Horizontal and vertical asymptotes of function

WEBVTT
Kind: captions
Language: en

00:00:00.030 --> 00:00:04.859
right over here I have the graph of f of
X and what I want to think about in this

00:00:04.859 --> 00:00:10.349
video is whether we could have sketched
this graph just by looking at the

00:00:10.349 --> 00:00:15.120
definition of our function which is
defined as a rational as a rational

00:00:15.120 --> 00:00:22.050
expression we have 2x plus 10 over 5x
minus 15 so there's a couple of ways to

00:00:22.050 --> 00:00:25.439
do this first you might just want to
pick out any numbers that are really

00:00:25.439 --> 00:00:29.670
easy to calculate so for example what
happens when X is equal to zero so we

00:00:29.670 --> 00:00:34.920
could say F of 0 F of 0 is going to be
equal to well all the X terms are going

00:00:34.920 --> 00:00:39.300
to be 0 so you're going to be left with
10 over negative 15 which is negative 10

00:00:39.300 --> 00:00:46.320
15 which is negative 2/3 so you could
plot that when x equals 0 f of x or y

00:00:46.320 --> 00:00:51.750
equals f of X is negative 2/3 and we see
that point let me do that in a darker

00:00:51.750 --> 00:00:54.750
color you see that point right over
there so we could have plotted that

00:00:54.750 --> 00:00:59.100
point we could have also said when does
this function equal zero well the

00:00:59.100 --> 00:01:02.699
function is equal to zero when the only
way to get the function equal to zero is

00:01:02.699 --> 00:01:07.950
if you get this numerator equal to zero
so you could try to solve 2x plus 10 is

00:01:07.950 --> 00:01:12.090
equal to zero well that's going to
happen when 2x is equal to negative 10 I

00:01:12.090 --> 00:01:16.380
just subtracted 10 from both sides and
if I divide both sides by 2 that's going

00:01:16.380 --> 00:01:22.439
to happen when X is equal to negative 5
and you see that you see this right over

00:01:22.439 --> 00:01:28.259
here when X is equal to negative 5 the
function intersects the x-axis so that's

00:01:28.259 --> 00:01:31.310
just two points but that still doesn't
give us enough to really form this

00:01:31.310 --> 00:01:34.590
interesting shape over here and you
could think about what other functions

00:01:34.590 --> 00:01:38.759
have this type of this type of shape so
now what I want to think about is the

00:01:38.759 --> 00:01:44.180
behavior of the function at different
points so first I want to think about

00:01:44.180 --> 00:01:49.770
when this function is undefined and what
type of behavior we might expect for

00:01:49.770 --> 00:01:53.369
that function when it's undefined so
this function is going to be undefined

00:01:53.369 --> 00:01:56.790
the only way I can think of to make this
undefined is if I make the denominator

00:01:56.790 --> 00:01:59.369
equal to zero we don't know what it
means to divide by zero that is

00:01:59.369 --> 00:02:04.890
undefined so this the function is going
to be undefined when five x this in blue

00:02:04.890 --> 00:02:11.250
when five X minus 15 is equal to zero or
adding 15 to both sides when five X is

00:02:11.250 --> 00:02:13.490
equal to 15 or dividing both sides by
five

00:02:13.490 --> 00:02:21.680
five when X is equal to three f is
undefined now there's a couple of ways

00:02:21.680 --> 00:02:26.060
for a function to be undefined at a
point you could have something like this

00:02:26.060 --> 00:02:30.800
so let's let me draw some axes right
over here so let's say that this is

00:02:30.800 --> 00:02:34.880
three you could have your function it
could look like this it could be defined

00:02:34.880 --> 00:02:39.680
it might kind of approach something but
just not be defined right at three and

00:02:39.680 --> 00:02:44.780
just keep on going like that or the
other possibility is it might have a

00:02:44.780 --> 00:02:49.580
vertical asymptote there and if it has a
vertical asymptote it's going to look

00:02:49.580 --> 00:02:54.410
something like this it might it might be
approach it might just pop up to

00:02:54.410 --> 00:02:59.180
infinity and then it might pop down from
infinity on this side or it might go

00:02:59.180 --> 00:03:03.890
from negative infinity right over here
that says that's what a vertical

00:03:03.890 --> 00:03:07.850
asymptote would look like that as we
approach from the left the graph is

00:03:07.850 --> 00:03:12.770
approaching a vertical but it never it
never quite gets two x equals three I

00:03:12.770 --> 00:03:15.350
guess one we could say it or the
function is defined not defined at x

00:03:15.350 --> 00:03:18.710
equals three and as you approach from
the right the same thing the function

00:03:18.710 --> 00:03:23.270
just in this case drops down it almost
becomes vertical its approaching

00:03:23.270 --> 00:03:28.160
negative infinity as X approaches three
from the positive direction so how would

00:03:28.160 --> 00:03:31.760
we know obviously when you look at here
when you look we know the graph ahead of

00:03:31.760 --> 00:03:35.870
time and if you say okay this is e this
is x equals ten so let's see how many

00:03:35.870 --> 00:03:40.880
this is one two three four five so each
of these are two so x equals three is

00:03:40.880 --> 00:03:43.400
right over here
when you look at the graph if you had

00:03:43.400 --> 00:03:47.920
the graph in front of you you'd see oh
look this is indeed a vertical asymptote

00:03:47.920 --> 00:03:52.640
just looking at the graph you see that
you have a vertical asymptote at x

00:03:52.640 --> 00:04:00.880
equals three and let me write that down
vertical asymptotes vertical asymptotes

00:04:00.880 --> 00:04:06.880
asymptote at x equals three at x equals
three but how would you have known that

00:04:06.880 --> 00:04:11.810
how would you have known that if you
didn't have the graph here if you just

00:04:11.810 --> 00:04:14.510
had this we know it's not defined at
three but how do we know it's not a

00:04:14.510 --> 00:04:20.330
point discontinuity and not a instead of
a vertical asymptote well there's a

00:04:20.330 --> 00:04:25.790
couple of ways to do it one ways you
could try values near three and see what

00:04:25.790 --> 00:04:27.620
happens
so for example you could get your

00:04:27.620 --> 00:04:34.430
calculator out and you could try let's
say three point zero one so if you say

00:04:34.430 --> 00:04:41.930
two times three point zero one plus ten
X G well that's the numerator and then

00:04:41.930 --> 00:04:48.190
I'm going to divide that the numerator
by five times three point zero one minus

00:04:48.190 --> 00:04:55.040
fifteen it gets us a fairly large number
it's kind of exploding on us and if

00:04:55.040 --> 00:05:03.500
we've got even closer so if we did two
times three point zero zero one plus let

00:05:03.500 --> 00:05:12.350
me plus ten plus ten divided by five
times three point zero zero one so now

00:05:12.350 --> 00:05:19.700
I'm just now I'm trying X is equal to
three point zero zero one minus 15 minus

00:05:19.700 --> 00:05:25.760
15 we see we get even a larger number so
as X gets closer and closer to three f

00:05:25.760 --> 00:05:30.440
of X seems to be exploding it seems to
be approaching positive infinity so

00:05:30.440 --> 00:05:34.250
that's one way to say okay this looks
like at least from this side we are

00:05:34.250 --> 00:05:37.820
approaching we are approaching positive
infinity so we would have been able to

00:05:37.820 --> 00:05:41.330
draw we would have been able to draw
something like that and then you could

00:05:41.330 --> 00:05:44.990
have tried values below you could have
tried values below so you could have

00:05:44.990 --> 00:05:50.810
said I actually let me just let me just
put the last entry here and let me just

00:05:50.810 --> 00:05:58.120
change the 3.00 ones to two point nine
nine nine two point nine nine nine and

00:05:58.120 --> 00:06:06.500
whoops let me go over here we have two
point nine nine nine and we get notice

00:06:06.500 --> 00:06:10.010
we're going really negative now we're
approaching negative infinity so if you

00:06:10.010 --> 00:06:13.520
just try that out that would give you a
pretty good indication that the graph is

00:06:13.520 --> 00:06:16.730
looking something like that right over
here which also seems to match

00:06:16.730 --> 00:06:20.690
connecting these two points that we've
already thought about but now let's see

00:06:20.690 --> 00:06:24.740
what's going on as X approaches really
large values or really negative really

00:06:24.740 --> 00:06:28.930
positive values are really negative
values it looks like there is a

00:06:28.930 --> 00:06:33.380
horizontal asymptote here just looking
at the graph just looking at the graph

00:06:33.380 --> 00:06:37.340
it seems like there's some value that as
X approaches really large values really

00:06:37.340 --> 00:06:39.980
positive values f of X is going to be
approaching

00:06:39.980 --> 00:06:44.240
that value that asymptotes from above
and as X becomes really negative it

00:06:44.240 --> 00:06:47.930
looks like f of X is approaching that
from below but how would we be able to

00:06:47.930 --> 00:06:52.310
figure that out just by looking at this
well one thought experiment is just to

00:06:52.310 --> 00:06:56.600
say well what happens to f of X as X
approaches infinity so let me write that

00:06:56.600 --> 00:07:06.680
down as X approaches infinity then then
f of X is going to approach what well as

00:07:06.680 --> 00:07:11.210
X approaches larger and larger values
the positive 10 and the negative 15

00:07:11.210 --> 00:07:14.450
start to matter a lot less that the
highest degree term in the numerator

00:07:14.450 --> 00:07:19.460
denominator start to dominate so we
could say as X approaches infinity f of

00:07:19.460 --> 00:07:28.790
X is getting closer and closer to 2 x
over 5 X which is which is 2/5 which is

00:07:28.790 --> 00:07:34.040
equal to 2/5 so you could say f of X is
approaching 2/5 and if you really want

00:07:34.040 --> 00:07:39.530
to kind of see that a little bit more a
little bit more concretely let's imagine

00:07:39.530 --> 00:07:48.140
different values for X as X gets larger
and larger and larger so if we have so X

00:07:48.140 --> 00:07:57.740
f of X if X is 1 that f of X is just
going to be 2 plus 10 over 5 minus 15 so

00:07:57.740 --> 00:08:01.760
here the 10 and the negative and the
subtracting the 15 matter a lot but if X

00:08:01.760 --> 00:08:10.160
was if X were a thousand then f of X
would be 2,000 plus 10 over 5,000 minus

00:08:10.160 --> 00:08:15.650
15 so now the 2,000 and 5,000 are really
kind of setting the agenda and then if X

00:08:15.650 --> 00:08:22.640
were if X were let's say 1 million 1
million and I just used blue for a more

00:08:22.640 --> 00:08:28.940
contrast then f of X would be 2 million
2 million plus 10 we move over to the

00:08:28.940 --> 00:08:36.320
right a little bit 2 million plus 10
over 5 million 5 million minus 15 so

00:08:36.320 --> 00:08:39.860
here the 10 and the 15 are almost
inconsequential you could imagine if X

00:08:39.860 --> 00:08:44.780
were a billion or trillion or googol
then these then the 10 and the negative

00:08:44.780 --> 00:08:50.060
15 starting to matter a lot and lot less
so as X approaches infinity these matter

00:08:50.060 --> 00:08:55.560
less the higher degree terms the
degree terms matter so f of X is going

00:08:55.560 --> 00:09:01.560
to approach 2x over 5x which is 2/5 so f
of X is approaching 2/5 and that's what

00:09:01.560 --> 00:09:06.240
this line looks like 2/5 is the same
thing as 0.4 so f of X and we see that

00:09:06.240 --> 00:09:10.890
in the graph f of X is approaching that
but it's not quite getting close to it

00:09:10.890 --> 00:09:14.250
it's it was it's not quite getting there
it's getting closer and closer to it as

00:09:14.250 --> 00:09:17.160
X goes to infinity but it's not quite
getting there because you're always

00:09:17.160 --> 00:09:21.120
going to have that plus 10 and that
minus 15 there's you're never going to

00:09:21.120 --> 00:09:26.430
be exactly at 2/5 and the same thing is
happening as X gets more and more and

00:09:26.430 --> 00:09:31.860
more negative you can make all of these
negative values so if if this were

00:09:31.860 --> 00:09:35.850
negative 1 then it would be negative 2
negative 5 this was negative 1000 would

00:09:35.850 --> 00:09:39.000
be negative 2000 over negative 5000 if
there were negative 1 million it would

00:09:39.000 --> 00:09:44.070
be negative 2 million over negative 5
million but you see even in this case f

00:09:44.070 --> 00:09:49.590
of X is approaching 2x over 5x which is
approaching 2/5 or you could say it's

00:09:49.590 --> 00:09:54.690
approaching negative 2 over negative 5
which is still 2/5 and you see that

00:09:54.690 --> 00:10:00.260
right over here so we would say that
this function has a horizontal asymptote

00:10:00.260 --> 00:10:10.800
horizontal horizontal asymptote
asymptote at y equals the horizontal

00:10:10.800 --> 00:10:16.110
line right over here is y is equal to
2/5 so hopefully this graph here is

00:10:16.110 --> 00:10:19.320
helping us appreciate what these
vertical and horizontal asymptotes

00:10:19.320 --> 00:10:23.100
actually are but if we didn't have the
graph we could have said okay we're

00:10:23.100 --> 00:10:26.910
undefined at x equals 3 we could test
some values around it and so we could

00:10:26.910 --> 00:10:30.240
say okay look it does look like we're
approaching negative infinity as X

00:10:30.240 --> 00:10:33.930
approaches negative 3 from the left it
looks like we're approaching positive

00:10:33.930 --> 00:10:38.460
infinity as X approaches negative 3 from
the right so we could we could put we

00:10:38.460 --> 00:10:41.940
could draw that kind of blue point right
there we could graph these two points

00:10:41.940 --> 00:10:45.510
when does F equal 0
what happens to F when x equals 0 and

00:10:45.510 --> 00:10:49.320
then we could think about the behavior
as X approaches infinity or negative

00:10:49.320 --> 00:10:54.270
infinity as X approaches negative as X
approaches infinity or negative infinity

00:10:54.270 --> 00:10:58.650
and draw this horizontal asymptote and
between all of those that would have

00:10:58.650 --> 00:11:04.100
been a pretty good way to be able to
sketch this actual graph


Matching rational functions to their graphs

WEBVTT
Kind: captions
Language: en

00:00:00.807 --> 00:00:02.830
Voiceover:So, we've got
three functions graphed here.

00:00:02.830 --> 00:00:04.452
The function f in magenta.

00:00:04.452 --> 00:00:07.358
We have the function
g in this green color.

00:00:07.358 --> 00:00:11.080
And we have the function h
in this dotted purple color.

00:00:11.080 --> 00:00:13.460
And then we have three
potential expressions,

00:00:13.460 --> 00:00:14.529
or three expressions that could be

00:00:14.529 --> 00:00:17.362
potential definitions, for f, g, and h.

00:00:17.362 --> 00:00:19.354
And what I want to do in this
video, is try to match them,

00:00:19.354 --> 00:00:22.863
try to match the function to a definition.

00:00:22.863 --> 00:00:25.795
I encourage you to pause this video

00:00:25.795 --> 00:00:26.997
and try to think about it on your own

00:00:26.997 --> 00:00:29.247
before I work through it.

00:00:29.879 --> 00:00:32.737
There's a couple of ways
that we could approach this.

00:00:32.737 --> 00:00:34.391
One, we could think about what the graphs

00:00:34.391 --> 00:00:36.229
of each of these look like,
and then think about which

00:00:36.229 --> 00:00:38.790
of these function's graphs look like that.

00:00:38.790 --> 00:00:40.197
Or, we could we could
look at the function graph

00:00:40.197 --> 00:00:42.830
and think about the vertical
and horizontal asymptotes,

00:00:42.830 --> 00:00:44.279
and think about which of
these expressions would have

00:00:44.279 --> 00:00:47.529
a vertical or horizontal
asymptotes at that point.

00:00:47.529 --> 00:00:48.811
So, I'm actually going
to do it the second way.

00:00:48.811 --> 00:00:50.496
Let's look at the graphs.

00:00:50.496 --> 00:00:52.311
I tend to be a little bit more visual,

00:00:52.311 --> 00:00:53.947
or I like to look at graphs.

00:00:53.947 --> 00:00:56.080
So it looks like - f, I'll start with f.

00:00:56.080 --> 00:01:00.860
f looks like it has a vertical
asymptote at x equals 5.

00:01:00.860 --> 00:01:03.294
Or, sorry, at x equals negative 5.

00:01:03.294 --> 00:01:05.461
We have a vertical
asymptote right over here,

00:01:05.461 --> 00:01:09.244
at x is equal to negative five.

00:01:09.244 --> 00:01:10.511
Let's think about which of these

00:01:10.511 --> 00:01:12.610
would have a vertical asymptote
at x equals negative 5.

00:01:12.610 --> 00:01:14.344
In order to have a vertical asymptote,

00:01:14.344 --> 00:01:16.636
it can't be defined there.

00:01:16.636 --> 00:01:18.161
So that'sthe first way
I would think about it.

00:01:18.161 --> 00:01:19.442
And then even if isn't defined there,

00:01:19.442 --> 00:01:22.523
we need to make sure that it's
an actual vertical asymptote,

00:01:22.523 --> 00:01:24.624
and not just a hole at that point,

00:01:24.624 --> 00:01:26.539
not just a point discontinuity.

00:01:26.539 --> 00:01:28.825
So, let's think about it.

00:01:28.825 --> 00:01:30.948
So, this first expression
is actually defined

00:01:30.948 --> 00:01:32.098
at x equals negative 5.

00:01:32.098 --> 00:01:33.965
The only reason why it
wouldn't be defined,

00:01:33.965 --> 00:01:35.781
if somehow we got a 0 in the denominator.

00:01:35.781 --> 00:01:37.195
But if you have negative 5 minus 5,

00:01:37.195 --> 00:01:38.244
that's negative 10.

00:01:38.244 --> 00:01:40.548
So this one's defined at
that point, so that's not f.

00:01:40.548 --> 00:01:42.937
This one's also defined
at x equals negative 5.

00:01:42.937 --> 00:01:45.753
The denominator does not
become 0, so that's not f.

00:01:45.753 --> 00:01:48.186
This one, when x is equal to negative 5,

00:01:48.186 --> 00:01:50.753
this denominator does become 0.

00:01:50.753 --> 00:01:54.007
So this seems, I just used
purely deductive reasoning,

00:01:54.007 --> 00:01:57.840
this looks like my best
candidate for f of x.

00:01:57.840 --> 00:01:59.122
But let's confirm that it's consistent

00:01:59.122 --> 00:02:02.023
with the other things that
we're seeing over here.

00:02:02.023 --> 00:02:05.561
So, let's look at f's
horizontal asymptote.

00:02:05.561 --> 00:02:06.425
If I look at the graph,

00:02:06.425 --> 00:02:08.211
it looks like there's
a horizontal asymptote.

00:02:08.211 --> 00:02:10.978
Especially as x gets to
larger and larger values,

00:02:10.978 --> 00:02:13.679
it looks like f of x is approaching 1.

00:02:13.679 --> 00:02:16.272
f of x is approaching 1.

00:02:16.272 --> 00:02:19.311
Now, is that the same case over here?

00:02:19.311 --> 00:02:22.077
As x gets larger and larger and larger,

00:02:22.077 --> 00:02:24.246
as x approaches infinity,

00:02:24.246 --> 00:02:26.545
well, then the negative 2
and the plus 5 don't matter.

00:02:26.545 --> 00:02:29.250
As x approaches infinity,
this is going to approximate,

00:02:29.250 --> 00:02:32.500
for very large x's, it's
going to be x over x.

00:02:32.500 --> 00:02:34.700
We look at the highest degree terms,

00:02:34.700 --> 00:02:36.583
and so that is going to approach 1,

00:02:36.583 --> 00:02:38.323
as x gets very very very large.

00:02:38.323 --> 00:02:41.066
Then the negative two,
subtracting the 2 in the numerator

00:02:41.066 --> 00:02:42.250
and adding 5 in the denominator

00:02:42.250 --> 00:02:43.466
are going to matter
less and less and less,

00:02:43.466 --> 00:02:45.368
because the x gets so big.

00:02:45.368 --> 00:02:46.634
So, this will approach 1.

00:02:46.634 --> 00:02:48.867
So, that seems consistent
from that point of view.

00:02:48.867 --> 00:02:51.283
And let's see, is there
anything else interesting?

00:02:51.283 --> 00:02:53.033
Well, when does this equal 0?

00:02:53.033 --> 00:02:56.214
Well, the numerator equals
0 when x is equal to 2.

00:02:56.214 --> 00:02:58.932
And we see that's indeed the
case for f right over there.

00:02:58.932 --> 00:03:02.200
So, I'm feeling really good
that this is our f of x.

00:03:02.200 --> 00:03:04.518
Now, let's look at g of x.

00:03:04.518 --> 00:03:07.885
g of x has, and actually they're
trying to trip us up here,

00:03:07.885 --> 00:03:10.900
because it looks like both h and g of x,

00:03:10.900 --> 00:03:16.067
have a vertical asymptote
at x is equal to 5.

00:03:16.067 --> 00:03:17.384
So the vertical asymptote isn't

00:03:17.384 --> 00:03:21.768
what's going to help us
resolve between g and h.

00:03:21.768 --> 00:03:23.268
Did I just say g and f?

00:03:23.268 --> 00:03:27.386
g and h both have a vertical
asymptote at x equals 5.

00:03:27.386 --> 00:03:28.502
And you see that over here.

00:03:28.502 --> 00:03:32.302
At x equals 5, both of them are undefined.

00:03:32.302 --> 00:03:35.317
When x equals 5, both of
their denominators are 0.

00:03:35.317 --> 00:03:39.233
So, let's see if the horizontal
asymptotes could help us.

00:03:39.233 --> 00:03:41.751
So, it looks like g has
a horizontal asymptote

00:03:41.751 --> 00:03:44.802
at y equals negative 2.

00:03:44.802 --> 00:03:46.667
y equals negative 2.

00:03:46.667 --> 00:03:49.385
if x becomes very
positive or very negative,

00:03:49.385 --> 00:03:52.018
it looks like y is approaching negative 2.

00:03:52.018 --> 00:03:55.384
So, what's happening up
here, for this top one?

00:03:55.384 --> 00:03:57.718
Well, if we expand out the numerator,

00:03:57.718 --> 00:04:03.917
this is equal to 2 x
minus 12, over x minus 5.

00:04:03.917 --> 00:04:07.733
For very large x's the negative 12

00:04:07.733 --> 00:04:09.680
and the negative 5 aren't
going to matter as much.

00:04:09.680 --> 00:04:12.864
So this is, for very very large x's,

00:04:12.864 --> 00:04:16.553
this is going to be
approximately 2 x over x.

00:04:16.553 --> 00:04:17.743
Let me make it very clear,

00:04:17.743 --> 00:04:24.621
for, maybe I'll write, "as
x approaches infinity."

00:04:25.050 --> 00:04:27.311
And 2 x over x would be 2,

00:04:27.311 --> 00:04:30.844
would be approximately equal
to 2 as x approaches infinity.

00:04:30.844 --> 00:04:33.898
g's asymptote isn't at
2, it's at negative 2.

00:04:33.898 --> 00:04:39.709
h's asymptote's the one
that looks like it's at 2.

00:04:39.709 --> 00:04:41.247
At y equals 2.

00:04:41.247 --> 00:04:44.826
So this top one looks like h of x.

00:04:44.826 --> 00:04:47.255
This top one looks like at h of x.

00:04:47.255 --> 00:04:48.310
And we can confirm that.

00:04:48.310 --> 00:04:49.913
When would h of x equal 0?

00:04:49.913 --> 00:04:52.389
Well, the numerator
equals 0 when x equals 6,

00:04:52.389 --> 00:04:54.440
and we see it right over here.

00:04:54.440 --> 00:04:55.888
Now, that might not
have helped us that much

00:04:55.888 --> 00:04:59.922
because g also equals 0 at six,

00:04:59.922 --> 00:05:02.108
but at least the horizontal asymptotes

00:05:02.108 --> 00:05:04.777
is what is cluing us in.

00:05:04.777 --> 00:05:06.897
As x gets really really really large,

00:05:06.897 --> 00:05:09.160
then the negative, subtracting
the 12 in the numerator,

00:05:09.160 --> 00:05:10.462
subtracting the 5 in the denominator,

00:05:10.462 --> 00:05:11.760
are going to matter
less and less and less.

00:05:11.760 --> 00:05:13.661
We want to look at the
highest degree terms.

00:05:13.661 --> 00:05:16.044
It's going to approach
2 x over x, which is 2.

00:05:16.044 --> 00:05:17.747
And we see that for h.

00:05:17.747 --> 00:05:21.676
Now, g, if we just use
our deductive reasoning,

00:05:21.676 --> 00:05:23.798
we'd say "okay, well,
hopefully this is our g of x."

00:05:23.798 --> 00:05:24.848
Now does this make sense?

00:05:24.848 --> 00:05:30.110
g of x is equal to 12
minus 2 x, over x minus 5.

00:05:30.110 --> 00:05:33.135
Which is approximately
going to be equal to,

00:05:33.135 --> 00:05:34.827
we look at the highest degree terms,

00:05:34.827 --> 00:05:39.549
negative 2 x over x,

00:05:39.549 --> 00:05:43.593
as x approaches infinity.

00:05:43.593 --> 00:05:46.010
Which is going to be equal to,

00:05:46.010 --> 00:05:48.053
or we're going to approach negative 2.

00:05:48.053 --> 00:05:50.952
And that is indeed where
g's horizontal asymptote is.

00:05:50.952 --> 00:05:53.772
As x gets very very large,
we approach negative 2,

00:05:53.772 --> 00:05:55.989
or frankly as x gets very very small,

00:05:55.989 --> 00:05:57.634
we're also going to approach negative 2.

00:05:57.634 --> 00:05:59.079
Negative 2 times the negative billion

00:05:59.079 --> 00:06:01.778
over a negative billion is
also going to be negative 2.

00:06:01.778 --> 00:06:05.331
So, we can feel pretty good
that this is our g of x.


Determining coefficient in binomial expansion

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:03.392
Voiceover:So we've got 3 Y
squared plus 6 X to the third

00:00:03.392 --> 00:00:05.754
and we're raising this
whole to the fifth power

00:00:05.754 --> 00:00:07.632
and we could clearly
use a binomial theorem

00:00:07.632 --> 00:00:10.592
or pascal's triangle in order
to find the expansion of that.

00:00:10.592 --> 00:00:13.400
But what I want to do
is really as an exercise

00:00:13.400 --> 00:00:16.730
is to try to hone in on
just one of the terms

00:00:16.730 --> 00:00:19.249
and in particular I want to
hone in on the term that has

00:00:19.249 --> 00:00:24.720
some coefficient times X to
the sixth, Y to the sixth.

00:00:24.720 --> 00:00:26.240
So in this expansion some term

00:00:26.240 --> 00:00:27.833
is going to have X to
the sixth, Y to sixth

00:00:27.833 --> 00:00:29.776
and I want to figure
out what the coefficient

00:00:29.776 --> 00:00:32.000
on that term is and I
encourage you to pause

00:00:32.000 --> 00:00:36.320
this video and try to
figure it out on your own.

00:00:36.320 --> 00:00:38.400
So I'm assuming you've had
a go at it and you might

00:00:38.400 --> 00:00:41.075
have at first found this to
be a little bit confusing.

00:00:41.075 --> 00:00:42.821
I'm only raising it to the fifth power,

00:00:42.821 --> 00:00:46.349
how do I get X to the
sixth, Y to the sixth?

00:00:46.349 --> 00:00:48.149
But then when you look at the actual terms

00:00:48.149 --> 00:00:49.677
of the binomial it starts
to jump out at you.

00:00:49.677 --> 00:00:51.445
Okay, I have a Y squared term,

00:00:51.445 --> 00:00:53.413
I have an X to the third term,

00:00:53.413 --> 00:00:55.701
so when I raise these to
powers I'm going to get,

00:00:55.701 --> 00:00:58.981
I could have powers higher
than the fifth power.

00:00:58.981 --> 00:01:00.389
But to actually think about which of

00:01:00.389 --> 00:01:02.765
these terms has the X to
the sixth, Y to the sixth,

00:01:02.765 --> 00:01:05.461
let's just look at the pattern in,

00:01:05.461 --> 00:01:08.581
in I guess the actual expansion

00:01:08.581 --> 00:01:10.605
without even thinking
about its coefficients.

00:01:10.605 --> 00:01:13.592
And we've seen this multiple times before

00:01:13.592 --> 00:01:16.653
where you could take your
first term in your binomial

00:01:16.653 --> 00:01:21.485
and you could start it off
it's going to start of at a,

00:01:21.485 --> 00:01:24.517
at the power we're taking
the whole binomial to

00:01:24.517 --> 00:01:25.909
and then in each term it's going

00:01:25.909 --> 00:01:27.473
to have a lower and lower power.

00:01:27.473 --> 00:01:29.589
So let me actually just
copy and paste this.

00:01:29.589 --> 00:01:33.021
So this is going to be, so copy

00:01:33.021 --> 00:01:37.245
and so that's first term, second term,

00:01:37.245 --> 00:01:38.605
let me make sure I have enough space here.

00:01:38.605 --> 00:01:46.070
Second term, third term,
fourth term, fourth term,

00:01:46.070 --> 00:01:52.570
fifth term, and sixth term it's
going to have 6 terms to it,

00:01:52.570 --> 00:01:55.539
you always have one more
term than the exponent.

00:01:55.539 --> 00:01:57.939
And then, actually before I
throw the exponents on it,

00:01:57.939 --> 00:02:00.027
let's focus on the second term.

00:02:00.027 --> 00:02:02.555
So the second term, actually
I'll write it like this.

00:02:02.555 --> 00:02:06.339
So the second term's
times 6 X to the third,

00:02:06.339 --> 00:02:10.651
let me copy and paste that, whoops.

00:02:10.651 --> 00:02:14.395
So let me copy and paste that.

00:02:14.395 --> 00:02:16.483
So we're going to put that there.

00:02:16.483 --> 00:02:18.955
That there.

00:02:18.955 --> 00:02:21.291
And that there.

00:02:21.291 --> 00:02:23.803
And that there.

00:02:23.803 --> 00:02:27.667
And then over to off your screen.

00:02:27.667 --> 00:02:29.395
I wrote it over there.

00:02:29.395 --> 00:02:31.157
We'll see if we have to go there.

00:02:31.157 --> 00:02:33.131
And then let's put the exponents.

00:02:33.131 --> 00:02:35.875
So this exponent, this is going to be

00:02:35.875 --> 00:02:38.251
the fifth power, fourth
power, third power,

00:02:38.251 --> 00:02:44.819
second power, first
power and zeroeth power.

00:02:44.819 --> 00:02:47.932
And for the blue expression,
for 6 X to the third,

00:02:47.932 --> 00:02:51.339
this is going to be the
zeroeth power, first power,

00:02:51.339 --> 00:02:58.067
first power, second power,
third power, fourth power,

00:02:58.067 --> 00:03:02.830
and then we're going to have
the fifth power right over here.

00:03:02.830 --> 00:03:04.643
Then and, of course, they're each going

00:03:04.643 --> 00:03:06.435
to have coefficients in front of them.

00:03:06.435 --> 00:03:07.531
They're each going to have

00:03:07.531 --> 00:03:08.827
coefficients in front of them.

00:03:08.827 --> 00:03:10.499
So there's going to be a
coefficient in front of this one,

00:03:10.499 --> 00:03:11.747
in front of this one, in front of this one

00:03:11.747 --> 00:03:13.291
and then we add them all together.

00:03:13.291 --> 00:03:14.779
But which of these terms is

00:03:14.779 --> 00:03:15.988
the one that we're talking about.

00:03:15.988 --> 00:03:18.451
Has X to the sixth, Y to the sixth.

00:03:18.451 --> 00:03:22.211
So here we have X, if we
take Y squared to the fourth

00:03:22.211 --> 00:03:25.079
it's going to be Y to the
eighth, so that's not it.

00:03:25.140 --> 00:03:29.773
if we go here we have Y
squared to the third power,

00:03:29.773 --> 00:03:31.927
that's Y to the sixth and here you have

00:03:31.927 --> 00:03:34.674
X to the third squared,
that X to the sixth.

00:03:34.674 --> 00:03:36.511
So what we really want to think about

00:03:36.511 --> 00:03:39.535
is what is the coefficient,
what is the coefficient

00:03:39.535 --> 00:03:44.341
in front of this term, in
front of this term going to be?

00:03:44.341 --> 00:03:46.445
Essentially if you put it
in this way it's going to be

00:03:46.445 --> 00:03:48.743
the third term that we
actually care about.

00:03:48.743 --> 00:03:50.679
So what is this coefficient going to be?

00:03:50.679 --> 00:03:52.399
Now we have to clear, this coefficient,

00:03:52.399 --> 00:03:54.287
whatever we put here that we can use

00:03:54.287 --> 00:03:57.993
the binomial theorem to figure
out isn't going to be this,

00:03:57.993 --> 00:04:01.785
this thing that we have to,
I guess our actual solution

00:04:01.785 --> 00:04:03.939
to the problem that we
posed is going to be

00:04:03.939 --> 00:04:06.858
the product of this coefficient

00:04:06.858 --> 00:04:09.178
and whatever other
coefficients we have over here.

00:04:09.178 --> 00:04:11.451
Cause we're going to have 3 to
the third power, six squared.

00:04:11.451 --> 00:04:13.394
So we're going to have to
figure out what that is.

00:04:13.394 --> 00:04:16.002
But let's first just figure
out what this term looks like,

00:04:16.002 --> 00:04:17.442
this term in the expansion.

00:04:17.442 --> 00:04:20.642
What this yellow part actually is.

00:04:20.642 --> 00:04:22.418
And there's a couple of
ways that we can do that.

00:04:22.418 --> 00:04:25.490
We could use Pascal's triangle
or we could use combinatorics.

00:04:25.490 --> 00:04:26.979
If we use combinatorics we know that

00:04:26.979 --> 00:04:34.523
the coefficient over here,
this is going to be 5 choose 0,

00:04:34.523 --> 00:04:36.466
this is going to be the coefficient,

00:04:36.466 --> 00:04:41.850
the coefficient over here
is going to be 5 choose 1.

00:04:41.850 --> 00:04:43.170
We've seen this multiple times.

00:04:43.170 --> 00:04:44.386
You could view it as essentially

00:04:44.386 --> 00:04:49.269
the exponent choose the the top,

00:04:49.269 --> 00:04:51.729
the 5 is the exponent that we're raising

00:04:51.729 --> 00:04:55.401
the whole binomial to and
we say choose this number,

00:04:55.401 --> 00:04:57.385
that's the exponent on the

00:04:57.385 --> 00:04:59.157
second term I guess you could say.

00:04:59.157 --> 00:05:02.337
So this would be 5 choose 1.

00:05:02.337 --> 00:05:05.953
And this one over here, the
coefficient, this thing in yellow.

00:05:05.953 --> 00:05:11.987
This is going to be 5, 5 choose 2.

00:05:11.987 --> 00:05:14.321
5 choose 2.

00:05:14.321 --> 00:05:16.915
Now what is 5 choose 2?

00:05:16.915 --> 00:05:22.953
Well that's equal to 5
factorial over 2 factorial,

00:05:22.953 --> 00:05:29.961
over 2 factorial, times,
times 5 minus 2 factorial.

00:05:29.961 --> 00:05:32.023
So let me just put that in here.

00:05:32.023 --> 00:05:37.775
Times 5 minus 2 factorial.

00:05:37.775 --> 00:05:40.007
And this is going to be equal to.

00:05:40.007 --> 00:05:45.071
Let's see 5 factorial is
5 times 4 times 3 times 2,

00:05:45.071 --> 00:05:47.831
we could write times 1 but
that won't change the value.

00:05:47.831 --> 00:05:50.199
Over 2 factorial.

00:05:50.199 --> 00:05:51.328
Actually let me just write that

00:05:51.328 --> 00:05:53.263
just so we make it clear
it is times 1 there.

00:05:53.263 --> 00:05:56.551
2 factorial is 2 times 1 and then what

00:05:56.551 --> 00:05:59.530
we have right over here,
this is 3 factorial,

00:05:59.530 --> 00:06:02.937
times 3 times 2 times 1.

00:06:02.937 --> 00:06:05.536
So let's see this 3
can cancel with that 3,

00:06:05.536 --> 00:06:08.325
that 2 can cancel with that
2, the 1's don't matter,

00:06:08.325 --> 00:06:12.632
won't change the value and
then 4 divided by 2 is 2.

00:06:12.632 --> 00:06:17.568
So that is just 2, so we're left
with 5 times 2 is equal to 10.

00:06:17.568 --> 00:06:19.792
So that's going to be this
number right over here.

00:06:19.792 --> 00:06:21.424
This is going to be a 10.

00:06:21.424 --> 00:06:24.704
Now another we could have done
it is using Pascal's triangle.

00:06:24.704 --> 00:06:29.188
We could have said okay
this is the binomial,

00:06:29.188 --> 00:06:30.544
now this is when I raise it to

00:06:30.544 --> 00:06:33.208
the second power as 1 2
1 are the coefficients.

00:06:33.208 --> 00:06:35.376
When I raise it to the third power,

00:06:35.376 --> 00:06:38.407
the coefficients are 1, 3, 3, 1.

00:06:38.407 --> 00:06:40.795
When I raise it to the fourth power

00:06:40.795 --> 00:06:45.416
the coefficients are 1, 4, 6, 4, 1

00:06:45.416 --> 00:06:48.027
and when I raise it to the fifth power

00:06:48.027 --> 00:06:50.631
which is the one we care
about, the coeffiencients

00:06:50.631 --> 00:07:03.994
are going to be 1, 5, 10, 5
or sorry 10, 10, 5, and 1.

00:07:03.994 --> 00:07:05.849
And we know that when we go,

00:07:05.849 --> 00:07:07.930
this is going to be the third term so

00:07:07.930 --> 00:07:09.840
this is going to be the
coefficient right over here.

00:07:09.840 --> 00:07:11.896
1, 2, 3, third term.

00:07:11.896 --> 00:07:14.016
So either way we know that this is 10.

00:07:14.016 --> 00:07:15.737
But now let's try to answer
our original question.

00:07:15.737 --> 00:07:16.792
What is this going to be?

00:07:16.792 --> 00:07:19.808
What are we multiplying times
X to the sixth, Y to the sixth?

00:07:19.808 --> 00:07:22.544
And now we just have to essentially
rewrite this expression.

00:07:22.544 --> 00:07:27.632
So it's going to be 10
times 3 to the third power,

00:07:27.632 --> 00:07:33.480
3 to the third power, times
Y squared to the third power,

00:07:33.480 --> 00:07:37.272
which is Y squared to the third
power is Y to the sixth power.

00:07:37.272 --> 00:07:38.992
Y to the sixth power.

00:07:38.992 --> 00:07:45.511
Times six squared so
times six squared times X

00:07:45.511 --> 00:07:48.864
to the third squared which
that's X to the 3 times 2

00:07:48.864 --> 00:07:53.608
or X to the sixth and so
this is going to be equal to.

00:07:53.608 --> 00:08:05.928
Let's see it's going to be
10 times 27 times 36 times 36

00:08:05.928 --> 00:08:07.536
and then we have, of course,

00:08:07.536 --> 00:08:12.617
our X to the sixth and Y to the sixth.

00:08:12.617 --> 00:08:14.856
So this is going to be, essentially,

00:08:14.856 --> 00:08:18.160
let's see 270 times 36 so let's see,

00:08:18.160 --> 00:08:19.358
let's get a calculator out.

00:08:19.358 --> 00:08:22.144
270, I could have done it by
hand but I'll just do this

00:08:22.144 --> 00:08:26.591
for the sake of time, times 36 is 9,720.

00:08:26.591 --> 00:08:28.920
So that's the coefficient right over here.

00:08:28.920 --> 00:08:35.019
It's going to be 9,720 X to
the sixth, Y to the sixth.

00:08:35.019 --> 23:59:59.999
9,720 X to the sixth, Y to
the sixth and we're done.


Intuitive connection between binomial expansion and combinatorics

WEBVTT
Kind: captions
Language: en

00:00:00.344 --> 00:00:01.813
Voiceover:What I want to do in this video

00:00:01.813 --> 00:00:03.370
is hopefully give more intuition

00:00:03.370 --> 00:00:05.457
as to why the binomial theorem

00:00:05.457 --> 00:00:09.399
or the binomial formula
involves combinatorics.

00:00:09.399 --> 00:00:11.830
Let's just think about what
this expansion would be.

00:00:11.830 --> 00:00:13.312
I'm just using a particular example

00:00:13.312 --> 00:00:14.265
that's pretty simple,

00:00:14.265 --> 00:00:15.687
x plus y to the third power

00:00:15.687 --> 00:00:18.117
which is x plus y, times
x plus y, times x plus y.

00:00:18.117 --> 00:00:20.662
We already know what the terms look like,

00:00:20.662 --> 00:00:25.630
we could pick an x from each of these.

00:00:25.630 --> 00:00:28.500
If we pick exactly an x from each of these

00:00:28.500 --> 00:00:30.695
that's going to be our
x to the third term,

00:00:30.695 --> 00:00:33.998
and we essentially picked no y [zen]

00:00:33.998 --> 00:00:35.729
because we picked an x from each of this

00:00:35.729 --> 00:00:37.608
when we multiplied out to get this term.

00:00:37.608 --> 00:00:40.704
How many ways are there to construct that?

00:00:40.704 --> 00:00:42.900
How many ways are there,

00:00:42.900 --> 00:00:43.923
if we're picking from three things.

00:00:43.923 --> 00:00:47.261
We're picking from three things,

00:00:47.261 --> 00:00:50.988
how many ways can we
choose exactly zero y's

00:00:50.988 --> 00:00:52.409
of not picking a y?

00:00:52.409 --> 00:00:54.129
There's only one way to do that,

00:00:54.129 --> 00:00:57.566
there's only one possible
way of doing that.

00:00:57.566 --> 00:01:01.399
By not picking a y from each of them

00:01:01.399 --> 00:01:03.659
or you can say by picking
an x from each of them.

00:01:03.659 --> 00:01:05.727
The coefficient right over here

00:01:05.727 --> 00:01:07.698
is one x to the third.

00:01:07.698 --> 00:01:08.742
There's only one way,

00:01:08.742 --> 00:01:10.100
when you take this product out,

00:01:10.100 --> 00:01:12.666
only one of the terms
initially is going to have

00:01:12.666 --> 00:01:14.806
an x to the third in it.

00:01:14.806 --> 00:01:17.318
Now, what about the next term?

00:01:17.318 --> 00:01:18.543
So plus.

00:01:18.543 --> 00:01:22.709
Now we're going to pick from three buckets

00:01:22.709 --> 00:01:24.645
but we want to pick one y.

00:01:24.645 --> 00:01:25.857
Another way of thinking about it

00:01:25.857 --> 00:01:27.710
you have three people

00:01:27.710 --> 00:01:28.962
or we have three people,

00:01:28.962 --> 00:01:31.706
this person, this person, and this person.

00:01:31.706 --> 00:01:34.919
One of them you're going to choose to be

00:01:34.919 --> 00:01:37.663
I guess be your friend

00:01:37.663 --> 00:01:39.461
that's another way of
saying which of these

00:01:39.461 --> 00:01:40.943
are you going to pick the y from?

00:01:42.912 --> 00:01:44.989
This is the exact same thing

00:01:44.989 --> 00:01:46.516
as a combinatorics problem,

00:01:46.516 --> 00:01:50.114
from three things

00:01:50.129 --> 00:01:53.308
you are choosing one of them to be y.

00:01:53.308 --> 00:01:54.713
Of course three choose one,

00:01:54.713 --> 00:01:56.593
this is equal to three.

00:01:56.593 --> 00:01:59.720
If you're picking one y

00:01:59.720 --> 00:02:01.630
that means you're picking two x's,

00:02:01.630 --> 00:02:03.446
and you're taking the
product of everything

00:02:03.446 --> 00:02:07.350
so it's x squared times y to the one

00:02:07.350 --> 00:02:10.148
and we could keep doing
that logic, so plus.

00:02:10.148 --> 00:02:12.391
Now we're going to think
about the situation

00:02:12.391 --> 00:02:16.950
where we are picking two
y's out of three things.

00:02:16.950 --> 00:02:20.657
It's x to the first, y squared.

00:02:20.657 --> 00:02:23.240
We're starting with three things

00:02:23.240 --> 00:02:26.121
and we're going to pick
two to two of this y's

00:02:26.121 --> 00:02:27.971
so it could be this y and this y

00:02:27.971 --> 00:02:31.577
to construct the product y.

00:02:31.577 --> 00:02:34.830
It could be y times y times that x

00:02:34.830 --> 00:02:39.213
or it could be y times y, times that x

00:02:39.213 --> 00:02:42.641
or it could be y times x, times y.

00:02:42.641 --> 00:02:44.954
This is three choose two,

00:02:44.954 --> 00:02:47.510
out of three things what
are the different ways,

00:02:47.510 --> 00:02:49.959
what are the combinations
of picking two things.

00:02:49.959 --> 00:02:51.342
If you have three friends

00:02:51.342 --> 00:02:53.476
how many combinations
can you put two of them

00:02:53.476 --> 00:02:55.464
in your two seater car
where you don't care

00:02:55.464 --> 00:02:57.640
about who sits in which seat.

00:02:57.640 --> 00:02:58.447
You're just thinking about

00:02:58.447 --> 00:02:59.363
how many different combinations

00:02:59.363 --> 00:03:00.524
can you pick from that

00:03:00.524 --> 00:03:03.461
and once again this evaluates to three

00:03:03.461 --> 00:03:08.363
and then finally, how many different ways

00:03:08.363 --> 00:03:10.371
from a set of three things,

00:03:10.371 --> 00:03:12.973
from three different things,

00:03:12.973 --> 00:03:14.593
so from each of these expressions,

00:03:14.593 --> 00:03:17.668
how many ways can you
pick exactly three y's?

00:03:17.668 --> 00:03:18.921
Well there's only one way to do it,

00:03:18.921 --> 00:03:22.420
you pick this y, this y and that y.

00:03:22.420 --> 00:03:23.858
If you're picking three y's

00:03:23.858 --> 00:03:26.640
that means you picked zero x's

00:03:26.640 --> 00:03:29.600
and you have picked three y's.

00:03:29.600 --> 00:03:32.200
That's why we're dealing
with combinatorics here.

00:03:32.200 --> 00:03:33.408
Three choose three.

00:03:33.408 --> 00:03:35.472
From three expressions,

00:03:35.472 --> 00:03:37.585
your three binomials, right over here

00:03:37.585 --> 00:03:39.257
that you are taking the product from

00:03:39.257 --> 00:03:42.792
you need to choose three y's,

00:03:42.792 --> 00:03:44.971
so you have to pick the
second term in all of them.

00:03:44.971 --> 00:03:46.283
There's only one way to do that,

00:03:46.283 --> 00:03:48.803
here you want to pick the
second term, the y term,

00:03:48.803 --> 00:03:51.313
in two of them, exactly two of them.

00:03:51.313 --> 00:03:52.825
Here you want to pick the y term

00:03:52.825 --> 00:03:54.390
in exactly one of them,

00:03:54.390 --> 00:03:54.990
there's one way to do this,

00:03:54.990 --> 00:03:55.978
three ways to do this,

00:03:55.978 --> 00:03:57.358
so there's three ways to do this,

00:03:57.358 --> 00:03:58.174
three ways to do this,

00:03:58.174 --> 00:03:58.938
one way to do that,

00:03:58.938 --> 00:03:59.869
one way to do that.

00:03:59.869 --> 00:04:01.860
This is one,

00:04:01.860 --> 00:04:03.834
this is three,

00:04:03.834 --> 00:04:06.957
when you evaluate, when
you apply the formula

00:04:06.957 --> 00:04:09.980
for three choose and choose k,

00:04:09.980 --> 00:04:13.310
three and this is one.


Connecting Pascal's triangle to binomial combinatorics

WEBVTT
Kind: captions
Language: en

00:00:00.319 --> 00:00:01.439
Voiceover:What I want to do in this video

00:00:01.439 --> 00:00:03.637
is further connect our understanding

00:00:03.637 --> 00:00:05.335
of the binomial theorem.

00:00:05.335 --> 00:00:08.269
Two combinatorics, two Pascal's triangle.

00:00:08.269 --> 00:00:10.564
Just to review the ideas again,

00:00:10.564 --> 00:00:12.830
if we're taking x plus
y to the third power

00:00:12.830 --> 00:00:14.199
and I'm just using this as an example

00:00:14.199 --> 00:00:16.017
that's a little bit easy to get around,

00:00:16.017 --> 00:00:17.142
get our heads around

00:00:17.142 --> 00:00:20.256
that's essentially taking
three equivalent expressions

00:00:20.256 --> 00:00:21.776
and multiplying them by each other.

00:00:21.776 --> 00:00:24.015
You're taking x plus y, times x plus y,

00:00:24.015 --> 00:00:25.575
times x plus y.

00:00:25.575 --> 00:00:27.752
Let's call this the first x plus y,

00:00:27.752 --> 00:00:31.222
the second x plus y,
and the third x plus y.

00:00:31.222 --> 00:00:33.851
If you're looking at the expansion of it

00:00:33.851 --> 00:00:37.817
and you're saying well how many ways

00:00:37.817 --> 00:00:41.149
can we construct x y squared

00:00:41.149 --> 00:00:42.690
or another way to think about it is

00:00:42.690 --> 00:00:44.390
you have these three expressions,

00:00:44.390 --> 00:00:46.279
you're going to choose two of them

00:00:46.279 --> 00:00:49.447
to contribute a y and take the product.

00:00:49.447 --> 00:00:52.704
For example, you could pick expression one

00:00:52.704 --> 00:00:54.918
and expression two to contribute the y.

00:00:54.918 --> 00:00:57.946
You could pick expression
one and expression three

00:00:57.946 --> 00:00:59.172
to contribute the y,

00:00:59.172 --> 00:01:00.914
or you could take expression two

00:01:00.914 --> 00:01:02.787
and expression three to contribute the y.

00:01:02.787 --> 00:01:03.657
Of course the other expressions

00:01:03.657 --> 00:01:05.345
what's going to contribute to the x.

00:01:05.345 --> 00:01:09.117
You have three expressions

00:01:09.117 --> 00:01:12.327
and you're going to choose two

00:01:12.327 --> 00:01:14.007
to contribute the y

00:01:14.007 --> 00:01:16.222
and so that's why you have
this combinatorics idea

00:01:16.222 --> 00:01:17.365
it's three choose two.

00:01:17.365 --> 00:01:19.372
It's the exact same mathematical problem,

00:01:19.372 --> 00:01:22.044
if you have three friends
and you're choosing two

00:01:22.044 --> 00:01:23.671
to ride in a car with you,

00:01:23.671 --> 00:01:25.965
you don't care which sit
they are going to sit in.

00:01:25.965 --> 00:01:27.211
You're just saying which two friends

00:01:27.211 --> 00:01:29.827
get to ride in my two
seater or get to ride,

00:01:29.827 --> 00:01:32.356
what are the two friends
that I'm going to pick?

00:01:32.356 --> 00:01:34.326
Same way here, out of three friends

00:01:34.326 --> 00:01:36.268
which are the two that I'm going to pick

00:01:36.268 --> 00:01:38.789
to contribute a y to this product

00:01:38.789 --> 00:01:41.039
and then the third is
going to contribute an x.

00:01:41.039 --> 00:01:43.188
Now let's go to Pascal's triangle

00:01:43.188 --> 00:01:45.142
and hopefully see that this is actually

00:01:45.142 --> 00:01:46.387
a very similar idea.

00:01:46.387 --> 00:01:48.179
We could go to the same term,

00:01:48.179 --> 00:01:50.328
so the x y squared term,

00:01:50.328 --> 00:01:51.895
so that's right over there.

00:01:51.895 --> 00:01:54.856
Pascal's triangle, I always
visualize it as a map.

00:01:54.856 --> 00:01:56.478
This is a node in the map

00:01:56.478 --> 00:01:58.201
and I think what are the different ways

00:01:58.201 --> 00:01:59.984
that I can get to this node on the map.

00:01:59.984 --> 00:02:05.653
I could have a y squared,

00:02:05.653 --> 00:02:07.215
and then multiplied by an x.

00:02:07.215 --> 00:02:08.399
Since we're multiplying an x

00:02:08.399 --> 00:02:09.974
that's why I made this line orange

00:02:09.974 --> 00:02:11.441
or I could have an x y

00:02:11.441 --> 00:02:14.409
and I could multiply by a blue

00:02:14.409 --> 00:02:16.812
so there's two immediate things above it

00:02:16.812 --> 00:02:18.063
that I can get to them

00:02:18.063 --> 00:02:19.738
but there's two ways to get to this one,

00:02:19.738 --> 00:02:21.271
there's one way to get to this one,

00:02:21.271 --> 00:02:23.664
and so the combined
ways to get to this one

00:02:23.664 --> 00:02:25.671
is two plus one is equal to three.

00:02:25.671 --> 00:02:27.695
Just to make it the connection

00:02:27.695 --> 00:02:29.071
between what we just said,

00:02:29.071 --> 00:02:30.663
what's really going on here

00:02:30.663 --> 00:02:33.830
is at each of these
nodes as we pick a path,

00:02:33.830 --> 00:02:36.421
we're essentially saying
are we picking an x or y

00:02:36.421 --> 00:02:37.944
from each of these expressions.

00:02:37.944 --> 00:02:40.121
We number the expressions,
let's number them here.

00:02:40.121 --> 00:02:41.787
Let's say this is expression one,

00:02:41.787 --> 00:02:43.126
this is expression two,

00:02:43.126 --> 00:02:45.115
this is expression three.

00:02:45.115 --> 00:02:47.339
We said that there's three paths

00:02:47.339 --> 00:02:48.562
that we could take us here,

00:02:48.562 --> 00:02:49.851
let's mark them out.

00:02:51.143 --> 00:02:53.263
There's this path,

00:02:53.263 --> 00:02:57.835
there's this path right
over here that gets us there

00:02:57.835 --> 00:03:01.119
and that's equivalent to picking the x

00:03:01.119 --> 00:03:02.982
from expression one.

00:03:02.982 --> 00:03:05.339
Notice we picked the left hand side

00:03:05.339 --> 00:03:06.939
then we go to the right hand side.

00:03:06.939 --> 00:03:09.246
We pick the y from expression two,

00:03:09.246 --> 00:03:13.078
and then we pick the y
from expression three,

00:03:13.078 --> 00:03:14.683
the last two, we pick the right path,

00:03:14.683 --> 00:03:16.154
we picked the blue path.

00:03:16.154 --> 00:03:18.127
That's essentially one way

00:03:18.127 --> 00:03:20.178
that we could choose two y's

00:03:20.178 --> 00:03:22.426
out of these three expressions.

00:03:22.426 --> 00:03:24.713
Once again, one way to choose two y's

00:03:24.713 --> 00:03:26.530
out of three expressions

00:03:26.530 --> 00:03:28.303
but we don't care about just that one,

00:03:28.303 --> 00:03:29.785
we care about all of the different paths.

00:03:29.785 --> 00:03:31.098
This was one way,

00:03:31.098 --> 00:03:33.292
another way would be to go like this.

00:03:33.292 --> 00:03:36.703
Pick a y from the first one,
an x from the second one

00:03:36.703 --> 00:03:38.818
and a y from the third one.

00:03:38.818 --> 00:03:42.147
Or pick a y from the first expression,

00:03:42.147 --> 00:03:44.180
a y from the second expression

00:03:44.180 --> 00:03:46.908
and an x from the third expression.

00:03:46.908 --> 00:03:51.100
It's really a kind of
fundamental mathematical level.

00:03:51.100 --> 00:03:54.377
It's solving the exact same problem.

00:03:54.377 --> 00:03:56.185
Hopefully this gives
you a little intuition

00:03:56.185 --> 00:03:59.136
on why Pascal's triangle is connected

00:03:59.136 --> 00:04:00.577
to two combinatorics

00:04:00.577 --> 00:04:02.178
and why they're both useful

00:04:02.178 --> 00:04:04.447
for finding binomial expansions.


Algorithm for mentally computing binomial expansion coefficients

WEBVTT
Kind: captions
Language: en

00:00:00.766 --> 00:00:02.439
Voiceover:What I want to
show you in this video is

00:00:02.439 --> 00:00:05.371
what could be described
as, I guess, a trick for

00:00:05.371 --> 00:00:07.374
finding binomial expansions, especially

00:00:07.374 --> 00:00:10.167
binomial expansions where
the exponent is fairly large.

00:00:10.167 --> 00:00:11.777
But what I want you to do after this video

00:00:11.777 --> 00:00:14.105
is think about how this
connects to the binomial theorem

00:00:14.105 --> 00:00:17.311
and how it connects to Pascal's Triangle.

00:00:17.311 --> 00:00:20.306
Now let me show you the trick.

00:00:20.306 --> 00:00:22.635
I'm going to take (X+Y)^7.

00:00:22.635 --> 00:00:24.435
That's going to have eight terms.

00:00:24.435 --> 00:00:25.167
How do I know that?

00:00:25.167 --> 00:00:28.370
Well (X+Y)^1 has two
terms, it's a binomial.

00:00:28.370 --> 00:00:30.306
(X+Y)^2 has three terms.

00:00:30.306 --> 00:00:32.714
(X+Y)^3 has four terms.

00:00:32.714 --> 00:00:34.694
So this is going to have eight terms.

00:00:34.694 --> 00:00:37.774
Let me just create little
buckets for each of the terms.

00:00:37.774 --> 00:00:38.978
This is the bucket, these aren't the

00:00:38.978 --> 00:00:40.513
coefficients these are just the buckets.

00:00:40.513 --> 00:00:45.971
1st term, 2nd term, 3rd term, 4th term,

00:00:45.971 --> 00:00:53.442
5th term, 6th term,
7th term, and 8th term.

00:00:53.442 --> 00:00:58.109
Now lets write out the actual X's and Y's.

00:00:58.109 --> 00:01:03.106
The first term, we're
going to start with X^7.

00:01:03.106 --> 00:01:07.312
Then each term after that our degree or

00:01:07.312 --> 00:01:09.709
our power on the X goes down by one.

00:01:09.709 --> 00:01:19.706
So you go X^6, X^5, X^4, X^3, X^2, X^1,

00:01:19.706 --> 00:01:20.973
we could just write that as X, and this is

00:01:20.973 --> 00:01:24.310
going to be X^0, which
is just going to be one.

00:01:24.310 --> 00:01:25.844
Now lets think about Y.

00:01:25.844 --> 00:01:27.313
This is going to start at Y^0,

00:01:27.313 --> 00:01:28.846
which is just one so I'm
not going to write it,

00:01:28.846 --> 00:01:40.235
Then it's going to be Y^1,
Y^2, Y^3, Y^4, Y^5, Y^6,

00:01:40.235 --> 00:01:43.233
and then Y^7 and you can
verify you got it right

00:01:43.233 --> 00:01:47.173
because for each term, the
exponents should add up to seven.

00:01:47.173 --> 00:01:48.108
You see that even here.

00:01:48.108 --> 00:01:50.312
This is X^1 times Y^6.

00:01:50.312 --> 00:01:53.637
Those add up to be seven.

00:01:53.637 --> 00:01:55.781
Now, lets get to the interesting part,

00:01:55.781 --> 00:01:58.842
which is actually
calculating the coefficient.

00:01:58.842 --> 00:02:02.577
And the algorithm is for
each term right over here,

00:02:02.577 --> 00:02:04.168
so lets just start, we
know the coefficient

00:02:04.168 --> 00:02:07.041
right over here is going to be one.

00:02:07.041 --> 00:02:08.236
Actually, let me write that down.

00:02:08.236 --> 00:02:12.514
The coefficient over
here is going to be one.

00:02:12.514 --> 00:02:16.703
So for each term, the
coefficient right over here,

00:02:16.703 --> 00:02:20.977
I'm going to try to color-code
it so we can see it,

00:02:20.977 --> 00:02:24.836
the coefficient is
going to be the exponent

00:02:24.836 --> 00:02:28.634
of the previous term, so the
exponent of the previous term

00:02:28.634 --> 00:02:33.447
in this case is the seven, the
exponent of the previous term,

00:02:33.447 --> 00:02:41.772
times the coefficient of the
previous term, divided by

00:02:41.772 --> 00:02:45.646
which term that actually
was, so divided by the term.

00:02:45.646 --> 00:02:47.711
That was the 1st term,
so now the coefficient

00:02:47.711 --> 00:02:49.634
on the 2nd term is seven
times one divided by seven,

00:02:49.634 --> 00:02:59.309
which is going to be equal to seven.

00:02:59.309 --> 00:03:00.436
Now what about this one?

00:03:00.436 --> 00:03:03.635
We use the exact same process.

00:03:03.635 --> 00:03:05.439
The exact same process.

00:03:05.439 --> 00:03:11.178
It is going to be the exponent
on the X term, I guess

00:03:11.178 --> 00:03:15.505
you could say the exponent on
X, I guess we could go with.

00:03:15.505 --> 00:03:18.647
The exponent on the X, which is six, times

00:03:18.647 --> 00:03:21.239
the coefficient of the previous term,

00:03:21.239 --> 00:03:26.104
so times seven, so we're
taking the X power times

00:03:26.104 --> 00:03:29.770
the coefficient of the
previous term, so times seven.

00:03:29.770 --> 00:03:30.912
So the X part of the previous term,

00:03:30.912 --> 00:03:32.707
times the coefficient
of the previous term,

00:03:32.707 --> 00:03:35.033
divided by the actual,
I guess you could say,

00:03:35.033 --> 00:03:39.100
index of the previous
term, so divided by two.

00:03:39.100 --> 00:03:41.238
So what is that going to be?

00:03:41.238 --> 00:03:43.112
So this is the equivalent
to three times seven,

00:03:43.112 --> 00:03:47.313
so this is going to be equal to 21.

00:03:47.313 --> 00:03:51.641
And now lets go to this
term, exact same idea.

00:03:51.641 --> 00:03:53.976
Go to the previous term,
what's our exponent on X?

00:03:53.976 --> 00:03:56.644
It is five.

00:03:56.644 --> 00:04:00.107
Lets multiply it times the coefficient,

00:04:00.107 --> 00:04:03.772
so lets multiply it times
21, and then lets divide it

00:04:03.772 --> 00:04:08.965
by which term that is,
so that was the 3rd term.

00:04:10.488 --> 00:04:12.443
So this is going to be, lets see,

00:04:12.443 --> 00:04:14.443
five times 21 over three is seven,

00:04:14.443 --> 00:04:18.911
so this is going to be
35, five times seven.

00:04:18.911 --> 00:04:22.038
And we can keep going, or we can

00:04:22.038 --> 00:04:24.303
recognize that there's a symmetry here.

00:04:24.303 --> 00:04:27.910
If this is one, then the last
term is also going to be one.

00:04:27.910 --> 00:04:32.179
If the second term is seven, then

00:04:32.179 --> 00:04:35.378
the second-to-last term is seven.

00:04:35.378 --> 00:04:38.768
If the third term is 21, then the

00:04:38.768 --> 00:04:41.378
third term to the last is 21.

00:04:41.378 --> 00:04:44.441
And then if the 4th term is 35,

00:04:44.441 --> 00:04:46.913
then the fourth from the last is 35.

00:04:46.913 --> 00:04:48.909
And just like that, we have figured

00:04:48.909 --> 00:04:55.833
out the expansion of (X+Y)^7.

00:04:55.833 --> 00:04:57.888
Pretty neat, in my mind.


Pascal's triangle for binomial expansion

WEBVTT
Kind: captions
Language: en

00:00:00.335 --> 00:00:03.403
In the previous video we were able
to apply the binomial theorem

00:00:03.403 --> 00:00:06.466
in order to figure out what
a plus b to fourth power is

00:00:06.466 --> 00:00:07.816
in order to expand this out.

00:00:07.816 --> 00:00:10.381
And we did it. And it was
a little bit tedious but

00:00:10.381 --> 00:00:11.781
hopefully you appreciated it.

00:00:11.781 --> 00:00:14.471
It would have been useful
if we did even a higher power--

00:00:14.471 --> 00:00:17.297
a plus b to the seventh power,
a plus b to the eighth power.

00:00:17.297 --> 00:00:19.557
But what I want to do
in this video is show you

00:00:19.557 --> 00:00:21.652
that there's another way
of thinking about it

00:00:21.652 --> 00:00:24.732
and this would be using
"Pascal's Triangle".

00:00:24.732 --> 00:00:27.095
And if we have time we'll also think about

00:00:27.095 --> 00:00:30.605
why these two ideas
are so closely related.

00:00:30.605 --> 00:00:36.474
So instead of doing a plus b to the fourth
using this traditional binomial theorem--

00:00:36.474 --> 00:00:40.026
I guess you could say-- formula right over
here, I'm going to calculate it

00:00:40.026 --> 00:00:42.280
using Pascal's triangle
and some of the patterns

00:00:42.280 --> 00:00:44.800
that we know about the expansion.

00:00:44.800 --> 00:00:49.634
So once again let me write down
what we're trying to calculate.

00:00:49.634 --> 00:00:59.014
We're trying to calculate a plus b

00:00:59.024 --> 00:01:03.059
to the fourth power--

00:01:03.059 --> 00:01:06.134
I'll just do this in a different color--
to the fourth power.

00:01:06.134 --> 00:01:10.867
So what I'm going to do is set up
Pascal's triangle. So Pascal's triangle--

00:01:10.867 --> 00:01:14.499
so we'll start with a one at the top.
And one way to think about it is,

00:01:14.499 --> 00:01:19.089
it's a triangle where if you start it
up here, at each level you're really

00:01:19.089 --> 00:01:22.358
counting the different ways
that you can get to the different nodes.

00:01:22.358 --> 00:01:26.387
So one-- and so I'm going to set up
a triangle. So if I start here

00:01:26.387 --> 00:01:28.147
there's only one way I can get here

00:01:28.147 --> 00:01:30.392
and there's only one way
that I could get there.

00:01:30.392 --> 00:01:32.284
But now this third level--

00:01:32.284 --> 00:01:34.524
if I were to say
how many ways can I get here--

00:01:34.524 --> 00:01:37.704
well, one way to get here,
one way to get here.

00:01:37.704 --> 00:01:39.335
So there's two ways to get here.

00:01:39.335 --> 00:01:41.945
One way to get there,
one way to get there.

00:01:41.945 --> 00:01:45.783
The only way I get there is like that,
the only way I can get there is like that.

00:01:45.783 --> 00:01:49.503
But the way I could get here, I could
go like this, or I could go like this.

00:01:49.503 --> 00:01:54.259
And then we could add a fourth level
where-- let's see, if I have--

00:01:54.259 --> 00:01:58.979
there's only one way to go there
but there's three ways to go here.

00:01:58.979 --> 00:02:01.269
One plus two. How are there three ways?

00:02:01.269 --> 00:02:04.269
You could go like this,
you could go like this,

00:02:04.269 --> 00:02:06.896
or you could go like that.

00:02:06.896 --> 00:02:10.476
Same exact logic:
there's three ways to get to this point.

00:02:10.476 --> 00:02:14.773
And then there's only one way
to get to that point right over there.

00:02:14.773 --> 00:02:19.227
And so let's add a fifth level because
this was actually what we care about

00:02:19.227 --> 00:02:21.774
when we think about
something to the fourth power.

00:02:21.774 --> 00:02:26.200
This is essentially zeroth power--
binomial to zeroth power, first power,

00:02:26.200 --> 00:02:31.945
second power, third power.
So let's go to the fourth power.

00:02:31.945 --> 00:02:33.915
So how many ways are there to get here?

00:02:33.915 --> 00:02:37.263
Well I just have to go all the way
straight down along this left side

00:02:37.263 --> 00:02:39.121
to get here, so there's only one way.

00:02:39.121 --> 00:02:44.041
There's four ways to get here. I could
go like that, I could go like that,

00:02:44.041 --> 00:02:47.803
I could go like that,
and I can go like that.

00:02:47.803 --> 00:02:49.932
There's six ways to go here.

00:02:49.932 --> 00:02:53.032
Three ways to get to this place,
three ways to get to this place.

00:02:53.032 --> 00:02:56.791
So six ways to get to that and, if you
have the time, you could figure that out.

00:02:56.791 --> 00:02:59.150
There's three plus one--
four ways to get here.

00:02:59.150 --> 00:03:01.880
And then there's one way to get there.

00:03:01.880 --> 00:03:04.333
And now I'm claiming that
these are the coefficients

00:03:04.333 --> 00:03:13.069
when I'm taking something to the-- if
I'm taking something to the zeroth power.

00:03:13.069 --> 00:03:16.609
This is if I'm taking a binomial
to the first power,

00:03:16.609 --> 00:03:18.699
to the second power.

00:03:18.699 --> 00:03:20.629
Obviously a binomial to the first power,

00:03:20.629 --> 00:03:23.439
the coefficients on a and b
are just one and one.

00:03:23.439 --> 00:03:27.641
But when you square it, it would be
a squared plus two ab plus b squared.

00:03:27.641 --> 00:03:31.238
If you take the third power, these
are the coefficients-- third power.

00:03:31.238 --> 00:03:34.394
And to the fourth power,
these are the coefficients.

00:03:34.394 --> 00:03:35.934
So let's write them down.

00:03:35.934 --> 00:03:38.169
The coefficients, I'm claiming,
are going to be

00:03:38.169 --> 00:03:44.469
one, four, six, four, and one.

00:03:44.469 --> 00:03:47.337
And how do I know what
the powers of a and b are going to be?

00:03:47.337 --> 00:03:49.262
Well I start a, I start this first term,

00:03:49.262 --> 00:03:52.072
at the highest power: a to the fourth.

00:03:52.072 --> 00:03:53.522
And then I go down from there.

00:03:53.522 --> 00:03:55.652
a to the fourth, a to the third,

00:03:55.652 --> 00:03:58.475
a squared, a to the first,

00:03:58.475 --> 00:03:59.965
and I guess I could write

00:03:59.965 --> 00:04:04.618
a to the zero which of course is just one.

00:04:04.618 --> 00:04:08.568
And then for the second term
I start at the lowest power, at zero.

00:04:08.568 --> 00:04:12.533
And then b to first, b squared,

00:04:12.533 --> 00:04:14.213
b to the third power,

00:04:14.213 --> 00:04:15.533
and then b to the fourth,

00:04:15.533 --> 00:04:19.456
and then I just add those terms together.

00:04:19.456 --> 00:04:20.956
And there you have it.

00:04:20.956 --> 00:04:23.198
I have just figured out the expansion

00:04:23.198 --> 00:04:28.338
of a plus b to the fourth power.

00:04:28.338 --> 00:04:30.398
It's exactly what I just wrote down.

00:04:30.398 --> 00:04:33.915
This term right over here,
a to the fourth, that's what this term is.

00:04:33.915 --> 00:04:36.945
One a to the fourth b to the zero:
that's just a to the fourth.

00:04:36.945 --> 00:04:39.555
This term right over here is equivalent to

00:04:39.555 --> 00:04:41.685
this term right over there.

00:04:41.685 --> 00:04:46.201
And so I guess you see that
this gave me an equivalent result.

00:04:46.201 --> 00:04:52.667
Now an interesting question is
'why did this work?'

00:04:52.667 --> 00:04:57.012
And I encourage you to pause this video
and think about it on your own.

00:04:57.012 --> 00:05:02.036
Well, to realize why it works let's just
go to these first levels right over here.

00:05:02.036 --> 00:05:06.068
If I just were to take
a plus b to the second power.

00:05:06.068 --> 00:05:16.531
a plus b to the second power.

00:05:16.531 --> 00:05:18.451
This is going to be,
we've already seen it,

00:05:18.451 --> 00:05:21.121
this is going to be
a plus b times a plus b

00:05:21.121 --> 00:05:24.691
so let me just write that down:
a plus b times a plus b.

00:05:24.691 --> 00:05:31.573
So we have an a, an a.
We have a b, and a b.

00:05:31.573 --> 00:05:35.875
We're going to add these together.
And then when you multiply it, you have--

00:05:35.875 --> 00:05:40.674
so this is going to be equal to a times a.
You get a squared.

00:05:40.674 --> 00:05:45.364
And that's the only way. That's the
only way to get an a squared term.

00:05:45.364 --> 00:05:50.246
There's only one way of getting
an a squared term.

00:05:50.246 --> 00:06:04.527
Then you're going to have
plus a times b. So-- plus a times b.

00:06:04.527 --> 00:06:09.929
And then you're going to have
plus this b times that a

00:06:09.929 --> 00:06:16.359
so that's going to be another a times b.

00:06:16.359 --> 00:06:21.599
Plus b times b which is b squared.

00:06:21.599 --> 00:06:23.723
Now this is interesting right over here.

00:06:23.723 --> 00:06:25.923
How many ways can you get
an a squared term?

00:06:25.923 --> 00:06:29.084
Well there's only one way. You're
multiplying this a times that a.

00:06:29.084 --> 00:06:33.035
There's one way of getting there.

00:06:33.035 --> 00:06:37.565
Now how many ways are there
of getting the b squared term?

00:06:37.565 --> 00:06:40.198
How many ways are there
of getting the b squared term?

00:06:40.198 --> 00:06:42.935
Well there's only one way.
Multiply this b times this b.

00:06:42.935 --> 00:06:44.896
There's only one way of getting that.

00:06:44.896 --> 00:06:47.896
But how many ways are there
of getting the ab term?

00:06:47.896 --> 00:06:50.076
The a to the first b to the first term.

00:06:50.076 --> 00:06:53.915
Well there's two ways. You can multiply
this a times that b,

00:06:53.915 --> 00:06:58.210
or this b times that a. There are--
just hit the point home--

00:06:58.210 --> 00:07:02.750
there are two ways,
two ways of getting an ab term.

00:07:02.750 --> 00:07:05.767
And so, when you take the sum of these two

00:07:05.767 --> 00:07:19.107
you are left with a squared plus
two times ab plus b squared.

00:07:19.107 --> 00:07:21.160
Notice the exact same coefficients:

00:07:21.161 --> 00:07:24.161
one two one, one two one.

00:07:24.161 --> 00:07:28.108
Why is that like that? Well there is only
one way to get an a squared,

00:07:28.108 --> 00:07:29.808
there's two ways to get an ab, and

00:07:29.808 --> 00:07:32.668
there's only one way to get a b squared.

00:07:32.668 --> 00:07:38.776
If you set it to the third power you'd say
okay, there's only one way to get to

00:07:38.776 --> 00:07:41.092
a to the third power.

00:07:41.092 --> 00:07:43.872
You just multiply
the first a's all together.

00:07:43.872 --> 00:07:48.902
And there is only one way
to get to b to the third power.

00:07:48.902 --> 00:07:51.256
But there's three ways to get to

00:07:51.256 --> 00:07:54.256
a squared b.

00:07:54.256 --> 00:07:56.456
And you could multiply it out,
and we did it.

00:07:56.456 --> 00:07:59.316
We did it all the way back over here.

00:07:59.316 --> 00:08:02.006
There's three ways to get a squared b.

00:08:02.006 --> 00:08:04.066
We saw that right over there.

00:08:04.066 --> 00:08:06.293
And there are three ways

00:08:06.293 --> 00:08:08.233
to get a b squared.

00:08:08.233 --> 00:08:11.763
Three ways to get a b squared.

00:08:11.763 --> 00:08:16.700
And if you sum this up you have the
expansion of a plus b to the third power.

00:08:16.700 --> 00:08:19.359
So hopefully you found that interesting.


Binomial theorem

WEBVTT
Kind: captions
Language: en

00:00:00.637 --> 00:00:02.500
Voiceover:It doesn't take long to realize

00:00:02.500 --> 00:00:05.166
that taking higher and
higher powers of binomials

00:00:05.166 --> 00:00:06.501
can get painful,

00:00:06.501 --> 00:00:07.634
but let's just work through a few

00:00:07.634 --> 00:00:11.500
just to realize how
quickly they get painful.

00:00:11.500 --> 00:00:14.633
If we take the binomial a plus b,

00:00:14.633 --> 00:00:17.500
it's a binomial because it
has two terms right over here,

00:00:17.500 --> 00:00:19.300
let's take that to the 0 power.

00:00:19.300 --> 00:00:21.500
Anything that's non-zero to the 0 power,

00:00:21.500 --> 00:00:23.168
that's just going to be equal to 1.

00:00:23.168 --> 00:00:24.300
That wasn't too bad.

00:00:24.300 --> 00:00:28.766
Now what about a plus b to the 1st power?

00:00:28.766 --> 00:00:31.766
That's just going to be a plus b.

00:00:31.766 --> 00:00:36.900
Now what about a plus b squared?

00:00:36.900 --> 00:00:40.169
Now, if you haven't been practicing

00:00:40.169 --> 00:00:41.633
taking powers of binomials,

00:00:41.633 --> 00:00:43.033
you might have been tempted to say

00:00:43.033 --> 00:00:44.833
a squared plus b squared,

00:00:44.833 --> 00:00:47.034
but that would be incorrect.

00:00:47.034 --> 00:00:48.300
If you did that, you should give yourself

00:00:48.300 --> 00:00:50.766
a very gentle but not overly discouraging

00:00:50.766 --> 00:00:53.900
slap on the wrist or
the brain or something.

00:00:53.900 --> 00:00:57.033
a plus b squared is not
a squared plus b squared.

00:00:57.033 --> 00:01:01.967
It is a plus b times a plus b.

00:01:01.967 --> 00:01:04.566
Then if you do this, it will be a times a,

00:01:04.566 --> 00:01:06.034
which is a squared,

00:01:06.034 --> 00:01:09.166
plus a times b, which is ab,

00:01:09.166 --> 00:01:12.233
plus b times a, which is another ab,

00:01:12.233 --> 00:01:15.633
plus b times b, which is b squared.

00:01:15.633 --> 00:01:17.902
You have two ab's here, so
you could add them together,

00:01:17.902 --> 00:01:24.068
so it's equal to a squared
plus 2ab plus b squared.

00:01:24.638 --> 00:01:28.433
Now things are going to get a
little bit more interesting.

00:01:28.433 --> 00:01:32.921
What is a plus b to the 3rd
power going to be equal to?

00:01:32.921 --> 00:01:34.033
I encourage you to pause this video

00:01:34.033 --> 00:01:35.813
and try to figure that out on your own.

00:01:36.167 --> 00:01:38.366
Well, we know that a
plus b to the 3rd power

00:01:38.366 --> 00:01:43.902
is just a plus b to the 2nd
power times another a plus b.

00:01:43.902 --> 00:01:45.693
Let's just multiply this times a plus b

00:01:45.693 --> 00:01:47.569
to figure out what it is.

00:01:47.569 --> 00:01:48.068
I'll do this.

00:01:48.068 --> 00:01:48.706
Let's see.

00:01:48.706 --> 00:01:52.733
Let's multiply that times a plus b.

00:01:52.900 --> 00:01:54.366
I'm just going to multiply it this way.

00:01:54.366 --> 00:01:57.386
First, I'll multiply b
times all of these things.

00:01:58.017 --> 00:02:00.566
I'll do it in this green color.

00:02:00.566 --> 00:02:03.966
b times b squared is b to the 3rd power.

00:02:03.966 --> 00:02:07.767
b times 2ab is 2a squared,

00:02:07.767 --> 00:02:11.367
so 2ab squared,

00:02:11.367 --> 00:02:14.501
and then b times a squared is ba squared,

00:02:14.501 --> 00:02:16.033
or a squared b,

00:02:16.033 --> 00:02:18.501
a squared b.

00:02:18.501 --> 00:02:21.167
I'll multiply b times all of this stuff.

00:02:21.167 --> 00:02:24.367
Now let's multiply a times all this stuff.

00:02:24.367 --> 00:02:28.633
a times b squared is ab squared,

00:02:28.633 --> 00:02:30.433
ab squared.

00:02:30.433 --> 00:02:34.700
a times 2ab is 2a squared b,

00:02:34.700 --> 00:02:37.033
2a squared b,

00:02:37.033 --> 00:02:40.200
and then a times a squared
is a to the 3rd power.

00:02:40.834 --> 00:02:43.567
Now when we add all of
these things together,

00:02:43.567 --> 00:02:44.833
we get,

00:02:44.833 --> 00:02:49.236
we get a to the 3rd power plus, let's see,

00:02:49.236 --> 00:02:52.233
we have 1 a squared b plus another,

00:02:52.233 --> 00:02:53.766
plus 2 more a squared b's.

00:02:53.766 --> 00:02:56.901
That's going to be 3a squared b

00:02:56.901 --> 00:03:00.007
plus 3ab squared.

00:03:00.007 --> 00:03:01.566
2ab squared plus another ab squared

00:03:01.566 --> 00:03:04.766
is going to be 3ab squared

00:03:04.766 --> 00:03:08.434
plus b to the 3rd power.

00:03:08.434 --> 00:03:09.889
Just taking some of the 3rd power,

00:03:09.889 --> 00:03:12.566
this already took us a little
reasonable amount of time,

00:03:12.566 --> 00:03:16.027
and so you can imagine
how painful it might get

00:03:16.027 --> 00:03:20.002
to do something like a
plus b to the 4th power,

00:03:20.002 --> 00:03:21.167
or even worse,

00:03:21.167 --> 00:03:25.367
if you're trying to find a
plus b to the 10th power,

00:03:25.367 --> 00:03:26.978
or to the 20th power.

00:03:26.978 --> 00:03:28.700
This would take you all day

00:03:28.700 --> 00:03:30.012
or maybe even longer than that.

00:03:30.012 --> 00:03:32.966
It would be incredibly,
incredibly painful.

00:03:32.966 --> 00:03:37.672
That's where the binomial
theorem becomes useful.

00:03:37.900 --> 00:03:40.833
What is the binomial theorem?

00:03:40.833 --> 00:03:43.976
The binomial theorem tells us,

00:03:43.976 --> 00:03:46.239
let me write this down,

00:03:46.239 --> 00:03:48.167
binomial theorem.

00:03:48.167 --> 00:03:51.913
Binomial theorem,

00:03:52.633 --> 00:03:54.433
it tells us that if we have a binomial,

00:03:54.433 --> 00:03:56.700
and I'll just stick with
the a plus b for now,

00:03:56.700 --> 00:03:58.234
if I have, and I'm going to try

00:03:58.234 --> 00:03:59.566
to color code this a little bit,

00:03:59.566 --> 00:04:03.834
if I have the binomial a plus b,

00:04:03.834 --> 00:04:05.166
a plus b,

00:04:05.166 --> 00:04:07.900
and I'm going to raise it the nth power,

00:04:07.900 --> 00:04:10.233
I'm going to raise this to the nth power,

00:04:10.233 --> 00:04:12.303
the binomial theorem tells us

00:04:12.303 --> 00:04:13.966
that this is going to be equal to,

00:04:13.966 --> 00:04:14.966
and the notation is going to look

00:04:14.966 --> 00:04:16.500
a little bit complicated at first,

00:04:16.500 --> 00:04:18.838
but then we'll work
through an actual example,

00:04:18.838 --> 00:04:25.233
is going to be equal to
the sum from k equals 0,

00:04:25.233 --> 00:04:29.615
k equals 0 to n,

00:04:29.768 --> 00:04:32.501
this n and this n are the same number,

00:04:32.501 --> 00:04:33.866
of ...

00:04:35.501 --> 00:04:36.167
I don't want to ...

00:04:36.167 --> 00:04:38.301
that's kind of a garish color ...

00:04:38.301 --> 00:04:42.104
of n choose k,

00:04:42.104 --> 00:04:43.301
n choose k,

00:04:43.301 --> 00:04:44.969
and we'll review that in a second;

00:04:44.969 --> 00:04:46.900
this comes straight out of combinatorics;

00:04:46.900 --> 00:04:53.833
n choose k times a to the n minus k,

00:04:53.833 --> 00:04:56.933
n minus k,

00:04:58.659 --> 00:05:00.834
times b,

00:05:00.834 --> 00:05:04.372
times b to the k,

00:05:04.372 --> 00:05:06.500
b to the k power.

00:05:06.500 --> 00:05:08.366
Now this seems a little bit unwieldy.

00:05:08.366 --> 00:05:10.034
Let's just review, remind ourselves

00:05:10.034 --> 00:05:12.700
what n choose k actually means.

00:05:12.700 --> 00:05:14.566
If we say n choose k,

00:05:14.566 --> 00:05:16.167
I'll do the same colors,

00:05:16.167 --> 00:05:18.900
n choose k,

00:05:18.900 --> 00:05:22.166
we remember from combinatorics
this would be equal to

00:05:22.166 --> 00:05:23.900
n factorial,

00:05:23.900 --> 00:05:29.500
n factorial over k factorial,

00:05:29.500 --> 00:05:35.299
over k factorial times
n minus k factorial,

00:05:35.299 --> 00:05:37.634
n minus k factorial,

00:05:37.634 --> 00:05:42.767
so n minus k minus k factorial,

00:05:42.767 --> 00:05:44.100
let me color code this,

00:05:44.100 --> 00:05:46.633
n minus k factorial.

00:05:46.633 --> 00:05:48.367
Let's try to apply this.

00:05:48.367 --> 00:05:50.235
Let's just start applying it to the thing

00:05:50.235 --> 00:05:51.766
that started to intimidate us,

00:05:51.766 --> 00:05:54.897
say, a plus b to the 4th power.

00:05:55.202 --> 00:05:56.694
Let's figure out what that's going to be.

00:05:56.694 --> 00:05:57.515
Let's try this.

00:05:57.515 --> 00:05:59.166
So a, and I'm going to
try to keep it color-coded

00:05:59.166 --> 00:06:00.834
so you know what's going on,

00:06:00.834 --> 00:06:02.238
a plus b,

00:06:02.238 --> 00:06:03.540
although it takes me
a little bit more time

00:06:03.540 --> 00:06:04.772
to keep switching colors,

00:06:04.772 --> 00:06:06.003
but hopefully it's worth it,

00:06:06.003 --> 00:06:07.033
a plus b.

00:06:07.033 --> 00:06:09.900
Let's take that to the 4th power.

00:06:09.900 --> 00:06:11.501
The binomial theorem tells us

00:06:11.501 --> 00:06:13.033
this is going to be equal to,

00:06:13.033 --> 00:06:14.967
and I'm just going to
use this exact notation,

00:06:14.967 --> 00:06:19.433
this is going to be the
sum from k equals 0,

00:06:19.433 --> 00:06:22.100
k equals 0 to 4,

00:06:22.100 --> 00:06:27.834
to 4 of 4 choose k,

00:06:27.834 --> 00:06:30.433
4 choose k,

00:06:30.433 --> 00:06:32.240
4 choose ...

00:06:32.240 --> 00:06:34.311
let me do that k in that purple color,

00:06:34.311 --> 00:06:42.900
4 choose k of a to the 4 minus k power,

00:06:42.900 --> 00:06:45.900
4 minus k power

00:06:45.900 --> 00:06:48.300
times b to the k power,

00:06:48.300 --> 00:06:50.233
b to the k power.

00:06:50.233 --> 00:06:51.900
Now what is that going to be equal to?

00:06:52.238 --> 00:06:54.167
Well, let's just actually just do the sum.

00:06:54.167 --> 00:06:56.236
This is going to be equal to,

00:06:56.236 --> 00:06:58.834
so we're going to start at k equals 0,

00:06:58.834 --> 00:07:03.634
so when k equals 0, it's
going to be 4 choose 0,

00:07:03.634 --> 00:07:07.234
4 choose 0,

00:07:07.234 --> 00:07:12.100
times a to the 4 minus 0 power,

00:07:12.100 --> 00:07:15.234
well, that's just going
to be a to the 4th power,

00:07:15.234 --> 00:07:17.436
times b to the 0 power.

00:07:17.436 --> 00:07:19.720
b to the 0 power is just
going to be equal to 1,

00:07:19.720 --> 00:07:21.048
so we could just put a
1 here if we want to,

00:07:21.048 --> 00:07:22.367
or we could just leave it like that.

00:07:22.367 --> 00:07:24.899
This is what we get when k equals 0.

00:07:24.899 --> 00:07:28.166
Then to that, we're going
to add when k equals 1.

00:07:28.166 --> 00:07:31.568
k equals 1 is going to be,

00:07:31.568 --> 00:07:35.966
the coefficient is going to be 4 choose 1,

00:07:35.966 --> 00:07:41.700
and it's going to be times
a to the 4 minus 1 power,

00:07:41.700 --> 00:07:44.100
so a to the 3rd power,

00:07:44.100 --> 00:07:45.766
and I'll just stick with that color,

00:07:45.766 --> 00:07:48.300
times b to the k power.

00:07:48.300 --> 00:07:51.300
Well, now, k is 1b to the 1st power.

00:07:51.300 --> 00:07:54.168
Then to that, we're going to add,

00:07:54.168 --> 00:07:57.433
we're going to add 4 choose 2,

00:07:57.433 --> 00:08:03.500
4 choose 2 times a to the ...

00:08:03.500 --> 00:08:04.700
well, now k is 2.

00:08:04.700 --> 00:08:06.167
4 minus 2 is 2.

00:08:06.167 --> 00:08:06.965
a squared.

00:08:06.965 --> 00:08:07.859
I think you see a pattern here.

00:08:07.859 --> 00:08:09.985
a to the 4th, a to the 3rd, a squared,

00:08:09.985 --> 00:08:11.566
and then times b to the k.

00:08:11.566 --> 00:08:13.907
Well, k is 2 now, so b squared,

00:08:13.907 --> 00:08:15.539
and you see a pattern again.

00:08:15.601 --> 00:08:18.966
You could say b to the
0, b to the 1, b squared,

00:08:18.966 --> 00:08:22.033
and we only have two
more terms to add here,

00:08:22.033 --> 00:08:25.566
plus 4 choose 3,

00:08:25.566 --> 00:08:32.433
4 choose 3 times 4 minus 3 is 1,

00:08:32.433 --> 00:08:35.700
times a, or a to the 1st,
I guess we could say,

00:08:35.700 --> 00:08:37.434
and then b to the 3rd power,

00:08:37.434 --> 00:08:40.500
times a to the 1st b to the third,

00:08:40.500 --> 00:08:42.433
and then only one more term,

00:08:42.433 --> 00:08:46.170
plus 4 choose,

00:08:46.170 --> 00:08:48.766
4 choose 4.

00:08:48.766 --> 00:08:50.033
k is now 4.

00:08:50.033 --> 00:08:52.166
This is going to be our
last term right now.

00:08:52.166 --> 00:08:54.372
We're getting k goes
from 0 all the way to 4,

00:08:54.372 --> 00:08:55.633
4 choose 4.

00:08:55.633 --> 00:08:58.766
a to the 4 minus 4,
that's just going to be 1,

00:08:58.766 --> 00:09:00.233
a to the 0, that's just 1,

00:09:00.233 --> 00:09:03.108
so we're going to be left
with just b to the k power,

00:09:03.108 --> 00:09:05.380
and b is 4 right over here.

00:09:05.380 --> 00:09:06.566
We're almost done.

00:09:06.566 --> 00:09:07.967
We've expanded it out.

00:09:07.967 --> 00:09:09.566
We just need it figure
out what 4 choose 0,

00:09:09.566 --> 00:09:11.902
4 choose 1, 4 choose 2,
et cetera, et cetera are,

00:09:11.902 --> 00:09:13.340
so let's figure that out.

00:09:13.739 --> 00:09:17.033
We could just apply this
over and over again.

00:09:17.033 --> 00:09:19.366
So 4 choose 0,

00:09:19.366 --> 00:09:26.766
4 choose 0 is equal to 4
factorial over 0 factorial

00:09:26.766 --> 00:09:28.917
times 4 minus 0 factorial.

00:09:28.917 --> 00:09:31.113
That's just going to be 4 factorial again.

00:09:31.113 --> 00:09:33.238
0 factorial, at least for these purposes,

00:09:33.238 --> 00:09:35.233
we are defining to be equal to 1,

00:09:35.233 --> 00:09:38.501
so this whole thing is
going to be equal to 1,

00:09:38.501 --> 00:09:41.233
so this coefficient is 1.

00:09:41.233 --> 00:09:41.701
Let's see.

00:09:41.701 --> 00:09:42.636
Let's keep going here.

00:09:42.636 --> 00:09:50.501
So 4 choose 1 is going to be
4 factorial over 1 factorial

00:09:51.624 --> 00:09:55.100
times 4 minus 1 factorial,

00:09:55.100 --> 00:09:58.176
4 minus 1 factorial, so 3 factorial.

00:09:58.176 --> 00:09:59.167
What's this going to be?

00:09:59.167 --> 00:10:01.567
1 factorial is just going to be 1.

00:10:01.567 --> 00:10:04.233
4 factorial is 4 times 3 times 2 times 1.

00:10:04.233 --> 00:10:06.304
3 factorial is 3 times 2 times 1.

00:10:06.304 --> 00:10:06.966
Let me make that clear.

00:10:06.966 --> 00:10:09.633
4 times 3 times 2 times 1

00:10:09.633 --> 00:10:12.033
over 3 times 2 times 1

00:10:12.033 --> 00:10:14.500
is just going to leave us with 4.

00:10:14.500 --> 00:10:17.124
This right over here
is just going to be 4.

00:10:17.124 --> 00:10:20.101
Then we need to figure
out what 4 choose 2 is.

00:10:20.101 --> 00:10:26.700
4 choose 2 is going to be 4
factorial over 2 factorial

00:10:26.700 --> 00:10:28.853
times what's 4 minus ...

00:10:28.853 --> 00:10:29.967
this is going to be n minus k,

00:10:29.967 --> 00:10:32.234
4 minus 2 over 2 factorial.

00:10:32.234 --> 00:10:34.100
So what is this going to be?

00:10:34.100 --> 00:10:36.433
Let me scroll over to
the right a little bit.

00:10:36.433 --> 00:10:40.567
This is going to be 4
times 3 times 2 times 1

00:10:40.567 --> 00:10:42.718
over 2 factorial is 2,

00:10:42.718 --> 00:10:44.676
over 2 times 2.

00:10:44.676 --> 00:10:46.500
This is 2, this is 2,

00:10:46.500 --> 00:10:48.104
so 2 times 2 is same thing as 4.

00:10:48.104 --> 00:10:49.643
We're left with 3 times 2 times 1,

00:10:49.643 --> 00:10:51.974
which is equal to 6.

00:10:52.189 --> 00:10:53.966
That's equal to 6.

00:10:53.966 --> 00:10:56.700
Then what is 4 choose 3?

00:10:56.700 --> 00:10:59.833
I'll use some space down here.

00:10:59.833 --> 00:11:02.633
So 4 choose 3,

00:11:02.633 --> 00:11:08.104
4 choose 3 is equal to 4 factorial over,

00:11:08.104 --> 00:11:12.033
over 3 factorial times
4 minus 3 factorial,

00:11:12.033 --> 00:11:14.311
so that's just going to be 1 factorial.

00:11:14.311 --> 00:11:15.844
Well, we already figured out what that is.

00:11:15.844 --> 00:11:17.367
That's the same thing
as this right over here.

00:11:17.367 --> 00:11:19.900
You just swap the 1 factorial
and the 3 factorial.

00:11:19.900 --> 00:11:23.566
We already figured out that
this is going to be equal to 4.

00:11:23.566 --> 00:11:25.500
That is equal to 4.

00:11:25.500 --> 00:11:27.366
4 choose 4?

00:11:27.366 --> 00:11:29.508
Well, this is just going to be,

00:11:29.508 --> 00:11:30.576
let me just do it over here,

00:11:30.576 --> 00:11:36.709
4 choose 4 is 4 factorial over 4 factorial

00:11:36.709 --> 00:11:39.300
times 0 factorial,

00:11:39.300 --> 00:11:40.833
which is the exact thing we had here,

00:11:40.833 --> 00:11:43.166
which we figured out was 1.

00:11:43.166 --> 00:11:44.700
Just like that, we're done.

00:11:44.700 --> 00:11:46.100
We were able to figure out

00:11:46.100 --> 00:11:48.700
what a plus b to the 4th power is.

00:11:48.700 --> 00:11:52.369
It's 1a to the 4th plus
4a to the 3rd b to the 1st

00:11:52.369 --> 00:11:56.766
plus 6a squared b squared plus 4ab cubed

00:11:56.766 --> 00:11:57.680
plus b to the 4th.

00:11:57.680 --> 00:11:58.766
Actually, let me just write that down,

00:11:58.766 --> 00:12:00.525
since we did all that work.

00:12:00.525 --> 00:12:04.300
This is equal to a to the 4th plus,

00:12:04.300 --> 00:12:06.999
plus 4,

00:12:06.999 --> 00:12:10.167
plus 4a to the 3rd,

00:12:10.167 --> 00:12:14.817
a to the 3rd b plus,

00:12:15.679 --> 00:12:17.321
plus 6,

00:12:17.721 --> 00:12:22.300
plus 6a squared b squared,

00:12:22.300 --> 00:12:25.033
a squared b squared,

00:12:25.033 --> 00:12:26.027
plus,

00:12:26.966 --> 00:12:28.089
plus,

00:12:28.535 --> 00:12:31.566
plus 4, I think you see a pattern here,

00:12:31.566 --> 00:12:36.904
plus 4a times b to the 3rd power

00:12:37.658 --> 00:12:39.586
plus b to the 4th power,

00:12:40.033 --> 00:12:41.710
plus b to the 4th power.

00:12:41.710 --> 00:12:42.767
There is an interesting pattern here.

00:12:42.767 --> 00:12:45.359
There is a symmetry where
you have the coefficient,

00:12:45.359 --> 00:12:47.952
you go 1, 4, 6 for the middle term,

00:12:47.952 --> 00:12:48.936
and then you go back to 4,

00:12:48.936 --> 00:12:51.500
and then you go back to 1.

00:12:51.500 --> 00:12:52.548
Then you also see that pattern,

00:12:52.548 --> 00:12:54.515
is that you start at a
to the 4th, a to the 3rd,

00:12:54.515 --> 00:12:55.515
a squared, a,

00:12:55.515 --> 00:12:57.966
and then you could say
there is an a to the 0 here,

00:12:57.966 --> 00:12:59.567
and then you started b to the 0,

00:12:59.567 --> 00:13:01.253
which we didn't write
it, but that's just 1,

00:13:01.253 --> 00:13:03.647
then b to the 1st, b
squared, b to the 3rd,

00:13:03.647 --> 00:13:05.067
b to the 4th.

00:13:05.227 --> 00:13:08.100
This is just one
application or one example.

00:13:08.100 --> 00:13:09.501
In future videos, we'll do more examples

00:13:09.501 --> 00:13:10.500
of the binomial theorem

00:13:10.500 --> 00:13:13.934
and also try to understand why it works.


Example factoring sum of squares

WEBVTT
Kind: captions
Language: en

00:00:00.727 --> 00:00:02.023
Voiceover:Let's see if we can factor

00:00:02.023 --> 00:00:06.242
36a to the eighth, plus
2b to the sixth power

00:00:06.242 --> 00:00:07.558
and I encourage you to pause the video

00:00:07.558 --> 00:00:09.789
and try it out on your own.

00:00:09.789 --> 00:00:11.885
So let's see if we can express this,

00:00:11.885 --> 00:00:15.960
or re-express this as
the difference of squares

00:00:15.960 --> 00:00:18.209
using imaginary numbers.

00:00:18.209 --> 00:00:22.088
So we can rewrite 36 as, six squared,

00:00:22.088 --> 00:00:24.367
and a to the 8th is the same thing as,

00:00:24.367 --> 00:00:27.318
a to the fourth squared,
and so let me actually

00:00:27.318 --> 00:00:28.307
just rewrite it this way.

00:00:28.307 --> 00:00:33.247
We can rewrite it as 6a
to the fourth squared.

00:00:33.247 --> 00:00:36.180
That's this first term right over here.

00:00:36.180 --> 00:00:41.210
And then the second
term we can write it as,

00:00:41.210 --> 00:00:42.435
actually just let me
write it this way first.

00:00:42.435 --> 00:00:44.378
Let me just write it as a square.

00:00:44.378 --> 00:00:51.944
Plus, the square root of 2, b
to the third power, squared.

00:00:51.944 --> 00:00:54.337
Now we wanted to write it
as a difference of squares.

00:00:54.337 --> 00:00:56.454
So instead of writing it this way,

00:00:56.454 --> 00:00:58.217
let's get rid of this plus, and let's -

00:00:58.217 --> 00:01:03.373
so let me clear that out,
and I could write it as

00:01:03.373 --> 00:01:08.826
subtracting a negative one times that,

00:01:08.826 --> 00:01:11.391
and negative one, we
know, is the same thing

00:01:11.391 --> 00:01:13.497
as i squared, so we can rewrite this

00:01:13.497 --> 00:01:18.130
whole thing as 6a to the fourth, squared.

00:01:18.130 --> 00:01:21.910
And then we have this minus
right over here, minus.

00:01:21.910 --> 00:01:24.177
And so this is i squared.

00:01:24.177 --> 00:01:25.356
Negative one is i squared.

00:01:25.356 --> 00:01:30.162
So we can rewrite this
in this pink color as

00:01:30.162 --> 00:01:34.650
i times the square root of
two, times b to the third,

00:01:34.650 --> 00:01:36.514
all of that squared.

00:01:36.514 --> 00:01:38.673
Notice i squared is negative one.

00:01:38.673 --> 00:01:41.463
Square root of two squared, is two.

00:01:41.463 --> 00:01:45.817
B to the third, squared,
is b to the sixth power.

00:01:45.817 --> 00:01:46.851
If I raise something to an exponent,

00:01:46.851 --> 00:01:47.860
and then raise it to another exponent,

00:01:47.860 --> 00:01:49.894
I would multiply the two exponents.

00:01:49.894 --> 00:01:51.432
And so now I've expressed it as

00:01:51.432 --> 00:01:54.881
a difference of squares,
so we're ready to factor.

00:01:54.881 --> 00:02:00.107
This is going to be equal
to 6a to the fourth,

00:02:00.107 --> 00:02:06.649
minus i times the square root of two,

00:02:06.649 --> 00:02:11.193
times b to the third, times -

00:02:11.193 --> 00:02:13.335
and let me get myself space here,

00:02:13.335 --> 00:02:24.369
times 6a to the fourth,
plus all of this business,

00:02:24.369 --> 00:02:26.139
i times the square root of 2,

00:02:26.139 --> 00:02:30.461
times b to the third
power, and we are done.


Possible number of real roots

WEBVTT
Kind: captions
Language: en

00:00:00.725 --> 00:00:03.010
Voiceover:So we have a
polynomial right over here.

00:00:03.010 --> 00:00:05.631
We have a function p(x)
defined by this polynomial.

00:00:05.631 --> 00:00:08.891
It's clearly a 7th degree polynomial,

00:00:08.891 --> 00:00:11.444
and what I want to do is think about,

00:00:11.444 --> 00:00:16.340
what are the possible number of real roots

00:00:16.340 --> 00:00:19.370
for this polynomial right over here.

00:00:19.370 --> 00:00:21.907
So what are the possible
number of real roots?

00:00:21.907 --> 00:00:23.949
For example, could you have 9 real roots?

00:00:23.949 --> 00:00:25.847
And so I encourage you to pause this video

00:00:25.847 --> 00:00:27.540
and think about, what are all the possible

00:00:27.540 --> 00:00:29.287
number of real roots?

00:00:30.301 --> 00:00:31.884
So I'm assuming you've given a go at it,

00:00:31.884 --> 00:00:34.819
so the Fundamental Theorem of Algebra

00:00:34.819 --> 00:00:40.208
tells us that we are definitely
going to have 7 roots

00:00:40.208 --> 00:00:43.676
some of which, could be actually real.

00:00:43.717 --> 00:00:45.760
So we're definitely not going to have

00:00:45.760 --> 00:00:47.948
8 or 9 or 10 real roots,

00:00:47.948 --> 00:00:50.214
at most we're going to have 7 real roots,

00:00:50.214 --> 00:00:52.380
so possible number of real roots,

00:00:52.380 --> 00:00:55.127
so possible - let me write this down -

00:00:55.127 --> 00:01:01.636
possible number of real roots.

00:01:01.636 --> 00:01:03.703
Well 7 is a possibility.

00:01:03.703 --> 00:01:05.549
If you graphed this out,

00:01:05.549 --> 00:01:09.947
it could potentially
intersect the x-axis 7 times.

00:01:09.947 --> 00:01:14.024
Now, would it be possible
to have 6 real roots?

00:01:14.024 --> 00:01:17.002
Is 6 real roots a possibility?

00:01:17.002 --> 00:01:18.892
Is this a possibility?

00:01:18.892 --> 00:01:20.584
Well, let's think about
what that would imply

00:01:20.584 --> 00:01:23.572
about the non-real complex roots.

00:01:23.572 --> 00:01:27.368
If you have 6 real, actually
let's do it this way.

00:01:28.337 --> 00:01:31.373
Let me write it this way.

00:01:31.373 --> 00:01:40.986
So real roots and then non-real, complex.

00:01:40.986 --> 00:01:42.766
The reason I'm not just saying complex

00:01:42.766 --> 00:01:44.848
is because real numbers

00:01:44.848 --> 00:01:46.719
are a subset of complex numbers,

00:01:46.719 --> 00:01:48.177
but this is being clear
that you're talking

00:01:48.177 --> 00:01:50.291
about complex numbers that are not real.

00:01:50.291 --> 00:01:55.167
So you could have 7 real roots,

00:01:55.167 --> 00:01:57.769
and then you would have no non-real roots,

00:01:57.769 --> 00:01:59.615
so this is absolutely possible.

00:01:59.615 --> 00:02:03.159
Now could you have 6 real roots,

00:02:03.159 --> 00:02:04.976
in which case that would imply that you

00:02:04.976 --> 00:02:07.927
have 1 non-real root.

00:02:07.927 --> 00:02:11.328
Well no, you can't have
this because the non-real

00:02:11.328 --> 00:02:15.117
complex roots come in
pairs, conjugate pairs,

00:02:15.117 --> 00:02:16.252
so you're always going to have

00:02:16.252 --> 00:02:18.781
an even number of non-real complex roots.

00:02:18.781 --> 00:02:22.751
So you can't just have 1,
so let's rule that out.

00:02:22.927 --> 00:02:26.501
Now what about having 5 real roots?

00:02:26.501 --> 00:02:30.630
That means that you would
have 2 non-real complex,

00:02:30.630 --> 00:02:34.328
adding up to 7, and that
of course is possible

00:02:34.328 --> 00:02:36.322
because now you have a pair here.

00:02:36.632 --> 00:02:38.432
So I think you're
starting to see a pattern.

00:02:38.432 --> 00:02:41.787
Essentially you can have
an odd number of real roots

00:02:41.787 --> 00:02:44.366
up to and including 7.

00:02:44.366 --> 00:02:46.408
So for example,this is possible

00:02:46.408 --> 00:02:47.985
and I could just keep going.

00:02:47.985 --> 00:02:51.517
I could have, let's see, 4 and 3.

00:02:51.517 --> 00:02:52.561
This is not possible

00:02:52.561 --> 00:02:53.686
because I have an odd number here.

00:02:53.686 --> 00:02:55.594
You're going to have
to have an even number

00:02:55.594 --> 00:02:57.710
of non-real complex roots.

00:02:57.710 --> 00:03:01.758
So rule that out, but
then if we go to 3 and 4,

00:03:01.758 --> 00:03:03.099
this is absolutely possible.

00:03:03.099 --> 00:03:06.261
You have two pairs of
non-real complex roots.

00:03:06.261 --> 00:03:09.256
And then we can go to 2 and 5,

00:03:09.256 --> 00:03:11.011
once again this is an odd number,

00:03:11.011 --> 00:03:13.881
these come in pairs,
so this is impossible.

00:03:13.881 --> 00:03:19.552
And then you could go to
1 real and 6 non-real.

00:03:19.552 --> 00:03:22.638
Completely possible,
this is an even number.

00:03:22.638 --> 00:03:24.951
And then finally, we could consider having

00:03:24.951 --> 00:03:28.755
0 real and 7 non-real complex

00:03:28.755 --> 00:03:30.397
and that's not possible because these

00:03:30.397 --> 00:03:31.990
are always going to
come in pairs, so you're

00:03:32.020 --> 00:03:33.596
always going to have an even number here.

00:03:33.907 --> 00:03:36.085
So the possible number of real roots,

00:03:36.085 --> 00:03:38.929
you could have 7 real roots, 5 real roots,

00:03:38.929 --> 00:03:41.591
3 real roots or 1 real root

00:03:41.591 --> 00:03:45.177
for this 7th degree polynomial.