Trig challenge problem: area of a triangle

WEBVTT
Kind: captions
Language: en

00:00:00.780 --> 00:00:04.860
Triangle ABC is a right
triangle with AC is equal to 7,

00:00:04.860 --> 00:00:08.950
BC is equal to 24, and
a right angle at C.

00:00:08.950 --> 00:00:11.289
So let's just try to draw that.

00:00:11.289 --> 00:00:12.580
And let's draw the right angle.

00:00:12.580 --> 00:00:14.288
Maybe we'll have to
use some coordinates.

00:00:14.288 --> 00:00:17.170
So let's draw the right
angle at the origin.

00:00:17.170 --> 00:00:21.780
So let's say that this
is C right over here.

00:00:21.780 --> 00:00:23.260
AC is equal to 7.

00:00:23.260 --> 00:00:24.630
So A, we'll put over here.

00:00:24.630 --> 00:00:27.760
This distance over
here is going to be 7.

00:00:27.760 --> 00:00:31.970
And then we have the
hypotenuse of our triangle.

00:00:31.970 --> 00:00:35.840
And then this can be B. And
this distance over here-- they

00:00:35.840 --> 00:00:40.720
tell us that BC is equal to 24.

00:00:40.720 --> 00:00:41.640
All right.

00:00:41.640 --> 00:00:44.840
Now, point M is
the midpoint of AB.

00:00:44.840 --> 00:00:47.260
So point M is right over here.

00:00:47.260 --> 00:00:48.930
Let me do that in
a different color.

00:00:48.930 --> 00:00:53.810
Point M is the midpoint of AB.

00:00:53.810 --> 00:00:57.130
So this distance is
equal to that distance.

00:00:57.130 --> 00:01:04.150
And D is on the same side of
AB as C. So C is on this side,

00:01:04.150 --> 00:01:08.300
on I guess you'd call it the
bottom left-hand side of AB,

00:01:08.300 --> 00:01:12.810
so that AD is equal
to BD is equal to 15.

00:01:12.810 --> 00:01:16.240
So D is going to be
someplace over here.

00:01:16.240 --> 00:01:18.680
It's going to be equal
distance between A and B.

00:01:18.680 --> 00:01:21.110
Obviously, all of the points
equal distant between A and B

00:01:21.110 --> 00:01:24.080
are going to sit on a line
that looks something like that.

00:01:24.080 --> 00:01:26.080
This is the midpoint of AB.

00:01:26.080 --> 00:01:28.070
So D is going to
sit right over here.

00:01:28.070 --> 00:01:30.970
And it's 15 away
from both A and B.

00:01:30.970 --> 00:01:33.200
So it would look
something like that.

00:01:33.200 --> 00:01:35.620
That distance over there is 15.

00:01:35.620 --> 00:01:41.800
And then this distance over
here is also going to be 15.

00:01:41.800 --> 00:01:49.320
Given that the area of
triangle CDM-- this is D. So

00:01:49.320 --> 00:01:55.390
CDM is the triangle that
they want to think about.

00:01:55.390 --> 00:01:58.790
Given that the area of triangle
CDM may be expressed as m times

00:01:58.790 --> 00:02:01.460
the square root of n over
p, where m, n, and p are

00:02:01.460 --> 00:02:04.722
positive integers and m
and p are relatively prime.

00:02:04.722 --> 00:02:06.430
So that just means
you can't simplify it.

00:02:06.430 --> 00:02:10.199
And n is not divisible by
the square of any prime.

00:02:10.199 --> 00:02:14.840
So you've simplified the
radical as much as possible--

00:02:14.840 --> 00:02:16.780
find m plus n plus p.

00:02:16.780 --> 00:02:19.960
So we essentially
need to find the area

00:02:19.960 --> 00:02:24.120
of this green triangle,
of CDM over here.

00:02:24.120 --> 00:02:26.285
And so let's see what we
can do to figure it out.

00:02:26.285 --> 00:02:27.910
So we could try some
of the coordinates

00:02:27.910 --> 00:02:29.560
for some of these points.

00:02:29.560 --> 00:02:31.620
So this point
right over here, A,

00:02:31.620 --> 00:02:33.730
is going to be-- its x
value is going to be 7.

00:02:33.730 --> 00:02:36.230
I could draw the coordinate,
just so you see what I'm doing.

00:02:36.230 --> 00:02:38.190
That could be the x-axis.

00:02:38.190 --> 00:02:41.240
And then this side
is on the y-axis.

00:02:41.240 --> 00:02:45.040
So the coordinate for
A would be 7 comma 0.

00:02:45.040 --> 00:02:47.570
Coordinate for C
would be 0 comma 0.

00:02:47.570 --> 00:02:52.575
And the coordinate
for B would be 0, 24.

00:02:55.390 --> 00:02:59.906
So the coordinate for M would
just be the average of B and A.

00:02:59.906 --> 00:03:04.460
So the coordinate for M, the
average of 0 and 7 is 7/2.

00:03:04.460 --> 00:03:07.640
And the coordinate
for the y-coordinate,

00:03:07.640 --> 00:03:11.590
the average of 24 and 0, is 12.

00:03:11.590 --> 00:03:12.552
That's fair enough.

00:03:12.552 --> 00:03:14.760
Now let's see what we can
figure out about the sides.

00:03:14.760 --> 00:03:16.260
So we know this is
a right triangle.

00:03:16.260 --> 00:03:19.090
So our gut reaction is always
to use the Pythagorean theorem.

00:03:19.090 --> 00:03:20.990
We know this side and that side.

00:03:20.990 --> 00:03:22.680
So if we wanted
to figure out AB,

00:03:22.680 --> 00:03:24.880
we could just say
that-- we just know

00:03:24.880 --> 00:03:37.060
that 24 squared plus 7 squared
is equal to AB squared.

00:03:37.060 --> 00:03:45.330
And 24 squared is 576, plus
49 is equal to AB squared.

00:03:45.330 --> 00:03:48.770
And let's see-- 576 plus 49.

00:03:48.770 --> 00:03:54.700
If it was plus 50, it would
get us to 626, but it's 1 less

00:03:54.700 --> 00:03:55.200
than that.

00:03:55.200 --> 00:03:56.900
So it's 625.

00:03:56.900 --> 00:04:02.060
So 625 is equal to AB squared.

00:04:02.060 --> 00:04:03.720
So AB is going to equal 25.

00:04:08.500 --> 00:04:11.300
So this distance, the distance
of this big hypotenuse here,

00:04:11.300 --> 00:04:12.400
is 25.

00:04:12.400 --> 00:04:14.810
Or half of the
distance, this is going

00:04:14.810 --> 00:04:22.070
to be 25/2, from B to M. From M
to A is also going to be 25/2.

00:04:22.070 --> 00:04:30.530
Now, the other thing we know
is that M right over here,

00:04:30.530 --> 00:04:33.710
the triangle CMA is
an isosceles triangle.

00:04:33.710 --> 00:04:35.470
How do we know that?

00:04:35.470 --> 00:04:37.480
Well, M, if you look
at its x-coordinate--

00:04:37.480 --> 00:04:40.720
its x-coordinate is directly
in between the x-coordinates

00:04:40.720 --> 00:04:43.650
for C and A. 7/2--
it's the average.

00:04:43.650 --> 00:04:44.660
This is 7.

00:04:44.660 --> 00:04:45.540
This is 0.

00:04:45.540 --> 00:04:47.081
Its M-coordinate is
right over there.

00:04:47.081 --> 00:04:50.866
It's directly above the
midpoint of this base over here.

00:04:50.866 --> 00:04:52.740
So this is going to be
an isosceles triangle.

00:04:52.740 --> 00:04:53.300
It's symmetric.

00:04:53.300 --> 00:04:54.680
You could flip
the triangle over.

00:04:54.680 --> 00:04:57.750
So this length-- and
this seems useful

00:04:57.750 --> 00:05:02.630
because this is kind of the base
of the triangle we care about.

00:05:02.630 --> 00:05:05.400
We could kind of view
this as the base of CDM--

00:05:05.400 --> 00:05:09.210
this is also going to be 25/2.

00:05:09.210 --> 00:05:10.460
It's an isosceles triangle.

00:05:10.460 --> 00:05:12.543
This is going to be the
same as that because we're

00:05:12.543 --> 00:05:15.539
symmetric around
this right over here.

00:05:15.539 --> 00:05:16.080
So let's see.

00:05:16.080 --> 00:05:18.702
We know one side
of this triangle.

00:05:18.702 --> 00:05:20.910
Let's see if we could figure
out this side over here.

00:05:20.910 --> 00:05:22.243
It seems pretty straightforward.

00:05:22.243 --> 00:05:25.880
This is going to be a right
triangle right over here,

00:05:25.880 --> 00:05:28.810
because this line
from D to M is going

00:05:28.810 --> 00:05:30.770
to be perpendicular to AB.

00:05:30.770 --> 00:05:34.650
All of the points that are
equidistant between A and B

00:05:34.650 --> 00:05:40.104
are going to be on a line
that is perpendicular to AB.

00:05:40.104 --> 00:05:41.770
So this is going to
be a right triangle.

00:05:41.770 --> 00:05:46.380
So we can figure out DM using
the Pythagorean theorem again.

00:05:46.380 --> 00:06:00.890
We get 25/2 squared
plus DM squared

00:06:00.890 --> 00:06:03.109
is going to be
equal to 15 squared.

00:06:03.109 --> 00:06:05.400
It's going to equal to the
hypotenuse of this triangle.

00:06:05.400 --> 00:06:09.500
So it's going to equal 225.

00:06:09.500 --> 00:06:10.650
And so what do we get?

00:06:10.650 --> 00:06:20.779
We get DM squared is
equal to 225 minus 625/4.

00:06:20.779 --> 00:06:21.820
So let's put this over 4.

00:06:21.820 --> 00:06:26.960
So 225 with 4 as the denominator
is the same thing as 900/4.

00:06:29.730 --> 00:06:33.800
And in the last video, I
actually mistakenly said 900/4

00:06:33.800 --> 00:06:35.812
was 125, obviously a
boneheaded mistake.

00:06:35.812 --> 00:06:37.270
But the numbers
show up again here.

00:06:37.270 --> 00:06:40.050
So 225 is obviously 900/4.

00:06:40.050 --> 00:06:42.100
And then we're going to
subtract from that 625/4.

00:06:47.930 --> 00:06:50.870
And this is equal
to-- let's see.

00:06:50.870 --> 00:06:54.510
In our numerator, we
have 900 minus 625

00:06:54.510 --> 00:06:55.860
would be 300 minus 20.

00:06:55.860 --> 00:06:56.870
It's going to be 275/4.

00:07:00.670 --> 00:07:05.390
And so DM is going to be equal
to the square root of this.

00:07:05.390 --> 00:07:08.660
So it's equal to the
square root of 275/4.

00:07:08.660 --> 00:07:14.380
275 is 25 times 11, because
25 times 12 would be 300.

00:07:14.380 --> 00:07:17.180
So 25 times 11 over 4.

00:07:17.180 --> 00:07:23.240
So this is going to be equal
to 5 square roots of 11 over 2.

00:07:23.240 --> 00:07:27.220
So DM is 5 square
roots of 11 over 2.

00:07:30.810 --> 00:07:32.560
So what we need to
do here is figure out

00:07:32.560 --> 00:07:34.680
the height of this
triangle right over here.

00:07:34.680 --> 00:07:37.552
If we could figure out the
height of that triangle,

00:07:37.552 --> 00:07:38.510
we're pretty much done.

00:07:38.510 --> 00:07:42.320
1/2 the base times the height,
but we don't know-- let's see,

00:07:42.320 --> 00:07:44.720
we could figure that if we
know the law of cosines.

00:07:44.720 --> 00:07:46.095
We could do
something over there.

00:07:46.095 --> 00:07:50.319
If we know the sine of
this angle right over here,

00:07:50.319 --> 00:07:51.860
the sine of that
angle is going to be

00:07:51.860 --> 00:07:55.820
this height over the
side we just figured out.

00:07:55.820 --> 00:07:58.390
So if we could figure out
the sine of that angle,

00:07:58.390 --> 00:08:01.430
then we'd be done, or we'll
be very close to being done.

00:08:01.430 --> 00:08:03.306
But it's not any obvious way.

00:08:03.306 --> 00:08:05.430
But one thing we can do,
if you look at this bigger

00:08:05.430 --> 00:08:08.120
triangle over here--
so let me highlight it.

00:08:08.120 --> 00:08:10.760
So if you look at triangle BMC.

00:08:10.760 --> 00:08:13.990
Let me draw triangle BMC.

00:08:13.990 --> 00:08:16.342
Well, I'll just draw it on
this drawing right over here.

00:08:16.342 --> 00:08:17.800
I don't want to
get too overloaded.

00:08:17.800 --> 00:08:22.200
But triangle BMC
right over here--

00:08:22.200 --> 00:08:24.830
we know this side is 25/2.

00:08:24.830 --> 00:08:27.890
We know this side
over here is 25/2.

00:08:27.890 --> 00:08:30.760
We know this side
over here is 24.

00:08:30.760 --> 00:08:33.470
And so what we want
to do is figure out

00:08:33.470 --> 00:08:36.440
the sine of this angle right
here, the sine of theta,

00:08:36.440 --> 00:08:39.081
of angle CMD.

00:08:39.081 --> 00:08:40.039
Now, that's hard to do.

00:08:40.039 --> 00:08:41.850
But what we can
do is use the law

00:08:41.850 --> 00:08:44.630
of cosines of theta plus 90.

00:08:44.630 --> 00:08:47.350
So let me redraw our triangle.

00:08:47.350 --> 00:08:52.000
So triangle BCM I
could redraw like this.

00:08:52.000 --> 00:08:54.450
It's actually an
isosceles triangle.

00:08:54.450 --> 00:08:55.535
So we have BCM.

00:08:58.600 --> 00:09:00.780
And what is this
angle right over here?

00:09:00.780 --> 00:09:04.950
It's going to be our theta that
we care about plus 90 degrees.

00:09:04.950 --> 00:09:10.550
So this angle right here
is theta plus 90 degrees.

00:09:10.550 --> 00:09:12.190
And this over here is 24.

00:09:12.190 --> 00:09:16.630
This is 25/2, and this is 25/2.

00:09:16.630 --> 00:09:18.790
And so using this, we could
use the law of cosines

00:09:18.790 --> 00:09:23.480
to figure out what
theta actually is here.

00:09:23.480 --> 00:09:24.750
So let's do that.

00:09:24.750 --> 00:09:26.875
We're going to use a little
bit of trig identities,

00:09:26.875 --> 00:09:27.960
but law of cosines.

00:09:27.960 --> 00:09:31.190
So we get the opposite
to the angle squared,

00:09:31.190 --> 00:09:33.660
so we get 24 squared.

00:09:33.660 --> 00:09:44.080
24 squared is equal to 25/2
squared, plus 25/2 squared,

00:09:44.080 --> 00:09:54.460
minus 2 times 25/2 times 25/2
times the cosine of this angle,

00:09:54.460 --> 00:09:57.840
times-- let me scroll
over-- the cosine

00:09:57.840 --> 00:10:03.676
of theta plus 90 degrees.

00:10:03.676 --> 00:10:05.300
Now, you're saying,
hey, Sal, you know,

00:10:05.300 --> 00:10:08.340
this has it in terms of cosine
of theta plus 90 degrees.

00:10:08.340 --> 00:10:10.190
How can we figure out
the sine of theta?

00:10:10.190 --> 00:10:11.720
That's what we
actually care about,

00:10:11.720 --> 00:10:13.390
to figure out the
area of this triangle,

00:10:13.390 --> 00:10:16.080
or actually to figure out
the height of this triangle.

00:10:16.080 --> 00:10:19.520
And to do that, you just
have to make the realization.

00:10:19.520 --> 00:10:24.660
We know the trig identity
that the cosine of theta

00:10:24.660 --> 00:10:26.780
is equal-- I won't use
theta because I don't want

00:10:26.780 --> 00:10:34.250
to overload theta-- cosine of x
is equal to sine of 90 minus x.

00:10:34.250 --> 00:10:41.170
So the cosine of
theta plus 90 degrees

00:10:41.170 --> 00:10:44.530
is going to be equal to
the sine of-- let me put it

00:10:44.530 --> 00:10:47.280
in parentheses-- 90
minus whatever is here.

00:10:47.280 --> 00:10:53.070
90 minus theta minus 90,
which is equal to-- the 90's

00:10:53.070 --> 00:10:56.844
cancel out-- sine
of negative theta.

00:10:56.844 --> 00:10:58.260
And we know sine
of negative theta

00:10:58.260 --> 00:11:03.010
is equal to the
negative sine of theta.

00:11:03.010 --> 00:11:05.400
So this over here
simplifies to-- this

00:11:05.400 --> 00:11:06.692
is the negative sine of theta.

00:11:06.692 --> 00:11:08.150
So we could write
the sine of theta

00:11:08.150 --> 00:11:10.990
here and then put the
negative out here.

00:11:10.990 --> 00:11:12.680
And this becomes a positive.

00:11:12.680 --> 00:11:13.930
So what does this simplify to?

00:11:13.930 --> 00:11:18.830
We have 24 squared,
which is 576.

00:11:18.830 --> 00:11:23.260
576 is equal to-- let's see.

00:11:23.260 --> 00:11:24.590
I won't skip any steps here.

00:11:24.590 --> 00:11:27.250
So we have 25 squared
plus 25 squared.

00:11:27.250 --> 00:11:40.620
This is 2 times 25/2
squared plus 2 times--

00:11:40.620 --> 00:11:48.250
this is 25/2 squared
again-- times sine of theta.

00:11:48.250 --> 00:11:51.020
Now we just have to
solve for sine of theta.

00:11:51.020 --> 00:11:53.250
So this is going to
be equal to-- well,

00:11:53.250 --> 00:12:02.210
this is 576 is equal to
2 times 25/2 squared.

00:12:02.210 --> 00:12:03.620
I'm just factoring that out.

00:12:03.620 --> 00:12:05.760
1 plus sine of theta.

00:12:08.870 --> 00:12:12.194
Or we can just divide both sides
of the equation by this here.

00:12:12.194 --> 00:12:13.610
Actually, let me
just simplify it.

00:12:13.610 --> 00:12:17.770
This thing over here is
625/4, but then we're

00:12:17.770 --> 00:12:19.150
going to multiply that by 2.

00:12:19.150 --> 00:12:23.300
So this thing over
here is 625/2.

00:12:23.300 --> 00:12:26.270
So let's divide both
sides of this by 625/2.

00:12:26.270 --> 00:12:35.660
And we get 576
times 2 over 625--

00:12:35.660 --> 00:12:38.230
just multiplying both sides
by the inverse-- is equal to.

00:12:38.230 --> 00:12:40.730
When you multiply both sides
by the inverse [? like this, ?]

00:12:40.730 --> 00:12:46.570
it actually cancels out-- is
equal to 1 plus sine of theta.

00:12:46.570 --> 00:12:49.880
Or we just subtract
1 from both sides.

00:12:49.880 --> 00:12:53.800
We get sine of theta is
equal to 576 times 2.

00:12:53.800 --> 00:12:57.460
Let's see, 76 times
2 is 152, plus 1,000.

00:12:57.460 --> 00:13:02.780
So it's 1,152/625.

00:13:02.780 --> 00:13:05.110
That's that part there, minus 1.

00:13:05.110 --> 00:13:07.750
But instead of 1,
let's say 625/625.

00:13:07.750 --> 00:13:10.650
So minus 625.

00:13:10.650 --> 00:13:14.140
And this is equal to-- I'll
have to do a little math

00:13:14.140 --> 00:13:20.550
on the side-- so
1,152 minus 625.

00:13:20.550 --> 00:13:21.700
You get a 12 there.

00:13:21.700 --> 00:13:22.535
This becomes a 4.

00:13:22.535 --> 00:13:23.970
12 minus 5 is 7.

00:13:23.970 --> 00:13:25.845
4 minus 2 is 2.

00:13:25.845 --> 00:13:27.990
11 minus 6 is 5.

00:13:27.990 --> 00:13:32.451
So this is equal to 527/625.

00:13:32.451 --> 00:13:34.950
Now, you might not realize it,
but we're in the home stretch

00:13:34.950 --> 00:13:36.050
now.

00:13:36.050 --> 00:13:39.510
Let me draw this CDM
triangle, the one that

00:13:39.510 --> 00:13:41.100
is really the focus
of the problem.

00:13:41.100 --> 00:13:43.440
And let me draw it a
little bit differently.

00:13:43.440 --> 00:13:47.050
Let me draw it
slightly differently.

00:13:47.050 --> 00:13:50.550
So now we know some very
interesting things about CDM.

00:13:50.550 --> 00:13:53.410
So this is C, this
is D, and this

00:13:53.410 --> 00:13:56.020
is M. We know this
side over here.

00:13:56.020 --> 00:13:58.675
We figured out it's
5/2 square root of 11.

00:14:02.030 --> 00:14:04.860
That's the length of DM.

00:14:04.860 --> 00:14:10.610
We also know that CM
right over here is 25/2.

00:14:10.610 --> 00:14:19.470
We also know that sine of
theta here is equal to 527/625.

00:14:19.470 --> 00:14:21.130
That's what we just figured out.

00:14:21.130 --> 00:14:25.930
And we can use that to figure
out the height of this triangle

00:14:25.930 --> 00:14:30.336
because we know that sine is the
opposite over the hypotenuse,

00:14:30.336 --> 00:14:31.960
if we draw a right
triangle right here.

00:14:31.960 --> 00:14:36.560
So sine of theta,
which is 527/625,

00:14:36.560 --> 00:14:38.910
is equal to the opposite,
is equal to the height

00:14:38.910 --> 00:14:42.920
of this triangle,
over the hypotenuse,

00:14:42.920 --> 00:14:46.950
over 5/2 square roots of 11.

00:14:46.950 --> 00:14:52.220
So we can multiply both sides
of this by 5/2 square roots 11.

00:14:52.220 --> 00:14:54.960
And we get the height
of the triangle

00:14:54.960 --> 00:15:05.730
is equal to 527/625 times
5/2 square roots of 11.

00:15:05.730 --> 00:15:11.150
And let's see, we can divide--
625 divided by 5 is 125.

00:15:11.150 --> 00:15:13.060
So you get a 1 here.

00:15:13.060 --> 00:15:16.840
And you have a 125 over here.

00:15:16.840 --> 00:15:21.370
So this is equal
to 527 square roots

00:15:21.370 --> 00:15:28.060
11 over 125 times
2, which is 250.

00:15:28.060 --> 00:15:29.070
That's the height.

00:15:29.070 --> 00:15:31.160
Now, what's the area
of the triangle?

00:15:31.160 --> 00:15:34.320
It's 1/2 base times height.

00:15:34.320 --> 00:15:42.560
Area is equal to 1/2 base,
which is 25/2, times the height,

00:15:42.560 --> 00:15:50.330
which we just figured out is
527 square roots of 11 over 250.

00:15:50.330 --> 00:15:51.480
Let's see.

00:15:51.480 --> 00:15:52.820
Divide the numerator by 25.

00:15:52.820 --> 00:15:54.220
Divide the denominator by 25.

00:15:54.220 --> 00:15:55.620
You get a 10 there.

00:15:55.620 --> 00:16:02.430
So this is equal to 527 square
roots of 11 over 2 times

00:16:02.430 --> 00:16:05.150
2 times 10, which is 40.

00:16:05.150 --> 00:16:06.090
So that's our area.

00:16:06.090 --> 00:16:07.180
And the whole
problem-- they didn't

00:16:07.180 --> 00:16:08.221
want us to find the area.

00:16:08.221 --> 00:16:12.340
They wanted us to find
m plus n plus p, so they

00:16:12.340 --> 00:16:16.183
want us to find essentially
527 plus 11 plus 40.

00:16:22.730 --> 00:16:29.610
So 527 plus 11 is-- let me
make sure I do this-- 538,

00:16:29.610 --> 00:16:33.320
and then plus 40 is 578.

00:16:33.320 --> 00:16:35.180
And we're done.


2003 AIME II problem 10

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:03.680
Two positive integers
differ by 60.

00:00:03.680 --> 00:00:05.810
The sum of their square
roots is the square root

00:00:05.810 --> 00:00:07.930
of an integer that is
not a perfect square.

00:00:07.930 --> 00:00:10.465
What is the maximum possible
sum of the two integers?

00:00:10.465 --> 00:00:12.340
So the second sentence
is a little confusing,

00:00:12.340 --> 00:00:13.810
but let's take a step-by-step.

00:00:13.810 --> 00:00:17.360
So two positive
integers differ by 60.

00:00:17.360 --> 00:00:19.620
So let's say they're a and b.

00:00:19.620 --> 00:00:20.670
So we have a and b.

00:00:20.670 --> 00:00:21.840
They're both greater than 0.

00:00:21.840 --> 00:00:22.964
They're also both integers.

00:00:27.730 --> 00:00:29.550
And they differ by 60.

00:00:29.550 --> 00:00:33.346
So a minus b is equal to 60.

00:00:33.346 --> 00:00:34.970
Or we could add b to
both sides, and we

00:00:34.970 --> 00:00:37.830
could get a is
equal to b plus 60.

00:00:37.830 --> 00:00:38.780
They're 60 apart.

00:00:38.780 --> 00:00:39.960
Fair enough.

00:00:39.960 --> 00:00:46.060
Now this second sentence, the
sum of their square roots--

00:00:46.060 --> 00:00:48.030
the square root of a
plus square root of b--

00:00:48.030 --> 00:00:50.180
the sum of the square
roots is the square root

00:00:50.180 --> 00:00:52.490
of an integer that is
not a perfect square.

00:00:52.490 --> 00:00:55.240
So it's equal to the square
root of some integer.

00:00:55.240 --> 00:01:00.390
So c is an integer, but
it's not a perfect square.

00:01:00.390 --> 00:01:04.709
So square root of c
is not an integer.

00:01:04.709 --> 00:01:05.960
So that's the second part.

00:01:05.960 --> 00:01:10.130
What is the maximum possible
sum of the two integers?

00:01:10.130 --> 00:01:13.130
So if we could maximize b
and find an a that fits it,

00:01:13.130 --> 00:01:15.010
or an a and a b that
meet these constraints

00:01:15.010 --> 00:01:16.551
we could get them
as big as possible,

00:01:16.551 --> 00:01:18.665
then the sum of a and b
will also be maximized.

00:01:18.665 --> 00:01:20.249
So let's think about
that in a second.

00:01:20.249 --> 00:01:22.748
But just to simplify-- this is
actually the really confusing

00:01:22.748 --> 00:01:24.850
part because c is an
integer and square root of c

00:01:24.850 --> 00:01:25.826
is not an integer.

00:01:25.826 --> 00:01:27.450
And to make it a
little less confusing,

00:01:27.450 --> 00:01:30.770
let's just square both sides
of this equation over here.

00:01:30.770 --> 00:01:33.630
So square root of a
squared is-- so we're just

00:01:33.630 --> 00:01:35.760
going to square both sides.

00:01:40.664 --> 00:01:42.080
So this is going
to be square root

00:01:42.080 --> 00:01:43.538
of a times square
root of a-- which

00:01:43.538 --> 00:01:46.230
is a-- plus 2 times the
square root of a times

00:01:46.230 --> 00:01:50.970
the square root of b-- which is
2 times the square root of ab--

00:01:50.970 --> 00:01:53.110
plus the square root of
b squared-- which is just

00:01:53.110 --> 00:01:58.970
going to be b-- is equal to c.

00:01:58.970 --> 00:02:02.070
Now, we know that
a is an integer.

00:02:02.070 --> 00:02:04.410
We know that b is an integer.

00:02:04.410 --> 00:02:06.610
And we know that
c is an integer.

00:02:06.610 --> 00:02:08.277
We know that these are integers.

00:02:08.277 --> 00:02:09.860
So if I have an
integer plus something

00:02:09.860 --> 00:02:11.749
plus an integer equaling
another integer,

00:02:11.749 --> 00:02:13.665
this thing in the middle
has to be an integer.

00:02:19.287 --> 00:02:21.620
And actually, I'll show you
right now that not only does

00:02:21.620 --> 00:02:24.906
this thing have to be an
integer, square root of ab

00:02:24.906 --> 00:02:25.780
has to be an integer.

00:02:25.780 --> 00:02:28.634
And you might be saying, wait,
wait, wait, wait a second.

00:02:28.634 --> 00:02:30.300
If this thing just
has to be an integer,

00:02:30.300 --> 00:02:32.591
can't I just say-- let me do
it over here on the side--

00:02:32.591 --> 00:02:35.570
can't I just say that 2
times the square root of ab

00:02:35.570 --> 00:02:38.320
has to be equal to some integer.

00:02:38.320 --> 00:02:39.940
You could divide
both sides by 2.

00:02:39.940 --> 00:02:42.180
And you could say, well,
maybe the square root of ab

00:02:42.180 --> 00:02:45.630
is equal to k/2, where
this is not an integer.

00:02:49.496 --> 00:02:51.620
I claim that this thing
does have to be an integer.

00:02:51.620 --> 00:02:53.220
But you say, hey look, just
straight from the math,

00:02:53.220 --> 00:02:54.520
this could be an integer.

00:02:54.520 --> 00:02:56.050
But maybe divide
both sides by 2.

00:02:56.050 --> 00:02:58.100
And this doesn't have
to be an integer.

00:02:58.100 --> 00:03:01.320
And if this is not
an integer, that

00:03:01.320 --> 00:03:04.140
means that k is odd,
because otherwise it

00:03:04.140 --> 00:03:06.960
would be divisible by 2.

00:03:06.960 --> 00:03:08.530
Now, if we square
both sides of this,

00:03:08.530 --> 00:03:13.256
we would get ab is equal
to k squared over 4.

00:03:13.256 --> 00:03:15.187
Well, this is going to
be an odd times an odd.

00:03:15.187 --> 00:03:16.520
It's going to be an odd squared.

00:03:16.520 --> 00:03:18.360
So this is still
going to be odd,

00:03:18.360 --> 00:03:20.150
which is not divisible by 4.

00:03:20.150 --> 00:03:23.650
So this thing on
the right hand side

00:03:23.650 --> 00:03:25.320
will still not be an integer.

00:03:27.890 --> 00:03:30.930
But remember, a
and b are integers.

00:03:30.930 --> 00:03:34.450
So a times b, this
is an integer.

00:03:34.450 --> 00:03:36.570
So if a and b are
integers, there's

00:03:36.570 --> 00:03:39.010
no way that the square
root of its product

00:03:39.010 --> 00:03:42.437
can be something divided by
2, where this doesn't reduce

00:03:42.437 --> 00:03:44.020
any more, where this
doesn't simplify,

00:03:44.020 --> 00:03:46.220
or some odd number divided by 2.

00:03:46.220 --> 00:03:47.720
So we just get a
contradiction here.

00:03:47.720 --> 00:03:51.380
Or another way to say it
is the square root of ab--

00:03:51.380 --> 00:03:54.920
because this whole thing has
to be an integer and both a

00:03:54.920 --> 00:03:59.190
and b are integers,
the square root of ab

00:03:59.190 --> 00:04:01.755
has to be an integer.

00:04:10.640 --> 00:04:13.930
But if the square root of
ab has to be an integer,

00:04:13.930 --> 00:04:16.670
then that means that ab
has to be a perfect square.

00:04:16.670 --> 00:04:19.986
So then I'll start over here
so I don't run out of space.

00:04:19.986 --> 00:04:21.610
Actually, let me do
it right under that

00:04:21.610 --> 00:04:23.290
because it's a straight result.

00:04:23.290 --> 00:04:28.635
So that means that a times
b is a perfect square.

00:04:37.000 --> 00:04:39.810
So let's see how we can
deal with this a little bit.

00:04:39.810 --> 00:04:42.075
We can actually do a
substitution for a. a

00:04:42.075 --> 00:04:43.910
is equal to b plus 60.

00:04:43.910 --> 00:04:48.060
So this means that
b plus 60 times

00:04:48.060 --> 00:04:51.650
b is equal to a
perfect square, so it

00:04:51.650 --> 00:04:56.205
could be equal to some number n
squared, where n is an integer.

00:04:58.762 --> 00:05:01.220
And let's see if we can play
around with this a little bit,

00:05:01.220 --> 00:05:03.450
put some constraints
on what we're doing.

00:05:03.450 --> 00:05:05.020
So if you multiply
this out, you get

00:05:05.020 --> 00:05:13.650
b squared plus 60b-- I'll
leave some space over here--

00:05:13.650 --> 00:05:16.191
is equal to n squared.

00:05:16.191 --> 00:05:18.440
And let's see, we could
complete the square over here.

00:05:18.440 --> 00:05:20.460
And maybe we could
subtract this over

00:05:20.460 --> 00:05:22.345
and we'll have a
difference of squares.

00:05:22.345 --> 00:05:23.970
So we want to complete
the square here.

00:05:23.970 --> 00:05:25.180
You take half of this.

00:05:25.180 --> 00:05:27.770
So half of 60, half
of the coefficient--

00:05:27.770 --> 00:05:29.580
that's 30-- and square it.

00:05:29.580 --> 00:05:33.034
So this would be
plus 900 minus 900.

00:05:33.034 --> 00:05:33.950
We haven't changed it.

00:05:33.950 --> 00:05:36.270
We're adding and
subtracting the same thing.

00:05:36.270 --> 00:05:38.970
But what that does for us
is that this part over here

00:05:38.970 --> 00:05:46.780
becomes b plus 30 squared,
and then we have the minus 900

00:05:46.780 --> 00:05:49.220
is equal to n squared.

00:05:49.220 --> 00:05:53.390
And then we could subtract
n squared from both sides

00:05:53.390 --> 00:05:56.270
and add 900 to both sides.

00:05:56.270 --> 00:05:59.642
And we get b plus 30--
and you really just

00:05:59.642 --> 00:06:01.100
have to play around
with this thing

00:06:01.100 --> 00:06:02.690
and see what
constraints you can get.

00:06:02.690 --> 00:06:09.010
b plus 30 squared minus n
squared is equal to 900.

00:06:09.010 --> 00:06:10.930
And this is just a
difference of squares.

00:06:10.930 --> 00:06:11.930
So this can be factored.

00:06:11.930 --> 00:06:14.440
This is b plus 30.

00:06:14.440 --> 00:06:16.380
This is a plus b.

00:06:16.380 --> 00:06:17.730
Or, I shouldn't use a and b.

00:06:17.730 --> 00:06:20.810
But it's the square root of this
plus the square root of this,

00:06:20.810 --> 00:06:22.131
so plus n.

00:06:22.131 --> 00:06:24.630
Times the square root of this
minus the square root of this.

00:06:24.630 --> 00:06:29.150
So, b plus 30 minus n.

00:06:29.150 --> 00:06:31.669
So we can factor
it just like that.

00:06:31.669 --> 00:06:33.960
And that's about as good of
a constraint as we can get.

00:06:33.960 --> 00:06:36.760
Remember, we want to
be able to maximize b.

00:06:36.760 --> 00:06:40.390
If we're maximizing b,
a is only 60 above b.

00:06:40.390 --> 00:06:43.010
So we're also going
to be maximizing a.

00:06:43.010 --> 00:06:45.610
And so we'll also be
maximizing the sum.

00:06:45.610 --> 00:06:50.540
So we want b to be
as high as possible.

00:06:50.540 --> 00:06:53.305
So b plus 30 plus n is going
to be some factor of 900.

00:06:53.305 --> 00:06:56.987
And b plus 30 minus n will
be the other factor of 900.

00:06:56.987 --> 00:06:59.320
And we could actually do a
little bit more of an insight

00:06:59.320 --> 00:06:59.820
here.

00:06:59.820 --> 00:07:01.170
We have two factors.

00:07:01.170 --> 00:07:03.070
Their product is an even number.

00:07:03.070 --> 00:07:04.095
So it's an even product.

00:07:08.200 --> 00:07:10.820
And if we were to take the
sum of these two factors,

00:07:10.820 --> 00:07:13.460
if you were to take
b plus 30 plus n

00:07:13.460 --> 00:07:16.640
and add it to b
plus 30 minus n--

00:07:16.640 --> 00:07:19.180
so we're just taking the
sum of these two factors--

00:07:19.180 --> 00:07:21.940
we get 2b plus 60.

00:07:21.940 --> 00:07:23.330
And these cancel out.

00:07:23.330 --> 00:07:25.160
And so this is also even.

00:07:25.160 --> 00:07:28.340
So if I take two numbers,
and their product is even,

00:07:28.340 --> 00:07:31.130
and their sum is
also even, that means

00:07:31.130 --> 00:07:32.640
the two numbers have to be even.

00:07:32.640 --> 00:07:38.330
So both this guy and this guy
over here have to be even.

00:07:38.330 --> 00:07:40.430
Both of them have to be even.

00:07:40.430 --> 00:07:43.430
And at this point, I think we've
used up all the information

00:07:43.430 --> 00:07:44.100
in the problem.

00:07:44.100 --> 00:07:46.580
We really just have to try
out factors of 900 that

00:07:46.580 --> 00:07:48.680
meet these constraints
where both are even.

00:07:48.680 --> 00:07:51.150
And let's try with the most
extreme things in an attempt

00:07:51.150 --> 00:07:54.625
to maximize our b.

00:07:54.625 --> 00:07:56.760
So let's think about
the factors of 900.

00:07:56.760 --> 00:07:59.040
You could do 1 and
900, but remember,

00:07:59.040 --> 00:08:01.250
both factors have to be even.

00:08:01.250 --> 00:08:02.670
1 is not even.

00:08:02.670 --> 00:08:07.549
We could try 2 and 450, where
this will be the higher factor.

00:08:07.549 --> 00:08:09.340
This will be the lower
factor because we're

00:08:09.340 --> 00:08:10.920
subtracting a number here.

00:08:10.920 --> 00:08:13.780
So let's try that out.

00:08:13.780 --> 00:08:16.239
Actually, let me
do it over here.

00:08:16.239 --> 00:08:17.530
Actually, I'll do it over here.

00:08:17.530 --> 00:08:20.400
Maybe this is 450 and this is 2.

00:08:20.400 --> 00:08:25.440
So we have b plus 30
plus n is equal to 450,

00:08:25.440 --> 00:08:30.260
and that b plus 30
minus n is equal to 2.

00:08:30.260 --> 00:08:31.775
And now we can solve for b.

00:08:31.775 --> 00:08:33.380
We can add these two equations.

00:08:33.380 --> 00:08:40.539
You get 2b plus 60-- these
cancel out-- is equal to 452.

00:08:40.539 --> 00:08:42.669
Subtract 60 from both sides.

00:08:42.669 --> 00:08:49.237
2b is equal to 392,
or b is equal to 196.

00:08:49.237 --> 00:08:50.570
So let's see how this works out.

00:08:50.570 --> 00:08:57.640
If b is equal to 196, then
a is just 60 plus that.

00:08:57.640 --> 00:09:02.060
So a is equal to 256.

00:09:02.060 --> 00:09:04.670
And let's see what it does
to this equation over here.

00:09:04.670 --> 00:09:07.070
Square root of a is 16.

00:09:07.070 --> 00:09:09.920
So we would have the
square root of 256

00:09:09.920 --> 00:09:12.520
plus the square root of 196.

00:09:12.520 --> 00:09:17.230
Well, that's equal to 16 plus
14, which is equal to 30.

00:09:17.230 --> 00:09:18.570
So it looks like it might work.

00:09:18.570 --> 00:09:22.380
But remember, this thing over
here, the square root of c,

00:09:22.380 --> 00:09:24.260
is not a perfect square.

00:09:24.260 --> 00:09:26.290
This value right over here.

00:09:26.290 --> 00:09:28.440
That's the square root of c.

00:09:28.440 --> 00:09:30.850
Or I should say the square
root of c is not an integer.

00:09:30.850 --> 00:09:31.960
It cannot be an integer.

00:09:31.960 --> 00:09:35.660
If it was, then c which
in this case would be 900,

00:09:35.660 --> 00:09:37.370
would be a perfect square.

00:09:37.370 --> 00:09:40.960
But they told us in the problem
c is not a perfect square.

00:09:40.960 --> 00:09:43.850
So this one actually does not
work because this would give us

00:09:43.850 --> 00:09:45.510
a c that is a perfect square.

00:09:45.510 --> 00:09:46.770
It gives us 900.

00:09:46.770 --> 00:09:48.800
Square root of c is 30.

00:09:48.800 --> 00:09:50.629
So we have to cross
these out too,

00:09:50.629 --> 00:09:52.420
although it was a very
tantalizing solution

00:09:52.420 --> 00:09:55.530
since it gave us
nice clean numbers.

00:09:55.530 --> 00:09:59.210
So let's keep trying
factors of 900.

00:09:59.210 --> 00:10:02.270
So remember, you
could try 3 and 300,

00:10:02.270 --> 00:10:04.110
but we know that they
both have to be even.

00:10:04.110 --> 00:10:07.750
So we could just reject
this because 3 is not even.

00:10:07.750 --> 00:10:14.877
So maybe it's 4 and-- 4
goes into 900 125 times.

00:10:14.877 --> 00:10:16.460
You might be tempted
to try these out.

00:10:16.460 --> 00:10:18.550
But 125 is not even.

00:10:18.550 --> 00:10:19.879
So we could keep going.

00:10:19.879 --> 00:10:20.920
So then we could go to 5.

00:10:20.920 --> 00:10:24.810
Well, once again 5 is not
even so that won't work.

00:10:24.810 --> 00:10:27.850
We could try 6.

00:10:27.850 --> 00:10:31.130
6 goes into 900-- it
goes into 90 15 times.

00:10:31.130 --> 00:10:34.790
So it goes into 900 150 times.

00:10:34.790 --> 00:10:36.580
They're both even numbers.

00:10:36.580 --> 00:10:39.390
So maybe, just maybe
these will work.

00:10:39.390 --> 00:10:40.990
So let's try it out again.

00:10:40.990 --> 00:10:43.510
So let me scroll
down a little bit.

00:10:43.510 --> 00:10:48.100
So you could have
b plus 30 plus n--

00:10:48.100 --> 00:10:50.656
that'll be the larger
number-- is equal to 150

00:10:50.656 --> 00:10:52.030
because we're
adding the n there.

00:10:52.030 --> 00:10:58.340
And then you have b plus 30
minus n would be equal to 6.

00:10:58.340 --> 00:11:08.320
You add both equations, you
get 2b plus 60 is equal to 156.

00:11:08.320 --> 00:11:11.520
Or subtracting 60
from both sides 2b

00:11:11.520 --> 00:11:15.430
is equal to-- this
is equal to 96.

00:11:15.430 --> 00:11:18.640
Or b is equal to 48.

00:11:18.640 --> 00:11:20.080
So here's our next candidate.

00:11:20.080 --> 00:11:23.660
Our next candidate
is b is equal to 48.

00:11:23.660 --> 00:11:25.610
Then a would be
60 more than this.

00:11:25.610 --> 00:11:30.450
a would be equal to
108, 60 more than that.

00:11:30.450 --> 00:11:33.450
If I were to take the
square root of 108

00:11:33.450 --> 00:11:38.900
plus the square root of
48, what does that give me?

00:11:38.900 --> 00:11:41.351
108 is 36 times 3.

00:11:41.351 --> 00:11:43.850
So this is the square root of
36 times the square root of 3,

00:11:43.850 --> 00:11:46.160
or 6 square roots of 3.

00:11:46.160 --> 00:11:47.900
Plus-- this is 16 times 3.

00:11:47.900 --> 00:11:50.290
So it's going to be the
square root of-- well 48

00:11:50.290 --> 00:11:51.206
is 16 times 3.

00:11:51.206 --> 00:11:53.580
So it's the square root of 16
times the square root of 3.

00:11:53.580 --> 00:11:56.150
So it's 4 square roots of 3.

00:11:56.150 --> 00:11:57.620
So when you add
these two together,

00:11:57.620 --> 00:12:00.080
you get 10 square roots of 3.

00:12:00.080 --> 00:12:03.050
So this fits our bill
for the square root of c.

00:12:03.050 --> 00:12:04.550
So over here we
have the square root

00:12:04.550 --> 00:12:06.410
of c is equal to 10 roots of 3.

00:12:06.410 --> 00:12:08.090
It's definitely not an integer.

00:12:08.090 --> 00:12:09.150
So it makes that.

00:12:09.150 --> 00:12:13.900
And if you square
it, you would get

00:12:13.900 --> 00:12:19.460
c as being equal to 100 times
3, which is equal to 300.

00:12:19.460 --> 00:12:20.470
And it's an integer.

00:12:20.470 --> 00:12:21.960
So it works.

00:12:21.960 --> 00:12:27.270
So our best shot, our best
values for a and b, b is 48,

00:12:27.270 --> 00:12:28.470
a is 108.

00:12:28.470 --> 00:12:37.500
If we take the sum, 108 plus
48, we get 16, 5, and 156.

00:12:37.500 --> 00:12:40.300
And you can see when you
keep trying-- obviously we

00:12:40.300 --> 00:12:44.080
didn't have the best result
when we had b being 196.

00:12:44.080 --> 00:12:46.985
But as we get our factors
closer and closer together,

00:12:46.985 --> 00:12:49.276
the b is going to go
lower and lower and lower.

00:12:49.276 --> 00:12:50.650
And we're obviously
going to have

00:12:50.650 --> 00:12:53.640
a less of a maximum
value for a plus b.

00:12:53.640 --> 00:12:57.990
So as we got from the most
extreme factors of 900,

00:12:57.990 --> 00:12:59.750
we keep going until
something met.

00:12:59.750 --> 00:13:01.210
This met our constraints.

00:13:01.210 --> 00:13:04.782
If we kept going, if
we tried eight and-- I

00:13:04.782 --> 00:13:06.740
don't know if that would
even work as a factor.

00:13:06.740 --> 00:13:09.360
But if we just tried and kept
looking at the other factors,

00:13:09.360 --> 00:13:10.380
we actually would
have gotten numbers

00:13:10.380 --> 00:13:11.630
that are closer to each other.

00:13:11.630 --> 00:13:14.970
And that would have led
to lower values for our b.

00:13:14.970 --> 00:13:18.800
So this right here
is our maximum b--

00:13:18.800 --> 00:13:21.980
or the b that will
maximize our result.

00:13:21.980 --> 00:13:24.170
And the sum is 156.


2003 AIME II problem 9

WEBVTT
Kind: captions
Language: en

00:00:00.561 --> 00:00:02.209
- [Voiceover] Consider the polynomials,

00:00:02.209 --> 00:00:04.832
P of X is equal to X to the
sixth minus X to the fifth

00:00:04.832 --> 00:00:07.201
minus X to the third
minus X squared minus X.

00:00:07.201 --> 00:00:09.431
And Q of X is equal to X to the fourth

00:00:09.431 --> 00:00:12.570
minus X to the third
minus X squared minus one.

00:00:12.570 --> 00:00:15.728
Given that Z one, Z
two, Z three, and Z four

00:00:15.728 --> 00:00:18.955
are the roots of Q of X equals zero,

00:00:18.955 --> 00:00:21.278
find P of Z one plus P of Z two,

00:00:21.278 --> 00:00:25.039
plus P of Z three, plus P of Z four.

00:00:25.039 --> 00:00:26.131
So a good place to start,

00:00:26.131 --> 00:00:28.197
and you might have already realized this,

00:00:28.197 --> 00:00:32.237
that if these guys are the
roots of Q of X equals zero,

00:00:32.237 --> 00:00:34.933
that means that Q of Z one

00:00:34.933 --> 00:00:37.923
is equal to Q of Z two,

00:00:37.923 --> 00:00:41.173
which is equal to Q of Z three,

00:00:41.173 --> 00:00:43.983
which is equal to Q of Z four,

00:00:43.983 --> 00:00:45.539
which is equal to zero.

00:00:45.539 --> 00:00:48.853
They're the roots, they are
solutions to this equation here.

00:00:48.853 --> 00:00:50.432
Now given that, and we want to evaluate

00:00:50.432 --> 00:00:53.033
P of Z one plus P of Z two,

00:00:53.033 --> 00:00:55.795
it seems like it would be useful if we can

00:00:55.795 --> 00:00:58.786
express P of X in terms of Q of X,

00:00:58.786 --> 00:01:01.108
because then when we evaluate these,

00:01:01.108 --> 00:01:03.741
the parts that would
be Q of Z one or Z two

00:01:03.741 --> 00:01:06.064
or whatever would evaluate to zero.

00:01:06.064 --> 00:01:07.899
So let's try to do that over there.

00:01:07.899 --> 00:01:09.269
So the first thing I'm gonna try to do,

00:01:09.269 --> 00:01:11.637
I'm gonna try to express
these higher degree terms

00:01:11.637 --> 00:01:13.100
in terms of Q of X.

00:01:13.100 --> 00:01:16.072
And to do that, if I multiple
Q of X times X squared,

00:01:16.072 --> 00:01:18.208
this X to the fourth will
become an X to the sixth,

00:01:18.208 --> 00:01:20.321
this X to the third will
become an X to the fifth.

00:01:20.321 --> 00:01:21.784
It won't be exact, but we might be able to

00:01:21.784 --> 00:01:25.029
adjust a little bit to get
something right over here.

00:01:25.029 --> 00:01:26.143
So let's try to do that.

00:01:26.143 --> 00:01:28.629
So let's think a little bit about what,

00:01:28.629 --> 00:01:30.522
let's think about what X squared,

00:01:31.247 --> 00:01:34.615
X squared times Q of X
is going to look like.

00:01:34.615 --> 00:01:36.866
That's going to be X squared
times X to the fourth.

00:01:36.866 --> 00:01:40.790
That's X to the sixth,
minus X to the fifth,

00:01:40.790 --> 00:01:43.090
minus X to the fourth,

00:01:43.090 --> 00:01:44.297
minus X squared.

00:01:44.297 --> 00:01:46.201
Just multiplied all of these terms

00:01:46.201 --> 00:01:48.616
times X squared.

00:01:48.616 --> 00:01:53.051
Now that isn't exactly what
we have up here for P of X.

00:01:53.051 --> 00:01:54.955
In order to get it exact,

00:01:54.955 --> 00:01:56.835
in order to get it exact,

00:01:56.835 --> 00:01:59.156
we need to, what are
we going to have to do?

00:01:59.156 --> 00:02:01.339
Let's see, we don't have
an X to the fourth up here,

00:02:01.339 --> 00:02:03.058
so we're gonna have to get
rid of this X to the fourth.

00:02:03.058 --> 00:02:05.417
So let's add X to the
fourth to both sides.

00:02:06.188 --> 00:02:10.159
So let's add X to the fourth
to both sides of this equation.

00:02:10.159 --> 00:02:12.050
So that would get rid
of an X to the fourth.

00:02:12.916 --> 00:02:14.565
And let's see, we have
no X to the third here,

00:02:14.565 --> 00:02:15.888
we do have an X to the third there,

00:02:15.888 --> 00:02:17.862
so let's subtract X to the third

00:02:17.862 --> 00:02:20.555
from both sides of this equation.

00:02:20.555 --> 00:02:22.807
So I'm gonna subtract X to the third

00:02:22.807 --> 00:02:25.036
from both sides of this equation.

00:02:25.036 --> 00:02:26.848
And then we do have a negative X squared,

00:02:26.848 --> 00:02:29.401
we do have a negative X squared over here.

00:02:29.401 --> 00:02:31.561
And then we have a negative X here.

00:02:31.561 --> 00:02:33.976
I don't see a negative X over here,

00:02:33.976 --> 00:02:36.484
so let's subtract a
negative X from both sides.

00:02:36.484 --> 00:02:37.482
So negative X.

00:02:37.482 --> 00:02:39.200
This is all for the left-hand side.

00:02:39.200 --> 00:02:40.785
And so we're left with

00:02:41.336 --> 00:02:43.276
the left-hand side of the equation we get

00:02:43.276 --> 00:02:45.913
X squared times Q of X,

00:02:46.731 --> 00:02:49.623
X squared times Q of X
plus X to the fourth,

00:02:50.301 --> 00:02:52.693
plus X to the fourth minus X to the third,

00:02:52.693 --> 00:02:54.992
minus X is equal to,

00:02:54.992 --> 00:02:55.874
what's this equal to?

00:02:55.874 --> 00:02:59.024
It's equal to X to the sixth.

00:02:59.024 --> 00:03:01.439
I didn't want to use that shade of purple,

00:03:01.439 --> 00:03:02.600
let me use the blue one.

00:03:02.600 --> 00:03:05.171
It's equal to X to the sixth

00:03:05.171 --> 00:03:07.564
minus X to the fifth,

00:03:07.564 --> 00:03:09.398
these guys cancel out,

00:03:09.398 --> 00:03:11.697
minus X to the third,

00:03:11.697 --> 00:03:14.608
minus X to third minus X squared,

00:03:15.412 --> 00:03:16.642
minus X squared,

00:03:16.642 --> 00:03:19.094
and then you have a minus X there.

00:03:19.094 --> 00:03:19.976
Minus X.

00:03:19.976 --> 00:03:21.486
Which is exactly, this is,

00:03:21.486 --> 00:03:24.481
we were actually able to
completely construct P of X.

00:03:24.481 --> 00:03:26.825
This is equal to P of X.

00:03:26.825 --> 00:03:29.613
It's equal to all of this
business on the left-hand side.

00:03:29.613 --> 00:03:31.577
Now we could just stop
there, but it looks,

00:03:31.577 --> 00:03:33.482
this part right here,

00:03:33.482 --> 00:03:35.897
this part right here
looks tantalizingly close

00:03:35.897 --> 00:03:37.987
to Q of X as well.

00:03:37.987 --> 00:03:40.355
In fact it is, that part right there,

00:03:40.355 --> 00:03:43.861
using the exact same
process would be Q of X,

00:03:43.861 --> 00:03:46.031
just the green part, it's Q of X

00:03:46.031 --> 00:03:47.634
with these things gotten rid of.

00:03:47.634 --> 00:03:49.724
So we would have to add an X squared.

00:03:49.724 --> 00:03:51.937
So it's Q of X plus X squared,

00:03:52.939 --> 00:03:55.239
plus X squared plus one.

00:03:55.239 --> 00:03:56.563
Right? If you start with Q of X,

00:03:56.563 --> 00:03:57.956
you add an X squared, you add a one,

00:03:57.956 --> 00:03:58.931
these two will go away,

00:03:58.931 --> 00:04:02.158
you just have an X to the
fourth minus X to the third.

00:04:02.158 --> 00:04:04.643
So now we can rewrite P of X.

00:04:04.643 --> 00:04:07.545
We can rewrite P of X,
I'll do it right over here,

00:04:07.545 --> 00:04:11.663
P of X is equal to X squared times Q of X,

00:04:12.498 --> 00:04:14.830
it's equal to, no,
that's not the same blue.

00:04:15.579 --> 00:04:18.759
It's equal to X squared

00:04:18.759 --> 00:04:23.474
times Q of X plus this business,

00:04:24.292 --> 00:04:25.853
plus Q of X,

00:04:26.764 --> 00:04:29.899
plus X squared plus one.

00:04:29.899 --> 00:04:32.360
And then we have this last negative X,

00:04:32.360 --> 00:04:35.262
or last minus X right over there.

00:04:35.262 --> 00:04:38.118
Now, when we evaluate, if I say,

00:04:38.118 --> 00:04:40.022
if I say that R is a root.

00:04:40.022 --> 00:04:42.925
So if R is a root of Q.

00:04:42.925 --> 00:04:46.524
Or if I write Q of R is equal to zero,

00:04:46.524 --> 00:04:48.659
then what is P of R?

00:04:48.659 --> 00:04:52.351
P of R is going to be equal to X squared

00:04:52.351 --> 00:04:55.987
times Q of R plus Q of R,

00:04:56.828 --> 00:04:59.336
plus R squared, I'll switch these,

00:04:59.336 --> 00:05:01.101
minus R, plus one.

00:05:01.101 --> 00:05:02.656
So this is going to be zero, obviously.

00:05:02.656 --> 00:05:05.350
Q of R is zero, so this
whole term is zero.

00:05:05.350 --> 00:05:06.766
This whole term is zero.

00:05:06.766 --> 00:05:08.833
So P of R is going to be equal to

00:05:08.833 --> 00:05:12.641
R squared minus R, plus one.

00:05:12.641 --> 00:05:15.218
Now, obviously all of these guys are roots

00:05:15.218 --> 00:05:17.401
so we can use that same logic.

00:05:17.401 --> 00:05:19.188
We can use that same logic.

00:05:19.188 --> 00:05:22.889
We can say P of, let
me do a new color here.

00:05:23.568 --> 00:05:24.904
A new color is warranted.

00:05:25.769 --> 00:05:30.667
P of Z one is equal to Z one squared,

00:05:30.667 --> 00:05:34.197
minus Z one plus one.

00:05:34.197 --> 00:05:36.078
P of Z two,

00:05:36.078 --> 00:05:40.304
P of Z two is equal to Z two squared,

00:05:40.304 --> 00:05:42.485
minus Z two plus one.

00:05:42.485 --> 00:05:44.692
I could, I'll go all four.

00:05:44.692 --> 00:05:46.318
P of Z three.

00:05:46.318 --> 00:05:50.334
P of Z three is equal to Z three squared,

00:05:50.334 --> 00:05:52.958
minus Z three plus one.

00:05:52.958 --> 00:05:56.580
And then finally we have P of Z four.

00:05:56.580 --> 00:06:00.319
P of Z four, in that slightly
different shade of orange,

00:06:00.319 --> 00:06:04.963
is Z sub four squared
minus Z sub four plus one.

00:06:04.963 --> 00:06:06.124
Now remember our problem.

00:06:06.124 --> 00:06:07.842
We just want to find
the sum of these things.

00:06:07.842 --> 00:06:10.791
We just want to take the sum.

00:06:10.791 --> 00:06:13.947
So this sum is going to be
the sum of these expressions

00:06:13.947 --> 00:06:17.036
and you might be saying,
Sal, we still don't know

00:06:17.036 --> 00:06:18.127
what these sums are.

00:06:18.127 --> 00:06:20.705
I could sum all the ones,
I'll get a four there,

00:06:20.705 --> 00:06:22.772
but now I have to sum
all the roots of Q of X

00:06:22.772 --> 00:06:25.442
and I have to sum all the
squares of the roots of Q of X?

00:06:25.442 --> 00:06:27.578
And this is not a simple thing

00:06:27.578 --> 00:06:30.643
and frankly it's not taught
in most algebra classes

00:06:30.643 --> 00:06:31.641
or precalculus classes.

00:06:31.641 --> 00:06:33.777
This is a problem to
really separate the people

00:06:33.777 --> 00:06:37.701
who study kind of quirky
properties of polynomials

00:06:37.701 --> 00:06:38.956
from the people who don't study

00:06:38.956 --> 00:06:40.626
quirky properties of polynomials,

00:06:40.626 --> 00:06:42.113
but the last two videos I just recorded

00:06:42.113 --> 00:06:44.296
in the competition math playlist,

00:06:44.296 --> 00:06:46.222
and I actually added them to
the algebra playlist as well,

00:06:46.222 --> 00:06:48.893
tell you how to find the sum
of the roots of a polynomial

00:06:48.893 --> 00:06:51.934
or the sum of the squares of
the roots of a polynomial.

00:06:51.934 --> 00:06:53.885
So let's think about this a little bit.

00:06:53.885 --> 00:06:56.138
So let's add up all the ones, that's easy,

00:06:56.138 --> 00:06:57.949
we get four.

00:06:57.949 --> 00:07:00.457
Now, we're also gonna
have to add up all of,

00:07:00.457 --> 00:07:03.358
when you add up all of
these guys over here.

00:07:03.358 --> 00:07:04.427
Let me do this in a new color.

00:07:04.427 --> 00:07:07.933
When I add up all of these guys over here,

00:07:07.933 --> 00:07:11.695
it's going to be negative Z one plus Z two

00:07:11.695 --> 00:07:15.363
plus Z three plus Z four.

00:07:15.363 --> 00:07:18.010
Now, two videos ago, I think it was,

00:07:18.010 --> 00:07:20.285
we show that the sum of
the roots of a polynomial,

00:07:20.285 --> 00:07:22.701
and this the sum of the roots of Q of X,

00:07:22.701 --> 00:07:25.742
that that is equal to the
negative of the coefficient

00:07:25.742 --> 00:07:28.319
on the second to highest degree term.

00:07:28.319 --> 00:07:30.409
Or, I should say that on the degree term

00:07:30.409 --> 00:07:32.174
that's one less than
the highest degree term.

00:07:32.174 --> 00:07:34.063
So if we go to Q of X,

00:07:34.063 --> 00:07:37.522
we go to Q of X, it's a
fourth degree polynomial.

00:07:37.522 --> 00:07:38.684
The sum of the roots,

00:07:38.684 --> 00:07:41.167
to find the sum of the roots we go

00:07:41.167 --> 00:07:42.329
one degree less than that,

00:07:42.329 --> 00:07:44.069
so we look for the third degree term,

00:07:44.069 --> 00:07:45.439
that's this over here.

00:07:45.439 --> 00:07:48.645
The coefficient on the
third degree term over here

00:07:48.645 --> 00:07:50.223
is negative one.

00:07:50.223 --> 00:07:51.965
It has a negative one coefficient,

00:07:51.965 --> 00:07:54.055
so the sum of the roots,

00:07:54.055 --> 00:07:55.146
the sum of the roots,

00:07:55.146 --> 00:07:56.511
this over here is going to be equal

00:07:56.511 --> 00:07:58.716
the negative of that coefficient.

00:07:58.716 --> 00:08:00.341
So negative negative one,

00:08:00.341 --> 00:08:02.617
so it's the negative of negative one,

00:08:02.617 --> 00:08:04.312
which is just plus one.

00:08:04.312 --> 00:08:06.239
And of course we have a
negative out front here,

00:08:06.239 --> 00:08:08.793
so this whole thing is going
to evaluate to negative one.

00:08:08.793 --> 00:08:13.437
So that whole business
evaluates to negative one.

00:08:13.437 --> 00:08:16.247
Now, we have to take
the sum of the squares,

00:08:16.247 --> 00:08:18.118
We have to do all of this over here.

00:08:19.190 --> 00:08:20.468
We have to do all of this over here.

00:08:20.468 --> 00:08:24.067
The sum of Z one squared
plus Z two squared

00:08:24.067 --> 00:08:28.339
plus Z three squared plus Z four squared.

00:08:28.339 --> 00:08:29.570
Now in the last video where I

00:08:29.570 --> 00:08:30.522
talk about the sum of the squares,

00:08:30.522 --> 00:08:32.101
I didn't prove it for
the fourth degree case.

00:08:32.101 --> 00:08:33.471
It's actually pretty
easy to extend it from

00:08:33.471 --> 00:08:35.607
the third degree case
that we did in that video

00:08:35.607 --> 00:08:37.302
to the fourth degree and it's little messy

00:08:37.302 --> 00:08:39.623
to do it generally do
the whole induction thing

00:08:39.623 --> 00:08:41.482
that I did with the straight up sum,

00:08:41.482 --> 00:08:42.726
not the sum of the squares.

00:08:43.405 --> 00:08:44.472
But hopefully you see,

00:08:44.472 --> 00:08:46.748
or at least you understood how to apply it

00:08:46.748 --> 00:08:48.792
and if you're interested,
you actually could extend it

00:08:48.792 --> 00:08:51.578
to the fourth power or to the nth power,

00:08:51.578 --> 00:08:55.246
but this is going to be
equal to the coefficient on,

00:08:55.246 --> 00:08:56.547
I should look at Q of X.

00:08:56.547 --> 00:09:00.401
The coefficient on the degree one less

00:09:00.401 --> 00:09:02.491
than the highest degree,
so it's going to be,

00:09:02.491 --> 00:09:03.790
so this is a negative one.

00:09:03.790 --> 00:09:07.599
So it's going to be equal
to negative one squared,

00:09:07.599 --> 00:09:11.198
negative one squared minus two times

00:09:11.198 --> 00:09:13.960
the coefficient on the degree below that.

00:09:13.960 --> 00:09:17.560
So minus two times the
coefficient over here.

00:09:17.560 --> 00:09:18.688
So over here,

00:09:19.436 --> 00:09:22.362
so we're looking, we
started at this degree,

00:09:22.362 --> 00:09:23.870
now we're going even one lower than that.

00:09:23.870 --> 00:09:26.099
Two lower than the
degree of the polynomial.

00:09:26.099 --> 00:09:27.679
So we go to the X squared term.

00:09:27.679 --> 00:09:29.396
This coefficient is negative one.

00:09:29.396 --> 00:09:31.487
So minus two times negative one.

00:09:31.487 --> 00:09:33.584
This is equal to one,

00:09:34.343 --> 00:09:38.210
one plus two, which is equal to three.

00:09:38.210 --> 00:09:40.531
So this thing over here in green

00:09:40.531 --> 00:09:41.553
when you take the sum of the squares,

00:09:41.553 --> 00:09:43.667
it's equal to three.

00:09:43.667 --> 00:09:45.547
And so when you take the total sum

00:09:45.547 --> 00:09:49.888
of P evaluated at each of
those possibly complex numbers,

00:09:49.888 --> 00:09:54.008
we get three minus two
plus four is equal to six.

00:09:54.008 --> 00:09:55.257
And we're done.


Sum of squares of polynomial roots

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:01.960
In the last video,
we figured out

00:00:01.960 --> 00:00:05.710
how to figured out the sum
of the roots of a polynomial.

00:00:05.710 --> 00:00:07.960
What we're going to try to
attempt to do in this video

00:00:07.960 --> 00:00:10.660
is think about the
sum of the squares

00:00:10.660 --> 00:00:12.690
of the roots of a polynomial.

00:00:12.690 --> 00:00:14.390
So let's think about
that a little bit.

00:00:14.390 --> 00:00:16.180
So let's say that I
have a second degree

00:00:16.180 --> 00:00:19.090
polynomial that looks like this.

00:00:19.090 --> 00:00:28.170
So it's x squared plus a1x,
plus a2 and this is equal to 0.

00:00:28.170 --> 00:00:29.890
We saw-- and actually,
I'll just redo

00:00:29.890 --> 00:00:31.473
some of what we did
in the last video,

00:00:31.473 --> 00:00:33.370
because it's going to
be useful over here.

00:00:33.370 --> 00:00:37.076
This thing is going to
have two roots, r1 and r2.

00:00:37.076 --> 00:00:40.080
Two roots, and that tells
us that this thing can

00:00:40.080 --> 00:00:47.130
be rewritten as x minus r1,
times x minus r2 is equal to 0.

00:00:47.130 --> 00:00:48.930
And if you expand
this out, you get

00:00:48.930 --> 00:00:58.860
x squared minus r1
plus r2, x, plus r1r2.

00:00:58.860 --> 00:01:00.567
So that's what you
get right over there.

00:01:00.567 --> 00:01:01.650
Now, let's think about it.

00:01:01.650 --> 00:01:09.160
We figured out from the last
video that r1 plus r2, which

00:01:09.160 --> 00:01:10.930
is right over here,
that needs to be

00:01:10.930 --> 00:01:13.250
equal to the negative
of this coefficient.

00:01:13.250 --> 00:01:17.460
This a1 over here is equal to
the negative of r1 plus r2.

00:01:17.460 --> 00:01:19.720
So r1 plus r2 is equal
to the negative of a1.

00:01:22.950 --> 00:01:25.660
So let's see if we can use that
information, right over there,

00:01:25.660 --> 00:01:29.250
to figure out what-- let
me do this in a new color--

00:01:29.250 --> 00:01:34.412
to figure out what r1
squared plus r2 squared is.

00:01:34.412 --> 00:01:35.870
So let's see how
we could get that.

00:01:35.870 --> 00:01:38.390
Well, one place that we
could start off with,

00:01:38.390 --> 00:01:42.005
we could start off with
r1 plus r2 squared.

00:01:42.005 --> 00:01:46.960
So r1 plus r2 squared, which
is, of course, the same thing

00:01:46.960 --> 00:01:50.350
as negative a1 squared, which
would be the same thing as a1

00:01:50.350 --> 00:01:51.450
squared.

00:01:51.450 --> 00:01:53.250
a1 squared-- I could
put a negative here,

00:01:53.250 --> 00:01:55.070
but a negative 1
times negative 1

00:01:55.070 --> 00:01:57.350
is obviously a
positive one-- this

00:01:57.350 --> 00:02:00.780
is going to be
equal to r1 squared

00:02:00.780 --> 00:02:07.030
plus 2r1r2, plus r2 squared.

00:02:07.030 --> 00:02:09.960
So this has the sum that
we care about in it.

00:02:12.920 --> 00:02:14.820
Let's call this
sum 1-- we're just

00:02:14.820 --> 00:02:18.330
raising the roots
to the first power.

00:02:18.330 --> 00:02:20.550
Let's call this sum 2.

00:02:20.550 --> 00:02:23.160
So this over here, we've
written, essentially,

00:02:23.160 --> 00:02:26.940
that sum 1 squared--
that's sum s

00:02:26.940 --> 00:02:32.670
sub 1 squared is equal
to-- this plus this

00:02:32.670 --> 00:02:38.140
is s sub 2-- is equal
to s sub 2 plus 2r1r2.

00:02:44.620 --> 00:02:47.180
Now, can we figure
out what r1r2 is

00:02:47.180 --> 00:02:50.789
based on looking at the
actual original polynomial?

00:02:50.789 --> 00:02:52.330
Well, you look over
here, if you just

00:02:52.330 --> 00:02:54.090
did an expansion--
of course, I forgot

00:02:54.090 --> 00:03:00.810
to write here that this is
equal to 0-- r1r2 is essentially

00:03:00.810 --> 00:03:03.790
represented by a2 over here.

00:03:03.790 --> 00:03:07.910
So to take the first degree
sum, or the roots raised

00:03:07.910 --> 00:03:10.330
to the first power sum, we
looked at this coefficient.

00:03:10.330 --> 00:03:11.970
Now all of a sudden, we're
involving the coefficient

00:03:11.970 --> 00:03:12.845
to the right of that.

00:03:12.845 --> 00:03:16.710
We're going two degrees below
the degree of the polynomial.

00:03:16.710 --> 00:03:19.510
So this, right over here, is a2.

00:03:19.510 --> 00:03:22.510
So r1r2 is a2.

00:03:22.510 --> 00:03:23.710
That's a2.

00:03:23.710 --> 00:03:28.470
This right here-- this whole
expression-- is a1 squared.

00:03:28.470 --> 00:03:30.100
So if we want to
solve for this sum--

00:03:30.100 --> 00:03:33.220
if we want to solve
for s sub 2, we

00:03:33.220 --> 00:03:39.020
would get s sub 2 is equal
to a1 squared-- that's

00:03:39.020 --> 00:03:45.540
this over here--
minus 2 times a sub 2.

00:03:45.540 --> 00:03:47.150
So we just figured
out a fast way

00:03:47.150 --> 00:03:51.920
to figure out the sum of the
squares of the root of just

00:03:51.920 --> 00:03:54.720
a second degree
quadratic right now.

00:03:54.720 --> 00:03:57.020
So for example,
if I were to give

00:03:57.020 --> 00:04:00.790
you-- let me give you
something strange.

00:04:00.790 --> 00:04:09.310
So if I were to give you 7x
squared minus pi x, plus e

00:04:09.310 --> 00:04:12.360
is equal to 0.

00:04:12.360 --> 00:04:16.130
And someone said, I
want you to figure out

00:04:16.130 --> 00:04:19.200
the sum of the squares of
the roots of this polynomial,

00:04:19.200 --> 00:04:21.160
first you want to
make the coefficient

00:04:21.160 --> 00:04:23.230
in front of the x squared 1.

00:04:23.230 --> 00:04:24.610
So divide everything by 7.

00:04:24.610 --> 00:04:31.126
So you get x squared minus
pi over 7 x, plus e over 7

00:04:31.126 --> 00:04:32.794
is equal to 0.

00:04:32.794 --> 00:04:35.210
And then you could just use
the result that we just found.

00:04:35.210 --> 00:04:38.581
The sum of the square of the
roots-- is this s2 over here.

00:04:38.581 --> 00:04:40.580
Whatever the roots are,
you square each of them,

00:04:40.580 --> 00:04:42.020
you take their
sum, and it's going

00:04:42.020 --> 00:04:43.750
to be equal to a1 squared.

00:04:43.750 --> 00:04:47.100
So it's going to be
pi squared over 49.

00:04:47.100 --> 00:04:49.494
I'm just squaring
this right over here,

00:04:49.494 --> 00:04:50.660
or I could just square that.

00:04:50.660 --> 00:04:52.576
The negative doesn't
matter what I'm squaring.

00:04:55.940 --> 00:04:59.339
minus 2 times e over seven.

00:04:59.339 --> 00:05:00.880
And that's about as
simplified as you

00:05:00.880 --> 00:05:02.100
could get this expression.

00:05:02.100 --> 00:05:05.020
So it's pi squared over
49, minus 2 e over 7,

00:05:05.020 --> 00:05:07.000
which by itself a
pretty neat result.

00:05:07.000 --> 00:05:09.437
Because it would be very hard
to find the roots of this.

00:05:09.437 --> 00:05:11.270
You'd get some crazy
numbers, and then you'd

00:05:11.270 --> 00:05:12.644
have to square
them and sum them.

00:05:12.644 --> 00:05:14.915
But we just figured out a
very fast way of doing it.

00:05:14.915 --> 00:05:17.290
Now, let's see if we can extend
this to the third degree.

00:05:17.290 --> 00:05:18.410
And it actually
turns out you can--

00:05:18.410 --> 00:05:20.410
not only to the third
degree you can extend this

00:05:20.410 --> 00:05:21.180
to the nth degree.

00:05:21.180 --> 00:05:23.370
And I might do an induction
proof it gets a little bit

00:05:23.370 --> 00:05:25.650
messier than when you just take
the first degree sum, right

00:05:25.650 --> 00:05:27.566
over here, where you
take the sum of the roots

00:05:27.566 --> 00:05:28.400
to the first power.

00:05:28.400 --> 00:05:32.900
But let's just see if it
works for the third power.

00:05:32.900 --> 00:05:35.220
So let's say, I have
the polynomial x

00:05:35.220 --> 00:05:44.800
to third plus a1x squared, plus
a2x, plus a3 is equal to 0.

00:05:44.800 --> 00:05:46.240
We have three roots.

00:05:46.240 --> 00:05:51.140
r1, r2, r3, which
tells us that we

00:05:51.140 --> 00:05:53.570
can rewrite this
polynomial as x minus--

00:05:53.570 --> 00:05:56.400
or this equation as
x minus r1, times

00:05:56.400 --> 00:06:05.910
x minus r2, times x
minus r3 is equal to 0.

00:06:05.910 --> 00:06:09.730
Now we already figured out
what this part right here

00:06:09.730 --> 00:06:12.510
is, it's this thing over here.

00:06:12.510 --> 00:06:13.750
We already multiplied it out.

00:06:13.750 --> 00:06:15.350
It's this thing.

00:06:15.350 --> 00:06:17.215
It is this over here.

00:06:17.215 --> 00:06:19.090
So essentially, to get
this whole expression,

00:06:19.090 --> 00:06:23.755
we just have to multiply this
times this thing over here.

00:06:23.755 --> 00:06:26.380
We can just use the distributive
property-- multiple this thing

00:06:26.380 --> 00:06:28.960
times each of these terms.

00:06:28.960 --> 00:06:32.410
So first, let's multiply
this whole thing by x.

00:06:32.410 --> 00:06:40.070
So you get x to the third, minus
r1 plus r2 times x squared,

00:06:40.070 --> 00:06:43.160
plus r1r2x.

00:06:43.160 --> 00:06:45.680
So I just multiplied this
thing over here times x.

00:06:45.680 --> 00:06:47.430
Now I can just multiply
this thing up here

00:06:47.430 --> 00:06:49.260
times negative r3.

00:06:49.260 --> 00:06:51.480
So negative r3 times x squared.

00:06:55.720 --> 00:07:00.670
Negative r3 times this term over
here, it will give us-- well

00:07:00.670 --> 00:07:02.990
it's going to be a
negative times a negative,

00:07:02.990 --> 00:07:05.290
so it's going to be a
positive-- so it's positive

00:07:05.290 --> 00:07:06.430
r3 three times this.

00:07:06.430 --> 00:07:07.750
We can distribute them.

00:07:07.750 --> 00:07:12.890
We're going to get r1-- so
let me make sure I'm doing.

00:07:12.890 --> 00:07:18.340
Negative r3 times this over
here is going to be negative--

00:07:18.340 --> 00:07:19.710
let me just write it like this.

00:07:19.710 --> 00:07:21.054
I don't want to skip steps.

00:07:21.054 --> 00:07:23.220
So it's going to be-- the
negative and the negatives

00:07:23.220 --> 00:07:23.840
cancel out.

00:07:23.840 --> 00:07:30.270
So negative r3
times r1 plus r2 x.

00:07:30.270 --> 00:07:34.570
And then finally, you have
the negative r3 times r1r2.

00:07:34.570 --> 00:07:36.280
So minus r1r2r3.

00:07:39.660 --> 00:07:41.440
And all of that's
going to be equal to 0.

00:07:41.440 --> 00:07:42.490
But let's simplify it.

00:07:42.490 --> 00:07:44.840
So this thing is
going to be equal to x

00:07:44.840 --> 00:07:49.920
to the third minus
r1 plus r2 plus r3--

00:07:49.920 --> 00:07:51.710
we did that in the last video.

00:07:51.710 --> 00:07:53.880
That shows us that
the sum of our roots

00:07:53.880 --> 00:07:56.470
is this second coefficient
here, the coefficient

00:07:56.470 --> 00:07:59.470
on the second degree
term, x squared.

00:07:59.470 --> 00:08:01.480
And then, we go over here.

00:08:01.480 --> 00:08:03.075
We have r1r2, x.

00:08:07.090 --> 00:08:08.990
Then you have r1, r3 x.

00:08:08.990 --> 00:08:12.060
Plus r1r3-- I'm just
distributing this.

00:08:12.060 --> 00:08:13.520
And then you have r3 times r2.

00:08:13.520 --> 00:08:16.270
So plus r2r3.

00:08:16.270 --> 00:08:21.960
All of that times x, and
then minus r1r2 and r3.

00:08:21.960 --> 00:08:25.880
Now if we want to take the sum
of the squares of the roots.

00:08:25.880 --> 00:08:32.320
So if we want to take r1
squared plus r2 squared plus r3

00:08:32.320 --> 00:08:34.970
squared, well, we could
try to do the same thing.

00:08:34.970 --> 00:08:39.380
We could take r1
plus r2 plus r3,

00:08:39.380 --> 00:08:42.620
which we know how to
calculate-- we can square that.

00:08:42.620 --> 00:08:44.150
And we're essentially
going to get--

00:08:44.150 --> 00:08:45.605
When you square
this you're going

00:08:45.605 --> 00:08:47.480
to get-- let me see if
I can write it neatly.

00:08:47.480 --> 00:08:51.444
This is going to be
equal to r1 times--

00:08:51.444 --> 00:08:53.360
So I'm just going to--
let me just rewrite it.

00:08:53.360 --> 00:08:55.070
This is equal to--
just to clarify,

00:08:55.070 --> 00:09:05.550
this is equal to r1 plus r2 plus
r3, times r1 plus r2 plus r3.

00:09:05.550 --> 00:09:07.620
So it's going to be-- I
could start over here.

00:09:07.620 --> 00:09:12.960
r1 times r1 is r1 squared,
plus r1 times r2-- so

00:09:12.960 --> 00:09:16.523
r1r2-- plus r1 times r3.

00:09:19.240 --> 00:09:21.660
And then, plus--
now we're in the r2.

00:09:21.660 --> 00:09:25.107
r2 times r1 is plus
r1r2-- same thing,

00:09:25.107 --> 00:09:27.190
I'm just switching the
order so it looks the same.

00:09:27.190 --> 00:09:30.860
r2 times r2 is r2 squared.

00:09:30.860 --> 00:09:35.340
And then, r2 times
r3 is plus r2r3.

00:09:35.340 --> 00:09:38.140
And then let's go to the
r3, r3 times r1 is just

00:09:38.140 --> 00:09:41.960
another r3r1-- I just
switched the order.

00:09:41.960 --> 00:09:45.750
r3 times r2 is another r2r3.

00:09:45.750 --> 00:09:49.475
And then, r3 times
r3 is r3 squared.

00:09:49.475 --> 00:09:50.850
So what did we
just get here when

00:09:50.850 --> 00:09:53.820
we squared just the
straight up sum?

00:09:53.820 --> 00:10:00.870
You got r1 squared plus r2
squared plus r3 squared,

00:10:00.870 --> 00:10:03.990
plus-- you essentially got all
of the different combinations,

00:10:03.990 --> 00:10:05.010
but you got them twice.

00:10:05.010 --> 00:10:08.690
You have two r1r2, so
you have 2 times r1r2.

00:10:11.790 --> 00:10:16.390
Then you have two r1r3, so plus
r1r3-- I have the 2 out front.

00:10:16.390 --> 00:10:17.270
Plus r2r3.

00:10:21.110 --> 00:10:23.700
And this over here is
equal to-- remember

00:10:23.700 --> 00:10:30.380
what we did-- this is equal
to r1 plus r2 plus r3 squared.

00:10:30.380 --> 00:10:32.770
So we got essentially
the same result.

00:10:32.770 --> 00:10:35.880
Remember, this thing right over
here-- let's make things clear.

00:10:35.880 --> 00:10:39.910
This thing over here
r1-- all the combinations

00:10:39.910 --> 00:10:41.840
of the products of the roots.

00:10:41.840 --> 00:10:43.780
If you look over here,
that's exactly what

00:10:43.780 --> 00:10:46.140
this thing is over
here, which must

00:10:46.140 --> 00:10:48.680
be what our a2 coefficient is.

00:10:48.680 --> 00:10:51.690
So this is our a2 coefficient.

00:10:51.690 --> 00:10:54.972
This r1 plus r2 plus
r3, that is-- we

00:10:54.972 --> 00:10:56.430
figured out multiple
times-- that's

00:10:56.430 --> 00:10:59.650
this thing right over here,
which is going to be a1.

00:10:59.650 --> 00:11:01.555
So this is going
to be a negative,

00:11:01.555 --> 00:11:03.080
or it's equal to negative a1.

00:11:03.080 --> 00:11:05.014
But you take
negative a1 squared,

00:11:05.014 --> 00:11:06.680
and that's the same
thing as a1 squared.

00:11:06.680 --> 00:11:08.680
So this is a1 squared.

00:11:08.680 --> 00:11:12.120
And this is the sum
that we care about.

00:11:12.120 --> 00:11:19.300
So we get r1 squared plus
r2 squared plus r3 squared

00:11:19.300 --> 00:11:22.510
is equal to this
business over here.

00:11:22.510 --> 00:11:29.560
It's equal to a1 squared
minus 2 times a2.

00:11:29.560 --> 00:11:34.260
So we got the exact
same result that we

00:11:34.260 --> 00:11:36.410
got for the second degree case.

00:11:36.410 --> 00:11:39.140
And it actually turns out this
will be true of any degree.

00:11:39.140 --> 00:11:41.060
I haven't proven to
you yet, although I

00:11:41.060 --> 00:11:42.560
can make an induction
argument now--

00:11:42.560 --> 00:11:44.050
we've proven some base cases.

00:11:44.050 --> 00:11:46.390
But just to make it
clear how to apply it,

00:11:46.390 --> 00:11:49.020
I give you crazy polynomial.

00:11:49.020 --> 00:11:51.560
I give you-- Let's do
a third degree case,

00:11:51.560 --> 00:11:56.980
let's say, 10x to
the third minus 5 x

00:11:56.980 --> 00:12:04.260
squared, plus 7x-- I'm making
this up on the fly-- plus 2

00:12:04.260 --> 00:12:06.130
is equal to 0.

00:12:06.130 --> 00:12:07.770
And if I were to
ask you the sum?

00:12:07.770 --> 00:12:11.354
If I were to ask you r1
squared plus r2 squared plus r3

00:12:11.354 --> 00:12:13.520
squared-- If I were to ask
you the square of the sum

00:12:13.520 --> 00:12:18.572
of the roots, you just take--
well, let me be careful.

00:12:18.572 --> 00:12:20.030
You have to make
sure that you have

00:12:20.030 --> 00:12:22.480
a one in front of the
highest degree term.

00:12:22.480 --> 00:12:24.920
So this thing over here,
has to be rewritten.

00:12:24.920 --> 00:12:30.430
Divide both sides by 10 x to
the third minus 5 over 10,

00:12:30.430 --> 00:12:35.150
so minus 1/2 x
squared, plus 7 over 10

00:12:35.150 --> 00:12:41.510
x, plus 2 over 10, which
is one fifth is equal to 0.

00:12:41.510 --> 00:12:44.540
So I just divided
both sides by 10,

00:12:44.540 --> 00:12:46.220
and yup, I did that part right.

00:12:46.220 --> 00:12:47.746
And now, we can apply this.

00:12:47.746 --> 00:12:49.120
I almost made a
careless mistake.

00:12:49.120 --> 00:12:51.370
You have to have a one
coefficient out here, at least,

00:12:51.370 --> 00:12:53.090
the way that we've derived this.

00:12:53.090 --> 00:12:58.920
So this sum is going to
be equal to a1 squared.

00:12:58.920 --> 00:13:01.600
So it's this-- 1/2 squared.

00:13:01.600 --> 00:13:05.660
So a1 is actually negative
1/2, but negative 1/2 half

00:13:05.660 --> 00:13:07.340
squared is just one fourth.

00:13:07.340 --> 00:13:17.070
So it is going to be one
fourth minus 2, times a2.

00:13:17.070 --> 00:13:21.700
So minus 2 times 7 over 10.

00:13:21.700 --> 00:13:28.720
So this is equal to one
fourth minus 14 over 10.

00:13:31.554 --> 00:13:33.220
We could find a common
denominator here.

00:13:33.220 --> 00:13:41.720
This is the same thing as
5 over 20 minus 14 over 10.

00:13:41.720 --> 00:13:48.940
And this is equal-- or sorry,
minus 28 over 20, I should do,

00:13:48.940 --> 00:13:52.340
which is equal to
negative 23 over 20.

00:13:52.340 --> 00:13:54.300
And you might say,
hey, Sal, hang on.

00:13:54.300 --> 00:13:57.060
I'm taking the square of a bunch
of roots and adding them up,

00:13:57.060 --> 00:13:58.980
and I'm getting a
negative number.

00:13:58.980 --> 00:14:01.470
I thought squares-- I thought
y squared numbers I always

00:14:01.470 --> 00:14:04.430
get a positive number that
would be true if you're dealing

00:14:04.430 --> 00:14:07.570
with real numbers, but remember
the words of a polynomial

00:14:07.570 --> 00:14:09.830
can be complex and
complex numbers

00:14:09.830 --> 00:14:12.970
squared can be negative numbers.

00:14:12.970 --> 00:14:17.200
So this obviously is involving
some type of complex roots.

00:14:17.200 --> 00:14:18.590
But we know that
the sum when you

00:14:18.590 --> 00:14:21.240
take the sum of the squares
of those complex roots you

00:14:21.240 --> 00:14:24.860
get negative 23 over 22.


Sum of polynomial roots (proof)

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:03.000
What I want to do
in this video, is

00:00:03.000 --> 00:00:05.850
figure out if
there's any fast way

00:00:05.850 --> 00:00:08.830
to figure out the sum of
the roots of any polynomial.

00:00:08.830 --> 00:00:10.555
And actually, there actually is.

00:00:10.555 --> 00:00:12.180
And so that's why
I'm doing this video.

00:00:12.180 --> 00:00:14.138
So let's start with a
second degree polynomial.

00:00:14.138 --> 00:00:22.970
So let's say it's x squared
plus a1x, plus a2 is equal to 0.

00:00:22.970 --> 00:00:26.390
So this is just a standard set
quadratic equation, right here,

00:00:26.390 --> 00:00:27.485
second degree equation.

00:00:27.485 --> 00:00:29.860
And you might be saying, hey,
but you put the coefficient

00:00:29.860 --> 00:00:31.640
on the x term being equal to 1.

00:00:31.640 --> 00:00:36.490
And in general, you can
always convert any polynomial.

00:00:36.490 --> 00:00:38.370
And I'll do it with
a second degree,

00:00:38.370 --> 00:00:42.960
but if you have ax squared
plus bx plus c is equal to 0,

00:00:42.960 --> 00:00:47.772
you can just divide both
sides of this equation by a.

00:00:47.772 --> 00:00:49.480
And you're going to
have something that's

00:00:49.480 --> 00:00:51.860
in this form, where the
coefficient on the x squared

00:00:51.860 --> 00:00:54.330
term is going to be equal to 1.

00:00:54.330 --> 00:00:56.320
And you can do that
with any polynomial.

00:00:56.320 --> 00:00:57.780
You can do that
with any polynomial

00:00:57.780 --> 00:01:00.160
that's set equal to 0.

00:01:00.160 --> 00:01:02.050
So with that out of
the way, let's think

00:01:02.050 --> 00:01:04.383
about what the sum of the
roots of this are going to be.

00:01:04.383 --> 00:01:07.990
So this is a second
degree polynomial.

00:01:07.990 --> 00:01:09.780
It's a quadratic equation.

00:01:09.780 --> 00:01:10.780
So it'll have two roots.

00:01:10.780 --> 00:01:12.780
They could be real or complex.

00:01:12.780 --> 00:01:16.282
So let's call the
roots r1 and r2.

00:01:16.282 --> 00:01:17.990
And that tells us,
since these are roots,

00:01:17.990 --> 00:01:25.160
that x minus r1 times x minus
r2 is going to be equal to 0.

00:01:25.160 --> 00:01:29.710
And if we multiply this out,
we get x times x is x squared.

00:01:29.710 --> 00:01:34.580
x times negative
r2 is negative r2x.

00:01:34.580 --> 00:01:37.660
And then we have negative
r1 times x, so negative r1x.

00:01:37.660 --> 00:01:40.080
And then we have negative
r1 times negative r2,

00:01:40.080 --> 00:01:44.285
which is plus r1r2
is equal to 0.

00:01:44.285 --> 00:01:46.410
Then we can simplify this
middle term a little bit.

00:01:46.410 --> 00:01:55.830
It becomes x squared minus
r1 plus r2 x plus r1r2

00:01:55.830 --> 00:01:57.930
is equal to 0.

00:01:57.930 --> 00:02:00.920
So let's think about what
the sum of the roots are.

00:02:00.920 --> 00:02:03.090
So based on what we're
looking at right over here,

00:02:03.090 --> 00:02:06.490
what is r1 plus r2?

00:02:06.490 --> 00:02:08.440
Well, we see it
right over here when

00:02:08.440 --> 00:02:11.460
we multiplied out
these two expressions.

00:02:11.460 --> 00:02:15.150
r1 plus r2 is the negative
of this second coefficient

00:02:15.150 --> 00:02:16.010
right over here.

00:02:16.010 --> 00:02:19.420
Or it's the negative of the
coefficient on the first degree

00:02:19.420 --> 00:02:19.920
term.

00:02:19.920 --> 00:02:23.210
So if we look at the first
degree term over here,

00:02:23.210 --> 00:02:29.530
a1 must be the same thing
as negative r1 plus r2.

00:02:29.530 --> 00:02:31.030
Or another way to
think about it is,

00:02:31.030 --> 00:02:35.980
r1 plus r2 must be
equal to negative a1.

00:02:35.980 --> 00:02:39.360
So that wasn't too bad for
the second degree case.

00:02:39.360 --> 00:02:41.100
Let's try the third degree case.

00:02:41.100 --> 00:02:47.730
Let's see, if we have x to
the third plus a1x squared

00:02:47.730 --> 00:02:53.510
plus a2x, plus a3 is equal to 0.

00:02:53.510 --> 00:02:57.130
Let's think about what the
sum of its roots might be.

00:02:57.130 --> 00:02:59.767
Well, this guy is now going
to have one more root.

00:02:59.767 --> 00:03:01.600
I'm not saying it's the
exact same equation,

00:03:01.600 --> 00:03:03.780
but let's say we're
keeping everything general.

00:03:03.780 --> 00:03:08.190
So now we could say we
have roots r1, r2, and r3.

00:03:08.190 --> 00:03:14.670
Or we could say that x minus
r1 times x minus r2 times x

00:03:14.670 --> 00:03:19.430
minus r3 is equal to 0.

00:03:19.430 --> 00:03:21.240
Now we could multiply
all of this out,

00:03:21.240 --> 00:03:26.880
but we already figured out what
x minus r1 times x minus r2 is.

00:03:26.880 --> 00:03:30.060
It's this business over here.

00:03:30.060 --> 00:03:34.424
So we just have to multiply
this times x minus r3.

00:03:34.424 --> 00:03:36.840
And actually, I'm not even
going to do the full expansion,

00:03:36.840 --> 00:03:40.110
because we have a hunch
here that it always deals,

00:03:40.110 --> 00:03:43.730
it seems, with the
coefficient on the term that's

00:03:43.730 --> 00:03:46.040
one degree lower than the
degree of the polynomial.

00:03:46.040 --> 00:03:48.290
This was a second degree,
we looked at the coefficient

00:03:48.290 --> 00:03:49.500
on the first degree term.

00:03:49.500 --> 00:03:51.800
Maybe to find the
sum, we only have

00:03:51.800 --> 00:03:53.550
to look at this
coefficient over here.

00:03:53.550 --> 00:03:56.510
So I'm only going to figure out
this product up to this point,

00:03:56.510 --> 00:03:59.350
and then we can just
ignore the rest of it

00:03:59.350 --> 00:04:01.960
and see if get something
that is useful.

00:04:01.960 --> 00:04:03.130
So let's do the expansion.

00:04:03.130 --> 00:04:08.540
Let's multiply this right over
here times this over here.

00:04:08.540 --> 00:04:11.360
And what do we get?

00:04:11.360 --> 00:04:13.480
To get the x to the
third term, the only way

00:04:13.480 --> 00:04:18.850
to get that is to multiply
this x times this x squared.

00:04:18.850 --> 00:04:20.500
So we're going to
get x to the third.

00:04:23.370 --> 00:04:25.910
And that's the only way to
get the x to the third term.

00:04:25.910 --> 00:04:28.100
That's the only way to
get the third degree term.

00:04:28.100 --> 00:04:29.740
Now how do we get
this term over here?

00:04:29.740 --> 00:04:31.573
How do we get-- let me
do this a new color--

00:04:31.573 --> 00:04:34.210
how do we get the x
squared term over here?

00:04:34.210 --> 00:04:37.390
Well, we could multiply
this x times this

00:04:37.390 --> 00:04:42.400
over here, because
the x times the x

00:04:42.400 --> 00:04:44.250
is going to give us x squared.

00:04:44.250 --> 00:04:50.929
So it'll be negative r1 plus
r2 x squared times that term.

00:04:50.929 --> 00:04:52.720
And then if you multiply
this x times that,

00:04:52.720 --> 00:04:54.750
you'll get r1r2 x,
and all of that.

00:04:54.750 --> 00:04:57.460
But I'll just write,
so on and so forth.

00:04:57.460 --> 00:05:00.240
And then, what other ways can
we get an x squared term here?

00:05:00.240 --> 00:05:02.240
There's no other way to
get x to the third term,

00:05:02.240 --> 00:05:04.580
but what other ways could
we get an x squared term?

00:05:04.580 --> 00:05:07.090
Well, when we multiply
this times this,

00:05:07.090 --> 00:05:09.420
you have the negative r3.

00:05:09.420 --> 00:05:12.850
Let me do this in another color.

00:05:12.850 --> 00:05:15.177
You have the negative
r3 three times this.

00:05:15.177 --> 00:05:16.760
Your obviously going
to take this term

00:05:16.760 --> 00:05:17.910
multiplied by everything.

00:05:17.910 --> 00:05:20.160
And then add it to this term
multiplied by everything.

00:05:20.160 --> 00:05:21.960
But we only care
about the things--

00:05:21.960 --> 00:05:24.650
I did this just to show you that
it's there-- but I only care,

00:05:24.650 --> 00:05:26.722
because we have a
hunch that what matters

00:05:26.722 --> 00:05:28.430
is the coefficient on
the x squared term.

00:05:28.430 --> 00:05:29.846
So we just want
to see, how do you

00:05:29.846 --> 00:05:33.550
build the x squared
term in our, I guess,

00:05:33.550 --> 00:05:36.325
our expanded polynomial?

00:05:36.325 --> 00:05:37.700
The other way to
get an x squared

00:05:37.700 --> 00:05:40.920
is to multiply the negative r3
times this x squared over here.

00:05:40.920 --> 00:05:43.959
So this is going to be
negative r3 times x squared.

00:05:43.959 --> 00:05:45.750
And then when you
multiply this times this,

00:05:45.750 --> 00:05:47.160
you're going to
get something else.

00:05:47.160 --> 00:05:48.360
Then when you multiply
this times this,

00:05:48.360 --> 00:05:49.960
you're just going to get
the negative product of all

00:05:49.960 --> 00:05:51.020
the roots.

00:05:51.020 --> 00:05:52.680
So that's just something else.

00:05:52.680 --> 00:05:56.390
And so when we add
everything together

00:05:56.390 --> 00:05:59.057
you get x to the third.

00:05:59.057 --> 00:06:00.890
And then what's the sum
of these two things?

00:06:00.890 --> 00:06:07.185
It's negative r1 plus
r2 plus r3 x squared.

00:06:07.185 --> 00:06:09.560
And then you're just going to
have a bunch of other stuff

00:06:09.560 --> 00:06:11.270
that we didn't take
the time to add.

00:06:11.270 --> 00:06:13.000
But it looks like
our hunch paid off,

00:06:13.000 --> 00:06:15.700
because what is now
the sum of the roots?

00:06:15.700 --> 00:06:19.074
r1 plus r2 plus r3.

00:06:19.074 --> 00:06:19.990
What is that equal to?

00:06:19.990 --> 00:06:22.560
Well, that's sitting
right over here.

00:06:22.560 --> 00:06:25.101
That's equal to the negative
of this coefficient right

00:06:25.101 --> 00:06:25.600
over here.

00:06:25.600 --> 00:06:27.450
That's once again
equal to negative a1.

00:06:27.450 --> 00:06:32.230
It's equal to the negative of
the coefficient on the degree

00:06:32.230 --> 00:06:37.144
term that's one less than
the degree of our polynomial.

00:06:37.144 --> 00:06:38.310
So we already see a pattern.

00:06:38.310 --> 00:06:41.610
Let's say if we can
prove it generally.

00:06:41.610 --> 00:06:44.180
So we've already proven,
essentially, two base cases.

00:06:44.180 --> 00:06:47.620
We proven it for a degree
of two, a degree of three.

00:06:47.620 --> 00:06:50.090
Let's assume that it's
true for degree of n,

00:06:50.090 --> 00:06:52.159
and then we can prove it
for degree of n plus 1.

00:06:52.159 --> 00:06:53.700
And essentially,
this will be a proof

00:06:53.700 --> 00:06:58.520
by induction for
polynomials of any degree.

00:06:58.520 --> 00:07:02.090
So let's just say we have
an nth degree polynomial.

00:07:02.090 --> 00:07:04.280
So let's just assume this.

00:07:04.280 --> 00:07:05.760
So we're going to
assume this step,

00:07:05.760 --> 00:07:08.960
and this gives a little proof
by mathematical induction

00:07:08.960 --> 00:07:09.970
practice.

00:07:09.970 --> 00:07:13.070
So let's just assume that we
have a polynomial x to the n--

00:07:13.070 --> 00:07:17.780
it's nth degrees-- so x to the
n, plus a1x to the n minus 1,

00:07:17.780 --> 00:07:20.150
and it just goes on and
on and on all the way down

00:07:20.150 --> 00:07:21.730
to the 0th degree term.

00:07:24.240 --> 00:07:31.220
It has roots, obviously, r1,
r2, all the way to r sub n.

00:07:31.220 --> 00:07:39.160
And we're going to assume
that r1 plus r2 plus all

00:07:39.160 --> 00:07:42.180
the way to r sub n is
equal to negative a1.

00:07:44.900 --> 00:07:47.260
Or another way to
think about it,

00:07:47.260 --> 00:07:54.260
if we were to multiply x
minus r1 times x minus r2,

00:07:54.260 --> 00:07:57.570
and just keep multiplying all
the way to x minus r sub n

00:07:57.570 --> 00:08:03.760
is equal to 0, this should
give us x to the n minus r1,

00:08:03.760 --> 00:08:10.960
plus r2, plus all the way
to rn x to the n minus 1,

00:08:10.960 --> 00:08:12.540
and then plus a
bunch of other stuff

00:08:12.540 --> 00:08:13.956
that we're not
going to calculate.

00:08:13.956 --> 00:08:15.850
So this tells us this.

00:08:15.850 --> 00:08:18.680
That this coefficient
right over here

00:08:18.680 --> 00:08:21.520
is going to be equal to
this right over here.

00:08:21.520 --> 00:08:24.190
So that is what we
are going to assume.

00:08:24.190 --> 00:08:28.630
So now let's think
about-- and we're not

00:08:28.630 --> 00:08:35.880
assuming this part-- given that,
let's think about the situation

00:08:35.880 --> 00:08:39.190
where we have a polynomial
x to the n plus 1,

00:08:39.190 --> 00:08:43.350
plus a1x to the n, plus--
and it just keeps going.

00:08:43.350 --> 00:08:45.210
You get all the
degree terms there.

00:08:45.210 --> 00:08:47.060
So now this is going
to have the roots

00:08:47.060 --> 00:08:51.260
r1, r2, all the way to rn.

00:08:51.260 --> 00:08:54.340
And then it'll have rn plus 1.

00:08:54.340 --> 00:08:57.810
It'll have these n roots, and
then it'll have an rn plus 1.

00:08:57.810 --> 00:09:01.650
So essentially, that tells
us that we're going to have,

00:09:01.650 --> 00:09:06.287
essentially, this product
times x minus rn plus 1

00:09:06.287 --> 00:09:06.870
is equal to 0.

00:09:06.870 --> 00:09:07.440
Let me write it out.

00:09:07.440 --> 00:09:08.460
I don't want to skip steps.

00:09:08.460 --> 00:09:10.360
So this tells us, if
these are all the roots,

00:09:10.360 --> 00:09:15.770
that x minus r1 times x
minus r2, blah, blah, blah,

00:09:15.770 --> 00:09:19.000
all the way to x
minus rn times x

00:09:19.000 --> 00:09:23.180
minus rn plus 1 is
going to be equal to 0.

00:09:23.180 --> 00:09:26.050
It has one more root than
the previous example,

00:09:26.050 --> 00:09:28.080
than the one where we
made the assumption.

00:09:28.080 --> 00:09:31.430
So if we were to expand this
thing out, how can we do that?

00:09:31.430 --> 00:09:34.980
Well, this thing, if we
were to multiply it out,

00:09:34.980 --> 00:09:38.620
it's just this thing over
here times this new binomial,

00:09:38.620 --> 00:09:41.570
because this thing
over here is--

00:09:41.570 --> 00:09:44.690
let me set it up, let me use
color coded-- this thing,

00:09:44.690 --> 00:09:45.960
let me do a different color.

00:09:45.960 --> 00:09:48.480
This thing over here
we already established.

00:09:48.480 --> 00:09:50.860
We assumed is this
thing over here,

00:09:50.860 --> 00:09:54.010
which is this thing over here.

00:09:54.010 --> 00:09:55.860
So if we want this
entire expansion,

00:09:55.860 --> 00:10:00.720
we just have to multiply
this times this.

00:10:00.720 --> 00:10:03.470
So how can we get an x
to the n plus 1 term?

00:10:03.470 --> 00:10:06.350
How can we get to the
x to the n plus 1 term?

00:10:06.350 --> 00:10:10.440
Well, there's only one way to
get the x to the n plus 1 term.

00:10:10.440 --> 00:10:15.000
And that's by multiplying
this x times this x to the n.

00:10:15.000 --> 00:10:20.520
So that will give us
x to the n plus 1.

00:10:20.520 --> 00:10:21.750
That's the only way to do it.

00:10:21.750 --> 00:10:23.800
That's the only way this
highest degree term.

00:10:23.800 --> 00:10:27.100
Now how can we get--
let me do it in purple--

00:10:27.100 --> 00:10:29.275
how can we get at
the x to the n term?

00:10:32.210 --> 00:10:35.700
Well, we can multiply this
x times this second term

00:10:35.700 --> 00:10:36.260
over here.

00:10:36.260 --> 00:10:39.350
So if you multiply this x
times this business over here,

00:10:39.350 --> 00:10:44.050
you're going to get negative
r1 plus r2, plus all the way

00:10:44.050 --> 00:10:46.020
to rn.

00:10:46.020 --> 00:10:48.667
x times x to the n minus 1, is
just going to be x to the n.

00:10:48.667 --> 00:10:50.500
And then obviously,
you're going to multiply

00:10:50.500 --> 00:10:51.930
this x times all
the other terms,

00:10:51.930 --> 00:10:54.055
so you're just going to
get a bunch of other stuff.

00:10:54.055 --> 00:10:56.320
So that's how we can get
these two terms using the x.

00:10:56.320 --> 00:10:58.069
And then can we get
either of these degree

00:10:58.069 --> 00:11:01.240
terms using the r sub n plus 1?

00:11:01.240 --> 00:11:05.690
Well, we can multiply
r sub n plus 1,

00:11:05.690 --> 00:11:09.550
or negative r sub n
plus 1 times x to the n.

00:11:09.550 --> 00:11:11.660
And then we'll also get
a similar degree term.

00:11:11.660 --> 00:11:16.569
So we'll have negative r sub
n plus 1 times x to the n.

00:11:16.569 --> 00:11:18.360
And then obviously,
we're going to multiply

00:11:18.360 --> 00:11:20.920
this times all of the other
characters in this polynomial.

00:11:20.920 --> 00:11:23.820
So you're going to have all
this plus so on and so forth.

00:11:23.820 --> 00:11:25.860
But this will suit our
purposes, because when

00:11:25.860 --> 00:11:28.516
you add these two
things, what do we get?

00:11:28.516 --> 00:11:33.510
We get x to the n
plus 1, and then we

00:11:33.510 --> 00:11:36.640
get minus-- well,
this and this are

00:11:36.640 --> 00:11:39.890
the coefficients on
the x to the n term.

00:11:39.890 --> 00:11:45.060
So this is going to be minus
r1 plus r2 plus all the way

00:11:45.060 --> 00:11:47.160
to plus rn.

00:11:47.160 --> 00:11:49.240
And then this is
minus rn plus 1.

00:11:49.240 --> 00:11:50.920
We have the minus out front.

00:11:50.920 --> 00:11:55.050
So this is plus r sub
n plus 1 x to the n.

00:11:55.050 --> 00:11:58.810
And then we're going to have
a bunch of other lower degree

00:11:58.810 --> 00:12:00.200
terms that don't matter.

00:12:00.200 --> 00:12:02.490
But we've just proven our case.

00:12:02.490 --> 00:12:09.360
What is r1 plus r2 plus all
the way to plus r sub n plus 1?

00:12:09.360 --> 00:12:10.710
What is this equal to?

00:12:10.710 --> 00:12:12.900
Well, in our expansion we
have it right over here.

00:12:12.900 --> 00:12:14.350
That's equal to that over there.

00:12:14.350 --> 00:12:19.170
That is equal to the
negative of this coefficient.

00:12:19.170 --> 00:12:23.080
It's equal to the
negative of a1.

00:12:23.080 --> 00:12:25.640
Now the reason why this
works out no matter what--

00:12:25.640 --> 00:12:27.470
you say, hey, if I
have complex roots,

00:12:27.470 --> 00:12:29.710
how does it always end
up being a real number?

00:12:29.710 --> 00:12:32.530
And that's because the
imaginary parts cancel out

00:12:32.530 --> 00:12:34.172
when you take their sum.

00:12:34.172 --> 00:12:35.130
So let's just apply it.

00:12:35.130 --> 00:12:36.140
We've proven it.

00:12:36.140 --> 00:12:37.770
We've proven, actually,
two base cases.

00:12:37.770 --> 00:12:40.900
We've proven it for a
degree 2, for degree 3.

00:12:40.900 --> 00:12:43.960
And then we showed that if
it's true for any degree n,

00:12:43.960 --> 00:12:46.800
then it's definitely true
for any degree n plus 1.

00:12:46.800 --> 00:12:49.150
So we know if it's
true for degree 3,

00:12:49.150 --> 00:12:51.600
then it's definitely
true for degree 4, which

00:12:51.600 --> 00:12:54.000
means it's definitely
true for degree 5.

00:12:54.000 --> 00:12:57.150
So A mathematical induction
is kind of this domino proof

00:12:57.150 --> 00:12:59.760
that if it's true for 3,
which makes it true for 4,

00:12:59.760 --> 00:13:02.810
then assuming it's true
for 4 makes it true for 5.

00:13:02.810 --> 00:13:05.620
And it just keeps
going to any n.

00:13:05.620 --> 00:13:07.300
So in general, if
someone gives you

00:13:07.300 --> 00:13:10.540
a polynomial-- let me think
of some crazy polynomial.

00:13:10.540 --> 00:13:18.020
If I were to give you x
to the seventh, minus pi x

00:13:18.020 --> 00:13:25.340
to the sixth plus ex to the
fifth, minus square root of 2 x

00:13:25.340 --> 00:13:28.410
to the fourth, minus-- I
won't write out all the terms.

00:13:28.410 --> 00:13:30.210
So minus 3 is equal to zero.

00:13:30.210 --> 00:13:32.970
If someone were to give you this
polynomial, and they were say,

00:13:32.970 --> 00:13:37.470
what do the seven roots of
this polynomial add up to?

00:13:37.470 --> 00:13:39.780
You say, oh, well,
the sum of these roots

00:13:39.780 --> 00:13:42.040
is going to be the negative
of this coefficient right

00:13:42.040 --> 00:13:42.740
over here.

00:13:42.740 --> 00:13:47.667
So r1 plus r2 plus all the
way to r7, all seven roots

00:13:47.667 --> 00:13:50.000
are going to add up to the
negative of this coefficient.

00:13:50.000 --> 00:13:52.314
They're going to be equal to pi.

00:13:52.314 --> 00:13:54.480
And one other thing, you
might be saying, hey, wait.

00:13:54.480 --> 00:13:56.431
Every polynomial you
did had a 1 coefficient

00:13:56.431 --> 00:13:57.930
in front of the
highest degree term.

00:13:57.930 --> 00:13:59.700
You know, not all
polynomials have that.

00:13:59.700 --> 00:14:01.200
And the answer there
is-- let's say,

00:14:01.200 --> 00:14:04.120
if I were to give you a
polynomial, I don't know,

00:14:04.120 --> 00:14:11.970
7x to the fifth, minus 6x to
the fourth, plus, I don't know,

00:14:11.970 --> 00:14:15.137
pi x to the third, plus a
bunch of stuff other stuff

00:14:15.137 --> 00:14:16.720
is equal to 0-- and
you wanted to find

00:14:16.720 --> 00:14:19.060
the sum of this
polynomials roots.

00:14:19.060 --> 00:14:21.210
You'd first divide
everything by 7.

00:14:21.210 --> 00:14:24.200
So it becomes x to
the fifth, minus 6

00:14:24.200 --> 00:14:29.200
over 7x to the fourth, plus
pi over 7x to the third,

00:14:29.200 --> 00:14:33.560
plus so on and so forth, is
equal to 0/7 which is just 0.

00:14:33.560 --> 00:14:35.060
And now you look
at this polynomial,

00:14:35.060 --> 00:14:38.460
well the sum is going to be the
negative of this coefficient

00:14:38.460 --> 00:14:38.960
here.

00:14:38.960 --> 00:14:41.490
So it's going to be the
sum of all five roots.

00:14:41.490 --> 00:14:44.390
Or it's going to be 6/7.

00:14:44.390 --> 00:14:46.850
Anyway, hopefully
you enjoyed that.