Interpreting change in exponential models

WEBVTT
Kind: captions
Language: en

00:00:00.125 --> 00:00:01.315
- [Voiceover] So I've
taken some screenshots

00:00:01.315 --> 00:00:04.966
of the Khan Academy exercise
"Interpreting Rate of Change

00:00:04.966 --> 00:00:08.577
"for Exponential Models
in Terms of Change"

00:00:08.577 --> 00:00:10.427
Maybe they're going to change the title.

00:00:10.427 --> 00:00:12.036
It seems a little bit too long.

00:00:12.036 --> 00:00:14.286
But anyway, let's actually
just tackle these together.

00:00:14.286 --> 00:00:16.832
So the first of spring an entire field

00:00:16.832 --> 00:00:19.167
of flowering trees blossoms.

00:00:19.167 --> 00:00:22.120
The population of locusts
consuming these flowers

00:00:22.120 --> 00:00:25.324
rapidly increases as the trees blossom.

00:00:25.324 --> 00:00:28.773
The relationship between
the elapsed time t

00:00:28.773 --> 00:00:31.228
in days since the beginning of spring

00:00:31.228 --> 00:00:34.557
and the total number of locusts L of t,

00:00:34.557 --> 00:00:36.239
so the number of locusts
is going to be a function

00:00:36.239 --> 00:00:39.055
of the number of days that have elapsed

00:00:39.055 --> 00:00:40.623
since the beginning of spring,

00:00:40.623 --> 00:00:42.919
is modeled by the following function.

00:00:42.919 --> 00:00:46.520
So locusts as a function
of time is going to be 750

00:00:46.520 --> 00:00:50.892
times 1.85 to the "t"th power.

00:00:50.892 --> 00:00:53.230
Complete the following sentence about

00:00:53.230 --> 00:00:56.474
the daily rate of change
of the locust population.

00:00:56.474 --> 00:00:59.386
Every day the locust population, well,

00:00:59.386 --> 00:01:01.435
every day think about
what's going to happen.

00:01:01.435 --> 00:01:03.117
I'll draw a little table, just to make it

00:01:03.117 --> 00:01:04.914
hopefully a little bit clearer.

00:01:04.914 --> 00:01:08.974
So, we draw a little bit of a table.

00:01:08.974 --> 00:01:13.077
So we'll put t and L of t.

00:01:13.077 --> 00:01:17.795
So when t is zero, so when
zero days have elapsed,

00:01:17.795 --> 00:01:20.126
well that's going to be
1.85 to the 0th power,

00:01:20.126 --> 00:01:21.357
that's just going to be one.

00:01:21.357 --> 00:01:25.790
So you're going to have 750
locusts right from the get go.

00:01:25.790 --> 00:01:28.818
Then when t equals one,
what's going to happen?

00:01:28.818 --> 00:01:31.032
Well then this is going to be 750

00:01:32.272 --> 00:01:35.328
times 1.85 to the 1st power.

00:01:35.328 --> 00:01:37.994
So it's going to be times 1.85.

00:01:37.994 --> 00:01:42.387
When t is equal to two, what's L of t?

00:01:42.387 --> 00:01:45.541
It's going to be 750

00:01:45.541 --> 00:01:48.902
times 1.85 squared.

00:01:48.902 --> 00:01:50.107
Well that's the same thing as

00:01:50.107 --> 00:01:54.917
1.85 times 1.85.

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So notice.

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And this is just comes out of

00:01:58.117 --> 00:01:59.552
this being an exponential function.

00:01:59.552 --> 00:02:02.388
Every day you have 1.85 times

00:02:02.388 --> 00:02:04.962
as many as you had the day before.

00:02:04.962 --> 00:02:05.892
1.85

00:02:05.892 --> 00:02:07.589
We essentially take what
we had the day before

00:02:07.589 --> 00:02:10.182
and we multiply it by 1.85.

00:02:10.182 --> 00:02:12.920
And since 1.85 is larger than one,

00:02:12.920 --> 00:02:15.554
that's going to grow the
number of locusts we have.

00:02:15.554 --> 00:02:17.530
So this is going to grow.

00:02:17.530 --> 00:02:20.602
I'm actually not using, I'm
not on the website right now

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so that's why, normally there
would be a dropdown here.

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So I'm going to grow by a factor of,

00:02:24.829 --> 00:02:29.575
well I'm going to grow by
a factor of 1.85 every day.

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Let's do another one of these.

00:02:32.735 --> 00:02:33.607
All right.

00:02:33.607 --> 00:02:38.236
So this one tells us
that Vera is an ecologist

00:02:38.236 --> 00:02:39.714
who studies the rate of change

00:02:39.714 --> 00:02:43.038
in the bear population
of Siberia over time.

00:02:43.038 --> 00:02:46.369
The relationship between
the elapsed time t in years

00:02:46.369 --> 00:02:48.750
since Vera began studying the population

00:02:48.750 --> 00:02:51.331
and the total number of bears N of t

00:02:51.331 --> 00:02:53.712
is modeled by the following function.

00:02:53.712 --> 00:02:55.269
All right, fair enough.

00:02:55.269 --> 00:02:58.021
We've got an exponential thing going on.

00:02:58.021 --> 00:03:00.401
Complete the following sentence about

00:03:00.401 --> 00:03:02.861
the yearly rate of change
of the bear population.

00:03:02.861 --> 00:03:04.502
Let's just thing about it.

00:03:04.502 --> 00:03:06.543
Every year that passes, t as in years now,

00:03:06.543 --> 00:03:07.891
every years that passes is going to be

00:03:07.891 --> 00:03:09.694
two thirds times the year before.

00:03:09.694 --> 00:03:11.744
I could do that same table that I just did

00:03:11.744 --> 00:03:14.906
just to make that clear,
so let me do that.

00:03:14.906 --> 00:03:16.166
Whoops.

00:03:16.166 --> 00:03:19.567
Let me, let me make this clear.

00:03:19.567 --> 00:03:22.477
So, table.

00:03:22.477 --> 00:03:25.855
So this is t and this is N of t.

00:03:25.855 --> 00:03:30.855
When t is zero, N of t you're
going to have 2187 bears.

00:03:30.982 --> 00:03:33.505
So that's the first year that she

00:03:33.505 --> 00:03:35.262
began studying that population.

00:03:35.262 --> 00:03:38.093
Zero years since Vera began
studying the population.

00:03:38.093 --> 00:03:41.621
The first year is going to be 2187

00:03:41.621 --> 00:03:45.634
times two thirds to the first
power, so times two thirds.

00:03:45.634 --> 00:03:50.047
The second year is going to be 2187

00:03:50.047 --> 00:03:51.895
times two thirds to the second power.

00:03:51.895 --> 00:03:55.208
So that's just two
thirds times two thirds.

00:03:55.208 --> 00:03:57.084
So each successive year
you're going to have

00:03:57.084 --> 00:04:00.006
two thirds the bear
population of the year before.

00:04:00.006 --> 00:04:02.712
You're multiplying the
year before by two thirds.

00:04:02.712 --> 00:04:05.450
So every year the bear population shrinks,

00:04:05.450 --> 00:04:10.009
shrinks by a factor of,

00:04:10.009 --> 00:04:13.743
by a factor of two thirds.

00:04:13.743 --> 00:04:15.589
All right, let's do one more of these.

00:04:15.589 --> 00:04:19.353
So they tell us that
Akiba started studying

00:04:19.353 --> 00:04:23.086
how the number of branches
on his tree change over time.

00:04:23.086 --> 00:04:24.278
All right.

00:04:24.278 --> 00:04:27.431
The relationship between
the elapsed time t in years

00:04:27.431 --> 00:04:29.768
since Akiba started studying his tree

00:04:29.768 --> 00:04:32.562
and the total number
of its branches N of t

00:04:32.562 --> 00:04:34.571
is modeled by the following function.

00:04:34.571 --> 00:04:36.374
Compete the following sentence

00:04:36.374 --> 00:04:39.696
about the yearly percent change
in the number of branches.

00:04:39.696 --> 00:04:42.683
Every year, blank percent of branches

00:04:42.683 --> 00:04:45.058
are added or subtracted from
the total number of branches.

00:04:45.058 --> 00:04:47.400
Well I'll draw another table,

00:04:47.400 --> 00:04:48.383
although you might get used to

00:04:48.383 --> 00:04:49.610
just being able to look at this and say

00:04:49.610 --> 00:04:50.881
well look, each year you're going to have

00:04:50.881 --> 00:04:55.016
1.75 times the number branches
you had the year before.

00:04:55.016 --> 00:04:57.109
And so we have 1.75 times

00:04:57.109 --> 00:04:58.992
the number of branches the year before,

00:04:58.992 --> 00:05:01.691
you have grown by 75%,

00:05:01.691 --> 00:05:02.966
and I'll make that a little bit clearer.

00:05:02.966 --> 00:05:06.253
So 75%, every year 75% percent of branches

00:05:06.253 --> 00:05:09.693
are added to the total number of branches.

00:05:09.693 --> 00:05:11.291
And I'll just draw that table again

00:05:11.291 --> 00:05:13.254
like I've done in the last two examples

00:05:13.254 --> 00:05:15.361
to make that hopefully clear.

00:05:15.361 --> 00:05:20.361
Okay, so this is t and this is N of t.

00:05:20.663 --> 00:05:23.946
So t equals zero, you have 42 branches.

00:05:23.946 --> 00:05:28.946
T equals one, it's going
to be 42 times 1.75.

00:05:30.511 --> 00:05:32.200
Times 1.75.

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When t equals two, it's going
to be 42 times 1.75 squared.

00:05:34.910 --> 00:05:39.425
42 times 1.75

00:05:39.425 --> 00:05:42.056
times 1.75.

00:05:42.056 --> 00:05:44.768
So every year you are multiplying

00:05:44.768 --> 00:05:49.285
times 1.75 so times 1.75,

00:05:49.285 --> 00:05:51.784
something funky is happening
with my pen right over there.

00:05:51.784 --> 00:05:53.665
But if you're multiplying by 1.75,

00:05:53.665 --> 00:05:55.765
if you're growing by a factor of 1.75,

00:05:55.765 --> 00:06:00.765
this is the same thing as adding 75%.

00:06:00.851 --> 00:06:05.258
Once again, you are adding 75%.

00:06:05.258 --> 00:06:06.283
Think about it this way.

00:06:06.283 --> 00:06:08.414
If you just grew by a factor of one,

00:06:08.414 --> 00:06:09.930
then you're not adding anything.

00:06:09.930 --> 00:06:11.252
You're staying constant.

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If you grow by 10%, then you're

00:06:14.004 --> 00:06:16.462
going to be 1.1 times as large.

00:06:16.462 --> 00:06:18.842
If you grow by 200% then you're

00:06:18.842 --> 00:06:21.429
going to be two times as large.

00:06:21.429 --> 00:06:23.153
So this right over here?

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This right over here is, is, is.

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If you, let me be very
careful, what I just said.

00:06:31.310 --> 00:06:32.620
I think I just mistake that.

00:06:32.620 --> 00:06:36.103
If you grow by 200%, you are going to be

00:06:36.103 --> 00:06:39.456
three times as large as you were before.

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One is constant, and then another 200%

00:06:42.522 --> 00:06:44.694
would be another two-fold so that

00:06:44.694 --> 00:06:46.255
would make you three times as large.

00:06:46.255 --> 00:06:47.246
Don't want to confuse you.

00:06:47.246 --> 00:06:48.679
My brain recognized that I said

00:06:48.679 --> 00:06:50.395
something weird right at that end.

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All right, hopefully you enjoyed that.


Solving exponential equations using exponent properties (advanced)

WEBVTT
Kind: captions
Language: en

00:00:00.041 --> 00:00:01.862
- [Voiceover] So, let's
get even more practice

00:00:01.862 --> 00:00:03.603
solving some exponential equations,

00:00:03.603 --> 00:00:05.867
and I have two different
exponential equations here.

00:00:05.867 --> 00:00:07.550
And like always, pause the video

00:00:07.550 --> 00:00:11.288
and see if you can solve
for x in both of them.

00:00:11.288 --> 00:00:13.889
All right, let's tackle
this one in purple first.

00:00:13.889 --> 00:00:16.239
And you might first notice that

00:00:16.239 --> 00:00:19.264
on both sides of the equation
I have different bases.

00:00:19.264 --> 00:00:21.562
So, it would be nice
to have a common base.

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And when you look at it, you're like,

00:00:22.967 --> 00:00:25.161
"Well, 32 is not a power of eight,

00:00:25.161 --> 00:00:27.483
"or at least it's not an
integer power of eight."

00:00:27.483 --> 00:00:30.652
But, they are both powers of two.

00:00:30.652 --> 00:00:35.017
32 is the same thing as
two to the fifth power,

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two to the fifth power,

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and eight is the same thing
as two to the third power,

00:00:40.653 --> 00:00:41.669
two to the third power.

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So, I can rewrite our original equation,

00:00:44.119 --> 00:00:46.255
as, instead of writing
32, I could write it

00:00:46.255 --> 00:00:48.911
as two to the fifth,

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and then that's going to be raised

00:00:50.620 --> 00:00:53.075
to the x over three power,

00:00:53.615 --> 00:00:55.112
X over three power,

00:00:55.112 --> 00:00:57.758
is equal to, instead of writing eight,

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I could write two to the third power,

00:01:00.347 --> 00:01:02.055
two to the third power,

00:01:02.055 --> 00:01:05.219
and I'm raising that to the x minus 12,

00:01:05.219 --> 00:01:07.267
x minus 12.

00:01:07.267 --> 00:01:08.718
Now if I raise something to a power,

00:01:08.718 --> 00:01:09.972
and then raise that to a power,

00:01:09.972 --> 00:01:12.154
I could just multiply these exponents.

00:01:12.154 --> 00:01:16.257
So, I can rewrite the
left-hand side as two

00:01:16.257 --> 00:01:20.675
to the five-x over three,

00:01:20.675 --> 00:01:23.612
five-x over three power.

00:01:23.612 --> 00:01:25.365
I just multiply these exponents.

00:01:25.365 --> 00:01:27.827
And that's going to be equal to

00:01:27.827 --> 00:01:29.916
two to the...

00:01:29.916 --> 00:01:34.856
And now I just multiply
the three times x minus 12.

00:01:34.856 --> 00:01:39.062
So, two to the three-x minus 36,

00:01:39.062 --> 00:01:40.746
and now things have simplified nicely.

00:01:40.746 --> 00:01:42.959
I have two to this power is equal to

00:01:42.959 --> 00:01:44.491
two to that power.

00:01:44.491 --> 00:01:47.343
So, these two exponents
must be equal to each other.

00:01:47.343 --> 00:01:50.827
Five-x over three must be equal to

00:01:50.827 --> 00:01:53.406
three-x minus 36.

00:01:53.406 --> 00:01:56.293
So, let's set them equal to
each other and solve for x.

00:01:56.299 --> 00:02:00.555
So five-x over three is equal to

00:02:00.555 --> 00:02:03.780
three-x minus 36.

00:02:03.780 --> 00:02:05.680
Let's see, we could,

00:02:05.680 --> 00:02:08.989
we could multiply, we could multiply

00:02:08.989 --> 00:02:11.439
everything by three.

00:02:11.439 --> 00:02:12.904
Let's do that.

00:02:12.904 --> 00:02:15.911
So if we multiply everything times three,

00:02:15.911 --> 00:02:19.371
here we're going to get five-x

00:02:19.371 --> 00:02:24.331
is equal to nine-x minus...

00:02:24.331 --> 00:02:24.908
What is this?

00:02:24.908 --> 00:02:27.648
Nine-x minus 108.

00:02:27.648 --> 00:02:31.049
And now we can subtract nine-x

00:02:31.049 --> 00:02:32.442
from both sides,

00:02:32.442 --> 00:02:34.787
and so we will get five-x minus nine-x

00:02:34.787 --> 00:02:37.938
is gonna be negative four-x is equal to

00:02:37.938 --> 00:02:40.260
negative 108.

00:02:40.260 --> 00:02:41.514
We're in the homestretch here.

00:02:41.514 --> 00:02:43.928
Divide, whoops, sorry about that.

00:02:43.928 --> 00:02:47.250
We could divide both
sides by negative four.

00:02:47.690 --> 00:02:49.210
Negative four.

00:02:49.210 --> 00:02:51.776
And we are left with x is equal to,

00:02:51.776 --> 00:02:53.773
what is this going to be, 27.

00:02:53.773 --> 00:02:56.095
X is equal to 27.

00:02:56.095 --> 00:02:58.869
And we are all done.

00:02:58.869 --> 00:02:59.811
We're all done.

00:02:59.811 --> 00:03:01.829
And if you had substituted
x back in there,

00:03:01.829 --> 00:03:04.786
you would get 32 to the 27 divided by 3,

00:03:04.786 --> 00:03:07.807
so 32 to the ninth
power, is the same thing

00:03:07.807 --> 00:03:11.674
as eight to the 27 minus 12th power.

00:03:11.674 --> 00:03:13.943
So eight to the 15th.

00:03:13.943 --> 00:03:16.871
Yeah, 27 minus 12,
eight to the 15th power.

00:03:16.871 --> 00:03:19.147
So, anyway, that was fun.

00:03:19.147 --> 00:03:21.473
Let's do the next one now.

00:03:21.473 --> 00:03:24.382
So this one looks
interesting in other ways.

00:03:24.382 --> 00:03:26.215
We have rational expressions.

00:03:26.215 --> 00:03:28.881
We have an exponential up
here, exponential down here.

00:03:28.881 --> 00:03:31.051
And the key realization here is...

00:03:31.051 --> 00:03:32.822
Well, the first thing I'd like to do.

00:03:32.822 --> 00:03:36.444
Let me write this 25 in terms of five.

00:03:36.444 --> 00:03:39.567
We know that 25 is the
same thing as five-squared.

00:03:39.567 --> 00:03:42.062
So, we can rewrite this as

00:03:42.872 --> 00:03:46.658
five to the four-x plus three

00:03:46.658 --> 00:03:49.864
over, instead of 25, I could rewrite that

00:03:49.864 --> 00:03:53.010
as five-squared, and then I'm gonna raise

00:03:53.010 --> 00:03:55.425
that to the nine minus x,

00:03:55.425 --> 00:03:58.222
to the nine minus x,

00:03:58.222 --> 00:03:59.546
and that, of course,
is going to be equal to

00:03:59.546 --> 00:04:03.087
five to the two-x plus five.

00:04:03.087 --> 00:04:04.944
Now five to the second, and then that

00:04:04.944 --> 00:04:06.662
to the nine minus x.

00:04:06.662 --> 00:04:08.880
I can just multiply these exponents.

00:04:08.880 --> 00:04:13.832
So, this is going to be five
to the four-x plus three

00:04:13.832 --> 00:04:18.068
over five to the...

00:04:18.068 --> 00:04:20.047
Two times nine is 18,

00:04:20.047 --> 00:04:23.704
two times negative-x is negative two-x,

00:04:23.704 --> 00:04:26.467
and that is going to be equal to,

00:04:26.467 --> 00:04:30.728
that is going to be equal to five

00:04:30.728 --> 00:04:34.048
to the two-x plus five.

00:04:34.048 --> 00:04:35.278
And now, let's see.

00:04:35.278 --> 00:04:37.147
There's multiple ways
that we could tackle it.

00:04:37.147 --> 00:04:39.254
We could multiply both
sides of this equation

00:04:39.254 --> 00:04:43.786
by five to the 18

00:04:43.786 --> 00:04:44.856
minus two-x.

00:04:44.856 --> 00:04:46.173
That's one way to do it.

00:04:46.173 --> 00:04:47.775
Or we could say, "Hey, look I have five

00:04:47.775 --> 00:04:49.353
"to some exponent divided by five

00:04:49.353 --> 00:04:51.687
"to some other exponent,
so I could just subtract

00:04:51.687 --> 00:04:54.589
"this blue exponent from this yellow one."

00:04:54.589 --> 00:04:57.223
So, the left-hand side will simplify to

00:04:57.793 --> 00:05:01.996
five to the four-x plus three

00:05:01.996 --> 00:05:06.430
minus, let me just do
minus in a neutral color,

00:05:06.430 --> 00:05:11.155
minus 18 minus two-x.

00:05:11.155 --> 00:05:13.210
18 minus two-x.

00:05:13.210 --> 00:05:15.648
and that, of course,
is going to be equal to

00:05:15.648 --> 00:05:17.505
what we've had on the right-hand side,

00:05:17.505 --> 00:05:20.849
five to the two-x plus five.

00:05:20.849 --> 00:05:23.554
Now we just have to simplify a little bit.

00:05:23.554 --> 00:05:25.574
Let's see, this is going to be...

00:05:25.574 --> 00:05:27.106
In fact, we could just say, "Look..."

00:05:27.106 --> 00:05:29.428
Now I'm having trouble
with my little pen tool.

00:05:29.428 --> 00:05:30.902
Whoops.

00:05:30.902 --> 00:05:31.773
All right.

00:05:31.773 --> 00:05:33.247
So now we could say this exponent needs

00:05:33.247 --> 00:05:34.269
to be equal to that exponent

00:05:34.269 --> 00:05:35.906
because we have the same base.

00:05:35.906 --> 00:05:40.182
And so, what we have here
on the left-hand side,

00:05:40.932 --> 00:05:44.891
that I could rewrite as four-x plus three

00:05:44.891 --> 00:05:48.792
minus 18 plus two-x.

00:05:48.792 --> 00:05:50.185
I'm just multiplying the negative times

00:05:50.185 --> 00:05:51.311
both of these terms.

00:05:51.311 --> 00:05:54.701
So, plus two-x, is going to be equal to

00:05:54.701 --> 00:05:56.929
two-x plus five.

00:05:56.929 --> 00:05:59.720
Two-x plus five.

00:05:59.720 --> 00:06:01.730
So, there's a bunch of different
things we could do here.

00:06:01.730 --> 00:06:04.766
One, we could subtract
two-x from both sides,

00:06:04.766 --> 00:06:06.205
that'll clean it up a little bit.

00:06:06.205 --> 00:06:07.749
Two-x from both sides.

00:06:07.749 --> 00:06:11.583
We could also subtract
five from both sides.

00:06:11.583 --> 00:06:13.345
So, let's just do that.

00:06:13.345 --> 00:06:17.083
So, well, let me just subtract it.

00:06:17.083 --> 00:06:19.265
Subtract five from both sides.

00:06:19.265 --> 00:06:20.821
I'm skipping some steps here,

00:06:20.821 --> 00:06:23.119
but I figure you're at this point

00:06:23.119 --> 00:06:25.956
reasonably comfortable
with linear equations.

00:06:25.956 --> 00:06:29.168
So then, on the left-hand side,

00:06:29.168 --> 00:06:31.722
we are going to have four-x,

00:06:31.722 --> 00:06:35.019
and then you have three
minus 18 minus five.

00:06:35.019 --> 00:06:39.296
Three minus 18 is negative 15,

00:06:39.296 --> 00:06:42.437
minus five, is negative 20,

00:06:42.437 --> 00:06:46.303
Is going to be equal to zero.

00:06:46.303 --> 00:06:48.462
And then, because those cancel out.

00:06:48.462 --> 00:06:49.843
And so add 20 to both sides,

00:06:49.843 --> 00:06:53.036
you get four-x is equal to 20.

00:06:53.036 --> 00:06:55.659
Divide both sides by four,

00:06:55.659 --> 00:06:58.724
and we get x is equal to five.

00:06:58.724 --> 00:07:00.846
And we are all done.


Solving exponential equations using exponent properties

WEBVTT
Kind: captions
Language: en

00:00:00.305 --> 00:00:01.385
- [Voiceover] Let's get some practice

00:00:01.385 --> 00:00:03.080
solving some exponential equations,

00:00:03.080 --> 00:00:04.485
and we have one right over here.

00:00:04.485 --> 00:00:09.041
We have 26 to the 9x plus five power

00:00:09.281 --> 00:00:10.681
equals one.

00:00:10.681 --> 00:00:13.644
So, pause the video and
see if you can tell me

00:00:13.644 --> 00:00:15.488
what x is going to be.

00:00:16.638 --> 00:00:20.006
Well, the key here is to realize that 26

00:00:20.006 --> 00:00:24.266
to the zeroth power, to the zeroth power

00:00:24.266 --> 00:00:25.677
is equal to one.

00:00:26.217 --> 00:00:29.107
Anything to the zeroth power is going

00:00:29.107 --> 00:00:30.907
to be equal to one.

00:00:30.907 --> 00:00:32.822
Zero to the zeroth power we can discuss

00:00:32.822 --> 00:00:34.505
at some other time, but anything other

00:00:34.505 --> 00:00:39.195
than zero to the zeroth
power is going to be one.

00:00:39.195 --> 00:00:41.947
So, we just have to
say, well, 9x plus five

00:00:41.947 --> 00:00:44.176
needs to be equal to zero.

00:00:44.886 --> 00:00:48.083
9x plus five

00:00:48.083 --> 00:00:51.143
needs to be equal to zero.

00:00:51.583 --> 00:00:53.115
And this is pretty
straightforward to solve.

00:00:53.115 --> 00:00:55.355
Subtract five from both sides.

00:00:55.355 --> 00:00:59.662
And we get 9x is equal to negative five.

00:00:59.662 --> 00:01:03.156
Divide both sides by nine,

00:01:03.156 --> 00:01:07.324
and we are left with x is
equal to negative five.

00:01:07.324 --> 00:01:08.868
Let's do another one of these,

00:01:08.868 --> 00:01:10.342
and let's make it a little bit more,

00:01:10.342 --> 00:01:12.362
a little bit more interesting.

00:01:12.362 --> 00:01:15.346
Let's say we have the exponential equation

00:01:15.346 --> 00:01:20.346
two to the 3x plus five power

00:01:21.044 --> 00:01:24.314
is equal to 64

00:01:24.314 --> 00:01:28.237
to the x minus seventh power.

00:01:28.667 --> 00:01:30.461
Once again, pause the video,

00:01:30.461 --> 00:01:33.943
and see if you can tell
me what x is going to be,

00:01:33.943 --> 00:01:38.390
or what x needs to be to satisfy
this exponential equation.

00:01:39.480 --> 00:01:41.811
All right, so you might, at first, say,

00:01:41.811 --> 00:01:43.305
oh, wait a minute,
maybe 3x plus five needs

00:01:43.305 --> 00:01:45.425
to be equal to x minus seven,
but that wouldn't work,

00:01:45.425 --> 00:01:47.038
because these are two different bases.

00:01:47.038 --> 00:01:49.140
You have two to the 3x plus five power,

00:01:49.140 --> 00:01:51.879
and then you have 64 to the x minus seven.

00:01:51.879 --> 00:01:54.619
So, the key here is to
express both of these

00:01:54.619 --> 00:01:57.662
with the same base, and lucky for us,

00:01:57.662 --> 00:02:01.271
64 is a power of two, two to the,

00:02:01.271 --> 00:02:03.780
let's see, two to the third is eight,

00:02:03.780 --> 00:02:07.123
so it's going to be two to the
third times two to the third.

00:02:07.123 --> 00:02:09.280
Eight times eight is 64,

00:02:09.280 --> 00:02:13.379
so it's two to the sixth is equal to 64,

00:02:13.379 --> 00:02:14.946
and you can verify that.

00:02:14.946 --> 00:02:16.607
Take six twos and multiply them together,

00:02:16.607 --> 00:02:17.918
you're going to get 64.

00:02:17.918 --> 00:02:19.346
This was just a little bit easier for me.

00:02:19.346 --> 00:02:22.063
Eight times eight, and
this is the same thing

00:02:22.063 --> 00:02:24.477
as two to the sixth power, is 64,

00:02:24.477 --> 00:02:26.056
and I knew it was two to the sixth power

00:02:26.056 --> 00:02:27.519
because I just added the exponents

00:02:27.519 --> 00:02:29.226
because I had the same base.

00:02:29.226 --> 00:02:31.884
All right, so I can rewrite 64.

00:02:31.884 --> 00:02:33.521
Let me rewrite the whole thing.

00:02:33.521 --> 00:02:37.236
So, this is two to the 3x plus five power

00:02:37.236 --> 00:02:40.498
is equal to, instead of writing 64,

00:02:40.498 --> 00:02:43.063
I'm going to write two to the sixth power,

00:02:43.063 --> 00:02:46.001
two to the sixth power, and then that

00:02:46.001 --> 00:02:48.798
to the x minus seventh power,

00:02:48.798 --> 00:02:51.376
x minus seven power.

00:02:52.006 --> 00:02:53.616
And to simplify this a little bit,

00:02:53.616 --> 00:02:55.381
we just have to remind ourselves

00:02:55.381 --> 00:02:59.339
that, if I raise something to one power,

00:02:59.339 --> 00:03:01.661
and then I raise that to another power,

00:03:01.661 --> 00:03:03.936
this is the same thing as raising my base

00:03:03.936 --> 00:03:05.573
to the product of these powers,

00:03:05.573 --> 00:03:07.361
a to the bc power.

00:03:07.361 --> 00:03:11.494
So, this equation I can rewrite as

00:03:11.494 --> 00:03:16.494
two to the 3x plus five is equal to

00:03:17.400 --> 00:03:20.433
two to the, and I just multiply six

00:03:20.433 --> 00:03:24.078
times x minus 7, so it's going

00:03:24.078 --> 00:03:26.339
to be 6x,

00:03:26.339 --> 00:03:30.629
6x minus six

00:03:30.629 --> 00:03:33.509
times seven is 42.

00:03:34.319 --> 00:03:35.943
I'll just write the whole thing in yellow.

00:03:35.943 --> 00:03:40.354
So, 6x minus 42, I just multiplied the six

00:03:40.354 --> 00:03:43.117
times the entire expression x minus 7.

00:03:43.117 --> 00:03:44.696
And so now it's interesting.

00:03:44.696 --> 00:03:46.380
I have two to the 3x plus 5 power

00:03:46.380 --> 00:03:50.140
has to be equal to two
to the 6x minus 42 power.

00:03:50.520 --> 00:03:52.903
So these need to be the same exponent.

00:03:54.003 --> 00:03:58.963
So, 3x plus five needs to
be equal to 6x minus 42.

00:03:59.773 --> 00:04:00.762
So there we go.

00:04:00.762 --> 00:04:03.073
It sets up a nice little
linear equation for us.

00:04:03.073 --> 00:04:07.101
3x plus five is equal to 6x minus 42.

00:04:07.921 --> 00:04:10.437
Let's see, we could get all of our,

00:04:10.857 --> 00:04:13.834
since, I'll put all my Xs
on the right hand side,

00:04:13.834 --> 00:04:15.692
since I have more Xs on the right already,

00:04:15.692 --> 00:04:19.012
so let me subtract 3x from both sides.

00:04:20.032 --> 00:04:22.715
And let me, I want to
get rid of this 42 here,

00:04:22.715 --> 00:04:25.696
so let's add 42 to both sides,

00:04:26.256 --> 00:04:28.832
and we are going to be left with

00:04:28.972 --> 00:04:32.696
five plus 42 is 47,

00:04:32.696 --> 00:04:35.936
is equal to, 47 is equal

00:04:35.936 --> 00:04:38.496
to 3x.

00:04:39.256 --> 00:04:42.081
Now we just divide both sides by three,

00:04:42.081 --> 00:04:47.006
and we are left with x is
equal to 47 over three.

00:04:47.736 --> 00:04:52.722
X is equal to 47 over three.

00:04:53.394 --> 00:04:55.072
And we are done.


Even & odd polynomials

WEBVTT
Kind: captions
Language: en

00:00:00.167 --> 00:00:01.525
- [Voiceover] So, we have
three functions here.

00:00:01.525 --> 00:00:02.791
What I want to do together

00:00:02.791 --> 00:00:04.010
is think about whether each of

00:00:04.010 --> 00:00:07.260
these functions are even or odd.

00:00:07.260 --> 00:00:09.443
And just as a little bit of a reminder,

00:00:09.443 --> 00:00:11.788
for an even function, if you were

00:00:11.788 --> 00:00:14.748
to input -x into the function,

00:00:14.748 --> 00:00:17.163
it's the same thing as inputing x.

00:00:17.163 --> 00:00:20.680
At f(-x) is the same thing as f(x).

00:00:20.680 --> 00:00:22.561
For an odd function,

00:00:22.561 --> 00:00:26.125
if you input -x into the function

00:00:26.125 --> 00:00:27.065
That's going to be the same thing

00:00:27.065 --> 00:00:31.987
as a negative of inputting
x into the function.

00:00:31.987 --> 00:00:33.845
And then, if it's neither, well then

00:00:33.845 --> 00:00:36.225
neither one of these are going to be true.

00:00:36.225 --> 00:00:37.803
So let's test this.

00:00:37.803 --> 00:00:39.742
So let's first look at h(x).

00:00:39.742 --> 00:00:41.042
And like always, pause the video

00:00:41.042 --> 00:00:45.407
and see if you can work
through this before I do.

00:00:45.407 --> 00:00:46.800
So, let's just see what happens

00:00:46.800 --> 00:00:50.283
if I try to evaluate h(-x).

00:00:50.283 --> 00:00:54.346
So it's going to be -10 times -x

00:00:54.346 --> 00:00:59.346
to the 11th power plus
-x to the ninth power

00:01:00.638 --> 00:01:05.638
minus -x to the third
power plus seven times -x.

00:01:07.743 --> 00:01:10.355
Well, what is that going to be equal to?

00:01:10.355 --> 00:01:14.697
Well, -x to the 11th power is going to be

00:01:14.697 --> 00:01:19.697
equal to the negative
of x to the 11th power.

00:01:19.840 --> 00:01:23.473
Because we have an odd
exponent right over here.

00:01:23.473 --> 00:01:25.714
Let me just restate what I just said.

00:01:25.714 --> 00:01:30.714
So negative, let me just, -x to the third,

00:01:31.681 --> 00:01:36.681
that's the same thing as the
negative of x to the third.

00:01:38.193 --> 00:01:41.374
We know it's going to
give you a negative value.

00:01:41.374 --> 00:01:46.157
Similarly, -x to the
seventh is going to be

00:01:46.157 --> 00:01:48.351
equal to the negative of x to the seventh.

00:01:48.351 --> 00:01:50.035
Try it out with some number here.

00:01:50.035 --> 00:01:53.540
If this was -1, -1 to the seventh power

00:01:53.540 --> 00:01:55.340
is the same thing as negative times

00:01:55.340 --> 00:01:57.615
one to the seventh power.

00:01:57.615 --> 00:02:01.690
So, let me write that over here.

00:02:01.690 --> 00:02:06.499
So this is going to be
-x to the 11th power.

00:02:08.547 --> 00:02:11.821
This is going to be -x to the ninth power.

00:02:11.821 --> 00:02:12.843
And here, while I'm writing this one,

00:02:12.843 --> 00:02:13.969
we're gonna do the exponents first

00:02:13.969 --> 00:02:15.942
and then put a negative in front of it.

00:02:15.942 --> 00:02:20.942
This is going to be negative
of x to the third power,

00:02:21.607 --> 00:02:24.127
and well this is just -x, so if you have

00:02:24.127 --> 00:02:27.423
- 10 times the negative of x to the 11th,

00:02:27.423 --> 00:02:30.326
negative times a negative
is going to be a positive.

00:02:30.326 --> 00:02:32.648
So this is going to be positive 10x

00:02:32.648 --> 00:02:37.495
to the 11th power, and then
minus x to the ninth power,

00:02:38.022 --> 00:02:40.809
minus x to the ninth power, and then

00:02:40.809 --> 00:02:43.781
you subtract a -x to the third,

00:02:43.781 --> 00:02:47.565
so that's going to be a
positive x to the third,

00:02:47.565 --> 00:02:50.776
and then this is going
to be minus seven x.

00:02:50.776 --> 00:02:52.599
And notice, what we have right over here

00:02:52.599 --> 00:02:54.224
is similar to what we have up here

00:02:54.224 --> 00:02:56.186
except all the signs are different.

00:02:56.186 --> 00:02:57.417
So this is the same thing as if I

00:02:57.417 --> 00:02:59.483
had took a negative and I multiplied

00:02:59.483 --> 00:03:03.836
it by h(x), so this is negative h(x).

00:03:03.836 --> 00:03:05.055
If I took each of these
and I multiplied them

00:03:05.055 --> 00:03:07.040
by a negative one, then I got what

00:03:07.040 --> 00:03:09.084
I just got here, which is h(-x).

00:03:09.084 --> 00:03:11.963
So we just saw that h(-x) is equal

00:03:11.963 --> 00:03:16.339
to negative h(x), and so we know

00:03:16.339 --> 00:03:20.553
that this is an odd function.

00:03:20.553 --> 00:03:22.318
And one telltale signature for it

00:03:22.318 --> 00:03:24.918
is it's made up of a
bunch of odd functions.

00:03:24.918 --> 00:03:27.298
We have an odd exponent over here.

00:03:27.298 --> 00:03:29.341
This is an odd, this is going to be an odd

00:03:29.341 --> 00:03:30.839
function if it was by itself.

00:03:30.839 --> 00:03:32.534
This is an odd function
if it was by itself.

00:03:32.534 --> 00:03:34.438
This is an odd function
if it was by itself.

00:03:34.438 --> 00:03:36.121
And so is that an odd function by itself.

00:03:36.121 --> 00:03:37.456
So if you add up a bunch of odd functions,

00:03:37.456 --> 00:03:40.103
you're going to get an odd function.

00:03:40.103 --> 00:03:43.934
All of these have odd exponents on them,

00:03:43.934 --> 00:03:46.232
which make them odd functions.

00:03:46.232 --> 00:03:49.332
So let's think about the
f(x) right over here.

00:03:49.332 --> 00:03:54.332
So, f(-x), well negative seven, well -x

00:03:54.951 --> 00:03:56.843
to the sixth power, you multiply

00:03:56.843 --> 00:04:00.117
a negative six times, it's
going to be a positive.

00:04:00.117 --> 00:04:02.172
So that's just going to be negative

00:04:02.172 --> 00:04:04.540
seven x to the sixth, and then plus three.

00:04:04.540 --> 00:04:06.537
Well negative x to the fourth power,

00:04:06.537 --> 00:04:08.440
I don't wanna skip a step here,

00:04:08.440 --> 00:04:11.772
so this is negative x to the sixth power

00:04:11.772 --> 00:04:16.772
plus three times negative
x to the fourth power

00:04:18.598 --> 00:04:23.283
minus nine times negative x squared.

00:04:23.416 --> 00:04:26.237
And then we have plus eight.

00:04:26.237 --> 00:04:28.153
Well, a negative number to the

00:04:28.153 --> 00:04:31.217
sixth power is going to
now be a positive number.

00:04:31.217 --> 00:04:32.309
So this is going to be equal to

00:04:32.309 --> 00:04:37.309
negative seven times the
same thing as x to the sixth.

00:04:37.509 --> 00:04:38.914
Plus, well, negative x to the fourth

00:04:38.914 --> 00:04:40.435
is same thing as x to the fourth,

00:04:40.435 --> 00:04:41.874
the negatives all cancel out, so you're

00:04:41.874 --> 00:04:43.848
gonna have three x to the fourth,

00:04:43.848 --> 00:04:46.112
then minus, negative x squared

00:04:46.112 --> 00:04:47.191
is the same thing as x squared,

00:04:47.191 --> 00:04:48.155
negative three squared is the same

00:04:48.155 --> 00:04:50.581
thing as three squared,
just giving an example.

00:04:50.581 --> 00:04:52.868
So it's going to be
negative nine x squared,

00:04:52.868 --> 00:04:54.145
the negatives here cancel out, plus eight.

00:04:54.145 --> 00:04:57.071
Well notice, this is
the same thing as f(x).

00:04:57.071 --> 00:05:02.071
So we have f(-x) is equal to f(x).

00:05:02.155 --> 00:05:05.313
So f(x) is even.

00:05:05.313 --> 00:05:06.648
And once again, this should not

00:05:06.648 --> 00:05:08.250
be a surprise, 'cause it's made

00:05:08.250 --> 00:05:12.406
up of a bunch of even
functions all added together.

00:05:12.406 --> 00:05:13.544
Each of these are symmetric about

00:05:13.544 --> 00:05:16.191
the y-axis, so you add 'em all together,

00:05:16.191 --> 00:05:17.711
you're going to get an even function.

00:05:17.711 --> 00:05:19.128
It's made up of a bunch of terms

00:05:19.128 --> 00:05:21.914
that all have even degrees.

00:05:21.914 --> 00:05:23.005
So it's the sixth degree, fourth degree,

00:05:23.005 --> 00:05:24.166
second degree; you could view this

00:05:24.166 --> 00:05:28.125
as a zero'th degree right over there.

00:05:28.125 --> 00:05:30.133
Now let's think about g(x).

00:05:30.133 --> 00:05:31.584
G(x) buried in here.

00:05:31.584 --> 00:05:32.989
And you might just be able to look

00:05:32.989 --> 00:05:34.103
at it, and say, "Okay, look, this is

00:05:34.103 --> 00:05:36.112
"an even function there, this is an

00:05:36.112 --> 00:05:40.047
"even function, but
this is an odd function,

00:05:40.047 --> 00:05:41.243
"and this is an odd function."

00:05:41.243 --> 00:05:44.203
Has a third degree term,
and a first degree term.

00:05:44.203 --> 00:05:46.107
So it's a mixture of
even and odd functions,

00:05:46.107 --> 00:05:50.089
so this is gonna be neither even nor odd.

00:05:50.089 --> 00:05:51.923
And you could test that out.

00:05:51.923 --> 00:05:55.951
We can look at g(-x) is going to be

00:05:55.951 --> 00:05:57.867
equal to, well -x to the fourth

00:05:57.867 --> 00:05:59.771
is the same thing as x to the fourth,

00:05:59.771 --> 00:06:02.046
so it's gonna be three x to the fourth.

00:06:02.046 --> 00:06:04.205
Negative x to the third is the

00:06:04.205 --> 00:06:07.108
negative of positive x to the third.

00:06:07.108 --> 00:06:12.108
So, that's going to be a
positive 10 x to the third.

00:06:13.307 --> 00:06:18.050
Actually let me just write
this, let me write it all out.

00:06:18.183 --> 00:06:20.806
So negative x to the fourth minus

00:06:20.806 --> 00:06:24.161
10 times negative x, to the third power,

00:06:24.161 --> 00:06:29.161
plus -x squared minus -x.

00:06:30.419 --> 00:06:33.181
So this is just x to the fourth,

00:06:33.181 --> 00:06:36.745
this is the negative of
positive x to the third,

00:06:36.745 --> 00:06:39.288
this is the same thing as x squared,

00:06:39.288 --> 00:06:41.377
and this, well, that's just -x,

00:06:41.377 --> 00:06:43.781
so it's going to be three
x to the fourth power.

00:06:43.781 --> 00:06:45.684
Negative 10 times the
negative of x to the third

00:06:45.684 --> 00:06:48.958
is positive 10 x to the third power

00:06:48.958 --> 00:06:53.323
plus x squared, and then you
subtract a negative plus x.

00:06:53.323 --> 00:06:56.620
So notice, this is neither g(x),

00:06:56.620 --> 00:06:57.920
it's definitely not g(x) 'cause

00:06:57.920 --> 00:07:00.126
we swapped the signs on the two

00:07:00.126 --> 00:07:05.126
odd-powered terms, we
swapped the signs over here,

00:07:05.280 --> 00:07:10.280
but it's also not the negative of g(x).

00:07:10.284 --> 00:07:12.141
We've only swapped signs on a few of them.

00:07:12.141 --> 00:07:13.140
And that's because we swapped signs

00:07:13.140 --> 00:07:16.344
on the odd terms, not on the even terms.

00:07:16.344 --> 00:07:17.795
So this one right over here is

00:07:17.795 --> 00:07:21.504
neither even nor odd.


Adding & subtracting rational expressions: like denominators

WEBVTT
Kind: captions
Language: en

00:00:00.408 --> 00:00:01.072
- [Voiceover] So let's add

00:00:01.072 --> 00:00:04.085
six over two X squared minus seven

00:00:04.085 --> 00:00:06.241
to negative 3 X minus eight

00:00:06.241 --> 00:00:09.238
over two X squared minus seven.

00:00:09.238 --> 00:00:10.438
And like always, pause the video

00:00:10.438 --> 00:00:12.807
and try to work it out before I do.

00:00:12.807 --> 00:00:13.659
When you look at this,

00:00:13.659 --> 00:00:15.507
we have these two rational expressions

00:00:15.507 --> 00:00:16.898
and we have the same denominator,

00:00:16.898 --> 00:00:18.361
two X squared minus seven.

00:00:18.361 --> 00:00:21.356
So you could say, we have six
two X squared minus sevenths

00:00:21.356 --> 00:00:24.087
and then we have negative
three X minus eight

00:00:24.087 --> 00:00:27.814
two X squared minus sevenths
is one way to think about it.

00:00:27.814 --> 00:00:29.243
So if you have the same denominator,

00:00:29.243 --> 00:00:30.882
this is going to be equal to,

00:00:30.882 --> 00:00:33.858
this is going to be equal to...

00:00:33.858 --> 00:00:37.761
our denominator is going to
be two X squared minus seven,

00:00:37.761 --> 00:00:41.103
two X squared minus seven,

00:00:41.103 --> 00:00:43.036
and then we just add the numerators.

00:00:43.036 --> 00:00:47.527
So it's going to be six
plus negative three X,

00:00:47.527 --> 00:00:51.673
negative three X minus eight.

00:00:51.673 --> 00:00:53.327
So if we want to simplify
this a little bit,

00:00:53.327 --> 00:00:55.972
we'd recognize that we can
add these two constant terms,

00:00:55.972 --> 00:00:58.586
the six and the negative eight.

00:00:58.586 --> 00:00:59.820
Six plus negative eight is going to be

00:00:59.820 --> 00:01:02.677
negative two, so it's
going to be negative two

00:01:02.677 --> 00:01:04.015
and then adding a negative three X,

00:01:04.015 --> 00:01:06.629
that's the same thing
as subtracting three X,

00:01:06.629 --> 00:01:10.663
so negative two minus
three X, all of that over,

00:01:10.663 --> 00:01:13.838
all of that with that same blue color,

00:01:13.838 --> 00:01:15.628
all of that over

00:01:15.628 --> 00:01:19.424
two X squared minus seven.

00:01:19.424 --> 00:01:20.192
And we're done.

00:01:20.192 --> 00:01:22.942
We've just added these
two rational expressions.

00:01:22.942 --> 00:01:24.824
Let's do another example.

00:01:24.824 --> 00:01:27.191
So here, we want to subtract

00:01:27.191 --> 00:01:29.213
one rational expression from another.

00:01:29.213 --> 00:01:31.580
So see if you can figure that out.

00:01:31.580 --> 00:01:33.845
Well, once again, both of
these rational expressions

00:01:33.845 --> 00:01:36.353
have the exact same denominator,

00:01:36.353 --> 00:01:37.703
the denominator for both of them is

00:01:37.703 --> 00:01:41.664
14 X squared minus nine,
14 X squared minus nine.

00:01:41.664 --> 00:01:45.040
So the denominator of the difference,

00:01:45.040 --> 00:01:46.225
I guess we can call it that,

00:01:46.225 --> 00:01:50.113
is going to be 14 X squared minus nine.

00:01:51.223 --> 00:01:54.367
So 14 X squared minus nine.

00:01:54.367 --> 00:01:55.382
Did I say four X squared before?

00:01:55.382 --> 00:01:57.120
14 X squared minus nine,

00:01:57.120 --> 00:01:58.477
that's the denominator of both of them,

00:01:58.477 --> 00:02:00.132
so that's going to be the denominator

00:02:00.132 --> 00:02:02.920
of our answer right over here.

00:02:02.920 --> 00:02:06.647
And so, we can just
subtract the numerators.

00:02:06.647 --> 00:02:07.897
So we're gonna have

00:02:07.897 --> 00:02:12.287
nine X squared plus three

00:02:12.287 --> 00:02:17.135
minus all of this business,

00:02:17.135 --> 00:02:22.001
minus negative three X squared plus five.

00:02:23.551 --> 00:02:25.849
And so we can distribute
the negative sign.

00:02:25.849 --> 00:02:27.679
This is going to be equal to

00:02:29.249 --> 00:02:33.043
nine X squared plus three, and then,

00:02:33.043 --> 00:02:36.353
if you distribute the negative sign,

00:02:36.353 --> 00:02:38.512
the negative of negative three X squared

00:02:38.512 --> 00:02:41.610
is going to be plus three X squared

00:02:41.610 --> 00:02:44.398
and then the negative of positive five

00:02:44.398 --> 00:02:45.284
is going to be negative five,

00:02:45.284 --> 00:02:47.270
so we're gonna subtract five from that,

00:02:47.270 --> 00:02:51.590
and all of that is going to be
over 14 X squared minus nine.

00:02:51.590 --> 00:02:55.071
14 X squared minus nine.

00:02:55.071 --> 00:02:57.615
And so in the numerator we
can do some simplification.

00:02:57.615 --> 00:03:01.654
We have nine X squared
plus three X squared,

00:03:01.654 --> 00:03:05.590
so that's going to be
equal to 12 X squared.

00:03:06.920 --> 00:03:09.561
And then, we have...

00:03:09.561 --> 00:03:13.672
we have three plus negative five,

00:03:13.672 --> 00:03:15.204
or we can say three minus five,

00:03:15.204 --> 00:03:17.085
so that's going to be negative two,

00:03:17.085 --> 00:03:20.044
and all of that is going to be over

00:03:20.044 --> 00:03:22.238
14 X squared minus nine.

00:03:22.238 --> 00:03:26.071
14 X squared minus nine.

00:03:26.071 --> 00:03:27.220
And we're all done.

00:03:27.220 --> 00:03:28.612
We have just subtracted.

00:03:28.612 --> 00:03:29.360
And we can think about it,

00:03:29.360 --> 00:03:31.347
is there any way we
can simplify this more,

00:03:31.347 --> 00:03:32.670
are there any common factors,

00:03:32.670 --> 00:03:35.420
but these both could be
considered differences of squares,

00:03:35.420 --> 00:03:37.301
but they're going to be
differences of squares

00:03:37.301 --> 00:03:38.835
of different things,

00:03:38.835 --> 00:03:40.367
so they're not going
to have common factors.

00:03:40.367 --> 00:03:43.404
So this is about as simple as we can get.


Multiplying & dividing rational expressions: monomials

WEBVTT
Kind: captions
Language: en

00:00:00.433 --> 00:00:02.435
- [Voiceover] So up
here, we are multiplying

00:00:02.435 --> 00:00:03.436
two rational expressions.

00:00:03.436 --> 00:00:05.438
And here, we're dividing
one rational expression

00:00:05.438 --> 00:00:06.639
by another one.

00:00:06.639 --> 00:00:08.641
Now what I encourage you
do is pause this video

00:00:08.641 --> 00:00:10.977
and think about what these
become when you multiply them.

00:00:10.977 --> 00:00:12.979
I don't know, maybe you
simplify it a little bit,

00:00:12.979 --> 00:00:15.749
and I also want you to
think about what constraints

00:00:15.749 --> 00:00:17.984
do you have to put on the x values

00:00:17.984 --> 00:00:20.286
in order for your resulting expression

00:00:20.286 --> 00:00:23.883
to be algebraically equivalent
to your original expression.

00:00:23.883 --> 00:00:26.426
So let's work it out
together just so you realize

00:00:26.426 --> 00:00:28.094
what I'm talking about.

00:00:28.094 --> 00:00:30.296
So this is going to be, in our numerator,

00:00:30.296 --> 00:00:34.367
we are going to get six x to
the third power times two,

00:00:34.367 --> 00:00:37.804
and our denominator,
we're going to have five

00:00:37.804 --> 00:00:40.106
times three x.

00:00:40.106 --> 00:00:42.609
And we can see both the
numerator and the denominator

00:00:42.609 --> 00:00:46.713
are divisible by x, so let's
divide the denominator by x.

00:00:46.713 --> 00:00:47.714
We get one there.

00:00:47.714 --> 00:00:49.849
Let's divide x to the third by x.

00:00:49.849 --> 00:00:51.851
We get x squared.

00:00:51.851 --> 00:00:54.587
And we can also see the both
the numerator and denominator

00:00:54.587 --> 00:00:57.257
are divisible by three,
so divide six by three,

00:00:57.257 --> 00:00:58.258
you get two.

00:00:58.258 --> 00:01:00.293
Divide three by three, you get one.

00:01:00.293 --> 00:01:05.063
And we are left with
two x squared times two,

00:01:05.063 --> 00:01:08.535
which is going to be four x squared

00:01:08.535 --> 00:01:11.638
over five times one times one over five.

00:01:11.638 --> 00:01:16.476
And we can also write
that as 4/5 x squared.

00:01:16.476 --> 00:01:18.678
Now someone just presented
you on the street

00:01:18.678 --> 00:01:20.680
with the expression 4/5 x squared and say,

00:01:20.680 --> 00:01:22.015
for what x is this defined?

00:01:22.015 --> 00:01:24.017
I could put any x here, x could be zero

00:01:24.017 --> 00:01:26.453
because zero squared is zero times 4/5

00:01:26.453 --> 00:01:28.521
is just going to be zero, so
it does seems to be defined

00:01:28.521 --> 00:01:30.123
for zero, and that is true.

00:01:30.123 --> 00:01:33.092
But if someone says, how
would I have to constrain this

00:01:33.092 --> 00:01:35.962
in order for it to be
algebraically equivalent

00:01:35.962 --> 00:01:38.531
to this first expression?

00:01:38.531 --> 00:01:40.533
Well then, you'd have to say,
well, this first expression

00:01:40.533 --> 00:01:42.936
is not defined for all x.

00:01:42.936 --> 00:01:45.605
For example, if x were equal
to zero, then you would be

00:01:45.605 --> 00:01:47.607
dividing by zero right over here,

00:01:47.607 --> 00:01:49.609
which would make this undefined.

00:01:49.609 --> 00:01:51.811
So you could explicitly
call it out, x can not be

00:01:51.811 --> 00:01:53.079
equal to zero.

00:01:53.079 --> 00:01:55.882
And so if you want this to
be algebraically equivalent,

00:01:55.882 --> 00:01:59.052
you would have to make
that same condition,

00:01:59.052 --> 00:02:01.855
x cannot be equal to zero.

00:02:01.855 --> 00:02:04.090
Another way to think about
it, if you had a function

00:02:04.090 --> 00:02:07.894
defined this way, if you
said, if you said f of x

00:02:07.894 --> 00:02:11.364
is equal to six x to the third over five

00:02:11.364 --> 00:02:15.969
times two over, times two over three x;

00:02:15.969 --> 00:02:18.771
and if someone said,
well what is f of zero,

00:02:18.771 --> 00:02:21.774
you would say f of zero is undefined.

00:02:22.785 --> 00:02:24.911
Undefined.

00:02:24.911 --> 00:02:25.912
Why is that?

00:02:25.912 --> 00:02:27.537
Because you put x equals zero
there, you're going to get

00:02:27.537 --> 00:02:29.916
two divided by zero and it's undefined.

00:02:29.916 --> 00:02:31.951
But if you said, okay,
well, can I simplify this

00:02:31.951 --> 00:02:34.220
a little bit to get the
exact same function?

00:02:34.220 --> 00:02:38.291
Well, we're saying you can
say f of x is equal to 4/5

00:02:38.291 --> 00:02:40.894
times x squared.

00:02:40.894 --> 00:02:43.963
But if you just left it at
that, you would get f of zero

00:02:43.963 --> 00:02:45.298
is equal to zero.

00:02:45.298 --> 00:02:48.001
So now it would be defined
at zero, but then this would

00:02:48.001 --> 00:02:49.002
make it a different function.

00:02:49.002 --> 00:02:51.004
These are two different
functions the way they're written

00:02:51.004 --> 00:02:52.005
right over here.

00:02:52.005 --> 00:02:55.475
Instead, to make them, to make it clear

00:02:55.475 --> 00:02:58.077
that this is equivalent to
that one, you would have to say

00:02:58.077 --> 00:03:01.381
x cannot be equal to zero.

00:03:01.381 --> 00:03:03.616
Now these functions are
equivalent because now,

00:03:03.616 --> 00:03:06.486
if u said f of zero, you'd say all right,

00:03:06.486 --> 00:03:08.621
x cannot be equal to zero, you know?

00:03:08.621 --> 00:03:11.524
This would be the case if x
is anything other than zero

00:03:11.524 --> 00:03:15.395
and it's not defined for
zero, and so you would say

00:03:15.395 --> 00:03:17.130
f or zero is undefined.

00:03:17.130 --> 00:03:20.033
So now, these two
functions are equivalent,

00:03:20.033 --> 00:03:23.303
or these two expressions are
algebraically equivalent.

00:03:23.303 --> 00:03:25.705
So thinking about that,
let's tackle this division

00:03:25.705 --> 00:03:27.140
situation here.

00:03:27.140 --> 00:03:29.175
So immediately, when you
look at this, you say, woah,

00:03:29.175 --> 00:03:30.910
what are constrains here?

00:03:30.910 --> 00:03:33.746
Well, x cannot be equal to
zero because if x was a zero,

00:03:33.746 --> 00:03:36.716
this second, this five x
to the fourth over four

00:03:36.716 --> 00:03:39.218
would be zero and you'd be
dividing, you'd be dividing

00:03:39.218 --> 00:03:40.219
by zero.

00:03:40.219 --> 00:03:44.791
So we can explicitly call
out that x cannot be equal

00:03:44.791 --> 00:03:45.792
to zero.

00:03:45.792 --> 00:03:47.208
And so if x cannot be equal to zero

00:03:47.208 --> 00:03:49.098
in the original expression, if the result,

00:03:49.098 --> 00:03:50.894
whatever we get for the
resulting expression,

00:03:50.894 --> 00:03:53.089
in order for it to be
algebraically equivalent,

00:03:53.089 --> 00:03:57.203
we have to give this same constraint.

00:03:57.203 --> 00:03:59.739
So let's multiply this,
or let's do the division.

00:03:59.739 --> 00:04:03.543
So this is going to be
the same thing as two x

00:04:03.543 --> 00:04:07.614
to the fourth power over
seven times the reciprocal,

00:04:07.614 --> 00:04:09.015
times...

00:04:10.423 --> 00:04:13.953
The reciprocal of this is
going to be four over five

00:04:13.953 --> 00:04:16.522
x to the fourth, which
is going to be equal to

00:04:16.522 --> 00:04:21.094
in the numerator, we're going
to have eight x to the fourth.

00:04:21.094 --> 00:04:23.730
So we're going to have
eight x to the fourth,

00:04:23.730 --> 00:04:25.732
four times two x to the fourth,

00:04:25.732 --> 00:04:29.102
over seven times five
x to the fourth is 35

00:04:29.102 --> 00:04:30.803
x to the fourth.

00:04:30.803 --> 00:04:31.804
And now, there's something.

00:04:31.804 --> 00:04:33.773
We can do a little bit
of simplification here,

00:04:33.773 --> 00:04:35.190
both the numerator and the
denominator are divisible

00:04:35.190 --> 00:04:37.777
by x to the fourth, so let's
divide by x to the fourth

00:04:37.777 --> 00:04:40.680
and we get eight over 35.

00:04:40.680 --> 00:04:42.052
So once again, you just look at eight 30,

00:04:42.052 --> 00:04:44.751
Well, this is going to
be defined for any x.

00:04:44.751 --> 00:04:46.886
X isn't even involved in the expression.

00:04:46.886 --> 00:04:50.390
But if we want this to be
algebraically equivalent

00:04:50.390 --> 00:04:52.625
to this first expression,
then we have to make

00:04:52.625 --> 00:04:56.868
the same constraint, x does
not, cannot be equal to zero.

00:04:57.196 --> 00:04:58.729
And to see, you know, this
even seems a little bit

00:04:58.729 --> 00:05:01.234
more nonsensical to say
x cannot be equal to zero

00:05:01.234 --> 00:05:03.236
for an expression that
does not even involve x.

00:05:03.236 --> 00:05:05.672
But one way to think about
it is imagine a function

00:05:05.672 --> 00:05:08.557
that was defined as g of x

00:05:09.142 --> 00:05:12.712
is equal to, is equal to
all of this business here.

00:05:12.712 --> 00:05:15.581
Well, g of zero would be undefined.

00:05:15.581 --> 00:05:20.053
But if you said g of x is equal to 8/35,

00:05:20.053 --> 00:05:22.522
well now, g of zero
would be defined as 8/35,

00:05:22.522 --> 00:05:24.524
which would make it a different function.

00:05:24.524 --> 00:05:27.160
So to make them algebraically equivalent,

00:05:27.160 --> 00:05:29.495
you could say g of x is equal to

00:05:29.495 --> 00:05:31.616
8/35,

00:05:31.616 --> 00:05:35.300
as long as x does not equal zero.

00:05:35.300 --> 00:05:38.004
And you could say it's
undefined if you want.

00:05:38.004 --> 00:05:40.239
Undefined for x equals zero.

00:05:40.239 --> 00:05:42.241
Or you don't even have to
include that second row,

00:05:42.241 --> 00:05:45.373
and that will literally
just make it undefined.

00:05:45.373 --> 00:05:48.281
But now, this expression,

00:05:48.281 --> 00:05:50.435
this algebraic expression, is equivalent

00:05:50.435 --> 00:05:54.085
to our original one even
though we had simplified it.


Extraneous solutions of radical equations (example 2)

WEBVTT
Kind: captions
Language: en

00:00:00.240 --> 00:00:02.175
- [Voiceover] We're
asked, which value for D,

00:00:02.175 --> 00:00:03.924
and we see D in this equation here,

00:00:03.924 --> 00:00:08.677
makes X equals negative
three an extraneous solution

00:00:08.677 --> 00:00:10.377
for this radical equation?

00:00:10.377 --> 00:00:14.954
Squared of 3x plus 25
is equal to D plus 2x.

00:00:15.564 --> 00:00:17.609
And I encourage you to
pause the video and try to

00:00:17.609 --> 00:00:19.099
think about it on your own before we work

00:00:19.099 --> 00:00:20.499
through it together.

00:00:20.979 --> 00:00:23.930
Alright, now let's work
through this together.

00:00:23.930 --> 00:00:25.868
So the first thing that
just to remind ourselves

00:00:25.868 --> 00:00:27.720
is what is an extraneous solution?

00:00:27.720 --> 00:00:30.829
Well that's a solution that
we get or we think we get

00:00:30.829 --> 00:00:33.489
but it's really just a
by-product of how we solved it

00:00:33.489 --> 00:00:35.780
but isn't going to be an actual solution

00:00:35.780 --> 00:00:37.596
of our original equation.

00:00:37.596 --> 00:00:41.236
Now how do these extraneous
solutions pop up?

00:00:41.236 --> 00:00:44.759
Well, it pops up when you
take the square of both sides.

00:00:45.288 --> 00:00:47.333
So for this equation right
over here, to get rid

00:00:47.333 --> 00:00:49.919
of the radical, I'd want
to square both sides of it.

00:00:49.919 --> 00:00:52.731
If I square both sides, the
lefthand side will become

00:00:52.731 --> 00:00:57.345
3x plus 25 and the righthand
side, if I square this,

00:00:57.725 --> 00:00:59.113
is going to be, what?

00:00:59.113 --> 00:01:04.113
It's going to be D squared
plus 4, 4dx plus X squared.

00:01:06.501 --> 00:01:08.787
So that's just squaring
both sides of this,

00:01:08.887 --> 00:01:11.862
but notice, there's actually
a different equation than

00:01:11.862 --> 00:01:13.786
this one that if you squared
both sides, you would also

00:01:13.786 --> 00:01:14.772
get this.

00:01:14.772 --> 00:01:16.692
What is that different equation?

00:01:16.692 --> 00:01:18.753
Well the different equation
is if you took the negative

00:01:18.753 --> 00:01:20.549
of one of these sides.

00:01:20.549 --> 00:01:22.544
So for example, if you had,
if you started the original

00:01:22.544 --> 00:01:27.425
equation, the negative
square root of 3x plus 25

00:01:27.869 --> 00:01:32.035
plus 25, is equal to D plus 2x.

00:01:32.606 --> 00:01:35.508
You square both sides of this,
you still get this purple

00:01:35.668 --> 00:01:37.922
equation because you square a negative,

00:01:37.922 --> 00:01:39.765
you get a positive.

00:01:39.765 --> 00:01:42.471
So both of them, when
you square both sides,

00:01:42.471 --> 00:01:44.485
get us over here.

00:01:45.016 --> 00:01:47.021
And so when you solve
this purple equation,

00:01:47.021 --> 00:01:49.185
this is a quadratic right over here.

00:01:49.185 --> 00:01:50.252
You just rearrange it a little bit.

00:01:50.252 --> 00:01:52.103
You get into standard quadratic form,

00:01:52.103 --> 00:01:53.702
you'll get two solutions.

00:01:53.702 --> 00:01:55.259
And it turns out, one of
the solutions is going to be

00:01:55.259 --> 00:01:57.284
for this yellow equation
and one of the solutions

00:01:57.284 --> 00:01:58.788
is going to be for the purple equation.

00:01:59.344 --> 00:02:04.184
And if the solution that
is for the purple equation,

00:02:04.401 --> 00:02:06.425
is going to be an extraneous solution

00:02:06.511 --> 00:02:07.817
for the yellow equation.

00:02:07.817 --> 00:02:09.087
This is actually not
going to be a solution

00:02:09.087 --> 00:02:10.744
for the yellow equation.

00:02:10.821 --> 00:02:13.242
So when they say, "Which
value for D makes X equals

00:02:13.242 --> 00:02:15.314
"negative three and extraneous solution

00:02:15.314 --> 00:02:16.726
"for this yellow equation?"

00:02:17.070 --> 00:02:18.201
That's the same thing as saying,

00:02:18.201 --> 00:02:20.658
"What value of D makes
X equals negative three

00:02:20.658 --> 00:02:23.285
"a solution for this?"

00:02:24.105 --> 00:02:28.196
So a solution for this.

00:02:31.026 --> 00:02:32.990
If it's a solution for this,
it's going to be an extraneous

00:02:32.990 --> 00:02:35.442
solution for that 'cause these
are two different equations.

00:02:36.902 --> 00:02:38.255
We're taking the negative of just one side

00:02:38.255 --> 00:02:39.864
of this equation to get this one.

00:02:39.981 --> 00:02:41.588
If you took the negative
of both sides of this

00:02:41.588 --> 00:02:42.964
and that becomes the same
thing 'cause you could

00:02:42.964 --> 00:02:45.342
multiply both sides of an
equation times a negative value.

00:02:45.867 --> 00:02:49.128
So a solution for this, which
is equivalently a solution

00:02:49.128 --> 00:02:51.643
which is equivalent to a
solution, if I, instead of putting

00:02:51.643 --> 00:02:53.723
the negative on the lefthand
side, if I multiply the

00:02:53.723 --> 00:02:54.937
righthand side by the negative.

00:02:55.136 --> 00:02:57.711
But anyway, let's think
about which value for D

00:02:57.711 --> 00:03:02.232
makes X equals negative
three a solution for this?

00:03:03.111 --> 00:03:05.286
Well, let's substitute X
equals negative three here

00:03:05.286 --> 00:03:07.180
and then we just have to solve for D.

00:03:07.180 --> 00:03:09.468
If X equals negative three,
this is going to be negative

00:03:09.468 --> 00:03:12.791
the square root of three
times negative three

00:03:12.791 --> 00:03:17.287
is negative nine, plus 25 is equal to D.

00:03:18.445 --> 00:03:21.022
Two times negative three is negative six.

00:03:21.444 --> 00:03:26.444
So D minus 6, and so now we can

00:03:26.932 --> 00:03:31.387
square both sides of, we can square,

00:03:31.545 --> 00:03:32.485
actually let's do it this way.

00:03:32.485 --> 00:03:34.172
We can, I don't wanna
square both sides 'cause

00:03:34.172 --> 00:03:35.486
we lose some information.

00:03:35.486 --> 00:03:38.237
It's going to be the negative
square root of negative nine

00:03:38.237 --> 00:03:42.392
plus 25 is 16, is equal to D minus 6,

00:03:43.012 --> 00:03:46.983
so this is going to be
equal to negative four.

00:03:47.392 --> 00:03:48.549
Principle over 16 is four.

00:03:48.549 --> 00:03:49.982
We have the negative out front,

00:03:49.982 --> 00:03:54.029
is equal to D minus six and
then add six to both sides.

00:03:54.763 --> 00:03:58.894
You get two is equal
to, two is equal to D.

00:03:59.621 --> 00:04:01.816
So if D is equal to two here,

00:04:01.816 --> 00:04:05.179
if D is equal to two, then
a solution to this purple

00:04:05.179 --> 00:04:08.041
equation is going to be
X equals negative three.

00:04:08.689 --> 00:04:10.454
And so that would be an
extraneous solution because

00:04:10.624 --> 00:04:13.612
if X equals negative three
satisfies this over here,

00:04:13.612 --> 00:04:16.280
it's definitely gonna
satisfy this over here,

00:04:16.280 --> 00:04:19.240
but it's not going to
satisfy this up here.

00:04:19.401 --> 00:04:21.205
And you could verify this if this is equal

00:04:23.045 --> 00:04:25.856
to two, try out X equals negative three.

00:04:25.856 --> 00:04:27.848
You're gonna get on the lefthand side,

00:04:27.848 --> 00:04:30.816
you're going to get 16
and on the righthand side,

00:04:30.816 --> 00:04:34.625
you're gonna get two minus six which is

00:04:34.625 --> 00:04:36.355
equal to negative four.

00:04:37.574 --> 00:04:39.209
Two minus six which is negative four,

00:04:39.209 --> 00:04:40.520
so this does not work out.

00:04:40.520 --> 00:04:42.776
X equals negative three
is not a solution to this,

00:04:42.776 --> 00:04:45.639
but it is a solution for
this and it is a solution

00:04:45.639 --> 00:04:47.570
to this quadratic right over here.

00:04:47.730 --> 00:04:50.662
So D equals two makes
X equal negative three,

00:04:50.662 --> 00:04:54.172
an extraneous solution for this equation.


Extraneous solutions of radical equations

WEBVTT
Kind: captions
Language: en

00:00:00.167 --> 00:00:02.384
- [Voiceover] So let's say
we have the radical equation

00:00:02.384 --> 00:00:06.006
two-x minus one is equal

00:00:06.006 --> 00:00:10.429
to the square root of eight minus x.

00:00:10.429 --> 00:00:13.041
So we already have the
radical isolated on one side

00:00:13.041 --> 00:00:13.842
of the equation.

00:00:13.842 --> 00:00:15.696
So, we might say, "Well, let's
just get rid of the radical.

00:00:15.696 --> 00:00:18.126
"Let's square both
sides of this equation."

00:00:18.126 --> 00:00:19.728
So we might say that
this is the same thing

00:00:19.728 --> 00:00:22.967
as two-x minus one squared

00:00:22.967 --> 00:00:27.776
is equal to the square
root of eight minus x,

00:00:27.776 --> 00:00:30.447
eight minus x squared,
and then we would get,

00:00:30.447 --> 00:00:31.999
let's see, two-x minus one squared

00:00:31.999 --> 00:00:35.006
is four x-squared minus four-x

00:00:35.006 --> 00:00:38.922
plus one is equal to eight minus x.

00:00:38.922 --> 00:00:42.788
Now we have to be very,
very, very careful here.

00:00:42.788 --> 00:00:45.187
We might feel, "Okay we
did legitimate operations.

00:00:45.187 --> 00:00:47.044
"We did the same thing to both sides.

00:00:47.044 --> 00:00:50.721
"That these are equivalent equations."

00:00:50.721 --> 00:00:52.907
But, they aren't quite equivalent.

00:00:52.907 --> 00:00:54.346
Because when you're squaring something,

00:00:54.346 --> 00:00:55.228
one way to think about it,

00:00:55.228 --> 00:00:58.084
is when you're squaring it,
you're losing information.

00:00:58.084 --> 00:01:00.615
So, for example, this would be true

00:01:00.615 --> 00:01:04.566
even if the original
equation were two-x...

00:01:04.566 --> 00:01:06.199
Let me, this in a different color.

00:01:06.199 --> 00:01:09.159
Even if the original
equation were two-x minus one

00:01:09.159 --> 00:01:10.819
is equal to the negative

00:01:10.819 --> 00:01:15.463
of the square root of eight minus x.

00:01:15.463 --> 00:01:17.959
Because if you squared both sides of this,

00:01:17.959 --> 00:01:19.642
you would also get,

00:01:19.642 --> 00:01:21.314
you would also get that right over there,

00:01:21.314 --> 00:01:25.036
because a negative squared
would be equal to a positive.

00:01:25.036 --> 00:01:27.293
So, when we're finding a solution to this,

00:01:27.293 --> 00:01:28.616
we need to test our solution

00:01:28.616 --> 00:01:30.032
to make sure it's truly the solution

00:01:30.032 --> 00:01:32.319
to this first yellow equation here,

00:01:32.319 --> 00:01:34.920
and not the solution to this up here.

00:01:34.920 --> 00:01:36.696
If it's a solution to
this right-hand side,

00:01:36.696 --> 00:01:37.880
and not the yellow one,

00:01:37.880 --> 00:01:41.363
then we would call that
an extraneous solution.

00:01:41.363 --> 00:01:43.661
So, let's see if we can solve this.

00:01:43.661 --> 00:01:46.935
So let's write this as kind
of a standard quadratic.

00:01:46.935 --> 00:01:50.557
Let's subtract eight from both sides.

00:01:50.557 --> 00:01:53.552
So let's subtract eight from both sides

00:01:53.552 --> 00:01:54.817
to get rid of this eight over here,

00:01:54.817 --> 00:01:57.731
and let's add x to both sides.

00:01:57.731 --> 00:02:00.494
So, plus x, plus x,

00:02:00.494 --> 00:02:02.816
and we are going to get,

00:02:02.816 --> 00:02:06.612
we are going to get four-x squared

00:02:06.612 --> 00:02:11.476
minus three-x, minus seven,

00:02:11.476 --> 00:02:14.750
minus seven, is equal to,

00:02:14.750 --> 00:02:17.571
is equal to zero.

00:02:17.571 --> 00:02:18.999
And let's see, we would want

00:02:18.999 --> 00:02:22.238
to factor this right over here,

00:02:22.238 --> 00:02:24.931
and, let's see, maybe I could do this

00:02:24.931 --> 00:02:27.857
by, if I do it by...

00:02:27.857 --> 00:02:30.573
Well, I'll just use the
quadratic formula, here.

00:02:30.573 --> 00:02:32.952
So the solutions are going to be...

00:02:33.452 --> 00:02:37.973
X is going to be equal
to negative b, so three,

00:02:37.973 --> 00:02:40.847
plus or minus the square root of

00:02:40.847 --> 00:02:43.715
b-squared, so negative
three squared is nine,

00:02:43.715 --> 00:02:47.755
minus four times a, which is four,

00:02:47.755 --> 00:02:50.146
times c, which is negative seven.

00:02:50.146 --> 00:02:51.992
So I could just say times...

00:02:51.992 --> 00:02:54.534
Well, I'll just write a seven here

00:02:54.534 --> 00:02:57.088
and then that negative is
gonna make this a positive.

00:02:57.088 --> 00:02:58.528
All of that over two-a.

00:02:58.528 --> 00:03:00.408
So two times four is eight.

00:03:00.408 --> 00:03:01.790
So, this is gonna be three

00:03:01.790 --> 00:03:04.855
plus or minus the square root of...

00:03:04.855 --> 00:03:09.713
Let's see, four times
four is 16 times seven.

00:03:09.713 --> 00:03:13.027
16 times seven is gonna be

00:03:13.027 --> 00:03:15.326
70 plus 42.

00:03:15.326 --> 00:03:17.090
Let me make sure I'm doing this right.

00:03:17.090 --> 00:03:19.354
So 16 times seven.

00:03:19.354 --> 00:03:21.827
The two, four.

00:03:21.827 --> 00:03:24.973
So, it's 112 plus nine.

00:03:24.973 --> 00:03:27.933
So, 121, that worked out nicely.

00:03:27.933 --> 00:03:30.232
So, plus or minus the square root of 121,

00:03:30.232 --> 00:03:31.927
all of that over eight.

00:03:31.927 --> 00:03:34.945
Well, that is equal to
three plus or minus 11,

00:03:34.945 --> 00:03:37.399
all of that over eight.

00:03:37.399 --> 00:03:40.250
So that is equal to, if we add 11,

00:03:40.250 --> 00:03:42.607
that is 14-eights.

00:03:42.607 --> 00:03:44.627
Or, if we subtract 11,

00:03:44.627 --> 00:03:48.969
three minus 11 is negative eight.

00:03:48.969 --> 00:03:51.523
Negative eight divided
by eight is negative one.

00:03:51.523 --> 00:03:52.451
So we have to think about...

00:03:52.451 --> 00:03:53.926
You might say, "Okay,
I found two solutions

00:03:53.926 --> 00:03:55.087
"to the radical equation."

00:03:55.087 --> 00:03:57.188
But remember, one of
these might be solutions

00:03:57.188 --> 00:03:58.860
to this alternate radical equation

00:03:58.860 --> 00:04:01.170
that got lost when we squared both sides.

00:04:01.170 --> 00:04:03.283
We have to make sure
that they're legitimate,

00:04:03.283 --> 00:04:05.686
or maybe one of these is
an extraneous solution.

00:04:05.686 --> 00:04:07.183
In fact, one is very likely a solution

00:04:07.183 --> 00:04:10.859
to this radical equation which
wasn't our original goal.

00:04:10.859 --> 00:04:12.326
So, let's see.

00:04:12.326 --> 00:04:15.205
Let's try out x equals negative one.

00:04:15.205 --> 00:04:16.621
If x equals negative one,

00:04:16.621 --> 00:04:19.249
we would have two times negative one

00:04:19.249 --> 00:04:22.426
minus one is equal to
the square root of eight

00:04:22.426 --> 00:04:24.239
minus negative one.

00:04:24.239 --> 00:04:27.988
So that would be negative
two minus one is equal

00:04:27.998 --> 00:04:30.784
to the square root of,

00:04:30.784 --> 00:04:32.561
is equal to the square root of nine.

00:04:32.561 --> 00:04:34.105
And so we'd have negative three

00:04:34.105 --> 00:04:36.101
is equal to the square root of nine.

00:04:36.101 --> 00:04:37.013
The principle root of nine.

00:04:37.013 --> 00:04:38.342
This is a positive square root.

00:04:38.342 --> 00:04:39.607
This is not true.

00:04:39.607 --> 00:04:40.548
So this, right over here,

00:04:40.548 --> 00:04:43.229
that is an extraneous solution.

00:04:43.229 --> 00:04:45.539
Extraneous.

00:04:45.539 --> 00:04:48.616
Extraneous solution.

00:04:48.616 --> 00:04:51.715
It is a solution to this
one right over here.

00:04:51.715 --> 00:04:53.996
Because notice, for that one,

00:04:53.996 --> 00:04:57.323
if you substitute two times negative one

00:04:57.323 --> 00:04:59.540
minus one is equal to the negative

00:04:59.540 --> 00:05:02.500
of eight minus negative one.

00:05:02.500 --> 00:05:04.453
So this is negative three is equal

00:05:04.453 --> 00:05:07.260
to the negative of three.

00:05:07.260 --> 00:05:09.640
So, it checks out for this one.

00:05:09.640 --> 00:05:12.493
So, this one right over here
is the extraneous solution.

00:05:12.493 --> 00:05:14.913
This one right over here is
gonna be the actual solution

00:05:14.913 --> 00:05:16.524
for our original equation,

00:05:16.524 --> 00:05:18.761
and you can test it out on your own.

00:05:18.761 --> 00:05:21.209
In fact, I encourage you to do so.


Analyzing positive and negative intervals of polynomials

WEBVTT
Kind: captions
Language: en

00:00:00.238 --> 00:00:02.002
- [Voiceover] So we have a function f of x

00:00:02.002 --> 00:00:04.162
it's written as the product of a bunch of

00:00:04.162 --> 00:00:05.961
first degree expressions.

00:00:05.961 --> 00:00:08.376
We obviously could also
view this as a (mumbles),

00:00:08.376 --> 00:00:09.723
specially if we expand this all out,

00:00:09.723 --> 00:00:11.766
we'll have a more traditional form.

00:00:11.766 --> 00:00:13.403
But what's nice about this form,

00:00:13.403 --> 00:00:15.006
and this is where we try to get

00:00:15.006 --> 00:00:16.387
when we're trying to find the zeros

00:00:16.387 --> 00:00:17.908
or the root of a (mumbles),

00:00:17.908 --> 00:00:21.333
is that is very easy
to pick out the zeros.

00:00:21.333 --> 00:00:24.572
The zeros here, these
are the x values that

00:00:24.572 --> 00:00:27.231
make our function equal to zero.

00:00:27.231 --> 00:00:29.924
Well that's gonna be x equals to two,

00:00:29.924 --> 00:00:31.920
x is equal to negative four,

00:00:31.920 --> 00:00:33.825
that would make this expression zero.

00:00:33.825 --> 00:00:35.555
X is equal to negative five,

00:00:35.555 --> 00:00:37.958
remember, if you make any
one of these equal to zero,

00:00:37.958 --> 00:00:39.975
then the entire function is
going to be equal to zero.

00:00:39.975 --> 00:00:42.263
Cause zero times anything is zero.

00:00:42.263 --> 00:00:44.515
X equals one,

00:00:44.515 --> 00:00:48.159
and x is equal to nine.

00:00:48.159 --> 00:00:49.879
Now, as we've talked
about in multiple videos,

00:00:49.879 --> 00:00:51.017
these are also going to be,

00:00:51.017 --> 00:00:53.350
these x values when you
input into the function

00:00:53.350 --> 00:00:55.277
give you f of two equals zero,

00:00:55.277 --> 00:00:57.112
f of negative four is equal to zero.

00:00:57.112 --> 00:00:59.004
Then that means that, there're also a way

00:00:59.004 --> 00:01:00.630
to find our x intercepts.

00:01:00.630 --> 00:01:01.988
What am I talking about?

00:01:01.988 --> 00:01:03.583
So, let me draw,

00:01:04.339 --> 00:01:06.405
let me draw a big axis here.

00:01:06.405 --> 00:01:08.238
See it goes lows x equals negative five

00:01:08.238 --> 00:01:09.761
as high as x equals 9.

00:01:09.761 --> 00:01:13.754
So, let me draw something like this.

00:01:13.754 --> 00:01:17.667
So, it's my y axis, this is my x axis.

00:01:17.667 --> 00:01:21.092
X's one, two, three,

00:01:21.092 --> 00:01:23.716
four, five, six, seven,

00:01:23.716 --> 00:01:25.678
eight, nine.

00:01:25.678 --> 00:01:30.241
One, two, three, four, five.

00:01:30.241 --> 00:01:31.390
Alright.

00:01:31.390 --> 00:01:32.737
Now let's just plot these.

00:01:32.737 --> 00:01:35.022
So when x is equal to two,

00:01:35.022 --> 00:01:36.823
f of two we already know it's zero.

00:01:36.823 --> 00:01:39.424
If you input two here, then
this expression's gonna be

00:01:39.424 --> 00:01:42.280
zero, zero, times a bunch
of stuff's gonna be zero.

00:01:42.280 --> 00:01:45.020
So f of two is equal to zero.

00:01:45.020 --> 00:01:47.319
So the point two comma zero

00:01:47.319 --> 00:01:50.860
is going to be on the
graph of y equals f of x.

00:01:50.860 --> 00:01:53.691
So one, two, comma, zero.

00:01:53.691 --> 00:01:55.283
So that's that point right over there.

00:01:55.283 --> 00:01:58.639
So notice, x equals two is a zero

00:01:58.639 --> 00:02:01.320
and makes our function equal to zero

00:02:01.320 --> 00:02:02.714
if we input it into it.

00:02:02.714 --> 00:02:05.198
And it's also the point two, comma, zero.

00:02:05.198 --> 00:02:09.300
It's going to be the x coordinate

00:02:09.300 --> 00:02:10.817
of this point right over here.

00:02:10.817 --> 00:02:12.314
It's going to define our x,

00:02:12.314 --> 00:02:14.451
or one of our x intercepts.

00:02:14.451 --> 00:02:15.751
Let's do it with the other ones.

00:02:15.751 --> 00:02:18.862
X is equal to negative
four, I can do that with a

00:02:18.862 --> 00:02:19.882
better color.

00:02:19.882 --> 00:02:21.324
X is equal to negative four.

00:02:21.324 --> 00:02:22.473
Negative one, negative
two, negative three,

00:02:22.473 --> 00:02:24.016
negative four.

00:02:24.016 --> 00:02:27.257
Right over there, x is
equal to negative five.

00:02:27.257 --> 00:02:30.101
X is equal to negative five.

00:02:31.778 --> 00:02:36.399
X is equal to one, and do that in,

00:02:36.399 --> 00:02:38.393
I will do that in pink.

00:02:38.393 --> 00:02:40.578
X is equal to one.

00:02:40.578 --> 00:02:43.189
And x is equal to nine.

00:02:43.189 --> 00:02:46.093
And x is equal to nine, that
was right over there, right?

00:02:46.093 --> 00:02:47.776
One, two, three, four, five,

00:02:47.776 --> 00:02:49.913
six, seven, eight, nine.

00:02:49.913 --> 00:02:51.619
Now I'm not gonna be able to exactly

00:02:51.619 --> 00:02:53.581
draw the graph of this function.

00:02:53.581 --> 00:02:55.868
But what's neat is, I
can realize, well look,

00:02:55.868 --> 00:02:57.714
if these are the points
where I'm going to intersect

00:02:57.714 --> 00:02:58.574
the x axis.

00:02:58.574 --> 00:03:00.048
And these are the only points

00:03:00.048 --> 00:03:01.999
where I'm going to intersect the x axis.

00:03:01.999 --> 00:03:04.460
The zeros, the real zeros define the

00:03:04.460 --> 00:03:07.351
all of the points that (mumbles)
to intersect the x axis,

00:03:07.351 --> 00:03:08.847
because, think about
it, there's a point that

00:03:08.847 --> 00:03:10.729
intersected the x axis that wasn't a zero,

00:03:10.729 --> 00:03:12.517
well, it would be a zero

00:03:12.517 --> 00:03:15.034
cause the function would take
on the value of zero there.

00:03:15.034 --> 00:03:18.309
So between those points,
if the function will

00:03:18.309 --> 00:03:20.934
stay positive or will stay negative.

00:03:20.934 --> 00:03:23.570
And we can test that a little bit.

00:03:23.570 --> 00:03:26.914
Let's think about what
the function is doing

00:03:26.914 --> 00:03:29.352
between x equals negative four

00:03:29.352 --> 00:03:33.810
and x equals one, so, over
this interval right over here.

00:03:33.810 --> 00:03:35.760
Well let's just pick a value there,

00:03:35.760 --> 00:03:36.828
and then think about whether the function

00:03:36.828 --> 00:03:39.603
is gonna be positive or
negative at that value.

00:03:39.603 --> 00:03:43.597
And an easy value to
think about is maybe zero.

00:03:43.597 --> 00:03:45.037
And I just want to think about the sign,

00:03:45.037 --> 00:03:47.103
is the function going to
be positive or negative

00:03:47.103 --> 00:03:48.891
when x is equal to zero.

00:03:48.891 --> 00:03:51.809
So f of zero, f of zero,

00:03:51.809 --> 00:03:53.814
I'm just going to think about the sign.

00:03:53.814 --> 00:03:57.053
So the f of zero is
gonna be zero minus two,

00:03:57.053 --> 00:03:59.062
well that's going to be negative.

00:03:59.062 --> 00:04:01.767
Zero plus four, that's gonna be positive.

00:04:01.767 --> 00:04:04.681
Zero plus five, that's gonna be positive.

00:04:04.681 --> 00:04:08.137
Zero minus one, is gonna be negative.

00:04:08.137 --> 00:04:12.873
Zero minus nine is going to be negative.

00:04:12.873 --> 00:04:15.858
So we can see, at x equals zero,

00:04:15.858 --> 00:04:17.203
we're going to get a, let's see,

00:04:17.203 --> 00:04:21.289
this is going to be a
negative times a negative,

00:04:21.289 --> 00:04:22.232
which would be a positive,

00:04:22.232 --> 00:04:24.159
then we multiply by a negative again.

00:04:24.159 --> 00:04:26.608
So this is going to be equal to,

00:04:26.608 --> 00:04:28.488
this is going to get us to a negative.

00:04:28.488 --> 00:04:31.102
So the function's gonna take
on a negative value here.

00:04:31.102 --> 00:04:32.983
And we know it's gonna
take a negative value

00:04:32.983 --> 00:04:35.281
over this entire interval,

00:04:35.281 --> 00:04:37.081
over this entire interval.

00:04:37.081 --> 00:04:39.728
Because if for some reason
it took on a positive value

00:04:39.728 --> 00:04:42.154
over that interval, it's
a continuous function.

00:04:42.154 --> 00:04:43.988
It would have to cross the x axis

00:04:43.988 --> 00:04:45.963
and that would define another zero.

00:04:45.963 --> 00:04:48.679
But we know there aren't any
zeros between these two points.

00:04:48.679 --> 00:04:50.932
So the function, I don't
know what it looks like,

00:04:50.932 --> 00:04:52.441
but it's going to stay positive

00:04:52.441 --> 00:04:55.285
between those two values.

00:04:55.285 --> 00:04:56.957
Now, as you go over to our next interval,

00:04:56.957 --> 00:04:59.290
it's probably going to change signs.

00:04:59.290 --> 00:05:02.414
And we can verify that if you look at f,

00:05:02.414 --> 00:05:07.414
if you look at, say, f of negative of 4.5.

00:05:07.638 --> 00:05:09.345
So we'll do the same drill.

00:05:09.345 --> 00:05:14.140
F of negative 4.5, well
that's going to be,

00:05:14.140 --> 00:05:16.589
that's going to be a see,

00:05:16.589 --> 00:05:20.930
negative times a negative,

00:05:20.930 --> 00:05:22.534
times a positive.

00:05:22.534 --> 00:05:25.831
Negative 4.5 plus five is a positive.

00:05:25.831 --> 00:05:30.614
Times a negative, times a negative.

00:05:30.614 --> 00:05:32.284
Well we have an even number of negatives

00:05:32.284 --> 00:05:33.970
being multiplied together here.

00:05:33.970 --> 00:05:35.758
So this gonna give us a positive.

00:05:35.758 --> 00:05:38.625
So this is going to give
us a positive value.

00:05:38.625 --> 00:05:39.972
So we're going to have a positive value

00:05:39.972 --> 00:05:41.481
between these two points.

00:05:41.481 --> 00:05:45.104
And then we're likely to flip
back to negative after that.

00:05:45.104 --> 00:05:47.971
So then the graph is likely to flip back

00:05:47.971 --> 00:05:51.733
flip back to negative, right over there.

00:05:51.733 --> 00:05:55.077
You can test that, try
a really negative value.

00:05:55.077 --> 00:05:58.676
So f of negative 10.

00:05:58.676 --> 00:06:01.729
Well that's going to be a negative

00:06:01.729 --> 00:06:06.093
times a negative, times a negative,

00:06:06.093 --> 00:06:09.554
let's see, times a
negative, times a negative,

00:06:09.554 --> 00:06:10.739
times a negative.

00:06:10.739 --> 00:06:12.492
So you have five negatives
being multiplied together,

00:06:12.492 --> 00:06:14.407
you're going to have a negative.

00:06:14.407 --> 00:06:16.207
Hopefully you see where
all of this is going,

00:06:16.207 --> 00:06:18.273
so this is, we're likely
to switch signs here.

00:06:18.273 --> 00:06:21.698
You can test 1.5 and see if
you get a positive value.

00:06:21.698 --> 00:06:23.451
Then we're likely to switch back here.

00:06:23.451 --> 00:06:24.531
And once again, the graph isn't gonna look

00:06:24.531 --> 00:06:25.669
exactly like this.

00:06:25.669 --> 00:06:30.669
It might deep, it might
be fairly shallow here,

00:06:31.741 --> 00:06:33.575
and then it might deep
down a lot right here.

00:06:33.575 --> 00:06:36.083
I'm just making up the
parts on the ups and downs.

00:06:36.083 --> 00:06:38.614
And then it probably
does another sign change

00:06:38.614 --> 00:06:40.529
right over there, and we can verify that.

00:06:40.529 --> 00:06:42.852
If f of positive 10

00:06:43.995 --> 00:06:46.572
well that's gonna be a positive

00:06:46.572 --> 00:06:48.569
times a positive,

00:06:48.569 --> 00:06:52.502
times a positive, times a
positive, times a positive.

00:06:53.648 --> 00:06:57.003
Positive times a positive,
so we're positive.

00:06:57.003 --> 00:06:59.360
After that, I don't know
exactly what it looks like.

00:06:59.360 --> 00:07:02.472
But the key realization here is

00:07:02.472 --> 00:07:04.468
that the zeros are the only places

00:07:04.468 --> 00:07:05.908
we're gonna intercept the x axis.

00:07:05.908 --> 00:07:08.404
And between those zeros,
if you were positive,

00:07:08.404 --> 00:07:09.783
if you're positive on the
interval, you're gonna be

00:07:09.783 --> 00:07:11.014
positive on any point.

00:07:11.014 --> 00:07:12.279
If you're positive at one
point in that interval,

00:07:12.279 --> 00:07:14.137
you're gonna be positive at
any point in that interval.

00:07:14.137 --> 00:07:16.250
And if you're negative at
any point of that interval,

00:07:16.250 --> 00:07:17.237
then you're going to be negative

00:07:17.237 --> 00:07:18.549
throughout the interval.

00:07:18.549 --> 00:07:19.988
And of course, at the end of the interval,

00:07:19.988 --> 00:07:22.879
you're going to be at zero.

00:07:22.879 --> 00:07:24.435
So, if someone were to say,

00:07:24.435 --> 00:07:26.223
what is the sign of f?

00:07:26.223 --> 00:07:28.101
If someone would say sing

00:07:29.741 --> 00:07:34.741
of f on the interval on

00:07:34.970 --> 00:07:38.834
say negative four is less
than x, is less than one.

00:07:38.834 --> 00:07:42.285
Well this interval goes from negative four

00:07:42.285 --> 00:07:45.615
to one, so that's the interval
we actually started off with.

00:07:45.615 --> 00:07:47.010
That's that interval right over there,

00:07:47.010 --> 00:07:48.275
you'd say, oh well look,

00:07:48.275 --> 00:07:51.793
f is always negative on this interval.

00:07:51.793 --> 00:07:52.942
If you don't include the end points.

00:07:52.942 --> 00:07:54.405
And we're not including
the end points here.

00:07:54.405 --> 00:07:57.238
We don't have a less and/or
equal either point here.

00:07:57.238 --> 00:07:58.550
So here's always negative.

00:07:58.550 --> 00:07:59.781
If someone gave you a different interval,

00:07:59.781 --> 00:08:01.813
if they give you an interval that includes

00:08:01.813 --> 00:08:04.018
where zero is part of the interval.

00:08:04.018 --> 00:08:05.446
So someone said,

00:08:05.446 --> 00:08:07.908
if someone said the interval from

00:08:07.908 --> 00:08:12.842
I don't know, the
interval from one to nine,

00:08:12.842 --> 00:08:15.094
well that's gonna include a zero in it.

00:08:15.094 --> 00:08:16.557
So it's gonna be positive,

00:08:16.557 --> 00:08:19.053
it's gonna be sometimes
positive and sometimes negative

00:08:19.053 --> 00:08:19.994
on this interval.

00:08:19.994 --> 00:08:21.085
So the key is,

00:08:21.085 --> 00:08:24.208
between two roots, not
including the roots,

00:08:24.208 --> 00:08:26.518
if you have an interval between two roots,

00:08:26.518 --> 00:08:28.025
then you're always going to be positive,

00:08:28.025 --> 00:08:30.312
or you're always going to be
negative over that interval.

00:08:30.312 --> 00:08:32.486
Whatever the sign is, at
any point, the interval's

00:08:32.486 --> 00:08:34.027
gonna be the sign throughout the interval,

00:08:34.027 --> 00:08:35.917
not including the end points.


Finding zeros of polynomials (example 2)

WEBVTT
Kind: captions
Language: en

00:00:00.098 --> 00:00:02.478
- So I have the polynomial p of x here,

00:00:02.478 --> 00:00:05.727
and p of x is being expressed
as a fourth degree polynomial

00:00:05.727 --> 00:00:09.479
times three x minus eight squared,

00:00:09.479 --> 00:00:10.745
so this would actually give you some,

00:00:10.745 --> 00:00:13.090
this would give you nine x squared

00:00:13.090 --> 00:00:14.321
and a bunch of other stuff,

00:00:14.321 --> 00:00:15.795
and then you multiply that times this.

00:00:15.795 --> 00:00:16.943
It would actually give you a

00:00:16.943 --> 00:00:19.576
sixth degree polynomial all in all,

00:00:19.576 --> 00:00:22.719
but our goal is to find the x values where

00:00:22.719 --> 00:00:25.957
that makes p of x equal to zero,

00:00:25.957 --> 00:00:27.770
or another way find the roots

00:00:27.770 --> 00:00:30.266
or the zeros of this polynomial,

00:00:30.266 --> 00:00:32.936
and in particular we're going
to focus on the real zeros,

00:00:32.936 --> 00:00:36.221
the real roots of this polynomial,

00:00:36.221 --> 00:00:39.162
and like always I encourage
you give a go at it,

00:00:39.162 --> 00:00:41.019
and then we'll do it together.

00:00:41.019 --> 00:00:42.900
Alright, let's tackle this.

00:00:42.900 --> 00:00:46.661
So, the way I want to solve
p of x is equal to zero.

00:00:46.661 --> 00:00:50.410
I want to solve p of x is equal to zero,

00:00:50.410 --> 00:00:51.561
and figure out what,

00:00:51.561 --> 00:00:52.885
and when I say solve it I want to say

00:00:52.885 --> 00:00:55.718
well what x values will make
the polynomial equal to zero.

00:00:55.718 --> 00:00:58.339
So I just need to set this right hand side

00:00:58.339 --> 00:01:00.942
to be equal to zero and then solve for x,

00:01:00.942 --> 00:01:03.090
and the best way that I
can think of doing that

00:01:03.090 --> 00:01:05.934
is by factoring this out as much as I can,

00:01:05.934 --> 00:01:08.278
and if I can rewrite it as

00:01:08.278 --> 00:01:11.482
a product of a bunch of
expressions equaling zero,

00:01:11.482 --> 00:01:13.679
well a product of a bunch
of things equaling zero,

00:01:13.679 --> 00:01:16.546
you can make it equal zero by
any one of them equaling zero,

00:01:16.546 --> 00:01:18.067
and so let's do that.

00:01:18.067 --> 00:01:19.599
So three x minus eight squared,

00:01:19.599 --> 00:01:21.654
this is already factored quite nicely,

00:01:21.654 --> 00:01:23.837
let's see if we can
factor if we can factor

00:01:23.837 --> 00:01:28.430
all of this business in white,

00:01:28.430 --> 00:01:31.272
and the way I will tackle it

00:01:31.272 --> 00:01:34.465
to see if I can factor by grouping.

00:01:34.465 --> 00:01:38.215
So let me group together
these first two terms,

00:01:38.215 --> 00:01:42.743
and then let me group together
these second two terms,

00:01:42.743 --> 00:01:44.995
and essentially factoring by grouping

00:01:44.995 --> 00:01:47.677
is doing the distributive
property in reverse twice.

00:01:47.677 --> 00:01:50.672
So from these first two terms,

00:01:50.672 --> 00:01:53.807
I could factor out,

00:01:53.807 --> 00:01:56.011
let's see what could I factor out?

00:01:56.011 --> 00:01:58.730
I could factor out a

00:01:58.730 --> 00:02:00.332
let me see,

00:02:00.332 --> 00:02:02.549
I'll just factor out an
x to the third power.

00:02:02.549 --> 00:02:06.125
So I get x to the third power times

00:02:06.125 --> 00:02:09.086
three x minus eight,

00:02:09.086 --> 00:02:11.814
interesting we have a three x
minus eight over there as well

00:02:11.814 --> 00:02:14.788
Now these second to two terms,

00:02:14.788 --> 00:02:19.067
I could factor out a five,

00:02:19.067 --> 00:02:21.354
so this is going to be plus five

00:02:21.354 --> 00:02:25.259
times x, sorry, times three x.

00:02:25.259 --> 00:02:28.195
Three x minus eight, very interesting,

00:02:28.195 --> 00:02:32.019
and of course I have these
parentheses around all of that,

00:02:32.019 --> 00:02:35.177
and then I have three x minus 8 squared.

00:02:35.177 --> 00:02:37.934
This three x minus eight
is showing up a lot,

00:02:37.934 --> 00:02:41.554
and so and of course this is
going to be equal to zero.

00:02:41.554 --> 00:02:43.528
So we're gonna be equal to zero,

00:02:43.528 --> 00:02:48.148
and now I can factor out a
three x minus eight over here,

00:02:48.148 --> 00:02:50.250
I could factor that
three x minus eight out,

00:02:50.250 --> 00:02:51.527
and I'm going to get

00:02:51.527 --> 00:02:55.380
three x minus eight times

00:02:56.560 --> 00:02:59.492
times x to the third power,

00:02:59.492 --> 00:03:02.624
x to the third power plus five.

00:03:02.624 --> 00:03:05.526
Once again I just factor
out a three x minus eight

00:03:05.526 --> 00:03:09.414
plus five, close the parentheses,

00:03:09.414 --> 00:03:13.145
and then times three x minus eight squared

00:03:13.145 --> 00:03:15.488
is all going to be equal to zero,

00:03:15.488 --> 00:03:17.266
is all equal zero.

00:03:17.266 --> 00:03:19.623
Now if what I just did
looks a little like voodoo,

00:03:19.623 --> 00:03:21.678
just realize I have two terms,

00:03:21.678 --> 00:03:23.652
both of them are multiples
of three x minus eight.

00:03:23.652 --> 00:03:25.254
I just factored out,

00:03:25.254 --> 00:03:29.120
I just factored out the
three x minus eight.

00:03:29.120 --> 00:03:31.523
I did distributive property in reverse,

00:03:31.523 --> 00:03:32.673
so I factored it out,

00:03:32.673 --> 00:03:33.950
and what you're left with this term

00:03:33.950 --> 00:03:35.342
you just look for the next of third,

00:03:35.342 --> 00:03:38.438
and in this term you're
just left with a plus five.

00:03:38.438 --> 00:03:41.672
Now three x minus eight times
x to the third plus five

00:03:41.672 --> 00:03:44.169
times three x minus eight squared.

00:03:44.169 --> 00:03:46.018
Well I could just rewrite this as

00:03:46.018 --> 00:03:47.902
three x minus eight to the third power

00:03:47.902 --> 00:03:49.306
times x to the third plus five,

00:03:49.306 --> 00:03:50.583
so let me do that.

00:03:50.583 --> 00:03:51.797
So I can just rewrite this as

00:03:51.797 --> 00:03:54.487
three x minus eight to the third power,

00:03:54.487 --> 00:03:56.472
that's that times that,

00:03:56.472 --> 00:03:58.167
and then times

00:03:58.167 --> 00:04:02.010
times, do this in a nicer color,

00:04:02.010 --> 00:04:06.399
times x to the third plus five

00:04:06.399 --> 00:04:08.920
is equal to zero,

00:04:09.470 --> 00:04:11.623
is equal to zero,

00:04:11.623 --> 00:04:14.282
and now in order to get
this to be equal zero,

00:04:14.282 --> 00:04:16.778
either the three x either

00:04:16.778 --> 00:04:18.759
this thing is going to be equal to zero,

00:04:18.769 --> 00:04:22.990
or this thing over here is
going to be equal to zero.

00:04:22.990 --> 00:04:26.682
So let's first think
about three minus eight

00:04:26.682 --> 00:04:29.154
to the third power equaling zero.

00:04:29.154 --> 00:04:31.291
So I can write this is as

00:04:31.291 --> 00:04:35.967
three x minus eight to the
third power is equal to zero

00:04:38.067 --> 00:04:42.499
or x to the third plus
five is equal to zero.

00:04:42.499 --> 00:04:46.554
So to make three x minus eight
to the third power equal zero

00:04:46.554 --> 00:04:48.816
well that means three x minus eight

00:04:48.816 --> 00:04:51.103
is going to be equal to zero

00:04:51.103 --> 00:04:54.110
or that three x is equal to eight,

00:04:54.110 --> 00:04:56.119
divide both sides by three,

00:04:56.119 --> 00:04:58.037
x is equal to eight thirds.

00:04:58.037 --> 00:05:00.060
So that's one way to make
this polynomial equal zero,

00:05:00.060 --> 00:05:01.679
x is equal to eight thirds,

00:05:01.679 --> 00:05:03.699
in fact just this right
over there will become zero,

00:05:03.699 --> 00:05:05.627
zero times anything is zero.

00:05:05.627 --> 00:05:08.366
So this is a zero of our polynomial,

00:05:08.366 --> 00:05:10.036
and let's see,

00:05:10.036 --> 00:05:11.684
so x to the third,

00:05:11.684 --> 00:05:13.104
so we could say,

00:05:13.104 --> 00:05:14.287
if we subtract five from both sides,

00:05:14.287 --> 00:05:15.551
we have x to the third is equal to the

00:05:15.551 --> 00:05:17.909
is equal to negative five,

00:05:17.909 --> 00:05:20.382
and so if we take both
to the one third power

00:05:20.382 --> 00:05:24.712
we could say x is equal to
cube root of negative five.

00:05:24.712 --> 00:05:25.792
Now at first you might say,

00:05:25.792 --> 00:05:27.302
"Wait, can I take the square
root of a negative number?"

00:05:27.302 --> 00:05:28.788
and I would say, "Of course you can!"

00:05:28.788 --> 00:05:31.922
The cube root of a negative
one is negative one.

00:05:31.922 --> 00:05:34.558
The cube root of negative
eight is negative two.

00:05:34.558 --> 00:05:37.820
In fact you could, even if
we're dealing with reals.

00:05:37.820 --> 00:05:39.457
This is going to be a negative number.

00:05:39.457 --> 00:05:42.214
This is not going to be an
imaginary number right over here,

00:05:42.214 --> 00:05:44.025
and so these are,

00:05:44.025 --> 00:05:48.170
these are the two zeros of the polynomial.

00:05:48.170 --> 00:05:49.447
There's gonna be negative, I think,

00:05:49.447 --> 00:05:50.771
negative one point something,

00:05:50.771 --> 00:05:51.988
I'm sure we could figure out it,

00:05:51.988 --> 00:05:54.048
figure out it exactly.

00:05:54.048 --> 00:05:56.091
So let's raise,

00:05:56.091 --> 00:05:58.289
so let's raise,

00:05:59.629 --> 00:06:03.849
five to the open parentheses,

00:06:03.849 --> 00:06:07.819
one divided by three close parentheses,

00:06:07.819 --> 00:06:09.326
is equal to,

00:06:09.326 --> 00:06:10.606
so that's five to the one third,

00:06:10.606 --> 00:06:12.382
so negative five to the one third power

00:06:12.382 --> 00:06:16.782
is gonna be negative one
point seven one approximately.

00:06:17.872 --> 00:06:21.985
So this is approximately equal
negative one point seven one.

00:06:21.985 --> 00:06:24.596
So we have two real roots,

00:06:24.596 --> 00:06:27.104
two real roots to this polynomial,

00:06:27.104 --> 00:06:30.679
or two zeros two real
zeros for this polynomial,

00:06:30.679 --> 00:06:32.943
and so those are going
to be the two places

00:06:32.943 --> 00:06:36.287
where we intercept the x-axis.

00:06:36.287 --> 00:06:37.680
The two x values for which

00:06:37.680 --> 00:06:40.269
where the two places where
we intercept the x-axis,

00:06:40.269 --> 00:06:43.094
is the easiest way to say it.