Quadratic inequality example 2

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:04.420
We've got the inequality negative x times
the expression

00:00:04.420 --> 00:00:07.664
2x minus 14 is greater than or equal to
24.

00:00:07.664 --> 00:00:09.710
So I encar, encourage you to pause this

00:00:09.710 --> 00:00:11.942
video now and think about what the
solution

00:00:11.942 --> 00:00:14.360
set to this inequality would actually be,
and

00:00:14.360 --> 00:00:16.990
actually plot the solution set on a number
line.

00:00:18.110 --> 00:00:19.860
So I'm assuming you've given a go at it,

00:00:19.860 --> 00:00:21.939
so let's just try to simplify this a
little bit.

00:00:21.939 --> 00:00:25.899
So on the left hand side we could
distribute, we could

00:00:25.899 --> 00:00:29.683
distribute this negative x, and so if we
did that we

00:00:29.683 --> 00:00:35.139
would get negative 2, negative 2x squared,
negative times a negative

00:00:35.139 --> 00:00:39.669
is a positive, plus 14x is greater than or
equal to 24.

00:00:39.669 --> 00:00:43.629
Now I'm gonna put the, I'm gonna subtract
24 from both sides, just so that we just

00:00:43.629 --> 00:00:47.386
have a 0 here and then we can think about
factoring what we have here on the left.

00:00:47.386 --> 00:00:52.090
So we have negative 2x squared, plus 14x,
I'm gonna subtract

00:00:52.090 --> 00:00:56.122
24 from both sides, so it's minus 24 is
greater than

00:00:56.122 --> 00:01:00.322
or equal to, I've subtracted 24 from the
right as well,

00:01:00.322 --> 00:01:03.940
so that's going to be greater than or
equal to 0.

00:01:03.940 --> 00:01:07.966
Now, I don't like, I don't like having
this negative 2 out front, so

00:01:07.966 --> 00:01:11.398
what I wanna do is I wanna divide this
left hand side by negative

00:01:11.398 --> 00:01:15.160
2, but I can't just divide the left hand
side only by negative 2,

00:01:15.160 --> 00:01:18.634
I have to do divide the right hand side by
negative 2 as well.

00:01:18.634 --> 00:01:21.914
And anytime I multiply or divide both
sides of an

00:01:21.914 --> 00:01:26.904
inequality by a negative number, it's
going to flip the inequality.

00:01:26.904 --> 00:01:32.121
So if I divide both sides by negative 2,
I'm going to be

00:01:32.121 --> 00:01:37.338
left with x squared, positive x squared
minus, so I'm

00:01:37.338 --> 00:01:42.358
dividing by negative 2, so minus 7x, plus
12.

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And now since I divided by negative 2, I'm
gonna flip this inequality,

00:01:46.388 --> 00:01:49.580
is less than or equal to zero divided by
negative 2 is 0.

00:01:49.580 --> 00:01:52.088
So that simplified things a good bit, and
now

00:01:52.088 --> 00:01:55.254
let's see if we can factor this quadratic
expression.

00:01:55.254 --> 00:01:59.797
So two numbers whose product is positive
12, so that means they're

00:01:59.797 --> 00:02:04.260
gonna have the same sign, and whose, and
whose sum is negative 7.

00:02:04.260 --> 00:02:07.040
So if they have the same sign, and their
sum is

00:02:07.040 --> 00:02:09.590
negative 7, that tells us that they're
both going to be negative.

00:02:09.590 --> 00:02:13.710
And let's see, negative 3 and negative 4
seem to fit the bill.

00:02:13.710 --> 00:02:15.240
Their product is positive 12.

00:02:15.240 --> 00:02:17.000
Their sum is negative 7.

00:02:17.000 --> 00:02:21.740
So we could write this as x minus 3 times
x

00:02:21.740 --> 00:02:26.630
minus 4, is going to be less than or equal
to 0.

00:02:26.630 --> 00:02:30.430
So now this is the point that we're gonna
do a little bit of interesting logic.

00:02:30.430 --> 00:02:33.820
If the product of two things is less than,
is less than or equal

00:02:33.820 --> 00:02:37.580
to 0, what does that tell us, tell what do
we know about it?

00:02:37.580 --> 00:02:41.170
Well, that tells us that either, either
one or

00:02:41.170 --> 00:02:45.220
both of them is 0, or they have different
signs.

00:02:45.220 --> 00:02:48.220
The only way that you're gonna get less
than 0 is if one is

00:02:48.220 --> 00:02:52.838
positive and the other is negative, or one
is negative and the other is positive.

00:02:52.838 --> 00:02:57.050
So our 1 is negative, is that they have
different signs.

00:02:57.050 --> 00:02:59.478
So let's write that down, let's write that
down.

00:02:59.478 --> 00:03:04.538
So either, either x minus 3, x minus 3 is
less than

00:03:04.538 --> 00:03:09.713
or, less than, either x minus 3 is less
than or equal

00:03:09.713 --> 00:03:15.118
to 0 and, and x minus 4 is greater than or
equal to 0, x

00:03:15.118 --> 00:03:20.523
minus 4 is greater than or equal to 0, so
notice, note

00:03:20.523 --> 00:03:25.583
this one is non positive, this one is non
negative,

00:03:25.583 --> 00:03:31.785
they're either equal to 0 or they are have
different signs.

00:03:31.785 --> 00:03:37.956
So that's one situation, or, or the other
way around, or x

00:03:37.956 --> 00:03:44.490
minus 3 is non negative, it's greater than
or equal to 0, and

00:03:44.490 --> 00:03:51.070
x minus 4 is non positive, x minus 4 is
less than or equal to 0.

00:03:51.070 --> 00:03:54.364
Once again, they're either zero or
different signs, that's all I'm

00:03:54.364 --> 00:03:57.527
doing with this little with this little
logic work right over here.

00:03:57.527 --> 00:04:00.504
So what are these, what is this, what is
this simplified to?

00:04:00.504 --> 00:04:03.780
So x minus 3 less than or equal to 0, add
3

00:04:03.780 --> 00:04:07.730
to both sides, you get x is less than or
equal to 3.

00:04:07.730 --> 00:04:12.980
And, and x minus 4 is greater than or
equal to 0.

00:04:12.980 --> 00:04:19.230
If you add 4 to both sides of this, you
get x is greater than or equal to 4.

00:04:19.230 --> 00:04:22.860
So what values of x are going to be less
than

00:04:22.860 --> 00:04:27.080
or equal to 3, and greater than or equal
to 4?

00:04:27.080 --> 00:04:28.670
Well anything that's less than or equal to
3 is

00:04:28.670 --> 00:04:30.210
not going to be greater than or equal to 4

00:04:30.210 --> 00:04:31.670
and anything that's greater than or equal
to 4 is

00:04:31.670 --> 00:04:34.220
not going to be less than or equal to 3.

00:04:34.220 --> 00:04:38.140
So there's no x, there's no x value that
can satisfy

00:04:38.140 --> 00:04:41.960
this situation right over here, there's no
X value that will

00:04:41.960 --> 00:04:46.110
result in this one being negative, and
this one being, or

00:04:46.110 --> 00:04:50.370
this one being non positive and this one
being non negative.

00:04:50.370 --> 00:04:52.332
So let's go to this one right over here.

00:04:52.332 --> 00:04:58.542
So if we add 3 to both sides, we get x is
greater than or equal to 3

00:04:58.542 --> 00:05:05.231
and, and we get adding 4 to both sides, x
is less than or equal to 4.

00:05:05.231 --> 00:05:07.616
Now does this make sense, that something
could be greater than

00:05:07.616 --> 00:05:09.220
or equal to 3 and less than or equal to 4?

00:05:09.220 --> 00:05:12.080
Sure, for example, well 3 is greater than
or equal

00:05:12.080 --> 00:05:14.030
to 3 and is less than or equal to 4,

00:05:14.030 --> 00:05:16.240
4 is greater than or equal to 3 and it's

00:05:16.240 --> 00:05:18.928
less than or equal to 4 and anything in
between.

00:05:18.928 --> 00:05:20.769
So we can plot the solution set here.

00:05:20.769 --> 00:05:22.955
So this is actually all that matters.

00:05:22.955 --> 00:05:26.080
Cuz this one, there's no situation which
that would've been true.

00:05:26.080 --> 00:05:27.909
So this is the only thing, this is the

00:05:27.909 --> 00:05:30.390
only thing that's gonna make this or part
true.

00:05:30.390 --> 00:05:32.610
This part is always going to be false.

00:05:32.610 --> 00:05:35.557
So if we wanted to make the solution set,
it would look something like this.

00:05:35.557 --> 00:05:38.940
So if this is our possible values of x.

00:05:38.940 --> 00:05:45.060
So let's say that that is 0, so this is 1,
2, 3,

00:05:45.060 --> 00:05:51.044
and 4, 3 and 4, 3 and 4, x could be
greater than

00:05:51.044 --> 00:05:57.149
or equal to 3, so it's greater than or
equal to 3.

00:05:58.710 --> 00:06:00.920
But it's also let, it also has to be less
than

00:06:00.920 --> 00:06:03.560
or equal to 4 so we can't just go all past
4.

00:06:03.560 --> 00:06:07.079
Also, less than or equal, less is since
it's, less than or

00:06:07.079 --> 00:06:10.501
equal we can color in these dots, less
than or equal to 4.

00:06:10.501 --> 00:06:13.765
So anything in this range, including 3 and
4, that's why

00:06:13.765 --> 00:06:17.310
we circled in the dots, this would satisfy
this equation here.

00:06:17.310 --> 00:06:20.105
And if you wanted to think about it
visually, hey,

00:06:20.105 --> 00:06:22.705
you know we know that this type of thing,
we know

00:06:22.705 --> 00:06:25.565
that this type of thing, or this type of
thing, this

00:06:25.565 --> 00:06:29.480
type of thing right over here, these are,
these are parabola's.

00:06:29.480 --> 00:06:31.890
So how would that relate to this little
solution

00:06:31.890 --> 00:06:34.760
set that we just thought about right over
here?

00:06:34.760 --> 00:06:37.780
Well if you look at let's, let's just look
at one of these.

00:06:37.780 --> 00:06:42.180
Let's say we went to, let's say we went to
this form right over here.

00:06:42.180 --> 00:06:46.487
So all, everything we did, this is just
another way of thinking about

00:06:46.487 --> 00:06:51.026
negative 2x squared plus 14x minus 24 is
greater than or equal to 0.

00:06:51.026 --> 00:06:53.917
So this right over here, we have a
negative coefficient on

00:06:53.917 --> 00:06:57.900
the x squared term, that's going to be a
downward opening parabola.

00:06:57.900 --> 00:06:59.915
So when is that greater than or equal to
0?

00:06:59.915 --> 00:07:01.451
So if, if we though about the

00:07:01.451 --> 00:07:04.651
downward opening parabola, it could might
look something

00:07:04.651 --> 00:07:06.827
like this, it might look something like

00:07:06.827 --> 00:07:09.377
this if we're now thinking in 2
dimensions.

00:07:09.377 --> 00:07:14.371
And this is the, this is the, if you think
of this as the Y axis right over here.

00:07:14.371 --> 00:07:16.179
So when is that greater than or equal to
0?

00:07:16.179 --> 00:07:19.189
Well it's greater than or equal to 0, it's
above

00:07:19.189 --> 00:07:21.915
the X axis in this range for x right over
here.

00:07:21.915 --> 00:07:23.833
So that's one way of thinking about it.

00:07:23.833 --> 00:07:26.313
If we though about it from the point of

00:07:26.313 --> 00:07:30.073
view, not of that parabola, not of that
parabola, but

00:07:30.073 --> 00:07:33.353
this parabola right over here, when is x
squared

00:07:33.353 --> 00:07:36.399
minus 7x plus 12 less than or equal to 0?

00:07:36.399 --> 00:07:39.450
Well this is gonna be an up, upward
opening parabola.

00:07:39.450 --> 00:07:43.720
So if it has a positive coefficient here,
so this problem might look something like

00:07:43.720 --> 00:07:47.602
this, might look something like this, when
is it less than or equal to 0?

00:07:47.602 --> 00:07:52.570
Well once again, once again it's less than
or equal to 0 in that same range.


Quadratic inequality example

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:05.540
Let's say that we want to
solve the inequality x squared

00:00:05.540 --> 00:00:09.030
plus 3x is greater than 10.

00:00:09.030 --> 00:00:11.200
We want to figure out
all of the x's that

00:00:11.200 --> 00:00:14.214
would satisfy this inequality.

00:00:14.214 --> 00:00:15.880
I encourage you to
pause this video now.

00:00:15.880 --> 00:00:17.100
And I'll give you a hint.

00:00:17.100 --> 00:00:19.600
Try to manipulate the
way that you would have

00:00:19.600 --> 00:00:22.150
if this was a
quadratic equation.

00:00:22.150 --> 00:00:23.910
But then as you
get to the end, try

00:00:23.910 --> 00:00:27.520
to reason through it, because
the reasoning might departure

00:00:27.520 --> 00:00:30.730
a little bit from
what you are used to.

00:00:30.730 --> 00:00:32.689
So I'm assuming you've
given a go at it.

00:00:32.689 --> 00:00:34.230
So the first thing
that we might want

00:00:34.230 --> 00:00:35.938
to do, just to get
into a form that we're

00:00:35.938 --> 00:00:38.790
more comfortable with, is
subtract 10 from both sides.

00:00:38.790 --> 00:00:40.236
If we subtract 10
from both sides,

00:00:40.236 --> 00:00:41.610
then on the left
hand side, we're

00:00:41.610 --> 00:00:46.430
going to have x squared
plus 3x minus 10 is still

00:00:46.430 --> 00:00:48.750
going to be greater than.

00:00:48.750 --> 00:00:50.930
If we add or subtract the
same thing to both sides,

00:00:50.930 --> 00:00:52.790
it won't change the inequality.

00:00:52.790 --> 00:00:54.740
But it's now going
to be greater than 0.

00:00:54.740 --> 00:00:57.097
10 minus 10 is 0.

00:00:57.097 --> 00:00:59.180
Now, this gets us into a
form that we're more used

00:00:59.180 --> 00:01:02.680
to seeing quadratic
expressions in.

00:01:02.680 --> 00:01:04.430
If this was an equal
sign right over here,

00:01:04.430 --> 00:01:05.721
we'd want to factor this thing.

00:01:05.721 --> 00:01:08.644
So let's just try to factor
here too and see what happens.

00:01:08.644 --> 00:01:09.810
So we're going to factor it.

00:01:09.810 --> 00:01:11.430
We're going to
think of two numbers

00:01:11.430 --> 00:01:16.630
whose product is negative 10
and whose sum is positive 3.

00:01:16.630 --> 00:01:18.840
And we've had a lot of
practice doing this.

00:01:18.840 --> 00:01:22.540
If you think about the factors
of 10, it's 1, 2, 5, and 10.

00:01:22.540 --> 00:01:25.660
2 and 5 seem tempting,
because their difference is 3.

00:01:25.660 --> 00:01:28.380
So if you have positive
5 and negative 2,

00:01:28.380 --> 00:01:29.670
that seems to work out.

00:01:29.670 --> 00:01:31.950
Positive 5 and negative 2.

00:01:31.950 --> 00:01:35.175
Their product is negative
10, their sum is positive 3.

00:01:37.980 --> 00:01:44.122
We could rewrite
this as x plus 5.

00:01:44.122 --> 00:01:45.580
Let me do that in
that yellow color

00:01:45.580 --> 00:01:47.520
so you see where this
5 is coming from.

00:01:47.520 --> 00:01:59.270
X plus 5 times x minus 2 is
going to be greater than 0.

00:01:59.270 --> 00:02:00.990
Now, if this was
an equality here,

00:02:00.990 --> 00:02:03.620
we would say well, how
do we get this equals 0?

00:02:08.180 --> 00:02:10.100
If either of these
things were equal to 0,

00:02:10.100 --> 00:02:12.460
then this entire expression
would be equal to 0,

00:02:12.460 --> 00:02:14.140
because 0 times anything is 0.

00:02:14.140 --> 00:02:15.690
But we don't have
an equality here.

00:02:15.690 --> 00:02:17.180
We have a greater than symbol.

00:02:17.180 --> 00:02:19.430
So let's think about how we
could reason through this.

00:02:19.430 --> 00:02:21.430
And I'll do a little
bit of an aside here.

00:02:21.430 --> 00:02:26.060
If I were to tell you
that numbers a and b,

00:02:26.060 --> 00:02:29.370
and if I were take
the product a times b,

00:02:29.370 --> 00:02:33.610
and if someone were to tell you
that product is greater than 0,

00:02:33.610 --> 00:02:36.610
what do we know about a and b?

00:02:36.610 --> 00:02:39.302
Well, we know that they
have to have the same sign.

00:02:39.302 --> 00:02:41.260
They're either both
positive-- a positive times

00:02:41.260 --> 00:02:44.200
a positive is going
to be a positive--

00:02:44.200 --> 00:02:48.060
or they're both going to be
negative-- a negative times

00:02:48.060 --> 00:02:50.050
a negative is a positive.

00:02:50.050 --> 00:02:52.450
It's going to be greater than 0.

00:02:52.450 --> 00:02:54.185
So we know the same thing here.

00:02:54.185 --> 00:02:55.060
Let me write it down.

00:02:55.060 --> 00:03:07.760
So we know either a is greater
than 0 and b is greater

00:03:07.760 --> 00:03:13.700
than zero-- so either
both of them are positive

00:03:13.700 --> 00:03:19.330
or both of them
are negative-- or a

00:03:19.330 --> 00:03:23.650
is less than zero and
b is less than zero.

00:03:26.420 --> 00:03:28.030
So we apply that
same logic here.

00:03:28.030 --> 00:03:30.220
You could view
this x plus 5 is a,

00:03:30.220 --> 00:03:34.140
you could view this x minus
2 as our b of the product

00:03:34.140 --> 00:03:34.930
of two things.

00:03:34.930 --> 00:03:36.210
The product is greater than 0.

00:03:36.210 --> 00:03:39.090
That means that either both of
these expressions are positive

00:03:39.090 --> 00:03:40.879
or they're both negative.

00:03:40.879 --> 00:03:41.920
So let's write that down.

00:03:45.370 --> 00:03:46.620
I'll write it this way.

00:03:46.620 --> 00:03:49.260
So either both of these
expressions are positive.

00:03:49.260 --> 00:03:57.010
So either x plus 5 is
greater than 0 and x minus 2

00:03:57.010 --> 00:04:12.800
is greater than 0-- let
me write it this way--

00:04:12.800 --> 00:04:14.720
or they're both negative.

00:04:14.720 --> 00:04:23.550
x plus 5 is less than 0 and
x minus 2 is less than 0.

00:04:28.350 --> 00:04:30.547
So now let's think about
all of these inequalities

00:04:30.547 --> 00:04:31.130
independently.

00:04:31.130 --> 00:04:33.904
But let's maintain this logic
here of the and and the or.

00:04:33.904 --> 00:04:34.820
So let's look at this.

00:04:34.820 --> 00:04:37.540
Either they're both positive,
so if both of these expressions

00:04:37.540 --> 00:04:39.870
are positive, what
do we know about x?

00:04:39.870 --> 00:04:42.770
Well, if you subtract 5 from
both sides of this inequality,

00:04:42.770 --> 00:04:46.010
you get x is greater
than negative 5.

00:04:46.010 --> 00:04:49.840
And if you add 2 to this
inequality, both sides

00:04:49.840 --> 00:04:53.950
of that inequality, you're going
to get x is greater than 2.

00:04:53.950 --> 00:04:58.380
So if x is greater than negative
5 and x is greater than 2,

00:04:58.380 --> 00:04:59.930
what do we know about x?

00:04:59.930 --> 00:05:01.720
Well, any x that's
greater than two

00:05:01.720 --> 00:05:03.840
is going to be greater
than negative 5.

00:05:03.840 --> 00:05:06.830
So we could just simplify
this right over here

00:05:06.830 --> 00:05:10.030
to say that x is greater than 2.

00:05:10.030 --> 00:05:12.000
So all of this, this
is equivalent to saying

00:05:12.000 --> 00:05:14.360
x is greater than 2,
because clearly anything

00:05:14.360 --> 00:05:16.710
that is greater than
2 will satisfy that.

00:05:16.710 --> 00:05:19.890
And both of these
things have to be true.

00:05:19.890 --> 00:05:21.940
For example, x
equal negative 4, it

00:05:21.940 --> 00:05:24.160
would satisfy this
inequality but not this one.

00:05:24.160 --> 00:05:25.670
And so the and would break down.

00:05:25.670 --> 00:05:27.710
And negative 4 does not
satisfy both of these.

00:05:27.710 --> 00:05:29.350
In order to satisfy
both of these,

00:05:29.350 --> 00:05:31.680
you essentially have
to satisfy that one.

00:05:31.680 --> 00:05:34.030
So this expression
simplified to that.

00:05:34.030 --> 00:05:35.460
Now what about this?

00:05:35.460 --> 00:05:38.120
What about this statement
right over here?

00:05:38.120 --> 00:05:43.100
Well, x plus 5 less than 0,
subtract 5 from both sides.

00:05:43.100 --> 00:05:46.950
That is, x is less
than negative 5.

00:05:46.950 --> 00:05:52.350
And add 2 to both sides
of that inequality,

00:05:52.350 --> 00:05:55.720
you get x is less than 2.

00:05:55.720 --> 00:06:00.430
Now if x is less than negative
5 and x is less than two,

00:06:00.430 --> 00:06:02.230
what do we know about x?

00:06:02.230 --> 00:06:04.815
Well, that just means that x
has to be less than negative 5.

00:06:10.490 --> 00:06:12.010
If it's less than
negative 5, it's

00:06:12.010 --> 00:06:14.430
definitely going
to be less than 2.

00:06:14.430 --> 00:06:17.300
And we got to remind ourselves
that we have this or here.

00:06:17.300 --> 00:06:19.790
And that's essentially
describing the solution

00:06:19.790 --> 00:06:25.290
set for this quadratic
inequality here.

00:06:25.290 --> 00:06:28.790
x is going to be
greater than 2 or x

00:06:28.790 --> 00:06:32.050
is going to be less
than negative 5.

00:06:32.050 --> 00:06:35.730
And we could actually plot this
solution set on a number line.

00:06:35.730 --> 00:06:39.130
So if this is our number line
right over here, and let's

00:06:39.130 --> 00:06:42.560
say that this is 0.

00:06:42.560 --> 00:06:44.350
Let's say that's 1.

00:06:44.350 --> 00:06:45.500
2 right over here.

00:06:45.500 --> 00:06:49.400
This is negative 1, negative
2, negative 3, negative 4,

00:06:49.400 --> 00:06:52.090
negative 5.

00:06:52.090 --> 00:06:56.100
So x could be greater than 2,
not greater than or equal to,

00:06:56.100 --> 00:06:58.380
so I'm going to put
an open circle here.

00:06:58.380 --> 00:07:01.680
So it could be greater than 2.

00:07:01.680 --> 00:07:04.360
Or it could be less
than negative 5,

00:07:04.360 --> 00:07:05.920
not less than or
equal to, so I'm

00:07:05.920 --> 00:07:08.090
going to put an
open circle here.

00:07:08.090 --> 00:07:10.200
And so it could
be less than that.

00:07:10.200 --> 00:07:13.190
So x could be any number.

00:07:13.190 --> 00:07:15.460
It could be negative 6,
negative 6 would satisfy this.

00:07:15.460 --> 00:07:16.850
You could verify that.

00:07:16.850 --> 00:07:23.800
Negative 6 squared is
36 plus negative 18,

00:07:23.800 --> 00:07:26.280
which is going to be 18,
which is greater than 10.

00:07:26.280 --> 00:07:29.310
Or you could have, say,
a positive 3 would work.

00:07:29.310 --> 00:07:34.290
3 squared is 9 plus
another 9 is going

00:07:34.290 --> 00:07:39.927
to be 18, which once again,
it is greater than 10.


Rewriting a quadratic function to find roots and vertex

WEBVTT
Kind: captions
Language: en

00:00:00.820 --> 00:00:05.880
I have a function here defined
as x squared minus 5x plus 6.

00:00:05.880 --> 00:00:07.600
And what I want
us to think about

00:00:07.600 --> 00:00:12.060
is what other forms we can
write this function in if we,

00:00:12.060 --> 00:00:14.719
say, wanted to find the
0s of this function.

00:00:14.719 --> 00:00:17.260
If we wanted to figure out where
does this function intersect

00:00:17.260 --> 00:00:20.370
the x-axis, what form
would we put this in?

00:00:20.370 --> 00:00:23.110
And then another form
for maybe finding out

00:00:23.110 --> 00:00:24.800
what's the minimum
value of this.

00:00:24.800 --> 00:00:27.780
We see that we have a positive
coefficient on the x squared

00:00:27.780 --> 00:00:28.280
term.

00:00:28.280 --> 00:00:31.229
This is going to be an
upward-opening parabola.

00:00:31.229 --> 00:00:32.770
But what's the
minimum point of this?

00:00:32.770 --> 00:00:35.364
Or even better, what's the
vertex of this parabola

00:00:35.364 --> 00:00:36.030
right over here?

00:00:36.030 --> 00:00:38.190
So if the function looks
something like this,

00:00:38.190 --> 00:00:41.180
we could use one
form of the function

00:00:41.180 --> 00:00:44.160
to figure out where does
it intersect the x-axis.

00:00:44.160 --> 00:00:46.277
So where does it
intersect the x-axis?

00:00:46.277 --> 00:00:48.360
And maybe we can manipulate
it to get another form

00:00:48.360 --> 00:00:49.985
to figure out what's
the minimum point.

00:00:49.985 --> 00:00:52.750
What's this point right
over here for this function?

00:00:52.750 --> 00:00:54.917
I don't even know if the
function looks like this.

00:00:54.917 --> 00:00:56.500
So I encourage you
to pause this video

00:00:56.500 --> 00:01:00.450
and try to manipulate this
into those two different forms.

00:01:00.450 --> 00:01:01.590
So let's work on it.

00:01:01.590 --> 00:01:03.590
So in order to find the
roots, the easiest thing

00:01:03.590 --> 00:01:06.020
I can think of doing
is trying to factor

00:01:06.020 --> 00:01:08.770
this quadratic
expression which is

00:01:08.770 --> 00:01:10.341
being used to define
this function.

00:01:10.341 --> 00:01:11.840
So we could think
about, well, let's

00:01:11.840 --> 00:01:17.170
think of two numbers whose
product is positive 6

00:01:17.170 --> 00:01:22.240
and whose sum is negative 5.

00:01:22.240 --> 00:01:25.090
So since their
product is positive,

00:01:25.090 --> 00:01:26.977
we know that they
have the same sign.

00:01:26.977 --> 00:01:29.560
And if they have the same sign
but we get to a negative value,

00:01:29.560 --> 00:01:32.280
that means they both
must be negative.

00:01:32.280 --> 00:01:35.800
So let's see-- negative 2
times negative 3 is positive 6.

00:01:35.800 --> 00:01:39.890
Negative 2 plus negative
3 is negative 5.

00:01:39.890 --> 00:01:41.847
So we could rewrite f of x.

00:01:41.847 --> 00:01:43.180
And so let me write it this way.

00:01:43.180 --> 00:01:47.440
We could write f of
x as being equal to x

00:01:47.440 --> 00:01:51.580
minus 2 times x minus 3.

00:01:51.580 --> 00:01:54.160
Now, how does this help
us find the zeroes?

00:01:54.160 --> 00:01:58.295
Well, in what situations is
this right-hand expression,

00:01:58.295 --> 00:02:00.861
is this expression on the right
hand going to be equal to 0?

00:02:00.861 --> 00:02:02.860
Well, it's the product
of these two expressions.

00:02:02.860 --> 00:02:06.786
If either one of these is equal
to 0, 0 times anything is 0.

00:02:06.786 --> 00:02:08.570
0 times anything else is 0.

00:02:08.570 --> 00:02:13.240
So this whole thing is going to
be 0 if x minus 2 is equal to 0

00:02:13.240 --> 00:02:17.160
or x minus 3 is equal to 0.

00:02:17.160 --> 00:02:19.110
Add 2 to both sides
of this equation.

00:02:19.110 --> 00:02:26.360
You get x is equal to
2 or x is equal to 3.

00:02:26.360 --> 00:02:31.190
So those are the two
zeroes for this function,

00:02:31.190 --> 00:02:32.240
I guess you could say.

00:02:32.240 --> 00:02:33.698
And we could already
think about it

00:02:33.698 --> 00:02:35.300
a little bit in
terms of graphing it.

00:02:37.980 --> 00:02:41.180
So let's try to
graph this thing.

00:02:41.180 --> 00:02:43.020
So this is x equals 1.

00:02:43.020 --> 00:02:44.480
This is x equals 2.

00:02:44.480 --> 00:02:47.860
This is x equals 3
right over there.

00:02:47.860 --> 00:02:49.840
So that's our x-axis.

00:02:49.840 --> 00:02:53.890
That, you could say, is our
y is equal to f of x axis.

00:02:53.890 --> 00:02:59.960
And we're seeing that we
intersect both here and here.

00:02:59.960 --> 00:03:03.590
When x is equal to 2,
this f of x is equal to 0.

00:03:03.590 --> 00:03:06.742
When x is equal to 3,
f of x is equal to 0.

00:03:06.742 --> 00:03:08.700
And you could substitute
either of these values

00:03:08.700 --> 00:03:09.730
into the original expression.

00:03:09.730 --> 00:03:11.229
And you'll see it's
going to get you

00:03:11.229 --> 00:03:13.390
to 0 because that is
the same thing as that.

00:03:13.390 --> 00:03:14.610
Now, what about the vertex?

00:03:14.610 --> 00:03:16.840
What form could we write
this original thing

00:03:16.840 --> 00:03:19.069
in order to pick out the vertex?

00:03:19.069 --> 00:03:20.610
Well, we're already
a little familiar

00:03:20.610 --> 00:03:21.735
with completing the square.

00:03:21.735 --> 00:03:27.520
And when you complete the
square with this expression,

00:03:27.520 --> 00:03:30.470
that seems to be a pretty good
way of thinking about what

00:03:30.470 --> 00:03:32.530
the minimum value
of this function is.

00:03:32.530 --> 00:03:34.690
So let's just do
that right over here.

00:03:34.690 --> 00:03:36.280
So I'm just going to rewrite it.

00:03:36.280 --> 00:03:43.759
So we get f of x is equal
to x squared minus 5x.

00:03:43.759 --> 00:03:46.050
And I'm just going to throw
the plus 6 right over here.

00:03:46.050 --> 00:03:47.990
And I'm giving myself
some real estate

00:03:47.990 --> 00:03:50.750
because what I need to do, what
I want to think about doing,

00:03:50.750 --> 00:03:53.504
is adding and subtracting
the same value.

00:03:53.504 --> 00:03:56.170
So I'm going to add it here, and
I'm going to subtract it there.

00:03:56.170 --> 00:03:58.530
And I can do that because
then I've just added 0.

00:03:58.530 --> 00:04:01.100
I haven't changed the value
of this right-hand side.

00:04:01.100 --> 00:04:04.660
But I want to do that so that
this part that I've underlined

00:04:04.660 --> 00:04:07.280
in this magenta color, so that
this part right over here,

00:04:07.280 --> 00:04:08.720
is a perfect square.

00:04:08.720 --> 00:04:10.160
And we've done
this multiple times

00:04:10.160 --> 00:04:11.270
when we've completed the square.

00:04:11.270 --> 00:04:12.730
I encourage you to
watch those videos

00:04:12.730 --> 00:04:14.490
if you need a little
bit of a review on it.

00:04:14.490 --> 00:04:16.050
But the general idea
is this is going

00:04:16.050 --> 00:04:19.230
to be a perfect
square if we take

00:04:19.230 --> 00:04:21.410
this coefficient
right over here.

00:04:21.410 --> 00:04:23.170
We take negative 5.

00:04:23.170 --> 00:04:28.220
We take 1/2 of that, which is
negative 5/2, and we square it.

00:04:28.220 --> 00:04:31.840
So we could write this
as plus negative--

00:04:31.840 --> 00:04:34.090
what's negative 5/2 squared?

00:04:34.090 --> 00:04:37.740
So I could write this--
negative 5/2 squared.

00:04:37.740 --> 00:04:39.320
Well, if we square
a negative number,

00:04:39.320 --> 00:04:40.695
it's just going
to be a positive.

00:04:40.695 --> 00:04:42.800
So it's going to be the
same thing as 5/2 squared.

00:04:42.800 --> 00:04:44.510
5 squared is 25.

00:04:44.510 --> 00:04:45.940
2 squared is 4.

00:04:45.940 --> 00:04:47.470
So this is going
to be plus 25/4.

00:04:51.520 --> 00:04:54.410
Now, once again, if we want
this equality to be true,

00:04:54.410 --> 00:04:56.830
we either have to add the
same thing to both sides.

00:04:56.830 --> 00:04:58.531
Or if we're just
operating on one side,

00:04:58.531 --> 00:05:00.280
if we added it to that
side, we could just

00:05:00.280 --> 00:05:01.820
subtract it from that side.

00:05:01.820 --> 00:05:04.950
And we haven't changed the
total value on that side.

00:05:04.950 --> 00:05:07.850
So we added 25/4, and
we subtracted 25/4.

00:05:10.460 --> 00:05:12.492
So what is this part
right over here?

00:05:12.492 --> 00:05:14.200
What does this become,
the part that I've

00:05:14.200 --> 00:05:15.117
underlined in magenta?

00:05:15.117 --> 00:05:17.116
Well, this is going to
be-- the whole reason why

00:05:17.116 --> 00:05:18.950
we engineered it in
this way is so that this

00:05:18.950 --> 00:05:25.140
could be x minus 5/2 squared.

00:05:25.140 --> 00:05:26.680
And I encourage
you to verify this.

00:05:26.680 --> 00:05:29.140
And we go into more
detail about why

00:05:29.140 --> 00:05:32.209
taking 1/2 the coefficient here
and then squaring it, adding it

00:05:32.209 --> 00:05:34.250
there and then subtracting
there, why that works.

00:05:34.250 --> 00:05:36.740
We do that in the completing
the square videos.

00:05:36.740 --> 00:05:38.369
But these two things,
you can verify

00:05:38.369 --> 00:05:39.410
that they are equivalent.

00:05:39.410 --> 00:05:41.050
So that's that part.

00:05:41.050 --> 00:05:43.160
And now we just have to
simplify 6 minus 25/4.

00:05:46.290 --> 00:05:50.210
So 6 could be rewritten as 24/4.

00:05:50.210 --> 00:05:58.670
24/4 minus 25/4 is negative 1/4,
so minus 1/4, just like that.

00:05:58.670 --> 00:06:01.960
So we've rewritten
our original function

00:06:01.960 --> 00:06:08.600
as f of x is equal to x
minus 5/2 squared minus 1/4.

00:06:08.600 --> 00:06:12.400
Now, why is this
form interesting?

00:06:12.400 --> 00:06:15.180
Well, one way to think
about it is this part

00:06:15.180 --> 00:06:17.790
is always going to
be non-negative.

00:06:17.790 --> 00:06:21.401
The minimum value of this part
in magenta is going to be 0.

00:06:21.401 --> 00:06:21.900
Why?

00:06:21.900 --> 00:06:23.700
Because we're
squaring this thing.

00:06:23.700 --> 00:06:25.310
If you're taking
something like this--

00:06:25.310 --> 00:06:27.060
and we're just dealing
with real numbers--

00:06:27.060 --> 00:06:29.000
and you're squaring
it, you're not

00:06:29.000 --> 00:06:31.090
going to be able to
get a negative value.

00:06:31.090 --> 00:06:35.460
At the minimum value,
this is going to be 0.

00:06:35.460 --> 00:06:37.962
And then it obviously could
be positive values, as well.

00:06:37.962 --> 00:06:40.170
So if we want to think about
when does this thing hit

00:06:40.170 --> 00:06:43.010
its minimum value-- well,
it hits its minimum value

00:06:43.010 --> 00:06:44.770
when you're squaring 0.

00:06:44.770 --> 00:06:46.240
And when are you squaring 0?

00:06:46.240 --> 00:06:51.510
Well, you're squaring 0 when
x minus 5/2 is equal to 0,

00:06:51.510 --> 00:06:54.660
or when x is equal
to 5/2 if you just

00:06:54.660 --> 00:06:58.250
want to add 5/2 to both
sides of that equation.

00:06:58.250 --> 00:07:01.970
So this thing hits its minimum
value when x is equal to 5/2.

00:07:01.970 --> 00:07:06.710
And then what is y, or what is
f of x, when x is equal to 5/2?

00:07:06.710 --> 00:07:10.920
f of 5/2-- and once again, you
could use any of those forms

00:07:10.920 --> 00:07:12.190
to evaluate 5/2.

00:07:12.190 --> 00:07:14.310
But it's really
easy in this form.

00:07:14.310 --> 00:07:17.710
When x is equal to 5/2, this
term right over here becomes 0.

00:07:17.710 --> 00:07:19.350
0 squared, 0.

00:07:19.350 --> 00:07:21.070
You're just left
with negative 1/4.

00:07:25.180 --> 00:07:27.170
So another way to think
about it is our vertex

00:07:27.170 --> 00:07:35.500
is at the point x equals
5/2, y equals negative 1/4.

00:07:35.500 --> 00:07:36.900
So x equals 5/2.

00:07:36.900 --> 00:07:38.970
That's the same
thing as 2 and 1/2.

00:07:38.970 --> 00:07:41.710
So x equals 5/2.

00:07:41.710 --> 00:07:44.500
And y is equal to negative 1/4.

00:07:44.500 --> 00:07:48.830
So if that is negative 1, 1/4
would be something like that.

00:07:48.830 --> 00:07:50.700
So that right over
there is the vertex.

00:07:50.700 --> 00:07:52.450
That is the point--
let me make it clear--

00:07:52.450 --> 00:07:57.830
that's the point 5/2
comma negative 1/4.

00:07:57.830 --> 00:07:59.804
And what's cool is we've
just used this form

00:07:59.804 --> 00:08:01.720
to figure out the minimum
point, to figure out

00:08:01.720 --> 00:08:03.550
the vertex in this case.

00:08:03.550 --> 00:08:07.350
And then we can use the
roots as two other points

00:08:07.350 --> 00:08:14.570
to get a rough sketch of what
this parabola will actually

00:08:14.570 --> 00:08:15.375
look like.

00:08:15.375 --> 00:08:17.875
So the interesting-- or I guess
the takeaway from this video

00:08:17.875 --> 00:08:21.310
is just to realize that we can
rewrite this in different forms

00:08:21.310 --> 00:08:26.000
depending on what we're
trying to understand

00:08:26.000 --> 00:08:28.550
about this function.


Simplifying with exponent properties

WEBVTT
Kind: captions
Language: en

00:00:00.810 --> 00:00:03.180
Let's see if we can
simplify 6 to the 1/2

00:00:03.180 --> 00:00:07.540
power times the fifth
root of 6 and all

00:00:07.540 --> 00:00:08.926
of that to the third power.

00:00:08.926 --> 00:00:10.550
And I encourage you
to pause this video

00:00:10.550 --> 00:00:12.650
and try it on your own.

00:00:12.650 --> 00:00:14.810
So let me actually color
code these exponents, just

00:00:14.810 --> 00:00:16.684
so we can keep track of
them a little better.

00:00:16.684 --> 00:00:18.950
So that's the 1/2 power in blue.

00:00:18.950 --> 00:00:22.750
This is the fifth
root here in magenta.

00:00:22.750 --> 00:00:23.660
And let's see.

00:00:23.660 --> 00:00:26.810
In green, let's think
about this third power.

00:00:26.810 --> 00:00:30.050
So one way to think
about this fifth root

00:00:30.050 --> 00:00:31.910
is that this is the
exact same thing

00:00:31.910 --> 00:00:35.560
as raising this 6
to the 1/5 power,

00:00:35.560 --> 00:00:37.140
so let's write it like that.

00:00:37.140 --> 00:00:40.460
So this part right over
here, we could rewrite as 6

00:00:40.460 --> 00:00:45.270
to the 1/5 power, and
then that whole thing

00:00:45.270 --> 00:00:50.520
gets raised to the third power.

00:00:50.520 --> 00:00:53.390
And of course, we have
this 6 to the 1/2 power

00:00:53.390 --> 00:00:59.870
out here, 6 to the 1/2 power
times all of this business

00:00:59.870 --> 00:01:01.290
right over here.

00:01:01.290 --> 00:01:04.319
Now, what happens if we raise
something to an exponent

00:01:04.319 --> 00:01:06.694
and then raise that whole
thing to another exponent?

00:01:06.694 --> 00:01:09.110
Well we've already seen in our
exponent properties, that's

00:01:09.110 --> 00:01:12.880
the equivalent of raising this
to the product of these two

00:01:12.880 --> 00:01:14.080
exponents.

00:01:14.080 --> 00:01:16.720
So this part right over
here could be rewritten

00:01:16.720 --> 00:01:27.850
as 6 to the-- 3 times 1/5
is 3/5-- 6 to the 3/5 power.

00:01:32.450 --> 00:01:36.100
And of course, we're multiplying
that times 6 to the 1/2 power.

00:01:36.100 --> 00:01:42.060
6 to the 1/2 power times
6 to the 3/5 power.

00:01:42.060 --> 00:01:43.810
And now, if you're
multiplying some base

00:01:43.810 --> 00:01:45.980
to this exponent and
then the same base

00:01:45.980 --> 00:01:47.750
again to another
exponent, we know

00:01:47.750 --> 00:01:49.450
that this is going
to be the same thing.

00:01:49.450 --> 00:01:51.440
And actually we could put these
equal signs the whole way,

00:01:51.440 --> 00:01:53.130
because these all
equal each other.

00:01:53.130 --> 00:01:58.050
This is the same thing as
6 being raised to the 1/2

00:01:58.050 --> 00:02:06.012
plus 3/5 power,
1/2 plus 3 over 5.

00:02:06.012 --> 00:02:09.232
Now, what's 1/2 plus 3 over 5?

00:02:09.232 --> 00:02:11.410
Well, we could find
a common denominator.

00:02:11.410 --> 00:02:15.440
It would be 10, so that's
the same thing as-- actually

00:02:15.440 --> 00:02:18.030
let me just write it this way--
this is the same thing as 6

00:02:18.030 --> 00:02:22.600
to the-- instead of 1/2,
we can write it as 5/10.

00:02:22.600 --> 00:02:30.200
Plus 3/5 is the same
thing as 6/10 power, which

00:02:30.200 --> 00:02:32.750
is the same thing-- and
we deserve a little bit

00:02:32.750 --> 00:02:36.820
of a drum roll here, this wasn't
that long of a problem-- 6

00:02:36.820 --> 00:02:40.770
to the 11/10 power.

00:02:40.770 --> 00:02:45.470
I'll just write it
all, 11/10 power.

00:02:45.470 --> 00:02:48.980
And so, that looks
pretty simplified to me.

00:02:48.980 --> 00:02:50.870
I guess we're done.


Exponential equation with rational answer

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:04.600
So right here, we've got 3 to
the a power-- or a-th power,

00:00:04.600 --> 00:00:05.286
I guess.

00:00:05.286 --> 00:00:07.160
I don't want to confuse
it with the number 8.

00:00:07.160 --> 00:00:14.010
3 to the a power is equal to
the fifth root of 3 squared.

00:00:14.010 --> 00:00:17.110
And what we need to figure out
is what would a be equal to.

00:00:17.110 --> 00:00:18.850
Let's solve for a.

00:00:18.850 --> 00:00:21.640
And I encourage you right
now to pause this video

00:00:21.640 --> 00:00:24.082
and try it on your own.

00:00:24.082 --> 00:00:26.040
Well, if you have a fifth
root right over here,

00:00:26.040 --> 00:00:27.498
one thing that you
might be tempted

00:00:27.498 --> 00:00:30.524
to do to undo the fifth root is
to raise it to the fifth power.

00:00:30.524 --> 00:00:32.940
And of course, we can't just
raise one side of an equation

00:00:32.940 --> 00:00:33.670
to the fifth power.

00:00:33.670 --> 00:00:35.128
Whatever we do to
one side, we have

00:00:35.128 --> 00:00:38.430
to do the other side if we
want this to still be equal.

00:00:38.430 --> 00:00:41.460
So let's raise both
sides of this equation

00:00:41.460 --> 00:00:42.540
to the fifth power.

00:00:45.860 --> 00:00:47.410
Now, this left-hand
side-- we just

00:00:47.410 --> 00:00:49.785
have to remember a little bit
of our exponent properties.

00:00:49.785 --> 00:00:52.110
3a to the fifth power.

00:00:52.110 --> 00:00:53.759
And if we want to
just remind ourselves

00:00:53.759 --> 00:00:55.800
where that comes from,
that's the same thing as 3

00:00:55.800 --> 00:01:01.310
to the a times 3 to the a
times 3 to the a times 3

00:01:01.310 --> 00:01:03.585
to the a times 3 to the a.

00:01:03.585 --> 00:01:05.330
Well, what's that
going to be equal to?

00:01:05.330 --> 00:01:10.980
That's going to be 3 to
the a plus a plus a plus a

00:01:10.980 --> 00:01:17.870
plus a power, which is the same
thing as 3 to the 5a power.

00:01:17.870 --> 00:01:19.590
So the exponent
property here is if you

00:01:19.590 --> 00:01:22.290
raise a base to some
exponent and then raise

00:01:22.290 --> 00:01:24.340
that whole thing to
another exponent, that's

00:01:24.340 --> 00:01:27.300
the equivalent of raising
the base to an exponent that

00:01:27.300 --> 00:01:29.680
is the product of
these two exponents.

00:01:29.680 --> 00:01:38.990
So we can rewrite this left-hand
side as 3 to the 5a power

00:01:38.990 --> 00:01:42.740
is going to be equal to-- Well,
if you take something that's

00:01:42.740 --> 00:01:45.030
a fifth root and you raise
it to the fifth power,

00:01:45.030 --> 00:01:48.330
then you're just left with
what you had under the radical.

00:01:48.330 --> 00:01:51.810
That's going to be
equal to 3 squared.

00:01:51.810 --> 00:01:53.290
So now things become
a lot clearer.

00:01:53.290 --> 00:01:58.619
3 to the 5a needs to
be equal to 3 squared.

00:01:58.619 --> 00:02:00.160
Or another way of
thinking about it--

00:02:00.160 --> 00:02:01.980
we have the same
base on both sides.

00:02:01.980 --> 00:02:05.540
So this exponent needs to be
equal to this exponent right

00:02:05.540 --> 00:02:06.440
over there.

00:02:06.440 --> 00:02:13.260
Or we could write that 5 times
a needs to be equal to 2.

00:02:13.260 --> 00:02:16.830
And of course, now we can
just divide both sides by 5.

00:02:16.830 --> 00:02:21.226
And we get a is equal to 2/5.

00:02:27.170 --> 00:02:28.830
And this is an
interesting result.

00:02:28.830 --> 00:02:30.960
And what's neat
about this example--

00:02:30.960 --> 00:02:32.490
it kind of shows
you the motivation

00:02:32.490 --> 00:02:35.370
for how we define
rational exponents.

00:02:35.370 --> 00:02:40.130
So let's just put this back
into the original expression.

00:02:40.130 --> 00:02:41.380
We've just solved for a.

00:02:41.380 --> 00:02:46.205
And we've gotten that 3 to
the 2/5 power-- and actually,

00:02:46.205 --> 00:02:47.580
let me color code
it a little bit

00:02:47.580 --> 00:02:49.210
because I think
that'll be interesting.

00:02:49.210 --> 00:03:01.510
3 to the 2/5 power is
equal to the fifth root--

00:03:01.510 --> 00:03:03.830
notice, the fifth root.

00:03:03.830 --> 00:03:06.340
So the denominator
here, that's the root.

00:03:06.340 --> 00:03:12.040
So the fifth root of 3 squared.

00:03:14.820 --> 00:03:17.610
So if you take this
base 3, you square it,

00:03:17.610 --> 00:03:19.410
but then you take the
fifth root of that,

00:03:19.410 --> 00:03:22.030
that's the same thing as
raising it to the 2/5 power.

00:03:22.030 --> 00:03:22.530
Notice.

00:03:22.530 --> 00:03:26.020
Take this 3, take it
to the second power,

00:03:26.020 --> 00:03:27.720
and then you find
the fifth root of it.

00:03:27.720 --> 00:03:31.790
Or if you use this property that
we just saw right over here,

00:03:31.790 --> 00:03:34.235
you could rewrite this.

00:03:34.235 --> 00:03:39.955
This is the same
thing as 3 squared.

00:03:42.480 --> 00:03:47.010
And then you raise
that to the 1/5 power.

00:03:47.010 --> 00:03:49.010
We saw that property
at play over here.

00:03:49.010 --> 00:03:50.880
You could just multiply
these two exponents.

00:03:50.880 --> 00:03:52.770
You'd get 3 to the 2/5 power.

00:03:52.770 --> 00:03:54.720
And that's the same
thing as 3 squared

00:03:54.720 --> 00:03:56.260
and then find the
fifth root of it.

00:03:56.260 --> 00:03:58.100
3 squared, and then
you're essentially

00:03:58.100 --> 00:04:00.731
finding the fifth root of it.


Quadratic inequality word problem

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:03.940
Lisa owns a "Random
Candy" vending machine,

00:00:03.940 --> 00:00:06.030
which is a machine
that picks a candy out

00:00:06.030 --> 00:00:08.910
of an assortment in
a random fashion.

00:00:08.910 --> 00:00:12.750
Lisa controls the probability
in which each candy is picked.

00:00:12.750 --> 00:00:15.160
She is running out
of "Honey Bunny,"

00:00:15.160 --> 00:00:17.700
so she wants to
program its probability

00:00:17.700 --> 00:00:21.000
so that the probability of
getting a different candy twice

00:00:21.000 --> 00:00:25.390
in a row is greater than 2
and 1/4 times the probability

00:00:25.390 --> 00:00:27.415
of getting "Honey
Bunny" in one try.

00:00:27.415 --> 00:00:28.940
So let me read that again.

00:00:28.940 --> 00:00:31.080
She wants to program
its probability

00:00:31.080 --> 00:00:33.670
so that the probability of
getting a different candy twice

00:00:33.670 --> 00:00:37.170
in a row, or really any
other candy twice in a row,

00:00:37.170 --> 00:00:41.450
is greater than 2 and 1/4 times
the probability of getting

00:00:41.450 --> 00:00:44.510
"Honey Bunny" in one try.

00:00:44.510 --> 00:00:48.430
Write an inequality that
models the situation.

00:00:48.430 --> 00:00:50.730
Use p to represent
the probability

00:00:50.730 --> 00:00:53.210
of getting "Honey
Bunny" in one try.

00:00:53.210 --> 00:00:55.994
Solve the inequality, and
complete the sentence.

00:00:55.994 --> 00:00:57.410
Remember that the
probability must

00:00:57.410 --> 00:00:59.140
be a number between 0 and 1.

00:00:59.140 --> 00:01:01.020
So we want to write
the inequality

00:01:01.020 --> 00:01:02.701
that models the problem here.

00:01:02.701 --> 00:01:04.450
And then we want to
complete the sentence,

00:01:04.450 --> 00:01:06.574
the probability of getting
"Honey Bunny" in one try

00:01:06.574 --> 00:01:08.800
must be-- so they give
us a bunch of options.

00:01:08.800 --> 00:01:11.300
Greater than, greater than or
equal to, less than, less than

00:01:11.300 --> 00:01:14.110
or equal to, and then we
have to put some number here.

00:01:14.110 --> 00:01:16.900
So to work through this, I've
copy and pasted this problem

00:01:16.900 --> 00:01:19.640
onto my little scratch
pad right over here.

00:01:19.640 --> 00:01:22.010
And so let's just think
about it a little bit.

00:01:22.010 --> 00:01:24.780
So they tell us,
use p to represent

00:01:24.780 --> 00:01:27.530
the probability of getting
"Honey Bunny" in one try.

00:01:27.530 --> 00:01:30.200
And they also say she wants
to program its probability so

00:01:30.200 --> 00:01:34.060
that the probability of
getting a different candy twice

00:01:34.060 --> 00:01:38.760
in a row is greater than 2 and
1/4 the probability of getting

00:01:38.760 --> 00:01:41.300
"Honey Bunny" in one try.

00:01:41.300 --> 00:01:46.800
So if p is the probability of
getting "Honey Bunny," what's

00:01:46.800 --> 00:01:51.980
the probability of getting
any other candy at once?

00:01:51.980 --> 00:01:55.569
Well, that's going
to be 1 minus p.

00:01:55.569 --> 00:01:57.860
If you have a probability of
p of getting "Honey Bunny,

00:01:57.860 --> 00:02:01.289
well, then it's 1 minus p of
anything but "Honey Bunny."

00:02:01.289 --> 00:02:03.330
Now, what's the probability
of getting this twice

00:02:03.330 --> 00:02:05.990
in a row, of getting
anything else twice in a row?

00:02:05.990 --> 00:02:07.610
Well, you're just
going to multiply

00:02:07.610 --> 00:02:09.280
this probability times itself.

00:02:09.280 --> 00:02:12.290
It's going to be 1
minus p times 1 minus p,

00:02:12.290 --> 00:02:16.560
or we could just write
that as 1 minus p squared.

00:02:16.560 --> 00:02:19.790
So this right over
here is the probability

00:02:19.790 --> 00:02:22.320
of getting a different
candy, any other candy,

00:02:22.320 --> 00:02:23.670
twice in a row.

00:02:23.670 --> 00:02:34.840
So prob of any other non-"Honey
Bunny" candy, any other candy,

00:02:34.840 --> 00:02:39.730
twice in a row.

00:02:39.730 --> 00:02:45.170
Now, they tell us
that this probability

00:02:45.170 --> 00:02:51.800
needs to be greater than 2
and 1/4 times the probability

00:02:51.800 --> 00:02:54.780
of getting "Honey
Bunny" in one try.

00:02:54.780 --> 00:03:00.312
So is greater than
2 and 1/4 times

00:03:00.312 --> 00:03:02.270
the probability of getting
"Honey Bunny" in one

00:03:02.270 --> 00:03:05.610
try, well, that is p-- times p.

00:03:05.610 --> 00:03:07.860
So we have just set
up the first part.

00:03:07.860 --> 00:03:11.710
We have written an inequality
that models the situation.

00:03:11.710 --> 00:03:15.440
Now let's actually
solve this inequality.

00:03:15.440 --> 00:03:19.200
And so to do that, I will just
expand 1 minus p squared out.

00:03:19.200 --> 00:03:24.307
1 minus p squared is the
same thing as-- well,

00:03:24.307 --> 00:03:25.390
I'll just multiply it out.

00:03:25.390 --> 00:03:34.600
So this is going to be 1
squared minus 2p plus p squared.

00:03:34.600 --> 00:03:39.010
And that's going to be
greater than 2 and 1/4 p.

00:03:42.720 --> 00:03:43.320
Now let's see.

00:03:43.320 --> 00:03:49.900
If we subtract 2 and
1/4 p from both sides,

00:03:49.900 --> 00:03:52.840
we're going to be left with--
and I'm going to reorder this.

00:03:52.840 --> 00:03:54.650
We're going to get p squared.

00:03:54.650 --> 00:03:58.280
So you have minus 2p
minus 2 and 1/4 p,

00:03:58.280 --> 00:04:01.190
so that's going to get
us minus 4 and 1/4 p.

00:04:01.190 --> 00:04:08.350
Or let me just write
that as 17/4 p plus 1 is

00:04:08.350 --> 00:04:11.880
greater than 0.

00:04:14.730 --> 00:04:21.160
And so let's think about solving
this quadratic right over here.

00:04:21.160 --> 00:04:24.350
And under which circumstances
is this greater than 0?

00:04:24.350 --> 00:04:26.354
To think about it,
let's factor it.

00:04:26.354 --> 00:04:27.770
And actually,
before we factor it,

00:04:27.770 --> 00:04:29.410
let's simplify it a little bit.

00:04:29.410 --> 00:04:32.050
I don't like having this
17/4 right over here,

00:04:32.050 --> 00:04:33.850
so let's multiply
both sides times 4.

00:04:33.850 --> 00:04:35.530
And since 4 is a
positive number,

00:04:35.530 --> 00:04:37.650
it's not going to
change the sign,

00:04:37.650 --> 00:04:39.830
the direction of
this inequality.

00:04:39.830 --> 00:04:42.870
So we could rewrite
this as 4p squared

00:04:42.870 --> 00:04:52.165
minus 17p plus 4
is greater than 0.

00:04:55.347 --> 00:04:55.930
And let's see.

00:04:55.930 --> 00:04:57.147
What are the roots of this?

00:04:57.147 --> 00:04:58.730
And we could use the
quadratic formula

00:04:58.730 --> 00:05:00.188
if we wanted to do
it really quick.

00:05:00.188 --> 00:05:02.160
We could probably
do it other ways.

00:05:02.160 --> 00:05:06.240
But negative b-- so it's
going to be 17 plus or minus

00:05:06.240 --> 00:05:16.810
the square root of negative 17
squared-- b squared-- so that's

00:05:16.810 --> 00:05:20.910
289 minus 4 times a times c.

00:05:20.910 --> 00:05:25.280
Well a times c is 16
times 4, so minus 64.

00:05:25.280 --> 00:05:29.170
All of that over 2 times
a-- all of that over 8.

00:05:29.170 --> 00:05:32.040
So that's 17 plus
or minus-- let's

00:05:32.040 --> 00:05:37.780
see, this is the square
root of 225 over 8,

00:05:37.780 --> 00:05:44.440
which is equal to 17
plus or minus 15 over 8,

00:05:44.440 --> 00:05:49.370
which is equal to-- let's
see, 17 minus 15 over 8

00:05:49.370 --> 00:05:53.250
is 2/8-- which is
equal to 2/8 or 1/4.

00:05:53.250 --> 00:05:54.750
So that's one of them.

00:05:54.750 --> 00:05:56.520
That's when we take the minus.

00:05:56.520 --> 00:06:03.290
And if we add 17 plus 15, that
gets us to 32 divided by 8

00:06:03.290 --> 00:06:06.600
is 4.

00:06:06.600 --> 00:06:09.990
So there's two situations
right over here.

00:06:09.990 --> 00:06:11.670
Let's factor this out.

00:06:11.670 --> 00:06:22.130
We could write this as p
minus 1/4 times p minus 4

00:06:22.130 --> 00:06:27.650
is greater than 0.

00:06:27.650 --> 00:06:32.070
So under what circumstances
is this going to be true?

00:06:32.070 --> 00:06:35.280
What constraints are
this going to be true?

00:06:35.280 --> 00:06:38.630
Well if you're taking
the product of two terms

00:06:38.630 --> 00:06:41.840
and they are going
to be greater than 0,

00:06:41.840 --> 00:06:45.800
that means that these two
things have to be the same sign.

00:06:45.800 --> 00:06:49.230
Or actually, in particular,
they both have to be positive,

00:06:49.230 --> 00:06:51.870
or they both have
to be negative.

00:06:51.870 --> 00:06:53.990
So let's look at
those two situations.

00:06:53.990 --> 00:06:56.660
And I'll switch colors
here just for fun--

00:06:56.660 --> 00:07:02.530
so both positive
or both negative.

00:07:02.530 --> 00:07:08.200
So if they are both
positive-- let me do it here.

00:07:08.200 --> 00:07:12.120
That means that p minus 1/4
has to be greater than 0,

00:07:12.120 --> 00:07:16.630
and p minus 4 is greater than 0.

00:07:16.630 --> 00:07:19.110
Add 1/4 on both sides
right over here.

00:07:19.110 --> 00:07:30.770
You get p is greater than
1/4, and p is greater than 4.

00:07:30.770 --> 00:07:34.650
So that's the situation
where they are both positive.

00:07:34.650 --> 00:07:37.000
Now, what about if
they are both negative?

00:07:37.000 --> 00:07:47.380
Well then you have p minus 1/4
is less than 0, and p minus 4

00:07:47.380 --> 00:07:48.580
is less than 0.

00:07:48.580 --> 00:07:50.240
Add 1/4 here.

00:07:50.240 --> 00:08:01.690
So p needs to be less than 1/4,
and p needs to be less than 4.

00:08:01.690 --> 00:08:07.320
Now, what does this
constraint simplify to?

00:08:07.320 --> 00:08:12.160
p has to be greater than 1/4,
and p has to be greater than 4.

00:08:12.160 --> 00:08:14.680
Well, if p is
greater than 4, it's

00:08:14.680 --> 00:08:16.610
definitely going to
be greater than 1/4.

00:08:16.610 --> 00:08:22.140
So all of this collapses into
p needs to be greater than 4.

00:08:22.140 --> 00:08:24.230
That's the situation
where both are positive.

00:08:24.230 --> 00:08:26.430
p must be greater than 4.

00:08:26.430 --> 00:08:28.230
Now what about here?

00:08:28.230 --> 00:08:31.140
Well if p is less than
1/4, it's definitely

00:08:31.140 --> 00:08:32.740
going to be less than four.

00:08:32.740 --> 00:08:36.580
And this is an and right over
here, so this collapses to p

00:08:36.580 --> 00:08:41.140
is a less than 1/4.

00:08:41.140 --> 00:08:44.260
So which one do we go with--
p needs to be greater than 4,

00:08:44.260 --> 00:08:47.406
or p needs to be less than 1/4?

00:08:47.406 --> 00:08:48.780
Well, we need to
remind ourselves

00:08:48.780 --> 00:08:50.290
that we're talking
about a probability.

00:08:50.290 --> 00:08:51.748
To go back to the
original problem,

00:08:51.748 --> 00:08:53.650
we're talking about a
probability of someone

00:08:53.650 --> 00:08:55.025
getting "Honey
Bunny" in one try.

00:08:55.025 --> 00:08:57.990
A probability must
be between 0 and 1,

00:08:57.990 --> 00:09:01.150
so the probability having
to be greater than 4,

00:09:01.150 --> 00:09:03.120
well that just doesn't
make any sense.

00:09:03.120 --> 00:09:06.310
That doesn't make any sense in
the context of this question.

00:09:06.310 --> 00:09:09.140
So we have to go with the
probability of getting

00:09:09.140 --> 00:09:14.510
"Honey Bunny" needs to be less
than 1/4, or less than 0.25,

00:09:14.510 --> 00:09:16.200
which makes complete sense.

00:09:16.200 --> 00:09:18.540
So let's fill in
this information.

00:09:18.540 --> 00:09:23.840
This was the inequality
that modeled the problem.

00:09:23.840 --> 00:09:27.010
And we got p has to
be less than 1/4.

00:09:27.010 --> 00:09:29.670
So let's go back to
the original problem.

00:09:29.670 --> 00:09:38.030
The inequality was 1
minus p squared needs

00:09:38.030 --> 00:09:41.750
to be greater than 2 and 1/4.

00:09:41.750 --> 00:09:43.370
So we could write
that multiple times.

00:09:43.370 --> 00:09:47.792
I could write that
as 2.25 times p.

00:09:47.792 --> 00:09:49.250
And then the
probability of getting

00:09:49.250 --> 00:09:57.750
"Honey Bunny" in one try
must be less than 0.25.

00:09:57.750 --> 00:09:59.640
And we're done.


Making more pizzas to spread cost per pizza

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:03.390
Dominique from Dominique's
Pizza bakes the same amount

00:00:03.390 --> 00:00:04.640
of pizza every day.

00:00:04.640 --> 00:00:07.810
She used to spend $8 each
day on using the oven,

00:00:07.810 --> 00:00:11.790
and $1.50 on ingredients
for each pizza.

00:00:11.790 --> 00:00:13.840
So $8 each day on
using the oven,

00:00:13.840 --> 00:00:17.190
and $1.50 on ingredients
for each pizza.

00:00:17.190 --> 00:00:20.350
One day the price for the
ingredients increased from

00:00:20.350 --> 00:00:23.300
$1.50 to $2 per pizza.

00:00:23.300 --> 00:00:25.150
Dominique made some
calculations and found

00:00:25.150 --> 00:00:28.200
that she should bake
8 pizzas more each day

00:00:28.200 --> 00:00:32.710
so the expenses for a single
pizza would remain the same.

00:00:32.710 --> 00:00:34.800
And assume they're
saying the total expenses

00:00:34.800 --> 00:00:38.350
for a single pizza, because
clearly the ingredients

00:00:38.350 --> 00:00:39.330
cost is not the same.

00:00:39.330 --> 00:00:41.080
We're talking about
the total ingredients.

00:00:41.080 --> 00:00:43.240
So if we were to spread
the cost of the oven

00:00:43.240 --> 00:00:45.640
across all of the pizzas.

00:00:45.640 --> 00:00:48.850
Write an equation to
find-- or the total cost

00:00:48.850 --> 00:00:52.130
for the oven per day, to spread
that across all the pizzas.

00:00:52.130 --> 00:00:56.030
Write an equation to find out
how many pizzas Dominique baked

00:00:56.030 --> 00:00:58.590
each day before the
change in price.

00:00:58.590 --> 00:01:01.720
Use p to represent
the number of pizzas.

00:01:01.720 --> 00:01:05.730
So let's just think about her
total cost per pizza before

00:01:05.730 --> 00:01:07.360
and then her total
cost per pizza

00:01:07.360 --> 00:01:11.260
after if she bakes
8 more pizzas.

00:01:11.260 --> 00:01:13.470
So before, we're
going to use p to say

00:01:13.470 --> 00:01:15.080
that's the number
of pizzas she baked

00:01:15.080 --> 00:01:17.560
per day before the
change in price.

00:01:17.560 --> 00:01:20.850
So before the change in
price, on a given day,

00:01:20.850 --> 00:01:25.070
she would spend $8 on the oven
and then $1.50 on ingredients

00:01:25.070 --> 00:01:25.970
for each pizza.

00:01:25.970 --> 00:01:29.760
So 1.5, or $1.50, times
the number of pizzas.

00:01:29.760 --> 00:01:32.810
This would be her total cost
on all the pizzas in that day.

00:01:32.810 --> 00:01:36.350
It's the oven cost plus
it's the ingredients cost.

00:01:36.350 --> 00:01:39.510
So if you wanted this
on a per pizza basis,

00:01:39.510 --> 00:01:43.580
you would just divide
by the number of pizzas.

00:01:43.580 --> 00:01:45.460
Now let's think
about what happens

00:01:45.460 --> 00:01:47.570
after the change in price.

00:01:47.570 --> 00:01:50.150
After the change in price,
her cost per day for the oven

00:01:50.150 --> 00:01:52.290
is still $8.

00:01:52.290 --> 00:01:55.770
But now she has to spend $2
per pizza on ingredients.

00:01:55.770 --> 00:01:58.250
So $2 per pizza.

00:01:58.250 --> 00:02:01.610
And instead of saying that
she's baking p pizzas,

00:02:01.610 --> 00:02:04.770
let's say that she's now
baking 8 more pizzas each day.

00:02:04.770 --> 00:02:09.604
So it's going to be p plus 8.

00:02:09.604 --> 00:02:11.645
And so this is going to
be her total cost for all

00:02:11.645 --> 00:02:13.840
of the pizzas she's now baking.

00:02:13.840 --> 00:02:16.830
And so if you want it on
a per pizza basis, well,

00:02:16.830 --> 00:02:18.890
she's now making
p plus 8 pizzas,

00:02:18.890 --> 00:02:21.220
you would divide by p plus 8.

00:02:21.220 --> 00:02:24.230
And the problem tells
us that these two things

00:02:24.230 --> 00:02:25.820
are equivalent.

00:02:25.820 --> 00:02:29.180
Here you had a higher cost
in ingredients per pizzas,

00:02:29.180 --> 00:02:32.850
but since you are now
baking more pizzas,

00:02:32.850 --> 00:02:37.670
you're spreading the oven cost
amongst more and more pizzas.

00:02:37.670 --> 00:02:40.340
So let's think about
what p has to be.

00:02:40.340 --> 00:02:43.490
p has to be some number,
some number of pizzas,

00:02:43.490 --> 00:02:46.100
so that these two
expressions are equal.

00:02:46.100 --> 00:02:49.660
Her total cost per pizza
before, when she only made p,

00:02:49.660 --> 00:02:52.380
is going to be the same as
her total cost per pizza

00:02:52.380 --> 00:02:55.760
when she's making
p plus 8 pizzas.

00:02:55.760 --> 00:02:57.787
So these two things
need to be equal.

00:02:57.787 --> 00:03:00.120
So we did that first part,
or we did what they asked us.

00:03:00.120 --> 00:03:03.320
We wrote an equation to find out
how many pizzas Dominique baked

00:03:03.320 --> 00:03:05.150
each day before the
change in price.

00:03:05.150 --> 00:03:07.430
And we used p to represent
the number of pizzas.

00:03:07.430 --> 00:03:11.330
But now for fun, let's
actually just solve for p.

00:03:11.330 --> 00:03:14.780
So let's just simplify
things a little bit.

00:03:14.780 --> 00:03:16.870
So this part right over here.

00:03:16.870 --> 00:03:20.860
Actually, let's just cross
multiply this on both sides.

00:03:20.860 --> 00:03:23.710
Or another way of thinking
is multiply both sides times

00:03:23.710 --> 00:03:27.150
p plus 8 and multiply
both sides times p.

00:03:27.150 --> 00:03:31.430
So if we multiply by p plus
8, and we multiply by p,

00:03:31.430 --> 00:03:35.870
we multiply by p plus
8, and we multiply

00:03:35.870 --> 00:03:38.656
by p, that cancels with that.

00:03:38.656 --> 00:03:40.260
That cancels with that.

00:03:40.260 --> 00:03:43.140
On the left-hand
side-- so let's see.

00:03:43.140 --> 00:03:45.230
We have to just do the
distributive property

00:03:45.230 --> 00:03:46.610
twice right over here.

00:03:46.610 --> 00:03:49.450
What is p times 8 plus 1.5p?

00:03:49.450 --> 00:03:52.090
Well, that's going to be 8p.

00:03:52.090 --> 00:03:56.290
I'm just multiplying the
p times this stuff first.

00:03:56.290 --> 00:03:59.820
Plus 1.5p squared.

00:03:59.820 --> 00:04:04.930
And now let's multiply the
8 times both of these terms.

00:04:04.930 --> 00:04:14.180
So plus 64 plus-- 8 times
1.5, that is 12-- plus 12p.

00:04:14.180 --> 00:04:17.660
And that's going to be equal
to-- let's see, let's multiply

00:04:17.660 --> 00:04:19.800
p times all of this business.

00:04:19.800 --> 00:04:24.740
So that's going to be equal
to 8p-- 8 times p is 8p--

00:04:24.740 --> 00:04:26.810
and let's see, I could
distribute these terms

00:04:26.810 --> 00:04:28.620
and then multiply by p.

00:04:28.620 --> 00:04:35.000
So 2 times p is 2p, times p is
2p squared, plus 2p squared.

00:04:35.000 --> 00:04:40.880
And then 2 times 8
is 16 times p is 16p.

00:04:40.880 --> 00:04:43.380
So now we have--
well, we essentially

00:04:43.380 --> 00:04:45.244
end up with a
quadratic equation,

00:04:45.244 --> 00:04:46.660
but let's simplify
it a little bit

00:04:46.660 --> 00:04:50.340
so that we can either factor it
or apply the quadratic formula.

00:04:50.340 --> 00:05:00.410
So let's see, let's subtract
1.5p squared from both sides.

00:05:00.410 --> 00:05:03.020
So subtract 1.5p squared.

00:05:06.950 --> 00:05:08.450
Actually, let me
just put everything

00:05:08.450 --> 00:05:10.120
on the left-hand side
just because that

00:05:10.120 --> 00:05:11.690
might be a little
bit more intuitive.

00:05:11.690 --> 00:05:14.485
So let's subtract 2p
squared from both sides.

00:05:18.320 --> 00:05:21.626
Let's subtract 16p
from both sides.

00:05:21.626 --> 00:05:23.250
We have an 8p and a
12p, and then we're

00:05:23.250 --> 00:05:26.050
going to subtract a
16p from both sides.

00:05:26.050 --> 00:05:29.224
And then, actually, let's
subtract an 8p as well

00:05:29.224 --> 00:05:29.890
from both sides.

00:05:29.890 --> 00:05:33.920
We have a 16p and an 8p, so that
actually works out quite well.

00:05:33.920 --> 00:05:38.240
So now we've subtracted
8p from both sides, 16p

00:05:38.240 --> 00:05:38.910
from both sides.

00:05:38.910 --> 00:05:40.951
So we've essentially
subtracted all of this stuff

00:05:40.951 --> 00:05:41.930
from both sides.

00:05:41.930 --> 00:05:46.910
And we are left with--
let's see, I'll do it

00:05:46.910 --> 00:05:47.830
in degree order.

00:05:47.830 --> 00:05:58.280
1.5p squared minus 2p squared
is negative 0.5p squared.

00:05:58.280 --> 00:06:01.090
Now let's see, these cancel out.

00:06:01.090 --> 00:06:05.420
12p minus 16p is minus 4p.

00:06:05.420 --> 00:06:09.220
And then we have plus 64.

00:06:09.220 --> 00:06:11.525
And then that is going
to be equal to 0.

00:06:16.530 --> 00:06:18.400
And just to simplify
this a little bit,

00:06:18.400 --> 00:06:20.149
or just to make this
a little bit cleaner,

00:06:20.149 --> 00:06:26.560
let's multiply both sides of
this equation by negative 2.

00:06:26.560 --> 00:06:28.840
I want the coefficient
over here to be 1.

00:06:28.840 --> 00:06:33.102
So then we get p
squared plus 8p.

00:06:33.102 --> 00:06:40.630
p squared plus 8p is going
to be equal to-- let's

00:06:40.630 --> 00:06:43.980
see, negative times negative 2.

00:06:43.980 --> 00:06:50.430
So minus 128-- is
going to be equal to 0.

00:06:50.430 --> 00:06:51.960
So let's see if we
can factor this.

00:06:51.960 --> 00:06:55.030
Can we think of two numbers
where if we take their product,

00:06:55.030 --> 00:06:57.640
we get negative 128?

00:06:57.640 --> 00:07:03.214
And if we were to add them
together, we get positive 8.

00:07:03.214 --> 00:07:04.880
So they're going to
have different signs

00:07:04.880 --> 00:07:05.970
right over here.

00:07:05.970 --> 00:07:12.070
So let's see, if we say 12
times-- well, let's see.

00:07:12.070 --> 00:07:14.430
What numbers could this be?

00:07:14.430 --> 00:07:21.800
So if we were to think about
128 is the same thing as-- 16,

00:07:21.800 --> 00:07:23.880
let's see, 16 goes into 128.

00:07:23.880 --> 00:07:25.090
Let me work through this.

00:07:25.090 --> 00:07:30.750
16 goes into 128,
does it go 8 times?

00:07:30.750 --> 00:07:37.290
8 times 6 is 48.

00:07:37.290 --> 00:07:40.860
8 times 10 is 80,
plus 40 is 128.

00:07:40.860 --> 00:07:42.760
Yep, it goes 8 times.

00:07:42.760 --> 00:07:44.930
So 16 and 8 seem to work.

00:07:44.930 --> 00:07:48.420
So if you have positive
16 and negative 8,

00:07:48.420 --> 00:07:50.870
their product would
be negative 128.

00:07:50.870 --> 00:07:57.980
So we can factor this out
as p plus 16 times p minus 8

00:07:57.980 --> 00:07:59.890
is equal to 0.

00:07:59.890 --> 00:08:04.740
Now, this is going to be equal
to 0 if at least one of these

00:08:04.740 --> 00:08:06.655
is going to be equal to 0.

00:08:06.655 --> 00:08:08.120
So we have two solutions.

00:08:08.120 --> 00:08:12.880
Either p plus 16 is going to
be equal to 0, or p minus 8

00:08:12.880 --> 00:08:14.580
is equal to 0.

00:08:14.580 --> 00:08:16.830
This one right over here,
subtract 16 from both sides,

00:08:16.830 --> 00:08:18.600
you get p is equal
to negative 16.

00:08:18.600 --> 00:08:23.720
Here, you get p is equal to
8, if you add 8 to both sides.

00:08:23.720 --> 00:08:26.690
Now, we're talking about
a number of pizzas made.

00:08:26.690 --> 00:08:28.400
So this one doesn't apply.

00:08:28.400 --> 00:08:31.220
This would be like
Dominique eating 16 pizzas,

00:08:31.220 --> 00:08:33.070
or somehow destroying
16 pizzas a day.

00:08:33.070 --> 00:08:35.950
We're not interested
in that solution.

00:08:35.950 --> 00:08:38.610
So if we want the solution
to the original question,

00:08:38.610 --> 00:08:41.820
the number of pizzas she made
before the increase in price,

00:08:41.820 --> 00:08:46.790
she made 8 pizzas per day.

00:08:46.790 --> 00:08:50.680
So p right over here
needs to be equal to 8.

00:08:50.680 --> 00:08:53.610
So before the change in price,
she made 8 pizzas a day.

00:08:53.610 --> 00:08:56.100
After the change in price,
she made 8 more pizzas a day,

00:08:56.100 --> 00:08:59.370
or 16 pizzas per day.


Dividing polynomials with remainders example

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:03.450
The quotient of two
polynomials-- a of x and b

00:00:03.450 --> 00:00:04.825
of x-- can be
written in the form

00:00:04.825 --> 00:00:09.700
a of x over b of x is equal to
q of x plus r of x over b of x--

00:00:09.700 --> 00:00:15.020
where q of x and r
of x are polynomials

00:00:15.020 --> 00:00:19.280
and the degree of r of x is
less than the degree of b of x.

00:00:19.280 --> 00:00:21.410
Write the quotient
7x to the sixth

00:00:21.410 --> 00:00:27.054
plus x to the third plus 2x plus
1 over x squared in this form.

00:00:27.054 --> 00:00:28.720
Well, this one is
pretty straightforward

00:00:28.720 --> 00:00:31.657
because we're
dividing by x squared.

00:00:31.657 --> 00:00:33.240
So you could literally
view this as 7x

00:00:33.240 --> 00:00:35.590
to the sixth
divided by x squared

00:00:35.590 --> 00:00:39.810
plus x to the third divided
by x squared plus 2x divided

00:00:39.810 --> 00:00:42.467
by x squared plus 1
divided by x squared.

00:00:42.467 --> 00:00:44.050
So we could just do
this term by term.

00:00:44.050 --> 00:00:48.030
What's 7x to the sixth
divided by x squared?

00:00:48.030 --> 00:00:51.680
Well, x to the sixth divided by
x squared is x to the fourth.

00:00:51.680 --> 00:00:56.570
So it's going to be 7x
to the fourth power.

00:00:56.570 --> 00:00:58.260
And then, same thing
right over here.

00:00:58.260 --> 00:01:01.850
Plus x to the third
divided by x squared.

00:01:01.850 --> 00:01:03.320
Well, that's just going to be x.

00:01:03.320 --> 00:01:05.450
So plus x.

00:01:05.450 --> 00:01:09.320
And then, we're going to
have 2x divided by x squared.

00:01:09.320 --> 00:01:10.750
But remember, we
want to write it

00:01:10.750 --> 00:01:14.570
in a form of r of x over
b of x-- where r of x

00:01:14.570 --> 00:01:16.290
has a lower degree than b of x.

00:01:16.290 --> 00:01:20.610
Well, 2x has a lower
degree than x squared.

00:01:20.610 --> 00:01:22.410
Here this is degree 1.

00:01:22.410 --> 00:01:23.890
This is degree 2.

00:01:23.890 --> 00:01:29.070
So you could write it as
plus 2x over x squared.

00:01:29.070 --> 00:01:30.380
Like that.

00:01:30.380 --> 00:01:33.220
And then, you could write
plus 1 over x squared.

00:01:33.220 --> 00:01:37.512
So you could do this--
plus 1 over x squared.

00:01:37.512 --> 00:01:38.845
So you could write it like that.

00:01:38.845 --> 00:01:40.803
But that's not exactly
the form that they want.

00:01:40.803 --> 00:01:44.020
They want us to write it q
of x-- and you could view

00:01:44.020 --> 00:01:46.560
that as 7x to the fourth plus x.

00:01:46.560 --> 00:01:49.920
And then, they want plus
r of x over b of x So

00:01:49.920 --> 00:01:54.500
plus some polynomial over
x squared in this case.

00:01:54.500 --> 00:01:58.500
So instead of writing it as 2x
over x squared plus 1 over x

00:01:58.500 --> 00:02:03.090
squared, we could just write
it as 2x plus 1 over x squared.

00:02:03.090 --> 00:02:05.880
So one way to think about it.

00:02:05.880 --> 00:02:07.620
So let me just put
some parentheses

00:02:07.620 --> 00:02:11.650
here so that it interprets
my typing correctly.

00:02:11.650 --> 00:02:14.810
So notice, this part
of the polynomial,

00:02:14.810 --> 00:02:19.410
these terms have an equal or
higher degree than x squared.

00:02:19.410 --> 00:02:20.620
So I just divided those.

00:02:20.620 --> 00:02:23.130
7x to the sixth divided by x
squared is 7x to the fourth.

00:02:23.130 --> 00:02:26.160
x to the third divided
by x squared is x.

00:02:26.160 --> 00:02:29.440
And then, once I got two terms
that had a lower degree than x

00:02:29.440 --> 00:02:30.690
squared, I just left on there.

00:02:30.690 --> 00:02:34.820
I just said plus whatever 2x
plus 1 divided by x squared is.

00:02:34.820 --> 00:02:37.735
And that's the form that they
wanted us to write it in.

00:02:37.735 --> 00:02:38.910
We'll check our answer.

00:02:38.910 --> 00:02:40.980
And we got it right.


Polynomial remainder theorem to determine coefficient example

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:05.280
For what value of c is x minus
3 a factor of the polynomial p

00:00:05.280 --> 00:00:06.170
of x?

00:00:06.170 --> 00:00:08.780
And they tell us that p of x is
negative x to the third power,

00:00:08.780 --> 00:00:12.030
plus cx squared,
minus 4x, plus 3.

00:00:12.030 --> 00:00:14.390
So I encourage you to
now pause this video

00:00:14.390 --> 00:00:17.550
and try to figure
this out on your own.

00:00:17.550 --> 00:00:22.400
So the remainder theorem
tells us that if x minus 3

00:00:22.400 --> 00:00:26.550
is a factor of p of x.

00:00:26.550 --> 00:00:28.870
Now that's another way
of saying that p of x

00:00:28.870 --> 00:00:37.570
could be written as x minus
3, times something else.

00:00:37.570 --> 00:00:49.660
This is true if and only
if p of 3 is equal to 0.

00:00:49.660 --> 00:00:55.760
p of three-- I would write
p in that yellow color--

00:00:55.760 --> 00:01:02.820
This is true if and only
if p of 3 is equal to 0.

00:01:02.820 --> 00:01:05.370
And this makes complete sense.

00:01:05.370 --> 00:01:07.670
Because if you put 3--
if you evaluate p of 3,

00:01:07.670 --> 00:01:10.010
we're saying that x minus
3 is a factor of p of x.

00:01:10.010 --> 00:01:12.400
That means that you could
write p of x like this.

00:01:12.400 --> 00:01:15.120
So, if you could write p of x
like this, what would p of 3

00:01:15.120 --> 00:01:15.762
be?

00:01:15.762 --> 00:01:17.220
You substitute a
3 right over here,

00:01:17.220 --> 00:01:20.390
you're going to have 3
minus 3 for this expression.

00:01:20.390 --> 00:01:23.260
That's just going to be
0 times something else.

00:01:23.260 --> 00:01:24.830
Anything times 0 is 0.

00:01:24.830 --> 00:01:27.810
So this should make just
a lot of common sense.

00:01:27.810 --> 00:01:30.590
So p of 3 needs
to be equal to 0.

00:01:30.590 --> 00:01:33.040
So let's just evaluate p
of 3 and then figure out

00:01:33.040 --> 00:01:37.470
what c needs to be in order
to make p of 3 equal to 0.

00:01:37.470 --> 00:01:38.720
So let's write this down.

00:01:38.720 --> 00:01:44.110
So p of-- I'll do the
three in green-- p of 3

00:01:44.110 --> 00:01:48.920
is going to be equal
to negative 3--

00:01:48.920 --> 00:01:51.810
wrote it all in one
color, actually--

00:01:51.810 --> 00:02:00.020
3 to the third power, plus c
times 3 squared, minus 4 times

00:02:00.020 --> 00:02:01.961
3, plus 3.

00:02:01.961 --> 00:02:03.460
So what's this going
to be equal to?

00:02:03.460 --> 00:02:08.370
This is going to be equal
to negative 27, plus 9

00:02:08.370 --> 00:02:13.550
c, minus 12, plus 3.

00:02:13.550 --> 00:02:14.790
So what does that equal to?

00:02:14.790 --> 00:02:20.270
This is negative 12 plus 3 is
negative 9, plus negative 27

00:02:20.270 --> 00:02:26.130
is negative 36 so this is
going to be 9 c minus 36.

00:02:26.130 --> 00:02:28.860
And remember, we want p
of 3 to be equal to 0.

00:02:28.860 --> 00:02:32.210
This is p of 3, so we want
this thing right over here

00:02:32.210 --> 00:02:33.017
to be equal to 0.

00:02:33.017 --> 00:02:34.350
So I'm going to make this clear.

00:02:34.350 --> 00:02:36.990
We have p of 3 is
equal to this, and we

00:02:36.990 --> 00:02:39.360
want to set this
to be equal to 0.

00:02:39.360 --> 00:02:42.550
So let's just solve this
equation right over here.

00:02:42.550 --> 00:02:48.320
So you add 36 to both sides,
you get 9 c is equal to 36.

00:02:48.320 --> 00:02:53.790
divide both sides by 9,
you get c is equal to 4.

00:02:53.790 --> 00:02:59.040
So if c is equal to
4, if p of x were

00:02:59.040 --> 00:03:01.940
to be equal to negative
x to the third,

00:03:01.940 --> 00:03:09.730
plus 4x squared, minus
4x, plus 3, then x minus 3

00:03:09.730 --> 00:03:13.220
is a factor of p of x.


Factoring 5th degree polynomial to find real zeros

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:04.280
Plot the real zeroes of the
given polynomial on the graph

00:00:04.280 --> 00:00:04.780
below.

00:00:04.780 --> 00:00:07.490
And they give us p of x
is equal to 2 x to the 5th

00:00:07.490 --> 00:00:10.526
plus x to the 4th
minus 2x minus 1.

00:00:10.526 --> 00:00:12.150
And when they say
plot it, they give us

00:00:12.150 --> 00:00:13.150
this little widget here.

00:00:13.150 --> 00:00:15.770
Where if we click at any point
on this, we get our point.

00:00:15.770 --> 00:00:17.830
And we get as many
points as we would like.

00:00:17.830 --> 00:00:19.960
And we can drag
these points around.

00:00:19.960 --> 00:00:21.990
Or if we don't want
these points anymore,

00:00:21.990 --> 00:00:24.510
we can just dump them
in this little trash

00:00:24.510 --> 00:00:26.950
bin at the bottom right.

00:00:26.950 --> 00:00:30.180
So let's think about what
the zeroes of this polynomial

00:00:30.180 --> 00:00:30.900
actually are.

00:00:30.900 --> 00:00:33.680
And to do that I'll
take my scratch pad out.

00:00:33.680 --> 00:00:36.450
And this is a little
daunting at first.

00:00:36.450 --> 00:00:40.530
This is a 5th degree
polynomial here.

00:00:40.530 --> 00:00:43.976
Factoring 5th degree polynomials
is really something of an art.

00:00:43.976 --> 00:00:46.350
You're really going to have
to sit and look for patterns.

00:00:46.350 --> 00:00:49.510
If they're actually expecting
you to find the zeroes here

00:00:49.510 --> 00:00:51.510
without the help of a
computer, without the help

00:00:51.510 --> 00:00:54.025
of a calculator, then there
must be some type of pattern

00:00:54.025 --> 00:00:55.150
that you can pick out here.

00:00:55.150 --> 00:00:57.550
So let me just rewrite p of x.

00:00:57.550 --> 00:01:04.860
So p of x is equal to 2 x to
the 5th plus x to the 4th minus

00:01:04.860 --> 00:01:07.310
2x minus 1.

00:01:07.310 --> 00:01:09.080
And one way that's
typically seen

00:01:09.080 --> 00:01:12.330
when you're trying to factor
this type of polynomial

00:01:12.330 --> 00:01:15.460
is to try to essentially undo
the distributive property

00:01:15.460 --> 00:01:16.800
a few times.

00:01:16.800 --> 00:01:19.740
And if you want to relate it
to techniques for factoring

00:01:19.740 --> 00:01:23.150
quadratics, it's essentially
factoring by grouping.

00:01:23.150 --> 00:01:27.880
So for example, you
see a 2x minus 1,

00:01:27.880 --> 00:01:32.090
or something that looks like
a 2x minus 1 right over here.

00:01:32.090 --> 00:01:37.660
And over here you have a 2x
to the 5th plus x to the 4th.

00:01:37.660 --> 00:01:39.540
So you have a 2x
of a higher degree

00:01:39.540 --> 00:01:44.110
term plus a 1 x of
a one degree lower.

00:01:44.110 --> 00:01:46.440
So there seems to be
some type of a pattern.

00:01:46.440 --> 00:01:48.010
2 times x of a
higher degree-- this

00:01:48.010 --> 00:01:50.750
is a first degree
term-- minus 1 times--

00:01:50.750 --> 00:01:53.882
you could do this as x to the
0-- of a lower degree term.

00:01:53.882 --> 00:01:55.590
And so let's think
about it a little bit.

00:01:55.590 --> 00:02:01.250
What happens if we essentially
try to group these two terms,

00:02:01.250 --> 00:02:04.450
and we group these two
terms right over here.

00:02:04.450 --> 00:02:06.214
And we try to
factor out anything

00:02:06.214 --> 00:02:07.880
to essentially clean
it up a little bit,

00:02:07.880 --> 00:02:09.690
to see if we can
make sense of it.

00:02:09.690 --> 00:02:12.750
Well these two terms, their
greatest common factor

00:02:12.750 --> 00:02:13.730
is x to the 4th.

00:02:13.730 --> 00:02:19.960
We could write this as x
to the 4th times 2x plus 1.

00:02:19.960 --> 00:02:22.300
And this should get us excited
because this looks pretty

00:02:22.300 --> 00:02:24.840
close to that,
especially if we were

00:02:24.840 --> 00:02:26.890
to factor out a negative 1 here.

00:02:26.890 --> 00:02:29.010
So we could factor
out a negative 1.

00:02:29.010 --> 00:02:32.090
And then this is
going to be 2x plus 1.

00:02:32.090 --> 00:02:36.460
And that's exciting because now
we can factor out a 2 x plus 1

00:02:36.460 --> 00:02:39.680
from each of these terms.

00:02:39.680 --> 00:02:41.140
So you have a 2x plus 1.

00:02:41.140 --> 00:02:46.030
We're gonna factor
both of these out,

00:02:46.030 --> 00:02:51.110
to get 2x plus 1-- which
we just factored out.

00:02:51.110 --> 00:02:55.410
And if you factored it out
of this term right over here,

00:02:55.410 --> 00:02:57.720
you're left with x to the 4th.

00:02:57.720 --> 00:02:59.180
And if you factor
it out this term,

00:02:59.180 --> 00:03:01.810
you're left with
just the minus 1.

00:03:01.810 --> 00:03:04.170
Minus 1.

00:03:04.170 --> 00:03:07.240
And now this is exciting
because 2x plus 1,

00:03:07.240 --> 00:03:08.790
this is pretty
easy to figure out

00:03:08.790 --> 00:03:10.670
when does this thing equal 0.

00:03:10.670 --> 00:03:12.330
And we'll do that
in a little bit.

00:03:12.330 --> 00:03:14.100
And this is pretty
easy to factor.

00:03:14.100 --> 00:03:16.720
This is a difference of squares.

00:03:16.720 --> 00:03:18.780
This right over here
can be rewritten

00:03:18.780 --> 00:03:29.700
as x squared plus 1
times x squared minus 1.

00:03:29.700 --> 00:03:32.880
And of course we still have
this 2x plus 1 out front.

00:03:32.880 --> 00:03:35.590
2x plus 1.

00:03:35.590 --> 00:03:38.220
And once again we have
another difference of squares.

00:03:38.220 --> 00:03:40.480
We have another difference
of squares right over here.

00:03:40.480 --> 00:03:45.490
That's the same thing as
x plus 1 times x minus 1.

00:03:45.490 --> 00:03:48.320
And let me just write all the
other parts of this expression.

00:03:48.320 --> 00:03:50.450
x squared plus 1.

00:03:50.450 --> 00:03:53.120
And you have 2x plus 1.

00:03:53.120 --> 00:03:54.160
2x plus 1.

00:03:54.160 --> 00:03:57.750
And I think I factored
p of x about as much

00:03:57.750 --> 00:04:01.830
as could be reasonably expected.

00:04:01.830 --> 00:04:04.099
So p of x is equal to
all of this business.

00:04:04.099 --> 00:04:06.640
And remember, the whole reason
why I wanted to factor it is I

00:04:06.640 --> 00:04:09.330
wanted to figure out when
does this thing equal 0?

00:04:09.330 --> 00:04:11.680
So if p of x can be
expressed as the product

00:04:11.680 --> 00:04:13.940
of a bunch of these
expressions, it's

00:04:13.940 --> 00:04:17.000
going to be 0 whenever at
least one of these expressions

00:04:17.000 --> 00:04:17.790
is equal to 0.

00:04:17.790 --> 00:04:19.250
If any of these
is equal to 0 then

00:04:19.250 --> 00:04:22.350
that's just going to make this
whole expression equal to 0.

00:04:22.350 --> 00:04:25.150
So when does 2x plus 1 equal 0?

00:04:25.150 --> 00:04:28.836
So 2x plus 1 is equal to zero.

00:04:28.836 --> 00:04:30.710
Well you could probably
do this in your head,

00:04:30.710 --> 00:04:33.000
or we could do it
systematically as well.

00:04:33.000 --> 00:04:36.330
Subtract 1 from both sides,
you get 2x equals negative 1.

00:04:36.330 --> 00:04:41.180
Divide both sides by 2, you
get x is equal to negative 1/2.

00:04:41.180 --> 00:04:43.730
So when x equals negative 1/2--
or one way to think about it,

00:04:43.730 --> 00:04:46.170
p of negative 1/2 is 0.

00:04:46.170 --> 00:04:48.300
So p of negative 1/2 is 0.

00:04:48.300 --> 00:04:52.200
So this right over here
is a point on the graph,

00:04:52.200 --> 00:04:54.910
and it is one of
the real zeroes.

00:04:54.910 --> 00:04:58.315
Now we could try to solve this.
x squared plus 1 equals 0,

00:04:58.315 --> 00:05:01.190
and I'll just write it
down just to show you.

00:05:01.190 --> 00:05:03.710
If we try to isolate
the x term on the left--

00:05:03.710 --> 00:05:05.210
subtract 1 from
both sides-- you get

00:05:05.210 --> 00:05:08.160
x squared is equal
to negative 1.

00:05:08.160 --> 00:05:12.090
Now if we were to start thinking
about imaginary numbers,

00:05:12.090 --> 00:05:15.230
we could think about
what x could be.

00:05:15.230 --> 00:05:17.320
But they want us to
find the real zeroes.

00:05:17.320 --> 00:05:18.380
The real zeroes.

00:05:18.380 --> 00:05:20.450
So there's no real
number where that number

00:05:20.450 --> 00:05:22.270
squared is equal to negative 1.

00:05:22.270 --> 00:05:25.530
So we're not going to
get any real zeroes

00:05:25.530 --> 00:05:28.550
by setting this
thing equal to 0.

00:05:28.550 --> 00:05:30.980
There's no real number
x where x squared

00:05:30.980 --> 00:05:33.840
plus 1 is going
to be equal to 0.

00:05:33.840 --> 00:05:37.770
Now let's think about when x
plus 1 could be equal to 0.

00:05:37.770 --> 00:05:39.340
We'll subtract 1
from both sides.

00:05:39.340 --> 00:05:41.260
You get x is equal
to negative 1.

00:05:41.260 --> 00:05:44.320
So p of negative 1
is going to be 0.

00:05:44.320 --> 00:05:47.720
So that's another one of
our zeroes right there.

00:05:47.720 --> 00:05:51.770
And then finally let's
think about when x minus 1

00:05:51.770 --> 00:05:52.630
is equal to 0.

00:05:52.630 --> 00:05:53.960
We'll add one to both sides.

00:05:53.960 --> 00:05:55.450
X is equal to 1.

00:05:55.450 --> 00:06:02.540
So we have another real
0 right over there.

00:06:02.540 --> 00:06:04.960
And so we could plot them.

00:06:04.960 --> 00:06:07.800
So it's negative 1,
negative 1/2, and 1.

00:06:07.800 --> 00:06:12.820
So it's negative 1,
negative 1/2 and 1.

00:06:12.820 --> 00:06:16.180
And we can check our
answer, and we got it right.

00:06:16.180 --> 00:06:19.736
Now one thing that might be
bugging you, is like hey Sal,

00:06:19.736 --> 00:06:22.110
you just happened to group
this in exactly the right way.

00:06:22.110 --> 00:06:23.984
What if I try to group
it in a different way?

00:06:23.984 --> 00:06:27.270
What if I tried to-- Well
actually let's try to do that.

00:06:27.270 --> 00:06:28.870
That could be interesting.

00:06:28.870 --> 00:06:31.280
Just to show you that
this isn't voodoo.

00:06:31.280 --> 00:06:35.150
And actually there's
several ways to get there.

00:06:35.150 --> 00:06:37.570
There's several
ways to get there.

00:06:37.570 --> 00:06:40.840
So what if instead of writing
it like this, where you're

00:06:40.840 --> 00:06:43.220
writing it kind of
in the highest degree

00:06:43.220 --> 00:06:45.690
term and the next highest
degree and so on and so forth,

00:06:45.690 --> 00:06:49.570
you were to write it like
this: p of x is equal to 2 x

00:06:49.570 --> 00:06:55.390
to the 5th minus 2x plus
x to the 4th minus 1.

00:06:55.390 --> 00:06:57.270
Well actually, even
this way you could

00:06:57.270 --> 00:06:58.734
do a fairly
interesting grouping.

00:06:58.734 --> 00:07:00.150
If you grouped
these two together,

00:07:00.150 --> 00:07:02.410
you see that they have
the common factor 2 x.

00:07:02.410 --> 00:07:08.780
You factor 2x out, you get 2x
times x to the 4th minus 1.

00:07:08.780 --> 00:07:10.730
And I think you see
what's going on.

00:07:10.730 --> 00:07:15.850
And then this can be
rewritten as plus 1 times x

00:07:15.850 --> 00:07:19.810
to the 4th minus 1.

00:07:19.810 --> 00:07:20.960
Minus 1.

00:07:20.960 --> 00:07:23.590
And now you can factor out
an x to the 4th minus 1.

00:07:23.590 --> 00:07:26.340
And you're left with-- I'll
do this in a neutral color--

00:07:26.340 --> 00:07:29.470
x to the fourth
minus 1 times 2 x

00:07:29.470 --> 00:07:31.680
plus 1, which is much
easier to factor now.

00:07:31.680 --> 00:07:32.680
A difference of squares.

00:07:32.680 --> 00:07:34.360
Exactly what we did
the last time around.

00:07:34.360 --> 00:07:35.776
So there are several
ways that you

00:07:35.776 --> 00:07:38.430
could have reasonably
grouped this, and reasonably

00:07:38.430 --> 00:07:40.580
undone the
distributive property.

00:07:40.580 --> 00:07:43.809
But I do admit it is
something of an art.

00:07:43.809 --> 00:07:45.600
You really just have
to play around in see,

00:07:45.600 --> 00:07:46.890
well let's group
the first two terms,

00:07:46.890 --> 00:07:48.706
let's see if there's
a common factor here.

00:07:48.706 --> 00:07:50.080
Let's group the
second two terms,

00:07:50.080 --> 00:07:51.830
let's see if there's
a common factor here.

00:07:51.830 --> 00:07:54.030
Hey, once we factor out
those common factors,

00:07:54.030 --> 00:07:56.260
it looks like both
of these two terms

00:07:56.260 --> 00:07:58.250
have this common
expression as a factor.

00:07:58.250 --> 00:08:01.590
And then you could start
to factor that out.


Identifying graph based on roots

WEBVTT
Kind: captions
Language: en

00:00:00.990 --> 00:00:04.120
Use the real 0's of the
polynomial function y

00:00:04.120 --> 00:00:07.230
equal to x to the third plus
3x squared plus x plus 3

00:00:07.230 --> 00:00:11.020
to determine which of the
following could be its graph.

00:00:11.020 --> 00:00:13.800
So there's several ways
of trying to approach it.

00:00:13.800 --> 00:00:17.320
One, we could just look at what
the 0's of these graphs are

00:00:17.320 --> 00:00:21.350
or what they appear to be and
then see if this function is

00:00:21.350 --> 00:00:23.870
actually 0 when x
is equal to that.

00:00:23.870 --> 00:00:26.554
So for example, in graph
A-- and first of all,

00:00:26.554 --> 00:00:28.470
as always, I encourage
you to pause this video

00:00:28.470 --> 00:00:34.470
and try it before I show
you how to solve it.

00:00:34.470 --> 00:00:36.470
So I'm assuming you've
given a go at it.

00:00:36.470 --> 00:00:38.180
So let's look at this
first graph here.

00:00:38.180 --> 00:00:41.630
Its 0, it clearly has a
0 right at this point.

00:00:41.630 --> 00:00:44.270
And just by trying to
inspect this graph,

00:00:44.270 --> 00:00:48.830
it looks like this is at
x is equal to negative 3,

00:00:48.830 --> 00:00:49.860
if I were to estimate.

00:00:49.860 --> 00:00:53.360
So that looks like the
point negative 3, 0.

00:00:53.360 --> 00:00:56.940
So let's see, if we substitute
x equals negative 3 here,

00:00:56.940 --> 00:01:00.650
whether we get y equaling 0.

00:01:00.650 --> 00:01:06.130
So let's see, negative 3
to the third power plus 3

00:01:06.130 --> 00:01:11.250
times negative 3 squared
plus negative 3 plus 3.

00:01:11.250 --> 00:01:12.730
What does this give us?

00:01:12.730 --> 00:01:15.500
This gives us negative 27.

00:01:15.500 --> 00:01:18.060
This gives us positive 27.

00:01:18.060 --> 00:01:19.650
This gives, of
course, negative 3.

00:01:19.650 --> 00:01:20.640
This is plus 3.

00:01:20.640 --> 00:01:21.660
These two cancel out.

00:01:21.660 --> 00:01:22.810
These two cancel out.

00:01:22.810 --> 00:01:25.650
This does indeed equal 0.

00:01:25.650 --> 00:01:27.500
So this was actually
pretty straightforward.

00:01:27.500 --> 00:01:30.130
Graph A does indeed work.

00:01:30.130 --> 00:01:32.265
You could try
graph B right here,

00:01:32.265 --> 00:01:36.790
and you would have to
verify that we have a 0 at,

00:01:36.790 --> 00:01:38.960
this looks like negative 2.

00:01:38.960 --> 00:01:43.965
Another one, this looks like at
1, another one that looks at 3.

00:01:43.965 --> 00:01:46.590
And since we already know that
A is the answer, none of these--

00:01:46.590 --> 00:01:50.260
if you were to input x equals
negative 2, x equals 1,

00:01:50.260 --> 00:01:54.100
or x equals 3 into this function
definition right over here,

00:01:54.100 --> 00:01:55.700
you should not get 0.

00:01:55.700 --> 00:01:57.430
And you'll see that
this doesn't work.

00:01:57.430 --> 00:01:58.429
Same thing for this one.

00:01:58.429 --> 00:02:01.310
If you tried 4 or
7 for your x's, you

00:02:01.310 --> 00:02:03.280
should not get 0
over here, because we

00:02:03.280 --> 00:02:07.160
see that the real function
does not equal 0 at 4 or 7.

00:02:07.160 --> 00:02:10.639
Another giveaway that this is
not going to be the function

00:02:10.639 --> 00:02:14.990
is that you are going to
have a total of three roots.

00:02:14.990 --> 00:02:16.470
Let me write this down.

00:02:16.470 --> 00:02:20.970
So you're going to have
a total of three roots.

00:02:20.970 --> 00:02:27.080
Now, , those three roots could
be real or complex roots.

00:02:27.080 --> 00:02:30.560
And the big key is complex
roots come in pairs.

00:02:37.090 --> 00:02:40.986
So you might have a situation
with three real roots.

00:02:40.986 --> 00:02:42.860
And this is an example
with three real roots,

00:02:42.860 --> 00:02:44.360
although we know
this actually isn't

00:02:44.360 --> 00:02:46.010
the function right over here.

00:02:46.010 --> 00:02:48.190
Or if you have one
complex root, you're

00:02:48.190 --> 00:02:49.940
going to have
another complex root.

00:02:49.940 --> 00:02:52.680
So if you have
any complex roots,

00:02:52.680 --> 00:02:57.735
the next possibility is one
real and two complex roots.

00:03:00.880 --> 00:03:02.859
And this right over
here has two real roots.

00:03:02.859 --> 00:03:03.900
That's not a possibility.

00:03:03.900 --> 00:03:08.980
That would somehow imply that
you have only one complex root,

00:03:08.980 --> 00:03:14.617
which that is not a possibility.

00:03:14.617 --> 00:03:16.950
Now another way that you could
have thought about this--

00:03:16.950 --> 00:03:18.300
and this would have
been the longer way.

00:03:18.300 --> 00:03:20.180
But let's say you didn't
have the graphs here for you,

00:03:20.180 --> 00:03:22.530
and someone asked you to
just find the roots-- well,

00:03:22.530 --> 00:03:24.550
you could have attempted
to factor this.

00:03:24.550 --> 00:03:27.040
And this one actually
is factorable.

00:03:27.040 --> 00:03:33.110
y is equal to x to the third
plus 3x squared plus x plus 3.

00:03:33.110 --> 00:03:36.330
As mentioned in previous videos,
factoring things of a degree

00:03:36.330 --> 00:03:39.490
higher than 2, there is
something of an art to it.

00:03:39.490 --> 00:03:41.241
But oftentimes, if
someone expects you to,

00:03:41.241 --> 00:03:43.490
you might be able to group
things in interesting ways,

00:03:43.490 --> 00:03:45.670
especially when you see
that several terms have

00:03:45.670 --> 00:03:47.210
some common factors.

00:03:47.210 --> 00:03:52.500
So for example, these first
two terms right over here

00:03:52.500 --> 00:03:54.334
have the common
factor x squared.

00:03:54.334 --> 00:03:55.750
So if you were to
factor that out,

00:03:55.750 --> 00:04:00.160
you would get x squared
times x plus 3, which is neat

00:04:00.160 --> 00:04:02.630
because that looks a lot
like the second two terms.

00:04:02.630 --> 00:04:06.710
We could write that as
plus 1 times x plus 3.

00:04:06.710 --> 00:04:09.720
And then you can factor
the x plus 3 out.

00:04:09.720 --> 00:04:12.510
We could factor
the x plus 3 out,

00:04:12.510 --> 00:04:21.250
and we would get x plus
3 times x squared plus 1.

00:04:24.840 --> 00:04:27.440
And now, your 0's
are going to happen,

00:04:27.440 --> 00:04:29.650
or this whole y--
remember this is

00:04:29.650 --> 00:04:34.030
equal to y-- y is
going to equal 0

00:04:34.030 --> 00:04:37.780
if either one of these
factors is equal to 0.

00:04:37.780 --> 00:04:40.790
So when does x plus 3 equal 0?

00:04:40.790 --> 00:04:42.630
Well, subtract 3
from both sides.

00:04:42.630 --> 00:04:45.575
That happens when x is
equal to negative 3.

00:04:51.550 --> 00:04:54.090
When does x squared plus
1 equal 0, I should say?

00:04:54.090 --> 00:04:58.180
Well, when x squared
is equal to negative 1.

00:04:58.180 --> 00:05:02.210
Well, there's no real
x's, no real valued x's.

00:05:02.210 --> 00:05:04.900
There's no real
number x's such that x

00:05:04.900 --> 00:05:06.440
squared is equal to negative 1.

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x is going to be
an imaginary-- or I

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guess I'll just say it in
more general terms-- it's

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going to be complex valued.

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So once again, you
see you're going

00:05:17.590 --> 00:05:20.050
to have a pair of
complex roots, and you

00:05:20.050 --> 00:05:27.000
have one real root at x
is equal to negative 3.