine

00:01:45.477 --> 00:01:49.245
times two X is going to be negative 18 X.

00:01:49.971 --> 00:01:51.897
Negative 18 X.

00:01:51.897 --> 00:01:55.543
And then you have negative nine times four

00:01:55.543 --> 00:01:58.353
is negative 36.

00:01:58.353 --> 00:02:00.234
And now we can simplify this a little bit.

00:02:00.234 --> 00:02:01.859
We have two first degree terms.

00:02:01.859 --> 00:02:04.297
So let's see, we have 10 X squared,

00:02:04.297 --> 00:02:06.016
10 X squared,

00:02:06.016 --> 00:02:08.476
and then these two first degree terms,

00:02:08.476 --> 00:02:09.846
let me circle them.

00:02:09.846 --> 00:02:11.983
So we have these two first degree terms.

00:02:11.983 --> 00:02:14.792
If I have 20 Xs and i were
to take away 18 of those

00:02:14.792 --> 00:02:17.138
Xs I'm going to have two X left over.

00:02:17.138 --> 00:02:19.715
20 minus 18, two X.

00:02:19.715 --> 00:02:21.549
And then of course we still have the

00:02:21.549 --> 00:02:22.803
minus 36.

00:02:22.803 --> 00:02:24.289
Now all I've been doing so far is

00:02:24.289 --> 00:02:27.888
simplifying or rewriting
the left-hand side.

00:02:27.888 --> 00:02:29.142
We have to remember, this was an equation

00:02:29.142 --> 00:02:31.975
so this needs to be equal
to the right-hand side.

00:02:31.975 --> 00:02:35.345
So this is going to be
equal to AX squared.

00:02:36.108 --> 00:02:39.227
So AX squared

00:02:40.115 --> 00:02:42.143
plus BX,

00:02:42.143 --> 00:02:45.525
plus BX minus 36.

00:02:45.525 --> 00:02:46.779
Minus 36.

00:02:46.779 --> 00:02:47.963
And now that I've wrote it

00:02:47.963 --> 00:02:50.401
a little color-coded it
might jump out at you

00:02:50.401 --> 00:02:52.537
what A and B are going to be.

00:02:52.537 --> 00:02:55.974
We have 10 X squared over here

00:02:55.974 --> 00:02:57.971
and then the second degree
term on the right-hand side

00:02:57.971 --> 00:02:59.502
is AX squared.

00:02:59.502 --> 00:03:02.126
So 10 must be equal to A,

00:03:02.126 --> 00:03:03.915
or these two coefficients must be equal.

00:03:03.915 --> 00:03:06.306
So we could write A is equal to 10.

00:03:06.306 --> 00:03:09.000
And then when we look at
the first degree term,

00:03:09.000 --> 00:03:12.251
we have two X here and we
have BX right over here.

00:03:12.251 --> 00:03:14.921
And so two must be equal to B,

00:03:14.921 --> 00:03:17.359
or B must be equal to two.

00:03:17.359 --> 00:03:20.006
And it all worked out
that our constant terms

00:03:20.006 --> 00:03:22.235
are the same on both sides.

00:03:22.235 --> 00:03:25.044
So there we have it, A
equals 10, B equals two.

00:03:25.044 --> 00:03:26.507
Now once you're practiced at this

00:03:26.507 --> 00:03:28.516
you might be able to
say well how can I get

00:03:28.516 --> 00:03:29.989
a faster way to do this?

00:03:29.989 --> 00:03:31.628
Although it might be a
little bit more prone

00:03:31.628 --> 00:03:33.484
to careless mistakes.

00:03:33.484 --> 00:03:35.134
Is you could say well how can I get,

00:03:35.134 --> 00:03:37.176
how do I get an X squared?

00:03:37.176 --> 00:03:39.498
How do I, when I multiply
these things out,

00:03:39.498 --> 00:03:40.358
how do I get an X squared?

00:03:40.358 --> 00:03:41.960
Well the only way that
I can get an X squared

00:03:41.960 --> 00:03:45.606
is when I multiply the
two X times the five X.

00:03:45.606 --> 00:03:47.940
And that's going to be 10 X squared.

00:03:48.828 --> 00:03:49.895
And then you could say, alright,

00:03:49.895 --> 00:03:51.893
A is going to be equal to 10.

00:03:51.893 --> 00:03:54.029
And then you could say
how could I get an X?

00:03:54.029 --> 00:03:56.003
Well there's two ways that you can get,

00:03:56.003 --> 00:03:57.674
two ways to get an X.

00:03:57.674 --> 00:04:01.040
You could multiply two X times

00:04:01.040 --> 00:04:02.713
negative nine.

00:04:02.713 --> 00:04:05.707
So that would be negative 18 X.

00:04:05.707 --> 00:04:08.795
Or you could multiply four times five X,

00:04:08.795 --> 00:04:11.698
which is going to be plus 20 X.

00:04:11.698 --> 00:04:13.928
If you add these two
together you're going to get,

00:04:13.928 --> 00:04:16.389
they're going to be equal to two X.

00:04:16.389 --> 00:04:18.131
So two.

00:04:18.131 --> 00:04:22.658
Two X, BX, B must be equal to two.

00:04:22.658 --> 00:04:23.865
And then you can just check,

00:04:23.865 --> 00:04:25.050
well how am I gonna get a constant term?

00:04:25.050 --> 00:04:26.907
Well I have to multiply
these two constant terms.

00:04:26.907 --> 00:04:29.531
Four times negative nine,
you're gonna get negative 36.

00:04:29.531 --> 00:04:30.993
So the second way I just did it will be

00:04:30.993 --> 00:04:32.132
a little bit faster, you're a little bit

00:04:32.132 --> 00:04:34.198
more prone to making careless mistakes,

00:04:34.198 --> 00:04:35.243
but hopefully you appreciate that,

00:04:35.243 --> 00:04:37.356
I'm really just doing the same thing.

00:04:37.356 --> 00:04:40.300
Maybe with different levels of clarity.


Multiplying binomials with distributive property

WEBVTT
Kind: captions
Language: en

00:00:00.190 --> 00:00:02.256
- [Voiceover] Let's see
if we can find the product

00:00:02.256 --> 00:00:05.437
of X minus four, and I'll
put that in parentheses,

00:00:05.437 --> 00:00:09.431
the product of that and X plus seven.

00:00:09.431 --> 00:00:11.939
And like always, I encourage
you to pause the video

00:00:11.939 --> 00:00:14.185
and see if you can figure
it out on your own.

00:00:15.120 --> 00:00:17.140
Well there are a bunch of
different ways to tackle

00:00:17.140 --> 00:00:19.344
multiplying two binomials like this,

00:00:19.344 --> 00:00:21.017
but the way that I like to tackle it,

00:00:21.017 --> 00:00:23.083
especially if we're just
learning how to do this

00:00:23.083 --> 00:00:24.638
is to just realized that we can apply

00:00:24.638 --> 00:00:27.889
the distributive property
that we all know and love.

00:00:27.889 --> 00:00:29.260
And how do you do that?

00:00:29.260 --> 00:00:32.766
Well you could just think
of this X minus four,

00:00:32.766 --> 00:00:35.111
we can distribute the entire binomial.

00:00:35.111 --> 00:00:38.223
We can multiply the X
minus four times the X

00:00:38.223 --> 00:00:41.079
and we can multiply it times the seven.

00:00:41.079 --> 00:00:42.379
So let me rewrite this.

00:00:42.379 --> 00:00:45.119
We can rewrite this as,
if we distribute it,

00:00:45.119 --> 00:00:48.980
it would be X minus four times this X,

00:00:50.031 --> 00:00:53.073
times that blue X and then plus,

00:00:53.073 --> 00:00:54.954
we're also distributing the X minus four,

00:00:54.954 --> 00:00:56.973
we have to distribute it on
every one of these terms.

00:00:56.973 --> 00:00:59.959
So plus X minus four,

00:01:01.008 --> 00:01:04.794
X minus four times this seven.

00:01:04.794 --> 00:01:06.975
Times the seven.

00:01:06.975 --> 00:01:10.645
Now what is this going to
be on the left-hand side?

00:01:10.645 --> 00:01:15.173
Well the X times the X
is going to be X squared.

00:01:15.173 --> 00:01:18.005
And then the X times, this is minus four,

00:01:18.005 --> 00:01:20.443
or X times the negative four is

00:01:20.443 --> 00:01:22.533
negative four X.

00:01:22.533 --> 00:01:25.969
And then over here I distribute the seven.

00:01:25.969 --> 00:01:28.476
Seven times X is seven X.

00:01:28.476 --> 00:01:30.938
So plus seven X.

00:01:30.938 --> 00:01:35.700
And then seven times
negative four is negative 28.

00:01:36.412 --> 00:01:37.831
And now we can simplify this a little bit.

00:01:37.831 --> 00:01:40.269
We see that we have
two first degree terms,

00:01:40.269 --> 00:01:43.543
two terms where I have
a certain number of Xs

00:01:43.543 --> 00:01:45.349
and so we can combine these.

00:01:45.349 --> 00:01:47.103
So we have X squared.

00:01:47.920 --> 00:01:50.219
If I take these two middle terms,

00:01:50.219 --> 00:01:54.375
if I take negative four
X and then add seven Xs,

00:01:54.375 --> 00:01:56.697
that's going to give me positive three X.

00:01:56.697 --> 00:01:59.995
Negative four plus
seven is positive three.

00:01:59.995 --> 00:02:02.061
So positive three X.

00:02:02.061 --> 00:02:04.614
And of course we have the,

00:02:04.614 --> 00:02:07.030
we have the minus 28.

00:02:07.030 --> 00:02:07.959
And we're done.

00:02:07.959 --> 00:02:11.233
We just multiplied these two binomials

00:02:11.233 --> 00:02:13.856
just using the distributive
property twice.

00:02:13.856 --> 00:02:16.271
First we distributed
the entire X minus four

00:02:16.271 --> 00:02:17.920
onto the X plus seven, so you got

00:02:17.920 --> 00:02:21.890
X minus four times X plus X
minus four times the seven.

00:02:21.890 --> 00:02:24.816
And then we distributed
the X onto the X minus four

00:02:24.816 --> 00:02:27.021
to get what we see in yellow here.

00:02:27.021 --> 00:02:29.528
And we distributed the
seven onto the X minus four

00:02:29.528 --> 00:02:31.479
to get what we got in green here.

00:02:31.479 --> 00:02:33.845
And then we just simplified a bit.


Multiplying binomials by polynomials: area model

WEBVTT
Kind: captions
Language: en

00:00:00.459 --> 00:00:01.667
- [Voiceover] What I
wanna do in this video

00:00:01.667 --> 00:00:05.126
is figure out multiple
ways to express the area

00:00:05.126 --> 00:00:07.402
of the entire large rectangle,

00:00:07.402 --> 00:00:11.535
which we see is made up of
these six smaller rectangles.

00:00:11.535 --> 00:00:13.369
So there's a couple of
ways that we can do it.

00:00:13.369 --> 00:00:15.104
One way is, we can just
multiply the height

00:00:15.104 --> 00:00:18.942
of this big rectangle times the
width of this big rectangle.

00:00:18.942 --> 00:00:20.242
So what's its height?

00:00:20.242 --> 00:00:22.842
Well, from here to here,

00:00:22.842 --> 00:00:25.420
that distance is going to be y squared,

00:00:25.420 --> 00:00:27.091
and then from there to there,

00:00:27.091 --> 00:00:30.481
that distance is going to be negative 6y.

00:00:30.481 --> 00:00:31.387
And I know what you're thinking.

00:00:31.387 --> 00:00:34.080
How can my distance be negative 6y?

00:00:34.080 --> 00:00:36.379
Isn't a distance always positive?

00:00:36.379 --> 00:00:39.342
Well, even negative 6y can be positive

00:00:39.342 --> 00:00:41.802
if y is negative,

00:00:41.802 --> 00:00:43.020
so it's completely reasonable to say,

00:00:43.020 --> 00:00:45.597
well, this distance could be negative 6y.

00:00:45.597 --> 00:00:49.621
So the entire height right
over here is going to be,

00:00:49.621 --> 00:00:53.986
it's going to be y squared minus 6y.

00:00:53.986 --> 00:00:57.540
Or you could do it as y
squared plus this distance,

00:00:57.540 --> 00:01:00.976
which is negative 6y, y
squared plus negative 6y,

00:01:00.976 --> 00:01:03.959
which is the same thing
as y squared minus 6y.

00:01:03.959 --> 00:01:06.705
So that's the height
of this big rectangle.

00:01:06.705 --> 00:01:08.726
What's its width?

00:01:08.726 --> 00:01:11.813
Well, the width is going to be the width

00:01:11.813 --> 00:01:16.410
of this purple rectangle,
it's going to be 3y squared,

00:01:16.410 --> 00:01:19.498
plus the width of this yellow rectangle,

00:01:19.498 --> 00:01:22.795
which is negative 2y, and that
can have a negative out here,

00:01:22.795 --> 00:01:24.769
the same logic why this
could have a negative,

00:01:24.769 --> 00:01:26.812
why the negative 6y could have a negative,

00:01:26.812 --> 00:01:29.437
and then plus the width

00:01:29.437 --> 00:01:31.847
of the blue rectangle.

00:01:31.847 --> 00:01:33.454
And so if you add them all together,

00:01:33.454 --> 00:01:36.476
the width of the entire rectangle is going

00:01:36.476 --> 00:01:41.394
to be 3y squared minus 2y,

00:01:41.394 --> 00:01:43.848
minus 2y plus one.

00:01:44.738 --> 00:01:47.594
And just like that, this
expression that I just wrote down

00:01:47.594 --> 00:01:51.401
will give us the area for the entire,

00:01:51.401 --> 00:01:54.884
the area for the entire big rectangle.

00:01:54.884 --> 00:01:56.138
Now, there's another way to do it,

00:01:56.138 --> 00:01:59.366
and a big clue was that we
subdivided the big rectangle

00:01:59.366 --> 00:02:01.595
into these six smaller rectangles,

00:02:01.595 --> 00:02:04.311
and we have the dimensions for
the six smaller rectangles.

00:02:04.311 --> 00:02:06.006
And so we could find the
area for each of these,

00:02:06.006 --> 00:02:08.167
and then we could add them all together.

00:02:08.167 --> 00:02:09.629
So let's look at this first one.

00:02:09.629 --> 00:02:10.837
Height times width.

00:02:10.837 --> 00:02:13.297
The area of this purple
rectangle is gonna be the height,

00:02:13.297 --> 00:02:16.245
y squared, times the width,

00:02:16.245 --> 00:02:18.605
which is 3y squared,

00:02:18.605 --> 00:02:21.447
which is going to be equal
to, it's gonna be three,

00:02:21.447 --> 00:02:25.859
and then y squared times y
squared is y to the fourth power.

00:02:25.859 --> 00:02:27.275
What's the area of this yellow rectangle?

00:02:27.275 --> 00:02:28.877
Height is y squared.

00:02:28.877 --> 00:02:31.223
It's going to be y
squared times the width,

00:02:31.223 --> 00:02:33.359
times negative 2y,

00:02:33.359 --> 00:02:37.886
which is going to give us
negative 2y to the third power.

00:02:37.886 --> 00:02:39.117
What about the blue one?

00:02:39.117 --> 00:02:42.949
Well, height times width, it's
gonna be y squared times one,

00:02:42.949 --> 00:02:46.338
which, of course, is just
going to be equal to y squared.

00:02:46.338 --> 00:02:48.474
Now, this green one,
it's gonna be the height,

00:02:48.474 --> 00:02:52.746
which is now negative 6y, times the width,

00:02:52.746 --> 00:02:57.274
which is 3y squared, which
is going to be equal to,

00:02:57.274 --> 00:03:01.012
let's see, negative six
times three is negative 18,

00:03:01.012 --> 00:03:05.656
and then y times y squared
is y to the third power.

00:03:05.656 --> 00:03:08.977
Now, the area of this gray
rectangle is gonna be the height,

00:03:08.977 --> 00:03:12.869
which is negative 6y, times the width,

00:03:12.869 --> 00:03:15.818
which is negative 2y,

00:03:15.818 --> 00:03:18.488
which gets us negative
six times negative two

00:03:18.488 --> 00:03:20.225
is positive 12,

00:03:20.225 --> 00:03:23.755
y times y is y squared.

00:03:23.755 --> 00:03:26.591
And then finally the area of
this rectangle right over here,

00:03:26.591 --> 00:03:30.259
it's gonna be the height,
which is negative 6y,

00:03:30.259 --> 00:03:32.907
times the width, which is just one,

00:03:32.907 --> 00:03:35.995
which is equal to negative 6y.

00:03:35.995 --> 00:03:38.410
And so if we want the area
of this entire rectangle,

00:03:38.410 --> 00:03:41.034
we can just add up the
areas of the smaller ones,

00:03:41.034 --> 00:03:44.401
so it's going to be equal to the three,

00:03:44.401 --> 00:03:47.976
it's going to be equal
to the 3y to the fourth,

00:03:47.976 --> 00:03:51.272
3y to the fourth,

00:03:51.272 --> 00:03:55.406
plus negative 2y to the third power.

00:03:55.406 --> 00:03:59.307
Let me write this in a color
that corresponds to that.

00:03:59.307 --> 00:04:03.278
Negative 2y to the third power,

00:04:03.278 --> 00:04:05.234
plus y squared,

00:04:06.644 --> 00:04:10.801
plus y squared,

00:04:10.801 --> 00:04:14.259
minus 18y to the third power,

00:04:14.259 --> 00:04:18.765
so minus 18y to the third power,

00:04:18.765 --> 00:04:21.783
plus 12y squared.

00:04:21.783 --> 00:04:23.362
Let's write that in black.

00:04:23.362 --> 00:04:26.404
So plus 12y squared,

00:04:26.404 --> 00:04:30.607
and then last but not
least, we have the minus 6y,

00:04:30.607 --> 00:04:32.302
minus 6y.

00:04:32.302 --> 00:04:35.134
So this is an expression for
the area of the entire thing,

00:04:35.134 --> 00:04:36.946
but we can simplify it more.

00:04:36.946 --> 00:04:40.800
So let's see, we only have
one fourth degree term,

00:04:40.800 --> 00:04:42.843
so I'll just rewrite that.

00:04:42.843 --> 00:04:47.719
So we have one fourth degree term,

00:04:47.719 --> 00:04:48.740
so I'll just rewrite that.

00:04:48.740 --> 00:04:50.877
3y to the fourth power.

00:04:50.877 --> 00:04:53.500
Now, how many third
degree terms do we have?

00:04:53.500 --> 00:04:57.309
We have negative 2y to the third power.

00:04:57.309 --> 00:05:00.676
We have negative 18y to the third power.

00:05:00.676 --> 00:05:01.953
So if we add these two together,

00:05:01.953 --> 00:05:03.671
how many y to the third powers do we have?

00:05:03.671 --> 00:05:06.782
Well, negative two plus
negative 18 is negative 20,

00:05:06.782 --> 00:05:10.438
negative 20y to the third power.

00:05:10.438 --> 00:05:13.913
And then how many second
degree terms do we have?

00:05:13.913 --> 00:05:16.117
Well, we have one y
squared right over here,

00:05:16.117 --> 00:05:18.253
and then we have another, 12y squareds.

00:05:18.253 --> 00:05:22.804
You add those together, you're
gonna have 13y squareds.

00:05:22.804 --> 00:05:27.703
And then finally we still
need to subtract the 6y.

00:05:27.703 --> 00:05:28.841
And there you have it,

00:05:28.841 --> 00:05:32.835
another expression for the
area of the entire rectangle.

00:05:32.835 --> 00:05:34.716
And the whole point of
doing this is to realize

00:05:34.716 --> 00:05:38.083
that this up here and this
down here are equivalent,

00:05:38.083 --> 00:05:41.147
and that the way that we multiply
this actually corresponds

00:05:41.147 --> 00:05:42.912
to exactly how we found the areas

00:05:42.912 --> 00:05:45.234
of the smaller rectangles right over here.

00:05:45.234 --> 00:05:48.601
You would say y squared times
3y squared is 3y fourth.

00:05:48.601 --> 00:05:51.085
Y squared times negative 2y is negative 2y

00:05:51.085 --> 00:05:52.153
to the third power.

00:05:52.153 --> 00:05:56.681
Y squared times one is y squared,

00:05:56.681 --> 00:05:58.840
which is exactly what we
did when we found the area

00:05:58.840 --> 00:06:00.984
of these rectangles in
this, I guess you could say,

00:06:00.984 --> 00:06:02.114
in this top row.

00:06:02.114 --> 00:06:03.716
And then you would take the negative six,

00:06:03.716 --> 00:06:07.153
and you would say negative
six times 3y squared

00:06:07.153 --> 00:06:09.149
is negative 18y to the third.

00:06:09.149 --> 00:06:13.190
Negative six times negative
2y is positive 12y squared.

00:06:13.190 --> 00:06:17.313
Negative 6y times one

00:06:17.313 --> 00:06:19.903
is negative 6y.

00:06:19.903 --> 00:06:21.726
And just to realize that
this isn't just some type

00:06:21.726 --> 00:06:22.707
of voodoo that we're doing.

00:06:22.707 --> 00:06:24.821
It completely makes sense
when you think about

00:06:24.821 --> 00:06:27.753
in terms of an area model like this.


Multiplying binomials: area model

WEBVTT
Kind: captions
Language: en

00:00:00.576 --> 00:00:02.770
- [Voiceover] So I have this
big rectangle here that's

00:00:02.770 --> 00:00:05.592
divided into four smaller
rectangles, and what I wanna

00:00:05.592 --> 00:00:09.238
do is I wanna express the
area of this larger rectangle

00:00:09.238 --> 00:00:10.933
and I wanna do it two ways.

00:00:10.933 --> 00:00:13.591
The first way I wanna
express it as the product

00:00:13.591 --> 00:00:16.969
of two binomials and
then I wanna express it

00:00:16.969 --> 00:00:20.313
as a trinomial, so let's
think about this a little bit.

00:00:20.313 --> 00:00:23.447
So one way to say well look,
the height of this larger

00:00:23.447 --> 00:00:27.951
rectangle from here to here,
we see that that distance is x,

00:00:27.951 --> 00:00:30.528
and then from here to here it's two,

00:00:30.528 --> 00:00:33.512
so the entire height right over here,

00:00:33.512 --> 00:00:38.133
the entire height right over
here is going to be x plus two.

00:00:38.133 --> 00:00:42.033
So the height is x plus
two, and what's the width?

00:00:42.033 --> 00:00:46.908
Well the width is, we go
from there to there is x,

00:00:46.908 --> 00:00:50.446
and then from there to there is three,

00:00:50.446 --> 00:00:53.521
so the entire width is x plus three.

00:00:53.521 --> 00:00:55.239
x plus three.

00:00:55.239 --> 00:00:57.365
So just like that, I've
expressed the area of the

00:00:57.365 --> 00:01:01.126
entire rectangle, and it's
the product of two binomials.

00:01:01.126 --> 00:01:03.901
But now let's express it as a trinomial.

00:01:03.901 --> 00:01:06.559
Well to do that, we can
break down the larger area

00:01:06.559 --> 00:01:09.624
into the areas of each of
these smaller rectangles.

00:01:09.624 --> 00:01:13.223
So what's the area of this
purple rectangle right over here?

00:01:13.223 --> 00:01:15.777
Well the purple rectangle, its height is x

00:01:15.777 --> 00:01:20.502
and its width is x, so
its area is x squared.

00:01:20.502 --> 00:01:23.520
Let me write that, that's x squared.

00:01:23.520 --> 00:01:27.235
What's the area of this yellow rectangle?

00:01:27.235 --> 00:01:30.799
Well it's height is x, same
height as right over here,

00:01:30.799 --> 00:01:34.316
its height is x and its width is three,

00:01:34.316 --> 00:01:38.843
so it's gonna be x times three, or 3x.

00:01:38.843 --> 00:01:41.258
It'll have an area of 3x.

00:01:41.258 --> 00:01:44.115
So that area is 3x, if
we're summing up the area

00:01:44.115 --> 00:01:47.156
of the entire thing,
this would be plus 3x.

00:01:47.156 --> 00:01:49.083
So this expression right
over here, that's the area

00:01:49.083 --> 00:01:52.241
of this purple region, plus
the area of this yellow region,

00:01:52.241 --> 00:01:54.156
and then we can move on
to this green region.

00:01:54.156 --> 00:01:56.083
What's the area going to be here?

00:01:56.083 --> 00:01:58.893
Well the height is two and the width is x,

00:01:58.893 --> 00:02:03.235
so multiply height times
width is gonna be two times x,

00:02:03.235 --> 00:02:07.356
and we can just add
that, plus two times x,

00:02:07.356 --> 00:02:10.757
and then finally this
little grey box here.

00:02:10.757 --> 00:02:13.578
Its height is two, we see
that right over there,

00:02:13.578 --> 00:02:18.321
its height is two, and its width is three,

00:02:18.321 --> 00:02:21.920
we see it right over there,
so it has an area of six.

00:02:21.920 --> 00:02:23.661
Two times three.

00:02:23.661 --> 00:02:26.274
So plus six, and you might say
well this isn't a trinomial,

00:02:26.274 --> 00:02:28.455
this has four terms right over here,

00:02:28.455 --> 00:02:31.219
but you might notice that we can add,

00:02:31.219 --> 00:02:34.574
that we can add these middle two terms,

00:02:34.574 --> 00:02:38.520
3x plus 2x, if I have three
x's and I add two x's to that,

00:02:38.520 --> 00:02:40.518
I'm gonna have five x's.

00:02:40.518 --> 00:02:44.941
So this entire thing
simplifies to x squared,

00:02:44.941 --> 00:02:49.886
x squared plus 5x plus six.

00:02:49.886 --> 00:02:51.646
Plus six.

00:02:51.646 --> 00:02:54.978
So this and this are two
ways of expressing the area,

00:02:54.978 --> 00:02:57.323
so they're going to be
equal, and that makes sense

00:02:57.323 --> 00:02:59.667
'cause if you multiply
it out, these binomials,

00:02:59.667 --> 00:03:01.757
and simplify it, you
would get this trinomial,

00:03:01.757 --> 00:03:03.022
and we can do that really fast.

00:03:03.022 --> 00:03:06.285
You multiply the x times the
x, actually let me do that

00:03:06.285 --> 00:03:07.747
in the same colors.

00:03:07.747 --> 00:03:11.632
You multiply the x times the x,

00:03:11.632 --> 00:03:13.907
you get the x squared.

00:03:13.907 --> 00:03:18.046
You multiply this x times the three,

00:03:18.046 --> 00:03:20.565
you get your 3x.

00:03:20.565 --> 00:03:24.792
You multiply the two times the x,

00:03:24.792 --> 00:03:26.775
you get your 2x.

00:03:26.775 --> 00:03:31.176
And then you multiply
the two times the three

00:03:31.176 --> 00:03:33.591
and you get your six.

00:03:33.591 --> 00:03:36.330
So what this, I guess you
can say this area model

00:03:36.330 --> 00:03:39.918
does for us is it hopefully
makes a visual representation

00:03:39.918 --> 00:03:43.760
of why it makes sense to
multiply binomials the way we do,

00:03:43.760 --> 00:03:45.711
and in other videos we talk
about it as applying the

00:03:45.711 --> 00:03:48.890
distribution property
twice, but this gives you

00:03:48.890 --> 00:03:51.073
a more visual representation for

00:03:51.073 --> 00:03:53.723
why it actually makes sense.


Multiplying monomials by polynomials challenge

WEBVTT
Kind: captions
Language: en

00:00:00.388 --> 00:00:01.665
- [Voiceover] So we have
an equation right here

00:00:01.665 --> 00:00:04.741
that says negative two-y
times the expression

00:00:04.741 --> 00:00:07.585
y-squared plus c-y minus three

00:00:07.585 --> 00:00:12.585
is equal to d-y to the third
plus 12-y-squared plus f-y.

00:00:12.600 --> 00:00:14.748
And what I'd like you to do
is pause the video and see

00:00:14.748 --> 00:00:19.748
if you can figure out what
the variables c, d, and f are.

00:00:21.423 --> 00:00:23.032
Alright, so now let's work through it.

00:00:23.032 --> 00:00:24.483
So at first this could
look kind of intimidating.

00:00:24.483 --> 00:00:26.468
We just have an equation,
you're trying to figure out

00:00:26.468 --> 00:00:29.986
what these three variables
are, how do you do it?

00:00:29.986 --> 00:00:32.772
And one reasonable approach
would be let's just try

00:00:32.772 --> 00:00:35.546
to simplify what we have
here on the left hand side.

00:00:35.546 --> 00:00:37.984
And to simplify that
we can just distribute

00:00:37.984 --> 00:00:41.255
the negative two-y onto this polynomial,

00:00:41.255 --> 00:00:44.041
onto the y-squared plus c-y minus three.

00:00:44.041 --> 00:00:45.376
And then we could set it equal to what

00:00:45.376 --> 00:00:46.653
we have here on the right and let's see

00:00:46.653 --> 00:00:48.673
if we can match up coefficients.

00:00:48.673 --> 00:00:50.112
So let's distribute it.

00:00:50.112 --> 00:00:51.394
So first we could think about what

00:00:51.394 --> 00:00:54.853
negative two-y times
y-squared is going to be.

00:00:54.853 --> 00:00:57.814
Negative two-y times y-squared,

00:00:57.814 --> 00:01:02.814
well that's going to be negative
two-y to the third power.

00:01:03.130 --> 00:01:07.484
Cause y to the first times
y-squared is y to the third power

00:01:07.484 --> 00:01:12.484
Alright, now let's multiply
the negative two-y times c-y.

00:01:14.246 --> 00:01:17.189
Well that's gonna be negative two times c.

00:01:17.189 --> 00:01:22.109
So negative two-c and then
you're gonna have y times y.

00:01:22.109 --> 00:01:25.348
So negative two-c-y-squared.

00:01:25.348 --> 00:01:29.724
And then we would multiply
the negative two-y times,

00:01:29.724 --> 00:01:31.710
and we gotta keep in mind this is a,

00:01:31.710 --> 00:01:33.103
we're subtracting three here,

00:01:33.103 --> 00:01:34.507
so we could view that as a negative three.

00:01:34.507 --> 00:01:37.363
Negative two-y times negative three,

00:01:37.363 --> 00:01:41.264
negative two times negative
three is positive six,

00:01:41.264 --> 00:01:43.319
and we still have that y over there.

00:01:43.319 --> 00:01:45.188
So there you have it we have simplified

00:01:45.188 --> 00:01:48.403
the left hand side of this equation.

00:01:48.403 --> 00:01:50.981
And now let's see if, let
me actually just write

00:01:50.981 --> 00:01:52.757
the right hand side in different colors

00:01:52.757 --> 00:01:55.102
and then things might jump out at us.

00:01:55.102 --> 00:01:58.248
So over her I wrote this,
the third degree term,

00:01:58.248 --> 00:02:00.279
the y to the third term,
I wrote that in blue,

00:02:00.279 --> 00:02:03.240
so let me write the y to the
third term here in blue as well

00:02:03.240 --> 00:02:05.945
So d-y to the third.

00:02:05.945 --> 00:02:08.719
And then I wrote the
y-squared term in magenta

00:02:08.719 --> 00:02:11.041
so let me write the y-squared
term in magenta here.

00:02:11.041 --> 00:02:13.966
So plus 12-y-squared.

00:02:13.966 --> 00:02:18.204
And then last but not least I
wrote the first degree term,

00:02:18.204 --> 00:02:21.257
this y term in green, so let
me write the first degree term

00:02:21.257 --> 00:02:25.355
in green right over here, so plus f-y.

00:02:25.355 --> 00:02:27.502
And when you see it like
that you see which terms

00:02:27.502 --> 00:02:29.743
match up with (stutters).

00:02:29.743 --> 00:02:33.504
With which terms match
up to which other terms.

00:02:33.504 --> 00:02:34.979
So we could see that, look,

00:02:34.979 --> 00:02:36.581
I have the third degree term here.

00:02:36.581 --> 00:02:39.123
That has to match up to this
third degree term there.

00:02:39.123 --> 00:02:44.123
So d, d needs to be equal to negative two.

00:02:44.197 --> 00:02:45.393
So let's write that down,

00:02:45.393 --> 00:02:48.344
d is equal to negative two.

00:02:48.344 --> 00:02:50.114
We could see the second degree term.

00:02:50.114 --> 00:02:51.554
This second degree term matches up

00:02:51.554 --> 00:02:54.108
to this second degree
term right over there.

00:02:54.108 --> 00:02:57.695
So that tells us that
this coefficient, the 12,

00:02:57.695 --> 00:03:01.073
must be equal to the negative two-c.

00:03:01.073 --> 00:03:05.578
So let's write that down,
negative two-c is equal to 12.

00:03:05.578 --> 00:03:09.954
To solve for c we can divide
both sides by negative two.

00:03:09.954 --> 00:03:12.694
And we get c is equal to 12 divided

00:03:12.694 --> 00:03:15.654
by negative two is negative six.

00:03:15.654 --> 00:03:17.709
And that makes sense,
if c is negative six,

00:03:17.709 --> 00:03:19.369
negative two times negative six,

00:03:19.369 --> 00:03:21.795
or subtracting two times negative six,

00:03:21.795 --> 00:03:23.908
is subtracting negative 12,

00:03:23.908 --> 00:03:26.613
which would be the same
thing as adding 12.

00:03:26.613 --> 00:03:28.749
So then we get the exact same coefficient

00:03:28.749 --> 00:03:30.932
for the second degree term.

00:03:30.932 --> 00:03:32.870
I think you see where this is going.

00:03:32.870 --> 00:03:35.215
Over here we have a six-y
on the first degree term.

00:03:35.215 --> 00:03:37.491
Here we have an f-y on
the first degree term.

00:03:37.491 --> 00:03:39.673
This f must be the same thing,

00:03:39.673 --> 00:03:42.007
must be equal to the coefficient here.

00:03:42.007 --> 00:03:45.791
So f must be equal to six.

00:03:45.791 --> 00:03:46.929
And we're done.

00:03:46.929 --> 00:03:49.785
And the key realization
here is you match up

00:03:49.785 --> 00:03:51.851
the corresponding degree terms.

00:03:51.851 --> 00:03:53.325
There's no way that,

00:03:53.325 --> 00:03:56.866
well I don't want to get too complex here,

00:03:56.866 --> 00:03:58.909
but the simplest way to address this,

00:03:58.909 --> 00:04:00.569
is to realize, "Okay, I have
a third degree term here,

00:04:00.569 --> 00:04:02.253
"I have a third degree term here.

00:04:02.253 --> 00:04:04.830
"Well if I look at it very
simply the way I'm gonna

00:04:04.830 --> 00:04:06.223
"get this third degree term here

00:04:06.223 --> 00:04:07.825
"is using this third degree term."

00:04:07.825 --> 00:04:09.265
And I just look at the
coefficients and say,

00:04:09.265 --> 00:04:11.923
"Okay, they must have
the same coefficient."

00:04:11.923 --> 00:04:14.721
And then we say, "D is
equal to negative two",

00:04:14.721 --> 00:04:16.555
and then we keep doing that.


Multiplying monomials by polynomials: area model

WEBVTT
Kind: captions
Language: en

00:00:00.391 --> 00:00:02.132
- [Voiceover] We're asked
to express the area of the

00:00:02.132 --> 00:00:05.731
entire rectangle below as a trinomial.

00:00:05.731 --> 00:00:08.169
We have our rectangle here
and it's broken up into

00:00:08.169 --> 00:00:10.979
these three smaller rectangles.

00:00:10.979 --> 00:00:12.983
And we see for all of these rectangles,

00:00:12.983 --> 00:00:17.256
the height here is four
units and then the widths

00:00:17.256 --> 00:00:19.531
are expressed in terms,
or at least the first two,

00:00:19.531 --> 00:00:22.375
are expressed in terms of
x and then this last one

00:00:22.375 --> 00:00:24.233
has a width of two.

00:00:24.233 --> 00:00:26.727
So what's the area of
the entire rectangle?

00:00:26.727 --> 00:00:29.550
I encourage you to pause the
video and think about it.

00:00:29.550 --> 00:00:31.802
What's the area of this blue, this blue,

00:00:31.802 --> 00:00:33.391
it looks like a square,
but let's just call it

00:00:33.391 --> 00:00:35.912
a rectangle, which all
squares are rectangles

00:00:35.912 --> 00:00:37.142
so that's safe.

00:00:37.142 --> 00:00:39.255
Well, it's going to be the
height times the width.

00:00:39.255 --> 00:00:41.531
So the area here is
going to be the height,

00:00:41.531 --> 00:00:43.899
which is four, times the width,

00:00:43.899 --> 00:00:46.987
which is x squared.

00:00:46.987 --> 00:00:51.143
And then to that, we want
to add the area of this,

00:00:51.143 --> 00:00:54.394
I guess we could say this
salmon colored rectangle

00:00:54.394 --> 00:00:56.414
and well that's going
to be the height four

00:00:56.414 --> 00:00:58.283
times the width 3x.

00:00:58.283 --> 00:01:01.139
So we could say four times 3x,

00:01:01.139 --> 00:01:02.532
we could write it like that,

00:01:02.532 --> 00:01:05.457
but what is 4 times 3x?

00:01:05.457 --> 00:01:07.570
Well, that's going to be 12x.

00:01:07.570 --> 00:01:10.589
You have 3x four times, I have 12 xs,

00:01:10.589 --> 00:01:12.191
so that's going to be 12x.

00:01:12.191 --> 00:01:15.279
12x is the area of this
salmon colored rectangle.

00:01:15.279 --> 00:01:17.949
And then, finally, the area
of this green rectangle,

00:01:17.949 --> 00:01:19.748
we actually can figure out
it exactly, we don't even

00:01:19.748 --> 00:01:21.455
have to express it in terms of a variable.

00:01:21.455 --> 00:01:24.265
Its height is four, its width is two,

00:01:24.265 --> 00:01:28.020
so the area's going to be
four times two, or eight.

00:01:28.814 --> 00:01:31.293
And we are done.


Multiplying monomials challenge

WEBVTT
Kind: captions
Language: en

00:00:00.305 --> 00:00:03.371
- [Voiceover] We have three
times X to the power of A

00:00:03.371 --> 00:00:05.855
times BX to the fourth power,

00:00:05.855 --> 00:00:09.337
and we have that equalling
negative 24 X to the sixth.

00:00:09.337 --> 00:00:11.126
And so what I'd like you
to do is pause this video

00:00:11.126 --> 00:00:15.282
and see if you can figure
out what A and B need to be.

00:00:15.282 --> 00:00:17.349
Well to figure this out
we can just multiply out

00:00:17.349 --> 00:00:19.787
this left-hand side, so let's do that.

00:00:19.787 --> 00:00:21.249
So what we have right over here,

00:00:21.249 --> 00:00:26.249
we can just rewrite that as
three times X to the power of A

00:00:26.381 --> 00:00:31.280
times B times X to the fourth.

00:00:31.280 --> 00:00:34.264
And we can rewrite this as,
we can just change the order,

00:00:34.264 --> 00:00:36.448
we're just multiplying
four different things.

00:00:36.448 --> 00:00:40.455
This is the same thing as three times B

00:00:41.342 --> 00:00:44.522
times X to the A,

00:00:44.522 --> 00:00:46.497
times X to the fourth.

00:00:46.497 --> 00:00:48.261
And if you were doing this on your own,

00:00:48.261 --> 00:00:49.445
you wouldn't have to
do all of these steps,

00:00:49.445 --> 00:00:52.325
but hopefully this makes it
clear what's actually going on.

00:00:52.325 --> 00:00:53.997
And what's this going to be?

00:00:53.997 --> 00:00:56.017
Well, we have our,

00:00:56.017 --> 00:00:57.433
let me do this in a new color,

00:00:57.433 --> 00:00:59.661
we have our three B here.

00:00:59.661 --> 00:01:02.193
So that is three B,

00:01:02.193 --> 00:01:05.428
and what's X to the A
times X to the fourth?

00:01:05.428 --> 00:01:07.930
X to the A times X to the fourth.

00:01:07.930 --> 00:01:10.160
Have the same base raised
to different exponents

00:01:10.160 --> 00:01:11.870
and I'm multiplying the two.

00:01:11.870 --> 00:01:14.237
Well we know from our exponent properties,

00:01:14.237 --> 00:01:19.162
this is going to be the same
thing as X to the A plus four.

00:01:19.778 --> 00:01:23.265
X to the A times X to the
fourth is X to the A plus four,

00:01:23.265 --> 00:01:25.077
we're just adding the
exponents because we have

00:01:25.077 --> 00:01:27.864
the same base and we're
multiplying these two.

00:01:27.864 --> 00:01:31.067
So we now have that three B is equal to X

00:01:31.067 --> 00:01:34.342
to the A plus fourth power.

00:01:34.342 --> 00:01:36.803
And this is going to be
equal to what we have

00:01:36.803 --> 00:01:38.683
on the right-hand side.

00:01:38.683 --> 00:01:41.447
So this is going to be equal to,

00:01:41.447 --> 00:01:42.910
put in that same color,

00:01:42.910 --> 00:01:46.452
negative 24 X to the sixth.

00:01:47.108 --> 00:01:48.778
So what could we do now?

00:01:48.778 --> 00:01:52.216
Well we can recognize that three B

00:01:52.216 --> 00:01:56.883
is going to have to be
equal to negative 24 and

00:01:56.883 --> 00:02:00.504
the A plus four right over
here is going to need to be

00:02:00.504 --> 00:02:03.036
equal to the six.

00:02:03.036 --> 00:02:04.521
So let's write that down.

00:02:04.521 --> 00:02:06.589
Three B is equal to negative 24.

00:02:06.589 --> 00:02:08.725
Three B is equal to negative 24,

00:02:08.725 --> 00:02:10.210
you might be able to do that in your head,

00:02:10.210 --> 00:02:12.300
or if you want to do it a
little bit more systematically

00:02:12.300 --> 00:02:16.016
you could divide both sides
by three to solve for your B.

00:02:16.016 --> 00:02:19.382
And you get B is equal to negative eight.

00:02:19.382 --> 00:02:24.382
And we can also say that A
plus four is equal to six.

00:02:25.279 --> 00:02:26.952
Subtract four from both sides,

00:02:26.952 --> 00:02:29.226
you get A is equal to two.

00:02:29.226 --> 00:02:30.712
And we are done.

00:02:30.712 --> 00:02:32.153
And you can check it if you want.

00:02:32.153 --> 00:02:33.987
We can rewrite it as,

00:02:33.987 --> 00:02:35.450
if we say A is equal to two,

00:02:35.450 --> 00:02:39.543
we can say three X squared times,

00:02:40.326 --> 00:02:42.475
instead of B we know
that's negative eight,

00:02:42.475 --> 00:02:45.378
negative eight X to the fourth.

00:02:45.378 --> 00:02:46.678
What is that going to be equal to?

00:02:46.678 --> 00:02:48.488
Three times negative eight is negative 24,

00:02:48.488 --> 00:02:52.181
X squared times X to the
fourth is X to the sixth.

00:02:52.181 --> 00:02:54.364
And once again as I said,
you wouldn't necessarily

00:02:54.364 --> 00:02:56.941
have to do all of these steps
if you're doing it on paper,

00:02:56.941 --> 00:02:58.938
but this is what you're doing
even if you look at this,

00:02:58.938 --> 00:03:00.307
if you looked at the
original problem you'd say

00:03:00.307 --> 00:03:03.326
okay, the coefficients
I can multiply those,

00:03:03.326 --> 00:03:06.345
three times B to get this
coefficient over here.

00:03:06.345 --> 00:03:08.619
So three times B is negative 24,

00:03:08.619 --> 00:03:10.803
B would have to be negative eight.

00:03:10.803 --> 00:03:13.891
So B would have to be negative eight here.

00:03:13.891 --> 00:03:16.654
And you can say X to the
A times X to the fourth

00:03:16.654 --> 00:03:18.094
is X to the sixth.

00:03:18.094 --> 00:03:20.091
So, well let's see, we're gonna exponents,

00:03:20.091 --> 00:03:22.087
A plus four is gonna B equal to six,

00:03:22.087 --> 00:03:25.059
A must be equal to two.

00:03:25.059 --> 00:03:26.383
So you might be able to do it that simply,

00:03:26.383 --> 00:03:27.660
but this is what you're doing,

00:03:27.660 --> 00:03:29.953
even if you're doing it
in your head like that.


Subtracting polynomials: two variables challenge

WEBVTT
Kind: captions
Language: en

00:00:00.213 --> 00:00:01.850
- [Voiceover] We're asked
to subtract negative six X

00:00:01.850 --> 00:00:04.870
to the fourth minus
three X squared Y squared

00:00:04.870 --> 00:00:09.421
plus Y to the fourth from
two X to the fourth minus

00:00:09.421 --> 00:00:13.483
eight X squared Y squared
mins Y to the fourth.

00:00:13.483 --> 00:00:17.055
And I encourage you to pause
this video and give it a try.

00:00:17.733 --> 00:00:18.685
Alright, let's work through it together.

00:00:18.685 --> 00:00:21.192
So We're gonna subtract
this green polynomial

00:00:21.192 --> 00:00:23.120
from this magenta one.

00:00:23.120 --> 00:00:25.791
So we can rewrite this as,

00:00:25.791 --> 00:00:27.707
we're to actually
perform it we could write

00:00:27.707 --> 00:00:30.086
two X to the fourth

00:00:30.086 --> 00:00:33.987
minus eight X squared Y squared

00:00:33.987 --> 00:00:36.490
minus Y to the fourth

00:00:36.490 --> 00:00:39.462
minus, and I'm gonna
write this in parenthesis,

00:00:39.462 --> 00:00:41.227
give us some space,

00:00:41.227 --> 00:00:45.685
minus negative six X to the fourth

00:00:45.685 --> 00:00:49.888
minus three X squared Y squared

00:00:49.888 --> 00:00:53.301
plus five Y to the fourth.

00:00:53.301 --> 00:00:55.994
So notice, I'm subtracting
this green polynomial

00:00:55.994 --> 00:00:58.758
in two variables from
this magenta polynomial

00:00:58.758 --> 00:00:59.918
in two variables.

00:00:59.918 --> 00:01:02.425
Which is exactly what
it says to do up here.

00:01:02.425 --> 00:01:04.005
So what's this going to be?

00:01:04.005 --> 00:01:06.118
Well I can just rewrite the magenta part.

00:01:06.118 --> 00:01:08.532
We're gonna have two X to the fourth

00:01:08.532 --> 00:01:12.225
minus eight X squared Y squared

00:01:12.225 --> 00:01:14.036
minus Y to the fourth.

00:01:14.036 --> 00:01:16.612
And then I can distribute
this negative sign.

00:01:16.612 --> 00:01:20.002
So if we say the negative of
negative six X to the fourth,

00:01:20.002 --> 00:01:22.835
that's gonna be positive
six X to the fourth.

00:01:22.835 --> 00:01:26.318
So that's gonna go positive
six X to the fourth.

00:01:26.318 --> 00:01:30.730
And then the negative of negative
three X squared Y squared

00:01:30.730 --> 00:01:34.538
is going to be positive
three X squared Y squared.

00:01:34.538 --> 00:01:38.763
So plus three X squared Y squared.

00:01:38.763 --> 00:01:40.319
And then last but not least,

00:01:40.319 --> 00:01:42.828
we have a negative, or we're subtracting

00:01:42.828 --> 00:01:45.382
positive five Y to the fourth.

00:01:45.382 --> 00:01:50.165
So that's going to be
subtracting five Y to the fourth,

00:01:50.165 --> 00:01:51.972
or negative five Y to the fourth.

00:01:51.972 --> 00:01:55.478
And now we can try to simplify.

00:01:55.478 --> 00:01:58.104
So let's first look at
this term right over here.

00:01:58.794 --> 00:02:01.221
We have two X to the fourths.

00:02:01.221 --> 00:02:05.228
And what we could look for is
another X to the fourth term.

00:02:05.888 --> 00:02:08.043
And we see it right over here.

00:02:08.043 --> 00:02:10.180
So we have two X to the fourths

00:02:10.180 --> 00:02:13.105
and we can add that to
six X to the fourths.

00:02:13.105 --> 00:02:14.172
So what's that going to be?

00:02:14.172 --> 00:02:16.100
Well if I have two of something,
and then I add another

00:02:16.100 --> 00:02:17.563
six of that something, that's going to be

00:02:17.563 --> 00:02:20.651
eight X to the fourths, two plus six.

00:02:20.651 --> 00:02:22.973
Two X to the fourth
plus six X to the fourth

00:02:22.973 --> 00:02:26.284
is going to be eight
X to the fourth power.

00:02:27.054 --> 00:02:30.551
And now we have this X
squared Y squared term.

00:02:31.252 --> 00:02:33.597
We could say we're
subtracting eight of them.

00:02:33.597 --> 00:02:35.688
And over here we're adding three of these

00:02:35.688 --> 00:02:38.032
X squared Y squared terms.

00:02:38.032 --> 00:02:39.426
So we could add these coefficients.

00:02:39.426 --> 00:02:41.562
If we're taking away eight,
but we're adding three

00:02:41.562 --> 00:02:45.039
we could view this as negative
eight X squared Y squareds

00:02:45.039 --> 00:02:48.313
plus three X squared Y squareds.

00:02:48.313 --> 00:02:51.308
Well what's negative eight plus three?

00:02:51.308 --> 00:02:53.491
Well that's going to be negative five.

00:02:53.491 --> 00:02:57.251
Negative five X squareds Y squareds.

00:02:57.251 --> 00:02:59.063
So that's that term.

00:02:59.063 --> 00:03:00.828
Let me write that a little bit neater.

00:03:00.828 --> 00:03:02.708
That's this term

00:03:02.708 --> 00:03:05.540
right over here, don't forget
to include this sign here.

00:03:05.540 --> 00:03:08.490
This is you're subtracting
eight X squared Y squareds,

00:03:08.490 --> 00:03:10.482
you could do that as negative
eight X squared Y squared

00:03:10.482 --> 00:03:13.895
plus three X squared Y squared.

00:03:13.895 --> 00:03:15.567
And then last but not least,

00:03:15.567 --> 00:03:19.213
you're subtracting one Y to the fourth,

00:03:19.213 --> 00:03:22.394
and then you're subtracting
five more Y to the fourths.

00:03:22.394 --> 00:03:23.986
So what's that going to be?

00:03:25.314 --> 00:03:27.869
You could view this as
negative one Y to the fourth

00:03:27.869 --> 00:03:29.680
minus five Y to the fourths.

00:03:29.680 --> 00:03:32.002
Well that's going to be
negative one minus five

00:03:32.002 --> 00:03:33.163
is negative six.

00:03:33.163 --> 00:03:37.272
Negative six Y to the fourths.

00:03:37.272 --> 00:03:39.269
And we're done, we have subtracted

00:03:39.269 --> 00:03:41.190
this from that.


Amy Jarich Interview

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.606
- [Voiceover] So I'm here with Amy Jarich,

00:00:01.606 --> 00:00:02.978
who's the Director of Admissions

00:00:02.978 --> 00:00:05.150
at the University of
California at Berkeley.

00:00:05.150 --> 00:00:08.126
And my question to you is when you imagine

00:00:08.126 --> 00:00:10.405
college admissions,
there's traditional notion

00:00:10.405 --> 00:00:12.968
of you get a GPA.

00:00:14.036 --> 00:00:15.875
You have to take certain
classes and all of these

00:00:15.875 --> 00:00:18.772
to be "qualified."

00:00:18.772 --> 00:00:21.970
Is that always the case, or
is there a path for students

00:00:21.970 --> 00:00:23.967
who might not have a traditional GPA

00:00:23.967 --> 00:00:26.912
or might not have all
the traditional classes?

00:00:26.912 --> 00:00:28.409
- So I would say

00:00:29.455 --> 00:00:30.969
particularly in selective admissions,

00:00:30.970 --> 00:00:34.409
our job is not so much to
focus on the admit rate

00:00:34.409 --> 00:00:36.923
and what those numbers
look like when you look

00:00:36.923 --> 00:00:39.937
at the incoming profile,
much more to think

00:00:39.937 --> 00:00:42.398
about who's a student who's
gonna graduate successfully

00:00:42.398 --> 00:00:43.906
from our institution.

00:00:43.906 --> 00:00:46.307
And I think if we're doing our job well,

00:00:46.307 --> 00:00:49.281
we're finding those students,
and it means that to keep up

00:00:49.281 --> 00:00:52.340
with the ways students learn
today, we have to look beyond

00:00:52.340 --> 00:00:55.623
those traditional measures,
so we do look past GPAs.

00:00:55.623 --> 00:00:58.253
We look beyond test scores.

00:00:59.272 --> 00:01:01.752
In terms of the tools that we use,

00:01:01.752 --> 00:01:04.817
we probably haven't tapped
into the very best ones

00:01:04.817 --> 00:01:08.832
available yet, but we
do read their writing.

00:01:08.832 --> 00:01:12.502
We look to see what extra
curriculars they're a part of.

00:01:12.502 --> 00:01:14.991
There's never going to
be a personal statement

00:01:14.991 --> 00:01:18.669
or an essay that trumps
the academic work though.

00:01:18.669 --> 00:01:22.480
Really it's thinking about
academically, are you prepared

00:01:22.480 --> 00:01:25.253
for a classroom like the one at Berkeley

00:01:25.253 --> 00:01:27.701
or the one at Virginia or
the one at Stanford, right,

00:01:27.701 --> 00:01:30.422
to make sure that you'll
succeed when you're there?

00:01:30.422 --> 00:01:32.973
I think for us we really
have started to see students

00:01:32.973 --> 00:01:35.508
who are branching out and
learning in different ways.

00:01:36.547 --> 00:01:38.430
And they're leading us, right.

00:01:38.430 --> 00:01:41.867
We're having to figure
out the ways to evaluate,

00:01:41.867 --> 00:01:44.477
and we don't have a
whole lot of experience.

00:01:44.477 --> 00:01:45.759
We certainly have

00:01:46.823 --> 00:01:48.835
a whole group of schools in the Northeast

00:01:48.835 --> 00:01:51.402
that are some of our
oldest secondary schools,

00:01:51.402 --> 00:01:55.138
who haven't given a letter
grade, and that's fine.

00:01:55.138 --> 00:01:57.103
We work with those, we always have.

00:01:57.103 --> 00:01:58.617
We've worked with homeschool students

00:01:58.617 --> 00:02:01.292
who typically bring us a different kind

00:02:01.292 --> 00:02:05.008
of high school record, so we're learning.

00:02:05.015 --> 00:02:07.369
But I do think that the process

00:02:07.369 --> 00:02:11.361
of changing secondary education is slowly

00:02:11.361 --> 00:02:13.221
also changing what we look at

00:02:13.227 --> 00:02:15.790
and how we evaluate for success

00:02:15.790 --> 00:02:18.249
once you get onto our campus.

00:02:18.249 --> 00:02:19.969
- [Voiceover] You mentioned
academics are very important,

00:02:19.969 --> 00:02:21.415
and I completely understand that,

00:02:21.415 --> 00:02:24.101
but how would you evaluate
someone's academic potential

00:02:24.101 --> 00:02:26.303
if they don't have a traditional GPA?

00:02:26.303 --> 00:02:29.314
- So a lot of what we're gonna
rely on the school to provide

00:02:29.314 --> 00:02:33.558
is something that articulates
in words or numbers,

00:02:33.558 --> 00:02:35.527
letters, whatever it is,

00:02:35.527 --> 00:02:39.827
the type of environment
the student is learning in,

00:02:39.836 --> 00:02:43.618
what the core mastery is supposed to be,

00:02:43.618 --> 00:02:47.793
and then just helping us
understand the evaluation scale

00:02:47.793 --> 00:02:51.079
and where does the student
fall on that scale.

00:02:51.079 --> 00:02:53.257
And I think sometimes that could be

00:02:53.257 --> 00:02:55.099
in the form of a paragraph.

00:02:55.099 --> 00:02:58.873
Sometimes it may be in
the form of a chart.

00:03:00.728 --> 00:03:02.259
What we do now is

00:03:03.326 --> 00:03:06.258
we read within context,
so it's not crazy to think

00:03:06.258 --> 00:03:09.722
that we have to investigate
the way a school teaches

00:03:09.722 --> 00:03:12.392
before we can make the
very best decisions about

00:03:12.392 --> 00:03:14.762
how the learns in that school.

00:03:14.762 --> 00:03:16.515
And that's something

00:03:16.515 --> 00:03:18.592
that's our responsibility to do.

00:03:18.602 --> 00:03:20.813
We're not in the business of building

00:03:20.813 --> 00:03:23.101
secondary school curriculums, right.

00:03:23.101 --> 00:03:26.619
We just need to learn what
type of curriculum you have

00:03:26.619 --> 00:03:29.880
and how it's articulated to us in terms

00:03:29.880 --> 00:03:32.168
of the measurement of success.

00:03:32.168 --> 00:03:34.089
- [Voiceover] So you
all take a closer look

00:03:34.089 --> 00:03:37.429
or as much context as you can
have on that particular school

00:03:37.429 --> 00:03:39.354
that might be in some ways nontraditional.

00:03:39.354 --> 00:03:42.116
And then there's also things
like standardized tests.

00:03:42.126 --> 00:03:45.424
- Sure, so we have standardized tests.

00:03:45.424 --> 00:03:48.078
Not all universities and colleges that use

00:03:48.078 --> 00:03:51.145
selective admissions look for
standardized tests anymore,

00:03:51.145 --> 00:03:53.239
but most still do.

00:03:54.371 --> 00:03:58.185
In most cases, they're simply
validating something else

00:03:58.185 --> 00:04:00.661
we see in the application though.

00:04:00.666 --> 00:04:04.137
For most schools, they're
not the decision maker.

00:04:04.137 --> 00:04:05.868
They're showing us that,

00:04:05.868 --> 00:04:07.724
"Oh look, this student is strong in math.

00:04:07.724 --> 00:04:08.919
"Let's look at the SAT.

00:04:08.919 --> 00:04:10.065
"Yup, turns out they're strong

00:04:10.065 --> 00:04:11.061
(interviewer laughs)
"in math."

00:04:11.061 --> 00:04:12.769
Right, there's no big surprise.

00:04:12.769 --> 00:04:14.998
Every once in a while there's a gap,

00:04:14.998 --> 00:04:18.511
and that sometimes leads to
really good conversations

00:04:18.511 --> 00:04:21.857
in our own environment
about why that happens.

00:04:21.857 --> 00:04:23.152
Is it test prep?

00:04:23.157 --> 00:04:25.630
Is it test anxiety?

00:04:25.643 --> 00:04:29.605
We have a provost at Berkeley
who's worked quite strongly

00:04:29.605 --> 00:04:33.615
in talking about the stereotype
threats and the whole idea

00:04:33.615 --> 00:04:36.480
that when you walk into a
space, someone has already

00:04:36.480 --> 00:04:38.072
prejudged the way you behave.

00:04:38.072 --> 00:04:39.961
And standardized test is usually

00:04:41.031 --> 00:04:43.401
the lab they use to show that,

00:04:43.401 --> 00:04:46.180
so we talk about that when there's a gap,

00:04:46.180 --> 00:04:50.135
but usually standardized tests
just validates what we see

00:04:50.138 --> 00:04:53.597
elsewhere in that academic
profile of the student.

00:04:53.597 --> 00:04:54.619
- [Voiceover] And there these notions

00:04:54.619 --> 00:04:56.470
of approved curricular.

00:04:56.470 --> 00:04:57.468
Correct me if I'm wrong.

00:04:57.468 --> 00:05:00.171
Especially here in California,
I think there's something

00:05:00.171 --> 00:05:02.132
to the effect of A to B, or there's some A

00:05:02.132 --> 00:05:03.215
to some type of,
- [Amy] G.

00:05:03.215 --> 00:05:04.808
- [Voiceover] A to G, A to G.

00:05:04.808 --> 00:05:05.886
What are those things,

00:05:05.886 --> 00:05:08.890
and how strictly are those enforced

00:05:08.890 --> 00:05:11.284
that you have to somehow align to that?

00:05:11.284 --> 00:05:15.657
- Sure so the university
system in California is really

00:05:15.657 --> 00:05:18.591
aligned, and it's been aligned
through the master plan

00:05:18.591 --> 00:05:22.581
that was developed in the 60s
to be able to allow students

00:05:22.581 --> 00:05:26.387
to build a path and stay in California,

00:05:26.387 --> 00:05:28.857
starting in the secondary
school and moving up.

00:05:28.857 --> 00:05:32.829
So the UC system, as the
most selective public school

00:05:32.829 --> 00:05:35.907
system in the state,
is looking to make sure

00:05:35.907 --> 00:05:39.267
that the curriculum in the high school is

00:05:40.271 --> 00:05:44.002
allowing the student to get the best prep.

00:05:44.002 --> 00:05:47.129
To do that, the UC system for years

00:05:47.129 --> 00:05:48.686
has actually evaluated

00:05:48.686 --> 00:05:50.086
the curriculum of a high school.

00:05:50.086 --> 00:05:52.734
So a high school, if they
come up with a new course,

00:05:52.734 --> 00:05:56.207
they need to submit it to the
UC system to get approval.

00:05:56.207 --> 00:05:58.683
And then when the application
is being filled out

00:05:58.683 --> 00:06:02.145
by the student, they'll see
that course is UC approved.

00:06:03.190 --> 00:06:06.231
Do we admit students from
schools that don't have

00:06:06.231 --> 00:06:07.422
UC-approved coursework?

00:06:07.422 --> 00:06:08.675
Absolutely.

00:06:08.675 --> 00:06:10.764
Think about all of the
out-of-state students

00:06:10.764 --> 00:06:12.454
that are applying, none of them have

00:06:12.454 --> 00:06:14.251
UC-approved coursework, right.

00:06:14.251 --> 00:06:16.513
So we are certainly thinking outside

00:06:16.513 --> 00:06:19.592
of that A through G box, but
we use the A through G language

00:06:19.592 --> 00:06:23.473
to help people understand
what the college prep path

00:06:23.473 --> 00:06:25.290
in California looks like.

00:06:25.290 --> 00:06:28.199
And the only way we can
make sure it's consistent

00:06:28.199 --> 00:06:31.858
is having a system that
evaluates each course.

00:06:32.961 --> 00:06:34.491
A bit like standardized testing, right,

00:06:34.491 --> 00:06:36.988
there's something about it
that feels standardized,

00:06:36.988 --> 00:06:40.201
so that you can trust
it a little bit more.

00:06:40.211 --> 00:06:41.426
- [Voiceover] Yeah.

00:06:41.427 --> 00:06:43.611
And you have, as you
mentioned, it could be

00:06:43.611 --> 00:06:45.519
out-of-state students who
obviously weren't participating

00:06:45.519 --> 00:06:48.786
in the A to G requirements,
and some of those might

00:06:48.786 --> 00:06:50.031
even be nontraditional.

00:06:50.031 --> 00:06:51.135
Might not have
- [Amy] Right, of course.

00:06:51.135 --> 00:06:52.157
- [Voiceover] a traditional GPA.

00:06:52.157 --> 00:06:53.516
- Yeah.

00:06:53.516 --> 00:06:56.478
And we also see a lot of students now.

00:06:57.488 --> 00:07:00.154
One of the hardest questions,
particularly for students

00:07:00.154 --> 00:07:01.976
out of state, I think
right now, when they think

00:07:01.976 --> 00:07:04.755
about A through G, it includes art.

00:07:05.812 --> 00:07:09.121
So if you're a student who
is incredibly interested

00:07:09.121 --> 00:07:12.364
in being admitted to one of
Berkeley's STEM programs,

00:07:12.364 --> 00:07:14.921
particularly engineering,
you have the opportunity

00:07:14.921 --> 00:07:18.563
to take a second year of
physics at an advanced level

00:07:18.563 --> 00:07:20.668
or complete the A through G requirements

00:07:20.668 --> 00:07:24.030
by taking art, what am I supposed to say?

00:07:24.030 --> 00:07:25.756
I want you to take the physics, right.

00:07:25.756 --> 00:07:27.869
I want you to have that background.

00:07:27.869 --> 00:07:29.769
I believe in art, course I do.

00:07:29.769 --> 00:07:32.293
I know why it's a part of A through G,

00:07:32.293 --> 00:07:34.014
but for that student's development

00:07:34.014 --> 00:07:36.498
and that student's interest,

00:07:36.498 --> 00:07:40.019
that A through G equation
just doesn't fit.

00:07:42.602 --> 00:07:45.165
As we adjust to see what
students are learning

00:07:45.165 --> 00:07:48.909
and how they're learning
and what are the topics

00:07:48.909 --> 00:07:52.879
that are most interesting
to them, we're just evolving

00:07:52.879 --> 00:07:54.994
in the admissions process
to be able to meet them

00:07:54.994 --> 00:07:56.756
where they are.

00:07:56.766 --> 00:07:59.268
And I'll be honest, we're
little bit behind, right.

00:07:59.268 --> 00:08:02.758
There plenty of schools
that challenge us with

00:08:03.770 --> 00:08:04.943
the way that they teach

00:08:04.943 --> 00:08:05.956
or what they're teaching.

00:08:05.956 --> 00:08:07.926
But when those students come to our campus

00:08:07.926 --> 00:08:10.820
as admitted students, and
they choose to attend,

00:08:10.820 --> 00:08:13.202
they're the ones who go to faculty hours.

00:08:13.202 --> 00:08:16.120
They're the ones who understand the value

00:08:16.120 --> 00:08:19.872
of having lab time and
doing hands-on activities

00:08:19.872 --> 00:08:23.286
and research, and I think
that's a really important part

00:08:23.286 --> 00:08:24.717
of the college experience.

00:08:24.717 --> 00:08:27.452
They don't have to
spend quite as much time

00:08:28.485 --> 00:08:30.935
building up that confidence
level in whether or not

00:08:30.935 --> 00:08:32.552
they should go to talk to the teacher

00:08:32.552 --> 00:08:34.707
or whether or not they

00:08:34.707 --> 00:08:38.469
could possibly participate
in hands-on learning.

00:08:38.469 --> 00:08:39.780
- [Voiceover] Yeah and
so your general sense,

00:08:39.780 --> 00:08:43.228
and obviously it's hard to
make overbroad statements.

00:08:43.228 --> 00:08:47.302
But a student, nontraditional
school or background,

00:08:47.302 --> 00:08:49.280
maybe doesn't have traditional GPA,

00:08:49.280 --> 00:08:51.717
but they're able to show that
their academic capabilities,

00:08:51.717 --> 00:08:53.765
maybe it's through standardized
tests or they take some

00:08:53.765 --> 00:08:56.915
other types of exams, they
can show that they are

00:08:56.915 --> 00:08:59.192
of the caliber that you all would expect,

00:08:59.192 --> 00:09:01.028
so that they could come
and do well at a place

00:09:01.028 --> 00:09:03.384
like Berkeley, and they
also show other things.

00:09:03.384 --> 00:09:05.112
They made cool projects, or they have

00:09:06.242 --> 00:09:08.693
really great portfolios of work.

00:09:08.693 --> 00:09:10.895
That's something that you
all take a serious look at.

00:09:10.895 --> 00:09:11.893
- Yeah.

00:09:12.890 --> 00:09:15.643
Through our process, we
actually ask the student

00:09:15.643 --> 00:09:16.718
to articulate that.

00:09:16.720 --> 00:09:19.121
So in their personal
statement, they might talk

00:09:19.121 --> 00:09:22.441
about an internship that
they had and why it mattered,

00:09:22.441 --> 00:09:24.083
what they did.

00:09:24.083 --> 00:09:27.166
Other schools actually might
have a letter of recommendation

00:09:27.166 --> 00:09:29.988
process or a portfolio review process

00:09:29.988 --> 00:09:33.547
where they'll look at the
things that were developed,

00:09:33.547 --> 00:09:35.391
or they'll hear the voice of the person

00:09:35.391 --> 00:09:39.538
who was this individual's
supervisor or mentor.

00:09:39.538 --> 00:09:43.951
And I think that's an
important piece of any type of

00:09:45.095 --> 00:09:48.740
less than traditional model because

00:09:48.750 --> 00:09:50.686
relying on a student

00:09:50.686 --> 00:09:53.411
who has only known that model

00:09:53.411 --> 00:09:56.671
to explain everything that's unique

00:09:56.671 --> 00:09:58.391
and cool about it doesn't make

00:09:58.391 --> 00:09:59.772
a lot of sense, right.

00:09:59.780 --> 00:10:01.942
You need someone from
the outside perspective

00:10:01.942 --> 00:10:05.845
to be able to compare and
contrast to maybe the way

00:10:05.845 --> 00:10:09.225
that they were educated and understanding

00:10:09.225 --> 00:10:12.789
that student's story is very
different and very powerful

00:10:12.789 --> 00:10:14.277
in a modern way.

00:10:14.277 --> 00:10:15.637
- Yeah, well thank you so much.

00:10:15.637 --> 00:10:17.376
This is really valuable.
- [Amy] Yeah, of course!

00:10:17.376 --> 00:10:19.485
(laughter)
Happy to help.


Polynomial subtraction

WEBVTT
Kind: captions
Language: en

00:00:00.644 --> 00:00:02.002
- [Voiceover] We're asked to subtract

00:00:02.002 --> 00:00:05.067
negative two x squared
plus four x minus one

00:00:05.067 --> 00:00:09.757
from six x squared plus
three x minus nine,

00:00:09.757 --> 00:00:11.406
and like always I encourage
you to pause the video

00:00:11.406 --> 00:00:13.797
and see if you can give it a go.

00:00:13.797 --> 00:00:16.096
All right, now let's work
through this together.

00:00:16.096 --> 00:00:20.020
So I could rewrite this as six x squared

00:00:20.020 --> 00:00:23.514
plus three x minus nine

00:00:23.514 --> 00:00:28.321
minus, minus this
expression right over here,

00:00:28.321 --> 00:00:29.905
so I'll put that in parentheses,

00:00:29.905 --> 00:00:34.895
minus negative two x squared,
negative two x squared

00:00:34.895 --> 00:00:39.130
plus four x minus one.

00:00:39.130 --> 00:00:40.999
Now what can we do from here?

00:00:40.999 --> 00:00:43.263
Well, we can distribute
this negative side,

00:00:43.263 --> 00:00:45.562
we can distribute this negative side,

00:00:45.562 --> 00:00:47.141
and then if we did that,

00:00:47.141 --> 00:00:50.184
we would get the six x squared
plus three x minus nine

00:00:50.184 --> 00:00:51.774
won't change so we still have that.

00:00:51.774 --> 00:00:56.464
Six x squared plus three x minus nine,

00:00:56.464 --> 00:00:59.064
but if I distribute the negative side,

00:00:59.064 --> 00:01:01.445
the negative of negative two x squared

00:01:01.445 --> 00:01:04.730
is positive two x squared.

00:01:04.730 --> 00:01:07.841
So that's going to be positive two...

00:01:07.841 --> 00:01:09.304
Get a little more space.

00:01:09.304 --> 00:01:13.274
Positive two x squared, and then subtract,

00:01:13.274 --> 00:01:15.875
and then the negative
of positive four x is...

00:01:15.875 --> 00:01:18.221
I'm going to subtract four x now,

00:01:18.221 --> 00:01:20.692
and then the negative of negative one,

00:01:20.692 --> 00:01:23.513
or the opposite of negative one
is going to be positive one.

00:01:23.513 --> 00:01:26.114
So I've just distributed
the negative side,

00:01:26.114 --> 00:01:29.910
and now I can add terms
that have the same degree

00:01:29.910 --> 00:01:31.763
on our x...

00:01:31.763 --> 00:01:33.871
The same degree terms,
I guess you could say,

00:01:33.871 --> 00:01:37.337
so I have an x squared term,

00:01:37.337 --> 00:01:38.945
here it's six x squared,

00:01:38.945 --> 00:01:41.290
here I have a two x squared term,

00:01:41.290 --> 00:01:42.660
so I can add those two together,

00:01:42.660 --> 00:01:44.401
six x squared plus two x squared.

00:01:44.401 --> 00:01:46.583
If I have six x squareds
and then I have another

00:01:46.583 --> 00:01:49.869
two x squareds, how many x
squareds am I now going to have?

00:01:49.869 --> 00:01:54.048
I'm now going to have eight
x squareds, eight x squareds,

00:01:54.048 --> 00:01:56.169
or six x squared plus two x squared.

00:01:56.169 --> 00:01:58.433
We add the coefficients,
the six and the two

00:01:58.433 --> 00:02:00.824
to get eight, eight x squared.

00:02:00.824 --> 00:02:04.324
Then we can add the x terms.

00:02:04.324 --> 00:02:06.212
You could view these as
the first-degree terms,

00:02:06.212 --> 00:02:08.120
three x...

00:02:08.120 --> 00:02:11.615
We have three x

00:02:11.615 --> 00:02:14.135
and then we have minus four x,

00:02:14.135 --> 00:02:16.189
so three x minus four x,

00:02:16.189 --> 00:02:18.244
if I have three of something
and I take away four of them,

00:02:18.244 --> 00:02:20.891
I'm now going to have
negative one of that thing,

00:02:20.891 --> 00:02:22.470
or you could say that the coefficients,

00:02:22.470 --> 00:02:24.339
three minus four would be negative one.

00:02:24.339 --> 00:02:26.730
So I now have negative one x.

00:02:26.730 --> 00:02:28.739
I could write it as negative one x,

00:02:28.739 --> 00:02:31.784
but I might as well just
write it as negative x.

00:02:31.784 --> 00:02:33.858
That's the same thing as negative one x.

00:02:33.858 --> 00:02:36.679
And then finally, I can deal
with our constant terms.

00:02:36.679 --> 00:02:40.603
I'm subtracting a nine
and then I'm adding a one.

00:02:40.603 --> 00:02:43.447
So you could say, "Well, what's
a negative nine plus one?"

00:02:43.447 --> 00:02:45.340
Well, that's going to be negative eight.

00:02:45.340 --> 00:02:49.535
That's going to be negative
eight and we are all done.

00:02:49.535 --> 00:02:52.514
And one thing that you
might find interesting

00:02:52.514 --> 00:02:56.519
is I had a polynomial here and from that

00:02:56.519 --> 00:02:58.322
I subtracted another polynomial,

00:02:58.322 --> 00:03:00.957
and notice, I got a polynomial,

00:03:00.957 --> 00:03:04.062
and this is actually always
going to be the case.

00:03:04.062 --> 00:03:07.354
If you think about the
set of all polynomials,

00:03:07.354 --> 00:03:08.550
if you just think about the set...

00:03:08.550 --> 00:03:10.536
Let me do this in a neutral color.

00:03:10.536 --> 00:03:12.022
So if we think about the set

00:03:12.022 --> 00:03:14.495
of all polynomials right over here,

00:03:14.495 --> 00:03:16.329
and if you take one polynomial,

00:03:16.329 --> 00:03:19.452
which you could imagine
this magenta polynomial.

00:03:19.452 --> 00:03:21.193
So this is a polynomial right over here,

00:03:21.193 --> 00:03:24.322
let's call this p of x.

00:03:24.322 --> 00:03:27.299
So this is p of x right over there,

00:03:27.299 --> 00:03:31.305
p of x, and then you
have another polynomial,

00:03:32.325 --> 00:03:34.929
this one right over here, let's call this,

00:03:34.929 --> 00:03:38.203
I don't know, we can call this q of x,

00:03:38.203 --> 00:03:40.443
q of x, just for kicks.

00:03:40.443 --> 00:03:43.833
So that's q of x, just like that.

00:03:43.833 --> 00:03:45.586
And if you apply the...

00:03:45.586 --> 00:03:48.693
In this case, we applied
the subtraction operator.

00:03:48.693 --> 00:03:50.450
If we apply subtraction...

00:03:50.450 --> 00:03:53.097
So you took these two,
you took these two...

00:03:53.097 --> 00:03:55.395
Let me see how I could depict this well.

00:03:55.395 --> 00:04:00.132
So we took p of x and you
subtracted from that q of x.

00:04:00.132 --> 00:04:02.561
We still get a polynomial,

00:04:02.561 --> 00:04:04.106
so that's going to give us...

00:04:04.106 --> 00:04:06.995
We stay in the set of polynomials.

00:04:06.995 --> 00:04:09.479
At any time you have a set of things,

00:04:09.479 --> 00:04:11.267
and you might be more
used to talking about this

00:04:11.267 --> 00:04:13.100
in terms of integers,

00:04:13.100 --> 00:04:16.944
or number sets, but you can
talk about this in general.

00:04:16.944 --> 00:04:18.685
Here we're talking about
the set of polynomials,

00:04:18.685 --> 00:04:21.123
and we just saw that if we
start with two polynomials,

00:04:21.123 --> 00:04:23.117
two members of the set of polynomials...

00:04:23.117 --> 00:04:25.316
Let me be clear, this is polynomials,

00:04:25.316 --> 00:04:28.245
polynomials, right over here.

00:04:28.245 --> 00:04:29.844
You take two members of the set

00:04:29.844 --> 00:04:32.534
and you perform the subtraction operation,

00:04:32.534 --> 00:04:35.695
you're still going to
get a member of the set.

00:04:35.695 --> 00:04:37.871
And when you have a situation like this,

00:04:37.871 --> 00:04:39.411
I could call this one,

00:04:39.411 --> 00:04:40.467
I don't know, I'm running out of letters.

00:04:40.467 --> 00:04:42.394
Well let me just call this one f of x,

00:04:42.394 --> 00:04:44.437
so we've got f of x here.

00:04:44.437 --> 00:04:45.746
When you have this
situation where you take

00:04:45.746 --> 00:04:48.280
two members of a set, you
apply an operator on them,

00:04:48.280 --> 00:04:50.451
or you take a certain
number of members of a set,

00:04:50.451 --> 00:04:51.391
you apply an operator on them,

00:04:51.391 --> 00:04:52.842
and you still get a member of the set,

00:04:52.842 --> 00:04:57.649
we would say that this set is
closed under that operation.

00:04:57.649 --> 00:05:00.297
So we could say that the set

00:05:00.297 --> 00:05:04.325
of polynomials, set of polynomials,

00:05:05.428 --> 00:05:08.662
polynomials closed,

00:05:09.863 --> 00:05:11.094
closed...

00:05:11.094 --> 00:05:12.188
I won't even put it in quotes.

00:05:12.188 --> 00:05:16.288
Closed under, under subtraction.

00:05:16.288 --> 00:05:19.244
And I didn't prove it here,
I just did one example

00:05:19.244 --> 00:05:21.520
where I subtracted two polynomials

00:05:21.520 --> 00:05:22.586
and I got another one,

00:05:22.586 --> 00:05:25.464
and there's clearly more
rigorous proofs that you can do.

00:05:25.464 --> 00:05:27.043
But this is actually the
case, as long as you have

00:05:27.043 --> 00:05:28.985
two polynomials, you apply subtraction,

00:05:28.985 --> 00:05:32.236
you're going to get another polynomial.

00:05:32.236 --> 00:05:35.289
And the fancy way of saying
that is the set of polynomials

00:05:35.289 --> 00:05:37.123
is closed under subtraction.

00:05:37.123 --> 00:05:39.352
This notion of closure
sometimes seems like this

00:05:39.352 --> 00:05:42.464
very fancy mathematical
idea, but it's not too fancy.

00:05:42.464 --> 00:05:43.822
It's just you take two members of a set,

00:05:43.822 --> 00:05:45.018
you apply an operation,

00:05:45.018 --> 00:05:46.515
if you still get a member of the set

00:05:46.515 --> 00:05:49.543
after that operation,
then that set is closed

00:05:49.543 --> 00:05:51.538
under that operation.


Finding an error in polynomial subtraction

WEBVTT
Kind: captions
Language: en

00:00:00.236 --> 00:00:02.512
- [Voiceover] Kenisha is asked
to subtract seven B squared

00:00:02.512 --> 00:00:05.484
minus five B plus one from two B squared

00:00:05.484 --> 00:00:07.410
plus three B plus 11.

00:00:07.410 --> 00:00:09.058
Her work is shown below and we see

00:00:09.058 --> 00:00:10.475
all the steps that she took.

00:00:10.475 --> 00:00:13.725
And then they say at which
step did Kenisha make an error?

00:00:13.725 --> 00:00:15.445
And it's pretty clear she made an error,

00:00:15.445 --> 00:00:16.791
because they're not giving us an option

00:00:16.791 --> 00:00:17.883
that she didn't make an error.

00:00:17.883 --> 00:00:19.577
So let's see what went on here.

00:00:19.577 --> 00:00:21.875
So the first thing is, we want to subtract

00:00:21.875 --> 00:00:24.918
seven B squared minus five B plus one,

00:00:24.918 --> 00:00:27.612
we want to subtract
that from two B squared

00:00:27.612 --> 00:00:30.166
plus three B plus 11.

00:00:30.166 --> 00:00:32.232
And we see that in step
one she set it up properly.

00:00:32.232 --> 00:00:34.483
She is indeed subtracting,

00:00:34.483 --> 00:00:37.410
she is indeed subtracting
seven B squared minus

00:00:37.410 --> 00:00:39.848
five B plus one from

00:00:39.848 --> 00:00:43.125
two B squared plus three B plus 11.

00:00:43.125 --> 00:00:44.195
That's fair.

00:00:44.195 --> 00:00:46.215
Alright, so step one seems to be okay.

00:00:46.215 --> 00:00:47.793
Now let's see step two,

00:00:47.793 --> 00:00:50.091
it looks like she's going to
distribute the negative sign,

00:00:50.091 --> 00:00:52.692
so the negative of seven B squared is,

00:00:52.692 --> 00:00:54.665
as you can see is
negative seven B squared.

00:00:54.665 --> 00:00:56.594
We're gonna subtract seven B squared.

00:00:56.594 --> 00:01:01.594
And then you have the
negative of negative five B.

00:01:02.306 --> 00:01:04.233
Well that's going to be positive five B.

00:01:04.233 --> 00:01:05.835
So this right over here,

00:01:05.835 --> 00:01:08.250
this one over here should,

00:01:08.250 --> 00:01:10.479
this over here should
be a positive five B.

00:01:10.479 --> 00:01:12.545
Let me do that in a new color.

00:01:12.545 --> 00:01:15.378
So this should be a positive five B.

00:01:15.378 --> 00:01:18.140
And then the negative of positive one,

00:01:18.140 --> 00:01:20.254
this should be a negative one.

00:01:20.254 --> 00:01:21.785
That should be a negative
one right over there.

00:01:21.785 --> 00:01:23.457
So she's making a mistake,

00:01:23.457 --> 00:01:26.174
she's making a mistake in step two

00:01:26.174 --> 00:01:29.263
when she distributes this negative sign.

00:01:29.263 --> 00:01:32.490
So step two is where she made,

00:01:32.490 --> 00:01:35.625
is where she actually made her error.

00:01:35.625 --> 00:01:37.506
From there it seem reasonable,

00:01:37.506 --> 00:01:41.452
she is subtracting the coefficients
on the B squared terms,

00:01:41.452 --> 00:01:44.309
so two B squared minus seven B squared,

00:01:44.309 --> 00:01:46.328
gets us this right over there.

00:01:46.328 --> 00:01:51.328
Then on the B terms, you
have three B and then,

00:01:51.368 --> 00:01:52.366
let's see,

00:01:52.366 --> 00:01:55.291
you have, it should be three B plus five B

00:01:55.291 --> 00:01:56.964
if she distributed the
negative sign right,

00:01:56.964 --> 00:01:58.774
but since she didn't distribute
the negative sign right,

00:01:58.774 --> 00:02:01.793
she has three B minus five B.

00:02:01.793 --> 00:02:04.626
And it should be 11 minus one,

00:02:04.626 --> 00:02:07.806
but since she didn't do
the distribution correctly,

00:02:07.806 --> 00:02:09.363
we have 11,

00:02:09.363 --> 00:02:12.637
it should be 11 minus
one, as we said before,

00:02:12.637 --> 00:02:14.423
because you're gonna
distribute this negative sign,

00:02:14.423 --> 00:02:16.374
but she left that as a plus one

00:02:16.374 --> 00:02:18.163
and so that's why she got that over there.

00:02:18.163 --> 00:02:20.903
So her real mistake was
clearly in step two.

00:02:20.903 --> 00:02:22.898
'Cause from there on she's
doing reasonable things,

00:02:22.898 --> 00:02:24.821
it's just she made that one error.


Beginning of the Civil War

WEBVTT
Kind: captions
Language: en

00:00:00.098 --> 00:00:02.025
- [Voiceover] Kim, we've been
doing a couple of videos where

00:00:02.025 --> 00:00:06.483
we saw how slavery became more
and more of a political issue

00:00:06.483 --> 00:00:09.594
as we're going through the early 1800s.

00:00:09.594 --> 00:00:11.963
You go all the way to
the Compromise of 1850,

00:00:11.963 --> 00:00:16.281
the Fugitive Slave Act,
it really starts to anger,

00:00:16.281 --> 00:00:19.625
especially, definitely
abolitionists, who are always

00:00:19.625 --> 00:00:23.131
very against slavery
under moral arguments,

00:00:23.131 --> 00:00:24.524
but, also, a lot of the anti-slave folks,

00:00:24.524 --> 00:00:27.380
Republican party forums, and then we have

00:00:27.380 --> 00:00:30.561
the election of Abraham Lincoln, in,

00:00:30.561 --> 00:00:32.976
let's see, if we zoom in on this timeline,

00:00:32.976 --> 00:00:36.018
this is from 1860 to
1870, we zoom in there,

00:00:36.018 --> 00:00:39.315
and if we really zoom in on
the early part of that decade,

00:00:39.315 --> 00:00:42.612
Lincoln gets elected in November of 1860,

00:00:42.612 --> 00:00:44.910
which seems, at least
based on this timeline,

00:00:44.910 --> 00:00:47.743
to start catalyzing a whole
chain reaction of things.

00:00:47.743 --> 00:00:49.601
- [Voiceover] Yeah, this
is a time when things

00:00:49.601 --> 00:00:54.546
happen really fast, Lincoln is
elected in November of 1860,

00:00:54.546 --> 00:00:58.842
but this is before we have
our modern election timeline,

00:00:58.842 --> 00:01:02.835
when presidents actually
take office in January.

00:01:02.835 --> 00:01:05.065
Presidents in this time
period didn't take office

00:01:05.065 --> 00:01:09.058
until March, so there is
this huge lame duck period

00:01:09.058 --> 00:01:11.705
where everyone's kind of
wondering what's going to happen

00:01:11.705 --> 00:01:13.888
when Lincoln actually
takes the presidency.

00:01:13.888 --> 00:01:17.287
- [Voiceover] The current
president during the election

00:01:17.287 --> 00:01:20.724
is Buchanan, and he's the lame duck.

00:01:20.724 --> 00:01:21.931
Someone else has been elected,

00:01:21.931 --> 00:01:24.787
and he has to kind of
pretend to be president

00:01:24.787 --> 00:01:29.477
for, what is it, five months,
until the actual inauguration.

00:01:29.477 --> 00:01:31.497
- [Voiceover] Right, this is
no president's favorite time,

00:01:31.497 --> 00:01:33.425
when everyone knows that a new president

00:01:33.425 --> 00:01:36.791
is going to be coming in, so
their influence is waning.

00:01:36.791 --> 00:01:39.740
What's important in this
time period is that,

00:01:39.740 --> 00:01:42.944
after Lincoln is elected,
and as we talked about

00:01:42.944 --> 00:01:45.893
in the last video, without
a single southern vote...

00:01:45.893 --> 00:01:47.588
- [Voiceover] He wasn't
even on the ballot.

00:01:47.588 --> 00:01:49.376
- [Voiceover] He wasn't even
on the ballot for most places.

00:01:49.376 --> 00:01:52.743
Southern states start to think about

00:01:52.743 --> 00:01:53.857
what they're going to do,

00:01:53.857 --> 00:01:58.083
so they know that Lincoln is
an anti-slavery president.

00:01:58.083 --> 00:02:00.196
They think that having Lincoln in office

00:02:00.196 --> 00:02:03.540
is going to be a direct challenge

00:02:03.540 --> 00:02:05.188
to the institution of slavery,

00:02:05.188 --> 00:02:08.764
so in this period of a few months,

00:02:08.764 --> 00:02:11.852
from December, at the end of 1860,

00:02:11.852 --> 00:02:16.798
to February in 1861, the southern states

00:02:16.798 --> 00:02:18.516
think about what they're going to do.

00:02:18.516 --> 00:02:20.373
They send commissioners around saying,

00:02:20.373 --> 00:02:22.417
"Do you think we should secede?"

00:02:22.417 --> 00:02:25.064
"Yeah, probably, we should secede."

00:02:25.064 --> 00:02:27.525
So in this two-month period,

00:02:27.525 --> 00:02:30.218
what they call the secession winter,

00:02:30.218 --> 00:02:33.260
seven states in the South secede.

00:02:33.260 --> 00:02:36.023
- [Voiceover] At this
point, did they think that,

00:02:36.023 --> 00:02:38.786
this is an act of war by seceding,

00:02:38.786 --> 00:02:42.246
or did they think, "Hey, we
can secede if we want to"?

00:02:42.246 --> 00:02:44.335
- [Voiceover] I think
they don't really know

00:02:44.335 --> 00:02:48.120
whether the United States
is going to act decisively

00:02:48.120 --> 00:02:50.837
to try to keep them in the Union,

00:02:50.837 --> 00:02:53.484
they're just waiting to find out.

00:02:53.484 --> 00:02:58.104
- [Voiceover] At this
point, Lincoln is waiting

00:02:58.104 --> 00:03:00.682
to become president, James
Buchanan is the president.

00:03:00.682 --> 00:03:03.723
He's not really in a position
to do much of anything.

00:03:03.723 --> 00:03:07.253
When does this become an all out war?

00:03:07.253 --> 00:03:09.946
- [Voiceover] Lincoln
actually takes office in

00:03:09.946 --> 00:03:14.946
March of 1861, and shortly
there after, a southern fort

00:03:16.354 --> 00:03:19.483
off the coast of
Charleston, South Carolina,

00:03:19.483 --> 00:03:21.563
Fort Sumpter, they let him know they're

00:03:21.563 --> 00:03:23.174
running out of supplies.

00:03:23.174 --> 00:03:25.643
So, this is a federal fort, which reports

00:03:25.643 --> 00:03:28.012
to Washington D.C., which is in a way,

00:03:28.012 --> 00:03:30.357
sort of trapped behind this enemy line

00:03:30.357 --> 00:03:33.050
in South Carolina, the
state that has seceded.

00:03:33.050 --> 00:03:34.908
So, Lincoln says, "All
right, well I'm going

00:03:34.908 --> 00:03:37.113
to send you a ship to resupply you."

00:03:37.113 --> 00:03:40.991
When the ship gets to
the Charleston harbor

00:03:40.991 --> 00:03:45.031
southern forces from South
Carolina fire on the ship.

00:03:45.031 --> 00:03:48.700
- [Voiceover] Lincoln
wasn't naive, he knew that

00:03:48.700 --> 00:03:49.837
that these states had now seceded

00:03:49.837 --> 00:03:53.808
for a few months,
especially South Carolina.

00:03:53.808 --> 00:03:58.336
He knew that it was
going to be viewed as him

00:03:58.336 --> 00:04:03.336
replenishing a fort, from their point

00:04:04.094 --> 00:04:05.464
of view, on their territory.

00:04:05.464 --> 00:04:06.996
- [Voiceover] Right,
from their point of view.

00:04:06.996 --> 00:04:09.411
You could also have just
said, "Here's some hungry men,

00:04:09.411 --> 00:04:11.361
I'm getting them some food, let me put

00:04:11.361 --> 00:04:13.985
my boat into the harbor,
and we'll get them

00:04:13.985 --> 00:04:15.750
their supplies and then I'll leave."

00:04:15.750 --> 00:04:18.141
I think it was really
up to the southerners

00:04:18.141 --> 00:04:20.951
whether they used this as an opportunity

00:04:20.951 --> 00:04:23.366
to reject federal authority.

00:04:23.366 --> 00:04:26.407
- [Voiceover] And was it just
food? Or were there arms?

00:04:26.407 --> 00:04:28.126
- [Voiceover] Well, it was a federal fort,

00:04:28.126 --> 00:04:30.981
so it was everything they needed
to keep this fort running.

00:04:30.981 --> 00:04:33.071
- [Voiceover] So, it
seems like on both sides,

00:04:33.071 --> 00:04:34.789
people knew where this was going.

00:04:34.789 --> 00:04:38.458
If war breaks out, South
Carolina is going to say,

00:04:38.458 --> 00:04:39.828
"Hey, we don't want
this thing replenished,

00:04:39.828 --> 00:04:41.384
in fact, we want to take that fort. "

00:04:41.384 --> 00:04:44.216
From their point of view,
it's on their territory.

00:04:44.216 --> 00:04:45.493
Lincoln's like, "I'm not going to let this

00:04:45.493 --> 00:04:49.023
whole secession thing go,
I want to replenish the

00:04:49.023 --> 00:04:50.903
force that I'm probably going to need."

00:04:50.903 --> 00:04:53.481
And then what happens?

00:04:53.481 --> 00:04:56.426
- [Voiceover] So, after
the southerners fire

00:04:56.426 --> 00:05:00.698
on this Union boat, Lincoln
realizes that this rebellion

00:05:00.698 --> 00:05:04.831
is very serious and so he calls for 75,000

00:05:04.831 --> 00:05:07.478
volunteers to come into the army

00:05:07.478 --> 00:05:09.963
and suppress the rebellion.

00:05:09.963 --> 00:05:12.504
- [Voiceover] This is shortly after the...

00:05:12.504 --> 00:05:16.080
- [Voiceover] Right, this
is just days after the April

00:05:16.080 --> 00:05:19.330
firing on Fort Sumpter.

00:05:19.330 --> 00:05:20.941
- [Voiceover] so 75,000 volunteers,

00:05:20.941 --> 00:05:22.659
which is massive by any standards,

00:05:22.659 --> 00:05:25.190
but the country is a lot
smaller then than it is today.

00:05:25.190 --> 00:05:27.843
- [Voiceover] Yeah, there
were only about 30 million

00:05:27.843 --> 00:05:29.236
Americans at that time,

00:05:29.236 --> 00:05:32.603
so the country is
literally ten times smaller

00:05:32.603 --> 00:05:36.922
than it is today. So,
75,000 volunteers is a lot.

00:05:36.922 --> 00:05:39.336
- [Voiceover] And then
what does that catalyze?

00:05:39.336 --> 00:05:43.887
- [Voiceover] Well, now war
is serious. The United States

00:05:43.887 --> 00:05:46.720
is gathering an army
and the southern states

00:05:46.720 --> 00:05:49.460
know that this US army is going to invade.

00:05:49.460 --> 00:05:52.850
- [Voiceover] From their
point of view its an invasion,

00:05:52.850 --> 00:05:56.054
from Abraham Lincoln's point of view,

00:05:56.054 --> 00:05:58.376
they're viewed as, this is a rebellion,

00:05:58.376 --> 00:05:59.978
and we're just going
to quell the rebellion.

00:05:59.978 --> 00:06:02.788
- [Voiceover] Right, so
the southern states think,

00:06:02.788 --> 00:06:05.109
"We're our own country."
They've made themselves

00:06:05.109 --> 00:06:07.129
the Confederate States of America.

00:06:07.129 --> 00:06:09.846
Lincoln doesn't accept that
they're their own country,

00:06:09.846 --> 00:06:12.540
these are just some states
that are in rebellion

00:06:12.540 --> 00:06:14.258
as part of the United States.

00:06:14.258 --> 00:06:19.258
- [Voiceover] At that point,
is it all out? We've only

00:06:19.737 --> 00:06:20.922
had seven states secede.

00:06:20.922 --> 00:06:24.033
- [Voiceover] Right, so
there are a few states that

00:06:24.033 --> 00:06:28.561
are still on the fence after
Lincoln's inauguration.

00:06:28.561 --> 00:06:32.607
They feel strong cultural
unity with the United States,

00:06:32.607 --> 00:06:36.043
but these are slave states,
and so after Lincoln calls

00:06:36.043 --> 00:06:38.737
for these 75,000 volunteers
that kind of pushes

00:06:38.737 --> 00:06:39.828
them over the edge and says,

00:06:39.828 --> 00:06:41.755
"All right, this is going to be war.

00:06:41.755 --> 00:06:44.100
It's going to be a war over
whether slavery exists,

00:06:44.100 --> 00:06:46.678
we care about the existence of slavery,

00:06:46.678 --> 00:06:48.930
and so we're going to join with the

00:06:48.930 --> 00:06:51.368
Confederates, in the
Confederate States of America,

00:06:51.368 --> 00:06:52.459
and join the rebellion."

00:06:52.459 --> 00:06:55.664
Most notable among these is Virginia.

00:06:55.664 --> 00:06:58.752
Virginia is not on of the
original states that secedes.

00:06:58.752 --> 00:07:01.236
Most of the action in
the Civil War is actually

00:07:01.236 --> 00:07:05.114
going to take place in
Virginia, and Virginia's

00:07:05.114 --> 00:07:07.366
greatest general, Robert E. Lee,

00:07:07.366 --> 00:07:11.824
is then pushed to, in his
mind, defend Virginia,

00:07:11.824 --> 00:07:13.449
and become the leading general

00:07:13.449 --> 00:07:15.632
for the Confederate States of America.

00:07:15.632 --> 00:07:17.280
- [Voiceover] Geographically,
it's right at the border,

00:07:17.280 --> 00:07:22.064
obviously their home to
a major military figure,

00:07:22.064 --> 00:07:26.011
and they were an
economically powerful state.

00:07:26.011 --> 00:07:27.311
- [Voiceover] Yes, absolutely.

00:07:27.311 --> 00:07:30.306
- [Voiceover] So then at
that point it's all out war.

00:07:30.306 --> 00:07:35.306
- [Voiceover] It's all out
war, and in July of that year,

00:07:35.693 --> 00:07:37.852
there's really the first
battle of the Civil War.

00:07:37.852 --> 00:07:42.852
First couple of months, everyone
is just gearing up for war.

00:07:43.286 --> 00:07:47.210
The forces of the United
States and the forces

00:07:47.210 --> 00:07:50.321
of the South, finally meet in battle,

00:07:50.321 --> 00:07:53.061
what we call the Battle of Bull Run,

00:07:53.061 --> 00:07:56.962
which is in Manassas, Virginia,
northern Virginia today.

00:07:56.962 --> 00:08:00.561
People really think at
this point that it's

00:08:00.561 --> 00:08:03.533
going to be a crushing
defeat for the South,

00:08:03.533 --> 00:08:06.808
in fact people don't even
really know what war is like.

00:08:06.808 --> 00:08:11.683
There are stories that
civilians from nearby

00:08:11.683 --> 00:08:16.683
packed a picnic to come
sit on a hill above

00:08:16.953 --> 00:08:18.695
this battle and watch it, sort of like

00:08:18.695 --> 00:08:20.622
we might watch a football game now.

00:08:20.622 --> 00:08:22.944
They were quickly disillusioned of what

00:08:22.944 --> 00:08:24.430
war was actually like, they had to

00:08:24.430 --> 00:08:26.868
run for their lives because this was

00:08:26.868 --> 00:08:30.281
a violent battle and it showed that it was

00:08:30.281 --> 00:08:31.813
not going to be a quick war.

00:08:31.813 --> 00:08:34.135
The South decisively wins this battle,

00:08:34.135 --> 00:08:37.084
and from that point forward they know

00:08:37.084 --> 00:08:39.290
it's going to be a very
long and bloody conflict.

00:08:39.290 --> 00:08:43.887
- [Voiceover] You mentioned
the first seven states secede,

00:08:43.887 --> 00:08:48.340
and then you have Lincoln
building up a real army,

00:08:48.340 --> 00:08:50.476
then you have four more states secede,

00:08:50.476 --> 00:08:52.380
even those some of these states feel

00:08:52.380 --> 00:08:53.936
an affinity to the United States,

00:08:53.936 --> 00:08:55.677
they are slave states,

00:08:55.677 --> 00:08:57.163
but not all the slave states left.

00:08:57.163 --> 00:08:59.346
- [Voiceover] Right, there are four

00:08:59.346 --> 00:09:00.878
more slave states that stick around.

00:09:00.878 --> 00:09:04.268
Most important of those is Maryland.

00:09:04.268 --> 00:09:06.288
Maryland, of course, is right on the

00:09:06.288 --> 00:09:08.099
border of Washington D.C.

00:09:08.099 --> 00:09:09.283
- [Voiceover] D.C. is sandwiched

00:09:09.283 --> 00:09:10.584
between Maryland and Virginia.

00:09:10.584 --> 00:09:12.650
- [Voiceover] Right, and
so if Maryland leaves

00:09:12.650 --> 00:09:15.878
and joins the rebellion,
then Washington D.C.

00:09:15.878 --> 00:09:17.874
is in real trouble because they will be

00:09:17.874 --> 00:09:19.894
completely surrounded by enemy territory.