Commutative property for addition

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:02.520
Rewrite the expression
below applying

00:00:02.520 --> 00:00:07.180
the commutative and associative
properties of addition.

00:00:07.180 --> 00:00:09.170
And then show that
both expressions

00:00:09.170 --> 00:00:12.080
yield the same result.

00:00:12.080 --> 00:00:14.349
So one, we could just evaluate
the expression the way

00:00:14.349 --> 00:00:14.890
it's written.

00:00:14.890 --> 00:00:16.348
And then we can
kind of mess around

00:00:16.348 --> 00:00:19.100
with it using the commutative
and associative properties

00:00:19.100 --> 00:00:20.480
of addition.

00:00:20.480 --> 00:00:24.860
So let's first add 17.5 plus 3.

00:00:24.860 --> 00:00:27.450
So that's going to give us 20.5.

00:00:27.450 --> 00:00:30.760
So this is going to be 20.5.

00:00:30.760 --> 00:00:34.440
And then to that, we're
going to add negative 7.5.

00:00:34.440 --> 00:00:39.070
Now, adding negative 7.5
is the exact same thing

00:00:39.070 --> 00:00:42.620
as subtracting 7.5.

00:00:42.620 --> 00:00:46.340
So this is going to be equal
to-- the 0.5's cancel out.

00:00:46.340 --> 00:00:51.350
And then 20 minus 7 is 13.

00:00:51.350 --> 00:00:53.660
So that's our first way
of getting the answer.

00:00:53.660 --> 00:00:56.000
And we kind of adhered
to the parentheses.

00:00:56.000 --> 00:00:58.260
Now let's use the
commutative property.

00:00:58.260 --> 00:00:59.910
The commutative
property tells us

00:00:59.910 --> 00:01:01.200
that the order doesn't matter.

00:01:01.200 --> 00:01:03.100
We can commute these
numbers around.

00:01:03.100 --> 00:01:04.480
They can move, commute.

00:01:04.480 --> 00:01:06.060
It's like you're going to work.

00:01:06.060 --> 00:01:07.190
They can move around.

00:01:07.190 --> 00:01:09.684
So let's just move
the numbers around.

00:01:09.684 --> 00:01:11.100
Let's make it--
actually, we could

00:01:11.100 --> 00:01:14.610
do all sorts of crazy things.

00:01:14.610 --> 00:01:16.110
We could just change
the order here.

00:01:16.110 --> 00:01:26.250
We could make this negative
7.5 plus 17.5 plus 3.

00:01:26.250 --> 00:01:28.582
We could keep the
parentheses, just like that.

00:01:28.582 --> 00:01:30.290
So we would have
essentially just changed

00:01:30.290 --> 00:01:32.710
the order of this
expression right over here.

00:01:32.710 --> 00:01:34.450
But let's use both
the commutative

00:01:34.450 --> 00:01:35.940
and the associative properties.

00:01:35.940 --> 00:01:38.260
So now we've commuted
everything around.

00:01:38.260 --> 00:01:40.000
And now we can
re-associate everything.

00:01:40.000 --> 00:01:41.999
So instead of putting the
parentheses like that,

00:01:41.999 --> 00:01:45.419
we could put the
parentheses like that.

00:01:45.419 --> 00:01:47.460
So that's what the
associative property tells us.

00:01:47.460 --> 00:01:48.830
So let me write this down.

00:01:48.830 --> 00:01:53.720
So associative
property of addition

00:01:53.720 --> 00:01:59.680
tells us that a plus b plus
c-- we do a plus b first--

00:01:59.680 --> 00:02:05.300
is the same thing as a plus b
plus c, where you do b plus c

00:02:05.300 --> 00:02:06.080
first.

00:02:06.080 --> 00:02:20.480
The commutative
property tells us

00:02:20.480 --> 00:02:24.090
that a plus b is
equal to b plus a,

00:02:24.090 --> 00:02:26.276
that you can move
these guys around.

00:02:26.276 --> 00:02:27.400
So let's evaluate this one.

00:02:27.400 --> 00:02:29.560
We actually got here
using both the commutative

00:02:29.560 --> 00:02:31.620
and the associative property.

00:02:31.620 --> 00:02:37.330
So we get negative
7.5 plus 17.5.

00:02:37.330 --> 00:02:42.330
This is the exact same
thing as 17.5 minus 7.5.

00:02:42.330 --> 00:02:45.750
It might be easier for
you to realize, OK,

00:02:45.750 --> 00:02:48.944
I'm adding two numbers
of different signs,

00:02:48.944 --> 00:02:51.360
so what I could do is take the
difference between the two.

00:02:51.360 --> 00:02:53.150
And since the larger
number is positive,

00:02:53.150 --> 00:02:54.230
I have a larger number.

00:02:54.230 --> 00:02:58.610
Or you could just view
this as 17.5 minus 7.5.

00:02:58.610 --> 00:03:02.590
So 17.5 minus 7.5--
the 0.5's cancel out.

00:03:02.590 --> 00:03:04.630
17 minus 7 is 10.

00:03:04.630 --> 00:03:07.330
So this part right
here becomes 10.

00:03:07.330 --> 00:03:11.300
And then we still
have the plus 3 there.

00:03:11.300 --> 00:03:15.280
And that, once again, is
going to be equal to 13.

00:03:15.280 --> 00:03:17.450
And we could keep
commuting this around

00:03:17.450 --> 00:03:18.750
and keep re-associating it.

00:03:18.750 --> 00:03:21.930
No matter how we do it,
we are going to get 13.


Forbes  Names You Need To Know:  Khan Academy

WEBVTT
Kind: captions
Language: en

00:00:00.244 --> 00:00:01.694
(whooshing sound)

00:00:01.694 --> 00:00:02.705
(lively music)

00:00:02.705 --> 00:00:06.634
Salman: So notice that when
the X term was positive,

00:00:06.634 --> 00:00:10.619
our hyperbola opened to
the right and the left.

00:00:10.619 --> 00:00:14.698
Voiceover: Sal Khan is the CEO of
nonprofit on-demand educational resource,

00:00:14.698 --> 00:00:15.999
Khan Academy.

00:00:15.999 --> 00:00:21.252
He started making educational videos
and compatible practice software in 2004

00:00:21.252 --> 00:00:26.134
for his cousins, that offer lessons and
many levels of mathematics and science,

00:00:26.134 --> 00:00:27.785
and they became viral,

00:00:27.785 --> 00:00:29.894
with many more people turning to him

00:00:29.894 --> 00:00:33.919
and asking him to create even more
lessons in a variety of areas.

00:00:33.919 --> 00:00:36.261
Now, he is doing it full time,

00:00:36.261 --> 00:00:41.366
and Khan Academy is a designated
Forbes Name You Need To Know in 2011.

00:00:41.366 --> 00:00:45.060
A recent blog post on Khan's
new approach to education,

00:00:45.060 --> 00:00:46.991
which takes the lectures
out of the classroom,

00:00:46.991 --> 00:00:49.710
leaving more room for
project-based learning at school,

00:00:49.710 --> 00:00:52.091
sparked quite the conversation online.

00:00:52.091 --> 00:00:53.330
Salman: Bruce Upbin brings up the point

00:00:53.330 --> 00:00:56.375
that there is not a chance on Khan
Academy to ask follow-up questions.

00:00:56.375 --> 00:00:58.510
I think, at least the
way I think about it is,

00:00:58.510 --> 00:01:00.454
let's say you have a question,
and you're doing calculus,

00:01:00.454 --> 00:01:02.188
you have a question on limits.

00:01:02.188 --> 00:01:03.517
There is a video on limits,

00:01:03.517 --> 00:01:05.945
and you can go back and you can
click that 10-minute video on limits,

00:01:05.945 --> 00:01:07.109
and just the act of that is, like,

00:01:07.109 --> 00:01:07.522
"You know what?

00:01:07.522 --> 00:01:08.671
"I don't understand limits.

00:01:08.671 --> 00:01:10.654
"Can you explain a
little bit more on that?"

00:01:10.654 --> 00:01:11.377
And if you're still stuck,

00:01:11.377 --> 00:01:12.562
if you don't even know how to navigate

00:01:12.562 --> 00:01:16.097
or know what you should be looking
at to answer your question,

00:01:16.097 --> 00:01:17.345
we have a pretty vibrant community.

00:01:17.345 --> 00:01:19.938
We have a million users a month.

00:01:19.938 --> 00:01:22.352
So to Bruce's second point
about hands-on learning,

00:01:22.352 --> 00:01:24.024
it's a completely fair point.

00:01:24.024 --> 00:01:25.685
And I think what Khan Academy enables

00:01:25.685 --> 00:01:28.291
is more time in the classroom
for actual hands-on learning.

00:01:28.291 --> 00:01:30.400
The students are getting the
lecture at home at their own pace.

00:01:30.400 --> 00:01:32.398
They can remediate if they need to.

00:01:32.398 --> 00:01:34.647
And now they can go to the classroom
and do project-based learning

00:01:34.647 --> 00:01:36.063
and spend more time with the professor.

00:01:36.063 --> 00:01:38.710
To Kerckhoff's point about
taking a relationship

00:01:38.710 --> 00:01:42.500
or connection with the teacher
to really get deep understanding,

00:01:42.500 --> 00:01:43.392
I think that's true.

00:01:43.392 --> 00:01:46.623
I think we'd be very optimistic
if we think a lot of the students

00:01:46.623 --> 00:01:49.988
are getting that deep intimate
connection in the classroom right now.

00:01:49.988 --> 00:01:51.530
I think most students aren't.

00:01:51.530 --> 00:01:53.885
And to some degree, and even though
it is virtual, it is strange,

00:01:53.885 --> 00:01:56.497
but I get a lot of feedback
from a lot of users

00:01:56.497 --> 00:01:58.799
that they feel a deep
intimate connection with me,

00:01:58.799 --> 00:02:00.931
that I feel like their
neighbor or their older brother

00:02:00.931 --> 00:02:04.374
and that I'm sitting next
to them at the kitchen table

00:02:04.374 --> 00:02:05.133
teaching them math.

00:02:05.133 --> 00:02:07.122
I love Christopher Helman's point,

00:02:07.122 --> 00:02:09.780
and I'm glad I was able to help his wife.

00:02:09.780 --> 00:02:11.303
To his point about replacing universities,

00:02:11.303 --> 00:02:13.904
I don't think that's going
to happen anytime soon,

00:02:13.904 --> 00:02:16.411
but I think it's an interesting idea.

00:02:16.411 --> 00:02:17.361
I think to think about that,

00:02:17.361 --> 00:02:19.165
we have to realize that a
university is two things:

00:02:19.165 --> 00:02:21.515
It's a credential and it's learning.

00:02:21.515 --> 00:02:25.885
I think Khan Academy is
going to be able to really

00:02:25.885 --> 00:02:29.539
change how the learning part is
done in a pretty significant way.

00:02:29.539 --> 00:02:33.759
The credentialing part, getting your
degree from Stanford or Harvard,

00:02:33.759 --> 00:02:35.693
that will probably be
there for a little while.

00:02:35.693 --> 00:02:38.575
I think if you fast
forward 10 years from now,

00:02:38.575 --> 00:02:40.175
I mean it's not going to happen overnight,

00:02:40.175 --> 00:02:41.828
I mean this is a social change happening,

00:02:41.828 --> 00:02:43.664
10 years, 20 years,

00:02:43.664 --> 00:02:47.623
I think an employer would rather see
your log from a site like Khan Academy,

00:02:47.623 --> 00:02:51.098
where it doesn't just get
a 3.2 GPA in psychology,

00:02:51.098 --> 00:02:53.161
it gets what you did, when you did it,

00:02:53.161 --> 00:02:55.661
how well you did it, how well
were you able to help your peers,

00:02:55.661 --> 00:02:57.166
how consistently did you work.

00:02:57.166 --> 00:03:00.662
"Wow, this guy worked three hours
everyday for 20 years on this stuff."

00:03:00.662 --> 00:03:04.092
"This is a persistent kind of
guy that I want working for me,"

00:03:04.092 --> 00:03:05.796
and we'll be able to give
people that type of analytics.

00:03:05.796 --> 00:03:08.649
I think that could be a
more powerful transcript

00:03:08.649 --> 00:03:11.175
than just a high-level degree right now.

00:03:11.267 --> 23:59:59.999
(whooshing sound)


2003 AIME II problem 15 (part 3)

WEBVTT
Kind: captions
Language: en

00:00:00.510 --> 00:00:02.240
Where we left off
in the last video,

00:00:02.240 --> 00:00:05.070
we were beginning
to try to figure out

00:00:05.070 --> 00:00:08.380
what the 24 roots of 1 were.

00:00:08.380 --> 00:00:10.380
And then we were going
to square those 24 roots,

00:00:10.380 --> 00:00:12.820
take the absolute value
of their imaginary part,

00:00:12.820 --> 00:00:14.350
and then sum them up.

00:00:14.350 --> 00:00:16.810
Once again, let's just think
about the roots that matter.

00:00:16.810 --> 00:00:20.470
So this over here, we figured
out this is pi over 12.

00:00:20.470 --> 00:00:23.355
So let me-- this is pi over 12.

00:00:23.355 --> 00:00:24.980
That's the angle, I
guess I should say.

00:00:24.980 --> 00:00:27.160
That's e to the pi over 12i.

00:00:27.160 --> 00:00:29.250
Let me just focus
on the angle, here.

00:00:29.250 --> 00:00:33.200
This over here is 2 pi
over 12, or pi over 6.

00:00:33.200 --> 00:00:36.920
This over here is 3 pi
over 12, or pi over 4.

00:00:36.920 --> 00:00:43.020
This over here is 4 pi
over 12, or pi over 3.

00:00:43.020 --> 00:00:45.425
And this right over
here is 5 pi over 12.

00:00:50.717 --> 00:00:53.050
And then this, obviously, is
6 pi over 12, or pi over 2.

00:00:53.050 --> 00:00:55.560
We could keep going, but
before we keep going--

00:00:55.560 --> 00:00:57.470
and this will
really just simplify

00:00:57.470 --> 00:00:59.070
the math a little
bit-- remember,

00:00:59.070 --> 00:01:00.350
we're going to have
to square these roots.

00:01:00.350 --> 00:01:01.850
So let's just think
about if we have

00:01:01.850 --> 00:01:07.460
some complex number,
a plus bi, let's

00:01:07.460 --> 00:01:09.660
think about what happens
when we square it.

00:01:09.660 --> 00:01:11.780
This is going to be
equal to a squared

00:01:11.780 --> 00:01:16.960
plus 2abi minus b squared.

00:01:16.960 --> 00:01:19.310
Or we could rewrite it as
a squared minus b squared--

00:01:19.310 --> 00:01:23.760
will be the real part--
and then, plus 2abi.

00:01:23.760 --> 00:01:26.690
2ab is the imaginary part.

00:01:26.690 --> 00:01:28.490
And the reason why I
even took the trouble

00:01:28.490 --> 00:01:30.590
of writing it this
way is to realize

00:01:30.590 --> 00:01:32.520
that what we're
going to do is we're

00:01:32.520 --> 00:01:34.680
going to square all of
these complex numbers,

00:01:34.680 --> 00:01:37.710
and then take the absolute
value of the imaginary part.

00:01:37.710 --> 00:01:43.200
So really, all that matters
is the absolute value of 2ab,

00:01:43.200 --> 00:01:49.840
or really, 2 times the
absolute value of ab.

00:01:49.840 --> 00:01:52.050
Now these guys all
have analogs where

00:01:52.050 --> 00:01:54.280
either a or b would be negative.

00:01:54.280 --> 00:01:58.780
So if this guy is--
this right over here--

00:01:58.780 --> 00:02:00.450
let's call this, a plus bi.

00:02:00.450 --> 00:02:05.510
If you had a minus bi,
you would be down here.

00:02:05.510 --> 00:02:08.979
So if this guy is a
plus bi, a minus bi

00:02:08.979 --> 00:02:10.474
is going to be over here.

00:02:10.474 --> 00:02:11.890
Or you could have
negative a minus

00:02:11.890 --> 00:02:16.760
bi is going to be
over here, or you

00:02:16.760 --> 00:02:18.820
could have negative
a plus bi over here.

00:02:21.592 --> 00:02:23.050
But the reason why
I did this is we

00:02:23.050 --> 00:02:25.410
showed you, just over
here, all of these guys

00:02:25.410 --> 00:02:27.310
when you square
them and you take

00:02:27.310 --> 00:02:29.430
the absolute value
of the imaginary part

00:02:29.430 --> 00:02:32.230
of that squared value, it's
all going to be the same.

00:02:32.230 --> 00:02:34.290
Because when you take--
it's going to be ab,

00:02:34.290 --> 00:02:36.100
absolute value is
going to be ab.

00:02:36.100 --> 00:02:38.650
Negative ab absolute value
just still going to be ab.

00:02:38.650 --> 00:02:40.490
So they're all going
to be the same.

00:02:40.490 --> 00:02:42.410
So what we can do is--
and each of these guys

00:02:42.410 --> 00:02:43.826
are going to have
the same analog.

00:02:43.826 --> 00:02:47.140
So what we could is just find
the values for these guys one,

00:02:47.140 --> 00:02:52.020
two, three, four, for
these five guys over here.

00:02:52.020 --> 00:02:55.620
And then just multiply whatever
we get by 4, because they

00:02:55.620 --> 00:02:58.740
each have-- I guess there's
four of these around the unit

00:02:58.740 --> 00:03:00.970
circle, and that'll
save a lot of our work.

00:03:00.970 --> 00:03:03.150
Now, the other thing we
want to think about-- we

00:03:03.150 --> 00:03:04.956
already said that
we have to ignore 1,

00:03:04.956 --> 00:03:06.080
because we added that root.

00:03:06.080 --> 00:03:07.913
But even if we didn't
remember to ignore it,

00:03:07.913 --> 00:03:10.950
it wouldn't matter, because 1
doesn't have an imaginary part.

00:03:10.950 --> 00:03:13.900
1 squared is still 1-- does
not have an imaginary part.

00:03:13.900 --> 00:03:21.850
We can also ignore an angle
of pi over 2, or 90 degrees,

00:03:21.850 --> 00:03:24.170
because this has no real part.

00:03:24.170 --> 00:03:26.680
And you see over here,
that when you square it,

00:03:26.680 --> 00:03:28.430
it's actually going
to take you over here,

00:03:28.430 --> 00:03:30.160
and it won't have
any imaginary part.

00:03:30.160 --> 00:03:32.435
When you square it it's
2 times the real times

00:03:32.435 --> 00:03:33.260
the imaginary part.

00:03:33.260 --> 00:03:35.330
This has no real part,
so this is going to be 0.

00:03:35.330 --> 00:03:38.060
So this guy will
also not contribute.

00:03:38.060 --> 00:03:42.100
So we really just care about
these angles right over here.

00:03:42.100 --> 00:03:44.590
And then we'll square them,
find the absolute value

00:03:44.590 --> 00:03:45.630
of their imaginary
parts, and then we'll

00:03:45.630 --> 00:03:47.350
multiply everything
by 4, because that'll

00:03:47.350 --> 00:03:49.970
correspond to these
other-- if you

00:03:49.970 --> 00:03:53.760
took the imaginary or the real
parts and made them negative.

00:03:53.760 --> 00:03:56.094
And it would take us all the
way around the unit circle.

00:03:56.094 --> 00:03:57.510
So let's think
about a little bit.

00:03:57.510 --> 00:03:58.930
Let me just write
these over here.

00:03:58.930 --> 00:04:02.320
So the z's that we'll
think about right now

00:04:02.320 --> 00:04:12.870
are going to be e to the pi
over 12, e to the pi over 6, e

00:04:12.870 --> 00:04:15.730
to the pi over 4-- oh,
e to the pi over 12i,

00:04:15.730 --> 00:04:22.590
I should say-- pi over 6i, pi
over 4i, e to the pi over 3i,

00:04:22.590 --> 00:04:26.470
and then we have e
to the 5 pi over 12i.

00:04:30.496 --> 00:04:32.120
Now we're going to
square each of them,

00:04:32.120 --> 00:04:34.419
and it's nice to leave it
in this exponential form

00:04:34.419 --> 00:04:35.210
when you square it.

00:04:35.210 --> 00:04:37.246
It's much easier to
square these values.

00:04:37.246 --> 00:04:39.120
If you square this,
you're just, essentially,

00:04:39.120 --> 00:04:40.800
multiplying the
exponent times 2.

00:04:40.800 --> 00:04:45.350
So this is going to be
e to the pi over 6i.

00:04:45.350 --> 00:04:51.070
This will be e to the pi over
3i-- we're just squaring them.

00:04:51.070 --> 00:04:53.180
So we're just taking
the square of each

00:04:53.180 --> 00:04:57.360
of these values,
each of these roots.

00:04:57.360 --> 00:04:59.830
So e to the pi over 3i,
and then here you're

00:04:59.830 --> 00:05:04.450
going to have e
to the pi over 2i.

00:05:08.300 --> 00:05:12.355
And over here, you'll have
e to the 2 pi over 3i.

00:05:15.240 --> 00:05:20.090
And then finally over here,
you'll have e to the 5 pi

00:05:20.090 --> 00:05:21.790
over 6i.

00:05:21.790 --> 00:05:25.582
Now these are the squared
values of these roots here.

00:05:25.582 --> 00:05:27.665
Now, let's just think about
their imaginary parts.

00:05:30.990 --> 00:05:32.510
So this guy right
over here can be

00:05:32.510 --> 00:05:39.900
rewritten as cosine of pi over
6 plus i sine of pi over 6.

00:05:39.900 --> 00:05:44.710
So his imaginary part
is sine of pi over 6.

00:05:44.710 --> 00:05:48.560
This guy's imaginary part--
if you expand it out,

00:05:48.560 --> 00:05:50.520
Euler's identity,
Euler's formula.

00:05:50.520 --> 00:05:54.340
Cosine of pi over 3 plus
i sine of pi over 3.

00:05:54.340 --> 00:05:59.180
So this imaginary part is just
going to be sine of pi over 3.

00:05:59.180 --> 00:06:02.270
Here is going to be
sine of pi over 2.

00:06:02.270 --> 00:06:05.270
Here is going to be
sine of 2 pi over 3.

00:06:05.270 --> 00:06:16.660
And here is going to be
a sine of 5 pi over 6.

00:06:16.660 --> 00:06:19.410
Now we just have to
evaluate these guys,

00:06:19.410 --> 00:06:23.370
and take their absolute
value, and then take the sum,

00:06:23.370 --> 00:06:25.240
and then multiply
everything times 4.

00:06:25.240 --> 00:06:26.990
And we're essentially
in the home stretch.

00:06:26.990 --> 00:06:31.150
So pi over 6 is 180--
if we were in degrees,

00:06:31.150 --> 00:06:34.137
and my brain has an easier
time processing that.

00:06:34.137 --> 00:06:36.470
So, actually, let me draw
another unit circle over here,

00:06:36.470 --> 00:06:39.920
just so we can
visualize these angles.

00:06:39.920 --> 00:06:42.100
So now we have
sine of pi over 6.

00:06:42.100 --> 00:06:48.260
Pi over 6 is the same
thing as 30 degrees.

00:06:48.260 --> 00:06:50.090
So it looks like this.

00:06:50.090 --> 00:06:55.290
And we know that the sine
of 30 degrees is 1/2.

00:06:55.290 --> 00:06:57.760
This is one, this is 1/2,
this the cosine square root

00:06:57.760 --> 00:06:58.320
of 3 over 2.

00:06:58.320 --> 00:07:01.480
But this right
over here, is 1/2.

00:07:01.480 --> 00:07:04.490
Sine of pi over 3.

00:07:04.490 --> 00:07:07.595
pi over 3 is the
equivalent of 60 degrees.

00:07:11.380 --> 00:07:15.750
The sin over here is
square root of 3 over 2.

00:07:15.750 --> 00:07:18.580
You can figure that out
from 30, 60, 90 triangles.

00:07:18.580 --> 00:07:21.330
Then you have sine of pi over 2.

00:07:21.330 --> 00:07:23.430
Pi over 2 is that
right over there.

00:07:23.430 --> 00:07:26.680
Well, the sine of
that is just 1.

00:07:26.680 --> 00:07:30.320
So this is the
imaginary part of this,

00:07:30.320 --> 00:07:32.540
or I guess, this would
be-- essentially,

00:07:32.540 --> 00:07:34.967
this evaluates just to i,
but the imaginary part,

00:07:34.967 --> 00:07:36.800
which is viewed as the
coefficient on the i,

00:07:36.800 --> 00:07:38.510
which is kind of non-intuitive.

00:07:38.510 --> 00:07:39.670
You would kind of think
it's the whole thing,

00:07:39.670 --> 00:07:41.294
but when people say
the imaginary part.

00:07:41.294 --> 00:07:44.480
So this will just be 1.

00:07:44.480 --> 00:07:49.990
And then sine of 2 pi over 3.

00:07:49.990 --> 00:07:53.770
Let's see, that is-- so
pi over 3 is 60 degrees.

00:07:53.770 --> 00:07:56.670
So you could view
that is 120 degrees.

00:07:56.670 --> 00:07:58.780
So it's 120 degrees.

00:07:58.780 --> 00:08:00.370
So it's right over here.

00:08:00.370 --> 00:08:09.470
So it's going to have the
same sin value as pi over 3,

00:08:09.470 --> 00:08:12.250
so it's going to be
square root of 3 over 2.

00:08:12.250 --> 00:08:15.230
And then, sine of 5 pi over 6.

00:08:15.230 --> 00:08:17.860
Pi over 6 is 30 degrees.

00:08:17.860 --> 00:08:20.620
So this is sine of 150 degrees.

00:08:20.620 --> 00:08:22.500
So it's going to
be just like that.

00:08:22.500 --> 00:08:26.450
And it's going to have the
same sine value as pi over 6.

00:08:26.450 --> 00:08:29.390
So it's going to be
1/2, and lucky for us,

00:08:29.390 --> 00:08:31.096
these are all positive values.

00:08:31.096 --> 00:08:32.220
So let's just take the sum.

00:08:32.220 --> 00:08:40.000
And so we have 1/2 plus
1/2 is 1, plus another 1

00:08:40.000 --> 00:08:43.590
over here is going to be
two, plus the square root

00:08:43.590 --> 00:08:48.520
of 3 over 2 plus square root of
3 over 2 is square root of 3.

00:08:48.520 --> 00:08:52.270
Now remember, we just did it
for this quadrant over here.

00:08:52.270 --> 00:08:54.020
We have to do it for
all of the quadrants,

00:08:54.020 --> 00:08:58.010
so we just need to multiply
everything times 4.

00:08:58.010 --> 00:09:01.520
So the sum of the absolute
value of the imaginary part

00:09:01.520 --> 00:09:10.540
of the square of the roots is
8 plus 4 square roots of 3.

00:09:10.540 --> 00:09:12.970
And going back to
the original problem,

00:09:12.970 --> 00:09:13.980
we got the answer here.

00:09:13.980 --> 00:09:17.934
This is 8 plus 4
square roots of 3.

00:09:17.934 --> 00:09:19.600
I want to make sure
I didn't mess it up.

00:09:19.600 --> 00:09:20.100
Right.

00:09:20.100 --> 00:09:22.000
8 plus 4 square roots of 3.

00:09:22.000 --> 00:09:26.040
So if we want find m plus
n plus p, it is 8 plus 4

00:09:26.040 --> 00:09:29.340
plus 3, which is equal to 15.

00:09:29.340 --> 00:09:31.130
And we're done.


2003 AIME II problem 15 (part 2)

WEBVTT
Kind: captions
Language: en

00:00:01.720 --> 00:00:03.610
Where we left off
in the last video,

00:00:03.610 --> 00:00:08.710
we had somewhat simplified our
polynomial p of x from the way

00:00:08.710 --> 00:00:11.600
that it was originally
described in the problem,

00:00:11.600 --> 00:00:13.100
but it still doesn't
seem like we're

00:00:13.100 --> 00:00:14.540
any closer to finding the roots.

00:00:14.540 --> 00:00:18.540
At least we were able to reduce
the degree from 47th degree

00:00:18.540 --> 00:00:21.760
polynomial to the product of a
first degree and a 23rd degree

00:00:21.760 --> 00:00:22.410
polynomial.

00:00:22.410 --> 00:00:23.785
But it still
doesn't seem like we

00:00:23.785 --> 00:00:26.630
are any closer to finding
the roots, especially

00:00:26.630 --> 00:00:28.799
the roots that matter.

00:00:28.799 --> 00:00:31.340
And here, we're going to have
to-- there's another little bit

00:00:31.340 --> 00:00:35.870
of an aha moment here-- that
this part in a blue, which

00:00:35.870 --> 00:00:39.080
is really just the sum of all
of the degrees of x from x

00:00:39.080 --> 00:00:41.437
to the 0 all the way to x to 23.

00:00:41.437 --> 00:00:42.520
This actually looks a lot.

00:00:42.520 --> 00:00:44.630
It is actually, if you
view the common ratio

00:00:44.630 --> 00:00:48.190
as x as a finite
geometric series.

00:00:48.190 --> 00:00:50.050
And I've done it
in other videos,

00:00:50.050 --> 00:00:53.767
there is a way to find the sum
of a finite geometric series.

00:00:53.767 --> 00:00:55.600
And we'll use a similar
technique right here

00:00:55.600 --> 00:00:58.030
to simplify this a little
bit-- to help us find the root.

00:00:58.030 --> 00:01:00.550
So let's just call
this whole thing S. So

00:01:00.550 --> 00:01:06.170
let's just call this S is
equal to x to the 23 plus x

00:01:06.170 --> 00:01:09.770
to the 22nd plus all the
way down to x plus 1.

00:01:09.770 --> 00:01:11.570
So that's this thing over here.

00:01:11.570 --> 00:01:14.800
Now let's multiply S times x.

00:01:14.800 --> 00:01:18.779
So x times S is going
to be equal to--

00:01:18.779 --> 00:01:20.320
and actually, let
me do it like this,

00:01:20.320 --> 00:01:21.610
because this will
help us a little bit.

00:01:21.610 --> 00:01:23.390
I'm going to write
it up here, and it's

00:01:23.390 --> 00:01:25.940
going to look a little
bit messy. x times S

00:01:25.940 --> 00:01:29.470
is going to be-- actually, let
me-- I'll just write it over

00:01:29.470 --> 00:01:35.610
here. x times S is equal
to-- let me save some space,

00:01:35.610 --> 00:01:36.560
actually.

00:01:36.560 --> 00:01:42.780
x times S is equal to-- x times
x to 23 is x to the 24th power,

00:01:42.780 --> 00:01:50.640
x times x to the 22nd is going
to be x to the 23rd power,

00:01:50.640 --> 00:01:53.030
and then there's going
to be an x to the 21st.

00:01:53.030 --> 00:01:55.450
When you multiply by x, you're
going to get x to the 22nd.

00:01:55.450 --> 00:02:00.100
All the way down to when
you multiply x times x,

00:02:00.100 --> 00:02:02.550
you're going to
get plus x squared.

00:02:02.550 --> 00:02:06.460
And then when you multiply the 1
times x, you're going to get x.

00:02:06.460 --> 00:02:10.729
Now, what I want to do, let's
subtract this from that.

00:02:10.729 --> 00:02:13.450
Let's subtract it.

00:02:13.450 --> 00:02:15.250
I won't put the
negative sign out there.

00:02:15.250 --> 00:02:16.916
So on the left hand
side of the equation

00:02:16.916 --> 00:02:22.510
you have S minus xS
is equal to-- what

00:02:22.510 --> 00:02:25.270
happens on the right hand
side of the equation?

00:02:25.270 --> 00:02:27.811
Well, you're going to have
this guy minus this guy.

00:02:27.811 --> 00:02:29.060
These are going to cancel out.

00:02:29.060 --> 00:02:30.310
These are going to cancel out.

00:02:30.310 --> 00:02:32.160
Everything is going
to cancel out,

00:02:32.160 --> 00:02:35.260
except you're going to have
a 1 minus x to the 24th.

00:02:35.260 --> 00:02:38.911
So you're going to have
1 minus x to the 24th.

00:02:38.911 --> 00:02:41.035
Now let's just multiply
both sides by a negative 1,

00:02:41.035 --> 00:02:44.760
because this will just get it in
a slightly easier form for us.

00:02:44.760 --> 00:02:46.180
So this is the same thing.

00:02:46.180 --> 00:02:47.940
If we multiply both
sides times negative 1

00:02:47.940 --> 00:02:53.610
it's xS minus S is equal
to x to the 24th minus 1.

00:02:53.610 --> 00:02:55.485
We can factor out an S here.

00:02:55.485 --> 00:02:56.360
I'll do it over here.

00:02:56.360 --> 00:02:58.401
So if we factor out an S
from the left hand side,

00:02:58.401 --> 00:03:05.390
we get S times x minus 1 is
equal to x to the 24th minus 1.

00:03:05.390 --> 00:03:12.130
Or we could say that the sum is
equal to x to the 24th minus 1

00:03:12.130 --> 00:03:14.400
over x minus 1.

00:03:14.400 --> 00:03:17.650
So this whole thing
in the blue over here

00:03:17.650 --> 00:03:20.770
can be rewritten as
this right over here--

00:03:20.770 --> 00:03:27.400
as x to the 24th minus
1 over x minus 1.

00:03:27.400 --> 00:03:28.780
Now let's be clear here.

00:03:28.780 --> 00:03:31.720
What we really care
about for this problem

00:03:31.720 --> 00:03:34.870
is to figure out the
roots that have--

00:03:34.870 --> 00:03:39.190
the roots whose squares
have complex parts.

00:03:39.190 --> 00:03:40.810
We already know
that one of the root

00:03:40.810 --> 00:03:43.944
is 0, because if you
set this equal to 0,

00:03:43.944 --> 00:03:44.860
x could be equal to 0.

00:03:44.860 --> 00:03:48.520
0 squared is still 0-- it
does not have a complex part.

00:03:48.520 --> 00:03:50.020
So it really doesn't matter.

00:03:50.020 --> 00:03:52.210
So for our problem--
and the other thing

00:03:52.210 --> 00:03:56.615
that we want to keep
in mind is that we

00:03:56.615 --> 00:03:57.990
don't want to
double count roots.

00:03:57.990 --> 00:03:59.987
Let's re-read the
problem a little bit.

00:03:59.987 --> 00:04:02.070
Because what we're going
to have is a bunch of-- I

00:04:02.070 --> 00:04:04.760
guess you could view
them as double roots.

00:04:04.760 --> 00:04:08.370
You're only going to
have 23 distinct roots

00:04:08.370 --> 00:04:11.660
from this part of
the polynomial.

00:04:11.660 --> 00:04:13.910
And they say, let
z1, z2, and zr be

00:04:13.910 --> 00:04:18.800
the distinct zeroes of p sub x.

00:04:18.800 --> 00:04:22.560
So what we only care about
right now are these 23 zeroes.

00:04:22.560 --> 00:04:24.950
We don't have to think
about those 23 zeroes twice.

00:04:24.950 --> 00:04:27.390
We're not going to add
their imaginary parts twice.

00:04:27.390 --> 00:04:32.790
So we can kind of substitute
p of x with-- let's call it

00:04:32.790 --> 00:04:35.220
q of x.

00:04:35.220 --> 00:04:36.720
We really just care
about the zeroes

00:04:36.720 --> 00:04:42.030
of q of x, which is just x to
the 23 plus x to the 22nd plus

00:04:42.030 --> 00:04:46.090
all the way down to x
squared plus x plus 1.

00:04:46.090 --> 00:04:47.177
That's this inside part.

00:04:47.177 --> 00:04:49.510
The fact that it's squared,
that just doubles the roots,

00:04:49.510 --> 00:04:51.218
but we just care about
the distinct root.

00:04:51.218 --> 00:04:55.012
So we really just care about
the sum of the imaginary part--

00:04:55.012 --> 00:04:57.470
of the absolute value of the
imaginary parts of the squares

00:04:57.470 --> 00:05:00.010
of the roots of this
guy, right over here.

00:05:00.010 --> 00:05:01.910
Now we just figured
out that this guy

00:05:01.910 --> 00:05:04.980
can be rewritten
as this over here.

00:05:04.980 --> 00:05:11.400
Or even better, let's define--
instead of our defining q

00:05:11.400 --> 00:05:19.500
of x like this, let's define
q of x as this times x--

00:05:19.500 --> 00:05:24.320
let me write it a little
bit-- as this times x minus 1.

00:05:24.320 --> 00:05:26.290
Let's just define q of x here.

00:05:26.290 --> 00:05:29.980
So this q of x is going to have
all of the same distinct roots,

00:05:29.980 --> 00:05:33.260
that matter as this
p of x, but it's also

00:05:33.260 --> 00:05:36.214
going to have the root x is
equal to 1, which is clearly

00:05:36.214 --> 00:05:36.880
not a root here.

00:05:36.880 --> 00:05:38.620
If you put x equals
1 here, you're

00:05:38.620 --> 00:05:40.640
actually going to get 24.

00:05:40.640 --> 00:05:43.040
So it's not going
to be equal to 0.

00:05:43.040 --> 00:05:46.530
But what makes this better,
easier to find the roots of,

00:05:46.530 --> 00:05:48.090
is that this thing
right over here

00:05:48.090 --> 00:05:55.740
is equivalent to-- this part in
yellow is x to the 24th minus 1

00:05:55.740 --> 00:06:01.700
over x minus 1 times
x minus 1, which

00:06:01.700 --> 00:06:06.320
is equal to x to
the 24th minus 1.

00:06:06.320 --> 00:06:10.800
So we've simplified the
problem a huge amount.

00:06:10.800 --> 00:06:14.110
What we've said is the roots
that we care about are actually

00:06:14.110 --> 00:06:16.180
the roots of this
polynomial right

00:06:16.180 --> 00:06:18.690
here, except for
x is equal to 1--

00:06:18.690 --> 00:06:21.120
we just kind of added that
there to simplify it--

00:06:21.120 --> 00:06:24.610
which is equal to the roots
of this polynomial here,

00:06:24.610 --> 00:06:28.190
except for 1, because
we added 1 as a root.

00:06:28.190 --> 00:06:30.700
So we just care about the
roots of this polynomial,

00:06:30.700 --> 00:06:32.490
except for 1.

00:06:32.490 --> 00:06:35.670
And the roots of this polynomial
are-- well, I won't say easy,

00:06:35.670 --> 00:06:39.760
but it's much more
straightforward to calculate.

00:06:39.760 --> 00:06:44.980
If we set this
polynomial to equal to 0,

00:06:44.980 --> 00:06:48.440
to figure out the roots,
we just get x to the 24th

00:06:48.440 --> 00:06:51.340
add 1 to both sides,
is equal to 1.

00:06:51.340 --> 00:06:53.029
So we're going to
have-- remember,

00:06:53.029 --> 00:06:54.320
this is going to have 24 roots.

00:06:54.320 --> 00:06:56.240
One of them is going to be 1.

00:06:56.240 --> 00:07:00.550
We want to ignore the 1, so we
care about the 23 non-1 roots.

00:07:00.550 --> 00:07:02.550
And we care about squaring
them, and then taking

00:07:02.550 --> 00:07:04.740
the absolute value of
the imaginary parts.

00:07:04.740 --> 00:07:07.410
So we essentially need
to think about all

00:07:07.410 --> 00:07:11.400
of the 24th root of
1, and then ignore 1.

00:07:11.400 --> 00:07:13.690
So let's think about how
to do that a little bit.

00:07:13.690 --> 00:07:18.720
And this is going to
delve a little bit

00:07:18.720 --> 00:07:22.450
into the complex plane.

00:07:22.450 --> 00:07:27.470
So just as a refresher, how do
we find all of the roots of 1?

00:07:27.470 --> 00:07:29.730
The 24th roots of 1?

00:07:29.730 --> 00:07:32.340
And just remember,
we can rewrite 1.

00:07:32.340 --> 00:07:34.430
So let's draw a little
complex plane over here.

00:07:38.240 --> 00:07:43.450
So this is the real axis, and
this is the imaginary axis.

00:07:43.450 --> 00:07:45.100
1 is sitting right over here.

00:07:45.100 --> 00:07:48.000
It only has a real
part-- it is 1.

00:07:48.000 --> 00:07:50.290
But we know that we
can rewrite this.

00:07:50.290 --> 00:07:56.140
We can rewrite 1 as--
let me write it this way.

00:07:56.140 --> 00:08:00.280
It could be e to the i times 0.

00:08:00.280 --> 00:08:02.790
Actually, so this is e to
the 0, obviously equal to 1.

00:08:02.790 --> 00:08:05.640
Or we can add 2 pi to it, if
we're dealing with radians.

00:08:05.640 --> 00:08:11.190
So 1 is equal to e to i0, which
is also equal to e to the 2 pi

00:08:11.190 --> 00:08:13.440
i, but you could keep
adding, because if you

00:08:13.440 --> 00:08:15.780
add 2 pi goes around
the complex plane once.

00:08:15.780 --> 00:08:19.340
If you add another 2 pi,
it goes around again.

00:08:19.340 --> 00:08:24.880
It also equals e to the 4 pi i.

00:08:24.880 --> 00:08:29.730
It also equals e to the 6 pi i.

00:08:29.730 --> 00:08:32.450
It also equals e to the 8 pi i.

00:08:32.450 --> 00:08:34.159
I think you get
the general idea.

00:08:34.159 --> 00:08:35.820
So if you want to
find the 24th roots,

00:08:35.820 --> 00:08:39.554
you want to just raise all of
these things to the 1 over 24.

00:08:42.230 --> 00:08:44.730
So essentially, you're going
to take this divide by 24,

00:08:44.730 --> 00:08:49.850
divide by 24, divide by 24,
divide by 24, divided by 24,

00:08:49.850 --> 00:08:51.197
and then keep going.

00:08:51.197 --> 00:08:52.530
You can actually go all the way.

00:08:52.530 --> 00:08:54.330
Well, I'll depict it over here.

00:08:54.330 --> 00:08:55.480
So what is this first root?

00:08:55.480 --> 00:08:57.622
The first root is
e to the 0 over 24.

00:08:57.622 --> 00:08:58.580
That's just e to the 0.

00:08:58.580 --> 00:08:59.750
That's just 1.

00:08:59.750 --> 00:09:01.810
That's the root that
we can ignore, right?

00:09:01.810 --> 00:09:04.190
Because we added that root
to q of x to simplify it

00:09:04.190 --> 00:09:05.120
a little bit.

00:09:05.120 --> 00:09:08.880
Now what's this root over here?

00:09:08.880 --> 00:09:15.880
This is e to the pi over
12i, which remember,

00:09:15.880 --> 00:09:19.690
is the same thing as
cosine of pi over 12--

00:09:19.690 --> 00:09:23.335
that's the real part--
plus i sine of pi over 12.

00:09:26.269 --> 00:09:27.810
And if we were to
draw that, we would

00:09:27.810 --> 00:09:29.990
have an angle of pi over 12.

00:09:29.990 --> 00:09:31.730
And if you wanted
to think in degrees,

00:09:31.730 --> 00:09:33.310
that would be 15 degrees.

00:09:33.310 --> 00:09:39.050
So it would look something
like this right over here.

00:09:39.050 --> 00:09:41.330
So we do care about that
root right over here.

00:09:41.330 --> 00:09:42.790
Now what would this root be?

00:09:42.790 --> 00:09:44.900
Well this root is
going to be pi over 6.

00:09:44.900 --> 00:09:46.400
It's going to have
twice the angles.

00:09:46.400 --> 00:09:48.570
So it's going to look like this.

00:09:48.570 --> 00:09:52.620
So this root right here is
going to look like this.

00:09:52.620 --> 00:09:55.750
This root over here-- and
notice, they're pi over 12

00:09:55.750 --> 00:09:56.930
apart.

00:09:56.930 --> 00:10:03.180
This root over here,
this is pi over 4.

00:10:03.180 --> 00:10:05.770
So it's going to look
like this-- pi over 4.

00:10:05.770 --> 00:10:07.070
That's 45 degrees.

00:10:07.070 --> 00:10:08.990
So it's going to look like this.

00:10:08.990 --> 00:10:12.220
So notice, each of the
roots we're just adding

00:10:12.220 --> 00:10:14.290
are just multiples
of this-- I shouldn't

00:10:14.290 --> 00:10:17.180
say multiples-- are powers
of this first root over here.

00:10:17.180 --> 00:10:21.680
We're just increasing
the degree by pi over 12.

00:10:21.680 --> 00:10:24.870
And remember, we just care
about the imaginary parts

00:10:24.870 --> 00:10:26.501
of their squares.

00:10:26.501 --> 00:10:28.750
So let's just first of all,
just think about the roots

00:10:28.750 --> 00:10:30.904
that matter.

00:10:30.904 --> 00:10:32.320
So one way to think
about it is we

00:10:32.320 --> 00:10:34.940
could literally just
draw a unit circle here

00:10:34.940 --> 00:10:37.790
and divide it into 24 sections.

00:10:37.790 --> 00:10:39.120
So let me draw that.

00:10:39.120 --> 00:10:41.232
Let me draw it a little bit.

00:10:41.232 --> 00:10:42.690
Let me redraw it
a little bit, so I

00:10:42.690 --> 00:10:44.666
don't overload this
graph over here.

00:10:44.666 --> 00:10:46.040
Actually, I'm
running out of time

00:10:46.040 --> 00:10:49.240
in this video-- I might
continue it in the next.

00:10:49.240 --> 00:10:51.060
So we could draw a
unit circle here.

00:10:51.060 --> 00:10:54.290
They're all going to
have magnitude one.

00:10:54.290 --> 00:10:56.970
We could draw the
unit circle over here.

00:10:56.970 --> 00:10:58.570
One of the roots is
1, but we're going

00:10:58.570 --> 00:11:04.360
to have 24 roots, so we can
divide it into 24 sections.

00:11:04.360 --> 00:11:06.310
So we have four
sections right here.

00:11:06.310 --> 00:11:08.660
We're going to have six
roots in each section.

00:11:08.660 --> 00:11:12.780
So we're going to have 1,
pi over 12, 2pi over 12,

00:11:12.780 --> 00:11:15.900
pi over 4, and then you have
the one symmetric to that,

00:11:15.900 --> 00:11:18.240
then you have the one
symmetric to that--

00:11:18.240 --> 00:11:22.410
let me make sure I'm
doing this right.

00:11:22.410 --> 00:11:25.219
And then you're going
to have pi over 2.

00:11:25.219 --> 00:11:27.510
So these are the roots that
are-- actually, let me just

00:11:27.510 --> 00:11:28.690
continue this in the next video.

00:11:28.690 --> 00:11:30.160
I don't want to-- when
I start rushing this--

00:11:30.160 --> 00:11:31.951
this is kind of a new
concept to everybody.

00:11:31.951 --> 00:11:33.950
I'll continue this
in the next video.


2003 AIME II problem 15 (part 1)

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:03.430
So I'll tell you ahead of
time, this problem is no joke.

00:00:03.430 --> 00:00:06.381
But if we do it step by step,
it's actually not too bad.

00:00:06.381 --> 00:00:08.630
But it's not an easy problem,
so don't get discouraged

00:00:08.630 --> 00:00:10.570
if you kind of don't
even know where

00:00:10.570 --> 00:00:12.580
to start on this
one right over here.

00:00:12.580 --> 00:00:15.910
So let p of x equal
24x to the 24th plus

00:00:15.910 --> 00:00:19.940
the sum from j equals 1 to
23 of 24 minus j times x

00:00:19.940 --> 00:00:24.280
to the 24th minus j plus
x to the 24th plus j.

00:00:24.280 --> 00:00:27.670
Let z1, z2, all the way
to zr be the distinct 0's

00:00:27.670 --> 00:00:30.090
of p of x-- fair enough.

00:00:30.090 --> 00:00:32.910
Let z sub k squared
equal a sub k plus b

00:00:32.910 --> 00:00:36.250
sub ki for k equals 1 through r.

00:00:36.250 --> 00:00:39.960
So there's z sub k
or-- I guess, they're

00:00:39.960 --> 00:00:43.640
saying a sub k plus
b sub ki is going

00:00:43.640 --> 00:00:45.790
to be the square of
each of these roots.

00:00:45.790 --> 00:00:47.774
So let's see, for k
is equal 1 through r

00:00:47.774 --> 00:00:49.690
where i is equal to
square root of negative 1.

00:00:49.690 --> 00:00:50.520
We know that.

00:00:50.520 --> 00:00:53.950
And a sub k and b sub
k are real numbers.

00:00:53.950 --> 00:00:58.500
Let the sum from k equal 1 to
r of the absolute value of b

00:00:58.500 --> 00:01:00.450
sub k equal this
number over here

00:01:00.450 --> 00:01:02.560
where m, n, and p
are integers and p

00:01:02.560 --> 00:01:04.510
is not divisible by the
square of any prime.

00:01:04.510 --> 00:01:05.884
So they're saying
that we've just

00:01:05.884 --> 00:01:07.880
simplified the radicals
as much as possible.

00:01:07.880 --> 00:01:10.270
Find m plus n plus p.

00:01:10.270 --> 00:01:13.220
So this sum right
here, this is the sum

00:01:13.220 --> 00:01:16.350
of the absolute value
of the imaginary parts

00:01:16.350 --> 00:01:18.330
of the square of the roots.

00:01:18.330 --> 00:01:24.340
Right? b sub k is the imaginary
part of the square of root k,

00:01:24.340 --> 00:01:25.700
I guess you could call it.

00:01:25.700 --> 00:01:27.500
And so what we're going to
do is take the absolute value

00:01:27.500 --> 00:01:29.500
of the imaginary part of
the square of the roots

00:01:29.500 --> 00:01:31.596
and sum them up.

00:01:31.596 --> 00:01:33.970
So before we even think about
how to do that, let's just,

00:01:33.970 --> 00:01:35.400
I don't know, let's
just try to visualize

00:01:35.400 --> 00:01:36.550
this polynomial a little better.

00:01:36.550 --> 00:01:37.925
In fact, I have
a feeling this is

00:01:37.925 --> 00:01:40.170
going to take us
multiple videos to do.

00:01:40.170 --> 00:01:43.520
So we have p of x
is equal to-- well,

00:01:43.520 --> 00:01:45.060
the first term here is just 24.

00:01:45.060 --> 00:01:45.970
Well, it's not the
first term, it's

00:01:45.970 --> 00:01:47.330
the first term as
the way it's written.

00:01:47.330 --> 00:01:48.704
The first term
could be anything.

00:01:48.704 --> 00:01:50.870
24x to the 24th.

00:01:50.870 --> 00:01:54.040
And now, let's write
out this over here.

00:01:54.040 --> 00:01:57.730
So if we set j equal
to 1, what do we get?

00:01:57.730 --> 00:02:07.810
We get 23 times x to
the 23 plus x to the 25.

00:02:07.810 --> 00:02:11.570
This is 24 plus j.

00:02:11.570 --> 00:02:13.654
And then when j is
equal to 2-- I'll

00:02:13.654 --> 00:02:15.070
just go down in a
big column here.

00:02:15.070 --> 00:02:19.240
When j is equal to 2, we
have plus 22, x to the 22

00:02:19.240 --> 00:02:21.912
plus x to the 26th.

00:02:21.912 --> 00:02:23.120
And then we could keep going.

00:02:23.120 --> 00:02:26.260
Plus-- well, you get
the general idea.

00:02:26.260 --> 00:02:28.656
This term over here,
the coefficients

00:02:28.656 --> 00:02:29.530
are going to go down.

00:02:29.530 --> 00:02:31.290
So is this exponent, but
this exponent over here

00:02:31.290 --> 00:02:32.110
is going to go up.

00:02:32.110 --> 00:02:34.960
So then we can keep
adding all the way-- let's

00:02:34.960 --> 00:02:37.810
go to j equal 22.

00:02:37.810 --> 00:02:42.200
So when j is equal to
22, 24 minus 22 is 2.

00:02:42.200 --> 00:02:46.810
It's going to be 2, x
squared-- 22 plus 24

00:02:46.810 --> 00:02:52.440
is 46-- so plus x to the 46th.

00:02:52.440 --> 00:02:53.430
Did I do that right?

00:02:53.430 --> 00:02:55.820
22, 46.

00:02:55.820 --> 00:02:58.952
And then, this last term over
here, when j is equal to 23,

00:02:58.952 --> 00:03:00.410
because that's
where we stop, we're

00:03:00.410 --> 00:03:06.860
going to get 1 times
x plus x to the 47th.

00:03:06.860 --> 00:03:08.080
So that's our polynomial.

00:03:08.080 --> 00:03:09.538
What I want to do
is I'm just going

00:03:09.538 --> 00:03:11.740
to rewrite this in
kind of the way we're

00:03:11.740 --> 00:03:14.377
used to seeing polynomials, and
that's start with the highest

00:03:14.377 --> 00:03:16.210
degree term and then
you go down from there.

00:03:16.210 --> 00:03:22.730
So we can write p of x as
being equal to-- what's

00:03:22.730 --> 00:03:24.470
the highest degree term here?

00:03:24.470 --> 00:03:29.440
Well, we have an x to the 47th
over here, right over here--

00:03:29.440 --> 00:03:31.096
x to the 47th.

00:03:31.096 --> 00:03:33.470
And then the coefficient here
is 1, if you distribute it.

00:03:33.470 --> 00:03:41.990
So it's going to be x to the
47th plus this term over here,

00:03:41.990 --> 00:03:43.800
2x to the 46th.

00:03:43.800 --> 00:03:46.730
I'm just distributing
this multiplying factor,

00:03:46.730 --> 00:03:48.220
or this, I guess,
its coefficient--

00:03:48.220 --> 00:03:50.800
it's going to be 2x squared
plus 2x to the 46th.

00:03:50.800 --> 00:03:55.550
And then we're just going
to keep going all the way up

00:03:55.550 --> 00:03:56.870
to 22x to the 26th.

00:03:59.934 --> 00:04:01.100
So you see what's happening?

00:04:01.100 --> 00:04:03.190
The coefficient is
increasing by 1 each time,

00:04:03.190 --> 00:04:05.410
and the exponent
is decreasing by 1.

00:04:05.410 --> 00:04:11.290
Plus 23x to the 25th.

00:04:11.290 --> 00:04:14.300
And then the next degree, we
don't have a x to the 24th

00:04:14.300 --> 00:04:16.480
here, but we have the x
to the 24th right there.

00:04:16.480 --> 00:04:18.519
So it seems almost by
design for the problem

00:04:18.519 --> 00:04:19.990
that they stuck it there.

00:04:19.990 --> 00:04:24.170
So plus 24x to the 24th.

00:04:24.170 --> 00:04:25.920
And now we can do all
of these terms here.

00:04:25.920 --> 00:04:36.680
So we have plus 23x to the
23rd, plus 22x to the 22nd,

00:04:36.680 --> 00:04:44.050
and then we go all the way
down to plus 2x squared plus x.

00:04:44.050 --> 00:04:46.430
So this is just another way
that, at least in my brain,

00:04:46.430 --> 00:04:48.820
simplifies this
polynomial a good bit.

00:04:48.820 --> 00:04:51.260
We now have a good sense
of what's happening here.

00:04:51.260 --> 00:04:56.090
It's really all of the powers
of x up to the 47th power.

00:04:56.090 --> 00:04:57.590
And then the
coefficients start at 1

00:04:57.590 --> 00:04:59.470
and they keep
increasing up to 24,

00:04:59.470 --> 00:05:01.661
then they start
decreasing again.

00:05:01.661 --> 00:05:03.910
Now, we're interested in the
roots of this polynomial.

00:05:03.910 --> 00:05:06.490
And there's one root
that's pretty easy to find.

00:05:06.490 --> 00:05:08.080
We can just factor
an x out of this.

00:05:08.080 --> 00:05:10.260
Every term here
is divisible by x.

00:05:10.260 --> 00:05:20.180
So we can rewrite p of x is
equal to x times x to the 46th

00:05:20.180 --> 00:05:24.240
plus 2x to the 45th-- I'm
just dividing all of these

00:05:24.240 --> 00:05:28.590
by x or factoring out an
x-- plus all the way to 22x

00:05:28.590 --> 00:05:35.710
to the 25th plus 23x
to the 24-- and then we

00:05:35.710 --> 00:05:41.304
have the one in magenta--
plus 24x to the 23rd--

00:05:41.304 --> 00:05:42.970
and then we have all
of these characters

00:05:42.970 --> 00:05:45.880
down here-- so then
we have, plus 23x

00:05:45.880 --> 00:05:55.500
to the 22nd plus 22x to the
21 all the way to 2x plus 1.

00:05:55.500 --> 00:05:57.960
Now x equals 0 is
clearly a root--

00:05:57.960 --> 00:06:00.316
or if you set p of x equals
0, x equals 0 is a root.

00:06:00.316 --> 00:06:01.940
It's actually not
going to matter much,

00:06:01.940 --> 00:06:04.280
because it has no
imaginary part.

00:06:04.280 --> 00:06:07.650
So if you take 0-- one of the
roots here is 0, you square 0,

00:06:07.650 --> 00:06:09.420
it's not going to have
an imaginary part.

00:06:09.420 --> 00:06:11.390
So the absolute value
of its imaginary part

00:06:11.390 --> 00:06:14.070
is not going to contribute
to this sum over here,

00:06:14.070 --> 00:06:15.340
but maybe it simplifies this.

00:06:15.340 --> 00:06:18.320
So we really care about the
sum of the roots other than 0.

00:06:18.320 --> 00:06:21.730
So we care about sum of the
roots of everything else, all

00:06:21.730 --> 00:06:23.670
the stuff in parentheses.

00:06:23.670 --> 00:06:25.850
And this part, I
guess the next step,

00:06:25.850 --> 00:06:28.399
it might not be obvious to
you, but once you see it--

00:06:28.399 --> 00:06:30.690
once you see the pattern, I
guess if you ever see again

00:06:30.690 --> 00:06:33.060
in your life, maybe in some
of these AMI type problems

00:06:33.060 --> 00:06:35.050
or competition problems-- you
should be able to recognize it,

00:06:35.050 --> 00:06:36.680
because it's an easy
pattern recognize.

00:06:36.680 --> 00:06:38.180
It's just not
something you normally

00:06:38.180 --> 00:06:40.840
see in your everyday curriculum.

00:06:40.840 --> 00:06:44.630
And so what I want to do
to see the pattern here

00:06:44.630 --> 00:06:47.750
is think about what happens when
we square different polynomial.

00:06:47.750 --> 00:06:51.057
So if we square where all
of the coefficients are 1.

00:06:51.057 --> 00:06:52.640
They don't all have
to be 1, but it'll

00:06:52.640 --> 00:06:54.391
help us see the pattern there.

00:06:54.391 --> 00:06:56.390
The pattern will obviously
be slightly different

00:06:56.390 --> 00:06:58.130
if the coefficients
are different.

00:06:58.130 --> 00:07:04.230
Now, if I just take x
plus 1-- x plus 1 squared

00:07:04.230 --> 00:07:07.270
is x squared plus 2x plus 1.

00:07:07.270 --> 00:07:09.200
We've seen that multiple times.

00:07:09.200 --> 00:07:15.492
Now what is x squared
plus x plus 1 squared?

00:07:15.492 --> 00:07:16.950
So all I'm doing
is I'm just taking

00:07:16.950 --> 00:07:19.680
all of the powers of
x and summing them up.

00:07:19.680 --> 00:07:22.459
Well, it's going to be-- well,
we've done enough practice

00:07:22.459 --> 00:07:24.375
for this, and if you're
watching this problem,

00:07:24.375 --> 00:07:25.470
you know how to do this.

00:07:25.470 --> 00:07:28.500
It's going to be x
squared times x squared.

00:07:28.500 --> 00:07:32.080
So it's going to be x to the
fourth plus x squared times x,

00:07:32.080 --> 00:07:35.290
which is going to be x to the
third, plus x squared times 1,

00:07:35.290 --> 00:07:40.420
which is going to be x squared,
plus x times x squared,

00:07:40.420 --> 00:07:43.630
which is going to
be x to the third,

00:07:43.630 --> 00:07:47.240
plus x times x, which is
going to be x squared,

00:07:47.240 --> 00:07:50.310
plus x times 1, which
is going to be x.

00:07:50.310 --> 00:07:52.196
And then finally, it's
going to be 1 times

00:07:52.196 --> 00:07:53.570
all of this stuff,
which is going

00:07:53.570 --> 00:07:58.864
to be plus x squared
plus x plus 1.

00:07:58.864 --> 00:08:00.280
And so if you take
the sum, you're

00:08:00.280 --> 00:08:03.290
going to get x to
the fourth plus 2x

00:08:03.290 --> 00:08:10.700
to the third plus 3x
squared plus 2x plus 1.

00:08:10.700 --> 00:08:13.000
And so you might already
see a pattern emerging.

00:08:13.000 --> 00:08:16.020
What happened over here when
you just took x plus 1 squared?

00:08:16.020 --> 00:08:19.130
You started with a 1,
you went up to a 2,

00:08:19.130 --> 00:08:20.440
and then you went back to a 1.

00:08:20.440 --> 00:08:21.690
So not clear that
you have a pattern,

00:08:21.690 --> 00:08:24.273
but it seems to have this-- the
coefficients increase and then

00:08:24.273 --> 00:08:25.000
decrease.

00:08:25.000 --> 00:08:26.990
What happens when you
took this thing squared?

00:08:26.990 --> 00:08:29.040
You started with a 1,
then the coefficients

00:08:29.040 --> 00:08:32.929
went 1, 2, 3, the
middle term, the 3,

00:08:32.929 --> 00:08:37.090
peaked out at the
middle term, and then

00:08:37.090 --> 00:08:39.100
you start going down
in coefficient again.

00:08:39.100 --> 00:08:41.039
1, 2, 3, and then 2, 1.

00:08:41.039 --> 00:08:42.539
And you can actually
prove that this

00:08:42.539 --> 00:08:43.830
is the case for any polynomial.

00:08:43.830 --> 00:08:46.117
So I could just write out--
I won't multiply it out.

00:08:46.117 --> 00:08:47.450
If you have time, you can do it.

00:08:47.450 --> 00:08:52.360
But if I were to take
x to the third plus

00:08:52.360 --> 00:08:55.700
x squared plus x plus 1.

00:08:55.700 --> 00:08:59.220
If I were to square this,
just based on the pattern--

00:08:59.220 --> 00:09:00.924
you could do it on
paper if you like.

00:09:00.924 --> 00:09:02.340
You hopefully know
how to multiply

00:09:02.340 --> 00:09:03.890
this type of polynomial.

00:09:03.890 --> 00:09:05.880
There are videos on
it, if you don't.

00:09:05.880 --> 00:09:08.402
This is going to be equal
to-- the highest degree

00:09:08.402 --> 00:09:09.860
term is going to
be x to the sixth.

00:09:09.860 --> 00:09:14.760
So it's going to be 1x to the
sixth plus 2x to the fifth plus

00:09:14.760 --> 00:09:18.190
3x to the fourth
plus-- and we have

00:09:18.190 --> 00:09:19.660
to be careful-- the middle term.

00:09:19.660 --> 00:09:23.220
So this right here
is going to have--

00:09:23.220 --> 00:09:25.830
you're going to have
each of these terms times

00:09:25.830 --> 00:09:28.740
all-- So the last term is going
to be-- so you're going to have

00:09:28.740 --> 00:09:30.740
x to fourth-- Let me not
write the coefficients.

00:09:30.740 --> 00:09:34.060
You're going to have x to the
fourth plus something times x

00:09:34.060 --> 00:09:38.960
to the third plus something
times x squared plus something

00:09:38.960 --> 00:09:41.465
times x plus 1.

00:09:41.465 --> 00:09:45.134
So you're going to have
1, 2, 3, 4, 5, 6, 7 terms.

00:09:45.134 --> 00:09:47.300
The middle term is going
to be this right over here.

00:09:47.300 --> 00:09:49.270
So that's where we peak
out our coefficients,

00:09:49.270 --> 00:09:50.950
and then they go
back down again.

00:09:50.950 --> 00:09:53.450
3, 2, oh, sorry.

00:09:53.450 --> 00:10:00.639
It should be 3, whoops, 4x, then
3, then 2, and then 1 again.

00:10:00.639 --> 00:10:02.430
And you can multiply
this out for yourself,

00:10:02.430 --> 00:10:04.760
but I think you now
see the pattern.

00:10:04.760 --> 00:10:06.260
So when you look
at this thing here,

00:10:06.260 --> 00:10:08.300
where the coefficients
are just going from 1,

00:10:08.300 --> 00:10:11.620
2, all the way to 24
is where they peak out.

00:10:11.620 --> 00:10:14.810
And notice they peaked
out-- the middle term

00:10:14.810 --> 00:10:16.720
where you're peaking
out-- in this case

00:10:16.720 --> 00:10:18.612
is, really, the highest.

00:10:18.612 --> 00:10:20.820
So we're peaking out here
at the x to the third term,

00:10:20.820 --> 00:10:22.750
and this was a third
degree polynomial

00:10:22.750 --> 00:10:24.140
that we were squaring.

00:10:24.140 --> 00:10:27.090
Over here, we peaked out
at the x squared term,

00:10:27.090 --> 00:10:29.960
and this was a second degree
polynomial we were squaring.

00:10:29.960 --> 00:10:33.100
Here we peak out at the x term,
and this was a first degree

00:10:33.100 --> 00:10:34.670
polynomial we're squaring.

00:10:34.670 --> 00:10:38.460
This is the exact same
pattern, but we peak out

00:10:38.460 --> 00:10:39.663
at x to the 23rd.

00:10:42.790 --> 00:10:45.220
So using the same pattern--
and you can actually

00:10:45.220 --> 00:10:46.470
prove it for the general case.

00:10:46.470 --> 00:10:49.610
It just gets messy, and I think
you get the general gist of it.

00:10:49.610 --> 00:10:52.040
This thing over here-- and
this is what I talked about,

00:10:52.040 --> 00:10:53.665
it's not the easiest
thing in the world

00:10:53.665 --> 00:10:57.090
to recognize-- p of x can
be rewritten as equal to x

00:10:57.090 --> 00:11:00.070
times-- this thing
over here is x

00:11:00.070 --> 00:11:06.020
to the 23rd-- x to the 23rd
plus x to the 22nd plus x

00:11:06.020 --> 00:11:13.000
to the 21st all the way down
to plus x to the third plus

00:11:13.000 --> 00:11:16.670
x squared plus x plus 1 squared.

00:11:19.940 --> 00:11:23.040
And we know that-- the only
way you'd really know that

00:11:23.040 --> 00:11:25.962
is if you've seen
this pattern before.

00:11:25.962 --> 00:11:27.170
But that's the way you do it.

00:11:27.170 --> 00:11:28.460
You look at-- this is a pattern.

00:11:28.460 --> 00:11:30.320
The coefficients just
keep increasing to 24,

00:11:30.320 --> 00:11:31.610
then keep decreasing.

00:11:31.610 --> 00:11:34.230
That's going to be whatever
the coefficient of the highest

00:11:34.230 --> 00:11:34.940
degree over here.

00:11:34.940 --> 00:11:35.950
So it's x to 23.

00:11:35.950 --> 00:11:37.698
And it's going to
be x to 23 plus all

00:11:37.698 --> 00:11:40.198
of the powers of x less than
that all the way down to x to 0

00:11:40.198 --> 00:11:41.054
squared.

00:11:41.054 --> 00:11:42.970
And we saw the pattern
here a couple of times.

00:11:42.970 --> 00:11:44.740
I'm going to leave
you in this video.

00:11:44.740 --> 00:11:46.490
This simplifies it a
good bit, but there's

00:11:46.490 --> 00:11:49.347
going to be a few more
kind of aha moments

00:11:49.347 --> 00:11:50.930
that we're going to
have to have to be

00:11:50.930 --> 00:11:52.898
able to solve this problem.


Trig challenge problem: area of a hexagon

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:03.920
Let A equal 0, 0 and
B equal lowercase b2

00:00:03.920 --> 00:00:05.820
be points on the
coordinate plane.

00:00:05.820 --> 00:00:10.950
Let ABCDEF be a convex
equilateral hexagon.

00:00:10.950 --> 00:00:13.130
Convex means that
it's not concave.

00:00:13.130 --> 00:00:15.320
A concave hexagon
would look like this.

00:00:15.320 --> 00:00:18.707
That's 2 sides-- 3, 4, 5, 6.

00:00:18.707 --> 00:00:20.040
This would be a concave hexagon.

00:00:20.040 --> 00:00:21.820
So it's going to be popped out.

00:00:21.820 --> 00:00:24.089
And all the sides are
going to be equal.

00:00:24.089 --> 00:00:25.380
So it's an equilateral hexagon.

00:00:25.380 --> 00:00:27.600
They're not telling us that
it's a regular hexagon.

00:00:27.600 --> 00:00:29.299
So we don't know that
all of the angles

00:00:29.299 --> 00:00:30.340
are going to be the same.

00:00:30.340 --> 00:00:32.280
But all the sides
will be the same.

00:00:32.280 --> 00:00:34.970
Such that FAB is
equal to 120 degrees.

00:00:34.970 --> 00:00:36.620
Then they show us
a bunch of sides

00:00:36.620 --> 00:00:38.359
that are parallel to each other.

00:00:38.359 --> 00:00:40.150
And then, the y-coordinates
of its vertices

00:00:40.150 --> 00:00:44.770
are distinct elements of
the set 0, 2, 4, 6, 8, 10.

00:00:44.770 --> 00:00:47.490
The area of the hexagon could
be written in the form m square

00:00:47.490 --> 00:00:50.910
root of n, where m and n are
positive integers and n is not

00:00:50.910 --> 00:00:52.570
divisible by the
square of any prime.

00:00:52.570 --> 00:00:53.944
That's just a
fancy way of saying

00:00:53.944 --> 00:00:56.420
that we've simplified this
radical as much as possible.

00:00:56.420 --> 00:00:58.460
Find m plus n.

00:00:58.460 --> 00:01:00.730
So really the first
part-- let's just

00:01:00.730 --> 00:01:03.100
make sure we can
visualize this hexagon.

00:01:03.100 --> 00:01:07.200
So let me draw-- we know
1.1 vertex for short 0, 0.

00:01:07.200 --> 00:01:10.440
So let me draw my x-axis.

00:01:10.440 --> 00:01:12.610
That is my x-axis.

00:01:12.610 --> 00:01:13.630
Right over there.

00:01:13.630 --> 00:01:15.230
And then my y-axis.

00:01:15.230 --> 00:01:18.390
My y-axis would look like that.

00:01:18.390 --> 00:01:19.850
Y-axis.

00:01:19.850 --> 00:01:23.650
We know that the vertex
A sits at the point 0,0.

00:01:23.650 --> 00:01:25.640
That is the vertex A.

00:01:25.640 --> 00:01:27.400
Now, we know that
all of the vertices

00:01:27.400 --> 00:01:32.060
have y-coordinates that are
either 0, 2, 4, 6, 8, 10.

00:01:32.060 --> 00:01:33.940
And they are distinct
members of the set.

00:01:33.940 --> 00:01:37.999
Which means no 2 of the vertices
share the same y-coordinate.

00:01:37.999 --> 00:01:40.290
So they're not going to be
on the same horizontal line.

00:01:40.290 --> 00:01:43.860
So let me draw these horizontal
lines-- the x-axis is 0.

00:01:43.860 --> 00:01:46.610
Then you have y is equal to 2.

00:01:46.610 --> 00:01:52.270
Then you have 4, then you
have 6, and then you have 8,

00:01:52.270 --> 00:01:55.880
and then you have 10 up here.

00:01:55.880 --> 00:01:57.572
Now, B we already know.

00:01:57.572 --> 00:01:59.530
So first of all, we've
already used up to the 0

00:01:59.530 --> 00:02:02.570
for A. A Is already
using up the 0.

00:02:02.570 --> 00:02:05.730
B uses of the 2-- they tell
us that the y-coordinate of B

00:02:05.730 --> 00:02:08.210
is 2.

00:02:08.210 --> 00:02:09.560
So we use that as well.

00:02:09.560 --> 00:02:13.910
Let me see if I can
draw B over here--

00:02:13.910 --> 00:02:16.310
It sits on this
horizontal someplace.

00:02:16.310 --> 00:02:22.464
And the hexagon
has side length S,

00:02:22.464 --> 00:02:24.130
where we don't know
what that length is,

00:02:24.130 --> 00:02:25.200
but they're all the same.

00:02:25.200 --> 00:02:26.700
So let's just call
this S it's going

00:02:26.700 --> 00:02:28.070
to help me think about it.

00:02:28.070 --> 00:02:31.170
Now that I know that this
is equilateral equaliateral

00:02:31.170 --> 00:02:36.080
hexagon, all the sides are
going to be the same length.

00:02:36.080 --> 00:02:38.260
We're going to go out here
the coordinate B comma 2.

00:02:38.260 --> 00:02:42.030
We don't know what B is,
but that is our vertex B.

00:02:42.030 --> 00:02:46.290
Now, F is the other vertex
that is connected to A.

00:02:46.290 --> 00:02:47.990
F cannot sit on
this horizontal--

00:02:47.990 --> 00:02:50.000
cannot sit on y is equal to 2.

00:02:50.000 --> 00:02:52.080
It can't sit on y is
equal to 6, because then

00:02:52.080 --> 00:02:53.800
this distance
would be super far.

00:02:53.800 --> 00:02:57.060
Clearly much further than
this distance over here.

00:02:57.060 --> 00:02:58.710
Or, actually, you
could have that.

00:02:58.710 --> 00:03:00.126
But then you would
you wouldn't be

00:03:00.126 --> 00:03:03.310
able to draw really
a convex hexagon.

00:03:03.310 --> 00:03:09.120
The next vertex is just going to
have to sit on this horizontal.

00:03:09.120 --> 00:03:12.100
So it's going to be S away.

00:03:12.100 --> 00:03:14.810
Maybe it will be
something like that.

00:03:17.510 --> 00:03:20.370
So let me draw it-- so
that is the next vertex.

00:03:20.370 --> 00:03:24.030
That is vertex F. Because
we're going A, B,C,D, E, F,

00:03:24.030 --> 00:03:26.540
and then back to A. Fair enough.

00:03:26.540 --> 00:03:28.090
Now what about vertex C?

00:03:28.090 --> 00:03:30.190
Well, vertex C can't
be on the 4 horizontal.

00:03:30.190 --> 00:03:32.330
So it's going to have to
be on the 6 horizontal.

00:03:32.330 --> 00:03:35.980
So vertex C is going to have
to be someplace like that.

00:03:38.720 --> 00:03:42.240
That's vertex C. And once again,
that length is S this length

00:03:42.240 --> 00:03:43.430
is S.

00:03:43.430 --> 00:03:45.510
Now what about vertex E?

00:03:45.510 --> 00:03:49.210
Can't be on the 6 horizontal,
already taken up by vertex C.

00:03:49.210 --> 00:03:50.990
So the 4 the 6 are taken up.

00:03:50.990 --> 00:03:54.540
So it has to be at
the 8 horizontal.

00:03:54.540 --> 00:03:56.290
And so this is length S.

00:03:56.290 --> 00:03:59.090
And we also know that we're
going back to the origin now.

00:03:59.090 --> 00:04:01.810
So this is the
vertex E right here.

00:04:01.810 --> 00:04:06.335
We know that we're going
back to the same x value,

00:04:06.335 --> 00:04:07.960
this is going to be
on the y-intercept.

00:04:07.960 --> 00:04:09.720
And the reason why
we know that is this

00:04:09.720 --> 00:04:12.040
is the length S and
in this is length S.

00:04:12.040 --> 00:04:15.420
And both of these
diagonals travel

00:04:15.420 --> 00:04:18.440
the same vertical distance.

00:04:18.440 --> 00:04:20.269
This base is 4.

00:04:20.269 --> 00:04:21.390
This base is 4.

00:04:21.390 --> 00:04:23.740
So you could kind of view
this as two right triangles.

00:04:23.740 --> 00:04:26.940
Both of them have base
4 and hypotenuse S

00:04:26.940 --> 00:04:29.300
and so they share this
side right over here.

00:04:29.300 --> 00:04:31.820
So this one goes out to the
left that distance and then

00:04:31.820 --> 00:04:34.360
this one's going to have
to come back that distance.

00:04:34.360 --> 00:04:36.650
Now, by the same
logic over here,

00:04:36.650 --> 00:04:38.560
this guy's going to
have to come back.

00:04:38.560 --> 00:04:44.540
So we can now use the
10 coordinate, the 10 y

00:04:44.540 --> 00:04:46.950
horizontal, or the
y-coordinate of 10.

00:04:46.950 --> 00:04:50.610
That's the only one we
haven't used yet for D.

00:04:50.610 --> 00:04:55.090
And since we came out when we
had a diagonal of length S,

00:04:55.090 --> 00:04:58.950
traveling 4 up, this
time, same logic--

00:04:58.950 --> 00:05:02.499
we had a diagonal of length
S. It traveled up 4 over here

00:05:02.499 --> 00:05:03.790
and it moved out this distance.

00:05:03.790 --> 00:05:06.590
When we go back in the other
direction and travel up 4

00:05:06.590 --> 00:05:08.890
you're going to go back
in the same direction.

00:05:08.890 --> 00:05:12.550
So this is going to be directly
on top of B. The coordinate

00:05:12.550 --> 00:05:16.570
for D is actually
going to be B comma 10.

00:05:16.570 --> 00:05:18.190
The y-coordinate here is 10.

00:05:18.190 --> 00:05:20.760
And there we have our hexagon.

00:05:20.760 --> 00:05:23.870
We're done drawing
our actual hexagon.

00:05:23.870 --> 00:05:26.360
And all this parallel line
information they told us--

00:05:26.360 --> 00:05:27.845
AB is parallel to DE.

00:05:31.840 --> 00:05:35.833
And this is kind of obvious
here, BC is parallel to EF.

00:05:42.040 --> 00:05:43.940
And then they say CD
is parallel to FA.

00:05:47.839 --> 00:05:49.380
And the way that we
drew it, it looks

00:05:49.380 --> 00:05:52.410
pretty clear that is the case.

00:05:52.410 --> 00:05:54.540
Now, we need to
find the area, we

00:05:54.540 --> 00:05:57.290
need to find the
area of this hexagon.

00:05:57.290 --> 00:05:59.120
And it seems like a
good starting point

00:05:59.120 --> 00:06:02.180
would be to figure
out what S is.

00:06:02.180 --> 00:06:04.040
And to figure out
what S is, it's

00:06:04.040 --> 00:06:06.320
really going to be a
function of how much we've

00:06:06.320 --> 00:06:07.390
inclined this thing.

00:06:07.390 --> 00:06:09.230
So let's draw--
and you could see

00:06:09.230 --> 00:06:11.410
that since this isn't an
equally angular hexagon,

00:06:11.410 --> 00:06:13.030
that this is kind of skwed.

00:06:13.030 --> 00:06:14.470
We distorted it a little bit.

00:06:14.470 --> 00:06:16.490
But all the sides
are the same length.

00:06:16.490 --> 00:06:18.440
So let's just call this theta.

00:06:18.440 --> 00:06:21.130
Let's call that angle
right over there theta.

00:06:21.130 --> 00:06:23.570
And then they tell us that
angle FAB is 120 degrees.

00:06:26.180 --> 00:06:28.340
That is 120 degrees.

00:06:28.340 --> 00:06:32.140
So this angle over
here on the left,

00:06:32.140 --> 00:06:36.330
is going to be 180
minus 120 minus theta.

00:06:36.330 --> 00:06:38.250
So 180 minus 120 is 60.

00:06:38.250 --> 00:06:41.064
So this angle over
here 60 minus theta.

00:06:41.064 --> 00:06:42.730
Now the reason why I
did that because we

00:06:42.730 --> 00:06:44.320
have some information.

00:06:44.320 --> 00:06:47.220
We know that we
traveled up 4 over here.

00:06:47.220 --> 00:06:49.759
And we know that we
traveled up 2 over here.

00:06:49.759 --> 00:06:51.300
And maybe we can
use that information

00:06:51.300 --> 00:06:54.069
to solve for S. Because S
is the hypotenuse of both

00:06:54.069 --> 00:06:56.110
of these right triangles
that I just constructed.

00:06:56.110 --> 00:06:58.430
Let me draw them-- So this
right triangle right over here,

00:06:58.430 --> 00:06:59.513
I could draw it like this.

00:07:01.860 --> 00:07:04.670
So I have S. I have theta.

00:07:04.670 --> 00:07:07.620
And I have 2, that's this
right triangle right over here.

00:07:07.620 --> 00:07:09.825
This right triangle
looks like this.

00:07:13.320 --> 00:07:16.940
This angle is 60 minus theta.

00:07:16.940 --> 00:07:20.330
And this height over here is 4.

00:07:20.330 --> 00:07:25.254
So let's see what we can do
to solve for S. This triangle

00:07:25.254 --> 00:07:27.170
on the left, or it was
on the right over here,

00:07:27.170 --> 00:07:29.710
this triangle says
predict the sine of theta.

00:07:29.710 --> 00:07:32.440
The sine of theta is
equal to the opposite

00:07:32.440 --> 00:07:33.430
over the hypotenuse.

00:07:33.430 --> 00:07:35.590
It's equal to 2 over S.

00:07:35.590 --> 00:07:38.140
This triangle tells us that
the sine-- and remember

00:07:38.140 --> 00:07:39.730
this hypotenuse
over here is also

00:07:39.730 --> 00:07:48.831
S-- the sine of 60 minus
theta is equal to 4 over S.

00:07:48.831 --> 00:07:50.830
And if we want to set
these equal to each other,

00:07:50.830 --> 00:07:53.110
we can multiply this
guy by 2 on both sides.

00:07:53.110 --> 00:07:55.450
You could say 2 sine
of theta is equal to 4

00:07:55.450 --> 00:07:59.315
over S. Sine of 60 minus 8
is also equal to 4 over S,

00:07:59.315 --> 00:08:01.270
so we could set them
equal to each other.

00:08:01.270 --> 00:08:12.010
So we have 2 sine of
theta is equal to sine

00:08:12.010 --> 00:08:16.167
of 60 minus theta.

00:08:16.167 --> 00:08:18.250
And then we could use some
of our trig identities,

00:08:18.250 --> 00:08:23.080
we know the sine of A
minus B is the same thing.

00:08:23.080 --> 00:08:28.680
The sine of A minus B, this is
equal to the sine of A times

00:08:28.680 --> 00:08:33.730
the cosine of B. Or I should
say theta in this case.

00:08:33.730 --> 00:08:36.937
So sine of 60 minus theta--
this is just a standard trig

00:08:36.937 --> 00:08:39.270
identity, it's called the
difference, sum and difference

00:08:39.270 --> 00:08:48.840
identity-- minus cosine of
60 times the sine of theta.

00:08:48.840 --> 00:08:50.940
And all this is equal
to 2 sine of theta.

00:08:53.520 --> 00:09:00.170
Well sine of 60 degrees-- This
is the square root of 3 over 2.

00:09:00.170 --> 00:09:04.970
Cosine of 60 degrees is 1/2.

00:09:04.970 --> 00:09:08.374
So we could add 1/2 sine
theta to both sides of this.

00:09:08.374 --> 00:09:09.540
And what we're going to get?

00:09:09.540 --> 00:09:11.740
So we're going to
add 1/2 sine theta.

00:09:11.740 --> 00:09:13.390
Then this guy's
going to go away.

00:09:13.390 --> 00:09:15.670
And then you add 1/2
sine theta to 2 sine

00:09:15.670 --> 00:09:18.810
of theta, which is
really 4/2 sine of theta.

00:09:18.810 --> 00:09:23.390
So that's going to be
5/2 the sine of theta.

00:09:23.390 --> 00:09:25.140
So I'm just adding 1/2
sine theta to this.

00:09:25.140 --> 00:09:31.740
So it's 5/2 sine of theta
is equal to square root

00:09:31.740 --> 00:09:33.290
of 3 over 2 cosine of theta.

00:09:37.670 --> 00:09:38.170
Right?

00:09:38.170 --> 00:09:41.180
I added 1/2 sine theta to both
sides of this to get this.

00:09:41.180 --> 00:09:44.050
I can multiply both sides
by 2 just to simplify it.

00:09:44.050 --> 00:09:46.750
So I get 5 sine of theta.

00:09:46.750 --> 00:09:49.210
5 sine of theta is
equal to the square root

00:09:49.210 --> 00:09:51.167
of 3 cosine of theta.

00:09:51.167 --> 00:09:53.000
Now, I want to use the
identity sine squared

00:09:53.000 --> 00:09:55.592
of theta plus cosine squared
of theta is equal to 1.

00:09:55.592 --> 00:09:57.300
So let me just square
both sides of that.

00:09:57.300 --> 00:09:59.070
That'll also help us
with this radical.

00:09:59.070 --> 00:10:03.030
So we'll get 25 sine
squared of theta

00:10:03.030 --> 00:10:06.200
is equal to square
root of 3 squared is 3.

00:10:06.200 --> 00:10:08.350
Instead of writing
cosine squared of theta,

00:10:08.350 --> 00:10:12.451
let's just write-- that's 1
minus sine squared of theta.

00:10:12.451 --> 00:10:12.950
Right?

00:10:12.950 --> 00:10:15.810
Cosine squared theta is 1
minus sine squared of theta.

00:10:15.810 --> 00:10:16.970
I just squared both sides.

00:10:16.970 --> 00:10:18.900
Let me just write
what I just did--

00:10:18.900 --> 00:10:21.070
I just squared to both sides.

00:10:21.070 --> 00:10:25.540
And so we get 25
sine squared theta

00:10:25.540 --> 00:10:30.190
is equal to 3 minus
3 sine squared theta.

00:10:30.190 --> 00:10:32.820
We can add 3 sine squared
theta to both sides--

00:10:32.820 --> 00:10:37.540
We get 28 sine squared
theta is equal to 3.

00:10:37.540 --> 00:10:41.770
Or, that the sine squared
of theta-- homestretch--

00:10:41.770 --> 00:10:44.260
is equal to 3 over 28.

00:10:44.260 --> 00:10:48.930
Or, we could even write
that sine of theta

00:10:48.930 --> 00:10:52.770
is equal to the square
root of 3 over 28.

00:10:52.770 --> 00:10:56.130
So it's equal to the
square root of 3 over 28.

00:10:56.130 --> 00:10:59.030
Now, we could simplify
that, 28 is 4 times 7,

00:10:59.030 --> 00:11:00.070
we could take it out.

00:11:00.070 --> 00:11:02.377
But that's good enough
for now, maybe we'll

00:11:02.377 --> 00:11:03.710
simplify it later if we have to.

00:11:03.710 --> 00:11:05.670
Sometimes these are
easier to deal with.

00:11:05.670 --> 00:11:06.920
So let's see over [INAUDIBLE].

00:11:06.920 --> 00:11:08.590
So we have the sine of theta.

00:11:08.590 --> 00:11:13.640
And now we can relate that,
actually, to S over here.

00:11:13.640 --> 00:11:15.870
We know that, before I
messed with this thing,

00:11:15.870 --> 00:11:21.760
that the sine of theta is equal
to 2 over S. Or that S over 2

00:11:21.760 --> 00:11:24.700
is equal to 1 over
sine of theta.

00:11:24.700 --> 00:11:28.260
Or that S is equal to 2
over the sine of theta.

00:11:28.260 --> 00:11:29.910
Well, we know what
sine of theta is,

00:11:29.910 --> 00:11:32.310
the square root of 3 over 28.

00:11:32.310 --> 00:11:35.470
So S is equal to 2
divided by sine of theta.

00:11:35.470 --> 00:11:38.130
That's like multiplying by
the inverse of sine of theta.

00:11:38.130 --> 00:11:42.535
So that's 2 times the
square root of 28 over 3.

00:11:46.130 --> 00:11:51.550
So we figured out our S. 2
times this thing over here.

00:11:51.550 --> 00:11:53.010
Now, given that we
know an S, let's

00:11:53.010 --> 00:11:55.700
see how we can
figure out the area.

00:11:55.700 --> 00:11:58.240
Well, what immediately
pops out is

00:11:58.240 --> 00:12:03.197
that we have this triangle
over here that has height,

00:12:03.197 --> 00:12:05.530
or I should say maybe its
base, if you view it sideways,

00:12:05.530 --> 00:12:06.595
its base is 8.

00:12:09.400 --> 00:12:11.140
And this distance
right over here,

00:12:11.140 --> 00:12:16.144
we should be able to figure out
using the Pythagorean theorem.

00:12:16.144 --> 00:12:18.310
Because we know that this
distance, right over here,

00:12:18.310 --> 00:12:18.945
is 4.

00:12:22.080 --> 00:12:25.470
We know that this distance,
the hypotenuse is S.

00:12:25.470 --> 00:12:28.530
So we could call this the
height of it, right over here.

00:12:28.530 --> 00:12:34.320
We could say that H squared
plus 4 squared plus 16 is

00:12:34.320 --> 00:12:37.180
equal to the hypotenuse
squared, is equal to S squared.

00:12:37.180 --> 00:12:39.150
S squared S is this
thing over here.

00:12:39.150 --> 00:12:49.480
So if we want to square S it
becomes 4 times 28 over 3.

00:12:49.480 --> 00:12:51.740
And we just subtract
16 from both sides.

00:12:51.740 --> 00:12:56.480
So H is equal to
4 times 28 over 3,

00:12:56.480 --> 00:12:59.197
minus, if I want
to write 16 over 3.

00:12:59.197 --> 00:13:01.030
Or if I want it, 16 to
something over three,

00:13:01.030 --> 00:13:06.250
that's going to be
minus 48 over 3.

00:13:06.250 --> 00:13:12.210
And let's see, I don't want to
have to multiply 4 times 28.

00:13:12.210 --> 00:13:14.860
So you could write
48 as 4 times 12.

00:13:18.090 --> 00:13:23.950
So this numerator is going to be
4 times 28 minus 12 over-- now,

00:13:23.950 --> 00:13:25.130
remember that's H squared.

00:13:25.130 --> 00:13:26.600
I should say-- H
squared is going

00:13:26.600 --> 00:13:29.470
to be 4 times 28
minus 12 over 3.

00:13:29.470 --> 00:13:33.740
Which is equal to
4 times 16 over 3.

00:13:33.740 --> 00:13:36.340
Which is equal to 64 over 3.

00:13:36.340 --> 00:13:37.510
That's H squared.

00:13:37.510 --> 00:13:39.780
So H is going to be the
square root of that,

00:13:39.780 --> 00:13:43.450
which is 8 over the
square root of 3.

00:13:43.450 --> 00:13:46.430
So H right over here is 8
over the square root of 3.

00:13:46.430 --> 00:13:49.815
So if I want to find the area
of this whole thing over here--

00:13:49.815 --> 00:13:51.940
well first, let's find the
area of this small thing

00:13:51.940 --> 00:13:52.690
right over here.

00:13:52.690 --> 00:13:54.990
That's just going
to be H times 4.

00:13:54.990 --> 00:13:57.610
So it's going to be-- well,
I could do it either way.

00:13:57.610 --> 00:14:01.720
But let's just say this
is H times 4 times 1/2.

00:14:01.720 --> 00:14:06.790
So it's going to be 2 times--
so the area of this triangle--

00:14:06.790 --> 00:14:09.610
let me do it in blue right
here-- this triangle's area is

00:14:09.610 --> 00:14:15.670
going to be H, which is 8 over
the square root of 3 times

00:14:15.670 --> 00:14:20.000
4 times 1/2.

00:14:20.000 --> 00:14:21.750
So this guy right
over here is just

00:14:21.750 --> 00:14:25.890
going to be 2 times 8
over the square root of 3,

00:14:25.890 --> 00:14:28.620
or that's going to be 16
over the square root of 3.

00:14:28.620 --> 00:14:32.100
So this guy over here is 16
over the square root of 3.

00:14:37.097 --> 00:14:38.930
Now we have a bunch of
it, we have this guy,

00:14:38.930 --> 00:14:41.900
and now this guy's going to have
the exact same area, right over

00:14:41.900 --> 00:14:42.610
there.

00:14:42.610 --> 00:14:44.193
And then you have
this guy who's going

00:14:44.193 --> 00:14:46.460
to have the exact
same area again.

00:14:46.460 --> 00:14:51.030
Same exact logic, same exact
logic, same base, same height.

00:14:51.030 --> 00:14:52.511
They're actually congruent.

00:14:52.511 --> 00:14:53.886
So you have 4 of
these triangles.

00:14:53.886 --> 00:14:55.370
So you are going
to multiply by 4

00:14:55.370 --> 00:14:57.745
If you want the
area of this area

00:14:57.745 --> 00:14:58.870
that I've are initiated in.

00:14:58.870 --> 00:15:00.280
So 4 times this.

00:15:00.280 --> 00:15:05.370
We're at 64 over the
square root of 3.

00:15:05.370 --> 00:15:08.520
Now the only area we
have left to figure out

00:15:08.520 --> 00:15:10.045
is the area of
this parallelogram.

00:15:12.730 --> 00:15:13.569
in the middle.

00:15:13.569 --> 00:15:15.360
Now, we know the base
of the parallelogram.

00:15:15.360 --> 00:15:18.260
The base of this
parallelogram is 8.

00:15:21.400 --> 00:15:22.990
We just have to
figure out its height.

00:15:25.820 --> 00:15:28.574
And once again, we can use
the Pythagorean theorem.

00:15:28.574 --> 00:15:30.490
So I'll call this, I
dunno, we already used H,

00:15:30.490 --> 00:15:31.600
well I'll use H again.

00:15:31.600 --> 00:15:33.224
But you just have to
remember that this

00:15:33.224 --> 00:15:34.920
is a different height over here.

00:15:34.920 --> 00:15:38.070
This base over here
is of length 2.

00:15:38.070 --> 00:15:39.530
I know it's hard to read now.

00:15:39.530 --> 00:15:47.320
So we can now write that H
squared plus 4 plus 2 squared

00:15:47.320 --> 00:15:51.094
is equal to S squared.

00:15:51.094 --> 00:15:53.510
Now we already figured out
what S squared was in the past.

00:15:53.510 --> 00:15:55.320
It's 4 times 28 over 3.

00:15:58.900 --> 00:16:00.730
Let's subtract 4
from both sides.

00:16:00.730 --> 00:16:01.820
You subtract 4 there.

00:16:01.820 --> 00:16:05.380
So minus 12 over 3.

00:16:05.380 --> 00:16:10.070
And now let's see, 12 is
the same thing as 4 times 3.

00:16:10.070 --> 00:16:18.230
So this is equal to
4 times 28 minus 3.

00:16:18.230 --> 00:16:21.670
So that's 4 times 25 over 3.

00:16:21.670 --> 00:16:24.650
Which is equal to 100 over 3.

00:16:24.650 --> 00:16:25.850
That's H squared.

00:16:25.850 --> 00:16:28.320
So this H is going to be equal
to the square root of this,

00:16:28.320 --> 00:16:30.550
which is 10 over the
square root of 3.

00:16:33.860 --> 00:16:35.610
So this distance
right over here is

00:16:35.610 --> 00:16:37.970
10 over the square root of 3.

00:16:37.970 --> 00:16:40.082
So if we want the area
of this parallelogram,

00:16:40.082 --> 00:16:41.540
it's going to be
that height, times

00:16:41.540 --> 00:16:43.392
the base of the parallelogram.

00:16:43.392 --> 00:16:45.850
So the parallelogram is going
to be 8 times 10 square roots

00:16:45.850 --> 00:16:51.294
of 3, or 80 square roots of 3.

00:16:51.294 --> 00:16:55.540
80-- oh no, let me be
very careful-- this

00:16:55.540 --> 00:16:59.767
was 10 over the square
root of 3 is this height.

00:16:59.767 --> 00:17:01.350
So the whole area
of this paralelagram

00:17:01.350 --> 00:17:03.310
is 8 times 10 over
the square root of 3.

00:17:03.310 --> 00:17:06.560
It's 80 over the
square root of 3.

00:17:06.560 --> 00:17:12.500
So our entire area now, if
we add everything together,

00:17:12.500 --> 00:17:15.520
we have 64 square roots of
3 for these 4 triangles,

00:17:15.520 --> 00:17:17.730
plus 80 over square root of 3.

00:17:17.730 --> 00:17:19.480
So let's add it together.

00:17:19.480 --> 00:17:23.109
So we have 80 over square roots
of 3 for the parallelogram,

00:17:23.109 --> 00:17:27.290
plus 64 over square roots
of 3 for the triangle parts.

00:17:27.290 --> 00:17:31.930
And this is equal to 144
over the square root of 3.

00:17:31.930 --> 00:17:33.630
We can rationalize
the denominator.

00:17:33.630 --> 00:17:38.066
So times the square root of
3, over the square root of 3.

00:17:38.066 --> 00:17:40.232
And in the denominator,
now, we're going to get a 3.

00:17:40.232 --> 00:17:45.690
144 over 3 is going to be what?

00:17:45.690 --> 00:17:47.900
That's 48.

00:17:47.900 --> 00:17:48.410
Right?

00:17:48.410 --> 00:17:51.350
Three times 40 is
120, 3 times 8 is 24.

00:17:51.350 --> 00:17:56.230
So it's going to be
48 square roots of 3

00:17:56.230 --> 00:18:00.220
for the area of
our entire hexagon.

00:18:00.220 --> 00:18:03.670
And so, we have in the form--
48 square roots of 3-- So

00:18:03.670 --> 00:18:09.640
if you want to find m plus
n is 48 plus 3, which is 51.

00:18:09.640 --> 00:18:11.760
That was a tiring problem.

00:18:11.760 --> 00:18:13.607
My brain started to
fry near the end of it.

00:18:13.607 --> 00:18:15.190
I had trouble keeping
track of things.

00:18:15.190 --> 00:18:17.710
Anyway, hopefully
you enjoyed that.