Graphing a parabola by finding the roots and vertex

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:03.870
We're asked to graph this
function, finding the roots

00:00:03.870 --> 00:00:06.660
and the vertex of
this parabola.

00:00:06.660 --> 00:00:10.150
So to find the roots, when
people talk about roots,

00:00:10.150 --> 00:00:12.870
they're talking about the points
of intersection of this

00:00:12.870 --> 00:00:14.470
function and the x-axis.

00:00:14.470 --> 00:00:17.740
So this is the x-axis here,
and this is the y-axis.

00:00:17.740 --> 00:00:20.110
And you're going to intersect
the x-axis when

00:00:20.110 --> 00:00:21.480
y is equal to 0.

00:00:21.480 --> 00:00:27.730
So to find the roots, we said
y is equal to 0, so we solve

00:00:27.730 --> 00:00:33.230
the equation, x squared plus
4x minus 12 is equal to 0.

00:00:33.230 --> 00:00:35.750
All I did is I said y is equal
to 0, and I swapped the left-

00:00:35.750 --> 00:00:37.270
and right-hand sides.

00:00:37.270 --> 00:00:40.080
Now, to factor this, we have a 1
as a leading coefficient, so

00:00:40.080 --> 00:00:42.760
we don't have to do the
all out grouping.

00:00:42.760 --> 00:00:46.610
We can just think, are there two
numbers whose products are

00:00:46.610 --> 00:00:49.850
negative 12, and if I
add them I get 4?

00:00:49.850 --> 00:00:50.960
So let's think about it.

00:00:50.960 --> 00:00:56.060
It looks like 2, positive 6, and
negative 2 would work, if

00:00:56.060 --> 00:00:58.600
I go through all of negative
12's factors.

00:00:58.600 --> 00:01:04.540
So this can be factored as x
plus 6, times x minus 2.

00:01:04.540 --> 00:01:06.310
6 minus 2 is 4.

00:01:06.310 --> 00:01:09.500
6 times negative 2 is negative
12, so that's going to be

00:01:09.500 --> 00:01:10.520
equal to 0.

00:01:10.520 --> 00:01:12.840
And if we have two numbers,
and if, when you multiply

00:01:12.840 --> 00:01:16.290
them, they equal 0, that means
that either x plus 6 is equal

00:01:16.290 --> 00:01:21.490
to 0, or x minus 2
is equal to 0.

00:01:21.490 --> 00:01:24.600
Subtract 6 from both sides of
this equation, you get x is

00:01:24.600 --> 00:01:26.430
equal to negative 6.

00:01:26.430 --> 00:01:29.790
Add 2 to both sides of this
equation, and you get x is

00:01:29.790 --> 00:01:31.140
equal to 2.

00:01:31.140 --> 00:01:35.010
So y is equal to 0 when x is
equal to negative 6, or x is

00:01:35.010 --> 00:01:35.910
equal to 2.

00:01:35.910 --> 00:01:38.680
So let's graph both
of those points.

00:01:38.680 --> 00:01:41.130
x is equal to negative 6,
y will be equal to 0.

00:01:41.130 --> 00:01:44.600
So x is 1, 2, 3, 4, 5, 6.

00:01:44.600 --> 00:01:47.600
So negative 6 comma 0.

00:01:47.600 --> 00:01:50.940
That will be on the graph of
this function, or the graph of

00:01:50.940 --> 00:01:52.380
this parabola.

00:01:52.380 --> 00:01:56.360
And also the point,
x is equal to 2.

00:01:56.360 --> 00:01:58.680
So 1, 2.

00:01:58.680 --> 00:02:01.840
Just like that. x
is equal to 2.

00:02:01.840 --> 00:02:03.580
So this is the point 2, 0.

00:02:03.580 --> 00:02:06.360
Now to find the x value of the
vertex, there's actually

00:02:06.360 --> 00:02:07.960
several ways of doing it.

00:02:07.960 --> 00:02:09.810
There is a formula that
gives it to you.

00:02:09.810 --> 00:02:12.480
But actually, the easiest way
to figure it out is it's

00:02:12.480 --> 00:02:16.240
actually going to be the
midpoint between the two 0's.

00:02:16.240 --> 00:02:19.280
If you already figured out the
0's, the x value where the

00:02:19.280 --> 00:02:24.840
vertex was going to lie--
because the two 0's are always

00:02:24.840 --> 00:02:28.630
symmetric around the axis of
symmetry, which is the same x

00:02:28.630 --> 00:02:30.340
value as the vertex.

00:02:30.340 --> 00:02:32.760
You can just find the midpoint
of these two points, and that

00:02:32.760 --> 00:02:34.500
will be the x value
of the vertex.

00:02:34.500 --> 00:02:35.900
So let's just do that.

00:02:35.900 --> 00:02:38.280
So if we take the midpoint, we
can literally just average

00:02:38.280 --> 00:02:39.100
these two numbers.

00:02:39.100 --> 00:02:44.265
So negative 6 plus 2, over 2 is
negative 4 over 2, which is

00:02:44.265 --> 00:02:45.720
equal to negative 2.

00:02:45.720 --> 00:02:48.680
So the vertex is x is
equal to negative 2.

00:02:48.680 --> 00:02:51.780
And when x is equal to negative
2, what is y?

00:02:51.780 --> 00:02:56.020
So x is the vertex, and then y
will be equal to negative 2

00:02:56.020 --> 00:02:58.560
squared, is positive 4.

00:02:58.560 --> 00:03:01.570
4 times negative 2
is negative 8.

00:03:01.570 --> 00:03:04.840
And then we have a
minus 12 there.

00:03:04.840 --> 00:03:09.280
So 4 minus 8 gets us to negative
4, minus another 12

00:03:09.280 --> 00:03:11.940
is negative 16.

00:03:11.940 --> 00:03:17.760
So the point negative
2, negative 16 is

00:03:17.760 --> 00:03:20.500
going to be our vertex.

00:03:20.500 --> 00:03:21.330
So let me graph that.

00:03:21.330 --> 00:03:23.060
I think it's going to fall off
of this graph a little bit.

00:03:23.060 --> 00:03:27.470
So negative 2, and then we're
going to go 1, 2, 3, 4, 5, 6,

00:03:27.470 --> 00:03:34.750
7, 8, 9, 10 11, 12,
13, 14, 15, 16.

00:03:34.750 --> 00:03:36.210
So it's going to be
right around here.

00:03:36.210 --> 00:03:37.830
We went off the graph paper.

00:03:37.830 --> 00:03:47.020
But negative 2 all the way
down to negative 16.

00:03:47.020 --> 00:03:48.630
So that right there
is the vertex.

00:03:48.630 --> 00:03:51.580
Negative 2, negative 16.

00:03:51.580 --> 00:03:52.550
All the way down here.

00:03:52.550 --> 00:03:54.540
We can even lower the y-axis.

00:03:54.540 --> 00:03:56.250
And so if I were going to graph
the parabola, it would

00:03:56.250 --> 00:03:57.690
look something like this.

00:03:57.690 --> 00:04:01.900
It would look something
like that.

00:04:01.900 --> 00:04:04.470
Just like that and obviously
it would keep going.

00:04:04.470 --> 00:04:06.980
Now, I figured out the vertex
in this problem just by

00:04:06.980 --> 00:04:09.530
averaging the x values
of the two routes.

00:04:09.530 --> 00:04:12.700
That gave me the x value of the
axis of symmetry, which is

00:04:12.700 --> 00:04:16.459
this right here. x is
equal to negative 2.

00:04:16.459 --> 00:04:19.399
That's the axis around which
the parabola is symmetric.

00:04:19.399 --> 00:04:21.860
The other way to figure out the
x value of the vertex, is

00:04:21.860 --> 00:04:25.555
to use the formula, negative
b over 2a.

00:04:25.555 --> 00:04:29.800
So the x value of the vertex
is equal to negative b--

00:04:29.800 --> 00:04:33.230
negative 4-- over 2 times a.

00:04:33.230 --> 00:04:36.860
a is 1, so over 2, which is also
going to give you x is

00:04:36.860 --> 00:04:39.150
equal to negative 2.

00:04:39.150 --> 00:04:43.380
Now, the last way to solve for
the vertex is to complete the

00:04:43.380 --> 00:04:48.790
square on this parabola
right here.

00:04:48.790 --> 00:04:50.710
So what we can do is
we can rewrite it.

00:04:50.710 --> 00:04:55.910
We can say this is y is equal
to x squared plus 4x.

00:04:55.910 --> 00:04:57.955
I'm going to put the
minus 12 out here.

00:04:57.955 --> 00:05:00.680
And what we want to do is add
and subtract the same number,

00:05:00.680 --> 00:05:02.890
so I can express this
as a perfect square.

00:05:02.890 --> 00:05:07.230
So if I take 1/2 of 4x, or if I
take 1/2 of 4, I get 2 if I

00:05:07.230 --> 00:05:08.440
square that.

00:05:08.440 --> 00:05:11.810
I could add a 4 here, and
then subtract a 4.

00:05:11.810 --> 00:05:18.110
And then what that does is it
turns this into x plus 2

00:05:18.110 --> 00:05:20.940
squared, right? x plus 2
squared is x squared

00:05:20.940 --> 00:05:21.570
plus 4x, plus 4.

00:05:21.570 --> 00:05:23.160
We've seen that multiple
times.

00:05:23.160 --> 00:05:24.870
So you have x plus 2 squared.

00:05:24.870 --> 00:05:28.740
And then you have the minus
4 minus 12, minus 16.

00:05:28.740 --> 00:05:30.540
So this is just another
way of writing

00:05:30.540 --> 00:05:31.910
this exact same function.

00:05:31.910 --> 00:05:37.790
But when you write it this way
you see that the lowest point,

00:05:37.790 --> 00:05:43.300
this part of the function right
here is always positive.

00:05:43.300 --> 00:05:44.650
It's a quantity squared.

00:05:44.650 --> 00:05:47.100
The lowest value that
it can be is 0.

00:05:47.100 --> 00:05:49.040
But then it can only
go up from 0.

00:05:49.040 --> 00:05:52.840
So the lowest point, the lowest
y value we can take on,

00:05:52.840 --> 00:05:58.650
is when this expression
right here is 0.

00:05:58.650 --> 00:06:01.990
And when that happens, the
y value is negative 16.

00:06:01.990 --> 00:06:05.070
Now, what x makes
this value 0?

00:06:05.070 --> 00:06:07.580
Well, x equals negative
2 will make this 0.

00:06:07.580 --> 00:06:12.310
If you wanted to say, x plus 2
is equal to 0, this expression

00:06:12.310 --> 00:06:15.170
right here is going to be 0 when
x plus 2 is equal to 0.

00:06:15.170 --> 00:06:18.460
Subtract 2 from both sides,
x is equal to negative 2.

00:06:18.460 --> 00:06:21.730
So that's another way to find
the vertex, all of them

00:06:21.730 --> 00:06:23.380
equally valid.


Dimensions from volume of box

WEBVTT
Kind: captions
Language: en

00:00:00.490 --> 00:00:04.670
The volume of a box is
405 cube units, or

00:00:04.670 --> 00:00:05.420
I guess cubic units.

00:00:05.420 --> 00:00:06.860
So they just want to
keep it general.

00:00:06.860 --> 00:00:09.850
It could've been in cubic feet,
or cubic meters, or

00:00:09.850 --> 00:00:12.310
cubic centimeters,
or cubic miles.

00:00:12.310 --> 00:00:12.790
Who knows?

00:00:12.790 --> 00:00:14.640
They just want to keep
it as units, keep it

00:00:14.640 --> 00:00:16.460
as general as possible.

00:00:16.460 --> 00:00:20.010
The length is x units, the width
is x plus 4 units, and

00:00:20.010 --> 00:00:21.320
the height is 9 units.

00:00:21.320 --> 00:00:22.840
So let me draw this box here.

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Let me draw a little box here,
so we have a nice little

00:00:31.340 --> 00:00:33.670
visualization.

00:00:33.670 --> 00:00:36.570
So they tell us, that
the length is x.

00:00:36.570 --> 00:00:39.360
Maybe we could call this
the length right there.

00:00:39.360 --> 00:00:45.060
They say the width is x
plus 4, and the height

00:00:45.060 --> 00:00:47.420
is 9 of this box.

00:00:47.420 --> 00:00:50.580
In units, what are the
dimensions of the box?

00:00:50.580 --> 00:00:53.400
Well, they also tell us that
the volume is 405.

00:00:53.400 --> 00:00:57.760
So the volume, 405-- let
me do it this way.

00:00:57.760 --> 00:01:00.340
So if we wanted to calculate the
volume, what would it be?

00:01:00.340 --> 00:01:04.739
Well it would be the width-- it
would be x plus 4 times the

00:01:04.739 --> 00:01:07.510
length -- times x-- times 9.

00:01:10.360 --> 00:01:13.010
That's, literally, the
volume of the box.

00:01:13.010 --> 00:01:16.880
Now they also tell us that the
volume of the box is 405 cubic

00:01:16.880 --> 00:01:20.300
units, is equal to 405.

00:01:20.300 --> 00:01:21.930
So now we just solve for x.

00:01:21.930 --> 00:01:22.630
So what do we get here?

00:01:22.630 --> 00:01:25.610
If we distribute this x
into this x plus 4.

00:01:25.610 --> 00:01:27.190
Actually, if we distribute
a 9x.

00:01:27.190 --> 00:01:28.300
Let me just rewrite it.

00:01:28.300 --> 00:01:31.890
This is the same thing
as 9x times x plus

00:01:31.890 --> 00:01:35.320
4 is equal to 405.

00:01:35.320 --> 00:01:40.970
9x times x is equal
to 9x squared.

00:01:40.970 --> 00:01:47.110
9x times 4 is equal to
36x, is equal to 405.

00:01:47.110 --> 00:01:50.420
Now we want our quadratic
expression to be equal to 0.

00:01:50.420 --> 00:01:58.260
So let's subtract 405 from both
sides of this equation.

00:01:58.260 --> 00:02:02.010
So when you do that, your
right-hand side equals 0, and

00:02:02.010 --> 00:02:11.520
your left-hand side is 9x
squared plus 36x minus 405.

00:02:11.520 --> 00:02:13.860
Now, is there any common
factor to these

00:02:13.860 --> 00:02:15.310
numbers right here?

00:02:15.310 --> 00:02:20.860
Well 405, 4 plus 0 plus 5 is 9,
so that is divisible by 9.

00:02:20.860 --> 00:02:23.060
So all of these are
divisible by 9.

00:02:23.060 --> 00:02:26.990
Let's just figure out what
405 divided by 9 is.

00:02:26.990 --> 00:02:32.700
So 9 goes into 405-- 9
goes into 40 4 times.

00:02:32.700 --> 00:02:36.200
4 times 9 is 36.

00:02:36.200 --> 00:02:38.940
Subtract you get 45.

00:02:38.940 --> 00:02:41.380
9 goes into 45 5 times.

00:02:41.380 --> 00:02:43.880
5 times 9 is 45.

00:02:43.880 --> 00:02:44.960
Subtract, you get 0.

00:02:44.960 --> 00:02:46.880
So it goes 45 times.

00:02:46.880 --> 00:02:52.090
So if we factor out a 9 here,
we get 9 times x squared--

00:02:52.090 --> 00:02:53.410
actually even better, you
don't even have to

00:02:53.410 --> 00:02:54.355
factor out of 9.

00:02:54.355 --> 00:02:56.990
If you think about it, you can
divide both sides of this

00:02:56.990 --> 00:02:58.460
equation by 9.

00:02:58.460 --> 00:03:02.640
So if you can divide all of
the terms by 9, it won't

00:03:02.640 --> 00:03:04.060
change the equation.

00:03:04.060 --> 00:03:06.465
You're doing the same thing to
both sides of equations, which

00:03:06.465 --> 00:03:10.020
we've learned long ago is a
very valid thing to do.

00:03:10.020 --> 00:03:14.150
So here you get x squared-- if
you just had this expression,

00:03:14.150 --> 00:03:16.600
here, and someone told you to
factor it, then you'd have to

00:03:16.600 --> 00:03:17.710
factor out the 9.

00:03:17.710 --> 00:03:20.120
But because this is an equation,
it equals 0, let's

00:03:20.120 --> 00:03:21.200
just divide everything by 9.

00:03:21.200 --> 00:03:22.610
It'll simplify things.

00:03:22.610 --> 00:03:30.730
So you get x squared plus 4x
minus 45 is equal to 0.

00:03:30.730 --> 00:03:33.560
And now we can try to factor
this right here.

00:03:33.560 --> 00:03:36.740
And this fits the pattern, where
we don't have a leading

00:03:36.740 --> 00:03:37.540
1 out here.

00:03:37.540 --> 00:03:39.500
So we don't even have to
do it by grouping.

00:03:39.500 --> 00:03:41.860
You just have to think, what 2
numbers, when I take their

00:03:41.860 --> 00:03:45.870
product I get negative 45, and
when I take their sum, I get

00:03:45.870 --> 00:03:46.960
positive 4.

00:03:46.960 --> 00:03:47.830
They are 4 apart.

00:03:47.830 --> 00:03:49.750
1 has to be positive, 1
has to be negative.

00:03:49.750 --> 00:03:51.650
Their positive versions
have to be 4 apart.

00:03:51.650 --> 00:03:53.430
Because when you take the sum,
you are really taking their

00:03:53.430 --> 00:03:55.950
difference because 1 of
them is negative.

00:03:55.950 --> 00:03:57.330
So let's think about it.

00:03:57.330 --> 00:04:00.800
When you have positive
9 and negative 5, I

00:04:00.800 --> 00:04:01.550
think that'll work.

00:04:01.550 --> 00:04:01.880
Right?

00:04:01.880 --> 00:04:04.520
Positive 9 plus negative
5 is 4.

00:04:04.520 --> 00:04:07.280
And when you take the product,
you get negative 45.

00:04:07.280 --> 00:04:14.240
So you have x plus 9 times
x minus 5 is equal to 0.

00:04:14.240 --> 00:04:15.560
Just factored it out.

00:04:15.560 --> 00:04:17.630
And we've seen this before.

00:04:17.630 --> 00:04:20.000
If you have 2 numbers, when you
take their product that

00:04:20.000 --> 00:04:23.000
equals 0, that means 1 of these
numbers at least has to

00:04:23.000 --> 00:04:24.960
be equal 0.

00:04:24.960 --> 00:04:29.180
So this means that x plus
9 is equal to 0.

00:04:29.180 --> 00:04:31.590
Scroll down a little bit.

00:04:31.590 --> 00:04:36.900
x plus 9 is equal 0, or
x minus 5 is equal 0.

00:04:36.900 --> 00:04:41.105
So if we subtract 9 from this
equation right there, you get

00:04:41.105 --> 00:04:45.500
x is equal to negative 9, or if
you add 5 to both sides of

00:04:45.500 --> 00:04:46.900
this equation, here.

00:04:46.900 --> 00:04:49.820
You get x is equal to 5.

00:04:49.820 --> 00:04:53.820
So these are both possible
values of x right here.

00:04:53.820 --> 00:04:57.950
So the box, if you take x is
equal to negative 9, well, x

00:04:57.950 --> 00:04:59.530
equal to negative
9 won't work.

00:04:59.530 --> 00:05:02.650
Because if you but negative 9
here, you're going to have a

00:05:02.650 --> 00:05:06.160
box that has a width of negative
5, a length of

00:05:06.160 --> 00:05:08.810
negative 9, and a height of 9.

00:05:08.810 --> 00:05:11.930
And if we're talking about our
reality, we don't have

00:05:11.930 --> 00:05:13.200
negative distances like this.

00:05:13.200 --> 00:05:14.850
That can't be the length
or the width.

00:05:14.850 --> 00:05:19.190
So x equals negative 9 isn't
appropriate for this problem

00:05:19.190 --> 00:05:21.340
Because in this problem we
need to have positive

00:05:21.340 --> 00:05:22.340
dimensions.

00:05:22.340 --> 00:05:24.525
So let's see what happens
with x equals 5.

00:05:24.525 --> 00:05:29.460
If x equals 5, x plus 4 is 9,
and this dimension right here

00:05:29.460 --> 00:05:31.290
is going to be 5.

00:05:31.290 --> 00:05:33.530
And that seems pretty reasonable
for our reality.

00:05:33.530 --> 00:05:36.620
And let's verify that
this does end up

00:05:36.620 --> 00:05:39.070
with a volume of 405.

00:05:39.070 --> 00:05:44.630
9 times 5 is 45 times
9 is indeed 405.

00:05:44.630 --> 00:05:51.810
We just figured that out over
here, that 45 times 9 is 405.

00:05:51.810 --> 00:05:53.350
So we're done.


Solving a quadratic equation by factoring

WEBVTT
Kind: captions
Language: en

00:00:00.470 --> 00:00:02.460
We're asked to solve for s.

00:00:02.460 --> 00:00:08.500
And we have s squared minus
2s minus 35 is equal to 0.

00:00:08.500 --> 00:00:11.570
Now if this is the first time
that you've seen this type of

00:00:11.570 --> 00:00:14.140
what's essentially a quadratic
equation, you might be tempted

00:00:14.140 --> 00:00:17.790
to try to solve for s using
traditional algebraic means,

00:00:17.790 --> 00:00:19.760
but the best way to solve this,
especially when it's

00:00:19.760 --> 00:00:23.690
explicitly equal to 0, is to
factor the left-hand side, and

00:00:23.690 --> 00:00:27.380
then think about the fact that
those binomials that you

00:00:27.380 --> 00:00:29.990
factor into, that they have
to be equal to 0.

00:00:29.990 --> 00:00:30.890
So let's just do that.

00:00:30.890 --> 00:00:32.000
So how can we factor this?

00:00:32.000 --> 00:00:33.390
We've seen it in several ways.

00:00:33.390 --> 00:00:36.040
I'll show you the standard
we've been doing it, by

00:00:36.040 --> 00:00:38.260
grouping, and then there's a
little bit of a shortcut when

00:00:38.260 --> 00:00:40.670
you have a 1 as a coefficient
over here.

00:00:40.670 --> 00:00:44.580
So when you do something by
grouping, when you factor by

00:00:44.580 --> 00:00:47.860
grouping, you think about two
numbers whose sum is going to

00:00:47.860 --> 00:00:49.710
be equal to negative 2.

00:00:49.710 --> 00:00:52.436
So you think about two numbers
whose sum, a plus b, is equal

00:00:52.436 --> 00:00:54.890
to negative 2 and whose product
is going to be equal

00:00:54.890 --> 00:00:56.540
to negative 35.

00:00:56.540 --> 00:00:59.530
a times b is equal
to negative 35.

00:00:59.530 --> 00:01:02.900
So if the product is a negative
number, one has to be

00:01:02.900 --> 00:01:05.910
positive, one has
to be negative.

00:01:05.910 --> 00:01:08.490
And so if you think about it,
ones that are about two apart,

00:01:08.490 --> 00:01:11.420
you have 5 and negative 7,
I think that'll work.

00:01:11.420 --> 00:01:16.410
5 plus negative 7 is equal
to negative 2.

00:01:16.410 --> 00:01:20.770
So to factor by grouping, you
split this middle term.

00:01:20.770 --> 00:01:23.540
We can split this into a--
let me write it this way.

00:01:23.540 --> 00:01:26.450
We have s squared, and then this
middle term right here,

00:01:26.450 --> 00:01:27.920
I'll do it in pink.

00:01:27.920 --> 00:01:34.440
This middle term right there I
can write it as plus 5s minus

00:01:34.440 --> 00:01:39.140
7s and then we have
the minus 35.

00:01:39.140 --> 00:01:42.560
And then, of course, all
of that is equal to 0.

00:01:42.560 --> 00:01:45.430
Now, we call it factoring by
grouping because we group it.

00:01:45.430 --> 00:01:49.120
So we can group these first two
terms. And these first two

00:01:49.120 --> 00:01:50.870
terms, they have a common
factor of s.

00:01:50.870 --> 00:01:51.900
So let's factor that out.

00:01:51.900 --> 00:01:56.090
You have s times s plus 5.

00:01:56.090 --> 00:01:59.020
That's the same thing as
s squared plus 5s.

00:01:59.020 --> 00:02:02.850
Now, in these second two terms
right here, you have a common

00:02:02.850 --> 00:02:06.510
factor of negative 7, so
let's factor that out.

00:02:06.510 --> 00:02:12.290
So you have negative
7 times s plus 5.

00:02:12.290 --> 00:02:15.670
And, of course, all of
that is equal to 0.

00:02:15.670 --> 00:02:19.580
Now, we have two terms here,
where both of them have s plus

00:02:19.580 --> 00:02:20.830
5 as a factor.

00:02:23.590 --> 00:02:25.240
So we can factor that out.

00:02:25.240 --> 00:02:26.320
So let's do that.

00:02:26.320 --> 00:02:37.930
So you have s plus 5 times
this s right here, right?

00:02:37.930 --> 00:02:39.930
S plus 5 times s will
give you this term.

00:02:39.930 --> 00:02:43.640
And then you have minus
that 7 right there.

00:02:43.640 --> 00:02:46.580
I undistributed the s plus 5.

00:02:46.580 --> 00:02:51.940
And then this is going
to be equal to 0.

00:02:51.940 --> 00:02:53.890
Now that we've factored it, we
just have to think a little

00:02:53.890 --> 00:02:55.390
bit about what happens
when you take the

00:02:55.390 --> 00:02:56.520
product of two numbers?

00:02:56.520 --> 00:02:58.200
I mean, s plus 5 is a number.

00:02:58.200 --> 00:03:00.180
s minus 7 is another number.

00:03:00.180 --> 00:03:01.930
And we're saying that the
product of those two numbers

00:03:01.930 --> 00:03:02.940
is equal to zero.

00:03:02.940 --> 00:03:05.570
If ever told you that I had
two numbers, if I told you

00:03:05.570 --> 00:03:10.110
that I had the numbers a times
b and that they equal to 0,

00:03:10.110 --> 00:03:13.770
what do we know about either
a or b or both of them?

00:03:13.770 --> 00:03:17.280
Well, at least one of them has
to be equal to 0, or both of

00:03:17.280 --> 00:03:19.150
them have to be equal to 0.

00:03:19.150 --> 00:03:22.090
So, the fact that this number
times that number is equal to

00:03:22.090 --> 00:03:30.820
zero tells us that either s plus
5 is equal to 0 or-- and

00:03:30.820 --> 00:03:36.120
maybe both of them-- s minus
7 is equal to 0.

00:03:36.120 --> 00:03:37.370
I'll do that in just green.

00:03:41.160 --> 00:03:42.925
And so you have these two
equations, and actually, we

00:03:42.925 --> 00:03:44.020
could say and/or.

00:03:44.020 --> 00:03:47.000
It could be or/and, either way,
and both of them could be

00:03:47.000 --> 00:03:48.470
equal to 0.

00:03:48.470 --> 00:03:52.390
So let's see how we can
solve for this.

00:03:52.390 --> 00:03:55.020
Well, we can just subtract
5 from both sides of this

00:03:55.020 --> 00:03:57.280
equation right there.

00:03:57.280 --> 00:04:00.150
And so you get, on the left-hand
side, you have s is

00:04:00.150 --> 00:04:01.620
equal to negative 5.

00:04:01.620 --> 00:04:07.840
That is one solution to the
equation, or you can add 7 to

00:04:07.840 --> 00:04:13.630
both sides of that equation, and
you get s is equal to 7.

00:04:13.630 --> 00:04:19.600
So if s is equal to negative 5,
or s is equal to 7, then we

00:04:19.600 --> 00:04:21.019
have satisfied this equation.

00:04:21.019 --> 00:04:22.250
We can even verify it.

00:04:22.250 --> 00:04:24.890
If you make s equal to negative
5, you have positive

00:04:24.890 --> 00:04:28.810
25 plus 10, which is minus 35.

00:04:28.810 --> 00:04:30.410
That does equal zero.

00:04:30.410 --> 00:04:36.890
If you have 7, 49 minus 14
minus 35 does equal zero.

00:04:36.890 --> 00:04:38.280
So we've solved for s.

00:04:38.280 --> 00:04:40.490
Now, I mentioned there's
an easier way to do it.

00:04:40.490 --> 00:04:45.270
And when you have something like
this, where you have 1 as

00:04:45.270 --> 00:04:47.980
the leading coefficient,
you don't have to do

00:04:47.980 --> 00:04:49.120
this two-step factoring.

00:04:49.120 --> 00:04:50.130
Let me just show
you an example.

00:04:50.130 --> 00:04:57.460
If I just have x plus a
times x plus b, what

00:04:57.460 --> 00:04:58.570
is that equal to?

00:04:58.570 --> 00:05:03.010
x times x is x squared,
x times b is bx.

00:05:03.010 --> 00:05:05.470
a times x is plus ax.

00:05:05.470 --> 00:05:08.590
a times b is ab.

00:05:08.590 --> 00:05:13.050
So you get x squared plus--
these two can be added-- plus

00:05:13.050 --> 00:05:17.350
a plus bx plus ab.

00:05:17.350 --> 00:05:19.260
And that's the pattern that
we have right here.

00:05:19.260 --> 00:05:22.300
We have 1 as a leading
coefficient here, we have 1 as

00:05:22.300 --> 00:05:23.770
a leading coefficient here.

00:05:23.770 --> 00:05:27.080
So once we have our two
numbers that add up to

00:05:27.080 --> 00:05:33.090
negative 2, that's our a plus
b, and we have our product

00:05:33.090 --> 00:05:38.050
that gets to negative 35, then
we can straight just factor it

00:05:38.050 --> 00:05:39.990
into the product of
those two things.

00:05:39.990 --> 00:05:44.410
So it will be-- or the product
of the binomials, where those

00:05:44.410 --> 00:05:45.610
will be the a's and the b's.

00:05:45.610 --> 00:05:47.110
So we figured it out.

00:05:47.110 --> 00:05:48.460
It's 5 and negative 7.

00:05:48.460 --> 00:05:51.050
5 plus negative 7
is negative 2.

00:05:51.050 --> 00:05:54.130
5 times negative 7
is negative 35.

00:05:54.130 --> 00:05:56.660
So we could have just straight
factored at this point.

00:05:56.660 --> 00:06:00.030
2, well, actually this
was the case of s.

00:06:00.030 --> 00:06:03.920
So we could have factored it
straight to the case of s plus

00:06:03.920 --> 00:06:07.860
5 times s minus 7.

00:06:07.860 --> 00:06:09.920
We could have done that straight
away and would've

00:06:09.920 --> 00:06:10.850
gotten to that right there.

00:06:10.850 --> 00:06:13.320
And, of course, that whole
thing was equal to zero.

00:06:13.320 --> 00:06:14.960
So that would've been a little
bit of a shortcut, but

00:06:14.960 --> 00:06:19.520
factoring by grouping is a
completely appropriate way to

00:06:19.520 --> 00:06:21.090
do it as well.


Recognizing a perfect square quadratic

WEBVTT
Kind: captions
Language: en

00:00:00.440 --> 00:00:02.320
We're asked to solve for x.

00:00:02.320 --> 00:00:07.060
And we have 9x squared minus
42x, plus 29 is equal to

00:00:07.060 --> 00:00:08.420
negative 20.

00:00:08.420 --> 00:00:11.560
Now, you might be tempted
immediately to try to factor

00:00:11.560 --> 00:00:14.080
the left-hand side of this
equation to get the product of

00:00:14.080 --> 00:00:16.870
two binomials, and that they
equal negative 20.

00:00:16.870 --> 00:00:18.180
But if you think about
it, that's not

00:00:18.180 --> 00:00:19.370
going to help you much.

00:00:19.370 --> 00:00:22.060
That product of two binomials
is only useful if they're

00:00:22.060 --> 00:00:22.810
equal to 0.

00:00:22.810 --> 00:00:25.750
Then you can say, well, one
or both of them need

00:00:25.750 --> 00:00:26.850
to be equal to 0.

00:00:26.850 --> 00:00:29.240
But you can't make any statement
like that if they

00:00:29.240 --> 00:00:30.960
both equal negative 20.

00:00:30.960 --> 00:00:33.790
So the first thing we want to do
here is not to factor this

00:00:33.790 --> 00:00:36.890
left-hand side, but to make sure
that we have a quadratic

00:00:36.890 --> 00:00:38.840
equation equaling 0.

00:00:38.840 --> 00:00:41.680
We want a 0 on one side
of this equation.

00:00:41.680 --> 00:00:43.830
And the best way I can think of
doing that is just to add

00:00:43.830 --> 00:00:46.210
20 to both sides of
this equation.

00:00:46.210 --> 00:00:51.000
So if we add 20 to both sides of
this equation, then we are

00:00:51.000 --> 00:01:00.320
left with, on the left-hand
side, 9x squared minus 42x,

00:01:00.320 --> 00:01:05.209
plus 49 is equal to 0.

00:01:05.209 --> 00:01:09.390
Now we have a quadratic
expression equaling 0.

00:01:09.390 --> 00:01:12.660
Now we have a nice quadratic
equation, and we can factor

00:01:12.660 --> 00:01:14.820
this thing on the
left-hand side.

00:01:14.820 --> 00:01:18.490
Now we could factor it by
grouping, but something might

00:01:18.490 --> 00:01:19.790
immediately pop out at you.

00:01:19.790 --> 00:01:22.170
And when you see numbers
like 9 and 49,

00:01:22.170 --> 00:01:23.510
those are perfect squares.

00:01:23.510 --> 00:01:25.440
So maybe this whole expression
is a perfect

00:01:25.440 --> 00:01:27.080
square of some binomial.

00:01:27.080 --> 00:01:28.760
So let's think about
it a little bit.

00:01:28.760 --> 00:01:33.055
9 squared is 3x squared.

00:01:37.160 --> 00:01:45.370
And 49 is 7 squared, or
negative 7 squared.

00:01:45.370 --> 00:01:51.810
And let's see, negative
42-- 3 times 7 is 21.

00:01:51.810 --> 00:01:53.690
21 times 2 is positive 42.

00:01:53.690 --> 00:01:58.020
But if we make this a negative
7, because negative 7 would

00:01:58.020 --> 00:02:01.820
also work, negative 7
squared is also 49.

00:02:01.820 --> 00:02:06.950
Then 3 times negative 7,
times 2 is negative 42.

00:02:06.950 --> 00:02:07.830
Let me write that down.

00:02:07.830 --> 00:02:14.750
42 is 2 times 3, times
negative 7.

00:02:14.750 --> 00:02:18.150
So this fits the pattern
of a perfect square.

00:02:18.150 --> 00:02:21.210
So the left-hand side of this
equation, right here, we can

00:02:21.210 --> 00:02:30.240
write it as 3x minus 7-- we
can write it as 3x minus 7

00:02:30.240 --> 00:02:34.350
times 3x minus 7, or just
3x minus 7 squared.

00:02:34.350 --> 00:02:38.510
And that is going to
be equal to 0.

00:02:38.510 --> 00:02:41.460
Now, we essentially only have
one expression equaling 0,

00:02:41.460 --> 00:02:42.400
times itself.

00:02:42.400 --> 00:02:45.080
So the only way that's going to
be equal to 0, is if that

00:02:45.080 --> 00:02:46.700
expression is equal to 0.

00:02:46.700 --> 00:02:50.400
3x minus 7 needs to
be equal to 0.

00:02:50.400 --> 00:02:52.700
And now we can just
solve for x.

00:02:52.700 --> 00:02:59.130
Let's add 7 to both sides of
this equation, and we get 3x,

00:02:59.130 --> 00:03:02.200
these cancel out,
is equal to 7.

00:03:02.200 --> 00:03:09.410
Divide both sides by 3, and we
get x is equal to 7 thirds.

00:03:09.410 --> 00:03:12.610
x is equal to-- let me just
write it here-- x is

00:03:12.610 --> 00:03:16.920
equal to 7 over 3.


Example 6: Factoring a difference of squares with two variables

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:05.960
We need to factor 49x squared
minus 49y squared.

00:00:05.960 --> 00:00:09.520
Now here there's a pattern
that you might already be

00:00:09.520 --> 00:00:10.190
familiar with.

00:00:10.190 --> 00:00:12.910
But just to make sure you are,
let's think about what happens

00:00:12.910 --> 00:00:16.870
if we multiply a plus b-- where
these are just two terms

00:00:16.870 --> 00:00:21.000
in a binomial-- times
a minus b.

00:00:21.000 --> 00:00:24.210
If you multiply this out, you
have a times a, which is a

00:00:24.210 --> 00:00:30.660
squared, plus a times negative
b, which is negative ab--

00:00:30.660 --> 00:00:34.280
that's a times negative b-- plus
b times a, which is the

00:00:34.280 --> 00:00:36.150
same thing as ab.

00:00:36.150 --> 00:00:38.780
And then you have b times
negative b, which

00:00:38.780 --> 00:00:40.830
is negative b squared.

00:00:40.830 --> 00:00:43.230
So when you do that, you have
a negative ab and a positive

00:00:43.230 --> 00:00:44.390
ab, they cancel out.

00:00:44.390 --> 00:00:45.840
And you're just going
to be left with an a

00:00:45.840 --> 00:00:48.720
squared minus a b squared.

00:00:48.720 --> 00:00:51.420
Now, this thing that we have
here is exactly that pattern.

00:00:51.420 --> 00:00:53.870
49x squared is a
perfect square.

00:00:53.870 --> 00:00:56.030
49y squared is a
perfect square.

00:00:56.030 --> 00:00:57.740
We can rewrite it like that.

00:00:57.740 --> 00:01:05.900
We could rewrite this over here
as 7x squared minus-- and

00:01:05.900 --> 00:01:10.560
I'll do it in blue--
minus 7y squared.

00:01:10.560 --> 00:01:11.620
And so you see it's a pattern.

00:01:11.620 --> 00:01:14.200
It's a squared minus
b squared.

00:01:14.200 --> 00:01:16.600
So if you wanted to factor
this-- if you would just use

00:01:16.600 --> 00:01:20.400
this pattern that we just
derived-- you would say that

00:01:20.400 --> 00:01:32.090
this is the same thing as a,
7x plus b plus 7y times 7x

00:01:32.090 --> 00:01:35.880
minus b, minus 7y.

00:01:35.880 --> 00:01:37.290
And you'd be done.

00:01:37.290 --> 00:01:39.980
Now there's one alternate way
that you could factor this and

00:01:39.980 --> 00:01:41.380
it'd be completely legitimate.

00:01:41.380 --> 00:01:43.530
You could start from
the beginning and

00:01:43.530 --> 00:01:44.350
say, you know what?

00:01:44.350 --> 00:01:47.470
49 is a common factor here, so
let me just factor that out.

00:01:47.470 --> 00:01:51.380
So you could say it's equivalent
to 49 times x

00:01:51.380 --> 00:01:54.615
squared minus y squared.

00:01:54.615 --> 00:01:58.820
And you say, oh, this fits the
pattern of-- this is a squared

00:01:58.820 --> 00:01:59.980
minus b squared.

00:01:59.980 --> 00:02:03.190
So this will be x plus
y times x minus y.

00:02:03.190 --> 00:02:06.880
So the whole thing would
be 49 times x plus y

00:02:06.880 --> 00:02:08.979
times x minus y.

00:02:08.979 --> 00:02:12.860
And to see that this, right
here, is the exact same thing

00:02:12.860 --> 00:02:17.140
as this right over here, you
could just factor 7 out of

00:02:17.140 --> 00:02:17.970
both of these.

00:02:17.970 --> 00:02:19.840
You'd factor out a 7 out
of that term, a factor

00:02:19.840 --> 00:02:20.530
7 out of that term.

00:02:20.530 --> 00:02:23.380
And when you multiply them,
you'd get the 49.

00:02:23.380 --> 00:02:27.650
So these are-- this or this--
these are both ways to factor

00:02:27.650 --> 00:02:29.110
this expression.


Example 3: Factoring quadratics as a perfect square of a sum: (a+b)^2

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:07.310
We need to factor negative 4t
squared minus 12t minus 9.

00:00:07.310 --> 00:00:09.290
And a good place to start is
to say, well, are there any

00:00:09.290 --> 00:00:11.460
common factors for all
of these terms?

00:00:11.460 --> 00:00:13.270
When you look at them, well
these first two are divisible

00:00:13.270 --> 00:00:16.980
by 4, these last 2 are divisible
by 3, but not all of

00:00:16.980 --> 00:00:18.920
them are divisible
any one number.

00:00:18.920 --> 00:00:21.270
Will, but you could factor out
a negative 1, but even if you

00:00:21.270 --> 00:00:23.550
factor out a negative 1-- so you
say this is the same thing

00:00:23.550 --> 00:00:30.990
as negative 1, times positive
4t squared plus 12t plus 9--

00:00:30.990 --> 00:00:34.050
you still end up with a non-one
coefficient out here

00:00:34.050 --> 00:00:37.120
and on the second degree term,
on the t squared term.

00:00:37.120 --> 00:00:39.200
So you might want
to immediately

00:00:39.200 --> 00:00:40.110
start grouping this.

00:00:40.110 --> 00:00:43.830
And if you did factor it by
grouping, it would work, you

00:00:43.830 --> 00:00:45.500
would get the right answer.

00:00:45.500 --> 00:00:51.550
But there is something about
this equation that might pop

00:00:51.550 --> 00:00:55.080
out at you that might make it a
little bit simpler to solve.

00:00:55.080 --> 00:01:00.540
And to understand that, let's
take a little bit of a break

00:01:00.540 --> 00:01:02.680
here on the right hand side,
and just think about what

00:01:02.680 --> 00:01:07.690
happens if you take a plus b
times a plus b, if you just

00:01:07.690 --> 00:01:10.060
have a binomial squared.

00:01:10.060 --> 00:01:13.060
Well you have a times a,
which is a squared.

00:01:13.060 --> 00:01:17.070
Then you have a times that
b, which is plus ab.

00:01:17.070 --> 00:01:21.310
Then you have b times a, which
is the same thing is ab.

00:01:21.310 --> 00:01:24.815
And then you have b times b,
or you have b squared.

00:01:24.815 --> 00:01:28.360
And so if you add these middle
two terms, right here, you're

00:01:28.360 --> 00:01:33.850
left with a squared plus
2ab plus b squared.

00:01:33.850 --> 00:01:36.150
This is the square
of a binomial.

00:01:36.150 --> 00:01:40.990
Now, does this right here, does
4t squared plus 12t plus

00:01:40.990 --> 00:01:43.400
9 fit this pattern?

00:01:43.400 --> 00:01:46.020
Well the 4t squared
is a squared.

00:01:46.020 --> 00:01:49.080
So this right here
is a squared.

00:01:49.080 --> 00:01:51.180
If that is a squared right
there, then what

00:01:51.180 --> 00:01:52.345
does a have to be?

00:01:52.345 --> 00:01:55.450
If this is a squared, then a
would be equal to the square

00:01:55.450 --> 00:01:56.150
root of this.

00:01:56.150 --> 00:01:57.940
It would be 2t.

00:01:57.940 --> 00:02:00.100
And if this is b squared,
let me do that

00:02:00.100 --> 00:02:01.940
in a different color.

00:02:01.940 --> 00:02:07.670
If this right here is b
squared, if the 9 is b

00:02:07.670 --> 00:02:10.110
squared, right there,
then that means that

00:02:10.110 --> 00:02:12.170
b is equal to 3.

00:02:12.170 --> 00:02:15.400
It's equal to the positive
square root of the 9.

00:02:15.400 --> 00:02:18.430
Now, this number, right here--
and actually it doesn't have

00:02:18.430 --> 00:02:20.150
to just be equal to 3,
it might have been

00:02:20.150 --> 00:02:21.430
negative 3 as well.

00:02:21.430 --> 00:02:23.240
It could be plus or minus 3.

00:02:23.240 --> 00:02:27.000
But this number here,
is it 2 times ab?

00:02:27.000 --> 00:02:27.330
Right?

00:02:27.330 --> 00:02:29.650
That's the middle term
that we care about.

00:02:29.650 --> 00:02:32.010
Is it 2 times ab?

00:02:32.010 --> 00:02:36.890
Well if we multiply 2t
times 3, we get 6t.

00:02:36.890 --> 00:02:40.260
And then if we multiply that
times 2, you get 12t.

00:02:40.260 --> 00:02:48.600
This right here, 12t, is equal
to 2 times 2t times 3.

00:02:48.600 --> 00:02:50.960
It is 2 times ab.

00:02:50.960 --> 00:02:53.060
And if this was a negative 3,
we would look to see if this

00:02:53.060 --> 00:02:56.010
was a negative 12, but this
does work for positive 3.

00:02:56.010 --> 00:02:59.030
So this it does fit the pattern
of a perfect square.

00:02:59.030 --> 00:03:05.060
This is a square
of a binomial.

00:03:05.060 --> 00:03:07.590
So if you wanted to factor
this-- the stuff on the

00:03:07.590 --> 00:03:09.940
inside, you still have that
negative 1 out there, the 4t

00:03:09.940 --> 00:03:13.490
squared plus 12t plus 9-- you
could immediately say, well

00:03:13.490 --> 00:03:18.880
that's going to be a plus
b times a plus b.

00:03:18.880 --> 00:03:29.570
Or 2t plus 3 times 2t plus 3, or
you could just say, it's 2t

00:03:29.570 --> 00:03:31.330
plus 3 squared.

00:03:31.330 --> 00:03:32.350
It fits this pattern.

00:03:32.350 --> 00:03:35.270
And, of course, you can't
forget about this

00:03:35.270 --> 00:03:36.450
negative 1 out here.

00:03:36.450 --> 00:03:38.750
You could have also solved it by
grouping, but this might be

00:03:38.750 --> 00:03:40.460
a quicker thing to recognize.

00:03:40.460 --> 00:03:41.820
This is a number squared.

00:03:41.820 --> 00:03:43.260
That's another number squared.

00:03:43.260 --> 00:03:45.600
If you take each of those
numbers that you're squaring,

00:03:45.600 --> 00:03:47.910
take their product and multiply
it by 2, you have

00:03:47.910 --> 00:03:48.650
that right there.

00:03:48.650 --> 00:03:51.330
So this is a perfect square.


Example 3: Factoring quadratics by taking a common factor and grouping

WEBVTT
Kind: captions
Language: en

00:00:00.460 --> 00:00:06.110
We're asked to factor 35k
squared plus 100k, minus 15.

00:00:06.110 --> 00:00:10.930
And because we have a non-1
coefficient out here, the best

00:00:10.930 --> 00:00:14.600
thing to do is probably to
factor this by grouping.

00:00:14.600 --> 00:00:16.870
But before we even do that,
let's see if there's a common

00:00:16.870 --> 00:00:19.250
factor across all of these
terms, and maybe we can get a

00:00:19.250 --> 00:00:20.280
1 coefficient, out there.

00:00:20.280 --> 00:00:22.140
If we can't get a 1 coefficient,
we'll at least

00:00:22.140 --> 00:00:23.660
have a lower coefficient here.

00:00:23.660 --> 00:00:25.950
And if we look at all of these
numbers, they all look

00:00:25.950 --> 00:00:27.140
divisible by 5.

00:00:27.140 --> 00:00:30.680
In fact their greatest
common factor is 5.

00:00:30.680 --> 00:00:32.270
So let's at least
factor out a 5.

00:00:32.270 --> 00:00:36.560
So this is equal to 5
times-- 35k squared

00:00:36.560 --> 00:00:41.430
divided by 5 is 7k squared.

00:00:41.430 --> 00:00:46.090
100k divided by 5 is 20k.

00:00:46.090 --> 00:00:50.610
And then negative 15 divided
by 5 is negative 3.

00:00:50.610 --> 00:00:53.710
So we were able to factor out a
5, but we still don't have a

00:00:53.710 --> 00:00:56.350
1 coefficient here, so we're
still going to have to factor

00:00:56.350 --> 00:00:57.300
by grouping.

00:00:57.300 --> 00:01:00.500
But at least the numbers here
are smaller so it'll be easier

00:01:00.500 --> 00:01:03.800
to think about it in terms of
finding numbers whose product

00:01:03.800 --> 00:01:06.200
is equal to 7 times negative
3, and whose

00:01:06.200 --> 00:01:08.140
sum is equal to 20.

00:01:08.140 --> 00:01:09.120
So let's think about that.

00:01:09.120 --> 00:01:12.970
Let's figure out two numbers
that if I were to add them, or

00:01:12.970 --> 00:01:16.680
even better if I were to take
their product, I get 7 times

00:01:16.680 --> 00:01:23.530
negative 3, which is equal
to negative 21.

00:01:23.530 --> 00:01:27.590
And if I were to take their
sum, if I add those two

00:01:27.590 --> 00:01:34.190
numbers, it needs to
be equal to 20.

00:01:34.190 --> 00:01:37.360
Now, once again, because their
product is a negative number,

00:01:37.360 --> 00:01:39.820
that means they have to be of
different signs, so when you

00:01:39.820 --> 00:01:42.440
add numbers of different signs,
you could view it as

00:01:42.440 --> 00:01:44.870
you're taking the difference
of the positive versions.

00:01:44.870 --> 00:01:46.930
So the difference between the
positive versions of the

00:01:46.930 --> 00:01:48.400
number has to be 20.

00:01:48.400 --> 00:01:50.550
So the number that immediately
jumps out is we're probably

00:01:50.550 --> 00:01:53.640
going to be dealing with 20
and 21, and 1 will be the

00:01:53.640 --> 00:01:56.230
negative, because we want
to get to a positive 20.

00:01:56.230 --> 00:01:57.710
So let's think about it.

00:01:57.710 --> 00:02:01.980
So if we think of 20 and
negative 1, their product is

00:02:01.980 --> 00:02:03.530
negative 21.

00:02:03.530 --> 00:02:04.100
Sorry.

00:02:04.100 --> 00:02:10.330
If we take 21 and negative 1,
their product is negative 21.

00:02:10.330 --> 00:02:13.060
21 times negative 1
is negative 21.

00:02:13.060 --> 00:02:16.200
and if you take their sum,
21 plus negative 1,

00:02:16.200 --> 00:02:17.820
that is equal to 20.

00:02:17.820 --> 00:02:21.940
So these two numbers right
there fit the bill.

00:02:21.940 --> 00:02:26.700
Now, let's break up this 20k
right here into a 21k and a

00:02:26.700 --> 00:02:29.430
negative 1k.

00:02:29.430 --> 00:02:29.770
So let's do that.

00:02:29.770 --> 00:02:31.560
So let's rewrite the
whole thing.

00:02:31.560 --> 00:02:36.720
We have 5 times 7k squared, and
I'm going to break this

00:02:36.720 --> 00:02:42.296
20k into a-- let me do it in
this color right here-- I'm

00:02:42.296 --> 00:02:48.940
going to break that 20k into
a plus 21k, minus k.

00:02:48.940 --> 00:02:51.270
Or you could say minus
1k if you want.

00:02:51.270 --> 00:02:53.370
I'm using those two factors
to break it up.

00:02:53.370 --> 00:02:57.760
And then we finally have the
minus 3 right there.

00:02:57.760 --> 00:02:59.940
Now, the whole point of doing
that is so that we can now

00:02:59.940 --> 00:03:03.420
factor each of the two groups.

00:03:03.420 --> 00:03:06.400
This could be our first
group right here.

00:03:06.400 --> 00:03:09.600
And so what can we factor out
of that group right there?

00:03:09.600 --> 00:03:12.530
Well, both of these are
divisible by 7k, so we can

00:03:12.530 --> 00:03:19.210
write this as 7k times-- 7k
squared divided by 7k, you're

00:03:19.210 --> 00:03:20.990
just going to have
a k left over.

00:03:20.990 --> 00:03:27.570
And then plus 21k divided by
7k is just going to be a 3.

00:03:27.570 --> 00:03:29.330
So that factors into that.

00:03:29.330 --> 00:03:33.290
And then we can look at
this group right here.

00:03:33.290 --> 00:03:34.310
They have a common factor.

00:03:34.310 --> 00:03:36.930
Well, we can factor out a
negative 1 if we like, so this

00:03:36.930 --> 00:03:42.160
is equal to negative 1 times--
k divided by negative 1 is k.

00:03:42.160 --> 00:03:46.050
Negative 3 divided by negative
1 is positive 3.

00:03:46.050 --> 00:03:48.215
And, of course, we have this
5 sitting out there.

00:03:51.850 --> 00:03:55.080
Now, ignoring that 5 for a
second, you see that both of

00:03:55.080 --> 00:04:02.380
these inside terms have
k plus 3 as a factor.

00:04:02.380 --> 00:04:04.360
So we can factor that out.

00:04:04.360 --> 00:04:06.150
So let's ignore this
5 for a second.

00:04:06.150 --> 00:04:08.860
This inside part right here,
the stuff that's inside the

00:04:08.860 --> 00:04:13.260
parentheses, we can factor k
plus 3 out, and it becomes k

00:04:13.260 --> 00:04:24.980
plus 3, times k plus 3,
times 7k minus 1.

00:04:24.980 --> 00:04:28.110
And if this seems a little
bizarre to you, distribute the

00:04:28.110 --> 00:04:29.270
k plus 3 on to this.

00:04:29.270 --> 00:04:31.970
K plus 3 times 7k is that term,
k plus 3 times negative

00:04:31.970 --> 00:04:33.050
1 is that term.

00:04:33.050 --> 00:04:35.060
And, of course, the whole
time you have

00:04:35.060 --> 00:04:36.640
that 5 sitting outside.

00:04:36.640 --> 00:04:37.300
You have that 5.

00:04:37.300 --> 00:04:38.650
We don't even have to put
parentheses there.

00:04:38.650 --> 00:04:42.770
5 times k plus 3, times
7k minus 1.

00:04:42.770 --> 00:04:45.050
And we factored it,
we're done.


U09_L1_T2_we3 Factoring Trinomials by Grouping 3

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:04.990
We're told to factor r squared
plus 4r, minus 45.

00:00:04.990 --> 00:00:07.740
And there are simpler ways to
factor this, because this has

00:00:07.740 --> 00:00:11.160
1 as a leading coefficient on
the r squared right here, but

00:00:11.160 --> 00:00:11.990
what we're going to do
is we're going to

00:00:11.990 --> 00:00:14.420
factor this by grouping.

00:00:14.420 --> 00:00:16.750
And when you factor it by
grouping, what you want to do

00:00:16.750 --> 00:00:20.110
is find two numbers whose
product-- so the two numbers

00:00:20.110 --> 00:00:22.920
are a and b-- whose product is
equal to the coefficient on r

00:00:22.920 --> 00:00:27.910
squared, 1 times negative 45,
which is just negative 45.

00:00:27.910 --> 00:00:30.900
And whose sum is equal to the
coefficient on the first

00:00:30.900 --> 00:00:33.450
degree term, is equal to 4.

00:00:33.450 --> 00:00:37.130
a plus b is equal to 4.

00:00:37.130 --> 00:00:38.950
So let's look at all of the
different combinations that

00:00:38.950 --> 00:00:41.180
goes to negative 45, and see
if when you add them,

00:00:41.180 --> 00:00:43.090
you ever get to 4.

00:00:43.090 --> 00:00:44.395
So let's take the
combinations.

00:00:46.980 --> 00:00:48.910
Let's look at their different
a's and b's, and let's see

00:00:48.910 --> 00:00:51.120
what happens when
you sum them.

00:00:51.120 --> 00:00:55.200
So I could try 1 and negative
45, but if you take their sum,

00:00:55.200 --> 00:00:57.260
that's not going to get you
anywhere close to 4, even if

00:00:57.260 --> 00:01:00.110
you swapped them, negative
1 and positive 45.

00:01:00.110 --> 00:01:05.030
Let's see, if you do 3 and 15, 3
and negative 15, or negative

00:01:05.030 --> 00:01:07.150
3 and 15, those are way
more than 4 apart.

00:01:07.150 --> 00:01:07.890
We're going to have
to find numbers

00:01:07.890 --> 00:01:09.930
that are closer together.

00:01:09.930 --> 00:01:13.660
So let's see, 3, 4, 5 and 9,
that looks interesting.

00:01:13.660 --> 00:01:16.120
5 and 9 are 4 apart.

00:01:16.120 --> 00:01:24.040
And if we go with 5 and negative
9, what's their sum?

00:01:24.040 --> 00:01:25.930
Their sum is negative 4.

00:01:25.930 --> 00:01:30.120
What happens if we go with
negative 5 and positive 9?

00:01:30.120 --> 00:01:33.120
Well, then their sum
is 4, so negative 5

00:01:33.120 --> 00:01:35.560
and positive 9 works.

00:01:35.560 --> 00:01:38.460
So what we want to do here, now
that we found the numbers

00:01:38.460 --> 00:01:41.350
whose product is equal to the 1
times negative 45, and whose

00:01:41.350 --> 00:01:46.550
sum is equal to 4, let's break
this 4r up into a sum of

00:01:46.550 --> 00:01:50.510
negative 5-- let me break
it up this way--

00:01:50.510 --> 00:01:55.390
into 9r minus 5r.

00:01:55.390 --> 00:01:59.040
I broke that 4r into that using
our two numbers, and

00:01:59.040 --> 00:02:03.430
then I could put the r squared
out front, r squared, and then

00:02:03.430 --> 00:02:04.810
we have a minus 45.

00:02:04.810 --> 00:02:07.370
And now we're ready
to group this and

00:02:07.370 --> 00:02:08.770
then factor by grouping.

00:02:08.770 --> 00:02:11.840
So let's look at this first
group right here.

00:02:11.840 --> 00:02:15.090
They both have the common
factor of r, so let's

00:02:15.090 --> 00:02:16.160
factor out an r.

00:02:16.160 --> 00:02:19.900
This is equal to r times-- r
square divided by r is r.

00:02:19.900 --> 00:02:22.890
9r divided by r is 9.

00:02:22.890 --> 00:02:24.700
So that's that first group.

00:02:24.700 --> 00:02:29.510
And the second group right here,
has a common factor of

00:02:29.510 --> 00:02:30.510
negative 5.

00:02:30.510 --> 00:02:31.760
So let's factor it out.

00:02:34.470 --> 00:02:37.210
Negative 5r divided by negative
5 is just an r.

00:02:37.210 --> 00:02:41.240
Negative 45 divided by
negative 5 is a 9.

00:02:41.240 --> 00:02:45.820
And then we have an r plus 9 in
both terms. r times r plus

00:02:45.820 --> 00:02:49.060
9, negative 5 times r plus 9.

00:02:49.060 --> 00:02:50.710
So we can factor that out.

00:02:50.710 --> 00:02:58.290
So you factor an r plus 9 out,
and you're left with r times r

00:02:58.290 --> 00:03:02.040
plus 9, divided by r plus 9
is just going to be an r.

00:03:02.040 --> 00:03:07.350
And then negative 5 times r plus
9, divided by r plus 9 is

00:03:07.350 --> 00:03:10.200
negative 5, and we're done!


Factoring Trinomials by Grouping 2

WEBVTT
Kind: captions
Language: en

00:00:00.510 --> 00:00:02.400
We're asked to factor
this expression.

00:00:02.400 --> 00:00:05.460
And there's going to be simpler
ways to factor it, but

00:00:05.460 --> 00:00:08.390
in this video I'm going to
factor it by grouping.

00:00:08.390 --> 00:00:10.460
And when you factor by grouping,
what you need to do

00:00:10.460 --> 00:00:13.170
is think about two numbers
whose products are

00:00:13.170 --> 00:00:14.050
going to be equal to.

00:00:14.050 --> 00:00:16.530
You have actually one
coefficient right here, right?

00:00:16.530 --> 00:00:19.020
t squared is the same
thing as 1t squared.

00:00:19.020 --> 00:00:21.070
So we're looking for two
numbers, let's call them a and

00:00:21.070 --> 00:00:23.510
b, a times b.

00:00:23.510 --> 00:00:25.750
the product of these two
numbers needs to be the

00:00:25.750 --> 00:00:29.820
product of the coefficient on
the t squared, which is 1, and

00:00:29.820 --> 00:00:32.090
the negative 15 right here.

00:00:32.090 --> 00:00:35.950
So a times b has to be equal
to 1 times negative 15, or

00:00:35.950 --> 00:00:38.510
just negative 15.

00:00:38.510 --> 00:00:43.770
And the sum of a and b, and a
plus b, needs to be equal to

00:00:43.770 --> 00:00:45.035
negative 2.

00:00:49.610 --> 00:00:51.940
And once we have these two
numbers, I can show you how we

00:00:51.940 --> 00:00:54.240
can use those to factor
by grouping.

00:00:54.240 --> 00:00:56.620
And in other videos I've
actually broken down to why

00:00:56.620 --> 00:00:58.430
this technique works.

00:00:58.430 --> 00:01:01.370
Now, let's think of the
different factors of negative

00:01:01.370 --> 00:01:04.290
15 when we take their product,
and if we take the sum, if we

00:01:04.290 --> 00:01:06.030
can somehow get to negative 2.

00:01:06.030 --> 00:01:10.590
So let's look at the different
factors of negative 15.

00:01:10.590 --> 00:01:15.260
So we could do-- let me do it in
this other, let me do it in

00:01:15.260 --> 00:01:19.250
pink-- see if 1 and negative 15,
these are-- so everything

00:01:19.250 --> 00:01:22.570
I list here, their product is
going to be negative 15.

00:01:22.570 --> 00:01:24.220
But let's think about
what happens when

00:01:24.220 --> 00:01:25.790
you take their sum.

00:01:25.790 --> 00:01:28.790
So 1 and negative 15, the
sum is negative 14.

00:01:28.790 --> 00:01:33.120
And if you did negative 1 and
15, you're just going to get

00:01:33.120 --> 00:01:33.930
the negative of that.

00:01:33.930 --> 00:01:34.730
You're going to get 14.

00:01:34.730 --> 00:01:36.240
It does not equal negative 2.

00:01:36.240 --> 00:01:42.090
So what happens if you take
3 and negative 5?

00:01:42.090 --> 00:01:45.430
So their product is definitely
negative 15, 3, plus negative

00:01:45.430 --> 00:01:48.150
5 is negative 2,
so that works.

00:01:48.150 --> 00:01:51.910
And if we tried negative 3 and
5 first, we would have gotten

00:01:51.910 --> 00:01:52.780
that to be positive 2.

00:01:52.780 --> 00:01:54.466
It's just like, oh, we just have
to swap the signs and we

00:01:54.466 --> 00:01:55.510
would have gotten
a negative 2.

00:01:55.510 --> 00:01:58.280
So these work, 3 and
negative 5 work.

00:01:58.280 --> 00:02:02.430
3 times negative 5 is negative
15, 3 plus negative 5 is

00:02:02.430 --> 00:02:03.420
negative 2.

00:02:03.420 --> 00:02:06.130
So what we want to do
here is break this

00:02:06.130 --> 00:02:07.230
middle term up here.

00:02:07.230 --> 00:02:10.550
We know that 3 plus negative
5 is equal to negative 2.

00:02:10.550 --> 00:02:14.010
So we can break up this middle
term here as a sum of-- and

00:02:14.010 --> 00:02:15.370
I'll do it right here; I'll
actually do it in the same

00:02:15.370 --> 00:02:18.910
color-- so this thing here, we
can rewrite as t squared.

00:02:18.910 --> 00:02:21.230
I'll put the minus
15 out here.

00:02:21.230 --> 00:02:27.320
But the negative 2t we can
rewrite as the sum of 3t.

00:02:27.320 --> 00:02:34.960
We could write it here
as plus 3t, minus 5t.

00:02:38.520 --> 00:02:41.070
And when you're trying to figure
out which one to put

00:02:41.070 --> 00:02:44.540
first or second, you should look
at these other terms, and

00:02:44.540 --> 00:02:46.290
say which ones have
common factors?

00:02:46.290 --> 00:02:51.570
The 3 and 5 both have a common
factor with 15, so it's not as

00:02:51.570 --> 00:02:53.460
obvious which one to put first,
so we're just going to

00:02:53.460 --> 00:02:53.890
go with this.

00:02:53.890 --> 00:02:58.250
3t minus 5t, I got that from
3t minus 5t is equal to

00:02:58.250 --> 00:02:59.390
negative 2t.

00:02:59.390 --> 00:03:02.300
Positive 3 times negative 5
is equal to negative 15.

00:03:02.300 --> 00:03:03.690
That's where it came from.

00:03:03.690 --> 00:03:06.480
Now, we're ready to factor
by grouping.

00:03:06.480 --> 00:03:09.530
So let's take the first group,
let's take these first two

00:03:09.530 --> 00:03:11.230
terms, right there.

00:03:11.230 --> 00:03:13.060
And what's the common
factor there?

00:03:13.060 --> 00:03:14.510
Well, the common factor
there is t.

00:03:14.510 --> 00:03:17.810
So if I factor a t out, that
becomes t times-- t squared

00:03:17.810 --> 00:03:19.270
divided by t is t.

00:03:19.270 --> 00:03:22.530
3t divided by t is 3.

00:03:22.530 --> 00:03:25.120
So these first two terms
are the same thing as t

00:03:25.120 --> 00:03:27.090
times t plus 3.

00:03:27.090 --> 00:03:32.500
Now, let's look at the
second two terms.

00:03:32.500 --> 00:03:34.150
What's a common factor?

00:03:34.150 --> 00:03:36.680
Well, they're both divisible by
negative 5, so let's factor

00:03:36.680 --> 00:03:37.930
out a negative 5.

00:03:40.680 --> 00:03:44.450
And negative 5t divided by
negative 5, if you factor out

00:03:44.450 --> 00:03:46.400
the negative 5, you're just
going to have a t there.

00:03:46.400 --> 00:03:49.620
And then negative 15, if you
factor out a negative 5, you

00:03:49.620 --> 00:03:51.700
divide negative 15 by negative
5, you're just going to have a

00:03:51.700 --> 00:03:54.970
positive 3.

00:03:54.970 --> 00:03:59.290
And then notice, you now have
two terms here, two products,

00:03:59.290 --> 00:04:05.660
and they both have the common
factor of t plus 3.

00:04:05.660 --> 00:04:10.200
So we can rewrite this right
here as a product of t plus 3.

00:04:10.200 --> 00:04:12.565
We're undistributing
the t plus 3.

00:04:12.565 --> 00:04:20.140
We're factoring out the t plus
3. t plus 3 times t, right?

00:04:20.140 --> 00:04:23.920
Times t minus 5.

00:04:23.920 --> 00:04:26.310
And I want you to really make
sure you feel good that these

00:04:26.310 --> 00:04:28.030
are really the same thing.

00:04:28.030 --> 00:04:31.405
If you take t times t plus 3 and
factor out the t plus 3,

00:04:31.405 --> 00:04:33.550
you're just left with that t.

00:04:33.550 --> 00:04:36.180
If you take negative 5 times t
plus 3 and you factor out the

00:04:36.180 --> 00:04:38.950
t plus 3, you're just left
with that negative 5.

00:04:38.950 --> 00:04:41.256
But once you factor out the t
plus 3, and you're just left

00:04:41.256 --> 00:04:44.470
with the t minus 5, you
have fully factored

00:04:44.470 --> 00:04:45.830
this expression here.

00:04:45.830 --> 00:04:47.580
And in the future we're going
to see easier ways of doing

00:04:47.580 --> 00:04:50.370
this, but factoring by grouping
is actually the

00:04:50.370 --> 00:04:55.130
easiest way to do it if you have
a coefficient higher than

00:04:55.130 --> 00:04:56.390
1, or a non-one coefficient.

00:04:56.390 --> 00:04:58.830
It could also be a negative
coefficient out front here.

00:04:58.830 --> 00:05:00.650
When you have 1 as your
coefficient here, there's

00:05:00.650 --> 00:05:03.710
actually much easier ways to
factor something like this,

00:05:03.710 --> 00:05:06.120
but it's really the same
thought process.


Example 7: Factor a polynomial with two variables by grouping

WEBVTT
Kind: captions
Language: en

00:00:00.450 --> 00:00:03.970
We're asked to factor this
expression by grouping.

00:00:03.970 --> 00:00:06.380
Now, they mention grouping,
we're going to see what

00:00:06.380 --> 00:00:08.130
grouping is, but we're going
to see very quickly that we

00:00:08.130 --> 00:00:10.230
have to do this thing called
grouping because you can't

00:00:10.230 --> 00:00:11.850
just factor this expression.

00:00:11.850 --> 00:00:15.610
If you look at these, each of
the terms, all but one of them

00:00:15.610 --> 00:00:16.750
is divisible by 5.

00:00:16.750 --> 00:00:19.590
So you can't just
factor out a 5.

00:00:19.590 --> 00:00:22.210
Not all of them are divisible
by either r or s.

00:00:22.210 --> 00:00:24.620
This is only divisible by r,
that's only divisible by s,

00:00:24.620 --> 00:00:26.060
that's divisible by neither.

00:00:26.060 --> 00:00:29.820
So there is no common factor
across all four of these

00:00:29.820 --> 00:00:32.400
terms. That's why we have to
group them into groups where

00:00:32.400 --> 00:00:34.940
there are common factors, and
then see if that simplifies

00:00:34.940 --> 00:00:36.010
the whole thing.

00:00:36.010 --> 00:00:38.890
And there is a little bit of an
art to recognizing when you

00:00:38.890 --> 00:00:41.230
can factor by grouping, but
they've set this problem up

00:00:41.230 --> 00:00:42.450
nicely for us.

00:00:42.450 --> 00:00:47.620
So if you look at these first
two terms right here.

00:00:47.620 --> 00:00:51.240
You have a 5rs and a 25r.

00:00:51.240 --> 00:00:54.180
These two guys clearly have
some common factors.

00:00:54.180 --> 00:00:58.770
They're both divisible by 5,
they're both divisible by r.

00:00:58.770 --> 00:01:02.415
So if I just wanted to factor
this one out, or if I wanted

00:01:02.415 --> 00:01:04.790
to rewrite it as a product
of two expressions, how

00:01:04.790 --> 00:01:05.269
could I write it?

00:01:05.269 --> 00:01:10.080
Well, I could write it as a
product of 5r times-- what's

00:01:10.080 --> 00:01:12.420
5rs divided by 5r?

00:01:12.420 --> 00:01:14.650
Well, you still have
an s left over, you

00:01:14.650 --> 00:01:15.810
just have an s there.

00:01:15.810 --> 00:01:19.010
Plus-- what's 25r
divided by 5r?

00:01:19.010 --> 00:01:24.070
Well, 25 divided by 5 is 5, and
r divided by r is just 1.

00:01:24.070 --> 00:01:28.000
So 25r over 5r is 5.

00:01:28.000 --> 00:01:31.210
So these first two terms can
be factored into these two

00:01:31.210 --> 00:01:32.230
expressions.

00:01:32.230 --> 00:01:36.450
And then let's look at these
second two terms. Well, they

00:01:36.450 --> 00:01:39.660
definitely have a common factor,
you have a negative 3

00:01:39.660 --> 00:01:42.210
or positive 3 common
to both of these.

00:01:42.210 --> 00:01:43.710
Let's just go with
the negative 3.

00:01:43.710 --> 00:01:46.770
And our goal is really to factor
it into a negative 3

00:01:46.770 --> 00:01:49.830
times, hopefully, something
very similar to s plus 5.

00:01:49.830 --> 00:01:51.240
And you might already be seeing
that it's going to

00:01:51.240 --> 00:01:52.210
factor into s plus it.

00:01:52.210 --> 00:01:52.950
5.

00:01:52.950 --> 00:01:54.580
So let's factor out
that negative 3.

00:01:54.580 --> 00:01:58.970
So these two terms you can
rewrite as negative 3 times--

00:01:58.970 --> 00:02:01.770
what's negative 3s divided
by negative 3?

00:02:01.770 --> 00:02:04.510
Well, you're just going to
have an s left over.

00:02:04.510 --> 00:02:08.220
And then what's negative 15
divided by negative 3?

00:02:08.220 --> 00:02:09.470
Well, that's just positive 5.

00:02:12.780 --> 00:02:14.570
And just like that, we've
grouped them and we're able to

00:02:14.570 --> 00:02:18.400
factor each of those groups, and
then something interesting

00:02:18.400 --> 00:02:19.620
might pop out at you.

00:02:19.620 --> 00:02:21.410
And one, you can always verify
that you factored this

00:02:21.410 --> 00:02:24.690
properly by distributing each
of these expressions.

00:02:24.690 --> 00:02:27.380
Distributing the 5r times s
plus 5, and the negative 3

00:02:27.380 --> 00:02:29.340
times s plus 5, you'll
get exactly this.

00:02:29.340 --> 00:02:32.590
But something maybe jumped
at out you just now.

00:02:32.590 --> 00:02:35.170
You have 5r times s plus 5.

00:02:35.170 --> 00:02:38.170
Then you have negative
3 times s plus 5.

00:02:38.170 --> 00:02:41.420
So now this expression, we
have two terms instead of

00:02:41.420 --> 00:02:44.360
four, right, this is one term,
this is another term.

00:02:44.360 --> 00:02:48.530
And they both have s plus 5 as a
common factor, so we can now

00:02:48.530 --> 00:02:50.445
factor out s plus 5.

00:02:53.940 --> 00:02:57.020
So this whole thing can
be rewritten as s

00:02:57.020 --> 00:03:04.490
plus 5 times 5r.

00:03:04.490 --> 00:03:04.720
Right?

00:03:04.720 --> 00:03:07.570
If you take 5r times s plus 5,
and you factor out the s plus

00:03:07.570 --> 00:03:09.670
5, you're just left
with the 5r.

00:03:09.670 --> 00:03:12.640
And then similarly, if you take
negative 3 times s plus

00:03:12.640 --> 00:03:15.820
5, and you factor out the s plus
5, or divide by s plus 5,

00:03:15.820 --> 00:03:19.800
you just have a negative
3, just like that.

00:03:19.800 --> 00:03:20.650
And then we're done!

00:03:20.650 --> 00:03:23.310
We've factored this expression
by grouping.

00:03:23.310 --> 00:03:26.520
It's s plus 5 times
5r minus 3.

00:03:26.520 --> 00:03:28.270
And you can verify it by
multiplying it out.

00:03:28.270 --> 00:03:32.860
If you distribute the s plus
5 onto each of these terms,

00:03:32.860 --> 00:03:34.660
you'll get this expression
up here, and then if you

00:03:34.660 --> 00:03:37.900
distribute the 5r over there
you're going to get that

00:03:37.900 --> 00:03:38.340
expression.

00:03:38.340 --> 00:03:41.180
If you distribute the negative
3, you're going to get that

00:03:41.180 --> 00:03:41.710
expression.

00:03:41.710 --> 00:03:45.040
So this does simplify to that,
so we have factored


Example 4: Factor a polynomial with two variables by taking a common factor

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:05.720
We're told to factor 4x to the
fourth y, minus 8x to the

00:00:05.720 --> 00:00:09.230
third y, minus 2x squared.

00:00:09.230 --> 00:00:12.160
So to factor this, we need to
figure out what the greatest

00:00:12.160 --> 00:00:14.190
common factor of each
of these terms are.

00:00:14.190 --> 00:00:15.030
So let me rewrite it.

00:00:15.030 --> 00:00:21.790
So we have 4x to the fourth y,
and we have minus 8x to the

00:00:21.790 --> 00:00:27.390
third y, and then we have
minus 2x squared.

00:00:27.390 --> 00:00:29.790
So in the other videos, we
looked at it in terms of

00:00:29.790 --> 00:00:31.630
breaking it down to its simplest
parts, but I think we

00:00:31.630 --> 00:00:34.830
have enough practice now to be
able to do a little bit more

00:00:34.830 --> 00:00:35.420
of it in our heads.

00:00:35.420 --> 00:00:37.460
So what is the largest
number that

00:00:37.460 --> 00:00:38.940
divides into all of these?

00:00:38.940 --> 00:00:41.700
When I say number I'm talking
about the actual, I guess,

00:00:41.700 --> 00:00:42.320
coefficients.

00:00:42.320 --> 00:00:44.840
We have a 4, an 8, a 2.

00:00:44.840 --> 00:00:46.950
We don't have to worry about the
negative signs just yet.

00:00:46.950 --> 00:00:50.290
And we say, well, the largest,
of, the largest common factor

00:00:50.290 --> 00:00:53.400
of 2, 8 and 4 is 2.

00:00:53.400 --> 00:00:56.180
2 goes into all of them, and
obviously that's the largest

00:00:56.180 --> 00:00:57.120
number that can go into 2.

00:00:57.120 --> 00:01:00.860
So that is the largest number
that's going to be part of the

00:01:00.860 --> 00:01:02.110
greatest common factor.

00:01:02.110 --> 00:01:03.460
So let's write that down.

00:01:03.460 --> 00:01:05.090
So it's going to be 2.

00:01:05.090 --> 00:01:08.490
And then what's the greatest,
I guess, factor, what's the

00:01:08.490 --> 00:01:10.640
greatest degree of x that's
divisible into

00:01:10.640 --> 00:01:12.290
all three of these?

00:01:12.290 --> 00:01:15.820
Well, x squared goes into all
three of these, and obviously

00:01:15.820 --> 00:01:18.590
that's the greatest degree of
x that can be divided into

00:01:18.590 --> 00:01:19.830
this last term.

00:01:19.830 --> 00:01:24.550
So x squared is going to be the
greatest common x degree

00:01:24.550 --> 00:01:25.450
in all of them.

00:01:25.450 --> 00:01:26.450
2x squared.

00:01:26.450 --> 00:01:29.630
And then what's the largest
degree of y that's divisible

00:01:29.630 --> 00:01:31.080
into all of them?

00:01:31.080 --> 00:01:34.050
Well, these two guys are
divisible by y, but this guy

00:01:34.050 --> 00:01:36.130
isn't, so there is no
degree of y that's

00:01:36.130 --> 00:01:37.560
divisible into all of them.

00:01:37.560 --> 00:01:41.520
So the greatest common factor
of all three of these guys

00:01:41.520 --> 00:01:44.480
right here is 2x squared.

00:01:44.480 --> 00:01:47.020
So what we can do now is we can
think about each of these

00:01:47.020 --> 00:01:50.320
terms as the product of the 2x
squared and something else.

00:01:50.320 --> 00:01:52.610
And to figure that something
else we can literally

00:01:52.610 --> 00:01:55.500
undistribute the 2x squared, say
this is the same thing, or

00:01:55.500 --> 00:01:58.020
even before we undistribute the
2x squared, we could say

00:01:58.020 --> 00:02:01.840
look, 4x to the fourth y is the
same thing as 2x squared,

00:02:01.840 --> 00:02:07.740
times 4x to the fourth
y, over 2x squared.

00:02:07.740 --> 00:02:08.060
Right?

00:02:08.060 --> 00:02:10.810
If you just multiply this
out, you get 4x to the fourth y.

00:02:10.810 --> 00:02:15.910
Similarly, you could say that
8x to the third y-- I'll put

00:02:15.910 --> 00:02:21.530
the negative out front-- is the
same thing as 2x squared,

00:02:21.530 --> 00:02:26.510
our greatest common factor,
times 8x to the third y, over

00:02:26.510 --> 00:02:28.340
2x squared.

00:02:28.340 --> 00:02:32.120
And then finally, 2x squared
is the same thing as if we

00:02:32.120 --> 00:02:35.560
factor out 2x squared-- so we
have that negative sign out

00:02:35.560 --> 00:02:41.140
front-- if we factor out 2x
squared, it's the same thing

00:02:41.140 --> 00:02:44.450
as 2x squared, times 2x squared,
over 2x squared.

00:02:44.450 --> 00:02:46.670
This is almost silly what I'm
doing here, but I'm just

00:02:46.670 --> 00:02:50.330
showing you that I'm just
multiplying and dividing both

00:02:50.330 --> 00:02:51.970
of these terms by 2x squared.

00:02:51.970 --> 00:02:54.030
Multiplying and dividing.

00:02:54.030 --> 00:02:55.420
Here it's trivially simple.

00:02:55.420 --> 00:02:59.500
This just simplifies to 2x
squared right there, or this

00:02:59.500 --> 00:03:01.510
2x squared times 1.

00:03:01.510 --> 00:03:05.920
That simplifies to 1, maybe
I should write it below.

00:03:05.920 --> 00:03:07.560
But what do these simplify to?

00:03:07.560 --> 00:03:12.220
So this first term over here,
this simplifies to 2x squared

00:03:12.220 --> 00:03:16.860
times-- now you get 4 divided
by 2 is 2, x to the fourth

00:03:16.860 --> 00:03:20.080
divided by x squared
is x squared.

00:03:20.080 --> 00:03:24.140
And then y divided by 1 is
just going to be a y.

00:03:24.140 --> 00:03:28.420
So it's 2x squared times 2x
squared y, and then you have

00:03:28.420 --> 00:03:35.382
minus 2x squared times,
8 divided by 2 is 4.

00:03:35.382 --> 00:03:38.880
x to the third divided
by x squared is x.

00:03:38.880 --> 00:03:42.400
And y divided by 1, you can
imagine, is just y.

00:03:42.400 --> 00:03:48.010
And then finally, of course,
you have minus 2x squared

00:03:48.010 --> 00:03:52.080
time-- this right here
simplifies to 1-- times 1.

00:03:52.080 --> 00:03:58.260
Now, if you were to undistribute
2x squared out of

00:03:58.260 --> 00:04:04.840
the expression, you'd
essentially get 2x squared

00:04:04.840 --> 00:04:09.250
times this term, minus this
term, minus this term.

00:04:09.250 --> 00:04:12.140
Right, if you distribute this
out, if you take that out of

00:04:12.140 --> 00:04:15.520
each of the terms, you're going
to get 2x squared times

00:04:15.520 --> 00:04:27.160
this 2x squared y, minus 4xy,
and then you have minus 1,

00:04:27.160 --> 00:04:29.130
minus 1, and we're done.

00:04:29.130 --> 00:04:30.760
We've factored the problem.

00:04:30.760 --> 00:04:32.220
Now, it looks like we
did a lot of steps.

00:04:32.220 --> 00:04:34.310
And the reason why I kind of of
went through great pains to

00:04:34.310 --> 00:04:36.820
show you exactly what we're
doing is so you know exactly

00:04:36.820 --> 00:04:37.770
what we're doing.

00:04:37.770 --> 00:04:39.660
In the future, you might
be able to do this

00:04:39.660 --> 00:04:40.490
a little bit quicker.

00:04:40.490 --> 00:04:43.190
You might be able to do many
of the steps in your head.

00:04:43.190 --> 00:04:45.190
You might say OK, let me
look at each of these.

00:04:45.190 --> 00:04:49.690
Well, the biggest coefficient
that divides all of these is a

00:04:49.690 --> 00:04:53.060
2, so let me put that 2,
let me factor 2 out.

00:04:53.060 --> 00:04:55.690
Well, all of these are divisible
by x squared.

00:04:55.690 --> 00:04:56.960
That's the largest
degree of x.

00:04:56.960 --> 00:04:59.130
Let me factor an
x squared out.

00:04:59.130 --> 00:05:02.360
And this guy doesn't have a y,
so I can't factor a y out.

00:05:02.360 --> 00:05:04.700
So let's see, it's going to be
2x squared times-- and what's

00:05:04.700 --> 00:05:06.480
this guy divided
by 2x squared?

00:05:06.480 --> 00:05:08.815
Well 4 divided by 2 is 2.

00:05:08.815 --> 00:05:12.450
x to the fourth divided by
x squared is x squared.

00:05:12.450 --> 00:05:15.720
y divided by 1, there's no
other y degree that we

00:05:15.720 --> 00:05:17.830
factored out, so it's just
going to be a y.

00:05:17.830 --> 00:05:22.540
And then you have minus
8 divided by 2 is 4.

00:05:22.540 --> 00:05:25.740
x to the third divided
by x squared is x.

00:05:25.740 --> 00:05:28.960
And then you have y divided
by say, 1, is just y.

00:05:28.960 --> 00:05:32.720
And then you have minus
2 divided by 2 is 1.

00:05:32.720 --> 00:05:35.170
x squared divided by x squared
1, so 2x squared divided by 2x

00:05:35.170 --> 00:05:36.570
squared is just 1.

00:05:36.570 --> 00:05:38.815
So in the future, you'll do it
more like this, where you kind

00:05:38.815 --> 00:05:41.280
of just factor it out in your
head, but I really want you to

00:05:41.280 --> 00:05:42.480
understand what we did here.

00:05:42.480 --> 00:05:43.370
There is no magic.

00:05:43.370 --> 00:05:45.840
And to realize that there's no
magic, you could just use the

00:05:45.840 --> 00:05:48.450
distributive property to
multiply this out again, to

00:05:48.450 --> 00:05:50.070
multiply it out again, and
you're going to see that you

00:05:50.070 --> 00:05:52.710
get exactly this.


Example 2: Factor a quadratic binomial with two variables by taking a common factor

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:06.480
We're asked to factor 20u
squared v minus 10uv squared.

00:00:06.480 --> 00:00:09.630
And when they say factor a
binomial like this, an

00:00:09.630 --> 00:00:12.470
expression that has two terms
like this, they really mean

00:00:12.470 --> 00:00:16.450
break it up into the product of
one or more terms. So let's

00:00:16.450 --> 00:00:17.210
see if we can do that.

00:00:17.210 --> 00:00:19.500
And the easiest way we can do
that, is say hey, is there any

00:00:19.500 --> 00:00:20.270
common factor?

00:00:20.270 --> 00:00:23.320
In particular, let's find the
greatest common factor of each

00:00:23.320 --> 00:00:25.610
of these terms and then
divide that out.

00:00:25.610 --> 00:00:27.800
Or you can almost imagine,
un-distribute it out, and I'll

00:00:27.800 --> 00:00:30.190
show you what I'm talking
about in a second.

00:00:30.190 --> 00:00:31.815
And eventually you'll be able
to do this in your head, but

00:00:31.815 --> 00:00:33.790
we'll work through it step
by step right now.

00:00:33.790 --> 00:00:37.970
So what is 20u squared v
if we factor it out?

00:00:37.970 --> 00:00:42.020
20u squared v, if we do into the
prime factorization, it is

00:00:42.020 --> 00:00:46.000
20 is 2 times 2 times 5.

00:00:46.000 --> 00:00:46.360
Right?

00:00:46.360 --> 00:00:48.660
That's 2 times 10, that's 20.

00:00:48.660 --> 00:00:54.560
u squared is times u times u
and then v is just one v.

00:00:54.560 --> 00:00:58.150
So we just rewrote 20u squared
v into kind of a product of

00:00:58.150 --> 00:01:01.310
its smallest components, its
most fundamental components,

00:01:01.310 --> 00:01:03.980
prime numbers and just
u's and v's.

00:01:03.980 --> 00:01:07.620
Now let's do the same exact
thing for the 10uv squared.

00:01:07.620 --> 00:01:09.520
So we'll put that minus sign
there so that we haven't

00:01:09.520 --> 00:01:11.440
fundamentally changed
the expression.

00:01:11.440 --> 00:01:16.250
10 is, if we break it down to
its prime factors, 2 times 5.

00:01:16.250 --> 00:01:20.890
Then we have one u times
one u times v times v.

00:01:20.890 --> 00:01:24.570
That's what v squared
is, times v times v.

00:01:24.570 --> 00:01:27.560
Now what's the greatest common
factor of these two terms

00:01:27.560 --> 00:01:28.490
right here?

00:01:28.490 --> 00:01:29.490
Well let's see.

00:01:29.490 --> 00:01:31.920
They both have one 2.

00:01:31.920 --> 00:01:34.090
They both have one
2 right there.

00:01:34.090 --> 00:01:35.880
Maybe I'll circle that one, I
could have circled that one.

00:01:35.880 --> 00:01:40.380
They both have one 5--
that's one 5, one 5.

00:01:40.380 --> 00:01:44.380
They both have one u, one
u there, one u there.

00:01:44.380 --> 00:01:46.600
This one has two, but only this
one has one, and they

00:01:46.600 --> 00:01:47.950
both have at least one v.

00:01:51.770 --> 00:01:57.610
So the greatest common factor is
2 times 5 times u times v.

00:01:57.610 --> 00:01:59.860
So I could rewrite this
expression, I can kind of

00:01:59.860 --> 00:02:02.520
un-distribute the 2 times
5 times u times v,

00:02:02.520 --> 00:02:03.660
and what'll we get?

00:02:03.660 --> 00:02:10.210
If we wrote 2 times 5 times u
times v, and we say that's

00:02:10.210 --> 00:02:13.750
going to be-- this expression
is equal to this times what?

00:02:13.750 --> 00:02:17.250
Well if you factor the 2 times
5 times u times v out, all

00:02:17.250 --> 00:02:20.090
you're going to be left with
in this first term is the 2

00:02:20.090 --> 00:02:26.060
times u, so 2u here.

00:02:26.060 --> 00:02:28.030
And in the second term,
all you're going to be

00:02:28.030 --> 00:02:29.650
left with is a v.

00:02:29.650 --> 00:02:29.910
Right?

00:02:29.910 --> 00:02:31.460
All this other stuff
gets factored out.

00:02:31.460 --> 00:02:33.520
All you're going to be
left with is a v.

00:02:33.520 --> 00:02:37.050
Hopefully you see, if I multiply
2 times 5 times u

00:02:37.050 --> 00:02:40.460
times v times 2u, I'm going to
get this first term here.

00:02:40.460 --> 00:02:43.220
So if I were to distribute it,
I would get this first term.

00:02:43.220 --> 00:02:47.150
And if I multiply 2 times 5
times u times v times this v

00:02:47.150 --> 00:02:49.350
over here, I'm going to
get this second term.

00:02:49.350 --> 00:02:51.860
So this expression, and
that expression is

00:02:51.860 --> 00:02:53.400
the exact same thing.

00:02:53.400 --> 00:02:54.850
We have factored it
out, now we can

00:02:54.850 --> 00:02:56.180
simplify it a little bit.

00:02:56.180 --> 00:03:01.510
2 times 5 times u times
v we rewrite as 10uv.

00:03:01.510 --> 00:03:04.530
And then inside the parentheses,
we of course have

00:03:04.530 --> 00:03:10.470
a 2u and then a minus v.

00:03:10.470 --> 00:03:11.090
And we're done!

00:03:11.090 --> 00:03:12.970
We have factored
the expression.

00:03:12.970 --> 00:03:16.240
Now you won't be doing it to
this granular level, but this

00:03:16.240 --> 00:03:18.850
is the best way to
think about it.

00:03:18.850 --> 00:03:20.230
Eventually you're going to
say, hey, wait, look, the

00:03:20.230 --> 00:03:23.360
largest number that divides
both of these is a 10.

00:03:23.360 --> 00:03:25.690
Because you could see 10 goes
into the 20, 10 goes into 10.

00:03:25.690 --> 00:03:28.120
And, let's see, a u goes into
both of these, and a v goes

00:03:28.120 --> 00:03:28.740
into both of these.

00:03:28.740 --> 00:03:32.510
So let me factor out a 10uv,
and then if I divide this

00:03:32.510 --> 00:03:36.390
thing by 10uv, I'm going
to be left with 2u.

00:03:36.390 --> 00:03:38.330
And if I divide this by 10uv,
I'm going to just

00:03:38.330 --> 00:03:40.020
be left with a v.

00:03:40.020 --> 00:03:41.360
So that's another way
to think about it.

00:03:41.360 --> 00:03:43.390
Let me do that right now, so we
could say that this is the

00:03:43.390 --> 00:03:44.730
same thing.

00:03:44.730 --> 00:03:47.030
Another way of approaching it,
you could have said that this

00:03:47.030 --> 00:03:49.280
is the same thing as-- Well,
the largest number that

00:03:49.280 --> 00:03:56.650
divides both of these is 10uv,
and that's going to be times

00:03:56.650 --> 00:04:06.370
20u squared v over 10uv
minus this thing.

00:04:06.370 --> 00:04:11.300
10uv squared over 10uv.

00:04:11.300 --> 00:04:13.910
This expression and this are
obviously the same thing.

00:04:13.910 --> 00:04:16.480
If I were to distribute the 10uv
it would cancel out with

00:04:16.480 --> 00:04:18.490
each of these in the denominator
right there.

00:04:18.490 --> 00:04:20.110
So they're the same thing,
but we can do is we

00:04:20.110 --> 00:04:21.329
can simplify this.

00:04:21.329 --> 00:04:25.740
We could say that 20 divided
by 10 is just 2, u squared

00:04:25.740 --> 00:04:30.570
divided by u is just a u, v
divided by v is just 1, 10

00:04:30.570 --> 00:04:34.280
divided by 10 is 1, u divide by
u is u, v squared divided

00:04:34.280 --> 00:04:36.900
by v is just a v to
the first power.

00:04:36.900 --> 00:04:42.620
So you're left with 10uv times
the quantity 2u minus v.

00:04:42.620 --> 00:04:45.180
Either way you get
the same answer.