Factorization with substitution

WEBVTT
Kind: captions
Language: en

00:00:00.480 --> 00:00:01.970
- [Instructor] We're told
that we want to factor

00:00:01.970 --> 00:00:04.400
the following expression
that they have right here

00:00:04.400 --> 00:00:06.800
and they say that we can
factor the expression

00:00:06.800 --> 00:00:09.650
as U plus V squared

00:00:09.650 --> 00:00:12.500
where U and V are either constant integers

00:00:12.500 --> 00:00:14.860
or single variable expressions.

00:00:14.860 --> 00:00:16.100
What are U and V?

00:00:16.100 --> 00:00:18.400
And then they ask us to
actually factor the expression.

00:00:18.400 --> 00:00:20.783
So pause this video and see
if you can work on that.

00:00:21.670 --> 00:00:23.750
All right, so let's go
with the first part of it.

00:00:23.750 --> 00:00:25.280
So they say they can factor the expression

00:00:25.280 --> 00:00:28.340
as U plus V squared.

00:00:28.340 --> 00:00:31.700
So how do we see this expression in terms

00:00:31.700 --> 00:00:33.970
of U plus V squared?

00:00:33.970 --> 00:00:36.040
Well one way is to just remind ourselves

00:00:36.040 --> 00:00:39.300
what U plus V squared even is.

00:00:39.300 --> 00:00:42.310
U plus V squared, this
is just going to be a,

00:00:42.310 --> 00:00:44.440
the square of a binomial
and you've seen this

00:00:44.440 --> 00:00:46.300
in many, many other videos.

00:00:46.300 --> 00:00:50.440
This is going to be U
squared plus two times

00:00:50.440 --> 00:00:51.950
the product of these two terms.

00:00:51.950 --> 00:00:55.240
So 2UV plus V squared.

00:00:55.240 --> 00:00:56.610
If you've never seen this before

00:00:56.610 --> 00:00:59.180
or not sure where this
came from, I encourage you

00:00:59.180 --> 00:01:00.660
to watch some of those early videos

00:01:00.660 --> 00:01:03.120
where we explain this out.

00:01:03.120 --> 00:01:06.150
But, does this match this pattern?

00:01:06.150 --> 00:01:10.790
Well can we express
this term as U squared?

00:01:10.790 --> 00:01:13.590
Well if this is U
squared, then U would have

00:01:13.590 --> 00:01:15.730
to be equal to X plus seven

00:01:15.730 --> 00:01:18.040
and when I say, actually I
should be a little careful.

00:01:18.040 --> 00:01:20.940
Can we express this entire
thing right over here

00:01:20.940 --> 00:01:22.640
as U squared?

00:01:22.640 --> 00:01:25.495
If U squared is equal
to X plus seven squared,

00:01:25.495 --> 00:01:30.020
that means that U is going
to be equal to X plus seven

00:01:30.020 --> 00:01:32.190
and then this right over here

00:01:32.190 --> 00:01:34.210
would have to be V squared.

00:01:34.210 --> 00:01:37.110
If this is V squared,
then that means that V

00:01:37.110 --> 00:01:40.130
is equal to Y squared
because Y squared squared

00:01:40.130 --> 00:01:41.850
is equal to Y to the fourth.

00:01:41.850 --> 00:01:46.010
So V is equal to Y squared.

00:01:46.010 --> 00:01:48.040
Now they already told us that this,

00:01:48.040 --> 00:01:51.080
this could be factored as the
expression U plus V squared,

00:01:51.080 --> 00:01:52.850
but let's make sure that
this actually works.

00:01:52.850 --> 00:01:54.860
Is this middle term right over here,

00:01:54.860 --> 00:01:59.860
is this truly equal to
two times U times V, 2UV?

00:02:01.010 --> 00:02:01.843
Well let's see.

00:02:01.843 --> 00:02:06.570
Two times U would be
two times X plus seven

00:02:06.570 --> 00:02:09.570
times V times Y squared

00:02:09.570 --> 00:02:11.860
and that's exactly what
we have right over here.

00:02:11.860 --> 00:02:15.470
It's 2Y squared times X plus seven.

00:02:15.470 --> 00:02:18.150
So this kind of hairy looking expression

00:02:18.150 --> 00:02:21.250
actually does fit this
pattern right over here.

00:02:21.250 --> 00:02:23.890
So you can view it as U plus V squared

00:02:23.890 --> 00:02:25.750
where U is equal to X plus seven

00:02:25.750 --> 00:02:29.020
and V is equal to Y squared.

00:02:29.020 --> 00:02:32.790
Now using that, we can now
actually factor the expression.

00:02:32.790 --> 00:02:35.640
We can write this thing as being equal

00:02:35.640 --> 00:02:38.320
to U plus V squared.

00:02:38.320 --> 00:02:40.290
And we know what U and V are.

00:02:40.290 --> 00:02:41.920
So this is, this whole expression

00:02:41.920 --> 00:02:46.920
is going to be equal to
U, which is X plus seven,

00:02:47.100 --> 00:02:48.580
and I'll put it in
parentheses just so you see it

00:02:48.580 --> 00:02:53.470
very clearly, plus V
plus Y squared squared,

00:02:53.470 --> 00:02:56.440
'cause that's exactly
what we wrote over there.

00:02:56.440 --> 00:02:58.340
And of course, you don't have
to write these parentheses.

00:02:58.340 --> 00:03:01.100
You could rewrite this as X plus seven

00:03:01.100 --> 00:03:04.070
plus Y squared squared.

00:03:04.070 --> 00:03:05.823
Let's do another example.

00:03:07.540 --> 00:03:10.170
So here, once again, we are told that

00:03:10.170 --> 00:03:12.010
we want to factor the following expression

00:03:12.010 --> 00:03:14.140
and they're saying that we
can factor the expression

00:03:14.140 --> 00:03:16.660
as U plus V times U minus V

00:03:16.660 --> 00:03:18.670
where U and V are either constant integers

00:03:18.670 --> 00:03:20.430
or single variable expressions.

00:03:20.430 --> 00:03:23.380
So pause this video and try
to figure out what U and V are

00:03:23.380 --> 00:03:26.100
and then actually factor the expression.

00:03:26.100 --> 00:03:28.300
All right, well let's just
remind ourselves in general

00:03:28.300 --> 00:03:33.300
what U plus V times U minus V is equal to.

00:03:33.980 --> 00:03:35.990
Well, if this is unfamiliar to you,

00:03:35.990 --> 00:03:37.330
I encourage you to watch the videos

00:03:37.330 --> 00:03:39.090
on difference of squares.

00:03:39.090 --> 00:03:40.690
When you multiply this all out

00:03:40.690 --> 00:03:42.960
this is going to give you
a difference of squares.

00:03:42.960 --> 00:03:45.810
U squared minus V squared.

00:03:45.810 --> 00:03:48.430
If you actually take the
trouble of multiplying this out,

00:03:48.430 --> 00:03:49.960
you're going to see that that middle term,

00:03:49.960 --> 00:03:52.830
that middle third term, or
the middle terms I should say,

00:03:52.830 --> 00:03:55.060
cancel out, so you're just
left with the U squared

00:03:55.060 --> 00:03:56.960
minus the V squared.

00:03:56.960 --> 00:03:59.080
And so does this fit this pattern?

00:03:59.080 --> 00:04:02.240
Well in order for this to be U squared

00:04:02.240 --> 00:04:05.600
and for this to be V squared,

00:04:05.600 --> 00:04:08.310
that means U squared
is equal to 4X squared

00:04:08.310 --> 00:04:09.950
so that means that U
would have to be equal

00:04:09.950 --> 00:04:12.840
to the square root of that
which would be two times X.

00:04:12.840 --> 00:04:17.750
Notice, U squared would be 2X
squared which is 4X squared

00:04:17.750 --> 00:04:20.060
and then V would have to be equal

00:04:20.060 --> 00:04:22.730
to the square root of 9Y to the sixth.

00:04:22.730 --> 00:04:24.491
With the square root of nine is three

00:04:24.491 --> 00:04:27.430
and the square of Y to
the sixth is going to be Y

00:04:27.430 --> 00:04:28.770
to the third power.

00:04:28.770 --> 00:04:31.130
And then we could use that
to factor the expression,

00:04:31.130 --> 00:04:32.833
'cause we could say hey,

00:04:32.833 --> 00:04:33.820
this right over here is the same thing

00:04:33.820 --> 00:04:36.160
as U squared minus V squared,

00:04:36.160 --> 00:04:37.540
so it's going to be equal to,

00:04:37.540 --> 00:04:40.260
we can factor it out as, or factor it as,

00:04:40.260 --> 00:04:45.260
U minus or U plus V times U minus V.

00:04:45.480 --> 00:04:47.250
So what's that going to be equal to?

00:04:47.250 --> 00:04:50.690
So U plus V is going to
be equal to 2X plus 3Y

00:04:53.670 --> 00:04:56.050
to the third and then U minus V

00:04:56.050 --> 00:05:00.440
is going to be equal to 2X,
which is our U right over here,

00:05:00.440 --> 00:05:05.060
minus our V, minus 3Y to the third.

00:05:05.060 --> 00:05:05.893
So there you have it.

00:05:05.893 --> 00:05:07.250
We factored the expression.

00:05:07.250 --> 00:05:08.780
You might want to write it down here

00:05:08.780 --> 00:05:12.593
but we just did it right
up there and we're done.


Factoring higher degree polynomials

WEBVTT
Kind: captions
Language: en

00:00:00.260 --> 00:00:02.220
- [Instructor] There are
many videos on Khan Academy

00:00:02.220 --> 00:00:05.030
where we talk about factoring polynomials.

00:00:05.030 --> 00:00:06.330
What we're going to do in this video

00:00:06.330 --> 00:00:07.650
is do a few more examples

00:00:07.650 --> 00:00:10.920
of factoring higher degree polynomials.

00:00:10.920 --> 00:00:13.770
So let's start with a
little bit of a warmup.

00:00:13.770 --> 00:00:18.120
Let's say that we wanted
to factor six x squared

00:00:18.120 --> 00:00:23.120
plus nine x times x squared
minus four x plus four.

00:00:25.080 --> 00:00:27.900
Pause this video and see
if you can factor this

00:00:27.900 --> 00:00:31.433
into the product of even more expressions.

00:00:32.440 --> 00:00:33.930
All right, now let's do this together

00:00:33.930 --> 00:00:35.910
and the way that this might
be a little bit different

00:00:35.910 --> 00:00:36.850
than what you've seen before

00:00:36.850 --> 00:00:39.290
is this is already partially factored.

00:00:39.290 --> 00:00:41.820
This polynomial, this
higher degree polynomial,

00:00:41.820 --> 00:00:43.830
is already expressed as the product

00:00:43.830 --> 00:00:46.260
of two quadratic expressions

00:00:46.260 --> 00:00:47.530
but as you might be able to tell,

00:00:47.530 --> 00:00:49.610
we can factor this further.

00:00:49.610 --> 00:00:53.350
For example, six x squared plus nine x,

00:00:53.350 --> 00:00:58.350
both six x squared and nine
x are divisible by three x.

00:00:58.550 --> 00:01:00.470
So let's factor out a three x here.

00:01:00.470 --> 00:01:03.180
So this is the same
thing as three x times,

00:01:03.180 --> 00:01:06.110
three x times what is six x squared?

00:01:06.110 --> 00:01:11.110
Well, three times two is six
and x times x is x squared

00:01:12.080 --> 00:01:15.160
and then three x times what is nine x?

00:01:15.160 --> 00:01:18.950
Well, three x times three
is nine x and you can verify

00:01:18.950 --> 00:01:21.140
that if we were to
distribute this three x,

00:01:21.140 --> 00:01:24.920
you would get six x squared plus nine x

00:01:24.920 --> 00:01:28.130
and then what about this second
expression right over here?

00:01:28.130 --> 00:01:30.020
Can we factor this?

00:01:30.020 --> 00:01:32.520
Well, you might recognize
this as a perfect square.

00:01:32.520 --> 00:01:33.353
Some of you might have said,

00:01:33.353 --> 00:01:34.760
hey, I need to come up with two numbers

00:01:34.760 --> 00:01:37.670
whose product is four and
whose sum is negative four

00:01:37.670 --> 00:01:40.410
and you might say, hey, that's
negative two and negative two

00:01:40.410 --> 00:01:44.460
and so this would be x minus two.

00:01:44.460 --> 00:01:46.040
We could write x minus two squared

00:01:46.040 --> 00:01:49.890
or we could write it as x
minus two times x minus two.

00:01:49.890 --> 00:01:51.810
If what I just did is unfamiliar,

00:01:51.810 --> 00:01:54.640
I encourage you to go
back and watch videos

00:01:54.640 --> 00:01:58.670
on factoring perfect square
quadratics and things like that

00:01:58.670 --> 00:01:59.503
but there you have it.

00:01:59.503 --> 00:02:03.410
I think we have factored
this as far as we can go.

00:02:03.410 --> 00:02:06.240
So now let's do a slightly trickier

00:02:06.240 --> 00:02:08.590
higher degree polynomial.

00:02:08.590 --> 00:02:11.910
So let's say we wanted
to factor x to the third

00:02:11.910 --> 00:02:16.890
minus four x squared plus six x minus 24

00:02:18.900 --> 00:02:21.050
and just like always, pause this video

00:02:21.050 --> 00:02:22.480
and see if you can have a go at it

00:02:22.480 --> 00:02:24.310
and I'll give you a little bit of a hint.

00:02:24.310 --> 00:02:27.350
You can factor in this case by grouping

00:02:27.350 --> 00:02:29.350
and in some ways it's a little bit easier

00:02:29.350 --> 00:02:30.710
than what we've done in the past.

00:02:30.710 --> 00:02:33.470
Historically, when we've
learned factoring by grouping,

00:02:33.470 --> 00:02:35.630
we've looked at a quadratic

00:02:35.630 --> 00:02:37.510
and then we looked at the middle term,

00:02:37.510 --> 00:02:39.910
the x term of the quadratic
and we broke it up

00:02:39.910 --> 00:02:41.420
so that we had four terms.

00:02:41.420 --> 00:02:43.430
Here, we already have four terms.

00:02:43.430 --> 00:02:45.080
See if you can have a go at that.

00:02:46.450 --> 00:02:48.410
All right, now let's do it together.

00:02:48.410 --> 00:02:51.910
So you can't always factor
a third degree polynomial

00:02:51.910 --> 00:02:53.610
by grouping but sometimes you can

00:02:53.610 --> 00:02:55.230
so it's good to look for it.

00:02:55.230 --> 00:02:57.430
So when we see it written like this,

00:02:57.430 --> 00:03:00.150
we say okay, x to the
third minus four x squared,

00:03:00.150 --> 00:03:02.450
is there a common factor here?

00:03:02.450 --> 00:03:05.620
Well, yeah, both x to the third
and negative four x squared

00:03:05.620 --> 00:03:07.160
are divisible by x squared.

00:03:07.160 --> 00:03:09.250
So what happens if we
factor out an x squared?

00:03:09.250 --> 00:03:13.520
So that's x squared times x minus four

00:03:13.520 --> 00:03:16.100
and what about these second two terms?

00:03:16.100 --> 00:03:20.150
Is there a common factor
between six x and negative 24?

00:03:20.150 --> 00:03:23.070
Yeah, they're both divisible by six.

00:03:23.070 --> 00:03:24.370
So let's factor out a six here.

00:03:24.370 --> 00:03:29.370
So plus six times x minus four

00:03:29.540 --> 00:03:33.450
and now you are probably
seeing the homestretch

00:03:33.450 --> 00:03:35.740
where you have something
times x minus four

00:03:35.740 --> 00:03:39.140
and then something else times
x minus four and so you can,

00:03:39.140 --> 00:03:42.170
sometimes I like to say
undistribute the x minus four

00:03:42.170 --> 00:03:44.130
or factor out the x minus four

00:03:44.130 --> 00:03:49.130
and so this is going to be x
minus four times x squared,

00:03:50.440 --> 00:03:55.440
x squared plus six and we are done.


Polynomial special products: perfect square

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.220
- [Instructor] What we're
going to do in this video

00:00:01.220 --> 00:00:04.190
is practice squaring binomials,
and this is something

00:00:04.190 --> 00:00:05.720
that we've done in the
past, but we're gonna do it

00:00:05.720 --> 00:00:07.900
with slightly more involved expressions.

00:00:07.900 --> 00:00:10.370
But let's just start with
a little bit of review.

00:00:10.370 --> 00:00:13.520
If I were to ask you,
what is a plus b squared,

00:00:13.520 --> 00:00:14.353
what would that be?

00:00:14.353 --> 00:00:16.690
Pause the video and try to figure it out.

00:00:16.690 --> 00:00:18.073
Well some of you might immediately know

00:00:18.073 --> 00:00:22.530
what a binomial like this
squared is, but I'll work it out.

00:00:22.530 --> 00:00:27.390
So this is the same thing
as a plus b times a plus b.

00:00:27.390 --> 00:00:30.720
And then we could multiply
this a times that a.

00:00:30.720 --> 00:00:32.710
So that's going to give us a squared.

00:00:32.710 --> 00:00:35.490
And then I can multiply
that a times that b,

00:00:35.490 --> 00:00:37.490
and that's going to give us ab.

00:00:37.490 --> 00:00:40.810
Then I could multiply this b times that a.

00:00:40.810 --> 00:00:42.570
I could write that as ba or ab,

00:00:42.570 --> 00:00:44.410
so I'll just write it as ab again.

00:00:44.410 --> 00:00:47.730
And then I multiply this b times that b,

00:00:47.730 --> 00:00:49.650
so plus b squared.

00:00:49.650 --> 00:00:51.020
And what I really just did

00:00:51.020 --> 00:00:52.800
is apply the distributive property twice.

00:00:52.800 --> 00:00:54.897
We go into a lot of
detail in previous videos.

00:00:54.897 --> 00:00:57.700
Some people also like to
call it the FOIL method.

00:00:57.700 --> 00:00:59.680
Either way, this should all be a review.

00:00:59.680 --> 00:01:00.900
If it's not, I encourage you to look

00:01:00.900 --> 00:01:02.559
at those introductory videos.

00:01:02.559 --> 00:01:05.310
But this is going to simplify to a squared

00:01:05.310 --> 00:01:07.720
plus we have an ab and another ab,

00:01:07.720 --> 00:01:12.428
so you add those together,
you get two ab plus b squared.

00:01:12.428 --> 00:01:14.587
Now why did I go through this review?

00:01:14.587 --> 00:01:18.650
Well now we can use this
idea that a plus b squared

00:01:18.650 --> 00:01:21.450
is equal to a squared
plus two ab plus b squared

00:01:21.450 --> 00:01:22.420
to tackle things

00:01:22.420 --> 00:01:24.960
that at least look a
little bit more involved.

00:01:24.960 --> 00:01:28.700
So if I were to ask you,
what is five x to the sixth

00:01:28.700 --> 00:01:32.010
plus four squared,
pause this video and try

00:01:32.010 --> 00:01:32.843
to figure it out.

00:01:32.843 --> 00:01:36.093
And try to keep this and this in mind.

00:01:37.330 --> 00:01:39.720
Well there is several ways
you could approach this.

00:01:39.720 --> 00:01:42.330
You could just expand this
out the way we just did,

00:01:42.330 --> 00:01:44.310
or you could recognize this pattern

00:01:44.310 --> 00:01:46.010
that we just established.

00:01:46.010 --> 00:01:47.890
That if I have an a
plus b and I square it,

00:01:47.890 --> 00:01:49.080
it's going to be this.

00:01:49.080 --> 00:01:51.360
And so what you might notice is,

00:01:51.360 --> 00:01:55.550
the role of a is being
played by five x to the sixth

00:01:55.550 --> 00:01:59.843
right over there, and the role
of b is being played by four

00:01:59.843 --> 00:02:01.340
right over there.

00:02:01.340 --> 00:02:05.183
So we could say, hey, this
is going to be equal to

00:02:05.183 --> 00:02:08.100
a squared, we have our a squared there.

00:02:08.100 --> 00:02:10.130
So what is a squared?

00:02:10.130 --> 00:02:12.460
Well five x to the sixth squared is going

00:02:12.460 --> 00:02:16.820
to be 25 x to the 12th power.

00:02:16.820 --> 00:02:21.520
And then it's going to be
plus two times a times b.

00:02:21.520 --> 00:02:26.060
So plus two times five x
to the sixth times four.

00:02:26.060 --> 00:02:27.150
Actually let me just write it out

00:02:27.150 --> 00:02:28.380
just so we don't confuse ourselves.

00:02:28.380 --> 00:02:32.095
Two times five x to the,
I'll color code it too,

00:02:32.095 --> 00:02:37.095
two times five x to the
sixth times four, times four,

00:02:40.510 --> 00:02:42.040
plus b squared.

00:02:42.040 --> 00:02:46.400
So plus four squared, so
that's going to be plus 16.

00:02:46.400 --> 00:02:48.750
And then we can simplify this.

00:02:48.750 --> 00:02:52.290
So this is going to be
equal to 25 x to the 12th.

00:02:52.290 --> 00:02:55.410
Two times five times four is 40.

00:02:55.410 --> 00:02:57.430
Two times five is 10 times four is 40.

00:02:57.430 --> 00:03:02.430
So plus 40 x to the sixth plus 16.

00:03:02.660 --> 00:03:03.820
Let's do another example.

00:03:03.820 --> 00:03:05.310
And I'll do this one
even a little bit faster,

00:03:05.310 --> 00:03:08.100
just because we're getting,
I think, pretty good at this.

00:03:08.100 --> 00:03:10.380
So let's say we're trying to determine

00:03:10.380 --> 00:03:13.060
what three t squared

00:03:13.060 --> 00:03:17.130
minus seven t to the
sixth power squared is.

00:03:17.130 --> 00:03:18.953
Pause the video and try to figure it out.

00:03:18.953 --> 00:03:21.740
All right we're going
to do it together now.

00:03:21.740 --> 00:03:25.160
So this is our a, and
our b now we should view

00:03:25.160 --> 00:03:28.060
as negative seven t to the sixth.

00:03:28.060 --> 00:03:31.010
Because this says plus b, so
you could view this as plus

00:03:31.010 --> 00:03:32.870
negative seven t to the sixth.

00:03:32.870 --> 00:03:34.150
We could even write that if we wanted.

00:03:34.150 --> 00:03:36.670
We could write this plus
negative seven t to the sixth

00:03:36.670 --> 00:03:39.540
if it helps us recognize
this whole thing as b.

00:03:39.540 --> 00:03:42.010
So this is going to be equal to a squared,

00:03:42.010 --> 00:03:45.377
which is nine t to the fourth,

00:03:45.377 --> 00:03:49.540
plus two times this times this,

00:03:49.540 --> 00:03:51.130
two times a times b.

00:03:51.130 --> 00:03:56.130
So two times three t squared
is going to be six t squared

00:03:57.570 --> 00:04:00.790
times negative seven t to the sixth.

00:04:00.790 --> 00:04:01.860
Actually let me write this out.

00:04:01.860 --> 00:04:03.600
This is getting a little bit complicated.

00:04:03.600 --> 00:04:08.600
So this is going to be plus
two times three t squared

00:04:09.370 --> 00:04:14.260
times negative seven t to the sixth power.

00:04:14.260 --> 00:04:17.550
And then last but not
least, we're going to square

00:04:17.550 --> 00:04:20.070
negative seven t to the sixth.

00:04:20.070 --> 00:04:24.030
So that's going to be negative
seven squared is positive 49,

00:04:24.030 --> 00:04:27.440
and t to the sixth
squared is t to the 12th,

00:04:27.440 --> 00:04:28.950
t to the 12th power.

00:04:28.950 --> 00:04:32.470
And so this is going to be
equal to nine t to the fourth,

00:04:32.470 --> 00:04:35.450
and let's see, two times three is six

00:04:35.450 --> 00:04:39.540
times negative seven is negative 42,

00:04:39.540 --> 00:04:42.840
and t squared times t to the
sixth, we add the exponents,

00:04:42.840 --> 00:04:45.760
we have the same base, so it's
going to be t to the eighth.

00:04:45.760 --> 00:04:50.760
And then we have plus
49 t to the 12th power.

00:04:52.200 --> 00:04:54.290
So it looks like we did
something really fancy.

00:04:54.290 --> 00:04:56.810
We have this higher degree polynomial.

00:04:56.810 --> 00:04:58.910
We were squaring this binomial that has

00:04:58.910 --> 00:05:00.320
these higher degree terms.

00:05:00.320 --> 00:05:02.770
But we're really just
applying the same idea

00:05:02.770 --> 00:05:04.890
that we learned many many many videos ago,

00:05:04.890 --> 00:05:09.057
many many lessons ago, in terms
of just squaring binomials.


Finding average rate of change of polynomials

WEBVTT
Kind: captions
Language: en

00:00:00.380 --> 00:00:01.360
- [Instructor] We are asked,

00:00:01.360 --> 00:00:04.630
What is the average rate of
change of the function f,

00:00:04.630 --> 00:00:05.653
and this function is f up here,

00:00:05.653 --> 00:00:08.370
this is the definition
of it, over the interval

00:00:08.370 --> 00:00:10.580
from negative two to three,
and it's a closed interval

00:00:10.580 --> 00:00:12.830
because they put these brackets around it

00:00:12.830 --> 00:00:13.910
instead of parentheses,

00:00:13.910 --> 00:00:16.710
so that means it includes
both of these boundaries.

00:00:16.710 --> 00:00:20.190
Pause this video and see if
you can work through that.

00:00:20.190 --> 00:00:21.900
All right, now let's
work through it together.

00:00:21.900 --> 00:00:24.380
So there's a couple of ways
that we can conceptualize

00:00:24.380 --> 00:00:27.216
average rate of change of a function.

00:00:27.216 --> 00:00:29.030
One way to think about it is

00:00:29.030 --> 00:00:32.583
it's our change in the
value of our function

00:00:32.583 --> 00:00:34.640
divided by our change in x,

00:00:34.640 --> 00:00:37.230
or it's our change in
the value of our function

00:00:37.230 --> 00:00:39.400
per x on average.

00:00:39.400 --> 00:00:41.910
So you can view it as change
in the value of the function,

00:00:41.910 --> 00:00:43.660
divided by your change in x.

00:00:43.660 --> 00:00:46.410
If you say that y is equal to f of x,

00:00:46.410 --> 00:00:49.540
you could also express it as change in y

00:00:49.540 --> 00:00:51.610
over change in x.

00:00:51.610 --> 00:00:53.846
On average, how much
does the function change

00:00:53.846 --> 00:00:57.620
per unit change in x on average?

00:00:57.620 --> 00:00:59.030
And we could do this with a table,

00:00:59.030 --> 00:01:01.580
or we could try to
conceptualize it visually.

00:01:01.580 --> 00:01:03.180
Let's just do this one with a table,

00:01:03.180 --> 00:01:04.960
and then we'll try to
connect the dots a little bit

00:01:04.960 --> 00:01:06.300
with a visual.

00:01:06.300 --> 00:01:09.407
So if we have x here, and
then if we have y is equal to

00:01:09.407 --> 00:01:14.010
f of x right over here, when
x is equal to negative two,

00:01:14.010 --> 00:01:16.330
what is y going to be equal to,

00:01:16.330 --> 00:01:18.925
or what is f of negative two?

00:01:18.925 --> 00:01:19.790
Well, let's see.

00:01:19.790 --> 00:01:24.030
f of, so why is equal
to f of negative two,

00:01:24.030 --> 00:01:26.350
which is going to be
equal to negative eight,

00:01:26.350 --> 00:01:28.080
that's negative two to the third power,

00:01:28.080 --> 00:01:29.840
minus four times negative two,

00:01:29.840 --> 00:01:32.570
so that's minus negative
eight, so that's plus eight.

00:01:32.570 --> 00:01:34.140
That equals zero.

00:01:34.140 --> 00:01:37.110
And then when x is equal to three,

00:01:37.110 --> 00:01:39.820
I'm going to the right
end of that interval.

00:01:39.820 --> 00:01:42.687
Well, now y is equal to f of three,

00:01:42.687 --> 00:01:46.860
which is equal to 27,
three to the third power,

00:01:46.860 --> 00:01:50.540
minus four times three, minus
12, which is equal to 15.

00:01:50.540 --> 00:01:52.519
So what is our change in y

00:01:52.519 --> 00:01:56.790
over our change in x over this interval?

00:01:56.790 --> 00:01:59.400
Well, our y went from zero to 15,

00:01:59.400 --> 00:02:02.581
so we have an increase of 15 in y.

00:02:02.581 --> 00:02:04.450
And what was our change in x?

00:02:04.450 --> 00:02:07.120
Well, we went from negative
two to positive three,

00:02:07.120 --> 00:02:09.480
so we had a plus five change in x.

00:02:09.480 --> 00:02:11.540
So our change in x is plus five,

00:02:11.540 --> 00:02:15.140
and so our average rate of change with y

00:02:15.140 --> 00:02:17.410
with respect to x, or the
rate of change of our function

00:02:17.410 --> 00:02:19.500
with respect to x, over the interval,

00:02:19.500 --> 00:02:22.460
is going to be equal to three.

00:02:22.460 --> 00:02:24.710
If you wanted to think
about this visually,

00:02:24.710 --> 00:02:27.250
I could try to sketch this.

00:02:27.250 --> 00:02:30.750
So this is the x-axis, the y-axis,

00:02:30.750 --> 00:02:34.140
and our function does something like this.

00:02:34.140 --> 00:02:37.620
So at x equals negative
two, f of x is zero,

00:02:37.620 --> 00:02:41.430
and then it goes up, and
then it comes back down,

00:02:41.430 --> 00:02:44.250
and then it does something like this,

00:02:44.250 --> 00:02:47.710
it does something like this, and it does,

00:02:47.710 --> 00:02:49.170
and it was going before this,

00:02:49.170 --> 00:02:51.430
and so the interval that we care about,

00:02:51.430 --> 00:02:54.640
where we're going from
negative two to three,

00:02:54.640 --> 00:02:56.420
which is right about there.

00:02:56.420 --> 00:02:59.150
So that's x equals negative
two to x equals three,

00:02:59.150 --> 00:03:01.830
and so what we want to do, at
the left end of the interval,

00:03:01.830 --> 00:03:04.672
our function is equal to
zero, so we're at this point,

00:03:04.672 --> 00:03:06.710
right over there, I'll
do this in a new color,

00:03:06.710 --> 00:03:08.350
at this point, right over there,

00:03:08.350 --> 00:03:10.150
and at the right end of our function,

00:03:10.150 --> 00:03:15.150
f of three is 15, so we
are up here someplace.

00:03:15.300 --> 00:03:17.233
Let me connect the curve a little bit.

00:03:18.950 --> 00:03:21.970
We are going to be up there.

00:03:21.970 --> 00:03:24.927
And so when we're thinking
about the average rate of change

00:03:24.927 --> 00:03:27.630
over this interval, we're
really thinking about

00:03:27.630 --> 00:03:31.360
the slope of the line that
connects these two points.

00:03:31.360 --> 00:03:33.170
So the line that connects these two points

00:03:33.170 --> 00:03:35.470
looks something like this.

00:03:36.410 --> 00:03:40.170
And we're just calculating
what is our change in y,

00:03:40.170 --> 00:03:43.090
which is going to be
this, our change in y,

00:03:43.090 --> 00:03:44.890
and we see that the value of our function

00:03:44.890 --> 00:03:49.070
increased by 15, divided
by our change in x.

00:03:49.070 --> 00:03:53.800
So this right over here
is our change in x,

00:03:53.800 --> 00:03:55.480
which we see went from
negative two to three.

00:03:55.480 --> 00:03:57.160
That's going to be equal to five.

00:03:57.160 --> 00:03:58.000
So that's all we're doing

00:03:58.000 --> 00:04:00.650
when we're thinking about
the average rate of change.


Introduction to factoring higher degree monomials

WEBVTT
Kind: captions
Language: en

00:00:00.160 --> 00:00:01.550
- [Instructor] In this
video we're going to dig

00:00:01.550 --> 00:00:04.030
a little bit deeper into our knowledge

00:00:04.030 --> 00:00:06.890
or our understanding of factoring.

00:00:06.890 --> 00:00:09.240
Now factoring is something
that we've been doing

00:00:09.240 --> 00:00:11.330
for many years now.

00:00:11.330 --> 00:00:13.660
You can go all the way back
to when you were thinking

00:00:13.660 --> 00:00:16.500
about how would I factor the number 12.

00:00:16.500 --> 00:00:21.000
Well I could write the number
12 as three times four.

00:00:21.000 --> 00:00:24.330
I could also write it as two times six.

00:00:24.330 --> 00:00:25.830
These are all legitimate factors.

00:00:25.830 --> 00:00:29.290
Or I could try to do a
prime factorization of 12,

00:00:29.290 --> 00:00:31.355
where I'm trying to
write it as the product

00:00:31.355 --> 00:00:34.650
of you could view it as its
most basic constituents,

00:00:34.650 --> 00:00:36.230
which would be the prime numbers.

00:00:36.230 --> 00:00:40.690
And so we've done stuff like,
well 12 can be expressed

00:00:40.690 --> 00:00:42.090
as two times six.

00:00:42.090 --> 00:00:44.820
Two is prime, but then
six can be expressed

00:00:44.820 --> 00:00:46.660
as two times three.

00:00:46.660 --> 00:00:51.140
And so 12 could be expressed
as two times two times three,

00:00:51.140 --> 00:00:52.370
which we see right over here.

00:00:52.370 --> 00:00:55.720
This is all review, and this
would be a prime factorization.

00:00:55.720 --> 00:00:57.540
And we saw an analogue

00:00:57.540 --> 00:00:59.790
when we first learned it in algebra I.

00:00:59.790 --> 00:01:03.180
In algebra I, we learned things like,

00:01:03.180 --> 00:01:05.170
and sometimes this might
be in a math I class

00:01:05.170 --> 00:01:07.070
or even in a pre-algebra class,

00:01:07.070 --> 00:01:10.148
you'll learn things like
hey, how do I factor

00:01:10.148 --> 00:01:13.770
x squared plus six x?

00:01:13.770 --> 00:01:15.320
And you might recognize that hey,

00:01:15.320 --> 00:01:18.470
x squared could be rewritten as x times x,

00:01:18.470 --> 00:01:22.220
and six x, that really
just means six times x.

00:01:22.220 --> 00:01:26.210
And so both of them have x as a factor,

00:01:26.210 --> 00:01:29.040
and so we might wanna factor that out.

00:01:29.040 --> 00:01:31.230
And so we could rewrite
this entire expression

00:01:31.230 --> 00:01:35.050
as x times x plus six.

00:01:35.050 --> 00:01:38.198
What we just did is we
factored out these x's

00:01:38.198 --> 00:01:40.920
that I am circling in blue.

00:01:40.920 --> 00:01:43.510
So in general, this idea of factoring,

00:01:43.510 --> 00:01:44.630
if you're thinking about numbers,

00:01:44.630 --> 00:01:46.840
you're writing one number as
the product of other numbers.

00:01:46.840 --> 00:01:48.260
If you're thinking about expressions,

00:01:48.260 --> 00:01:50.220
you're writing an
expression as the product

00:01:50.220 --> 00:01:51.920
of other expressions.

00:01:51.920 --> 00:01:54.770
Well now as we go a little bit
more advanced into algebra,

00:01:54.770 --> 00:01:56.330
we're gonna start
thinking about doing this

00:01:56.330 --> 00:01:58.390
with higher order expressions.

00:01:58.390 --> 00:02:01.610
So we've done it with just
an x or just an x squared,

00:02:01.610 --> 00:02:02.670
but now we're gonna start thinking about,

00:02:02.670 --> 00:02:04.500
well what happens if we have
something to the third power,

00:02:04.500 --> 00:02:08.500
fourth power, sixth power,
10th power, 100th power?

00:02:08.500 --> 00:02:10.650
But it's really the same ideas.

00:02:10.650 --> 00:02:13.280
And we could start with monomials,

00:02:13.280 --> 00:02:15.540
which is fancy word
for just a single term.

00:02:15.540 --> 00:02:20.130
So let's say I had something
like six x to the seventh.

00:02:20.130 --> 00:02:23.610
What are the different ways
that I could factor this?

00:02:23.610 --> 00:02:24.830
Pause this video and think about it.

00:02:24.830 --> 00:02:28.343
Can I express this as the
product of two other things?

00:02:29.190 --> 00:02:33.043
Well, I could rewrite this
as this as being equal to

00:02:33.043 --> 00:02:38.043
two x to the third times what?

00:02:38.480 --> 00:02:40.610
Well let's see, what do
I have to multiply two by

00:02:40.610 --> 00:02:41.443
to get to six?

00:02:41.443 --> 00:02:42.760
I have to multiply it by three.

00:02:42.760 --> 00:02:44.590
And what do I have to
multiply x to the third by

00:02:44.590 --> 00:02:45.660
to get to x to the seventh?

00:02:45.660 --> 00:02:48.810
I could multiply it times
three x to the fourth.

00:02:48.810 --> 00:02:50.860
Notice, two times three is six,

00:02:50.860 --> 00:02:54.100
x to the third times x to the
fourth is x to the seventh.

00:02:54.100 --> 00:02:56.440
We add exponents when we're multiplying,

00:02:56.440 --> 00:02:59.180
when we're multiplying
things with the same base.

00:02:59.180 --> 00:03:00.740
But this isn't the only way to factor.

00:03:00.740 --> 00:03:03.320
Just as we saw that three
times four wasn't the only way

00:03:03.320 --> 00:03:04.500
to factor 12.

00:03:04.500 --> 00:03:08.840
You could also express this
as maybe being equal to

00:03:08.840 --> 00:03:11.728
x to the sixth times what?

00:03:11.728 --> 00:03:15.570
Well, we would still have
to multiply by six then,

00:03:15.570 --> 00:03:17.830
and then we'd have to
multiply by another x.

00:03:17.830 --> 00:03:20.970
So we could write this as
x to the sixth times six x.

00:03:20.970 --> 00:03:24.530
So there is oftentimes
multiple ways to factor

00:03:24.530 --> 00:03:28.600
a higher degree monomial like this.

00:03:28.600 --> 00:03:31.700
And there is also an
analogue to doing something

00:03:31.700 --> 00:03:33.090
like a prime factorization.

00:03:33.090 --> 00:03:36.410
When you're trying to really decompose,

00:03:36.410 --> 00:03:40.030
rewrite this expression as a
product of it simplest parts.

00:03:40.030 --> 00:03:42.760
How would you do that
for six x to the seventh?

00:03:42.760 --> 00:03:44.670
Well, you could rewrite that.

00:03:44.670 --> 00:03:48.470
You could say six x to the seventh,

00:03:48.470 --> 00:03:51.020
well that's equal to, we could
think about the six first.

00:03:51.020 --> 00:03:53.340
We know that the prime
factorization of six

00:03:53.340 --> 00:03:56.280
is two times three, two times three.

00:03:56.280 --> 00:03:59.720
And then x to the
seventh is just seven x's

00:03:59.720 --> 00:04:00.950
multiplied by each other.

00:04:00.950 --> 00:04:04.300
So times x times x

00:04:04.300 --> 00:04:06.600
times x times x

00:04:06.600 --> 00:04:08.160
times x, how many is that?

00:04:08.160 --> 00:04:12.200
That's five, six, and seven.

00:04:12.200 --> 00:04:13.690
And so some of what we were doing,

00:04:13.690 --> 00:04:15.680
when I said two x to the third,

00:04:15.680 --> 00:04:16.920
what we really thought about is okay,

00:04:16.920 --> 00:04:20.550
I had a two, and then I
had x times x times x.

00:04:20.550 --> 00:04:22.370
And then what do I have to multiply that?

00:04:22.370 --> 00:04:27.370
Well I have to multiply that
by three x to the fourth.

00:04:27.690 --> 00:04:31.200
And as we will see, being
able to think about monomials

00:04:31.200 --> 00:04:34.380
in this way will be useful for factoring

00:04:34.380 --> 00:04:36.930
higher degree things
that aren't monomials,

00:04:36.930 --> 00:04:38.660
things that are binomials, trinomials,

00:04:38.660 --> 00:04:40.350
or polynomials in general.

00:04:40.350 --> 00:04:42.100
And we'll do that in future videos.


Polynomial special products: difference of squares

WEBVTT
Kind: captions
Language: en

00:00:00.450 --> 00:00:03.120
- [Instructor] Earlier in
our mathematical adventures,

00:00:03.120 --> 00:00:07.680
we had expanded things like x plus y

00:00:07.680 --> 00:00:10.610
times x minus y.

00:00:10.610 --> 00:00:11.950
Just as a but of review,

00:00:11.950 --> 00:00:16.850
this is going to be equal to
x times x, which is x squared;

00:00:16.850 --> 00:00:20.473
plus x times negative
y, which is negative xy;

00:00:21.580 --> 00:00:24.623
plus y times x, which is plus xy;

00:00:26.030 --> 00:00:28.730
and then minus y times y.

00:00:28.730 --> 00:00:31.260
Or you could say y times a negative y,

00:00:31.260 --> 00:00:33.770
so it's going to be minus y squared.

00:00:33.770 --> 00:00:37.170
Negative xy, positive xy, so
this is just going to simplify

00:00:37.170 --> 00:00:40.870
to x squared minus y squared.

00:00:40.870 --> 00:00:42.150
And this is all review.

00:00:42.150 --> 00:00:44.750
We covered it, and when we
thought about factoring things

00:00:44.750 --> 00:00:46.400
that are differences of squares,

00:00:46.400 --> 00:00:48.270
we thought about this when we were

00:00:48.270 --> 00:00:51.430
first learning to multiply binomials.

00:00:51.430 --> 00:00:53.560
And what we're going to do now is

00:00:53.560 --> 00:00:55.060
essentially just do the same thing,

00:00:55.060 --> 00:00:58.320
but do it with slightly more
complicated expressions.

00:00:58.320 --> 00:01:00.780
And so, another way of
expressing what we just did

00:01:00.780 --> 00:01:02.200
is we could also write something like

00:01:02.200 --> 00:01:07.200
a plus b times a minus b is
going to be equal to what?

00:01:07.940 --> 00:01:11.280
Well, it's going to be equal
to a squared minus b squared.

00:01:11.280 --> 00:01:12.470
The only difference between what I did

00:01:12.470 --> 00:01:13.590
up here and what I did over here

00:01:13.590 --> 00:01:15.000
is instead of an x, I wrote an a;

00:01:15.000 --> 00:01:17.390
and instead of a y, I wrote a b.

00:01:17.390 --> 00:01:20.440
So, given that, let's see if we can expand

00:01:20.440 --> 00:01:23.200
and then combine like terms for,

00:01:23.200 --> 00:01:25.520
if I'm multiplying these two expressions.

00:01:25.520 --> 00:01:30.090
Say I'm multiplying three
plus 5x to the fourth

00:01:30.090 --> 00:01:34.670
times three minus 5x to the fourth.

00:01:34.670 --> 00:01:37.923
Pause this video, and see
if you can work this out.

00:01:39.260 --> 00:01:41.070
Alright, well, there's
two ways to approach it.

00:01:41.070 --> 00:01:43.000
You could just approach it exactly the way

00:01:43.000 --> 00:01:44.700
that I approached it up here,

00:01:44.700 --> 00:01:47.320
but we already know that
when we have this pattern

00:01:47.320 --> 00:01:49.720
where we have something plus something

00:01:49.720 --> 00:01:51.940
times that same original something

00:01:51.940 --> 00:01:54.010
minus the other something,

00:01:54.010 --> 00:01:56.220
well that's going to be of the form

00:01:56.220 --> 00:02:00.320
of this thing squared
minus this thing squared.

00:02:00.320 --> 00:02:02.620
And remember, the only
reason why I'm applying that

00:02:02.620 --> 00:02:05.110
is I have a three right
over here and here,

00:02:05.110 --> 00:02:07.730
so the three is playing the role of the a.

00:02:07.730 --> 00:02:11.080
So, let me write that down, that is our a.

00:02:11.080 --> 00:02:13.920
And then the role of the b

00:02:13.920 --> 00:02:17.100
is being played by 5x to the fourth.

00:02:17.100 --> 00:02:19.590
So, that is our b right over there.

00:02:19.590 --> 00:02:23.090
So, this is going to be equal
to a squared minus b squared.

00:02:23.090 --> 00:02:24.360
But our a is three,

00:02:24.360 --> 00:02:26.730
so it's going to be
equal to three squared,

00:02:26.730 --> 00:02:31.000
minus, and then our b is 5x to the fourth,

00:02:31.000 --> 00:02:34.360
minus 5x to the fourth squared.

00:02:34.360 --> 00:02:36.710
Now, what does all of this simplify to?

00:02:36.710 --> 00:02:38.320
Well, this is going to be equal to,

00:02:38.320 --> 00:02:41.010
three squared is nine, and then minus

00:02:41.010 --> 00:02:42.400
5x to the fourth squared.

00:02:42.400 --> 00:02:45.260
Let's see, 5 squared is 25.

00:02:45.260 --> 00:02:47.640
And then x to the fourth squared,

00:02:47.640 --> 00:02:49.410
well, that is just going
to be x to the fourth

00:02:49.410 --> 00:02:52.170
times x to the fourth, which
is just x to the eighth.

00:02:52.170 --> 00:02:54.367
Another way to think about
it are exponent properties.

00:02:54.367 --> 00:02:56.160
This is the same thing as 5 squared

00:02:56.160 --> 00:02:57.480
times x to the fourth squared.

00:02:57.480 --> 00:02:58.620
If I raise something to an exponent

00:02:58.620 --> 00:02:59.980
and then raise that to another exponent,

00:02:59.980 --> 00:03:01.400
I multiply the exponents.

00:03:01.400 --> 00:03:03.570
And there you have it.

00:03:03.570 --> 00:03:05.500
Let's do another example.

00:03:05.500 --> 00:03:08.180
Let's say that I were to ask you,

00:03:08.180 --> 00:03:10.990
what is 3y squared

00:03:10.990 --> 00:03:14.060
plus 2y to the fifth

00:03:14.060 --> 00:03:16.340
times 3y squared

00:03:16.340 --> 00:03:19.300
minus 2y to the fifth?

00:03:19.300 --> 00:03:22.660
Pause this video, and see
if you can work that out.

00:03:22.660 --> 00:03:23.930
Well, we're going to do it the same way.

00:03:23.930 --> 00:03:25.700
You could, of course, always

00:03:25.700 --> 00:03:28.430
just try to expand it out
the way we did originally.

00:03:28.430 --> 00:03:30.330
But we could recognize here that, hey,

00:03:30.330 --> 00:03:34.800
I have an a plus a b
times the a minus a b.

00:03:34.800 --> 00:03:37.210
So, that's going to be
equal to our a squared.

00:03:37.210 --> 00:03:39.320
So, what's 3y squared?

00:03:39.320 --> 00:03:42.780
Well, that's going to be 9y to the fourth

00:03:42.780 --> 00:03:45.310
minus our b squared.

00:03:45.310 --> 00:03:48.480
Well, what's 2y to the fifth squared?

00:03:48.480 --> 00:03:52.860
Well, 2 squared is four,
and y to the fifth squared

00:03:52.860 --> 00:03:56.550
is y to the five times
two, y to the 10th power.

00:03:56.550 --> 00:03:59.610
And there's no further
simplification that I could do here.

00:03:59.610 --> 00:04:01.470
I can't combine any like terms.

00:04:01.470 --> 00:04:04.813
And so, we are done here as well.


Khan Stories - Sean

WEBVTT
Kind: captions
Language: en

00:00:00.531 --> 00:00:03.948
(happy electronic music)

00:00:09.060 --> 00:00:10.490
- Gonna lift up the top card.

00:00:10.490 --> 00:00:12.133
This is your card, remember this card?

00:00:18.915 --> 00:00:19.748
- Stop.

00:00:19.748 --> 00:00:22.554
- Right there, where you said
stop was where your card was.

00:00:22.554 --> 00:00:23.637
- Wait, what?

00:00:28.130 --> 00:00:29.330
- I'm learning more stuff,

00:00:29.330 --> 00:00:32.230
it's like, it's basically like magic

00:00:32.230 --> 00:00:35.200
because like you start off here and then

00:00:35.200 --> 00:00:37.850
at some point, when you don't expect it,

00:00:37.850 --> 00:00:39.740
you'll get to another level.

00:00:39.740 --> 00:00:42.110
- Get mine, where do I get mine?

00:00:43.363 --> 00:00:45.500
(kids mumble)

00:00:45.500 --> 00:00:47.620
- There's constantly something going on.

00:00:47.620 --> 00:00:50.119
Always busy, always noise.

00:00:50.119 --> 00:00:52.786
- I walk away with the pink one.

00:01:00.554 --> 00:01:02.760
- Any mom who is raising
her kids alone is worried

00:01:02.760 --> 00:01:04.810
that their kids are gonna fall behind

00:01:04.810 --> 00:01:08.030
or not have enough of something to reach

00:01:09.210 --> 00:01:11.483
where they should in life as they grow.

00:01:15.434 --> 00:01:18.200
- There was like some
things actually like,

00:01:18.200 --> 00:01:21.440
in third grade that I
went back and relearned

00:01:21.440 --> 00:01:24.103
and it actually helped me
a lot in the sixth grade.

00:01:28.100 --> 00:01:29.390
One thing that really helped me

00:01:29.390 --> 00:01:32.120
is where if you can't get it,

00:01:32.120 --> 00:01:34.850
you can get a hint or watch the videos.

00:01:34.850 --> 00:01:36.590
- When he's using the Khan Academy,

00:01:36.590 --> 00:01:39.510
he tells me that he gets
to watch his progress

00:01:39.510 --> 00:01:42.100
so he gets to see each step that he takes,

00:01:42.100 --> 00:01:43.520
how far he's getting in it,

00:01:43.520 --> 00:01:46.550
and that makes him motivated to keep going

00:01:46.550 --> 00:01:48.800
and to get to the next level.

00:01:48.800 --> 00:01:53.643
And it's like instant
gratification for him.

00:01:55.009 --> 00:01:56.040
(Sean mumbles)

00:01:56.040 --> 00:01:57.274
- Oh.

00:01:57.274 --> 00:01:58.574
- That was the second one.

00:01:59.560 --> 00:02:01.270
- I am taking control of his learning

00:02:01.270 --> 00:02:04.280
and just seeing his success,

00:02:04.280 --> 00:02:07.850
it gives him his own boost of confidence.

00:02:07.850 --> 00:02:10.493
Not just in the classroom, but at home.

00:02:16.582 --> 00:02:18.150
I'm extremely proud,

00:02:18.150 --> 00:02:21.160
just seeing the little
person he's growing into

00:02:21.160 --> 00:02:22.960
and the pride he has in himself

00:02:22.960 --> 00:02:24.260
always makes me feel good

00:02:24.260 --> 00:02:26.420
'cause then I feel like
I'm doing something right

00:02:26.420 --> 00:02:27.533
as a parent.

00:02:27.533 --> 00:02:28.366
- Need help?

00:02:28.366 --> 00:02:30.040
I really like helping a lot.

00:02:30.040 --> 00:02:32.710
Besides math and learning and all that,

00:02:32.710 --> 00:02:35.383
that's one of my favorite things to do.