Estimating when subtracting large numbers

WEBVTT
Kind: captions
Language: en

00:00:00.400 --> 00:00:02.930
- [Narrator] Let's say that
you have a jar of jelly beans,

00:00:02.930 --> 00:00:07.920
and you know that there
are exactly 282 jelly beans

00:00:07.920 --> 00:00:09.970
in that jar of jelly beans.

00:00:09.970 --> 00:00:12.850
And then the next day you come
and you see there are fewer

00:00:12.850 --> 00:00:15.070
and you say, what happened,
and let's say someone

00:00:15.070 --> 00:00:18.040
who lives with you or
your friend says, oh yeah,

00:00:18.040 --> 00:00:23.040
I counted exactly and I ate
59 of those jelly beans.

00:00:23.340 --> 00:00:28.340
So you started with 282, 59 are taken out,

00:00:28.720 --> 00:00:31.630
and you don't need to know
exactly how many jelly beans

00:00:31.630 --> 00:00:34.270
are left, but you wanna estimate roughly

00:00:34.270 --> 00:00:36.640
how many jelly beans are
left, and that's why,

00:00:36.640 --> 00:00:38.460
as this title of the video says,

00:00:38.460 --> 00:00:42.140
we're going to estimate when
subtracting large numbers.

00:00:42.140 --> 00:00:44.360
And we can debate how large is large,

00:00:44.360 --> 00:00:46.230
but these are reasonably large.

00:00:46.230 --> 00:00:49.010
So let's do that, let's
see if we can estimate

00:00:49.010 --> 00:00:52.420
what 282 minus 59 is.

00:00:52.420 --> 00:00:54.140
And sometimes when we're estimating,

00:00:54.140 --> 00:00:56.920
we will use this squiggly equal sign.

00:00:56.920 --> 00:00:58.990
This means approximately,
which is another,

00:00:58.990 --> 00:01:02.670
just a fancy way of saying
roughly what is this equal to.

00:01:02.670 --> 00:01:05.680
So the way that I would
approach this, I would say hey,

00:01:05.680 --> 00:01:10.260
are these close to numbers that
are easier to subtract with.

00:01:10.260 --> 00:01:14.730
So for example, 282,
I'd say, you know what,

00:01:14.730 --> 00:01:17.830
maybe I can round to the nearest 10 here,

00:01:17.830 --> 00:01:22.640
and so I could round it
down to 280, if I round down

00:01:22.640 --> 00:01:23.473
to the nearest 10.

00:01:23.473 --> 00:01:27.490
If I rounded up, it would
be 290, but 282 is closer

00:01:27.490 --> 00:01:30.420
to 280 than it is to 290.

00:01:30.420 --> 00:01:34.953
So I would round this,
when I'm estimating to 280.

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And then what about 59?

00:01:38.470 --> 00:01:40.840
If I were to round to the nearest 10,

00:01:40.840 --> 00:01:42.040
what would I round that to?

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Pause the video and think about that.

00:01:44.610 --> 00:01:47.790
Well, 59 if I round
down to the nearest 10,

00:01:47.790 --> 00:01:50.190
it would be 50, and if I
round up to the nearest 10,

00:01:50.190 --> 00:01:54.240
it would be 60, and 59 is
definitely closer to 60.

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So I would say that this is
roughly, if I'm rounding,

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I could say, hey I'm
gonna round that to 60.

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So I could say the 282
minus 59 is roughly going

00:02:03.930 --> 00:02:07.580
to be 280 minus 60.

00:02:07.580 --> 00:02:08.700
And now what is this?

00:02:08.700 --> 00:02:10.410
You might be able to do this in your head.

00:02:10.410 --> 00:02:15.290
You could view this as
28 tens minus six tens,

00:02:15.290 --> 00:02:18.410
and so if you have 28 of
something minus six of it,

00:02:18.410 --> 00:02:21.770
you're going to be left
with 22 of that something.

00:02:21.770 --> 00:02:25.270
And so you could say that
this is going to be 22 tens.

00:02:25.270 --> 00:02:28.400
22 tens, or another way to think about it.

00:02:28.400 --> 00:02:31.610
You have two 100s and then 8 tens,

00:02:31.610 --> 00:02:33.500
and then you're gonna take
six tens away from it,

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so then you'll be left
with two 100s and two tens.

00:02:37.870 --> 00:02:39.410
Just like we have here.

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So the whole point here, this is not,

00:02:41.290 --> 00:02:46.010
this is actually not exactly
what 282 minus 59 is,

00:02:46.010 --> 00:02:49.730
but if you just said, hey
ballpark do I, I wanna know,

00:02:49.730 --> 00:02:52.080
do I have roughly how many
jelly beans do I have,

00:02:52.080 --> 00:02:53.650
well this could be useful.

00:02:53.650 --> 00:02:55.260
And you'll see later in
life this is actually

00:02:55.260 --> 00:02:57.120
a very, very usefully
skill where you might have

00:02:57.120 --> 00:02:59.770
these numbers that get
a lot larger than this,

00:02:59.770 --> 00:03:01.490
but you'll round to the nearest 10,

00:03:01.490 --> 00:03:03.110
or you'll round to other places,

00:03:03.110 --> 00:03:05.983
so that you can do that
computation in your head.


Place value tables

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.630
- [Instructor] What were
going to do in this video

00:00:01.630 --> 00:00:04.210
is think a little bit
about place value tables

00:00:04.210 --> 00:00:07.310
and sometimes they're referred
to as place value charts.

00:00:07.310 --> 00:00:09.280
In previous videos, we've
already talked about

00:00:09.280 --> 00:00:11.091
the idea of place value,

00:00:11.091 --> 00:00:14.470
and a place value table
or a place value chart

00:00:14.470 --> 00:00:17.920
is just a way to say how much we have,

00:00:17.920 --> 00:00:20.240
how much value we have in each place

00:00:20.240 --> 00:00:21.950
in a very, very clear way.

00:00:21.950 --> 00:00:25.667
So, here they say use the place
value chart to write 60,229.

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And, this is the place value
table or place value chart

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right over here.

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They say, enter a digit,

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one-digit number in each box
of the place value chart.

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So really, all they're saying is

00:00:37.590 --> 00:00:39.160
how many 10 thousands do we have?

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How many thousands?

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How many hundreds?

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How many tens?

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And, how many ones?

00:00:44.470 --> 00:00:46.740
And like always, it's always a good idea

00:00:46.740 --> 00:00:49.030
to try to have a go at it
before I do it with you.

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So, why don't you do that?

00:00:49.940 --> 00:00:51.060
Why don't you pause the video

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and on a little piece of scratch paper

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say how many 10 thousands are in 60,229?

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How many thousands are there?

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How many hundreds?

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How many tens?

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And, how many ones?

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All right, now let's do this together.

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So, we could start at the left
in the 10 thousands place.

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This is the ones place.

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This is the tens place.

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Let me do that a little bit neater.

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So, this is ones, tens, hundreds,

00:01:15.880 --> 00:01:17.644
thousands, 10 thousands.

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So, we can see that we
have six 10 thousands.

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So, all we would do is
say, we'd put a six there.

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We have six 10 thousands.

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And then, how many thousands?

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Well, that would be one
place over to the right.

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This is interesting.

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We can see very clearly,

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we actually have zero thousands.

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So, zero thousands.

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You might say, hey.

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When you say this out you say 60,229.

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Don't we have 60 thousands?

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Well, you already accounted for that

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in the six 10 thousands.

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Six 10 thousands is 60,000.

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But just in the thousands
place, you have none.

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And then, we move on
to the hundreds place.

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You can see very clearly
we have two hundreds,

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and then we can go to the tens place.

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And, I've written over it a little bit,

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but we have two tens.

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And then last but not least,

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we have nine ones.

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And, we're done.

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And so, the whole point of this video

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is to just get you comfortable
with this type of format

00:02:13.080 --> 00:02:15.400
with a place value table
or a place value chart,

00:02:15.400 --> 00:02:16.410
and all it is,

00:02:16.410 --> 00:02:18.507
is a way of really clearly writing out

00:02:18.507 --> 00:02:20.808
how much value you have in each place

00:02:20.808 --> 00:02:23.643
in a number like this.


Geometric series introduction

WEBVTT
Kind: captions
Language: en

00:00:00.370 --> 00:00:01.960
- [Instructor] In this
video we're gonna study

00:00:01.960 --> 00:00:05.650
geometric series, and to
understand that I'm gonna construct

00:00:05.650 --> 00:00:07.380
a little bit of a table to understand

00:00:07.380 --> 00:00:11.327
how our money could grow if
we keep depositing, let's say,

00:00:11.327 --> 00:00:14.100
$1,000 a year in a bank account.

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So, let's say this is the year,

00:00:17.560 --> 00:00:19.220
and we're gonna think
about how much we have

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at the beginning of the year,

00:00:21.250 --> 00:00:26.040
and then this is the
dollars in our account.

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And let's say that the bank
is always willing to give us

00:00:29.330 --> 00:00:32.460
5% per year, which is pretty good.

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It's very hard to find a bank account

00:00:34.040 --> 00:00:37.210
that will actually give
you 5% growth per year.

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So that means if you put
$100 in at the end of a year,

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or exactly a year later it'd be $105.

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If you put $1,000 in, a
year later, it'd be 1,050.

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It'd be 5% larger.

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And so, let's say that we want
to put $1,000 in per year,

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and I want to think about, well,

00:00:53.540 --> 00:00:55.840
what is going to be my balance

00:00:55.840 --> 00:00:59.140
at the beginning of year one,
at the beginning of year two,

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at the beginning of
year three, and then see

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if we can come up with
a general expression

00:01:03.320 --> 00:01:07.050
for the beginning of year n.

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So, year one, right at
the beginning of the year,

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I put in $1,000 in the account.

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That's pretty straightforward.

00:01:15.620 --> 00:01:17.840
But then what happens in year two?

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I'm going to deposit $1,000,
but then that original $1,000

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that I have would have grown.

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So, I'm going to deposit $1,000,

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and then the original $1,000 that I put

00:01:29.790 --> 00:01:33.610
at the beginning of year one,
that has now grown by 5%.

00:01:33.610 --> 00:01:38.033
Growing by 5% is the same
thing as multiplying by 1.05.

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So, this is now going to be plus $1,000

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times 1.05.

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Fairly straightforward.

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Now, what about the
beginning of year three?

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How much would I have in the bank account

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right when I've made that
first, that year three deposit?

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Pause this video.

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See if you can figure that out.

00:02:00.170 --> 00:02:02.400
Well, just like at the
beginning of year two

00:02:02.400 --> 00:02:03.650
and the beginning of year one,

00:02:03.650 --> 00:02:06.870
we're going to make $1,000 deposit,

00:02:06.870 --> 00:02:11.610
but now the money from
year two has grown by 5%,

00:02:11.610 --> 00:02:16.610
so this is now going to
be $1,000 times 1.05.

00:02:18.040 --> 00:02:20.590
And then that money that
we originally deposited

00:02:20.590 --> 00:02:25.590
from year one, that was
1,000 times 1.05 in year two,

00:02:25.650 --> 00:02:28.080
that's going to grow by another 5%.

00:02:28.080 --> 00:02:30.990
And so, this is going to be plus 1,000

00:02:31.890 --> 00:02:35.710
times 1.05 times 1.05.

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We're growing by another 5%.

00:02:37.830 --> 00:02:39.600
Well, we could just rewrite this part

00:02:39.600 --> 00:02:43.760
right over here as 1.05 squared.

00:02:43.760 --> 00:02:45.410
So, do you see a general pattern

00:02:45.410 --> 00:02:47.340
that's going to happen here?

00:02:47.340 --> 00:02:49.870
Well, as you go to year n, in
fact, pause the video again,

00:02:49.870 --> 00:02:51.460
see if you can write a general expression.

00:02:51.460 --> 00:02:52.380
You're gonna have to do a little bit

00:02:52.380 --> 00:02:55.150
of this dot, dot, dot
action in order to do it.

00:02:55.150 --> 00:02:58.100
But see if you could write a
general expression for year n.

00:02:58.970 --> 00:03:03.333
Well, for year n, you're going
to make that original $1,000

00:03:05.230 --> 00:03:08.590
at the beginning of year n,
and then you're going to have

00:03:08.590 --> 00:03:12.660
1,000, 1,000, times 1.05

00:03:14.100 --> 00:03:15.800
for that $1,000 that you deposit

00:03:15.800 --> 00:03:18.360
at the beginning of year n minus one.

00:03:18.360 --> 00:03:21.980
And then this is just going to keep going,

00:03:21.980 --> 00:03:26.450
and it's going to go all
the way to plus $1,000

00:03:26.450 --> 00:03:31.450
to times 1.05 to the power

00:03:31.810 --> 00:03:34.420
of the number of years
you've been compounding.

00:03:34.420 --> 00:03:36.160
So, you could do this $1,000 as the one

00:03:36.160 --> 00:03:37.260
that you put in year one,

00:03:37.260 --> 00:03:40.230
and then how many years has it compounded?

00:03:40.230 --> 00:03:41.470
Well, when you go from one to two,

00:03:41.470 --> 00:03:42.680
you've compounded one year.

00:03:42.680 --> 00:03:45.110
When you go from one to three,
you've compounded two years.

00:03:45.110 --> 00:03:47.640
So, when we're talking about
the beginning of year n,

00:03:47.640 --> 00:03:50.320
you go up to the exponent
that is one less than that.

00:03:50.320 --> 00:03:54.360
And so, this is going to be
to the n minus one power.

00:03:54.360 --> 00:03:56.810
So, what we just did here
is we've just constructed

00:03:56.810 --> 00:03:58.720
each one of these when we're saying,

00:03:58.720 --> 00:04:00.560
okay, how much money do we
have in our bank account

00:04:00.560 --> 00:04:02.430
at the beginning of year three?

00:04:02.430 --> 00:04:03.880
Or how much do we have in our bank account

00:04:03.880 --> 00:04:05.720
at the beginning of year n?

00:04:05.720 --> 00:04:08.600
These are geometric series,

00:04:08.600 --> 00:04:11.293
and I will write that word down.

00:04:11.293 --> 00:04:16.293
Geo, geometric series.

00:04:16.570 --> 00:04:18.610
Now, just as a little bit of a review,

00:04:18.610 --> 00:04:20.970
or it might not be review,
it might be a primer,

00:04:20.970 --> 00:04:23.750
series are related to sequences,

00:04:23.750 --> 00:04:27.700
and you can really view
series as sums of sequences.

00:04:27.700 --> 00:04:30.210
Sequences, and let me go down a little bit

00:04:30.210 --> 00:04:33.340
so that you can, so we have
a little bit more space,

00:04:33.340 --> 00:04:36.350
a sequence is an ordered list of numbers.

00:04:36.350 --> 00:04:38.730
A sequence might be something like,

00:04:38.730 --> 00:04:42.160
well, let's say we have
a geometric sequence,

00:04:42.160 --> 00:04:44.780
and a geometric sequence, each successive

00:04:44.780 --> 00:04:48.560
term is the previous term
times a fixed number.

00:04:48.560 --> 00:04:49.700
So, let's say we start at two,

00:04:49.700 --> 00:04:51.610
and every time we multiply by three.

00:04:51.610 --> 00:04:54.570
So, we'll go from two,
two times three is six,

00:04:54.570 --> 00:04:59.170
six times three is 18,
18 times three is 54.

00:04:59.170 --> 00:05:03.040
This is a geometric sequence,
ordered list of numbers.

00:05:03.040 --> 00:05:06.890
Now, if we want to think
about the geometric series,

00:05:06.890 --> 00:05:09.090
or the one that's analogous to this,

00:05:09.090 --> 00:05:10.840
is that we would sum the terms here.

00:05:10.840 --> 00:05:15.840
So, this would be two
plus six plus 18 plus 54.

00:05:16.330 --> 00:05:18.310
Or we could even write it,
and this would look similar

00:05:18.310 --> 00:05:20.560
to what we had just done with
our little savings example

00:05:20.560 --> 00:05:24.480
is this is two plus two times three,

00:05:24.480 --> 00:05:27.330
plus two times three squared,

00:05:27.330 --> 00:05:32.000
plus two times three to the third power.

00:05:32.000 --> 00:05:34.310
And so, with the geometric series,

00:05:34.310 --> 00:05:38.600
you're going to have a sum
where each successive term

00:05:38.600 --> 00:05:42.020
in the expression is equal to,
if you put 'em all in order,

00:05:42.020 --> 00:05:44.430
is going to be equal to the term before it

00:05:44.430 --> 00:05:46.910
times a fixed amount.

00:05:46.910 --> 00:05:49.720
So, the second term is equal
to the first term times three,

00:05:49.720 --> 00:05:51.630
and we're summing them in a sequence.

00:05:51.630 --> 00:05:53.010
You're just looking at it.

00:05:53.010 --> 00:05:54.630
It's an ordered list, so to speak,

00:05:54.630 --> 00:05:58.270
but here you are actually
adding up the ordered list.

00:05:58.270 --> 00:06:02.810
So, what we just saw
in this example is one,

00:06:02.810 --> 00:06:05.560
what a geometric series is, but also,

00:06:05.560 --> 00:06:07.740
a famous example of how it's useful,

00:06:07.740 --> 00:06:09.210
and this is just scratching the surface.

00:06:09.210 --> 00:06:12.140
If you were to go further
in finance or in business,

00:06:12.140 --> 00:06:14.330
you'll actually see geometric series

00:06:14.330 --> 00:06:16.553
popping up all over the place.


Plessy v. Ferguson

WEBVTT
Kind: captions
Language: en

00:00:00.140 --> 00:00:02.760
- [Kim] Long before Rosa
Parks refused to move

00:00:02.760 --> 00:00:06.190
to the back of the bus, Homer
Plessy boarded a train car

00:00:06.190 --> 00:00:10.010
in New Orleans to protest
Jim Crow segregation laws.

00:00:10.010 --> 00:00:12.930
Plessy was arrested and
convicted in Louisiana,

00:00:12.930 --> 00:00:16.240
but his test case for
segregated public transportation

00:00:16.240 --> 00:00:19.430
reached the Supreme Court in 1896.

00:00:19.430 --> 00:00:21.090
This is Kim from Khan Academy.

00:00:21.090 --> 00:00:24.200
And today, we're learning
more about the landmark case,

00:00:24.200 --> 00:00:26.670
Plessy versus Ferguson, which asks whether

00:00:26.670 --> 00:00:29.120
separate but equal accommodations

00:00:29.120 --> 00:00:31.190
for Black and White Americans

00:00:31.190 --> 00:00:34.520
violated the 14th Amendment
of the Constitution.

00:00:34.520 --> 00:00:37.990
To learn more about this case,
I spoke with two experts.

00:00:37.990 --> 00:00:40.890
Jamal Greene is the
Dwight Professor of Law

00:00:40.890 --> 00:00:42.580
at Columbia Law School.

00:00:42.580 --> 00:00:45.710
Earl Maltz is a Distinguished
Professor of Law

00:00:45.710 --> 00:00:47.300
at Rutgers Law School.

00:00:47.300 --> 00:00:50.120
So Professor Greene, could
you kind of set the stage

00:00:50.120 --> 00:00:51.510
for us in this time period?

00:00:51.510 --> 00:00:53.770
After the Civil War, what was the legal

00:00:53.770 --> 00:00:56.410
and social status of former slaves?

00:00:56.410 --> 00:01:00.930
- [Jamal] Well, of course,
the Civil War ended in 1865.

00:01:00.930 --> 00:01:04.610
And it was fought in large
part over the institution

00:01:04.610 --> 00:01:09.540
of chattel slavery, so slavery
of, generally speaking,

00:01:09.540 --> 00:01:11.970
Black or African American slaves.

00:01:11.970 --> 00:01:14.430
And right at the end of the Civil War,

00:01:14.430 --> 00:01:16.040
the 13th Amendment was passed.

00:01:16.040 --> 00:01:18.920
And the 13th Amendment basically said that

00:01:18.920 --> 00:01:21.786
there shouldn't be any slavery
or involuntary servitude

00:01:21.786 --> 00:01:23.630
in the United States.

00:01:23.630 --> 00:01:27.130
So the institution of
slavery itself had ended,

00:01:27.130 --> 00:01:30.320
but the passage of the
13th Amendment did not mean

00:01:30.320 --> 00:01:32.960
that former slaves had equal rights.

00:01:32.960 --> 00:01:35.970
A number of the former
states of the Confederacy,

00:01:35.970 --> 00:01:38.610
the, generally speaking, Southern states,

00:01:38.610 --> 00:01:41.600
passed a number of racially
discriminatory laws

00:01:41.600 --> 00:01:44.730
immediately after the end of slavery

00:01:44.730 --> 00:01:47.770
that prevented Black
Americans from participating

00:01:47.770 --> 00:01:50.970
in civil society on
equal terms with Whites.

00:01:50.970 --> 00:01:53.820
For example, laws restricting
the ability of Blacks

00:01:53.820 --> 00:01:56.390
to enter into and enforce contracts,

00:01:56.390 --> 00:01:59.260
restricting the ability
of Blacks to own property,

00:01:59.260 --> 00:02:02.860
to sit on juries, to
vote, to testify in court,

00:02:02.860 --> 00:02:03.693
and so forth.

00:02:03.693 --> 00:02:07.610
So there were a number of
openly discriminatory laws.

00:02:07.610 --> 00:02:11.200
There were also laws that
required Blacks to be employed

00:02:11.200 --> 00:02:15.520
on pain of having their labor forced,

00:02:15.520 --> 00:02:18.470
so ways of essentially re-instituting

00:02:18.470 --> 00:02:19.447
the institution of slavery.

00:02:19.447 --> 00:02:22.030
- [Kim] And these were
known as the Black Codes.

00:02:22.030 --> 00:02:24.160
- [Jamal] Those were known
as the Black Codes, exactly.

00:02:24.160 --> 00:02:26.710
- [Earl] In the Reconstruction period

00:02:26.710 --> 00:02:31.710
between say the late
1860s and the mid 1870s,

00:02:32.870 --> 00:02:35.450
there was a concerted effort
by the federal government

00:02:35.450 --> 00:02:39.260
to improve the social status

00:02:39.260 --> 00:02:43.040
and political rights of African Americans.

00:02:43.040 --> 00:02:45.840
In 1876, there was part of the settlement

00:02:45.840 --> 00:02:49.620
of the presidential election of 1876,

00:02:49.620 --> 00:02:52.740
the federal government drew back some.

00:02:52.740 --> 00:02:57.740
But most of the so-called
Redeemer movement

00:02:57.820 --> 00:03:00.190
really took off in the 1890s.

00:03:00.190 --> 00:03:04.540
I think 1891 is when the
last real effort is made

00:03:05.430 --> 00:03:06.470
by the federal government

00:03:06.470 --> 00:03:08.940
to have a serious Voting Rights Act.

00:03:08.940 --> 00:03:12.440
After that, the South is
pretty much under control

00:03:12.440 --> 00:03:17.440
of the people who sympathized
with ex-Confederates.

00:03:17.440 --> 00:03:19.400
- [Jamal] In both Northern
and Southern states,

00:03:19.400 --> 00:03:22.110
there was widespread racial segregation.

00:03:22.110 --> 00:03:25.050
So there were laws that
were basically codifying

00:03:25.050 --> 00:03:29.160
long-existing social practices
of segregated housing,

00:03:29.160 --> 00:03:32.530
segregated schools, and
segregated public conveyances

00:03:32.530 --> 00:03:35.150
like steamships and rail cars.

00:03:35.150 --> 00:03:38.060
But much of that changed in the years

00:03:38.060 --> 00:03:40.390
immediately following the Civil War.

00:03:40.390 --> 00:03:44.230
Congress passed a number of federal laws

00:03:44.230 --> 00:03:46.740
that banned racial discrimination,

00:03:46.740 --> 00:03:49.340
particularly in
contracting and in housing.

00:03:49.340 --> 00:03:53.270
Quite significantly, in addition
to the actual federal laws

00:03:53.270 --> 00:03:55.340
that Congress passed, the country passed

00:03:55.340 --> 00:03:57.450
and ratified the 14th Amendment.

00:03:57.450 --> 00:03:59.800
- [Kim] So I think one
thing that is very hard

00:03:59.800 --> 00:04:03.420
for me to understand and
that I've seen students

00:04:03.420 --> 00:04:07.520
struggle with is you have the
passage of the 14th Amendment

00:04:07.520 --> 00:04:11.550
and the 15th Amendment in 1868, 1870.

00:04:11.550 --> 00:04:15.510
And these are supposed to
guarantee equal protection

00:04:15.510 --> 00:04:19.020
and citizenship and voting
rights for African Americans,

00:04:19.020 --> 00:04:22.130
specifically men in the 15th Amendment.

00:04:22.130 --> 00:04:25.060
And then, you have Jim Crow.

00:04:25.060 --> 00:04:30.060
So how did we get from this
moment after the Civil War

00:04:30.570 --> 00:04:33.150
where things really seem
like they're looking up

00:04:33.150 --> 00:04:35.710
in terms of African American citizenship

00:04:35.710 --> 00:04:38.410
to the system of Jim Crow
that's going to persist

00:04:38.410 --> 00:04:40.430
into the 1960s and 70s.

00:04:40.430 --> 00:04:42.510
- [Jamal] The Civil
War did not end racism.

00:04:42.510 --> 00:04:44.820
It simply ended slavery.

00:04:44.820 --> 00:04:48.760
And so, we're still
living in a racist society

00:04:48.760 --> 00:04:51.800
in which residential and
school segregation remained

00:04:51.800 --> 00:04:53.730
both in Southern states
and in Northern states

00:04:53.730 --> 00:04:57.810
notwithstanding the
Civil Rights Amendments.

00:04:57.810 --> 00:05:00.400
And in Southern states, Reconstruction,

00:05:00.400 --> 00:05:04.310
the process of trying
to bring former slaves

00:05:04.310 --> 00:05:08.180
fully into civil society,
was enforced by the presence

00:05:08.180 --> 00:05:10.500
of federal troops in Southern states

00:05:10.500 --> 00:05:15.020
on the theory, the very
well-founded theory,

00:05:15.020 --> 00:05:17.940
that states that had
just gone to war in order

00:05:17.940 --> 00:05:21.270
to perpetuate the institution
of slavery were not going

00:05:21.270 --> 00:05:25.880
to willingly adopt equal
rights for the former slaves

00:05:25.880 --> 00:05:28.520
that they had just been
holding in bondage.

00:05:28.520 --> 00:05:31.570
And so, there was a
federal military occupation

00:05:31.570 --> 00:05:34.650
of a number of former Confederate states

00:05:34.650 --> 00:05:38.680
for a good decade plus after the Civil War

00:05:38.680 --> 00:05:41.630
really ending in 1877.

00:05:41.630 --> 00:05:43.610
- [Earl] And at that point,

00:05:43.610 --> 00:05:45.530
again through the Redeemer movement,

00:05:45.530 --> 00:05:48.973
people who were the White power structure,

00:05:49.950 --> 00:05:54.350
most of which had sympathized

00:05:54.350 --> 00:05:57.460
with the Secessionist Movement,

00:05:57.460 --> 00:06:00.640
the White power structure
and its successors

00:06:00.640 --> 00:06:05.110
took power back in the
beginning and mid 1870s

00:06:07.020 --> 00:06:09.000
in the Southern states.

00:06:09.000 --> 00:06:11.723
And as part of their campaign,

00:06:12.800 --> 00:06:15.420
they imposed the Jim Crow system.

00:06:15.420 --> 00:06:18.250
- [Jamal] Slavery was
not just about labor.

00:06:18.250 --> 00:06:21.540
It was really a system
of racial hierarchy.

00:06:21.540 --> 00:06:23.610
And many in the United
States remained committed

00:06:23.610 --> 00:06:27.160
to that system even after
bondage itself ended.

00:06:27.160 --> 00:06:29.130
And if you don't have the political will

00:06:29.130 --> 00:06:31.680
within the Northern states to enforce

00:06:31.680 --> 00:06:34.800
the Reconstruction
Amendments, you had really

00:06:34.800 --> 00:06:38.330
a retrenchment of deep racial inequality

00:06:38.330 --> 00:06:39.630
within the Southern states,

00:06:39.630 --> 00:06:41.420
but not just within the Southern states,

00:06:41.420 --> 00:06:44.430
but also within a number
of Northern states as well.

00:06:44.430 --> 00:06:47.330
- [Kim] So let's kind
of dial into the case

00:06:47.330 --> 00:06:48.940
Plessy versus Ferguson.

00:06:48.940 --> 00:06:50.990
Who was Homer Plessy?

00:06:50.990 --> 00:06:54.663
And why did he take
issue with segregation?

00:06:55.700 --> 00:06:57.810
- [Earl] Well the law that was at issue

00:06:57.810 --> 00:07:01.960
required what was in
theory separate but equal

00:07:01.960 --> 00:07:06.350
accommodation of African
Americans and Whites

00:07:09.730 --> 00:07:11.783
on public transportation.

00:07:12.840 --> 00:07:15.500
Homer Plessy objected to it because A,

00:07:15.500 --> 00:07:19.820
the facilities weren't really equal.

00:07:19.820 --> 00:07:23.850
And he objected to it in part because

00:07:23.850 --> 00:07:26.510
he was classified as
Black, but also in general

00:07:26.510 --> 00:07:29.530
because he thought that
that was demeaning.

00:07:29.530 --> 00:07:34.000
Obviously, a majority of
members of the Supreme Court

00:07:36.080 --> 00:07:40.377
believed that the Southern
states should have

00:07:41.430 --> 00:07:45.990
at least some leeway
and to establish their

00:07:47.500 --> 00:07:49.280
the system of racial segregation.

00:07:49.280 --> 00:07:53.880
- [Jamal] And so, Plessy was
in league with the railroad

00:07:53.880 --> 00:07:55.830
and with the civil rights organization

00:07:55.830 --> 00:07:58.660
that recruited him to set up a case.

00:07:58.660 --> 00:08:03.430
So he agreed with the railroad to board

00:08:03.430 --> 00:08:08.370
the White area of the railway car

00:08:08.370 --> 00:08:12.620
on a car going from New Orleans
to a town called Covington.

00:08:12.620 --> 00:08:16.150
And it was agreed that the
railway would ask him to leave.

00:08:16.150 --> 00:08:17.810
He would refuse.

00:08:17.810 --> 00:08:19.590
And then, he would be arrested.

00:08:19.590 --> 00:08:23.030
And once he was arrested,
that would enable him

00:08:23.030 --> 00:08:25.810
to challenge the law under
which he was arrested

00:08:25.810 --> 00:08:26.960
under the Constitution.

00:08:26.960 --> 00:08:29.100
The railroads didn't like this law.

00:08:29.100 --> 00:08:31.430
They didn't like this law
because they didn't want

00:08:31.430 --> 00:08:35.531
to be subject to fines or liability

00:08:35.531 --> 00:08:39.040
for not properly
maintaining separate cars.

00:08:39.040 --> 00:08:41.620
It was really up to the
conductors to make sure

00:08:41.620 --> 00:08:43.407
that separate cars were maintained.

00:08:43.407 --> 00:08:46.030
And the conductors
themselves could be fined

00:08:46.030 --> 00:08:48.480
by the state for not doing so.

00:08:48.480 --> 00:08:50.730
And they could also be fined by passengers

00:08:50.730 --> 00:08:54.300
for mistakenly putting
someone in the wrong car.

00:08:54.300 --> 00:08:56.300
So the railroads didn't
really wanna be bothered

00:08:56.300 --> 00:08:57.840
with this kind of law.

00:08:57.840 --> 00:09:00.620
And so, this particular
railroad, Eastern Railway,

00:09:00.620 --> 00:09:03.790
was willing to agree to set up a situation

00:09:03.790 --> 00:09:05.030
to challenge the law.

00:09:05.030 --> 00:09:08.870
- [Kim] So Homer Plessy,
he gets on this train

00:09:08.870 --> 00:09:10.810
and he challenges the statute.

00:09:10.810 --> 00:09:13.540
I believe he sat in a Whites-only car

00:09:13.540 --> 00:09:15.650
and announced that he
was African American.

00:09:15.650 --> 00:09:17.210
And then, he was arrested.

00:09:17.210 --> 00:09:19.340
So what happened next?

00:09:19.340 --> 00:09:20.190
- [Jamal] He's arrested.

00:09:20.190 --> 00:09:23.630
And then, he is eventually
charged with a crime,

00:09:23.630 --> 00:09:25.723
with a violation of the statute.

00:09:27.308 --> 00:09:30.880
And there's a fine associated
with violating the state law.

00:09:30.880 --> 00:09:35.830
And his lawyers bring a claim

00:09:35.830 --> 00:09:39.560
that the law violates
the Federal Constitution.

00:09:39.560 --> 00:09:43.280
So initially, it goes through
the state courts of Louisiana.

00:09:43.280 --> 00:09:46.310
And then eventually, they
rule against Homer Plessy

00:09:46.310 --> 00:09:47.480
in favor of the law.

00:09:47.480 --> 00:09:49.689
And then, his lawyers appealed the case

00:09:49.689 --> 00:09:51.810
to the U.S. Supreme Court.

00:09:51.810 --> 00:09:53.220
- [Kim] So how did the court rule?

00:09:53.220 --> 00:09:56.120
- [Jamal] In the years
leading up to the case,

00:09:56.120 --> 00:10:00.250
the lawyers for Homer
Plessy were quite concerned

00:10:00.250 --> 00:10:02.090
about the composition of the court

00:10:02.090 --> 00:10:04.570
because they weren't
sure if they could count

00:10:04.570 --> 00:10:07.340
five votes in favor of Black civil rights.

00:10:07.340 --> 00:10:09.960
Because none of the justices on the court

00:10:09.960 --> 00:10:12.330
were considered to be particularly friends

00:10:12.330 --> 00:10:13.660
of Black Americans.

00:10:13.660 --> 00:10:18.020
- [Earl] In the civil
rights cases in 1883,

00:10:19.810 --> 00:10:24.020
the court had already held
that Congress lacked authority

00:10:24.020 --> 00:10:28.610
to prohibit segregation
in public accommodations,

00:10:28.610 --> 00:10:32.420
which meant they viewed public
accommodations as something

00:10:32.420 --> 00:10:35.220
fairly private rather than a civil right,

00:10:35.220 --> 00:10:38.560
a quasi-governmental right
or a quasi-public right.

00:10:38.560 --> 00:10:40.200
And that's one of the big distinctions

00:10:40.200 --> 00:10:42.930
between the majority
and the dissent in both

00:10:42.930 --> 00:10:46.210
the civil rights cases and
in Plessy versus Ferguson.

00:10:46.210 --> 00:10:50.210
The court ruled that in fact
that so long as the state

00:10:50.210 --> 00:10:53.230
of Louisiana maintained
separate but equal facilities,

00:10:53.230 --> 00:10:55.710
they could do that, that
that was not prohibited

00:10:55.710 --> 00:10:57.600
by the 14th Amendment.

00:10:57.600 --> 00:11:02.470
- [Jamal] The Supreme Court
in 1896 rules seven to one

00:11:02.470 --> 00:11:05.370
that the Separate Car
Act is constitutional.

00:11:05.370 --> 00:11:09.850
So a state is allowed to
segregate its public conveyances,

00:11:09.850 --> 00:11:12.830
including rail cars, by race.

00:11:12.830 --> 00:11:17.830
The court denies that the
Separate Car Act violates

00:11:17.900 --> 00:11:20.520
the 14th Amendment to the Constitution.

00:11:20.520 --> 00:11:23.390
What the court basically says is:

00:11:23.390 --> 00:11:26.870
look, the law says the
railway cars have to be equal,

00:11:26.870 --> 00:11:28.140
even if they're separate.

00:11:28.140 --> 00:11:32.390
And all the 14th Amendment requires is

00:11:32.390 --> 00:11:35.310
that basic equality in civil rights.

00:11:35.310 --> 00:11:36.710
- [Earl] John Marshall Harlan, of course,

00:11:36.710 --> 00:11:38.260
dissented from that.

00:11:38.260 --> 00:11:39.930
I think that it's important to understand

00:11:39.930 --> 00:11:42.230
what was the actual
nature of Harlan's dissent

00:11:42.230 --> 00:11:44.650
rather than the way that
it is actually portrayed.

00:11:44.650 --> 00:11:49.650
Harlan is famous talking about
the colorblind Constitution,

00:11:51.140 --> 00:11:53.967
but in fact, what he says is:

00:11:53.967 --> 00:11:57.710
"With respect to civil rights
common to all citizens,"

00:11:57.710 --> 00:12:00.423
I don't have the exact language before me,

00:12:01.257 --> 00:12:05.260
"that the Constitution which
required to be colorblind."

00:12:05.260 --> 00:12:09.020
So one of the big distinctions
between the majority

00:12:09.020 --> 00:12:11.960
and the dissent is that
Harlan does in fact

00:12:11.960 --> 00:12:16.070
believe that the right to
use public transportation

00:12:16.070 --> 00:12:20.250
counted as a civil right, and therefore,

00:12:20.250 --> 00:12:22.750
was protected against
segregation by Section 1

00:12:22.750 --> 00:12:23.943
of the 14th Amendment.

00:12:25.710 --> 00:12:29.530
I wanna make that point
because it's pretty clear

00:12:29.530 --> 00:12:32.940
that Harlan believed, for
example, that maintenance

00:12:32.940 --> 00:12:35.820
of segregated schools
would be constitutional.

00:12:35.820 --> 00:12:38.800
And it's also true that Harlan voted for

00:12:38.800 --> 00:12:43.490
to say that miscegenation
laws were constitutional.

00:12:43.490 --> 00:12:44.910
- [Jamal] The dissenting judge,

00:12:44.910 --> 00:12:48.340
Justice Harlan himself a
former supporter of slavery,

00:12:48.340 --> 00:12:51.320
who changed his views and
eventually became known

00:12:51.320 --> 00:12:53.920
as a champion of Black civil rights.

00:12:53.920 --> 00:12:57.050
So Justice Harlan, the lone dissenter,

00:12:57.050 --> 00:13:00.380
one of the only Southern
judges on the court.

00:13:00.380 --> 00:13:02.550
But the others were basically Northern,

00:13:02.550 --> 00:13:04.130
both Republican and Democrat.

00:13:04.130 --> 00:13:07.283
They didn't have strong views about race.

00:13:10.150 --> 00:13:12.430
They didn't have unusually
strong views about race

00:13:12.430 --> 00:13:14.050
for their time.

00:13:14.050 --> 00:13:17.890
And they maintained this distinction

00:13:17.890 --> 00:13:21.140
between social and civil rights.

00:13:21.140 --> 00:13:23.660
It's important to understand
in trying to understand

00:13:23.660 --> 00:13:26.120
the context of Plessy versus Ferguson

00:13:26.120 --> 00:13:29.860
that the Supreme Court used to distinguish

00:13:29.860 --> 00:13:31.940
between what it called civil rights

00:13:31.940 --> 00:13:33.840
and what it called social rights.

00:13:33.840 --> 00:13:37.320
Civil rights were basically
rights to participate

00:13:37.320 --> 00:13:40.440
in civil society, and
included rights like the right

00:13:40.440 --> 00:13:43.270
to enter into contracts,
the right to buy property,

00:13:43.270 --> 00:13:45.830
the right to testify in court.

00:13:45.830 --> 00:13:48.660
The court understood
social rights as something

00:13:48.660 --> 00:13:52.020
very different from that,
which is really the right

00:13:52.020 --> 00:13:55.580
to do all of those things in the company

00:13:55.580 --> 00:13:57.890
of people of a different race.

00:13:57.890 --> 00:13:58.723
- [Kim] That's fascinating.

00:13:58.723 --> 00:14:02.330
So what was the effect of this ruling

00:14:02.330 --> 00:14:04.310
in Plessy versus Ferguson?

00:14:04.310 --> 00:14:06.720
- [Earl] There are two ways
that you could look at it.

00:14:06.720 --> 00:14:10.320
That until 1954,

00:14:10.320 --> 00:14:13.410
the effect of the ruling

00:14:16.787 --> 00:14:20.120
was to say that the state
governments were allowed

00:14:20.120 --> 00:14:24.000
to segregate their citizenry
on the basis of race.

00:14:24.000 --> 00:14:25.320
That's one way you could look at it.

00:14:25.320 --> 00:14:28.260
Now one of the interesting questions is

00:14:28.260 --> 00:14:29.990
how much difference it would have made

00:14:29.990 --> 00:14:33.400
given the sort of culture
of the Southern states

00:14:33.400 --> 00:14:37.320
even if the court had held
that it was unconstitutional

00:14:37.320 --> 00:14:39.760
for the state to formally
require segregation

00:14:39.760 --> 00:14:41.020
among the races.

00:14:41.020 --> 00:14:44.740
That is, that there were a
lot of informal pressures

00:14:47.312 --> 00:14:50.170
which would have pushed toward segregation

00:14:50.170 --> 00:14:52.334
even if the court had said that

00:14:52.334 --> 00:14:54.930
the statute was unconstitutional.

00:14:54.930 --> 00:14:56.220
But we'll never know that.

00:14:56.220 --> 00:14:59.670
So in other words, the question in Plessy

00:14:59.670 --> 00:15:04.670
is not whether the federal government

00:15:05.200 --> 00:15:08.560
was going to mandate segregation,

00:15:08.560 --> 00:15:10.920
but rather whether the federal government

00:15:10.920 --> 00:15:14.960
was simply going to leave the
states and their citizenry

00:15:14.960 --> 00:15:18.490
to their own devices in
determining whether to segregate

00:15:18.490 --> 00:15:22.240
their public transportation
and some other thing.

00:15:22.240 --> 00:15:26.308
- [Kim] So this concept
of separate but equal

00:15:26.308 --> 00:15:29.560
is I think the most important
thing that comes out

00:15:29.560 --> 00:15:33.500
of Plessy versus Ferguson,
and then later will be

00:15:33.500 --> 00:15:35.630
at issue in the 20th century.

00:15:35.630 --> 00:15:40.400
So was separate ever equal
in theory or in practice?

00:15:40.400 --> 00:15:42.990
- [Jamal] It was very clear at the time.

00:15:42.990 --> 00:15:47.470
And Justice Harlan says so
in his dissenting opinion

00:15:47.470 --> 00:15:52.460
in Plessy that the practice
of separating railway cars

00:15:52.460 --> 00:15:57.460
or any number of other
public accommodations by race

00:15:57.910 --> 00:16:02.690
was not designed for the
comfort of Black Americans.

00:16:02.690 --> 00:16:06.750
It was designed in order to maintain

00:16:06.750 --> 00:16:11.230
their social inferiority
through legal institutions.

00:16:11.230 --> 00:16:14.660
So once you no longer have
the institution of slavery,

00:16:14.660 --> 00:16:18.760
there was a felt need
among many in the South

00:16:18.760 --> 00:16:22.000
to maintain the system of social relations

00:16:22.000 --> 00:16:23.430
that slavery represented.

00:16:23.430 --> 00:16:26.000
And that's what Jim Crow was all about.

00:16:26.000 --> 00:16:28.800
And everyone knew that's
what Jim Crow was all about.

00:16:28.800 --> 00:16:32.960
So Jim Crow was really
kind of in its infancy

00:16:32.960 --> 00:16:35.550
when Plessy versus Ferguson was decided.

00:16:35.550 --> 00:16:38.670
Laws that prevented Blacks from voting

00:16:38.670 --> 00:16:41.810
through a number of literacy requirements

00:16:41.810 --> 00:16:44.200
and property requirements

00:16:44.200 --> 00:16:47.170
and good character
requirements and so forth,

00:16:47.170 --> 00:16:51.500
those kinds of laws were
very much in their infancy

00:16:51.500 --> 00:16:54.400
at the time Plessy versus
Ferguson was decided.

00:16:54.400 --> 00:16:57.890
And so, the whole system of segregation

00:16:57.890 --> 00:17:01.800
is really revving up in the 1890s.

00:17:01.800 --> 00:17:05.200
And the court just gives it carte blanche

00:17:05.200 --> 00:17:07.529
to continue after that.

00:17:07.529 --> 00:17:11.110
And it's important to
remember that as of the 1890s,

00:17:11.110 --> 00:17:15.070
the Supreme Court had not
admitted to ever having

00:17:15.070 --> 00:17:16.520
reversed one of its own decisions.

00:17:16.520 --> 00:17:19.692
Whereas your Plessy case
was quite clear about this.

00:17:19.692 --> 00:17:22.420
The assumption was that
once the court ruled,

00:17:22.420 --> 00:17:24.660
it was going to be an awfully long time

00:17:24.660 --> 00:17:27.220
before you could get the
court to reverse itself.

00:17:27.220 --> 00:17:28.680
And that's in fact what happened.

00:17:28.680 --> 00:17:31.770
So the court does not reverse
Plessy versus Ferguson

00:17:31.770 --> 00:17:35.770
until Brown versus Board
of Education in 1954.

00:17:35.770 --> 00:17:38.250
And so, you had an almost 60-year period

00:17:38.250 --> 00:17:42.400
in which practices of
institutionalized segregation

00:17:42.400 --> 00:17:44.670
had the blessing of the Supreme Court.

00:17:44.670 --> 00:17:48.100
- [Kim] So we've learned that
in Plessy versus Ferguson,

00:17:48.100 --> 00:17:50.870
the Supreme Court took a narrow view

00:17:50.870 --> 00:17:52.900
of the Equal Protection Clause,

00:17:52.900 --> 00:17:55.730
ruling that separate
but equal accommodations

00:17:55.730 --> 00:17:58.930
for White and Black
Americans did not violate

00:17:58.930 --> 00:18:00.740
the 14th Amendment.

00:18:00.740 --> 00:18:03.560
Earl Maltz suggests that it's difficult

00:18:03.560 --> 00:18:07.330
to tell if a different outcome
in Plessy versus Ferguson

00:18:07.330 --> 00:18:09.570
would have made much
difference in the actions

00:18:09.570 --> 00:18:11.970
of Southern states if
there was no political will

00:18:11.970 --> 00:18:13.960
to enforce integration anyway.

00:18:13.960 --> 00:18:17.670
Jamal Greene, by contrast,
reminds us that segregation

00:18:17.670 --> 00:18:20.730
was just getting started at
the time of the Plessy case.

00:18:20.730 --> 00:18:24.860
And this ruling by the court
legitimized Jim Crow laws

00:18:24.860 --> 00:18:27.890
that would continue to
spread for nearly 60 years.

00:18:27.890 --> 00:18:30.170
To learn more about
Plessy versus Ferguson,

00:18:30.170 --> 00:18:32.330
check out the National
Constitution Center's

00:18:32.330 --> 00:18:35.320
Interactive Constitution
in Khan Academy's resources

00:18:35.320 --> 00:18:37.233
on U.S. government and history.


Polynomial identities introduction

WEBVTT
Kind: captions
Language: en

00:00:00.160 --> 00:00:01.530
- [Instructor] What we're
going to do in this video

00:00:01.530 --> 00:00:04.280
is talk a little bit about
polynomial identities,

00:00:04.280 --> 00:00:06.630
and this is really just a fancy way

00:00:06.630 --> 00:00:08.730
of seeing whether an expression

00:00:08.730 --> 00:00:12.450
that involves a polynomial is
equal to another expression.

00:00:12.450 --> 00:00:14.410
So for example, you're familiar

00:00:14.410 --> 00:00:16.930
with x squared plus two x plus one,

00:00:16.930 --> 00:00:19.130
we've seen polynomials
like this multiple times,

00:00:19.130 --> 00:00:21.770
this is a quadratic,
and you might recognize

00:00:21.770 --> 00:00:26.060
that this would be equal
to x plus one squared,

00:00:26.060 --> 00:00:27.740
that for any value of x,

00:00:27.740 --> 00:00:28.810
x squared plus two x plus one

00:00:28.810 --> 00:00:30.700
is the same thing as adding one to that x,

00:00:30.700 --> 00:00:32.430
and then squaring the whole thing.

00:00:32.430 --> 00:00:34.580
And we saw this when we first got,

00:00:34.580 --> 00:00:37.290
when we first learned
how to multiply binomials

00:00:37.290 --> 00:00:40.810
and we took the square of binomials,

00:00:40.810 --> 00:00:42.500
but now we're going to do this

00:00:42.500 --> 00:00:44.660
with slightly complicated expressions,

00:00:44.660 --> 00:00:46.890
things that aren't just simple quadratics

00:00:46.890 --> 00:00:49.810
or that might not be as obvious as this.

00:00:49.810 --> 00:00:51.950
And the way that we're gonna
prove whether they are true

00:00:51.950 --> 00:00:55.090
or not is just with a little
bit of algebraic manipulation.

00:00:55.090 --> 00:00:58.480
So for example, if someone
walked up to you on the street

00:00:58.480 --> 00:01:01.250
and said, all right, m
to the third minus one,

00:01:01.250 --> 00:01:06.250
is it equal to m minus one
times one plus m plus m squared?

00:01:08.800 --> 00:01:11.610
Pause this video and see what
you would tell that person,

00:01:11.610 --> 00:01:13.750
whether you could prove whether it is

00:01:13.750 --> 00:01:16.773
or is not a true polynomial identity.

00:01:17.620 --> 00:01:18.670
Okay, let's do it together,

00:01:18.670 --> 00:01:21.150
and the way I would tackle
this is I would expand out,

00:01:21.150 --> 00:01:24.100
I would multiply out what we
have on the right-hand side,

00:01:24.100 --> 00:01:26.040
so this is going to be equal to,

00:01:26.040 --> 00:01:27.250
so first, I could take this m

00:01:27.250 --> 00:01:28.930
and then multiply it times every term

00:01:28.930 --> 00:01:30.220
in this second expression,

00:01:30.220 --> 00:01:35.220
so m times one is m, m
times m is m squared,

00:01:36.230 --> 00:01:40.950
and then m times m squared
is m to the third power,

00:01:40.950 --> 00:01:45.260
and then I would take this negative one,

00:01:45.260 --> 00:01:48.580
and then multiply and then
distribute that times every term

00:01:48.580 --> 00:01:49.860
in that other expression,

00:01:49.860 --> 00:01:52.500
so negative one times one is negative one,

00:01:52.500 --> 00:01:55.130
negative one times m is negative m,

00:01:55.130 --> 00:02:00.130
and negative one times m
squared is negative m squared,

00:02:00.200 --> 00:02:03.600
and now let's see if we can simplify this.

00:02:03.600 --> 00:02:05.600
We have an m and a negative m,

00:02:05.600 --> 00:02:07.330
so those are going to cancel out,

00:02:07.330 --> 00:02:10.850
we have an m squared and
a negative m squared,

00:02:10.850 --> 00:02:13.000
so those cancel out, and so we are

00:02:13.000 --> 00:02:18.000
going to be left with m to
the third power, minus one.

00:02:18.340 --> 00:02:20.580
Now, clearly, m to the
third power minus one is

00:02:20.580 --> 00:02:21.920
going to be equal to m to the third power

00:02:21.920 --> 00:02:24.410
minus one for any value of m,

00:02:24.410 --> 00:02:27.300
these are identical expressions.

00:02:27.300 --> 00:02:31.600
So this is, this is indeed
a polynomial identity.

00:02:31.600 --> 00:02:33.150
Let's do another example.

00:02:33.150 --> 00:02:35.640
Let's say someone were to
walk up to you on the street

00:02:35.640 --> 00:02:40.640
and said, quick, n plus
three squared plus two n,

00:02:41.240 --> 00:02:44.410
is that equal to eight n plus 13,

00:02:44.410 --> 00:02:46.550
is this a polynomial identity?

00:02:46.550 --> 00:02:49.590
Pause this video and see
if you can figure that out.

00:02:49.590 --> 00:02:52.600
All right, now we're gonna
work on that together.

00:02:52.600 --> 00:02:54.040
And I would do it the exact same way.

00:02:54.040 --> 00:02:57.180
I would try to simplify with
a little bit of algebra,

00:02:57.180 --> 00:02:58.750
maybe the easiest thing to do first,

00:02:58.750 --> 00:03:00.950
and you could do this in multiple ways is,

00:03:00.950 --> 00:03:03.430
I have this, I have these n terms,

00:03:03.430 --> 00:03:05.480
two n's here, eight n's over here,

00:03:05.480 --> 00:03:07.490
well, what if I were to
get these two n's out

00:03:07.490 --> 00:03:08.950
of the left-hand side, so if I were

00:03:08.950 --> 00:03:13.100
to just subtract two n from
both sides of this equation,

00:03:13.100 --> 00:03:15.630
I'm going to get on the left-hand side,

00:03:15.630 --> 00:03:19.800
n plus three squared, and
on the right-hand side,

00:03:19.800 --> 00:03:24.800
I'm going to get six n,
eight n minus two n, plus 13.

00:03:25.640 --> 00:03:27.230
Now, what's n plus three squared?

00:03:27.230 --> 00:03:29.890
Well, that's going to be n squared

00:03:29.890 --> 00:03:32.160
plus two times three times n.

00:03:32.160 --> 00:03:34.430
And if what I just did does
not seem familiar to you,

00:03:34.430 --> 00:03:37.050
I encourage you to look

00:03:37.050 --> 00:03:39.670
at the videos about squaring binomials,

00:03:39.670 --> 00:03:41.870
but this is going to be plus six n,

00:03:41.870 --> 00:03:43.900
plus three squared, which is nine,

00:03:43.900 --> 00:03:48.270
and is this going to be
equal to six n plus 13?

00:03:48.270 --> 00:03:50.440
Well, already this is
starting to look a little bit,

00:03:50.440 --> 00:03:52.370
a little bit sketchy, but let's

00:03:52.370 --> 00:03:54.720
just keep going with the algebra.

00:03:54.720 --> 00:03:57.570
So let's see if we subtract six n

00:03:57.570 --> 00:04:00.783
from both sides, what do you get?

00:04:01.740 --> 00:04:02.810
Well, on the left-hand side,

00:04:02.810 --> 00:04:05.340
you're just going to
have n squared plus nine,

00:04:05.340 --> 00:04:09.810
and on the right-hand side,
you're going to get 13.

00:04:09.810 --> 00:04:12.160
Now, are there values of n for

00:04:12.160 --> 00:04:15.520
which this is not always true?

00:04:15.520 --> 00:04:16.353
Well, sure.

00:04:16.353 --> 00:04:18.230
I can find a lot of values of n for

00:04:18.230 --> 00:04:19.720
which this is not always true.

00:04:19.720 --> 00:04:21.760
If n is a zero, this is
not going to be true.

00:04:21.760 --> 00:04:24.270
If n is one, this is not going to be true.

00:04:24.270 --> 00:04:27.060
If n is two, this actually would be true,

00:04:27.060 --> 00:04:29.810
but if n is three, this
is not going to be true.

00:04:29.810 --> 00:04:31.310
If n is four, five, et cetera,

00:04:31.310 --> 00:04:32.610
so for actually most values of n,

00:04:32.610 --> 00:04:34.090
this is not going to be true.

00:04:34.090 --> 00:04:36.420
So in order for it to be
a polynomial identity,

00:04:36.420 --> 00:04:38.680
it has to be true for all of the values

00:04:38.680 --> 00:04:42.350
that are legitimate values that
you can evaluate for those,

00:04:42.350 --> 00:04:44.410
for the variable in question.

00:04:44.410 --> 00:04:48.820
So this one right over
here is not a polynomial,

00:04:50.379 --> 00:04:55.379
polynomial identity, and we are done.


What was the Articles of Confederation?

WEBVTT
Kind: captions
Language: en

00:00:00.164 --> 00:00:01.012
- So, John.

00:00:01.012 --> 00:00:03.606
People are always talking
about the Constitution.

00:00:03.606 --> 00:00:04.770
But, the Constitution

00:00:04.770 --> 00:00:08.690
was not the first founding
document of the United States.

00:00:08.690 --> 00:00:10.600
What were the Articles of Confederation,

00:00:10.600 --> 00:00:11.909
and why did they need to get replaced?

00:00:11.909 --> 00:00:13.580
- Well, the Articles of Confederation

00:00:13.580 --> 00:00:15.320
were the first loose set of rules

00:00:15.320 --> 00:00:17.430
to govern these 13 states,

00:00:17.430 --> 00:00:18.470
but they were a mess.

00:00:18.470 --> 00:00:20.300
Essentially, they allowed the states

00:00:20.300 --> 00:00:22.773
to be kind of their own
little sovereign islands.

00:00:22.773 --> 00:00:25.740
So, it was not a united nation.

00:00:25.740 --> 00:00:29.150
It was like an Archipelago of islands.

00:00:29.150 --> 00:00:30.470
And, the reason they were a mess.

00:00:30.470 --> 00:00:31.303
There's a couple of things.

00:00:31.303 --> 00:00:32.383
One,

00:00:32.383 --> 00:00:36.410
when they tried to repay the
Revolutionary War soldiers,

00:00:36.410 --> 00:00:38.190
Congress and Washington
with very little power,

00:00:38.190 --> 00:00:39.700
had to go to the states and say,

00:00:39.700 --> 00:00:41.970
please give us some money so
we can repay the soldiers.

00:00:41.970 --> 00:00:43.450
A lot of the states said, no thank you.

00:00:43.450 --> 00:00:44.790
We're not going to do that.

00:00:44.790 --> 00:00:45.770
Then if you had a river

00:00:45.770 --> 00:00:48.000
that rolled through
several different states,

00:00:48.000 --> 00:00:49.360
and you wanted to have a trade agreement

00:00:49.360 --> 00:00:50.670
with the Spanish, for example,

00:00:50.670 --> 00:00:52.199
to use that river and trade along it,

00:00:52.199 --> 00:00:54.280
then the government didn't have one way

00:00:54.280 --> 00:00:55.520
to negotiate with the Spanish.

00:00:55.520 --> 00:00:57.080
Individual states had to do it,

00:00:57.080 --> 00:00:59.700
and individual states
had different interests.

00:00:59.700 --> 00:01:01.760
Some wanted to trade with the Spanish.

00:01:01.760 --> 00:01:03.190
Some didn't want to trade at all.

00:01:03.190 --> 00:01:06.300
And so, how do you get those
states to agree on something?

00:01:06.300 --> 00:01:08.330
There was also not universal coinage.

00:01:08.330 --> 00:01:10.242
The states all made their
own money and differently.

00:01:10.242 --> 00:01:12.750
Well, states might print a bunch of money

00:01:12.750 --> 00:01:14.200
in order to payoff some debts,

00:01:14.200 --> 00:01:15.338
and then the money in one state

00:01:15.338 --> 00:01:17.390
is worth less than the money in another,

00:01:17.390 --> 00:01:18.940
who regulates all of that?

00:01:18.940 --> 00:01:21.800
So, commerce, and
industry, and self-defense.

00:01:21.800 --> 00:01:23.811
There was no way to raise
an army an pay for it.

00:01:23.811 --> 00:01:27.351
So, the nation was
crumbling before they got to

00:01:27.351 --> 00:01:29.930
Philadelphia in 1787.

00:01:29.930 --> 00:01:30.763
- And to your point,

00:01:30.763 --> 00:01:33.300
in most countries the parts of the country

00:01:33.300 --> 00:01:35.560
are called things like provinces,

00:01:35.560 --> 00:01:36.450
but ours are states,

00:01:36.450 --> 00:01:40.940
because they view themselves
as individual countries.

00:01:40.940 --> 00:01:41.773
- Absolutely,

00:01:41.773 --> 00:01:44.523
and because of course, the
Articles of Confederation

00:01:44.523 --> 00:01:47.070
had been formed in the wake of this fear,

00:01:47.070 --> 00:01:49.270
and the experience of
the fear of a monarchy.

00:01:49.270 --> 00:01:51.720
So, they wanted personal liberty,

00:01:51.720 --> 00:01:54.863
and get the monarchy and national control,

00:01:54.863 --> 00:01:56.110
throw it all away,

00:01:56.110 --> 00:01:59.300
because they believe that
once you consolidate control,

00:01:59.300 --> 00:02:02.300
in a national government of any kind,

00:02:02.300 --> 00:02:03.610
that it would trample liberty.

00:02:03.610 --> 00:02:05.680
And so, after having fought a revolution

00:02:05.680 --> 00:02:07.280
for the purposes of liberating the people,

00:02:07.280 --> 00:02:08.690
you're not gonna design a government

00:02:08.690 --> 00:02:11.060
that then stomps down on that liberty.

00:02:11.060 --> 00:02:12.868
So, they created something
that gave the states

00:02:12.868 --> 00:02:14.560
lots of flexibility,

00:02:14.560 --> 00:02:16.390
and then that flexibility
allowed everybody

00:02:16.390 --> 00:02:17.930
to go off in their different directions.

00:02:17.930 --> 00:02:19.520
- So, Articles of Confederation,

00:02:19.520 --> 00:02:20.920
maybe too much independence

00:02:20.920 --> 00:02:22.600
for the individual states.

00:02:22.600 --> 00:02:25.380
So, it seems like there
was a consensus to fix it.

00:02:25.380 --> 00:02:26.472
What was the central debate

00:02:26.472 --> 00:02:28.670
when they decided to fix it?

00:02:28.670 --> 00:02:31.820
- Well, there was a
consensus it had to be fixed,

00:02:31.820 --> 00:02:34.310
but when they got to Philadelphia.

00:02:34.310 --> 00:02:36.010
First of all, Rhode Island was invited

00:02:36.010 --> 00:02:36.990
and said no thank you,

00:02:36.990 --> 00:02:39.140
so 12 of the 13 states showed up.

00:02:39.140 --> 00:02:41.520
And, they knew they wanted
to centralize things.

00:02:41.520 --> 00:02:43.200
But, what did that mean?

00:02:43.200 --> 00:02:45.960
And, did it mean one president,
or a council of presidents?

00:02:45.960 --> 00:02:47.560
Did it mean a strong Congress?

00:02:47.560 --> 00:02:48.393
How strong?

00:02:48.393 --> 00:02:49.890
Could they tell states what to do?

00:02:49.890 --> 00:02:50.723
Well, if they did that

00:02:50.723 --> 00:02:52.938
then they were acting just
like George III had acted.

00:02:52.938 --> 00:02:56.000
So, they had to iron
out all of these issues

00:02:56.000 --> 00:02:57.449
to bring enough central control,

00:02:57.449 --> 00:02:59.510
and enough quick movement of government,

00:02:59.510 --> 00:03:01.326
that it could address national problems,

00:03:01.326 --> 00:03:02.680
but not so much that it

00:03:02.680 --> 00:03:04.800
trampled and stomped on that liberty.

00:03:04.800 --> 00:03:06.840
And, that was the constant debate.

00:03:06.840 --> 00:03:08.420
Constantly trying to figure out

00:03:08.420 --> 00:03:09.690
how to keep the balance between

00:03:09.690 --> 00:03:11.540
giving enough national power,

00:03:11.540 --> 00:03:12.690
but enough liberty.

00:03:12.690 --> 00:03:14.690
And that, some of the
biggest fights included

00:03:14.690 --> 00:03:16.880
fights over slavery, North versus South,

00:03:16.880 --> 00:03:18.970
fights over big states
versus small states.

00:03:18.970 --> 00:03:21.200
Who has representation in
this national government,

00:03:21.200 --> 00:03:22.430
and how do you figure that out?

00:03:22.430 --> 00:03:23.812
And then of course, the question of

00:03:23.812 --> 00:03:25.910
do we want a president?

00:03:25.910 --> 00:03:27.160
Will it be a single person,

00:03:27.160 --> 00:03:29.300
and how the dickens do
we elect that person?

00:03:29.300 --> 00:03:30.900
Which led us to the Electoral College,

00:03:30.900 --> 00:03:33.810
which has had some bumpy history.

00:03:33.810 --> 00:03:34.940
- And, where do you think we ended up?

00:03:34.940 --> 00:03:36.473
If on a scale of zero to 10.

00:03:36.473 --> 00:03:40.440
If was zero was a complete,
you know, independent states,

00:03:40.440 --> 00:03:42.340
and 10 is a federal government

00:03:42.340 --> 00:03:43.450
that just controls everything.

00:03:43.450 --> 00:03:44.780
Where do you think the US Constitution

00:03:44.780 --> 00:03:46.780
ended up relative to the
Articles of Confederation?

00:03:46.780 --> 00:03:49.770
- Well, in September of 1787,

00:03:49.770 --> 00:03:52.698
when it gets September 17th, 1787.

00:03:52.698 --> 00:03:55.670
When the new Constitution gets voted on,

00:03:55.670 --> 00:03:57.810
it is a stronger national document

00:03:57.810 --> 00:04:00.140
that has basically three main parts.

00:04:00.140 --> 00:04:02.418
One, the people are at the
heart and center of it.

00:04:02.418 --> 00:04:04.812
It is the people who
are the representatives,

00:04:04.812 --> 00:04:07.041
or at the center of the
republican government.

00:04:07.041 --> 00:04:09.410
The second thing is that
the national government

00:04:09.410 --> 00:04:11.630
can tell states what to
do in some instances.

00:04:11.630 --> 00:04:14.488
Those instances are
circumscribed, but it can happen.

00:04:14.488 --> 00:04:16.390
The states have to fall in line.

00:04:16.390 --> 00:04:17.280
That was very new,

00:04:17.280 --> 00:04:18.510
and there is this thing called

00:04:18.510 --> 00:04:20.040
a presidency which is created.

00:04:20.040 --> 00:04:21.130
Single person,

00:04:21.130 --> 00:04:23.810
created really in the
mold of George Washington.

00:04:23.810 --> 00:04:25.321
So, it is a nationalized government,

00:04:25.321 --> 00:04:29.369
but with a strong
attention to this question

00:04:29.369 --> 00:04:32.800
of protecting liberty through
a balance of power system,

00:04:32.800 --> 00:04:36.007
so that both the national government has

00:04:36.007 --> 00:04:37.383
checks and balances,

00:04:37.383 --> 00:04:38.950
and also the relationship

00:04:38.950 --> 00:04:40.490
between the federal government
and the state government

00:04:40.490 --> 00:04:42.760
has a number of checks and balances.

00:04:42.760 --> 00:04:45.330
So, even though they went in
a more centralized direction,

00:04:45.330 --> 00:04:48.780
they were constantly attentive
to this idea of liberty,

00:04:48.780 --> 00:04:50.048
keeping it free in the states,

00:04:50.048 --> 00:04:51.964
and not messing with them too much

00:04:51.964 --> 00:04:54.900
in their effort to get some
kind of centralized control.

00:04:54.900 --> 00:04:56.010
- So, it sounds like they might have gone

00:04:56.010 --> 00:04:57.858
from a one or a two with the
Articles of Confederation,

00:04:57.858 --> 00:05:00.300
to maybe a seven?

00:05:00.300 --> 00:05:01.133
Six?

00:05:01.133 --> 00:05:01.966
- Well, they went,

00:05:01.966 --> 00:05:04.160
I think they went from a one or two

00:05:04.160 --> 00:05:07.220
to maybe a five or a six,

00:05:07.220 --> 00:05:08.759
which has now moved.

00:05:08.759 --> 00:05:09.820
- Over time.

00:05:09.820 --> 00:05:11.200
- Over time,

00:05:11.200 --> 00:05:13.389
closer to maybe an eight or nine.

00:05:13.389 --> 00:05:14.423
- Fascinating.


Why was George Washington the first president?

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:03.550
- So in the early debates
about the Constitution,

00:00:03.550 --> 00:00:06.830
there are folks that wanted
a strong central leadership

00:00:06.830 --> 00:00:07.730
and other folks who didn't,

00:00:07.730 --> 00:00:10.130
because they felt it felt
a lot like George III.

00:00:11.050 --> 00:00:14.540
How did the existence of
Washington as a person

00:00:14.540 --> 00:00:15.930
affect the debate?

00:00:15.930 --> 00:00:16.763
- It's amazing.

00:00:16.763 --> 00:00:19.100
So George Washington, his
friends have to plead with him

00:00:19.100 --> 00:00:20.660
to come to the Constitutional Convention.

00:00:20.660 --> 00:00:22.760
He thinks, frankly,
that rewriting the rules

00:00:22.760 --> 00:00:24.750
of the country, it's not gonna work.

00:00:24.750 --> 00:00:26.370
But they finally convince him to come

00:00:26.370 --> 00:00:29.320
and they make him president
of the convention.

00:00:29.320 --> 00:00:31.350
And they put him at the front of the room,

00:00:31.350 --> 00:00:34.660
and he says almost nothing
during the entire four months.

00:00:34.660 --> 00:00:38.090
But what he does is at the
front of the room he is a model,

00:00:38.090 --> 00:00:40.460
because, remember, he
resigned his commission

00:00:40.460 --> 00:00:43.970
as commander of the Continental
Army and gave up power.

00:00:43.970 --> 00:00:45.850
He didn't seize the power that he had as,

00:00:45.850 --> 00:00:48.340
he was basically the biggest
celebrity in America,

00:00:48.340 --> 00:00:51.340
and he gave his commission
back to the government,

00:00:51.340 --> 00:00:54.130
which was an act of sublimating

00:00:54.130 --> 00:00:55.410
his own personal self interest

00:00:55.410 --> 00:00:57.320
for the benefit of the republic.

00:00:57.320 --> 00:01:00.530
And that's the model they
wanted for the Constitution.

00:01:00.530 --> 00:01:03.610
So he didn't say much, but
they designed basically,

00:01:03.610 --> 00:01:05.970
as one writer said that what
they were essentially doing

00:01:05.970 --> 00:01:08.140
was writing his future job description,

00:01:08.140 --> 00:01:11.030
because everybody in the
room knew he would ultimately

00:01:11.030 --> 00:01:12.140
become the president,

00:01:12.140 --> 00:01:14.370
because there was nobody
else in America like him.

00:01:14.370 --> 00:01:16.700
And what was important was
not that he was a general,

00:01:16.700 --> 00:01:18.990
but that he had this virtue inside of him,

00:01:18.990 --> 00:01:21.220
which was that he would know how far to go

00:01:21.220 --> 00:01:24.810
and when to stop, and when
to protect those liberties.

00:01:24.810 --> 00:01:28.640
And so he sat on a chair
with a sun on the back of it,

00:01:28.640 --> 00:01:30.090
and at the end of the proceedings,

00:01:30.090 --> 00:01:33.440
Benjamin Franklin, who was
the only other great, kind of,

00:01:33.440 --> 00:01:36.370
superstar in America who
participated in the convention

00:01:36.370 --> 00:01:39.650
said that he looked at that
sun on Washington's chair,

00:01:39.650 --> 00:01:41.650
and he wasn't sure whether
it was a rising sun

00:01:41.650 --> 00:01:42.550
or a setting sun,

00:01:42.550 --> 00:01:44.400
but now after they finished their work,

00:01:44.400 --> 00:01:47.050
he had decided that the sun was rising,

00:01:47.050 --> 00:01:50.040
which was basically anointing
and blessing everything

00:01:50.040 --> 00:01:51.410
that had happened there,

00:01:51.410 --> 00:01:54.340
and this new office of
presidency that they had created,

00:01:54.340 --> 00:01:56.410
that George Washington
was to go walk into,

00:01:56.410 --> 00:01:58.410
and that's why Washington's statue

00:01:58.410 --> 00:02:00.210
is in front of Independence Hall

00:02:00.210 --> 00:02:03.420
where the Constitutional
Convention took place.

00:02:03.420 --> 00:02:05.820
But in the dialogue over four months,

00:02:05.820 --> 00:02:07.840
you almost never see his words.

00:02:07.840 --> 00:02:11.630
He was there as a symbol and participant,

00:02:11.630 --> 00:02:14.440
but not like James
Madison, or James Wilson,

00:02:14.440 --> 00:02:16.910
or Gouverneur Morris who were
in there in the nitty gritty

00:02:16.910 --> 00:02:17.970
of every little detail.

00:02:17.970 --> 00:02:20.640
- Do we know what Washington was thinking,

00:02:20.640 --> 00:02:23.130
did he want the job, did he have a view,

00:02:23.130 --> 00:02:24.207
or he just said,

00:02:24.207 --> 00:02:26.810
"Hey, I'm just gonna do
what everyone else decides"?

00:02:26.810 --> 00:02:28.780
- Fortunately, they did
everything in secret.

00:02:28.780 --> 00:02:31.780
Washington was so virtuous
that he didn't even write

00:02:31.780 --> 00:02:33.270
about it in his diary.

00:02:33.270 --> 00:02:36.550
Now, fortunately, we have other
people who did keep diaries,

00:02:36.550 --> 00:02:39.337
and James Madison took notes and said,

00:02:39.337 --> 00:02:42.790
"Publish them only after
all 55 members are dead."

00:02:42.790 --> 00:02:45.120
What Washington thought is he
wanted a central government,

00:02:45.120 --> 00:02:47.230
because as a general he
knew there had to be an army

00:02:47.230 --> 00:02:48.850
to handle rebellions,

00:02:48.850 --> 00:02:51.210
and there had been Shays' Rebellion

00:02:51.210 --> 00:02:53.980
that had tested under the
Articles of Confederation.

00:02:53.980 --> 00:02:55.580
So he wanted a strong government,

00:02:55.580 --> 00:02:58.570
but he was, and he was very worried

00:02:58.570 --> 00:03:01.030
that having been
successful revolutionaries,

00:03:01.030 --> 00:03:03.810
they could do what he
knew was hard by history,

00:03:03.810 --> 00:03:05.570
which is that revolutionaries
aren't very good

00:03:05.570 --> 00:03:06.880
at creating governments.

00:03:06.880 --> 00:03:09.060
And so he knew that what they were doing

00:03:09.060 --> 00:03:11.550
was a real risk and a real gamble.

00:03:11.550 --> 00:03:14.470
He believed that it
could be done, however.

00:03:14.470 --> 00:03:17.600
He believed that a strong
national government was required.

00:03:17.600 --> 00:03:19.120
But then was he was given the job,

00:03:19.120 --> 00:03:20.750
he was incredibly nervous,

00:03:20.750 --> 00:03:23.980
this incredible military
leader basically thought

00:03:23.980 --> 00:03:25.350
that he might fail.

00:03:25.350 --> 00:03:27.670
It was more likely than not
that this whole darn thing

00:03:27.670 --> 00:03:29.499
would fail and that he might fail.

00:03:29.499 --> 00:03:32.060
And as he rode to his inauguration,

00:03:32.060 --> 00:03:34.510
he kept writing letters and in his diary

00:03:34.510 --> 00:03:36.850
talking about how the
expectations of his countrymen

00:03:36.850 --> 00:03:38.450
were just too much for him.

00:03:38.450 --> 00:03:40.800
And that was both a personal worry,

00:03:40.800 --> 00:03:42.560
and he also worried about monarchy.

00:03:42.560 --> 00:03:43.770
How does monarchy build?

00:03:43.770 --> 00:03:46.580
It's either when a monarch demands power

00:03:46.580 --> 00:03:49.920
or when the mob hands all
of the power to the person

00:03:49.920 --> 00:03:51.540
and says, "Do everything for us,"

00:03:51.540 --> 00:03:52.980
and gives them ultimate power.

00:03:52.980 --> 00:03:56.670
And what they knew in the
convention was that human beings

00:03:56.670 --> 00:03:58.880
were sinful and could not handle power.

00:03:58.880 --> 00:04:00.950
If they were given the
power, they would abuse it

00:04:00.950 --> 00:04:03.300
as surely as the sun
comes up in the morning.

00:04:03.300 --> 00:04:06.230
And he was worried that
if given too much power,

00:04:06.230 --> 00:04:08.740
perhaps he could be susceptible to that.

00:04:08.740 --> 00:04:12.216
And so he was for as confident
and as much of a model

00:04:12.216 --> 00:04:13.049
of strength that he was,

00:04:13.049 --> 00:04:14.620
he was a very nervous guy.

00:04:14.620 --> 00:04:18.030
- And it sounds like an
unusual person where the power,

00:04:18.030 --> 00:04:20.450
at least as the history I've read,

00:04:20.450 --> 00:04:22.980
didn't corrupt him and he
didn't try to do a power grab.

00:04:22.980 --> 00:04:25.470
- That's right, Washington was constantly,

00:04:25.470 --> 00:04:29.640
he was a man of a rigid code,

00:04:29.640 --> 00:04:32.470
and he believed that the
standards were necessary

00:04:32.470 --> 00:04:34.310
for the proper kind of human behavior.

00:04:34.310 --> 00:04:37.480
He wrote a list of a
hundred different things

00:04:37.480 --> 00:04:40.560
that a gentleman should
do to comport himself

00:04:40.560 --> 00:04:43.290
in the proper way in society,

00:04:43.290 --> 00:04:45.040
and that's what made
him such a good model.

00:04:45.040 --> 00:04:45.953
- I'd like a copy of that.

00:04:45.953 --> 00:04:46.960
- (laughs) Yes, exactly.
- I have a feeling

00:04:46.960 --> 00:04:48.540
I don't check most of them off. (laughing)

00:04:48.540 --> 00:04:49.890
- We'd all be better.

00:04:49.890 --> 00:04:51.400
But things like clearing your throat,

00:04:51.400 --> 00:04:54.370
how you behave in the presence of a lady.

00:04:54.370 --> 00:04:58.330
And he believed in these codes,

00:04:58.330 --> 00:05:01.770
because he believed if
everybody maintained them,

00:05:01.770 --> 00:05:03.390
then the system would work.

00:05:03.390 --> 00:05:05.560
This was during the
period where Newton's laws

00:05:05.560 --> 00:05:08.020
were making people think
about a clockwork universe

00:05:08.020 --> 00:05:09.850
where if everything runs,

00:05:09.850 --> 00:05:12.790
if the machine is put
together with tension, right,

00:05:12.790 --> 00:05:15.330
so it recognized that
people were not angels,

00:05:15.330 --> 00:05:16.330
tension in the machine,

00:05:16.330 --> 00:05:18.390
but if everybody did their thing

00:05:18.390 --> 00:05:21.560
and the pieces stayed in
their lane, as it were,

00:05:21.560 --> 00:05:22.890
then the clock would work.

00:05:22.890 --> 00:05:25.510
And so he had that code
which tried to keep him

00:05:25.510 --> 00:05:28.470
in his place and keep
everybody else in his place,

00:05:28.470 --> 00:05:30.070
and he set a standard,

00:05:30.070 --> 00:05:31.610
and then everybody tried to live up to it,

00:05:31.610 --> 00:05:33.517
as opposed to saying,
"Well, that standard's nice,

00:05:33.517 --> 00:05:35.130
"but I'm now gonna go do this."

00:05:35.130 --> 00:05:37.980
And that's why when he resigned
his military commission,

00:05:37.980 --> 00:05:41.400
and he also undid a coup that
some of his men were planning

00:05:41.400 --> 00:05:43.250
back when he was leader of the army.

00:05:43.250 --> 00:05:45.300
They were basically gonna
go to Congress and say,

00:05:45.300 --> 00:05:47.040
this is in Newburgh, New York,

00:05:47.040 --> 00:05:48.957
they were gonna go to Congress and say,

00:05:48.957 --> 00:05:50.167
"Unless you give us our money,

00:05:50.167 --> 00:05:51.560
"we're gonna stage a coup."

00:05:51.560 --> 00:05:54.487
He found out about the plot,
went to his men and said,

00:05:54.487 --> 00:05:56.607
"This is a sin both against the revolution

00:05:56.607 --> 00:05:58.247
"and my own personal virtue,

00:05:58.247 --> 00:06:00.320
"because I put myself on the line."

00:06:00.320 --> 00:06:02.138
They backed off.

00:06:02.138 --> 00:06:03.537
And what he coulda done is say,

00:06:03.537 --> 00:06:06.247
"Let's go, let's ride to
Washington and get you your money.

00:06:06.247 --> 00:06:07.827
"You fought in this war.

00:06:07.827 --> 00:06:10.527
"Your wives and children
are begging and poor.

00:06:10.527 --> 00:06:11.547
"You deserve this money.

00:06:11.547 --> 00:06:13.930
"Let's use our power and
authority and take it,"

00:06:13.930 --> 00:06:15.250
and he said, "No."

00:06:15.250 --> 00:06:17.250
Joseph Ellis, the
historian, writes about it

00:06:17.250 --> 00:06:20.090
as being the last temptation
of Washington as a general.

00:06:20.090 --> 00:06:22.220
He has the chance to grab the power,

00:06:22.220 --> 00:06:24.230
and he says, "No, we shouldn't do it."

00:06:24.230 --> 00:06:25.600
And that's why he was a model,

00:06:25.600 --> 00:06:28.390
because he could resist the pull of power,

00:06:28.390 --> 00:06:31.330
and that's the image in
which they wanted to create

00:06:31.330 --> 00:06:32.330
this new government.

00:06:32.330 --> 00:06:33.163
- Wow.


What was the Gilded Age?

WEBVTT
Kind: captions
Language: en

00:00:00.043 --> 00:00:03.660
- So what was the Gilded
Age and why did it happen?

00:00:03.660 --> 00:00:06.070
- Ah, the Gilded Age is
this fascinating period

00:00:06.070 --> 00:00:08.640
from about 1870 to 1900,

00:00:08.640 --> 00:00:11.000
you can change the dates
a little bit but that's,

00:00:11.000 --> 00:00:12.600
so we're talking post Civil War,

00:00:12.600 --> 00:00:16.430
America becomes an industrial
powerhouse, the cities rise.

00:00:16.430 --> 00:00:19.880
So in 1850 fewer people live in the cities

00:00:19.880 --> 00:00:22.404
than in the rural part of the country,

00:00:22.404 --> 00:00:26.000
by 1900 more people live in the city.

00:00:26.000 --> 00:00:28.320
And basically you have the
birth of the railroads,

00:00:28.320 --> 00:00:30.380
the railroads get connected in 1869,

00:00:31.367 --> 00:00:34.160
continental railroad going all
the way across the country.

00:00:34.160 --> 00:00:37.320
You have the rise of
oil and John Rockefeller

00:00:37.320 --> 00:00:38.520
and basically, this period,

00:00:38.520 --> 00:00:42.640
think of Rockefeller and Vanderbilt, as,

00:00:42.640 --> 00:00:46.720
and Carnegie and JP Morgan as powerful

00:00:46.720 --> 00:00:47.840
the way we think of presidents.

00:00:47.840 --> 00:00:51.170
Presidents during this period
kind of were on the descent

00:00:51.170 --> 00:00:54.780
but the magnets of industry,
railroad, oil, steel,

00:00:54.780 --> 00:00:56.220
those were all banking,

00:00:56.220 --> 00:00:58.650
those were the superheroes in America,

00:00:58.650 --> 00:01:02.237
and they led to this amazing
growth in industrialization

00:01:02.237 --> 00:01:05.135
but then also huge disparities.

00:01:05.135 --> 00:01:09.280
No labor laws that we would
be familiar with today.

00:01:09.280 --> 00:01:11.960
So you had this
industrialization with people

00:01:11.960 --> 00:01:14.920
working in the industries
who had no protections,

00:01:14.920 --> 00:01:17.490
child labor, working 22 hours a day,

00:01:17.490 --> 00:01:19.263
horrible health conditions.

00:01:19.263 --> 00:01:23.328
So this gurgling, booming America,

00:01:23.328 --> 00:01:25.680
but where there's great disparities

00:01:25.680 --> 00:01:28.740
between who's doing well and who's not.

00:01:28.740 --> 00:01:31.200
- And it was called Gilded
because those who were doing well

00:01:31.200 --> 00:01:33.000
were living very well.

00:01:33.000 --> 00:01:36.140
- Right, Gilded as in a gilded
frame, covered with cold

00:01:36.140 --> 00:01:39.350
which came from a novel by Mark Twain.

00:01:39.350 --> 00:01:41.500
And that's right, it takes on

00:01:41.500 --> 00:01:43.793
the caste of this extraordinary wealth.

00:01:45.580 --> 00:01:48.920
Rockefeller was the first
billionaire in America,

00:01:48.920 --> 00:01:50.710
Vanderbilt built the biggest house,

00:01:50.710 --> 00:01:54.460
still the biggest house in
America during this period.

00:01:54.460 --> 00:01:58.680
So the wealth was, people would
make eight to $10 in a week,

00:01:58.680 --> 00:02:02.890
some of these tycoons were
making eight to $10 in a minute.

00:02:02.890 --> 00:02:05.660
And so that kind of vast wealth,

00:02:05.660 --> 00:02:08.510
because you could only
make so much wealth before,

00:02:08.510 --> 00:02:10.930
this is just mountains of wealth

00:02:10.930 --> 00:02:13.290
and then also, again,
this great disparity.

00:02:13.290 --> 00:02:16.480
- And it seems like it came
from, really, technology.

00:02:16.480 --> 00:02:18.580
Technology allowed all
of this productivity,

00:02:18.580 --> 00:02:21.480
the railroads, steel,
et cetera, et cetera,

00:02:21.480 --> 00:02:24.270
and then of course finance
was able to get in there

00:02:24.270 --> 00:02:26.680
and help move capital more efficiently.

00:02:26.680 --> 00:02:28.610
What parallels do you
see with our current age

00:02:28.610 --> 00:02:30.800
where technology seems to
be doing something similar

00:02:30.800 --> 00:02:33.380
where we have all of these new
industries, and new wealth,

00:02:33.380 --> 00:02:36.800
but some fear that it might
be causing some inequality.

00:02:36.800 --> 00:02:38.810
- You have, yes, you
have a couple of things.

00:02:38.810 --> 00:02:41.990
You have innovation in these
very different industries,

00:02:41.990 --> 00:02:44.480
both innovation in the creation of things

00:02:44.480 --> 00:02:47.340
but then also innovations in
the structures of business.

00:02:47.340 --> 00:02:50.220
Buying up small business,
creating big conglomerates.

00:02:50.220 --> 00:02:52.582
Then using that leverage and power

00:02:52.582 --> 00:02:55.760
to then crowd out competitors for sure

00:02:55.760 --> 00:02:57.600
but then also raise prices

00:02:57.600 --> 00:02:59.729
because you're the only game in town.

00:02:59.729 --> 00:03:03.580
And you also have business
practices that are not

00:03:03.580 --> 00:03:08.340
the sort of laissez-faire
economic belief said essentially

00:03:08.340 --> 00:03:12.110
that in the economy it was
like in the American system,

00:03:12.110 --> 00:03:13.990
which was let it operate.

00:03:13.990 --> 00:03:15.640
Don't get in the way, don't mess with it,

00:03:15.640 --> 00:03:16.810
because when it operates

00:03:16.810 --> 00:03:19.300
it runs the most efficiently for America.

00:03:19.300 --> 00:03:20.820
- In the end markets can be messy

00:03:20.820 --> 00:03:22.260
but they're gonna have the best outcome.

00:03:22.260 --> 00:03:23.093
- That's right.

00:03:23.093 --> 00:03:24.330
- Is the argument behind laissez-faire.

00:03:24.330 --> 00:03:27.930
- That's exactly right, and
it got this wonderful assist

00:03:27.930 --> 00:03:29.360
from Charles Darwin who said,

00:03:29.360 --> 00:03:31.140
we can explain the growth of,

00:03:31.140 --> 00:03:32.930
or you can explain the species

00:03:32.930 --> 00:03:34.840
and we can explain our natural world

00:03:34.840 --> 00:03:38.550
with this theory about the
competition among the species

00:03:38.550 --> 00:03:42.880
and this term survival of the
fittest, social Darwinism,

00:03:42.880 --> 00:03:44.470
survival of the fittest, which some,

00:03:44.470 --> 00:03:46.870
and I certainly thought might've
come from Darwin didn't,

00:03:46.870 --> 00:03:48.280
it came from Herbert Spencer

00:03:48.280 --> 00:03:50.220
who basically had an economic theory

00:03:50.220 --> 00:03:52.510
of the survival of the fittest
and then it went this way.

00:03:52.510 --> 00:03:54.190
Some people have more talent than others

00:03:54.190 --> 00:03:56.780
and when they exercise their
talent they do very well

00:03:56.780 --> 00:03:58.920
and that's the best thing for society.

00:03:58.920 --> 00:04:01.390
It believed that society
was ever increasing,

00:04:01.390 --> 00:04:03.900
it didn't mean that every
single person was increasing,

00:04:03.900 --> 00:04:05.830
but that if you followed
survival of the fittest

00:04:05.830 --> 00:04:07.410
and the best people did the best

00:04:07.410 --> 00:04:09.365
then ultimately society will always be

00:04:09.365 --> 00:04:11.619
on an evolutionary plane of moving upward.

00:04:11.619 --> 00:04:14.440
And so that was the theory behind

00:04:14.440 --> 00:04:18.430
get out of the way of these
big companies and big tycoons

00:04:18.430 --> 00:04:20.900
and they will do the best for America

00:04:20.900 --> 00:04:23.260
and so the reason that was important

00:04:24.607 --> 00:04:26.060
was A, it kept government out of the way,

00:04:26.060 --> 00:04:31.060
B, it kept religious,
it created a religion,

00:04:31.245 --> 00:04:33.900
it was a secular religion of course,

00:04:33.900 --> 00:04:36.490
but it created a theory that said,

00:04:36.490 --> 00:04:38.150
wow that looks like what you're doing

00:04:38.150 --> 00:04:41.280
is totally self interested but
there's this theory behind it

00:04:41.280 --> 00:04:44.030
and everybody will
improve so okay, go ahead.

00:04:44.030 --> 00:04:45.630
- Yeah, that was the beginning

00:04:45.630 --> 00:04:47.850
of Gordon Gekko's famous greed is good.

00:04:47.850 --> 00:04:49.330
- Right, right, yeah.

00:04:49.330 --> 00:04:53.860
Yeah, greed is good, so the
comparison to our current moment

00:04:53.860 --> 00:04:58.860
is you have huge disparity
and technologies that are,

00:04:59.030 --> 00:05:01.830
to use a cliche of the day, disruptive

00:05:01.830 --> 00:05:03.350
that are completely changing

00:05:03.350 --> 00:05:05.010
the way everybody does business,

00:05:05.010 --> 00:05:07.980
changing when we think about the way

00:05:07.980 --> 00:05:11.983
in which Americans behave,
that rapid sense of change,

00:05:11.983 --> 00:05:14.110
it's changing culture rapidly

00:05:14.110 --> 00:05:16.550
and it's making big winners and losers

00:05:16.550 --> 00:05:19.440
and those big disparities exist as well.

00:05:19.440 --> 00:05:21.950
- How did the Gilded Age play out?

00:05:21.950 --> 00:05:24.920
Were some of these forces
moderated eventually,

00:05:24.920 --> 00:05:26.370
maybe around 1900,

00:05:26.370 --> 00:05:29.750
and do you think similar
things might happen for us?

00:05:29.750 --> 00:05:31.960
- There were two big moderation

00:05:31.960 --> 00:05:34.010
in response to the Gilded Age.

00:05:34.010 --> 00:05:37.200
You had government came awake again

00:05:38.587 --> 00:05:40.815
and then you also had labor movements

00:05:40.815 --> 00:05:45.230
that came into formation,

00:05:45.230 --> 00:05:47.877
basically to slow down the growth

00:05:47.877 --> 00:05:50.430
and the rapacious demands
of the Gilded Age.

00:05:50.430 --> 00:05:52.970
Politics during the
Gilded Age kind of went,

00:05:52.970 --> 00:05:57.330
became an offshoot of
the titans of industry.

00:05:57.330 --> 00:06:02.040
If you look at the presidency
between 1876 and 1892

00:06:02.040 --> 00:06:03.860
they're all one term presidents

00:06:03.860 --> 00:06:06.670
and none of them get
more than 50% of the vote

00:06:06.670 --> 00:06:09.400
and basically what the
presidents spend their time doing

00:06:09.400 --> 00:06:11.550
is using the spoil system,

00:06:11.550 --> 00:06:13.328
which is essentially
putting people in jobs

00:06:13.328 --> 00:06:16.840
to pay off the local bosses
who helped them get elected.

00:06:16.840 --> 00:06:20.870
So getting elected became
a job of staying elected

00:06:20.870 --> 00:06:22.862
and that meant dolling out patronage,

00:06:22.862 --> 00:06:26.010
basically giving people
jobs who were your friends

00:06:26.010 --> 00:06:27.380
so that they would go and vote for you

00:06:27.380 --> 00:06:29.870
because these elections
were all very close

00:06:29.870 --> 00:06:32.930
and that's not getting a lot
of work done for the people.

00:06:32.930 --> 00:06:35.010
Some of 'em, Rutherford B. Hayes

00:06:35.010 --> 00:06:37.760
tried these little efforts
at civil service reform,

00:06:37.760 --> 00:06:40.110
which essentially meant
putting people in jobs

00:06:40.110 --> 00:06:42.120
who could monitor the factories,

00:06:42.120 --> 00:06:46.100
make sure that people
weren't getting abused

00:06:46.100 --> 00:06:47.700
or that health wasn't declining

00:06:47.700 --> 00:06:49.240
or that anything that
a government might do

00:06:49.240 --> 00:06:51.070
that we think of today

00:06:51.070 --> 00:06:53.212
but his political
patrons didn't want that.

00:06:53.212 --> 00:06:55.590
So that was all very hard to do.

00:06:55.590 --> 00:06:58.542
What happened on the worker's
end is they realized,

00:06:58.542 --> 00:07:02.570
and the most famous moment was in 1911,

00:07:02.570 --> 00:07:04.690
the Triangle Shirtwaist Fire where

00:07:04.690 --> 00:07:07.127
a number of women working
in a garment factory are,

00:07:07.127 --> 00:07:12.127
there's a fire and the
bosses lock the doors

00:07:12.861 --> 00:07:17.830
and I think 40 some odd
women die, mostly women,

00:07:17.830 --> 00:07:21.720
and it highlighted the labor issues

00:07:21.720 --> 00:07:23.460
but labor unions start to organize

00:07:23.460 --> 00:07:26.940
and there are huge clashes and
strikes and consumer boycotts

00:07:26.940 --> 00:07:29.468
and those start to put
some pressure on business

00:07:29.468 --> 00:07:34.040
to change their practices, at
least in terms of worker hours

00:07:34.040 --> 00:07:35.730
and the kinds of things that we now

00:07:35.730 --> 00:07:37.770
would certainly take for granted.

00:07:37.770 --> 00:07:39.640
- Fascinating.

00:07:39.640 --> 00:07:41.490
- It's an incredible period.
- Yeah.


Identifying quadratic patterns

WEBVTT
Kind: captions
Language: en

00:00:00.810 --> 00:00:02.130
- [Instructor] We're
told that we wanna factor

00:00:02.130 --> 00:00:03.800
the following expression and they ask us

00:00:03.800 --> 00:00:07.190
which pattern can we use
to factor the expression?

00:00:07.190 --> 00:00:09.060
And U and V are either constant integers

00:00:09.060 --> 00:00:11.250
or single-variable expressions.

00:00:11.250 --> 00:00:12.610
So we'll do this one together and then

00:00:12.610 --> 00:00:13.730
we'll have a few more examples where

00:00:13.730 --> 00:00:15.830
I'll encourage you to pause the video.

00:00:15.830 --> 00:00:17.760
So when they're talking about patterns

00:00:17.760 --> 00:00:20.287
they're really saying, "Hey, can we say

00:00:20.287 --> 00:00:22.727
"that some of these can generally form

00:00:22.727 --> 00:00:25.027
"a pattern that matches what we have here

00:00:25.027 --> 00:00:27.137
"and then we can use
that pattern to factor

00:00:27.137 --> 00:00:28.080
"it into one of these forms?"

00:00:28.080 --> 00:00:29.410
What do I mean by that?

00:00:29.410 --> 00:00:30.990
Well, let's just imagine something like

00:00:30.990 --> 00:00:33.760
U plus V squared.

00:00:33.760 --> 00:00:36.380
We've squared binomials in the past.

00:00:36.380 --> 00:00:38.790
This is going to be equal to U squared

00:00:39.650 --> 00:00:41.980
plus two times the product of these terms

00:00:41.980 --> 00:00:46.213
so two U V and then plus V squared.

00:00:47.220 --> 00:00:49.230
Now, when you look at
this polynomial right

00:00:49.230 --> 00:00:51.830
over here, it actually has this form

00:00:51.830 --> 00:00:53.300
if you look at it carefully.

00:00:53.300 --> 00:00:54.950
How can it have this form?

00:00:54.950 --> 00:00:57.730
Well, if we view U squared as nine X

00:00:57.730 --> 00:01:01.480
to the eighth then that means that U is,

00:01:01.480 --> 00:01:02.810
and let me write it as a capital U,

00:01:02.810 --> 00:01:07.110
U is equal to three X to the fourth.

00:01:07.110 --> 00:01:08.830
'Cause notice if you square this

00:01:08.830 --> 00:01:10.240
you're gonna get nine X to the eighth.

00:01:10.240 --> 00:01:13.960
So this right over here is U squared.

00:01:13.960 --> 00:01:16.940
And if we said that V squared is equal

00:01:16.940 --> 00:01:20.810
to Y squared, so if this
is capital V squared

00:01:20.810 --> 00:01:25.450
then that means that V is equal to Y.

00:01:25.450 --> 00:01:28.200
And then this would have
to be two times U V.

00:01:28.200 --> 00:01:29.240
Is it?

00:01:29.240 --> 00:01:32.100
Well, see if I multiply
U times V I get three

00:01:32.100 --> 00:01:36.460
X to the fourth Y and then two times that

00:01:36.460 --> 00:01:38.490
is indeed six X to the fourth Y.

00:01:38.490 --> 00:01:42.460
So this right over here is two U V.

00:01:42.460 --> 00:01:45.210
So notice this polynomial,
this higher-degree

00:01:45.210 --> 00:01:47.750
polynomial can be
expressed in this pattern,

00:01:47.750 --> 00:01:51.490
which means it can be factored this way.

00:01:51.490 --> 00:01:53.410
So when they say which pattern can we use

00:01:53.410 --> 00:01:55.820
to factor this expression,
well, I would use

00:01:55.820 --> 00:01:58.430
a pattern for U plus V squared.

00:01:58.430 --> 00:02:01.320
So I would go with that
choice right over there.

00:02:01.320 --> 00:02:03.923
Let's do a few more examples.

00:02:05.930 --> 00:02:08.850
So here once again we're
told the same thing.

00:02:08.850 --> 00:02:10.440
We're given a different expression

00:02:10.440 --> 00:02:12.850
and they're asking us
what pattern can we use

00:02:12.850 --> 00:02:14.780
to factor the expression?

00:02:14.780 --> 00:02:17.240
So I have these three terms here.

00:02:17.240 --> 00:02:20.320
It looks like maybe I could use, I can see

00:02:20.320 --> 00:02:21.153
a perfect square here.

00:02:21.153 --> 00:02:22.410
Let's see if that works.

00:02:22.410 --> 00:02:27.410
If this is U squared, if this is U squared

00:02:27.660 --> 00:02:29.730
then that means that
U is going to be equal

00:02:29.730 --> 00:02:32.940
to two X to the third power.

00:02:32.940 --> 00:02:37.940
And if this is V squared then that means

00:02:38.130 --> 00:02:40.750
that V is equal to five.

00:02:40.750 --> 00:02:43.510
Now is this equal to two times U V?

00:02:43.510 --> 00:02:47.340
Well, let's see, two times
U V would be equal to,

00:02:47.340 --> 00:02:49.490
well, you're not gonna have any Y in it

00:02:49.490 --> 00:02:52.360
so this is not going to be two U V.

00:02:52.360 --> 00:02:55.360
So this actually is not fitting
the perfect square pattern.

00:02:55.360 --> 00:02:57.760
So we could rule this out.

00:02:57.760 --> 00:03:00.210
And both of these are
perfect squares of some form.

00:03:00.210 --> 00:03:02.010
One just has a, I guess
you'd say adding V.

00:03:02.010 --> 00:03:03.970
The other one is subtracting V.

00:03:03.970 --> 00:03:05.920
This right over here,
if I were to multiply

00:03:05.920 --> 00:03:07.430
this out, this is going to be equal to,

00:03:07.430 --> 00:03:09.520
this is a difference of squares
and we've seen this before.

00:03:09.520 --> 00:03:11.840
This is U squared minus V squared.

00:03:11.840 --> 00:03:14.480
So you wouldn't have a
three-term polynomial like that.

00:03:14.480 --> 00:03:15.970
So we could rule that one out.

00:03:15.970 --> 00:03:19.200
So I would pick that we can't
use any of the patterns.

00:03:19.200 --> 00:03:21.430
Let's do yet another example.

00:03:21.430 --> 00:03:23.140
And I encourage you,
pause the video and see

00:03:23.140 --> 00:03:25.190
if you can work this one out on your own.

00:03:26.340 --> 00:03:28.460
So the same idea, they wanna factor

00:03:28.460 --> 00:03:30.230
the following expression.

00:03:30.230 --> 00:03:32.500
And this one essentially has two terms.

00:03:32.500 --> 00:03:35.763
We have a term here and
we have a term here.

00:03:36.938 --> 00:03:38.780
They both look like they are the square

00:03:38.780 --> 00:03:41.820
of something and we have
a difference of squares.

00:03:41.820 --> 00:03:43.570
So this is making me feel pretty good

00:03:43.570 --> 00:03:45.690
about this pattern but
let's see if that works out.

00:03:45.690 --> 00:03:48.380
Remember, U plus V
times U minus V is equal

00:03:48.380 --> 00:03:51.430
to U squared minus V squared.

00:03:51.430 --> 00:03:56.300
So if this is equal to U
squared then that means

00:03:56.300 --> 00:04:00.670
that capital U is equal to six X squared.

00:04:01.600 --> 00:04:02.433
That works.

00:04:02.433 --> 00:04:05.570
And if this is equal to V squared, well,

00:04:05.570 --> 00:04:09.000
that means that V is
equal to Y plus three.

00:04:09.000 --> 00:04:10.870
So this is fitting this
pattern right over here.

00:04:10.870 --> 00:04:12.500
And they're just asking
us to say what pattern

00:04:12.500 --> 00:04:14.350
can we use to factor the expression.

00:04:14.350 --> 00:04:16.460
They're not asking us
to actually factor it.

00:04:16.460 --> 00:04:18.030
So we'll just pick this choice.

00:04:18.030 --> 00:04:19.690
But once you identify
the pattern it's actually

00:04:19.690 --> 00:04:21.300
pretty straightforward to factor it

00:04:21.300 --> 00:04:23.210
because if you say this
is just going to factor

00:04:23.210 --> 00:04:27.670
into U plus V times U
minus V, well, U plus V

00:04:27.670 --> 00:04:32.490
is going to be six X squared plus V plus Y

00:04:32.490 --> 00:04:35.720
plus three times U minus V.

00:04:35.720 --> 00:04:38.150
U is six X squared.

00:04:38.150 --> 00:04:41.570
Minus V is minus, we could write minus Y

00:04:41.570 --> 00:04:44.170
plus three or we could
distribute the negative sign.

00:04:44.170 --> 00:04:45.710
But either way this might
make it a little bit

00:04:45.710 --> 00:04:47.290
clearer what we just did.

00:04:47.290 --> 00:04:50.530
We used a pattern to
factor this higher-degree

00:04:50.530 --> 00:04:52.260
polynomial which is essentially just

00:04:52.260 --> 00:04:54.090
a difference of squares.

00:04:54.090 --> 00:04:55.793
Let's do one last example.

00:04:57.730 --> 00:05:00.930
So here once again we
wanna factor an expression.

00:05:00.930 --> 00:05:03.200
Which pattern can we use?

00:05:03.200 --> 00:05:04.093
Pause the video.

00:05:04.940 --> 00:05:07.450
All right, so we have two terms here.

00:05:07.450 --> 00:05:08.850
So it looks like it might be a difference

00:05:08.850 --> 00:05:13.020
of squares if we set U is equal to seven

00:05:13.020 --> 00:05:17.250
and then this would be U U squared.

00:05:17.250 --> 00:05:19.720
But then what can we square to get 10 X

00:05:19.720 --> 00:05:21.110
to the third power?

00:05:21.110 --> 00:05:24.060
Remember, we wanna have
integer exponents here.

00:05:24.060 --> 00:05:26.890
And the square root of
10 X to the third power,

00:05:26.890 --> 00:05:29.070
if I were to take the square root of 10 X

00:05:29.070 --> 00:05:31.280
to the third power it would be something

00:05:31.280 --> 00:05:33.890
a little bit involved
like the square root of 10

00:05:33.890 --> 00:05:38.160
times X times the square root of X

00:05:38.160 --> 00:05:39.720
to the third power and
I'm not going to get

00:05:39.720 --> 00:05:41.230
an integer exponent here.

00:05:41.230 --> 00:05:44.350
So it doesn't look like I can
express this as V squared.

00:05:44.350 --> 00:05:48.330
So I would go with that we
can't use any of the patterns.

00:05:48.330 --> 00:05:49.163
And we're done.