Adding Rational Expressions Example 1

WEBVTT
Kind: captions
Language: en

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Let's do a few examples adding
rational expressions.

00:00:03.820 --> 00:00:10.650
So let's say we had negative
x squared over x squared

00:00:10.650 --> 00:00:14.460
minus 7x plus 6.

00:00:14.460 --> 00:00:21.970
And to that I want to
add plus x minus 4.

00:00:21.970 --> 00:00:25.250
So to some degree what I want to
do in this first example is

00:00:25.250 --> 00:00:28.170
kind of our algebraic long
division or polynomial

00:00:28.170 --> 00:00:29.710
division in reverse.

00:00:29.710 --> 00:00:32.770
I want to put this x minus
4 in the numerator with x

00:00:32.770 --> 00:00:37.020
squared minus 7x plus 6
in the denominator.

00:00:37.020 --> 00:00:41.010
The easiest way to do that is
to just literally multiply x

00:00:41.010 --> 00:00:51.080
minus 4 times x squared minus
7x plus 6 over x squared

00:00:51.080 --> 00:00:53.480
minus 7x plus 6.

00:00:53.480 --> 00:00:57.800
I'm not changing the number,
I'm multiplying it by 1.

00:00:57.800 --> 00:01:00.540
So if we do that
what do we get?

00:01:00.540 --> 00:01:04.180
Well we just have to multiply
this numerator right here, and

00:01:04.180 --> 00:01:06.870
we can do that with algebraic
multiplication.

00:01:06.870 --> 00:01:18.730
So if we have x squared minus 7x
plus 6 times x minus 4, we

00:01:18.730 --> 00:01:20.380
can multiply these two.

00:01:20.380 --> 00:01:24.590
6 times negative 4
is negative 24.

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Negative 4 times negative
7x is positive 28x.

00:01:29.550 --> 00:01:34.520
Negative 4 times x squared
is negative 4x squared.

00:01:34.520 --> 00:01:39.950
And if x times 6 is 6x, x times
negative 7x is negative

00:01:39.950 --> 00:01:41.870
7x squared.

00:01:41.870 --> 00:01:45.800
x times x squared is
x to the third.

00:01:45.800 --> 00:01:50.000
So the new numerator is going
to be x to the third minus,

00:01:50.000 --> 00:01:56.590
this is 11x squared
plus 34x minus 24.

00:01:56.590 --> 00:01:59.960
So now I can rewrite this-- let
me copy and paste this--

00:01:59.960 --> 00:02:07.200
this is going to be
equal to that.

00:02:07.200 --> 00:02:14.330
Plus I can rewrite this thing
now as being over a

00:02:14.330 --> 00:02:21.220
denominator of x squared
minus 7x plus 6.

00:02:21.220 --> 00:02:24.950
And our numerator, x minus 4
times this thing over here, we

00:02:24.950 --> 00:02:25.880
get this whole thing.

00:02:25.880 --> 00:02:30.760
We multiplied them out. x to
the third minus 11x squared

00:02:30.760 --> 00:02:37.000
plus 34x minus 24.

00:02:37.000 --> 00:02:40.130
And now, since we have the exact
same denominator we can

00:02:40.130 --> 00:02:42.840
add these two rational
expressions.

00:02:42.840 --> 00:02:45.300
So if you add them, you
literally can just say well we

00:02:45.300 --> 00:02:51.000
have the common denominator, so
our common denominator is x

00:02:51.000 --> 00:02:54.310
squared minus 7x plus 6.

00:02:54.310 --> 00:02:58.540
That was the whole point of
multiplying this expression

00:02:58.540 --> 00:03:00.800
over this expression so that
we could get the common

00:03:00.800 --> 00:03:01.740
denominator.

00:03:01.740 --> 00:03:03.170
And now we just add
the numerator.

00:03:03.170 --> 00:03:06.130
So this is going to add to the
x squared side, so you're

00:03:06.130 --> 00:03:09.250
still going to have
an x to the third.

00:03:09.250 --> 00:03:13.940
And then negative or minus 11x
squared minus another x

00:03:13.940 --> 00:03:23.120
squared is minus 12x squared
plus 34x minus 24.

00:03:23.120 --> 00:03:26.320
And we are done.

00:03:26.320 --> 00:03:29.210
Now we could have checked
whether we could have factored

00:03:29.210 --> 00:03:33.440
this, but this just factors
into x minus 6

00:03:33.440 --> 00:03:35.930
times x minus 1.

00:03:35.930 --> 00:03:39.480
So it really doesn't factor
with the minus 4 in any

00:03:39.480 --> 00:03:41.850
meaningful way, so there really
isn't any further

00:03:41.850 --> 00:03:45.170
simplification that we can
do for this problem.

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Hopefully you enjoyed that.


Adding Rational Expressions Example 3

WEBVTT
Kind: captions
Language: en

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Let's do one more fairly
involved example of adding, or

00:00:04.100 --> 00:00:05.970
in this case actually
subtracting rational

00:00:05.970 --> 00:00:07.020
expressions.

00:00:07.020 --> 00:00:15.070
Let's say I have 4 over
9x squared minus 49.

00:00:15.070 --> 00:00:18.730
And from that I want to
subtract-- maybe I'll do this

00:00:18.730 --> 00:00:28.170
in different color-- I want to
subtract 1 over 3x squared

00:00:28.170 --> 00:00:33.190
plus 5x, minus 28.

00:00:33.190 --> 00:00:35.630
So just like we saw in the last
video, we want to find

00:00:35.630 --> 00:00:38.550
the least common multiple of
these two expressions.

00:00:38.550 --> 00:00:41.660
And the way to figure out that
is to factor these two things.

00:00:41.660 --> 00:00:44.470
Now this first expression,
you might recognize, is a

00:00:44.470 --> 00:00:46.190
difference of squares.

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This is 3x squared
minus 7 squared.

00:00:49.710 --> 00:00:56.100
So we can rewrite it as 4 over
3x-- well, let me write it

00:00:56.100 --> 00:01:03.660
this way-- 3x plus 7,
times 3x minus 7.

00:01:03.660 --> 00:01:06.300
3x squared is 9x squared.

00:01:06.300 --> 00:01:08.580
7 squared is 49.

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Difference of squares.

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And from that we're going to
subtract this thing over here.

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Let's see how we can
factor that out.

00:01:14.830 --> 00:01:18.700
Well, once again, this has a
non-1 coefficient here, so you

00:01:18.700 --> 00:01:20.390
can always use the Pythagorean
theorem, but that actually

00:01:20.390 --> 00:01:23.050
gets messy if you actually just
want to factor as opposed

00:01:23.050 --> 00:01:23.930
to getting roots.

00:01:23.930 --> 00:01:26.810
So the best thing here
is to do grouping.

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So let me factor it over here.

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I need to find two numbers.

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Let me just write
it over here.

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So we have 3x-- let
me write it.

00:01:35.560 --> 00:01:37.890
I want to write it someplace
where I can-- well, actually

00:01:37.890 --> 00:01:38.420
I'll write it here.

00:01:38.420 --> 00:01:43.690
So I have 3x squared
plus 5x, minus 28.

00:01:43.690 --> 00:01:46.390
I need to find two numbers that
when I multiply them I

00:01:46.390 --> 00:01:51.550
get 3 times negative 28.

00:01:51.550 --> 00:01:55.140
So it's negative 60 and negative
24, that's negative

00:01:55.140 --> 00:01:57.910
84, if I multiply those
two numbers.

00:01:57.910 --> 00:02:05.930
And a plus b should
be equal to 5.

00:02:05.930 --> 00:02:07.420
So what numbers?

00:02:07.420 --> 00:02:09.680
Well, it's 7 and 12.

00:02:09.680 --> 00:02:10.419
Your times table.

00:02:10.419 --> 00:02:11.910
7 times 12 is 84.

00:02:11.910 --> 00:02:18.970
So if you make this a 12 and
you make this a negative 7,

00:02:18.970 --> 00:02:19.460
then it works.

00:02:19.460 --> 00:02:22.360
12 minus 7 is 5, 12 times
negative 7 is 84.

00:02:22.360 --> 00:02:26.650
So I can rewrite this
as 3x squared.

00:02:26.650 --> 00:02:29.200
I'll group the 12 with the 3,
because they have a common

00:02:29.200 --> 00:02:30.520
factor of 3.

00:02:30.520 --> 00:02:37.090
So plus 12x minus
7x, minus 28.

00:02:37.090 --> 00:02:39.300
Once again, 7 and 28 have
a common factor, so

00:02:39.300 --> 00:02:40.540
it's looking good.

00:02:40.540 --> 00:02:44.220
And just to reiterate, all I
did is I rewrote this 5x as

00:02:44.220 --> 00:02:46.730
12x minus 7x.

00:02:46.730 --> 00:02:48.610
These are, obviously,
the same thing.

00:02:48.610 --> 00:02:51.390
12x minus 7x is 5x.

00:02:51.390 --> 00:02:57.090
But then we can group
them just like that.

00:02:57.090 --> 00:03:00.470
If we factor out a 3x here, this
first expression is the

00:03:00.470 --> 00:03:06.210
same thing as 3x
times x plus 4.

00:03:06.210 --> 00:03:10.020
And in the second expression
over here-- I'll do it in red,

00:03:10.020 --> 00:03:13.270
or in white I guess-- let's
factor out a negative 7.

00:03:13.270 --> 00:03:17.250
So minus 7 times x plus 4.

00:03:17.250 --> 00:03:20.930
And lucky for us, we have a
common factor is x plus 4, so

00:03:20.930 --> 00:03:23.270
we can factor that out.

00:03:23.270 --> 00:03:31.200
So if you factor out the x plus
4, you get x plus 4 times

00:03:31.200 --> 00:03:32.875
3x minus 7.

00:03:37.290 --> 00:03:40.490
So that's what this guy over
here will factor into.

00:03:40.490 --> 00:03:43.420
So I can rewrite
it right here.

00:03:43.420 --> 00:03:49.730
So this expression is the exact
same thing as 1 over--

00:03:49.730 --> 00:03:54.490
well, I can write the 3x minus
7 first-- 3x minus 7

00:03:54.490 --> 00:03:57.520
times x plus 4.

00:03:57.520 --> 00:04:00.400
This and this is the exact
same statement.

00:04:00.400 --> 00:04:02.840
So let's figure out what our
common denominator is here.

00:04:06.490 --> 00:04:10.710
So our common denominator-- I'll
write it like that-- it

00:04:10.710 --> 00:04:13.170
needs to have all of the factors
of this guy right

00:04:13.170 --> 00:04:16.790
here, so it needs to
have a 3x plus 7.

00:04:16.790 --> 00:04:20.680
It needs to have a 3x minus 7.

00:04:20.680 --> 00:04:23.060
And also it has to have all of
the factors of this guy.

00:04:23.060 --> 00:04:26.160
Well, it already has
a 3x minus 7, so

00:04:26.160 --> 00:04:27.020
this is already done.

00:04:27.020 --> 00:04:29.270
But it doesn't have an
x plus 4 in it yet.

00:04:29.270 --> 00:04:31.940
So now I have to multiply
it by x plus 4.

00:04:31.940 --> 00:04:34.070
So clearly, if you look at
this, this is clearly

00:04:34.070 --> 00:04:37.650
divisible by this guy, because
it has all of its factors.

00:04:37.650 --> 00:04:39.040
You could divide this out.

00:04:39.040 --> 00:04:40.510
And it's clearly divisible
by this guy.

00:04:40.510 --> 00:04:42.330
You have 3x minus 7
times x plus 4.

00:04:42.330 --> 00:04:44.340
3x minus 7 times x plus 4.

00:04:44.340 --> 00:04:47.820
So this is the least common
multiple of these two

00:04:47.820 --> 00:04:48.970
expressions.

00:04:48.970 --> 00:04:55.020
Now, how do we get-- so what
do we have to multiply this

00:04:55.020 --> 00:04:58.420
guy by to have this as
the denominator?

00:04:58.420 --> 00:05:00.280
Well, I just have to multiply
the numerator and the

00:05:00.280 --> 00:05:02.460
denominator by x plus 4.

00:05:02.460 --> 00:05:04.540
So if I multiply this guy-- let
me scroll to the left a

00:05:04.540 --> 00:05:11.250
little bit-- by x plus
4 over x plus 4, my

00:05:11.250 --> 00:05:12.860
denominator will work out.

00:05:12.860 --> 00:05:16.390
And so my numerator will become
4 times x plus 4, which

00:05:16.390 --> 00:05:19.670
is 4x plus 16.

00:05:19.670 --> 00:05:21.740
That's this first
term over here.

00:05:21.740 --> 00:05:24.290
And then what do I have to
multiply this guy by to have

00:05:24.290 --> 00:05:26.870
the same denominator as this?

00:05:26.870 --> 00:05:29.100
Well, I have a 3x minus
7 times x plus 4.

00:05:29.100 --> 00:05:31.290
3x minus 7 times x plus 4.

00:05:31.290 --> 00:05:33.420
I have to multiply
it by 3x plus 7.

00:05:37.540 --> 00:05:42.030
So I also have to multiply the
numerator by 3x plus 7.

00:05:42.030 --> 00:05:49.650
So this is 4x plus 16 minus 1
times 3x plus 7, or minus 3x

00:05:49.650 --> 00:05:53.010
plus 7 is the same thing
as minus 3x minus 7.

00:05:53.010 --> 00:05:55.080
I'm just distributing
the negative sign.

00:05:55.080 --> 00:05:58.170
And then what is this going
to be equal to?

00:05:58.170 --> 00:06:04.860
The numerator, we have a 4x
minus a 3x will just be an x.

00:06:04.860 --> 00:06:09.540
And then we have a 16 minus
a 7, which is a 9.

00:06:09.540 --> 00:06:12.630
And then the denominator, we,
of course, have all of this

00:06:12.630 --> 00:06:19.490
stuff over here, which is 3x
plus 7, times 3x minus 7,

00:06:19.490 --> 00:06:21.600
times x plus 4.

00:06:21.600 --> 00:06:23.150
And we're done.


Adding Rational Expressions Example 2

WEBVTT
Kind: captions
Language: en

00:00:00.370 --> 00:00:02.095
Let's add some more rational
expressions.

00:00:02.095 --> 00:00:07.360
So let's say I have 1 over
x squared minus 9.

00:00:07.360 --> 00:00:14.390
And to that I'm going to
add 2 over x squared

00:00:14.390 --> 00:00:17.860
plus 5x, plus 6.

00:00:17.860 --> 00:00:20.930
So just like we did when
we first learned to add

00:00:20.930 --> 00:00:24.650
fractions, or to add rational
numbers, we always had to come

00:00:24.650 --> 00:00:25.870
up with a common denominator.

00:00:25.870 --> 00:00:27.440
So once again, we're going to
have to come up with a common

00:00:27.440 --> 00:00:28.280
denominator.

00:00:28.280 --> 00:00:30.520
And what we really want to do
is find the least common

00:00:30.520 --> 00:00:33.780
multiple of these
two expressions.

00:00:33.780 --> 00:00:35.390
And when I say that, let
me just remind you.

00:00:35.390 --> 00:00:41.120
Let's say we wanted to
add 1/4 plus 1/6.

00:00:41.120 --> 00:00:48.300
Well, we know that 1/4 is 1 over
2 times 2, and that 1/6

00:00:48.300 --> 00:00:51.650
is 1 over 2 times 3.

00:00:51.650 --> 00:00:55.240
So to be able to add these two
things, we incrementally-- we

00:00:55.240 --> 00:00:58.840
both have a common 2.

00:00:58.840 --> 00:01:00.980
We both have this common 2.

00:01:00.980 --> 00:01:04.590
But in order for this guy to
go into whatever our common

00:01:04.590 --> 00:01:05.980
denominator is, we're
going to have to be

00:01:05.980 --> 00:01:07.220
divisible by another 2.

00:01:07.220 --> 00:01:07.880
So let me do that.

00:01:07.880 --> 00:01:09.990
So our common denominator's
going to have to be-- we have

00:01:09.990 --> 00:01:11.330
this 2 in each of them.

00:01:11.330 --> 00:01:13.400
So that takes care
of that first 2.

00:01:13.400 --> 00:01:15.440
But then we need the second 2.

00:01:15.440 --> 00:01:17.800
We need the second 2 there.

00:01:17.800 --> 00:01:19.820
2 times a second 2.

00:01:19.820 --> 00:01:22.120
And then we need this 3 here.

00:01:22.120 --> 00:01:23.820
We need this 3 here.

00:01:23.820 --> 00:01:26.270
We have to have all of the prime
factors of both of them.

00:01:26.270 --> 00:01:28.990
This has two 2's, so this
guy has to have two 2's.

00:01:28.990 --> 00:01:31.630
This guy has a 2 and a
3, so this guy has to

00:01:31.630 --> 00:01:33.170
have a 2 and a 3.

00:01:33.170 --> 00:01:36.070
And so our least common multiple
here is 2 times 2,

00:01:36.070 --> 00:01:38.750
times 3, which is 12.

00:01:38.750 --> 00:01:40.580
And then we can add
accordingly.

00:01:40.580 --> 00:01:42.350
That is our common
denominator.

00:01:42.350 --> 00:01:43.730
That's the least common
multiple.

00:01:43.730 --> 00:01:45.506
You could have just multiplied
these two out, but you would

00:01:45.506 --> 00:01:48.360
have gotten 24, and that's
larger than necessary.

00:01:48.360 --> 00:01:50.210
You could have still worked it
out, but you would have had

00:01:50.210 --> 00:01:52.370
more simplifying left
for you to do.

00:01:52.370 --> 00:01:55.550
The same idea we can
do right here.

00:01:55.550 --> 00:01:57.090
The same idea we can do here.

00:01:57.090 --> 00:02:01.000
We can factor both of these
denominators, so x squared,

00:02:01.000 --> 00:02:05.560
that is x plus 3,
times x minus 3.

00:02:05.560 --> 00:02:07.610
It's just a difference
of squares.

00:02:07.610 --> 00:02:10.770
And x squared plus 5x, plus 6.

00:02:10.770 --> 00:02:11.510
That is what?

00:02:11.510 --> 00:02:16.070
That is x plus 2,
times x plus 3.

00:02:16.070 --> 00:02:17.740
2 times 3 is 6.

00:02:17.740 --> 00:02:20.500
2 plus 3 is 5.

00:02:20.500 --> 00:02:23.010
So our common denominator's
going to be what?

00:02:23.010 --> 00:02:23.960
What's going to be our
common denominator?

00:02:23.960 --> 00:02:28.080
Let me clear up this real
estate over here.

00:02:28.080 --> 00:02:31.080
What's our common denominator
going to be?

00:02:31.080 --> 00:02:35.780
Our common denominator-- so this
is going to be equal to--

00:02:35.780 --> 00:02:38.520
our common denominator's going
to be-- our common

00:02:38.520 --> 00:02:43.530
denominator's going to have
to have an x plus 3 in it.

00:02:46.850 --> 00:02:49.890
And that takes care of this
x plus 3 as well.

00:02:49.890 --> 00:02:52.800
It's going to have an
x minus 3 in it.

00:02:52.800 --> 00:02:56.190
It's going to have to have
an x minus 3 in it.

00:02:56.190 --> 00:02:58.180
So that's an x minus 3.

00:02:58.180 --> 00:03:00.040
And it's going to have
that x plus 2 in it.

00:03:02.610 --> 00:03:04.310
Another way you could have
thought about is, like, look,

00:03:04.310 --> 00:03:06.810
it's going to have to have all
of the factors of this guy,

00:03:06.810 --> 00:03:09.680
which are x plus 3, times
x minus 3. x plus 3

00:03:09.680 --> 00:03:12.100
times x minus 3.

00:03:12.100 --> 00:03:13.900
And then once you write those
down, you say, well, it also

00:03:13.900 --> 00:03:15.320
has to have all of the
factors for this guy.

00:03:15.320 --> 00:03:17.750
Well, we already have an x plus
3 there, so we just have

00:03:17.750 --> 00:03:20.080
to only add on the x plus 2.

00:03:20.080 --> 00:03:22.330
And that saves us from
multiplying both of these

00:03:22.330 --> 00:03:23.560
expressions together.

00:03:23.560 --> 00:03:26.180
This is the least common
denominator.

00:03:26.180 --> 00:03:29.430
Now, if this is the least common
denominator, what do we

00:03:29.430 --> 00:03:31.820
have to change the 1 by?

00:03:31.820 --> 00:03:34.870
Well, let me actually rewrite
this a little bit.

00:03:34.870 --> 00:03:36.170
Well, I'll write it like this.

00:03:36.170 --> 00:03:41.810
So what do you have to do to
go from-- if I start with 1

00:03:41.810 --> 00:03:47.050
over x squared-- let me write it
this way-- 1 over x plus 3,

00:03:47.050 --> 00:03:50.040
times x minus 3.

00:03:50.040 --> 00:03:52.370
This and this is the
exact same thing.

00:03:52.370 --> 00:03:54.820
If I want an x plus 2 in
the denominator, what

00:03:54.820 --> 00:03:55.720
do I have to do?

00:03:55.720 --> 00:03:58.730
I have to multiply the numerator
and the denominator

00:03:58.730 --> 00:04:00.280
by x plus 2.

00:04:02.830 --> 00:04:06.400
So this expression right here,
which is this expression right

00:04:06.400 --> 00:04:14.170
here, will become x plus 2 over
x plus 3, times x minus

00:04:14.170 --> 00:04:17.300
3, times x plus 2.

00:04:17.300 --> 00:04:19.510
Those would cancel out and
you're just left with a 1.

00:04:19.510 --> 00:04:21.310
That's that right there.

00:04:21.310 --> 00:04:24.540
So this is the first term.

00:04:24.540 --> 00:04:26.720
That is this term.

00:04:26.720 --> 00:04:30.890
This term over here, the second
term over here, this is

00:04:30.890 --> 00:04:38.150
2 over x plus 2,
times x plus 3.

00:04:38.150 --> 00:04:40.010
That's in its original form.

00:04:40.010 --> 00:04:48.740
Now, if we also want an
x minus 3 here in the

00:04:48.740 --> 00:04:51.010
denominator, we can multiply
it by x minus 3

00:04:51.010 --> 00:04:53.840
over x minus 3.

00:04:53.840 --> 00:04:55.270
Or essentially, we
can multiply the

00:04:55.270 --> 00:04:57.010
numerator by x minus 3.

00:04:57.010 --> 00:05:00.900
So this is going to be equal
to 2 times x minus 3 is 2x

00:05:00.900 --> 00:05:10.990
minus 6 over x plus 2, times
x plus 3, times x minus 3.

00:05:10.990 --> 00:05:16.080
So this expression right here
is the exact same thing as

00:05:16.080 --> 00:05:20.390
this expression right
over here.

00:05:20.390 --> 00:05:22.730
So if we were to add these two
expressions, they have the

00:05:22.730 --> 00:05:24.100
same denominator.

00:05:24.100 --> 00:05:26.410
I just switched the order
that I'm multiplying.

00:05:26.410 --> 00:05:29.160
But they're the exact same
denominator, which is exactly

00:05:29.160 --> 00:05:30.700
the same thing as this.

00:05:30.700 --> 00:05:33.190
So this, this first expression,
we're going to get

00:05:33.190 --> 00:05:36.160
our x plus 2 here.

00:05:36.160 --> 00:05:38.690
x plus 2 over all of this would
cancel, and you'd get 1

00:05:38.690 --> 00:05:42.800
over x plus 3, times x minus
3, over 1 over x

00:05:42.800 --> 00:05:44.230
squared minus 9.

00:05:44.230 --> 00:05:52.920
And then to that you're going
to add plus 2x minus 6.

00:05:52.920 --> 00:05:54.320
And why am I adding
2x minus 6?

00:05:54.320 --> 00:05:58.590
Because the second term is 2x
minus 6 over all of this

00:05:58.590 --> 00:06:01.080
business, because we were
just multiplying.

00:06:01.080 --> 00:06:04.420
This would also be written as 2
times x minus 3, over all of

00:06:04.420 --> 00:06:08.070
this. x minus 3's would cancel
out and you'd get this again.

00:06:08.070 --> 00:06:11.380
So this is what we get when we
add the 2, and so this will be

00:06:11.380 --> 00:06:14.420
equal to-- let me scroll down
a little bit-- this will be

00:06:14.420 --> 00:06:17.760
equal to x plus 2x is 3x.

00:06:17.760 --> 00:06:21.000
2 minus 6 is negative 4.

00:06:21.000 --> 00:06:22.950
All of that over this thing.

00:06:22.950 --> 00:06:25.970
We can either keep it multiplied
out, or factor it

00:06:25.970 --> 00:06:28.200
out, or we could multiply it
if you like, but it'll take

00:06:28.200 --> 00:06:29.160
some time to multiply it.

00:06:29.160 --> 00:06:33.670
So I'll just say x plus
3, times x minus 3,

00:06:33.670 --> 00:06:36.880
times x plus 2.

00:06:36.880 --> 00:06:39.370
And we are done.


Multiplying and Dividing Rational Expressions

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:04.120
Let's do some more work with
rational expressions.

00:00:04.120 --> 00:00:13.650
So let's say I had a squared
plus 2ab plus b squared and

00:00:13.650 --> 00:00:22.980
all of that is over ab squared
minus a squared b.

00:00:22.980 --> 00:00:30.480
And we're going to take this
and divide it by a plus b.

00:00:34.040 --> 00:00:36.580
So the first thing we might want
to do is just factor this

00:00:36.580 --> 00:00:39.240
numerator and this
denominator, and

00:00:39.240 --> 00:00:40.170
then we could divide.

00:00:40.170 --> 00:00:41.420
Or actually, we could
go the other way.

00:00:41.420 --> 00:00:42.980
We could divide and
then factor.

00:00:42.980 --> 00:00:45.630
So if we divide and then factor,
we could say, well,

00:00:45.630 --> 00:00:47.350
this is the same thing.

00:00:47.350 --> 00:00:50.070
This is equal to this
whole expression.

00:00:50.070 --> 00:00:51.300
Let me just copy and paste it.

00:00:51.300 --> 00:00:54.060
That's probably the easiest
thing to do.

00:00:54.060 --> 00:00:56.640
Copy and then paste it.

00:00:56.640 --> 00:01:01.710
It's equal to that expression
times the inverse of this.

00:01:01.710 --> 00:01:05.870
If I divide by something,
that's the same thing as

00:01:05.870 --> 00:01:09.990
multiplying by the inverse,
as multiplying by

00:01:09.990 --> 00:01:12.190
1 over a plus b.

00:01:12.190 --> 00:01:19.640
And this is just going to be the
same thing as that with a

00:01:19.640 --> 00:01:21.550
plus b in the denominator,
because your numerator's going

00:01:21.550 --> 00:01:23.790
to be that times 1, which
is just that.

00:01:23.790 --> 00:01:27.490
Your denominator is going to be
ab squared minus a squared

00:01:27.490 --> 00:01:34.090
b times a plus b, so that
times a plus b.

00:01:34.090 --> 00:01:37.910
So this would be a legitimate
answer, but I have a suspicion

00:01:37.910 --> 00:01:40.400
that this can be further
simplified.

00:01:40.400 --> 00:01:42.530
So let's see if we
can simplify it.

00:01:42.530 --> 00:01:47.510
So our numerator, a squared plus
2 ab plus b squared, you

00:01:47.510 --> 00:01:48.850
might recognize that.

00:01:48.850 --> 00:01:51.590
That is a perfect square.

00:01:51.590 --> 00:01:52.860
That is our numerator.

00:01:52.860 --> 00:01:54.810
Let me color code it.

00:01:54.810 --> 00:02:00.440
So that numerator right there,
that is the same thing as a

00:02:00.440 --> 00:02:05.420
plus b squared, or a plus b
times a plus b, because you

00:02:05.420 --> 00:02:08.270
have a squared, you have a b
squared, and then you have 2

00:02:08.270 --> 00:02:10.430
times a and b right
in the middle.

00:02:10.430 --> 00:02:11.470
That is a perfect square.

00:02:11.470 --> 00:02:13.240
Multiply it out if you
don't believe me.

00:02:13.240 --> 00:02:16.180
So that numerator is
that right there.

00:02:16.180 --> 00:02:20.930
And then the denominator,
what is this?

00:02:20.930 --> 00:02:25.280
Well, what happens if we factor
out an ab down here?

00:02:25.280 --> 00:02:27.980
Let's factor out an ab.

00:02:27.980 --> 00:02:33.600
So ab squared divided by ab, a
cancels out, you're just left

00:02:33.600 --> 00:02:40.240
with ab, right? ab times b is
ab squared, and then minus a

00:02:40.240 --> 00:02:46.580
squared b divided
by ab is just a.

00:02:46.580 --> 00:02:50.030
This right here is the
same thing as that.

00:02:50.030 --> 00:02:52.820
And you can multiply it out.
ab times negative a is

00:02:52.820 --> 00:02:54.200
negative a squared b.

00:02:54.200 --> 00:02:55.380
You got that right there.

00:02:55.380 --> 00:02:56.150
So multiply it out.

00:02:56.150 --> 00:02:57.760
You should get this
right there.

00:02:57.760 --> 00:03:02.570
And, of course, you also have
your a plus b sitting here.

00:03:02.570 --> 00:03:05.970
Now, we have an a plus b squared
in the numerator, we

00:03:05.970 --> 00:03:08.420
have an a plus b into the
denominator, or we could say a

00:03:08.420 --> 00:03:10.320
plus b to the first power.

00:03:10.320 --> 00:03:13.060
And then we know our exponent
rules here.

00:03:13.060 --> 00:03:16.270
We can just essentially cancel
out an a plus b in the

00:03:16.270 --> 00:03:19.490
numerator and the denominator,
which decrements this exponent

00:03:19.490 --> 00:03:21.180
by 1, so this just
becomes a 1.

00:03:21.180 --> 00:03:24.850
And then this becomes a zero,
or it cancels out.

00:03:24.850 --> 00:03:37.010
And so we get this is being
equal to a plus b over ab

00:03:37.010 --> 00:03:43.250
times b minus a with the caveat
because we canceled

00:03:43.250 --> 00:03:45.390
this thing out in
the denominator.

00:03:45.390 --> 00:03:48.150
We have to say, look, this thing
could still not equal

00:03:48.150 --> 00:03:49.870
zero, because this
will still make

00:03:49.870 --> 00:03:52.080
this expression undefined.

00:03:52.080 --> 00:03:55.000
We have to say the only way that
this is going to be equal

00:03:55.000 --> 00:03:58.580
to zero is if we subtract b from
both sides, a is equal to

00:03:58.580 --> 00:03:59.710
negative b.

00:03:59.710 --> 00:04:03.160
So we have to add the caveat
that a cannot be equal to

00:04:03.160 --> 00:04:03.820
negative b.

00:04:03.820 --> 00:04:07.270
We have to add this condition.

00:04:07.270 --> 00:04:10.210
So this is our answer.

00:04:10.210 --> 00:04:12.690
Let's do another one.

00:04:12.690 --> 00:04:19.180
Let's say I have-- let's do a
more involved one, just to

00:04:19.180 --> 00:04:21.540
really get our juices flowing.

00:04:21.540 --> 00:04:32.210
Let's say we have x squared plus
8x plus 16 divided by, or

00:04:32.210 --> 00:04:50.120
over, 7x squared plus 9x plus 2
divided by 7x over 4 over x

00:04:50.120 --> 00:04:55.020
squared plus 4x.

00:04:55.020 --> 00:04:57.000
So what is this going
to be equal to?

00:04:57.000 --> 00:04:59.100
So here we could divide first.
That's the same thing as

00:04:59.100 --> 00:05:01.420
multiplying by the inverse
of this thing.

00:05:01.420 --> 00:05:04.155
But just for fun, let's factor
these two, the numerator and

00:05:04.155 --> 00:05:05.990
the denominator here.

00:05:05.990 --> 00:05:09.730
So this numerator over here,
that's pretty straightforward.

00:05:09.730 --> 00:05:10.960
That is a perfect square.

00:05:10.960 --> 00:05:12.660
That is x plus 4 squared.

00:05:12.660 --> 00:05:14.400
4 times 4 is 16.

00:05:14.400 --> 00:05:15.860
4 plus 4 is 8.

00:05:15.860 --> 00:05:22.720
So our numerator is
x plus 4 squared.

00:05:22.720 --> 00:05:26.370
And then our denominator,
what does that become?

00:05:26.370 --> 00:05:29.930
Here, we're going to have to do
a little bit of grouping to

00:05:29.930 --> 00:05:31.360
factor that out.

00:05:31.360 --> 00:05:36.750
So we have 7x squared
plus 9x plus 2.

00:05:36.750 --> 00:05:37.810
We have to find two numbers.

00:05:37.810 --> 00:05:41.700
When I multiply them, I get 7
times 2, which is equal to 14.

00:05:41.700 --> 00:05:46.050
And when I add them, a
plus b, they equal 9.

00:05:46.050 --> 00:05:48.960
And the easiest two numbers
are 7 and 2.

00:05:48.960 --> 00:05:50.480
So let's rewrite this.

00:05:50.480 --> 00:05:54.720
This is the same thing as 7x
squared-- I'll group the 7x

00:05:54.720 --> 00:05:58.780
with the 7 because they have a
common factor of 7-- plus 7x

00:05:58.780 --> 00:06:02.055
plus 2x plus 2.

00:06:02.055 --> 00:06:05.260
I put parentheses around there
so you can see the grouping.

00:06:05.260 --> 00:06:11.140
Factor out a 7x over here, you
get 7x times x plus 1 plus--

00:06:11.140 --> 00:06:12.280
factor out a 2 here.

00:06:12.280 --> 00:06:26.240
2 times x plus 1, and you get 7x
plus 2 times your x plus 1.

00:06:26.240 --> 00:06:29.010
We're undistributing
that expression.

00:06:29.010 --> 00:06:31.710
So this is our denominator
right there.

00:06:34.980 --> 00:06:37.880
And actually, I just realized
that I made a slight error

00:06:37.880 --> 00:06:39.520
when I wrote down
this problem.

00:06:39.520 --> 00:06:43.330
The numerator here is actually
7x plus 2, lucky for us.

00:06:43.330 --> 00:06:46.820
Obviously, this is going to
lead to interesting things

00:06:46.820 --> 00:06:48.550
later on, potentially.

00:06:48.550 --> 00:06:50.480
So let's see what we get.

00:06:50.480 --> 00:06:52.250
And actually, I think
there's a typo here.

00:06:52.250 --> 00:06:56.330
I think this is supposed to be
a 4x-- oh, sorry, just a 4.

00:06:56.330 --> 00:06:59.150
So let me just put it like that
just in case that is a

00:06:59.150 --> 00:07:00.850
typo because I think it's
going to make it more

00:07:00.850 --> 00:07:01.400
interesting.

00:07:01.400 --> 00:07:02.790
So that's our original
problem.

00:07:02.790 --> 00:07:05.380
We haven't messed with this part
yet, so it won't change

00:07:05.380 --> 00:07:06.390
any of our math.

00:07:06.390 --> 00:07:09.020
But so far, we factored
this bottom part.

00:07:09.020 --> 00:07:14.600
We factored this bottom part
using grouping as 7x plus 2

00:07:14.600 --> 00:07:17.780
times x plus 1.

00:07:17.780 --> 00:07:21.050
And so this expression is the
exact same thing as this

00:07:21.050 --> 00:07:23.650
expression over here, and we're
going to divide it by

00:07:23.650 --> 00:07:25.180
that expression.

00:07:25.180 --> 00:07:29.200
So dividing by that expression,
that is the same

00:07:29.200 --> 00:07:32.360
exact thing as multiplying
by the inverse of this

00:07:32.360 --> 00:07:35.290
expression.

00:07:35.290 --> 00:07:35.850
Actually, you know what?

00:07:35.850 --> 00:07:38.750
I just realized that it wasn't
x squared plus 4x.

00:07:38.750 --> 00:07:40.010
I just saw the problem.

00:07:40.010 --> 00:07:41.680
That is an x squared plus 4x.

00:07:41.680 --> 00:07:42.560
This was the original problem.

00:07:42.560 --> 00:07:43.370
Sorry for all that.

00:07:43.370 --> 00:07:45.860
We haven't done any work
here so it's all cool.

00:07:45.860 --> 00:07:50.890
So this right over here,
multiplying by the inverse of

00:07:50.890 --> 00:07:54.350
this, we multiply-- dividing by
this is the same thing as

00:07:54.350 --> 00:07:57.240
multiplying by the inverse, so
we just flip the numerator and

00:07:57.240 --> 00:07:58.120
the denominator.

00:07:58.120 --> 00:08:08.080
So x squared plus 4x
over 7x plus 2.

00:08:08.080 --> 00:08:11.120
And then another thing we can
do, if we look at this up

00:08:11.120 --> 00:08:13.060
here, we can factor out an x.

00:08:13.060 --> 00:08:15.380
So this right here, if we factor
out an x is the same

00:08:15.380 --> 00:08:19.180
thing as x times x plus 4.

00:08:19.180 --> 00:08:21.080
So if we do all of the
multiplication, what does our

00:08:21.080 --> 00:08:21.920
numerator become?

00:08:21.920 --> 00:08:28.840
We have x-- I'll use a new
color-- we have x plus 4

00:08:28.840 --> 00:08:33.809
squared times x times
x plus 4.

00:08:33.809 --> 00:08:36.130
That is our numerator.

00:08:36.130 --> 00:08:44.530
And our denominator is 7x plus
2 times another 7x plus 2

00:08:44.530 --> 00:08:46.890
times an x plus 1.

00:08:46.890 --> 00:08:50.170
So here nothing is canceling
out, but we have the same term

00:08:50.170 --> 00:08:52.186
being multiplied
multiple times.

00:08:52.186 --> 00:08:55.680
x to the fourth squared times
x to the fourth to the first

00:08:55.680 --> 00:08:58.610
power, this is going to be equal
to-- we still have our x

00:08:58.610 --> 00:09:03.450
out front, so it's x times x
plus 4, we could say to the

00:09:03.450 --> 00:09:08.480
third power, all of that over--
we have two 7x's plus

00:09:08.480 --> 00:09:16.290
2, so 7x plus 2 squared
times x plus 1.

00:09:16.290 --> 00:09:18.020
And we are done!


Simplifying rational expressions introduction

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:04.570
When we first started learning
about fractions or rational

00:00:04.570 --> 00:00:07.600
numbers, we learned about the
idea of putting things in

00:00:07.600 --> 00:00:12.790
lowest terms. So if we saw
something like 3, 6, we knew

00:00:12.790 --> 00:00:15.490
that 3 and 6 share
a common factor.

00:00:15.490 --> 00:00:18.940
We know that the numerator,
well, 3 is just 3, but that 6

00:00:18.940 --> 00:00:22.250
could be written as 2 times 3.

00:00:22.250 --> 00:00:26.650
And since they share a common
factor, the 3 in this case, we

00:00:26.650 --> 00:00:30.920
could divide the numerator by 3
and the denominator by 3, or

00:00:30.920 --> 00:00:34.330
we could say that this
is just 3/3, and they

00:00:34.330 --> 00:00:35.480
would cancel out.

00:00:35.480 --> 00:00:39.880
And in lowest terms, this
fraction would be 1/2.

00:00:39.880 --> 00:00:48.050
Or just to kind of hit the point
home, if we had 8/24,

00:00:48.050 --> 00:00:52.270
once again, we know that this is
the same thing as 8 over 3

00:00:52.270 --> 00:00:56.380
times 8, or this is the
same thing as 1 over 3

00:00:56.380 --> 00:00:58.240
times 8 over 8.

00:00:58.240 --> 00:01:04.090
The 8's cancel out and we get
this in lowest terms as 1/3.

00:01:04.090 --> 00:01:09.470
The same exact idea applies
to rational expressions.

00:01:09.470 --> 00:01:10.790
These are rational numbers.

00:01:10.790 --> 00:01:14.090
Rational expressions are
essentially the same thing,

00:01:14.090 --> 00:01:16.360
but instead of the numerator
being an actual number and the

00:01:16.360 --> 00:01:18.980
denominator be an actual number,
they're expressions

00:01:18.980 --> 00:01:20.670
involving variables.

00:01:20.670 --> 00:01:23.200
So let me show you what
I'm talking about.

00:01:23.200 --> 00:01:37.030
Let's say that I had 9x plus
3 over 12x plus 4.

00:01:37.030 --> 00:01:39.100
Now, this numerator up here,
we can factor it.

00:01:39.100 --> 00:01:40.950
We can factor out a 3.

00:01:40.950 --> 00:01:47.420
This is equal to 3
times 3x plus 1.

00:01:47.420 --> 00:01:49.590
That's what our numerator
is equal to.

00:01:49.590 --> 00:01:51.720
And our denominator, we
can factor out a 4.

00:01:51.720 --> 00:01:57.380
This is the same thing
as 4 times 3x.

00:01:57.380 --> 00:01:59.710
12 divided by 4 is 3.

00:01:59.710 --> 00:02:01.760
12x over 4 is 3x.

00:02:01.760 --> 00:02:04.380
Plus 4 divided by 4 is 1.

00:02:04.380 --> 00:02:07.810
So here, just like there, the
numerator and the denominator

00:02:07.810 --> 00:02:09.620
have a common factor.

00:02:09.620 --> 00:02:11.370
In this case, it's 3x plus 1.

00:02:11.370 --> 00:02:15.220
In this case, it's a variable
expression.

00:02:15.220 --> 00:02:16.870
It's not an actual number,
but we can do

00:02:16.870 --> 00:02:18.440
the exact same thing.

00:02:18.440 --> 00:02:20.150
They cancel out.

00:02:20.150 --> 00:02:23.160
So if we were to write this
rational expression in lowest

00:02:23.160 --> 00:02:26.260
terms, we could say that
this is equal to 3/4.

00:02:31.290 --> 00:02:34.230
Let's do another one.

00:02:34.230 --> 00:02:40.020
Let's say that we had
x squared-- let

00:02:40.020 --> 00:02:40.740
me see a good one.

00:02:40.740 --> 00:02:53.820
So let's say we had x squared
minus 9 over 5x plus 15.

00:02:53.820 --> 00:02:56.230
So what is this going
to be equal to?

00:02:58.910 --> 00:03:00.875
So the numerator
we can factor.

00:03:00.875 --> 00:03:02.210
It's a difference of squares.

00:03:02.210 --> 00:03:06.750
We have x plus 3 times
x minus 3.

00:03:06.750 --> 00:03:09.210
And in the denominator we
can factor a 5 out.

00:03:09.210 --> 00:03:12.680
This is 5 times x plus 3.

00:03:12.680 --> 00:03:15.360
So once again, a common factor
in the numerator and in the

00:03:15.360 --> 00:03:17.970
denonminator, we can
cancel them out.

00:03:17.970 --> 00:03:21.070
But we touched on this a
couple of videos ago.

00:03:21.070 --> 00:03:22.750
We have to be very careful.

00:03:22.750 --> 00:03:24.010
We can cancel them out.

00:03:24.010 --> 00:03:28.230
We can say that this is going to
be equal to x minus 3 over

00:03:28.230 --> 00:03:33.240
5, but we have to exclude the
values of x that would have

00:03:33.240 --> 00:03:36.620
made this denominator equal to
0, that would have made the

00:03:36.620 --> 00:03:39.170
entire expression undefined.

00:03:39.170 --> 00:03:42.730
So we could write this as being
equal to x minus 3 over

00:03:42.730 --> 00:03:47.880
5, but x cannot be equal
to negative 3.

00:03:47.880 --> 00:03:51.190
Negative 3 would make this zero
or would make this whole

00:03:51.190 --> 00:03:52.340
thing zero.

00:03:52.340 --> 00:03:56.200
So this and this whole
thing are equivalent.

00:03:56.200 --> 00:03:59.970
This is not equivalent to this
right here, because this is

00:03:59.970 --> 00:04:03.630
defined that x is equal to
negative 3, while this isn't

00:04:03.630 --> 00:04:05.590
defined that x is equal
to negative 3.

00:04:05.590 --> 00:04:10.540
So to make them the same, I
also have to add the extra

00:04:10.540 --> 00:04:13.680
condition that x cannot
equal negative 3.

00:04:13.680 --> 00:04:18.990
So likewise, over here, if this
was a function, let's say

00:04:18.990 --> 00:04:23.840
we wrote y is equal to 9x plus
3 over 12x plus 4 and we

00:04:23.840 --> 00:04:27.360
wanted to graph it, when we
simplify it, the temptation is

00:04:27.360 --> 00:04:30.230
oh, well, we factored out a 3x
plus 1 in the numerator and

00:04:30.230 --> 00:04:30.960
the denonminator.

00:04:30.960 --> 00:04:31.960
They cancel out.

00:04:31.960 --> 00:04:34.030
The temptation is to say, well,
this is the same graph

00:04:34.030 --> 00:04:38.840
as y is equal to the constant
3/4, which is just a

00:04:38.840 --> 00:04:42.010
horizontal line at y
is equal to 3/4.

00:04:42.010 --> 00:04:43.760
But we have to add
one condition.

00:04:43.760 --> 00:04:47.080
We have to eliminate-- we have
to exclude the x-values that

00:04:47.080 --> 00:04:51.110
would have made this thing right
here equal to zero, and

00:04:51.110 --> 00:04:55.660
that would have been zero if
x is equal to negative 1/3.

00:04:55.660 --> 00:04:58.930
If x is equal to negative 1/3,
this or this denominator would

00:04:58.930 --> 00:05:00.140
be equal to zero.

00:05:00.140 --> 00:05:03.650
So even over here, we'd have
to say x cannot be equal to

00:05:03.650 --> 00:05:05.260
negative 1/3.

00:05:05.260 --> 00:05:14.100
That condition is what really
makes that equal to that, that

00:05:14.100 --> 00:05:17.830
x cannot be equal
to negative 1/3.

00:05:17.830 --> 00:05:20.330
Let's do a couple
more of these.

00:05:20.330 --> 00:05:22.770
And I'll do these in pink.

00:05:22.770 --> 00:05:35.380
Let's say that I had x squared
plus 6x plus 8 over x

00:05:35.380 --> 00:05:37.830
squared plus 4x.

00:05:37.830 --> 00:05:40.460
Or actually, even better, let
me do this a little bit. x

00:05:40.460 --> 00:05:52.220
squared plus 6x plus 5 over
x squared minus x minus 2.

00:05:52.220 --> 00:05:54.540
So once again, we want to factor
the numerator and the

00:05:54.540 --> 00:05:56.640
denonminator, just like we did
with traditional numbers when

00:05:56.640 --> 00:06:01.380
we first learned about fractions
and lowest terms. So

00:06:01.380 --> 00:06:03.330
if we factor the numerator,
what two numbers when I

00:06:03.330 --> 00:06:06.980
multiply them equal 5 and
I add them equal 6?

00:06:06.980 --> 00:06:09.370
Well, the numbers that pop
in my head are 5 and 1.

00:06:09.370 --> 00:06:14.960
So the numerator is x plus
5 times x minus 1.

00:06:14.960 --> 00:06:17.280
And then our denonminator,
two numbers.

00:06:17.280 --> 00:06:20.940
Multiply negative 2,
add a negative 1.

00:06:20.940 --> 00:06:24.830
Negative 2 and positive
1 pop out of my head.

00:06:24.830 --> 00:06:27.210
So this is a positive
1, right? x plus 5

00:06:27.210 --> 00:06:29.050
times x plus 1, right?

00:06:29.050 --> 00:06:30.600
1 times 5 is 5.

00:06:30.600 --> 00:06:33.250
5x plus 1x is 6x.

00:06:33.250 --> 00:06:38.410
So here we have a positive
1 and a negative 2.

00:06:38.410 --> 00:06:43.635
So x minus 2 times x plus 1.

00:06:46.760 --> 00:06:49.970
So we have a common factor
in the numerator and the

00:06:49.970 --> 00:06:50.350
denonminator.

00:06:50.350 --> 00:06:51.820
These cancel out.

00:06:51.820 --> 00:06:56.120
So you could say that this
is equal to x plus 5

00:06:56.120 --> 00:06:57.940
over x minus 2.

00:06:57.940 --> 00:06:59.630
But for them to really
be equal, we

00:06:59.630 --> 00:07:00.950
have to add the condition.

00:07:00.950 --> 00:07:04.970
We have to add the condition
that x cannot be equal to

00:07:04.970 --> 00:07:08.025
negative 1 because
if x is equal to

00:07:08.025 --> 00:07:09.910
negative 1, this is undefined.

00:07:09.910 --> 00:07:13.390
We have to add that condition
because this by itself is

00:07:13.390 --> 00:07:14.940
defined at x is equal
to negative 1.

00:07:14.940 --> 00:07:16.250
You could put negative
1 here and you're

00:07:16.250 --> 00:07:17.360
going to get a number.

00:07:17.360 --> 00:07:20.350
But this is not defined at x is
equal to negative 1, so we

00:07:20.350 --> 00:07:23.820
have to add this condition
for this to

00:07:23.820 --> 00:07:27.710
truly be equal to that.

00:07:27.710 --> 00:07:30.090
Let's do a harder one here.

00:07:30.090 --> 00:07:40.810
Let's say we have 3x squared
plus 3x minus 18, all of that

00:07:40.810 --> 00:07:49.390
over 2x squared plus
5x minus 3.

00:07:49.390 --> 00:07:52.840
So it's always a little bit more
painful to factor things

00:07:52.840 --> 00:07:55.490
that have a non-one coefficient
out here, but

00:07:55.490 --> 00:07:56.500
we've learned how to do that.

00:07:56.500 --> 00:08:00.155
We can do it by grouping, and
this is a good practice for

00:08:00.155 --> 00:08:01.980
our grouping, so let's do it.

00:08:01.980 --> 00:08:08.500
So remember, let's factor 3x
squared plus 3x minus 18.

00:08:08.500 --> 00:08:09.840
So you need to think
of two numbers.

00:08:09.840 --> 00:08:12.010
This is just a review
of grouping.

00:08:12.010 --> 00:08:14.310
You need to think of two numbers
that when we multiply

00:08:14.310 --> 00:08:18.330
them are equal to 3 times
negative 18, or it's equal to

00:08:18.330 --> 00:08:19.990
negative 54, right?

00:08:19.990 --> 00:08:21.920
That's 3 times negative 18.

00:08:21.920 --> 00:08:26.960
And when we add them, a plus
b, needs to be equal to 3x

00:08:26.960 --> 00:08:31.930
because we're going to split up
the 3x into an ax and a bx.

00:08:31.930 --> 00:08:34.929
Or even better, not
3x, equal to 3.

00:08:34.929 --> 00:08:36.620
So what two numbers
could there be?

00:08:36.620 --> 00:08:38.100
Let's see, our times tables.

00:08:38.100 --> 00:08:39.280
Let's see, they are
three apart.

00:08:39.280 --> 00:08:41.510
One's going to have to be
positive and one's negative.

00:08:41.510 --> 00:08:43.940
9 times 6 is 54.

00:08:43.940 --> 00:08:47.650
If we make the 9 positive
and we make the b

00:08:47.650 --> 00:08:48.970
negative 6, it works.

00:08:48.970 --> 00:08:50.770
9 minus 6 is 3.

00:08:50.770 --> 00:08:54.520
9 times negative 6
is negative 54.

00:08:54.520 --> 00:08:57.620
So we can rewrite
this up here.

00:08:57.620 --> 00:09:03.050
We can rewrite this as 3x
squared, and I'm going to say

00:09:03.050 --> 00:09:11.900
plus 9x minus 6x minus 18.

00:09:11.900 --> 00:09:16.300
Notice, all I did here
is I split this 3x

00:09:16.300 --> 00:09:20.200
into a 9x minus 6x.

00:09:20.200 --> 00:09:22.610
The only difference between
this expression and this

00:09:22.610 --> 00:09:25.910
expression is that I split the
3x into a 9x minus 6x.

00:09:25.910 --> 00:09:28.670
You add these two together,
you get 3x.

00:09:28.670 --> 00:09:30.360
The way I wrote it right here
you can actually ignore the

00:09:30.360 --> 00:09:31.460
parentheses.

00:09:31.460 --> 00:09:35.350
And the whole reason why I did
that is so I can now group it.

00:09:35.350 --> 00:09:38.510
And normally, I decide which
term goes with which based on

00:09:38.510 --> 00:09:40.920
what's positive or negative or
which has common factors.

00:09:40.920 --> 00:09:45.430
They both have a common
factor with 3.

00:09:45.430 --> 00:09:48.150
Actually, it probably wouldn't
matter in this situation, but

00:09:48.150 --> 00:09:50.920
I like the 9 on this side
because they're both positive.

00:09:50.920 --> 00:09:53.400
So let's factor out
a 3x out of this

00:09:53.400 --> 00:09:56.280
expression on the left.

00:09:56.280 --> 00:09:58.780
If we factor out a 3x out of
this expression this becomes

00:09:58.780 --> 00:10:02.920
3x times x plus 3.

00:10:02.920 --> 00:10:07.140
And then on this expression, if
we factor out a negative 6,

00:10:07.140 --> 00:10:12.260
we get negative 6
times x plus 3.

00:10:12.260 --> 00:10:15.640
And now this is very clear our
grouping was successful.

00:10:15.640 --> 00:10:18.990
This is the same thing as-- we
can kind of undistribute this

00:10:18.990 --> 00:10:29.260
as 3x minus 6 times x plus 3.

00:10:29.260 --> 00:10:33.580
If we were to multiply this
times each of these terms, you

00:10:33.580 --> 00:10:36.120
get that right there.

00:10:36.120 --> 00:10:42.950
So the top term, we can rewrite
it as 3x minus 6-- let

00:10:42.950 --> 00:10:45.012
me do it in the same color.

00:10:45.012 --> 00:10:52.720
So we can rewrite it as 3x
minus 6 times x plus 3.

00:10:52.720 --> 00:10:54.750
That's this term right here.

00:10:54.750 --> 00:10:55.920
I don't want to make it look
like a negative sign.

00:10:55.920 --> 00:10:57.350
That's that term right there.

00:10:57.350 --> 00:11:00.000
Now let's factor this bottom
part over here.

00:11:00.000 --> 00:11:02.210
Scroll to the left
a little bit.

00:11:02.210 --> 00:11:09.600
So if I want to factor 2x
squared plus 5x plus 3, I need

00:11:09.600 --> 00:11:12.410
to think of two numbers that
when I take their product, I

00:11:12.410 --> 00:11:16.630
get 2 times 3, which is equal
to 6, and they need to

00:11:16.630 --> 00:11:19.840
add up to be 5.

00:11:19.840 --> 00:11:25.110
And the two obvious numbers
here are 2 and 3.

00:11:25.110 --> 00:11:32.900
I can rewrite this up here as
2x squared plus 2x plus 3x

00:11:32.900 --> 00:11:36.290
plus 3, just like that.

00:11:36.290 --> 00:11:40.050
And then if I put parentheses
over here, and I decided to

00:11:40.050 --> 00:11:42.220
group the 2 with the 2 because
they have a common factor of

00:11:42.220 --> 00:11:44.930
2, and I grouped the 3 with
the 3 because they have a

00:11:44.930 --> 00:11:46.200
common factor of 3.

00:11:46.200 --> 00:11:50.000
This right here is 2 and a 3.

00:11:50.000 --> 00:11:52.270
So here we can factor
out a 2x.

00:11:52.270 --> 00:11:58.870
If you factor out a 2x, you get
2x times x plus 1 plus--

00:11:58.870 --> 00:12:02.880
you factor out a 3 here--
plus 3 times x plus 1.

00:12:02.880 --> 00:12:05.950
And our grouping
was successful.

00:12:05.950 --> 00:12:09.110
This is clearly-- let me switch
colors-- this is the

00:12:09.110 --> 00:12:16.325
same thing as 2x plus
3 times x plus 1.

00:12:20.900 --> 00:12:23.990
So here we've been able
to factor it as well.

00:12:23.990 --> 00:12:30.940
We were able to factor out
the denominator as well.

00:12:30.940 --> 00:12:33.340
Actually, I just realized
that I made a mistake.

00:12:33.340 --> 00:12:34.540
I wrote here minus 3.

00:12:34.540 --> 00:12:36.040
I wrote a plus 3 over here.

00:12:36.040 --> 00:12:38.085
Let me backtrack this.

00:12:42.080 --> 00:12:44.320
That would have been
a horrible mistake.

00:12:44.320 --> 00:12:46.330
I would have had to
redo the video.

00:12:46.330 --> 00:12:50.670
Let me clear all of this, all
of this business over here.

00:12:50.670 --> 00:12:53.220
Let me clear that.

00:12:53.220 --> 00:13:00.150
This is 2x squared
plus 5x minus 3.

00:13:00.150 --> 00:13:05.830
So once again, a times b needs
to be equal to negative 3

00:13:05.830 --> 00:13:08.330
times 2, which is negative 6.

00:13:08.330 --> 00:13:12.960
And a plus b needs
to be equal to 5.

00:13:12.960 --> 00:13:17.430
So in this situation, it looks
like if we went with 6 and

00:13:17.430 --> 00:13:22.680
negative 1, that seems to
be a better situation.

00:13:22.680 --> 00:13:23.830
6 minus 1 is 5.

00:13:23.830 --> 00:13:26.800
6 times negative 1
is negative 6.

00:13:26.800 --> 00:13:28.890
So that would have been
a horrible mistake.

00:13:28.890 --> 00:13:34.790
So we can rewrite this up here
as 2x squared, and I'll group

00:13:34.790 --> 00:13:36.800
the 6 with the 2x squared
because they

00:13:36.800 --> 00:13:37.880
share a common factor.

00:13:37.880 --> 00:13:46.190
So plus 6x minus x, this is the
same thing as 5x minus 3.

00:13:46.190 --> 00:13:50.200
I just had to find the numbers
to split this 5x into.

00:13:50.200 --> 00:13:53.190
But 6x minus x is 5x.

00:13:53.190 --> 00:13:57.420
And if I put some parentheses
here, I can factor out of 2x

00:13:57.420 --> 00:13:58.530
out of this first term.

00:13:58.530 --> 00:14:02.780
I get 2x times x plus 3.

00:14:02.780 --> 00:14:06.870
And here I can factor out
a negative 1, so minus 1

00:14:06.870 --> 00:14:09.540
times x plus 3.

00:14:09.540 --> 00:14:11.330
And then our grouping
was successful.

00:14:11.330 --> 00:14:18.880
We get 2-- let me do this in a
different color-- we get 2x

00:14:18.880 --> 00:14:25.590
minus 1 times x plus 3.

00:14:28.870 --> 00:14:37.720
So our denominator here
is equal to 2x minus 1

00:14:37.720 --> 00:14:40.750
times x plus 3.

00:14:40.750 --> 00:14:44.080
And once again, we have a common
factor in our numerator

00:14:44.080 --> 00:14:47.200
and our denonminator,
the x plus 3.

00:14:47.200 --> 00:14:49.550
But we have to add the condition
that x cannot be

00:14:49.550 --> 00:14:51.880
equal to negative 3, because
that would make this whole

00:14:51.880 --> 00:14:53.670
thing equal to zero.

00:14:53.670 --> 00:14:56.810
Or not equal to zero, it would
make us divide by zero, which

00:14:56.810 --> 00:14:58.020
is undefined.

00:14:58.020 --> 00:15:02.960
So we have to say that x cannot
be equal to negative 3.

00:15:02.960 --> 00:15:09.850
So this expression up here is
the same thing as 3x minus 6

00:15:09.850 --> 00:15:14.560
over 2x minus 1, granted that we
also imposed the condition

00:15:14.560 --> 00:15:19.750
that x does not equal
negative 3.

00:15:19.750 --> 00:15:22.220
Hopefully, you found
that interesting.


Polynomial division

WEBVTT
Kind: captions
Language: en

00:00:00.380 --> 00:00:03.670
In this video, we're going to
learn to divide polynomials,

00:00:03.670 --> 00:00:07.230
and sometimes this is called
algebraic long division.

00:00:07.230 --> 00:00:08.550
But you'll see what
I'm talking about

00:00:08.550 --> 00:00:10.330
when we do a few examples.

00:00:10.330 --> 00:00:14.720
Let's say I just want to
divide 2x plus 4 and

00:00:14.720 --> 00:00:17.230
divide it by 2.

00:00:17.230 --> 00:00:19.680
We're not really changing
the value.

00:00:19.680 --> 00:00:22.790
We're just changing how we're
going to express the value.

00:00:22.790 --> 00:00:24.600
So we already know how
to simplify this.

00:00:24.600 --> 00:00:27.810
We've done this in the past. We
could divide the numerator

00:00:27.810 --> 00:00:30.490
and the denominator by
2, and then this

00:00:30.490 --> 00:00:32.100
would be equal to what?

00:00:32.100 --> 00:00:37.250
This would be equal to x plus
2-- let me write it this way--

00:00:37.250 --> 00:00:39.490
it would be equal to, if
you divide this by 2,

00:00:39.490 --> 00:00:40.340
it becomes an x.

00:00:40.340 --> 00:00:42.650
You divide the 4 by
2, it becomes a 2.

00:00:42.650 --> 00:00:44.880
If you divide the 2
by 2, you get a 1.

00:00:44.880 --> 00:00:48.380
So this is equal to x plus
2, which is pretty

00:00:48.380 --> 00:00:49.910
straightforward, I think.

00:00:49.910 --> 00:00:52.500
The other way is you could have
factored a 2 out of here,

00:00:52.500 --> 00:00:54.175
and then those would
have canceled out.

00:00:54.175 --> 00:00:56.830
But I'll also show you how to
do it using algebraic long

00:00:56.830 --> 00:00:59.100
division, which is a bit of
overkill for this problem.

00:00:59.100 --> 00:01:01.130
But I just want to show you that
it's not fundamentally

00:01:01.130 --> 00:01:01.640
anything new.

00:01:01.640 --> 00:01:04.300
It's just a different way of
doing things, but it's useful

00:01:04.300 --> 00:01:06.180
for more complicated problems.

00:01:06.180 --> 00:01:11.610
So you could have also written
this as 2 goes into 2x plus 4

00:01:11.610 --> 00:01:13.070
how many times?

00:01:13.070 --> 00:01:15.300
And you would perform this
the same way you would do

00:01:15.300 --> 00:01:17.020
traditional long division.

00:01:17.020 --> 00:01:19.780
You'd say 2-- you always start
with the highest degree term.

00:01:19.780 --> 00:01:21.840
2 goes into the highest
degree term.

00:01:21.840 --> 00:01:23.350
You would ignore the 4.

00:01:23.350 --> 00:01:26.050
2 goes into 2x how many times?

00:01:26.050 --> 00:01:28.770
Well, it goes into 2x
x times and you put

00:01:28.770 --> 00:01:30.962
the x in the x place.

00:01:30.962 --> 00:01:34.070
x times 2 is 2x.

00:01:34.070 --> 00:01:36.673
And just like traditional long
division, you now subtract.

00:01:39.440 --> 00:01:44.490
So 2x plus 4 minus 2x is what?

00:01:44.490 --> 00:01:46.550
It's 4, right?

00:01:46.550 --> 00:01:48.740
And then 2 goes into
4 how many times?

00:01:48.740 --> 00:01:51.810
It goes into it two times,
a positive two times.

00:01:51.810 --> 00:01:54.050
Put that in the constants
place.

00:01:54.050 --> 00:01:56.240
2 times 2 is 4.

00:01:56.240 --> 00:01:59.280
You subtract, remainder 0.

00:01:59.280 --> 00:02:02.530
So this might seem overkill
for what was probably a

00:02:02.530 --> 00:02:04.140
problem that you already
knew how to do and

00:02:04.140 --> 00:02:05.410
do it in a few steps.

00:02:05.410 --> 00:02:07.240
We're now going to see
that this is a very

00:02:07.240 --> 00:02:08.970
generalizable process.

00:02:08.970 --> 00:02:13.140
You can do this really for any
degree polynomial dividing

00:02:13.140 --> 00:02:15.070
into any other degree
polynomial.

00:02:15.070 --> 00:02:17.850
Let me show you what
I'm talking about.

00:02:17.850 --> 00:02:29.980
So let's say we wanted to divide
x plus 1 into x squared

00:02:29.980 --> 00:02:36.260
plus 3x plus 6.

00:02:36.260 --> 00:02:37.780
So what do we do here?

00:02:37.780 --> 00:02:40.050
So you look at the highest
degree term here, which is an

00:02:40.050 --> 00:02:43.030
x, and you look at the highest
degree term here,

00:02:43.030 --> 00:02:44.550
which is an x squared.

00:02:44.550 --> 00:02:46.140
So you can ignore
everything else.

00:02:46.140 --> 00:02:48.530
And that really simplifies
the process.

00:02:48.530 --> 00:02:52.280
You say x goes into x squared
how many times?

00:02:52.280 --> 00:02:56.040
Well, x squared divided by x is
just x, right? x goes into

00:02:56.040 --> 00:02:57.900
x squared x times.

00:02:57.900 --> 00:02:59.730
You put it in the x place.

00:02:59.730 --> 00:03:02.270
This is the x place right
here or the x to

00:03:02.270 --> 00:03:04.000
the first power place.

00:03:04.000 --> 00:03:06.350
So x times x plus 1 is what?

00:03:06.350 --> 00:03:10.722
x times x is x squared.

00:03:10.722 --> 00:03:15.810
x times 1 is x, so it's
x squared plus x.

00:03:15.810 --> 00:03:19.080
And just like we did over
here, we now subtract.

00:03:19.080 --> 00:03:20.220
And what do we get?

00:03:20.220 --> 00:03:24.350
x squared plus 3x plus 6 minus
x squared-- let me be very

00:03:24.350 --> 00:03:27.630
careful-- this is minus
x squared plus x.

00:03:27.630 --> 00:03:30.030
I want to make sure that
negative sign only-- it

00:03:30.030 --> 00:03:32.030
applies to this whole thing.

00:03:32.030 --> 00:03:35.810
So x squared minus x squared,
those cancel out.

00:03:35.810 --> 00:03:38.020
3x, this is going
to be a minus x.

00:03:38.020 --> 00:03:40.020
Let me put that sign there.

00:03:40.020 --> 00:03:43.410
So this is minus x squared minus
x, just to be clear.

00:03:43.410 --> 00:03:45.250
We're subtracting
the whole thing.

00:03:45.250 --> 00:03:48.720
3x minus x is 2x.

00:03:48.720 --> 00:03:51.880
And then you bring down the 6,
or 6 minus 0 is nothing.

00:03:51.880 --> 00:03:55.030
So 2x plus 6.

00:03:55.030 --> 00:03:58.220
Now, you look at the highest
degree term, an x and a 2x.

00:03:58.220 --> 00:04:01.530
How many times does
x go into 2x?

00:04:01.530 --> 00:04:04.290
It goes into it two times.

00:04:04.290 --> 00:04:07.660
2 times x is 2x.

00:04:07.660 --> 00:04:11.673
2 times 1 is 2.

00:04:15.090 --> 00:04:18.790
So we get 2 times x plus
1 is 2x plus 2.

00:04:18.790 --> 00:04:21.120
But we're going to want to
subtract this from this up

00:04:21.120 --> 00:04:22.900
here, so we're going
to subtract it.

00:04:22.900 --> 00:04:26.240
Instead of writing 2x plus 2,
we could just write negative

00:04:26.240 --> 00:04:28.330
2x minus 2 and then add them.

00:04:28.330 --> 00:04:29.850
These guys cancel out.

00:04:29.850 --> 00:04:32.780
6 minus 2 is 4.

00:04:32.780 --> 00:04:35.620
And how many times
does x go into 4?

00:04:35.620 --> 00:04:38.320
We could just say that's zero
times, or we could say that 4

00:04:38.320 --> 00:04:39.570
is the remainder.

00:04:45.410 --> 00:04:55.680
So if we wanted to rewrite x
squared plus 3x plus 6 over x

00:04:55.680 --> 00:04:58.160
plus 1-- notice, this is the
same thing as x squared plus

00:04:58.160 --> 00:05:02.980
3x plus 6 divided by x plus 1,
this thing divided by this, we

00:05:02.980 --> 00:05:06.640
can now say that this is
equal to x plus 2.

00:05:06.640 --> 00:05:13.130
it is equal to x plus 2 plus
the remainder divided by x

00:05:13.130 --> 00:05:19.820
plus 1 plus 4 over x plus 1.

00:05:19.820 --> 00:05:24.530
This right here and this right
here are equivalent.

00:05:24.530 --> 00:05:26.690
And if you wanted to check that,
if you wanted to go from

00:05:26.690 --> 00:05:33.130
this back to that, what you
could do is multiply this by x

00:05:33.130 --> 00:05:35.120
plus 1 over x plus 1
and it add the two.

00:05:35.120 --> 00:05:40.780
So this is the same
thing as x plus 2.

00:05:40.780 --> 00:05:45.240
And I'm just going to multiply
that times x plus

00:05:45.240 --> 00:05:48.770
1 over x plus 1.

00:05:48.770 --> 00:05:50.500
That's just multiplying
it by 1.

00:05:50.500 --> 00:05:53.440
And then to that, add
4 over x plus 1.

00:05:53.440 --> 00:05:55.870
I did that so I have the same
common denominator.

00:05:55.870 --> 00:05:58.820
And when you perform this
addition right here, when you

00:05:58.820 --> 00:06:02.050
multiply these two binomials
and then add the 4 up here,

00:06:02.050 --> 00:06:06.690
you should you get x squared
plus 3x plus 6.

00:06:06.690 --> 00:06:08.540
Let's do another one of these.

00:06:08.540 --> 00:06:10.070
They're kind of fun.

00:06:10.070 --> 00:06:18.020
So let's say that we have-- we
want to simplify x squared

00:06:18.020 --> 00:06:26.310
plus 5x plus 4 over x plus 4.

00:06:26.310 --> 00:06:29.930
So once again, we can do our
algebraic long division.

00:06:29.930 --> 00:06:43.410
We can divide x plus 4 into
x squared plus 5x plus 4.

00:06:43.410 --> 00:06:45.610
And once again, same
exact process.

00:06:45.610 --> 00:06:49.690
Look at the highest degree terms
in both of them. x goes

00:06:49.690 --> 00:06:51.120
into x squared how many times?

00:06:51.120 --> 00:06:53.390
It goes into it x times.

00:06:53.390 --> 00:06:54.790
Put it in the x place.

00:06:54.790 --> 00:06:56.230
This is our x place
right here.

00:06:56.230 --> 00:06:59.820
X times x is x squared.

00:06:59.820 --> 00:07:03.440
x times 4 is 4x.

00:07:03.440 --> 00:07:05.060
And then, of course, we're
going to want to subtract

00:07:05.060 --> 00:07:05.990
these from there.

00:07:05.990 --> 00:07:08.750
So let me just put a negative
sign there.

00:07:08.750 --> 00:07:10.240
And then these cancel out.

00:07:10.240 --> 00:07:12.540
5x minus 4x is x.

00:07:12.540 --> 00:07:16.800
4 minus 0 is plus 4.

00:07:16.800 --> 00:07:19.940
x plus 4, and then you could
even see this coming.

00:07:19.940 --> 00:07:22.190
You could say x plus 4 goes into
x plus 4 obviously one

00:07:22.190 --> 00:07:25.770
time, or if you were not looking
at the constant terms,

00:07:25.770 --> 00:07:27.770
you would completely just
say, well, x goes

00:07:27.770 --> 00:07:29.300
into x how many times?

00:07:29.300 --> 00:07:30.770
Well, one time.

00:07:30.770 --> 00:07:31.580
Plus 1.

00:07:31.580 --> 00:07:34.140
1 times x is x.

00:07:34.140 --> 00:07:36.260
1 times 4 is 4.

00:07:36.260 --> 00:07:39.430
We're going to subtract them
from up here, so it cancels

00:07:39.430 --> 00:07:41.540
out, so we have no remainder.

00:07:41.540 --> 00:07:45.410
So this right here simplifies
to-- this is

00:07:45.410 --> 00:07:48.000
equal to x plus 1.

00:07:48.000 --> 00:07:49.600
And there's other ways you
could have done this.

00:07:49.600 --> 00:07:51.980
We could have tried to factor
this numerator.

00:07:51.980 --> 00:07:58.200
x squared plus 5x plus
4 over x plus 4.

00:07:58.200 --> 00:07:59.270
This is the same
thing as what?

00:07:59.270 --> 00:08:03.190
We could have factored this
numerator as x plus 4

00:08:03.190 --> 00:08:05.910
times x plus 1.

00:08:05.910 --> 00:08:07.380
4 times 1 is 4.

00:08:07.380 --> 00:08:12.980
4 plus 1 is 5, all of
that over x plus 4.

00:08:12.980 --> 00:08:15.700
That cancels out and you're
left just with x plus 1.

00:08:15.700 --> 00:08:18.400
Either way would have worked,
but the algebraic long

00:08:18.400 --> 00:08:22.630
division will always work, even
if you can't cancel out

00:08:22.630 --> 00:08:24.920
factors like that, even if
you did have a remainder.

00:08:24.920 --> 00:08:26.870
In this situation, you didn't.

00:08:26.870 --> 00:08:30.830
So this was equal to x plus 1.

00:08:30.830 --> 00:08:37.159
Let's do another one of these
just to make sure that you

00:08:37.159 --> 00:08:39.690
really-- because this is
actually a very, very useful

00:08:39.690 --> 00:08:41.650
skill to have in your toolkit.

00:08:41.650 --> 00:08:49.030
So let's say we have
x squared-- let me

00:08:49.030 --> 00:08:49.730
just change it up.

00:08:49.730 --> 00:08:53.850
Let's say we had 2x squared--
I could really make these

00:08:53.850 --> 00:08:54.950
numbers up on the fly.

00:08:54.950 --> 00:09:01.280
2x squared minus 20x plus 12
divided by-- actually, let's

00:09:01.280 --> 00:09:03.800
make it really interesting, just
to show you that it'll

00:09:03.800 --> 00:09:04.510
always work.

00:09:04.510 --> 00:09:06.830
I want to go above quadratic.

00:09:06.830 --> 00:09:15.860
So let's say we have 3x to the
third minus 2x squared plus 7x

00:09:15.860 --> 00:09:24.590
minus 4, and we want to divide
that by x squared plus 1.

00:09:24.590 --> 00:09:25.880
I just made this up.

00:09:25.880 --> 00:09:27.980
But we can just do the algebraic
long division to

00:09:27.980 --> 00:09:30.470
figure out what this is going
to be or what this

00:09:30.470 --> 00:09:32.280
simplified will be.

00:09:32.280 --> 00:09:39.250
x squared plus 1 divided into
this thing up here, 3x to the

00:09:39.250 --> 00:09:46.140
third minus 2x squared
plus 7x minus 4.

00:09:46.140 --> 00:09:48.790
Once again, look at the
highest degree term.

00:09:48.790 --> 00:09:52.010
x squared goes into 3x to the
third how many times?

00:09:52.010 --> 00:09:55.930
Well, it's going to go
into it 3x times.

00:09:55.930 --> 00:09:58.830
You multiply 3x times this,
you get 3x to the third.

00:09:58.830 --> 00:10:00.930
So it's going to go
into it 3x times.

00:10:00.930 --> 00:10:03.885
So you have to write the 3x over
here in the x terms. So

00:10:03.885 --> 00:10:09.000
it's going to go into it 3x
times, just like that.

00:10:09.000 --> 00:10:10.380
Now let's multiply.

00:10:10.380 --> 00:10:17.130
3x times x squared is 3x
to the third, right?

00:10:17.130 --> 00:10:20.820
3x times squared plus
3x times 1.

00:10:20.820 --> 00:10:24.350
So we have a 3x over here.

00:10:24.350 --> 00:10:26.380
I'm making sure to put
it in the x place.

00:10:26.380 --> 00:10:28.440
And we're going to want
to subtract them.

00:10:28.440 --> 00:10:30.090
And what do we have?

00:10:30.090 --> 00:10:32.000
What do we have when
we do that?

00:10:32.000 --> 00:10:33.060
These cancel out.

00:10:33.060 --> 00:10:36.090
We have a minus 2x squared.

00:10:36.090 --> 00:10:39.890
And then 7x minus-- because I
just subtracted 0 from there--

00:10:39.890 --> 00:10:45.830
7x minus 3x is plus 4x,
and we have a minus 4.

00:10:45.830 --> 00:10:48.140
Once again, look at the
highest degree term.

00:10:48.140 --> 00:10:51.450
x squared and a negative
2x squared.

00:10:51.450 --> 00:10:53.210
So x squared goes
into negative 2x

00:10:53.210 --> 00:10:54.960
squared negative 2 times.

00:10:54.960 --> 00:10:57.450
Negative 2, put it in
the constants place.

00:10:57.450 --> 00:11:02.430
Negative 2 times x squared
is negative 2x squared.

00:11:02.430 --> 00:11:06.370
Negative 2 times 1
is negative 2.

00:11:09.490 --> 00:11:11.820
Now, we're going to want to
subtract these from there, so

00:11:11.820 --> 00:11:14.100
let's multiply them
by negative 1, or

00:11:14.100 --> 00:11:15.460
those become a positive.

00:11:15.460 --> 00:11:16.800
These two guys cancel out.

00:11:16.800 --> 00:11:24.520
4x minus 0 is-- let me switch
colors-- 4x minus 0 is 4x.

00:11:24.520 --> 00:11:29.470
Negative 4 minus negative 2 or
negative 4 plus 2 is equal to

00:11:29.470 --> 00:11:31.340
negative 2.

00:11:31.340 --> 00:11:36.385
And then x squared, now it has
a higher degree than 4x, and

00:11:36.385 --> 00:11:39.463
the highest degree here, so we
view this as the remainder.

00:11:45.450 --> 00:11:49.430
So this expression we could
rewrite it as being equal to

00:11:49.430 --> 00:11:57.490
3x minus 2-- that's the 3x minus
2-- plus our remainder

00:11:57.490 --> 00:12:03.425
4x minus 2, all of that
over x squared plus 1.

00:12:06.150 --> 00:12:09.150
Hopefully, you found that
as fun as I did.


A third example of graphing a rational function

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:03.410
Let's graph another rational
function, because you really

00:00:03.410 --> 00:00:05.270
can't get enough
practice here.

00:00:05.270 --> 00:00:12.690
So let's say we have y is equal
to x over x squared

00:00:12.690 --> 00:00:16.510
minus x minus 6.

00:00:16.510 --> 00:00:19.000
So the first thing we might want
to do is just factor this

00:00:19.000 --> 00:00:22.680
denominator so we can identify
our vertical asymptotes, if

00:00:22.680 --> 00:00:23.880
there are any.

00:00:23.880 --> 00:00:26.190
So what two numbers when I
take their product I get

00:00:26.190 --> 00:00:29.880
negative 6 and if I add them
up I get negative 1?

00:00:29.880 --> 00:00:31.810
So they have to be of
different signs.

00:00:31.810 --> 00:00:35.510
So one's going to be a plus and
one-- let me write my x's

00:00:35.510 --> 00:00:38.940
a little bit neater than that--
so one is going to be a

00:00:38.940 --> 00:00:42.470
positive and one is going
to be a negative.

00:00:42.470 --> 00:00:45.240
A 2 and a 3 seem to be pretty
close, because they're one

00:00:45.240 --> 00:00:47.890
apart, and I'm going to subtract
the larger number

00:00:47.890 --> 00:00:49.830
because when I add them,
I get a negative.

00:00:49.830 --> 00:00:53.780
So x minus 3 times x plus
2 seems to work.

00:00:53.780 --> 00:00:55.250
That gets negative 6.

00:00:55.250 --> 00:00:57.150
Negative 3x plus 2x.

00:00:57.150 --> 00:00:59.830
negative 3 times x
plus 2 times x is

00:00:59.830 --> 00:01:01.970
negative x, so that works.

00:01:01.970 --> 00:01:10.140
So this is equal to x over
x plus 2 times x minus 3.

00:01:10.140 --> 00:01:12.930
And like we saw in the last
video, since these

00:01:12.930 --> 00:01:15.590
expressions, since the x plus
2 doesn't cancel out with

00:01:15.590 --> 00:01:19.290
anything in the numerator and
the x minus 3 doesn't cancel

00:01:19.290 --> 00:01:22.820
out with anything in the
numerator, we know that these

00:01:22.820 --> 00:01:25.100
can be used to find our
vertical asymptotes.

00:01:25.100 --> 00:01:27.940
The vertical asymptotes are when
either that term is equal

00:01:27.940 --> 00:01:31.070
to zero or when that term is
equal to zero, because at

00:01:31.070 --> 00:01:33.670
those points, our equation
is undefined.

00:01:33.670 --> 00:01:37.640
So this is equal to zero when x
is equal to negative 2, and

00:01:37.640 --> 00:01:41.900
this is equal to zero when
x is equal to positive 3.

00:01:41.900 --> 00:01:43.060
And you could try it out here.

00:01:43.060 --> 00:01:45.890
If x is equal to negative 2 or
positive 3, you're going to

00:01:45.890 --> 00:01:46.490
get a zero in the

00:01:46.490 --> 00:01:49.310
denonminator, y will be undefined.

00:01:49.310 --> 00:01:52.940
So vertical asymptotes at x
is equal to negative 2.

00:01:52.940 --> 00:01:56.930
So there's a vertical asymptote,
a vertical

00:01:56.930 --> 00:01:59.650
asymptote right there.

00:01:59.650 --> 00:02:01.940
Another vertical asymptote
is x is equal to 3.

00:02:01.940 --> 00:02:04.050
One, two, three.

00:02:04.050 --> 00:02:09.410
There is our other vertical
asymptote.

00:02:09.410 --> 00:02:11.840
Now let's think about horizontal
asymptotes, or if

00:02:11.840 --> 00:02:13.650
there are any.

00:02:13.650 --> 00:02:17.470
So what happens as x gets
super-positive or

00:02:17.470 --> 00:02:18.290
super-negative?

00:02:18.290 --> 00:02:20.090
And as we said before, you
just have to look at the

00:02:20.090 --> 00:02:23.560
highest degree term on the
numerator and the highest

00:02:23.560 --> 00:02:26.350
degree term on the
denominator.

00:02:26.350 --> 00:02:29.990
Now, notice the highest degree
term on the denominator is x

00:02:29.990 --> 00:02:32.400
squared, while the highest
degree term on the numerator

00:02:32.400 --> 00:02:33.870
is only an x.

00:02:33.870 --> 00:02:36.680
So when x gets really large,
what's going to happen?

00:02:36.680 --> 00:02:38.850
You could imagine, this is going
to be like a million

00:02:38.850 --> 00:02:44.260
over a million squared, which
is still one over a million.

00:02:44.260 --> 00:02:46.470
These terms over here
don't matter much.

00:02:46.470 --> 00:02:48.850
But this term right here is
going to grow faster than

00:02:48.850 --> 00:02:50.350
everything.

00:02:50.350 --> 00:02:52.060
This is an x squared term.

00:02:52.060 --> 00:02:54.480
As x gets large, it's going
to get way larger than

00:02:54.480 --> 00:02:57.550
everything, including this
term on the top, so it's

00:02:57.550 --> 00:02:59.230
essentially going
to go to zero.

00:02:59.230 --> 00:03:02.080
When the denominator just gets
bigger, faster than the

00:03:02.080 --> 00:03:05.340
numerator as you're going
to approach zero.

00:03:05.340 --> 00:03:08.745
So we have a horizontal
asymptote at y is equal to 0.

00:03:11.900 --> 00:03:16.490
I could draw it as a dotted
line over our x-axis.

00:03:16.490 --> 00:03:22.530
So that right there is the
line, y is equal to 0.

00:03:22.530 --> 00:03:25.760
Once again, we identify that
looking at the highest degree

00:03:25.760 --> 00:03:26.630
term there.

00:03:26.630 --> 00:03:29.700
The denominator has a
higher-degree term, so it's

00:03:29.700 --> 00:03:31.590
going to grow faster
than the numerator.

00:03:31.590 --> 00:03:33.250
You could try it out
on your calculator.

00:03:33.250 --> 00:03:35.450
And that's true whether you
go in the super-negative

00:03:35.450 --> 00:03:37.120
direction or the super-positive
direction.

00:03:37.120 --> 00:03:41.030
This thing is going to overwhelm
this thing up here,

00:03:41.030 --> 00:03:44.240
the denominator grows faster
than the numerator, which

00:03:44.240 --> 00:03:45.540
essentially we're going
to approach zero.

00:03:45.540 --> 00:03:48.310
You're going to get smaller
and smaller fractions.

00:03:48.310 --> 00:03:55.000
Just remember, 1/10 and then--
let me actually just-- as x

00:03:55.000 --> 00:03:56.700
gets larger and larger
and larger,

00:03:56.700 --> 00:03:57.430
what's going to happen?

00:03:57.430 --> 00:04:01.730
Let me just show you
on my calculator.

00:04:01.730 --> 00:04:03.920
Let's say x is equal to 10.

00:04:03.920 --> 00:04:11.570
10 divided by 10 squared minus
10, and normally you wouldn't

00:04:11.570 --> 00:04:12.090
have to do this.

00:04:12.090 --> 00:04:15.010
I just really want to show
you the intuition.

00:04:15.010 --> 00:04:15.800
Whoops!

00:04:15.800 --> 00:04:17.050
I'm not trying to graph.

00:04:21.220 --> 00:04:24.460
Let me exit from here.

00:04:24.460 --> 00:04:30.400
So if we have 10 over 10 squared
minus 10, once again,

00:04:30.400 --> 00:04:31.800
you normally wouldn't
have to do this.

00:04:31.800 --> 00:04:34.430
I just want to show you,
give you the intuition.

00:04:34.430 --> 00:04:36.670
Let me put some parentheses
there.

00:04:36.670 --> 00:04:39.270
Let me put some parentheses
over here.

00:04:39.270 --> 00:04:43.360
So let me insert the parentheses
there and put a

00:04:43.360 --> 00:04:45.840
parentheses over here.

00:04:45.840 --> 00:04:47.830
You get a small number.

00:04:47.830 --> 00:04:49.380
What happens if x gets
even larger?

00:04:51.960 --> 00:04:57.110
Let me make, instead of a 10,
let me make it all 100.

00:04:57.110 --> 00:05:03.040
Let me make these tens into
hundreds, into 100.

00:05:03.040 --> 00:05:04.850
Insert 100 there,
what do we get?

00:05:04.850 --> 00:05:06.470
We get even a smaller number.

00:05:06.470 --> 00:05:08.090
And if you try x is equal to
1,000, it's going to be be

00:05:08.090 --> 00:05:09.100
even smaller than that.

00:05:09.100 --> 00:05:11.510
That's because this term right
here is growing faster than

00:05:11.510 --> 00:05:13.620
every single other term.

00:05:13.620 --> 00:05:17.210
That's why our horizontal
asymptote is y is equal to 0.

00:05:17.210 --> 00:05:19.610
Now, the last thing we want to
do, we've drawn all of our

00:05:19.610 --> 00:05:21.930
asymptotes, is just try
out some points.

00:05:21.930 --> 00:05:26.130
So let's draw like a
little table here.

00:05:26.130 --> 00:05:27.650
There's our table.

00:05:27.650 --> 00:05:30.570
When x is equal to
0, what is y?

00:05:30.570 --> 00:05:32.950
x is 0, we have 0 over
all of this.

00:05:32.950 --> 00:05:37.200
0 minus 6, 0 over negative
6 is just 0.

00:05:37.200 --> 00:05:39.990
When x is equal to-- I don't
know, let's just try when x is

00:05:39.990 --> 00:05:42.900
equal to 1, what do we have?

00:05:42.900 --> 00:05:44.980
We have 1 over-- I'll
write it here.

00:05:44.980 --> 00:05:48.650
1 over 1 squared minus 1.

00:05:48.650 --> 00:05:52.410
Now that's just 0, so
we have negative 6.

00:05:52.410 --> 00:05:54.910
When x is equal to negative
1, what do we have?

00:05:54.910 --> 00:06:02.790
When x is equal to negative
1, we have negative 1 over

00:06:02.790 --> 00:06:08.310
negative 1 squared, which
is 1 minus negative 1.

00:06:08.310 --> 00:06:14.350
So that's plus 1-- right, minus
negative 1-- minus 6.

00:06:14.350 --> 00:06:16.630
So what is this right here?

00:06:16.630 --> 00:06:22.460
This is negative 1, so this is
going to be negative 1 over 2

00:06:22.460 --> 00:06:24.400
minus 6 over negative 4.

00:06:24.400 --> 00:06:28.410
This is going to be equal to
1/4, so we're going to get a

00:06:28.410 --> 00:06:29.990
positive value.

00:06:29.990 --> 00:06:32.870
So we have-- let me draw this--
negative 1, we're at

00:06:32.870 --> 00:06:35.130
1/4 right here.

00:06:35.130 --> 00:06:36.170
That's about right there.

00:06:36.170 --> 00:06:38.490
I'll do it in a darker color.

00:06:38.490 --> 00:06:43.630
We had the point 0, 0, and then
at x is equal to 1, we

00:06:43.630 --> 00:06:45.600
had negative 1/6.

00:06:45.600 --> 00:06:51.240
So you could keep graphing more
and more points, but it

00:06:51.240 --> 00:06:54.150
looks like as we approach this
vertical asymptote from the

00:06:54.150 --> 00:06:57.210
right, we go to positive
infinity.

00:07:00.100 --> 00:07:02.060
And that should make sense.

00:07:02.060 --> 00:07:04.810
Let's see, if we were to put
in-- we're approaching

00:07:04.810 --> 00:07:06.310
negative 2 from the right.

00:07:06.310 --> 00:07:12.350
So if you were to put in
negative 1.9999999, this term

00:07:12.350 --> 00:07:14.700
is going to be a very small
positive number.

00:07:14.700 --> 00:07:17.010
This term's going to be
a negative number.

00:07:17.010 --> 00:07:18.640
This term's going to
be negative number.

00:07:18.640 --> 00:07:19.970
The negatives cancel out.

00:07:19.970 --> 00:07:22.870
You have a very small positive
number in the denominator.

00:07:22.870 --> 00:07:26.280
1 over that gives you a
very positive number.

00:07:26.280 --> 00:07:30.390
Now, as we approach the other
vertical asymptote from the

00:07:30.390 --> 00:07:33.830
left, we're going to
go super-negative.

00:07:33.830 --> 00:07:37.375
My gut tells me that because
when I tried x is equal to 1,

00:07:37.375 --> 00:07:39.560
I already went to a
negative value.

00:07:39.560 --> 00:07:44.530
But you could imagine if
you did 2.99999, right?

00:07:44.530 --> 00:07:45.920
Let me draw that a little
bit better.

00:07:48.580 --> 00:07:49.590
You get the idea.

00:07:49.590 --> 00:07:52.520
If x is equal to 2.999, so we
get really close to the

00:07:52.520 --> 00:07:55.970
asymptote, this is going to be
positive, this is going to be

00:07:55.970 --> 00:07:59.250
negative, that's going to be
positive, and this is going to

00:07:59.250 --> 00:08:00.510
be a small number.

00:08:00.510 --> 00:08:05.210
So you're going to have 1 over
a very small negative number,

00:08:05.210 --> 00:08:09.120
which is a very, I guess,
negative number.

00:08:09.120 --> 00:08:11.520
It's a negative of 1 over a very
small number, so you're

00:08:11.520 --> 00:08:13.370
going to approach negative
infinity.

00:08:13.370 --> 00:08:16.810
Now, let's try some point out
here to see what happens.

00:08:16.810 --> 00:08:21.630
So what happens when
x is equal to 4?

00:08:21.630 --> 00:08:31.910
When x is equal to 4, you have
4 over 16 minus 4 minus 6.

00:08:31.910 --> 00:08:32.490
What is that?

00:08:32.490 --> 00:08:34.360
That's 16 minus 10.

00:08:34.360 --> 00:08:35.034
That is 6.

00:08:35.034 --> 00:08:42.419
So this is equal to 4/6,
which is equal to 2/3.

00:08:42.419 --> 00:08:46.390
So the point 4, 2/3 is here,
so one, two, three, four.

00:08:46.390 --> 00:08:48.510
2/3, just like that.

00:08:48.510 --> 00:08:50.870
So that gives me the sense,
look, I have to approach this

00:08:50.870 --> 00:08:53.680
horizontal asymptote as we go
further and further out.

00:08:53.680 --> 00:08:56.330
We're going to probably
approach positive

00:08:56.330 --> 00:09:00.130
infinity like this.

00:09:00.130 --> 00:09:02.250
Let me draw it a little
neater than that.

00:09:02.250 --> 00:09:04.250
Like that.

00:09:04.250 --> 00:09:05.730
You get the idea.

00:09:05.730 --> 00:09:10.540
Then over here, we're going to
get closer and closer to our

00:09:10.540 --> 00:09:13.530
horizontal asymptote as
we approach infinity.

00:09:13.530 --> 00:09:15.530
This should be a
smoother-looking curve right

00:09:15.530 --> 00:09:16.620
around there.

00:09:16.620 --> 00:09:18.230
I'm making a mess here.

00:09:18.230 --> 00:09:20.170
This should be a
smoother-looking curve.

00:09:20.170 --> 00:09:22.550
You get the idea, I think.

00:09:22.550 --> 00:09:25.150
Now, let's see what happens when
x is equal to negative 3.

00:09:28.440 --> 00:09:34.200
So when x is equal to negative
3, we have negative 3 over

00:09:34.200 --> 00:09:38.860
negative 3 squared, which is 9
minus negative 3, so that's

00:09:38.860 --> 00:09:42.290
plus 3 minus 6.

00:09:42.290 --> 00:09:43.770
So what is this equal to?

00:09:43.770 --> 00:09:49.380
This is equal to negative 3
over-- this is 12 minus 6 over

00:09:49.380 --> 00:09:55.630
6, right, which is equal to
negative 1/2, So negative 3,

00:09:55.630 --> 00:09:57.700
negative 1/2.

00:09:57.700 --> 00:09:59.120
Negative 1/2 is right there.

00:09:59.120 --> 00:10:01.670
So we're going to approach
this asymptote as we get

00:10:01.670 --> 00:10:03.570
really negative.

00:10:03.570 --> 00:10:07.340
And we're probably going to go
straight down like that as we

00:10:07.340 --> 00:10:10.180
approach this vertical asymptote
right there.

00:10:10.180 --> 00:10:13.270
And you could try more points
if you don't believe me.

00:10:13.270 --> 00:10:19.350
But let us graph it just to
verify it for ourselves.

00:10:19.350 --> 00:10:36.220
So our equation is x divided by
x squared minus x minus 6.

00:10:36.220 --> 00:10:39.420
And let's graph it.

00:10:39.420 --> 00:10:40.730
And there you go.

00:10:40.730 --> 00:10:41.670
There you go.

00:10:41.670 --> 00:10:43.240
All right, looks pretty good.

00:10:46.220 --> 00:10:48.390
Our asymptote is zero,
we go down.

00:10:48.390 --> 00:10:49.850
Vertical asymptote, bam!

00:10:49.850 --> 00:10:52.740
Go up there, then we go back
down here, then we go just

00:10:52.740 --> 00:10:53.920
like that again.

00:10:53.920 --> 00:10:58.550
So once again, this looks just
about exactly what we got.

00:10:58.550 --> 00:11:01.250
Obviously, the graphing
calculator, it kind of a

00:11:01.250 --> 00:11:04.020
pitters out as you get close
to these values and it does

00:11:04.020 --> 00:11:06.600
weird things, but it has
the same general shape.

00:11:06.600 --> 00:11:10.010
We could actually close the
range a little bit if we want,

00:11:10.010 --> 00:11:11.190
if we want to graph it.

00:11:11.190 --> 00:11:15.600
Let's make our x minimum
value, let's make it 5.

00:11:15.600 --> 00:11:17.550
And let's take our x maximum
value, let's

00:11:17.550 --> 00:11:18.990
make that 5 as well.

00:11:18.990 --> 00:11:20.970
We're kind of zooming
in a little bit.

00:11:20.970 --> 00:11:22.520
So let's graph it now.

00:11:22.520 --> 00:11:23.930
Bam!

00:11:23.930 --> 00:11:24.900
bam!

00:11:24.900 --> 00:11:25.590
There you go!

00:11:25.590 --> 00:11:27.750
It's the same shape as we
graphed right here.

00:11:27.750 --> 00:11:30.010
Hopefully, you found
that satisfying.


Another rational function graph example

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:02.150
Let's do a couple more examples

00:00:02.150 --> 00:00:04.310
graphing rational functions.

00:00:04.310 --> 00:00:11.530
So let's say I have y is equal
to 2x over x plus 1.

00:00:11.530 --> 00:00:14.600
So the first thing we might
want to do is identify our

00:00:14.600 --> 00:00:18.180
horizontal asymptotes,
if there are any.

00:00:18.180 --> 00:00:20.440
And I said before, all you
have to do is look at the

00:00:20.440 --> 00:00:23.510
highest degree term in the
numerator and the denominator.

00:00:23.510 --> 00:00:26.210
The highest degree term here,
there's only one term.

00:00:26.210 --> 00:00:28.140
It is 2x.

00:00:28.140 --> 00:00:30.450
And the highest degree
term here is x.

00:00:30.450 --> 00:00:32.780
They're both first-degree
terms.

00:00:32.780 --> 00:00:39.630
So you can say that as x
approaches infinity, y is

00:00:39.630 --> 00:00:43.150
going to be-- as x gets
super-large values, these two

00:00:43.150 --> 00:00:44.300
terms are going to dominate.

00:00:44.300 --> 00:00:46.280
This isn't going to
matter so much.

00:00:46.280 --> 00:00:50.710
So then our expression, then y
is going to be approximately

00:00:50.710 --> 00:00:57.570
equal to 2x over x, which
is just equal to 2.

00:00:57.570 --> 00:01:00.140
That actually would
also be true as x

00:01:00.140 --> 00:01:01.990
approaches negative infinity.

00:01:01.990 --> 00:01:06.440
So as x gets really large or
super-negative, this is going

00:01:06.440 --> 00:01:07.340
to approach 2.

00:01:07.340 --> 00:01:09.210
This term won't matter much.

00:01:09.210 --> 00:01:12.330
So let's graph that horizontal
asymptote.

00:01:12.330 --> 00:01:14.980
So it's y is equal to 2.

00:01:14.980 --> 00:01:15.810
Let's graph it.

00:01:15.810 --> 00:01:18.800
So this is our horizontal
asymptote right there.

00:01:24.000 --> 00:01:30.340
y is equal to 2, that right
there-- let me write it down--

00:01:30.340 --> 00:01:36.340
horizontal asymptote.

00:01:36.340 --> 00:01:39.820
That is what our graph
approaches but never quite

00:01:39.820 --> 00:01:43.950
touches as we get to more and
more positive values of x or

00:01:43.950 --> 00:01:45.890
more and more negative
values of x.

00:01:45.890 --> 00:01:49.800
Now, do we have any vertical
asymptotes here?

00:01:49.800 --> 00:01:50.700
Well, sure.

00:01:50.700 --> 00:01:55.410
We have when x is equal to
negative 1, this equation or

00:01:55.410 --> 00:01:57.630
this function is undefined.

00:01:57.630 --> 00:02:07.030
So we say y undefined when
x is equal to negative 1.

00:02:07.030 --> 00:02:09.800
That's definitely true because
when x is equal to negative 1,

00:02:09.800 --> 00:02:11.130
the denominator becomes zero.

00:02:11.130 --> 00:02:12.830
We don't know what 1/0 is.

00:02:12.830 --> 00:02:15.200
It's not defined.

00:02:15.200 --> 00:02:17.670
And this is a vertical asymptote
because the x

00:02:17.670 --> 00:02:19.160
doesn't cancel out.

00:02:19.160 --> 00:02:21.490
The x plus 1-- sorry--
doesn't cancel out

00:02:21.490 --> 00:02:22.610
with something else.

00:02:22.610 --> 00:02:25.050
Let me give you a quick
example right here.

00:02:25.050 --> 00:02:28.170
Let's say I have the equation
y is equal to x plus

00:02:28.170 --> 00:02:31.070
1 over x plus 1.

00:02:31.070 --> 00:02:33.430
In this circumstance, you might
say, hey, when x is

00:02:33.430 --> 00:02:36.730
equal to negative 1, my
graph is undefined.

00:02:36.730 --> 00:02:39.470
And you would be right because
if you put a negative 1 here,

00:02:39.470 --> 00:02:40.770
you get a 0 down here.

00:02:40.770 --> 00:02:42.420
In fact, you'll also
get a 0 on top.

00:02:42.420 --> 00:02:45.330
You'll get a 0 over a 0.

00:02:45.330 --> 00:02:47.370
It's undefined.

00:02:47.370 --> 00:02:52.040
But as you can see, if you
assume that x does not equal

00:02:52.040 --> 00:02:55.630
negative 1, if you assume that
this term and that term are

00:02:55.630 --> 00:02:58.310
not equal to zero, you can
divide the numerator and the

00:02:58.310 --> 00:03:02.340
denominator by x plus 1, or you
could say, well, that over

00:03:02.340 --> 00:03:04.710
that, if it was anything
else over itself, it

00:03:04.710 --> 00:03:05.950
would be equal to 1.

00:03:05.950 --> 00:03:14.740
You would say this would be
equal to 1 when x does not

00:03:14.740 --> 00:03:18.340
equal negative 1 or when these
terms don't equal zero.

00:03:18.340 --> 00:03:21.810
It equals 0/0, which we don't
know what that is, when x is

00:03:21.810 --> 00:03:23.090
equal to negative 1.

00:03:23.090 --> 00:03:25.570
So in this situation,
you would not

00:03:25.570 --> 00:03:28.140
have a vertical asymptote.

00:03:28.140 --> 00:03:36.610
So this graph right here,
no vertical asymptote.

00:03:36.610 --> 00:03:38.350
And actually, you're probably
curious, what does

00:03:38.350 --> 00:03:39.850
this graph look like?

00:03:39.850 --> 00:03:42.470
I'll take a little aside here
to draw it for you.

00:03:42.470 --> 00:03:47.170
This graph right here, if I had
to graph this right there,

00:03:47.170 --> 00:03:49.810
what this would be is this would
be y is equal to 1 for

00:03:49.810 --> 00:03:53.250
all the values except for x
is equal to negative 1.

00:03:53.250 --> 00:03:56.760
So in this situation the graph,
it would be y is equal

00:03:56.760 --> 00:04:04.090
to 1 everywhere, except for
y is equal to negative 1.

00:04:04.090 --> 00:04:06.355
And y is equal to negative
1, it's undefined.

00:04:06.355 --> 00:04:08.710
So we actually have
a hole there.

00:04:08.710 --> 00:04:11.070
We actually draw a little circle
around there, a little

00:04:11.070 --> 00:04:15.760
hollowed-out circle, so that we
don't know what y is when x

00:04:15.760 --> 00:04:16.990
is equal to negative 1.

00:04:16.990 --> 00:04:19.360
So this looks like
that right there.

00:04:19.360 --> 00:04:21.940
It looks like that
horizontal line.

00:04:21.940 --> 00:04:23.580
No vertical asymptote.

00:04:23.580 --> 00:04:28.270
And that's because this term and
that term cancel out when

00:04:28.270 --> 00:04:31.370
they're not equal to zero,
when x is not equal to

00:04:31.370 --> 00:04:32.310
negative 1.

00:04:32.310 --> 00:04:34.640
So when your identifying
vertical asymptotes-- let me

00:04:34.640 --> 00:04:36.160
clear this out a little bit.

00:04:36.160 --> 00:04:39.680
when you're identifying vertical
asymptotes, you want

00:04:39.680 --> 00:04:42.160
to be sure that this expression
right here isn't

00:04:42.160 --> 00:04:44.080
canceling out with something
in the numerator.

00:04:44.080 --> 00:04:45.410
And in this case, it's not.

00:04:45.410 --> 00:04:47.160
In this case, it did,
so you don't

00:04:47.160 --> 00:04:48.400
have a vertical asymptote.

00:04:48.400 --> 00:04:51.410
In this case, you aren't
canceling out, so this will

00:04:51.410 --> 00:04:52.560
define a vertical asymptote.

00:04:52.560 --> 00:04:56.610
x is equal to negative 1 is a
vertical asymptote for this

00:04:56.610 --> 00:04:57.690
graph right here.

00:04:57.690 --> 00:05:01.720
So x is equal to negative 1--
let me draw the vertical

00:05:01.720 --> 00:05:04.510
asymptote-- will
look like that.

00:05:08.250 --> 00:05:10.080
And then to figure out what the
graph is doing, we could

00:05:10.080 --> 00:05:12.630
try out a couple of values.

00:05:12.630 --> 00:05:16.270
So what happens when
x is equal to 0?

00:05:16.270 --> 00:05:21.080
So when x is equal to 0 we have
2 times 0, which is 0

00:05:21.080 --> 00:05:22.800
over 0 plus 1.

00:05:22.800 --> 00:05:25.700
So it's 0/1, which is 0.

00:05:25.700 --> 00:05:31.970
So the point 0, 0
is on our curve.

00:05:31.970 --> 00:05:34.990
What happens when
x is equal to 1?

00:05:34.990 --> 00:05:38.860
We have 2 times 1, which
is 2 over 1 plus 1.

00:05:38.860 --> 00:05:41.040
So it's 2/2.

00:05:41.040 --> 00:05:43.310
So it's 1, 1 is also
on our curve.

00:05:43.310 --> 00:05:45.380
So that's on our curve
right there.

00:05:45.380 --> 00:05:47.630
So we could keep plotting
points, but the curve is going

00:05:47.630 --> 00:05:48.730
to look something like this.

00:05:48.730 --> 00:05:52.020
It looks like it's going
approach negative infinity as

00:05:52.020 --> 00:05:56.600
it approaches the vertical
asymptote from the right.

00:05:56.600 --> 00:05:58.500
So as you go this way it, goes
to negative infinity.

00:05:58.500 --> 00:06:03.010
And then it'll approach our
horizontal asymptote from the

00:06:03.010 --> 00:06:03.665
negative direction.

00:06:03.665 --> 00:06:07.100
So it's going to look
something like that.

00:06:07.100 --> 00:06:12.850
And then, let's see, what
happens when x is equal to--

00:06:12.850 --> 00:06:14.765
let me do this in
a darker color.

00:06:14.765 --> 00:06:16.520
I'll do it in this red color.

00:06:16.520 --> 00:06:20.540
What happens when x is
equal to negative 2?

00:06:20.540 --> 00:06:25.420
We have negative 2 times
2 is negative 4.

00:06:25.420 --> 00:06:28.420
And then we have negative-- so
it's going to be negative 4

00:06:28.420 --> 00:06:32.950
over negative 2 plus 1,
which is negative 1,

00:06:32.950 --> 00:06:34.760
which is just 4.

00:06:34.760 --> 00:06:36.460
So it's just equal
to negative 2, 4.

00:06:36.460 --> 00:06:39.940
So negative 2-- 1, 2, 3, 4.

00:06:39.940 --> 00:06:43.460
Negative 2, 4 is on our line.

00:06:43.460 --> 00:06:46.560
And what about-- well, let's
just do one more point.

00:06:46.560 --> 00:06:49.040
What about negative 3?

00:06:49.040 --> 00:06:51.536
So the point negative 3-- on the
numerator, we're going to

00:06:51.536 --> 00:06:55.010
get 2 times negative 3 is
negative 6 over negative 3

00:06:55.010 --> 00:06:57.570
plus 1, which is negative 2.

00:06:57.570 --> 00:07:01.290
Negative 6 over negative
2 is positive 3.

00:07:01.290 --> 00:07:04.150
So negative 3, 3.

00:07:04.150 --> 00:07:05.890
1, 2, 3.

00:07:05.890 --> 00:07:07.090
1, 2, 3.

00:07:07.090 --> 00:07:08.660
So that's also there.

00:07:08.660 --> 00:07:11.393
So the graph is going to look
something like that.

00:07:14.390 --> 00:07:17.570
So as we approach negative
infinity, we're going to

00:07:17.570 --> 00:07:20.170
approach our horizontal
asymptote from above.

00:07:20.170 --> 00:07:24.115
As we approach negative 1, x is
equal to negative 1, we're

00:07:24.115 --> 00:07:26.270
going to pop up to positive
infinity.

00:07:26.270 --> 00:07:30.340
So let's verify that once again
this is indeed the graph

00:07:30.340 --> 00:07:31.180
of our equation.

00:07:31.180 --> 00:07:35.780
Let's get our graphing
calculator out.

00:07:35.780 --> 00:07:46.660
We're going to define y as 2x
divided by x plus 1 is equal

00:07:46.660 --> 00:07:51.530
to-- delete all of that out--
and then we want to graph it.

00:07:51.530 --> 00:07:52.650
And there we go.

00:07:52.650 --> 00:07:54.520
It looks just like
what we drew.

00:07:54.520 --> 00:07:57.680
And that vertical asymptote, it
connected the dots, but we

00:07:57.680 --> 00:07:59.240
know that it's not
defined there.

00:07:59.240 --> 00:08:01.450
It just tried to connect
the super-positive

00:08:01.450 --> 00:08:02.490
value all the way down.

00:08:02.490 --> 00:08:04.660
Because it's just trying out--
all the graphing calculator's

00:08:04.660 --> 00:08:07.740
doing is actually just making
a very detailed table of

00:08:07.740 --> 00:08:09.460
values and then just connecting
all the dots.

00:08:09.460 --> 00:08:11.740
So it doesn't know that this
is an asymptote, so it

00:08:11.740 --> 00:08:13.340
actually tried to connect
the dots.

00:08:13.340 --> 00:08:16.350
But there should be no
connection right there.

00:08:16.350 --> 00:08:19.190
Hopefully, you found this
example useful.