End behavior of rational functions

WEBVTT
Kind: captions
Language: en

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- [Voiceover] So, we're
given this function, f of x,

00:00:02.239 --> 00:00:05.014
and it equals this rational
expression over here

00:00:05.014 --> 00:00:07.522
and we're asked "What does f of x approach

00:00:07.522 --> 00:00:10.283
"as x approaches negative infinity?"

00:00:10.283 --> 00:00:13.424
So, as x becomes more and more
and more and more negative,

00:00:13.424 --> 00:00:14.887
what does f of x approach?

00:00:14.887 --> 00:00:16.126
And, like always, pause the video

00:00:16.126 --> 00:00:18.317
and see if you can think
about that on your own.

00:00:19.527 --> 00:00:20.636
Well, one thing that I like to do

00:00:20.636 --> 00:00:23.330
when I'm trying to consider
the behavior of a function

00:00:23.330 --> 00:00:26.294
as x gets really positive
or really negative

00:00:26.294 --> 00:00:27.464
is to rewrite it.

00:00:27.464 --> 00:00:28.869
So, f of x,

00:00:28.869 --> 00:00:29.983
I'm just rewriting it once,

00:00:29.983 --> 00:00:34.902
is equal to 7x-squared, minus 2x

00:00:34.902 --> 00:00:39.269
over 15x minus five.

00:00:39.269 --> 00:00:41.923
Now, an interesting technique
to think about what happens

00:00:41.923 --> 00:00:44.544
to the different terms
as x gets very positive

00:00:44.544 --> 00:00:46.676
or x gets very negative,

00:00:46.676 --> 00:00:50.007
is to divide both the
numerator and the denominator

00:00:50.007 --> 00:00:54.560
by the highest degree term
of x in the denominator.

00:00:54.560 --> 00:00:57.228
And the highest degree term
of x in the denominator

00:00:57.228 --> 00:00:59.672
is the first-degree term.

00:00:59.672 --> 00:01:01.679
We have just a single x there.

00:01:01.679 --> 00:01:05.454
So, let's multiply both the
numerator and the denominator

00:01:05.454 --> 00:01:07.612
by one over x, or another
way of thinking about it

00:01:07.612 --> 00:01:10.422
is we're dividing both the
numerator and the denominator

00:01:10.422 --> 00:01:11.624
by x.

00:01:11.624 --> 00:01:13.321
And if we're doing the
same thing to the numerator

00:01:13.321 --> 00:01:14.244
and the denominator,

00:01:14.244 --> 00:01:16.207
if we're multiplying or
dividing them by the same value,

00:01:16.207 --> 00:01:17.344
I should say,

00:01:17.344 --> 00:01:19.843
well then, I'm just really
just multiplying it by one.

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So, I'm not changing its value.

00:01:21.782 --> 00:01:23.532
This will make it a little
bit more interesting,

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and a little bit easier for
us to think about what happens

00:01:25.672 --> 00:01:28.446
when x becomes very, very, very negative.

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So, 7x-squared divided by x,

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or being multiplied by one
over x, is going to be 7x.

00:01:34.949 --> 00:01:39.914
2x times one over x, or 2x
divided by x, is just two.

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And then all of that over

00:01:43.118 --> 00:01:47.160
15x divided by x, or 15x over x,

00:01:47.160 --> 00:01:48.923
is just going to be 15.

00:01:50.003 --> 00:01:52.777
And then you have five over x.

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Five times one over x
is equal to five over x.

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Minus five over x.

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Now, this is equivalent, for our purposes,

00:02:02.396 --> 00:02:04.968
to what we started with but it
makes it a little bit easier

00:02:04.968 --> 00:02:07.160
to think about what happens when x

00:02:07.160 --> 00:02:09.906
gets very, very, very, very negative.

00:02:09.906 --> 00:02:11.888
Well, when x gets very,
very, very, very, very, very,

00:02:11.888 --> 00:02:13.318
very negative,

00:02:13.318 --> 00:02:17.379
this is going to become a
very large negative number.

00:02:17.379 --> 00:02:19.950
You subtract two from it,
it really won't matter much.

00:02:19.950 --> 00:02:22.703
You divide that by 15, well,
that's not gonna matter much.

00:02:22.703 --> 00:02:25.476
And this is just going to
become very, very, very small.

00:02:25.476 --> 00:02:27.290
You're taking five and
you're dividing it by

00:02:27.290 --> 00:02:29.701
ever-larger negative numbers,

00:02:29.701 --> 00:02:31.104
or more and more negative numbers.

00:02:31.104 --> 00:02:33.509
So, this right over here
is gonna go to zero.

00:02:33.509 --> 00:02:37.102
This thing over here is
gonna go towards infinity.

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Or, I should say, it's gonna
go towards negative infinity.

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Seven times a negative trillion,

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seven times a negative googol,

00:02:44.095 --> 00:02:45.438
seven times a negative googolplex,

00:02:45.438 --> 00:02:47.326
we're getting more and
more negative numbers,

00:02:47.326 --> 00:02:48.610
this is gonna get, this is going

00:02:48.610 --> 00:02:50.293
to approach negative infinity.

00:02:50.293 --> 00:02:52.149
Doesn't matter that you're
subtracting two from that.

00:02:52.149 --> 00:02:53.990
In fact, that'll get even more negative.

00:02:53.990 --> 00:02:56.430
And it doesn't matter if
you then divide that by 15,

00:02:56.430 --> 00:02:58.015
you're still approaching
negative infinity.

00:02:58.015 --> 00:03:01.631
If you had a arbitrarily negative number,

00:03:01.631 --> 00:03:02.882
you divide it by 15, you still have

00:03:02.882 --> 00:03:04.955
an arbitrarily negative number.

00:03:04.955 --> 00:03:07.023
And, so, you could say
that this is going to go

00:03:07.023 --> 00:03:09.513
to negative infinity.

00:03:09.513 --> 00:03:12.024
Now, another way that you
could've thought about it.

00:03:12.024 --> 00:03:13.515
This is actually how I do think about it

00:03:13.515 --> 00:03:15.031
when I'm trying to,

00:03:15.031 --> 00:03:17.110
when I see these types of problems.

00:03:17.110 --> 00:03:19.477
I say, well which terms in the numerator

00:03:19.477 --> 00:03:21.125
and the denominator are going to dominate?

00:03:21.125 --> 00:03:22.708
And what do I mean by "dominate"?

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Well, as x gets very positive
or x gets very negative,

00:03:26.213 --> 00:03:28.579
another way to think about
it is the magnitude of x

00:03:28.579 --> 00:03:32.597
gets large, the absolute
value of x gets large.

00:03:32.597 --> 00:03:36.335
The higher degree terms are
going to grow much faster

00:03:36.335 --> 00:03:38.093
than the lesser degree terms.

00:03:38.093 --> 00:03:42.277
And so, we could say that for large x,

00:03:47.477 --> 00:03:48.907
for large x,

00:03:48.907 --> 00:03:53.519
and when I say "large" I
mean high absolute value.

00:03:53.519 --> 00:03:56.740
High absolute value.

00:03:56.740 --> 00:03:58.111
And if we're going to negative infinity,

00:03:58.111 --> 00:04:00.640
that's high absolute value.

00:04:00.640 --> 00:04:03.677
So, f of x is going to
be approximately equal to

00:04:03.677 --> 00:04:06.308
the highest degree term on the top,

00:04:06.308 --> 00:04:09.804
which is 7x-squared, divided by

00:04:09.804 --> 00:04:11.891
the highest degree term on the bottom.

00:04:11.891 --> 00:04:14.833
15x is going to grow, in
fact, this is right over here,

00:04:14.833 --> 00:04:15.758
this constant.

00:04:15.758 --> 00:04:17.928
So, as this becomes larger
and larger and larger,

00:04:17.928 --> 00:04:20.074
this is going to matter a lot, lot less.

00:04:20.074 --> 00:04:22.601
So, it's going to be approximately that.

00:04:22.601 --> 00:04:25.178
Which is equal to

00:04:25.178 --> 00:04:28.008
7x over 15.

00:04:28.008 --> 00:04:29.678
Well, even here, if you
think about what happens

00:04:29.678 --> 00:04:31.686
when x becomes very, very negative here.

00:04:31.686 --> 00:04:33.333
Well, you're just gonna get larger,

00:04:33.333 --> 00:04:34.824
you're gonna get more and more and more

00:04:34.824 --> 00:04:37.008
negative values for f of x.

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So, once again, f of x
itself is going to approach,

00:04:39.863 --> 00:04:42.413
is going to go to, negative infinity

00:04:42.413 --> 00:04:45.114
as x goes to negative infinity.

00:04:45.114 --> 00:04:47.512
Let's do another one of these.

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So, here they're telling us to find

00:04:48.820 --> 00:04:51.306
the horizontal asymptote of q.

00:04:53.281 --> 00:04:55.608
A horizontal asymptote,
you can think about it as

00:04:55.608 --> 00:04:58.859
what is the function
approaching as x becomes,

00:04:58.859 --> 00:05:00.411
as x approaches infinity,

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or as x approaches negative infinity.

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And just as a couple of examples here.

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It's not necessarily the q
of x that we're focused on.

00:05:08.652 --> 00:05:09.863
But you could imagine a function,

00:05:09.863 --> 00:05:12.155
let's say it has a horizontal asymptote

00:05:12.155 --> 00:05:16.596
at y is equal to two,

00:05:16.596 --> 00:05:18.236
so that's y is equal to two there.

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Let me draw that line.

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So, let's say it has a
horizontal asymptote like that.

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Well then the graph could
look something like this.

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It could look, let me
draw a couple of them

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that have horizontal asymptotes.

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So, maybe it's over
here, it does some stuff,

00:05:32.659 --> 00:05:37.587
but as x gets really large,
it starts approaching,

00:05:37.587 --> 00:05:39.941
the function starts
approaching that y equals two

00:05:39.941 --> 00:05:41.688
without ever quite getting there.

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And it could do that on this side as well.

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As x becomes more and more negative.

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As it gets more negative, it approaches it

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without ever getting there.

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Or, it could do something like this.

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You could have, if it has
a vertical asymptote, too,

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it could look something like this.

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Where it approaches the
horizontal asymptote

00:06:05.012 --> 00:06:07.998
from below, as x becomes more negative,

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and from above,

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as x becomes more positive.

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Or vice versa.

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Or vice versa.

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So, this is just a sense of
what a horizontal asymptote is.

00:06:20.665 --> 00:06:22.591
It'll show you what's
the behavior, what value

00:06:22.591 --> 00:06:25.551
is this function approaching,
as x becomes really positive

00:06:25.551 --> 00:06:27.897
or x becomes really negative.

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Well, let's just think about it.

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We could essentially do what we just did

00:06:30.352 --> 00:06:32.033
in that last example.

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What happens if we were to,

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if we were to divide all of these terms

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by the highest degree
term in the denominator?

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Well, if we divide, so q of
x is going to be equal to,

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the highest degree term in the denominator

00:06:47.706 --> 00:06:49.893
is x to the ninth power.

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So, we could say six,

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6x to the fifth divided by x to the ninth

00:06:55.967 --> 00:06:59.406
is going to be six over x to the fourth.

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And then minus two times x to the ninth.

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All of that over three over,

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I'm gonna divide this by x to the ninth,

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x to the seventh, plus one.

00:07:13.753 --> 00:07:16.658
Well, if x approaches
positive or negative infinity,

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six divided by arbitrarily large numbers,

00:07:18.808 --> 00:07:20.622
that's gonna go to zero.

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Two divided by arbitrarily large numbers,

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whether they are positive or negative,

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that's going to go to zero.

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So your numerator's
clearly gonna go to zero.

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This term of the
denominator, three divided

00:07:31.845 --> 00:07:33.288
by arbitrarily large numbers,

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whether we're going in the positive

00:07:35.392 --> 00:07:37.285
or the negative direction,

00:07:37.285 --> 00:07:40.105
it is gonna approach zero.

00:07:40.105 --> 00:07:42.186
It'll approach zero from
the negative direction,

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or we could say from below.

00:07:45.749 --> 00:07:49.288
If we're dealing with very negative x's.

00:07:49.288 --> 00:07:51.261
If we're dealing with very positive x's,

00:07:51.261 --> 00:07:53.230
then we're going to
approach zero from above.

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We're gonna get smaller and
smaller positive values.

00:07:55.434 --> 00:07:57.130
So, all of these things go to zero

00:07:57.130 --> 00:08:00.250
and this right over here is
going to be, would stay at, one.

00:08:00.250 --> 00:08:02.496
And so if you're approaching
zero in your numerator

00:08:02.496 --> 00:08:04.345
and approaching one in your denominator,

00:08:04.345 --> 00:08:06.934
the whole thing is going to approach zero.

00:08:06.934 --> 00:08:08.549
So, in the case of q of x,

00:08:08.549 --> 00:08:11.170
you have a horizontal asymptote

00:08:11.170 --> 00:08:14.259
at y is equal to zero.

00:08:14.259 --> 00:08:16.401
I don't know exactly
what the graph looks like

00:08:16.401 --> 00:08:19.175
but we could draw a horizontal
line at y equals zero

00:08:19.175 --> 00:08:20.608
and it would approach it.

00:08:20.608 --> 00:08:23.301
It would approach it from above or below.

00:08:24.761 --> 00:08:26.007
Let's do one more.

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What does f of x approach as x
approaches negative infinity?

00:08:32.652 --> 00:08:36.373
Well, let's divide all of these terms

00:08:36.373 --> 00:08:38.437
by the highest degree that
we see in the denominator.

00:08:38.437 --> 00:08:40.184
We see an x to the fourth.

00:08:40.184 --> 00:08:43.348
So, 3x to the fourth divided
by x to the fourth is three.

00:08:43.348 --> 00:08:46.013
Minus seven over x-squared,
I'm just dividing

00:08:46.013 --> 00:08:50.147
by x to the fourth, minus
one over x to the fourth,

00:08:50.147 --> 00:08:53.835
over, x to the fourth divided
by x to the fourth is one,

00:08:53.835 --> 00:08:57.312
minus two over x, plus
three x to the fourth.

00:08:57.312 --> 00:08:59.955
This is an equivalent,
this right over here is,

00:08:59.955 --> 00:09:02.084
for our purposes, for thinking
about what's happening

00:09:02.084 --> 00:09:04.147
on a kind of an end
behavior as x approaches

00:09:04.147 --> 00:09:06.015
negative infinity, this will do.

00:09:06.015 --> 00:09:08.453
I've just divided everything
by x to the fourth.

00:09:08.453 --> 00:09:09.458
And so what's gonna happen

00:09:09.458 --> 00:09:11.024
as x approaches negative infinity?

00:09:11.024 --> 00:09:13.125
This is going to approach zero.

00:09:13.125 --> 00:09:15.299
This is going to approach zero.

00:09:15.299 --> 00:09:17.393
This is going to approach zero.

00:09:17.393 --> 00:09:18.966
And this is going to approach zero.

00:09:18.966 --> 00:09:20.938
And so, as all of that
stuff approaches zero,

00:09:20.938 --> 00:09:23.347
what we're left with is
we're going to approach,

00:09:23.347 --> 00:09:25.469
we're going to approach three over one,

00:09:25.469 --> 00:09:27.561
or we could say just three.

00:09:27.561 --> 00:09:29.918
Another way you could
think about doing these

00:09:29.918 --> 00:09:31.655
is look at the highest degree terms.

00:09:31.655 --> 00:09:33.869
3x to the fourth, x to the fourth.

00:09:33.869 --> 00:09:35.209
Ignore everything else because they're

00:09:35.209 --> 00:09:37.893
going to be overwhelmed by
these higher degree terms.

00:09:37.893 --> 00:09:40.482
So, you could say f of
x is approximately equal

00:09:40.482 --> 00:09:43.986
to 3x to the fourth over x to the fourth

00:09:43.986 --> 00:09:48.103
for large magnitude x.

00:09:49.533 --> 00:09:50.563
Magnitude x.

00:09:50.563 --> 00:09:52.795
And very negative is still
a very large magnitude,

00:09:52.795 --> 00:09:54.513
large absolute value.

00:09:54.513 --> 00:09:57.571
And so, 3x to the fourth
divided by x to the fourth,

00:09:57.571 --> 00:10:01.015
f of x is going to be
approximately equal to three.

00:10:01.015 --> 00:10:02.560
Or it's going to approach three.

00:10:02.560 --> 00:10:04.812
So, that's another way that
you could think about it.


Equations with rational expressions (example 2)

WEBVTT
Kind: captions
Language: en

00:00:00.317 --> 00:00:02.199
- [Voiceover] So we have a
nice, little equation here

00:00:02.199 --> 00:00:04.451
that has some rational expressions in it

00:00:04.451 --> 00:00:06.033
and like always, pause the video

00:00:06.033 --> 00:00:07.497
and see if you can figure out

00:00:07.497 --> 00:00:10.720
which x's satisfy this equation.

00:00:10.720 --> 00:00:12.105
All right. Let's work through it together.

00:00:12.105 --> 00:00:15.448
Now, when I see things at
the denominator like this,

00:00:15.448 --> 00:00:18.768
my instinct is to try to not
have denominators like this.

00:00:18.768 --> 00:00:21.811
And so what we could do is to
get rid of this x minus one

00:00:21.811 --> 00:00:23.243
in the denominator on the left-hand side.

00:00:23.243 --> 00:00:25.909
We can multiply both sides of the equation

00:00:25.909 --> 00:00:28.324
times x minus one,

00:00:28.324 --> 00:00:30.009
x minus one.

00:00:30.009 --> 00:00:33.977
We're gonna multiply both
sides by x minus one.

00:00:33.977 --> 00:00:37.140
And once again, the
whole point of doing that

00:00:37.140 --> 00:00:38.952
is so that we get rid of this x minus one

00:00:38.952 --> 00:00:40.854
in the denominator right over here.

00:00:40.854 --> 00:00:42.996
And then to get rid of this x plus one

00:00:42.996 --> 00:00:44.911
in the denominator over here,

00:00:44.911 --> 00:00:49.308
we can multiply both sides of
the equation times x plus one.

00:00:49.308 --> 00:00:53.026
So, x plus one.

00:00:53.026 --> 00:00:56.156
Multiply both sides times x plus one.

00:00:56.156 --> 00:00:58.945
And so what is that going to give us?

00:00:58.945 --> 00:01:03.163
Well, on the left-hand
side, that is going to,

00:01:03.163 --> 00:01:06.229
x minus one divided by x
minus one is just gonna be one

00:01:06.229 --> 00:01:09.781
for the x's where that's defined,

00:01:09.781 --> 00:01:11.967
for x not being equal to one.

00:01:11.967 --> 00:01:13.620
And so, we're gonna have x plus one

00:01:13.620 --> 00:01:15.425
times negative two x plus four.

00:01:15.425 --> 00:01:16.770
So let me write that down.

00:01:16.770 --> 00:01:19.282
So we have x plus,

00:01:19.282 --> 00:01:20.413
I'm gonna need some space,

00:01:20.413 --> 00:01:22.569
so let me make sure I don't write too big.

00:01:22.569 --> 00:01:26.924
x plus one times negative two x,

00:01:26.924 --> 00:01:29.796
negative two x plus four

00:01:29.796 --> 00:01:31.778
is going to be equal to.

00:01:31.778 --> 00:01:34.404
Now, if we multiply both of these

00:01:34.404 --> 00:01:36.950
times three over x plus one,

00:01:36.950 --> 00:01:39.339
the x plus one is going to
cancel with the x plus one

00:01:39.339 --> 00:01:42.805
and we're gonna be left with
three times x minus one.

00:01:42.805 --> 00:01:46.005
So that is going to be
three x minus three,

00:01:46.005 --> 00:01:48.514
three x minus three,

00:01:48.514 --> 00:01:51.845
and then minus,

00:01:51.845 --> 00:01:55.698
minus one times both of these.

00:01:55.698 --> 00:01:59.409
So one times x minus one times x plus one.

00:01:59.409 --> 00:02:02.278
So, minus one times

00:02:02.278 --> 00:02:05.508
x minus one times x plus one.

00:02:05.508 --> 00:02:06.706
All I did is I multiplied,

00:02:06.706 --> 00:02:08.268
took the x minus one times x plus one,

00:02:08.268 --> 00:02:10.440
multiplied times each of these terms.

00:02:10.440 --> 00:02:12.392
When I multiply it times this first term,

00:02:12.392 --> 00:02:14.584
the x plus one and the
x plus one canceled,

00:02:14.584 --> 00:02:17.557
so I just have to multiply
three times x minus one.

00:02:17.557 --> 00:02:18.782
And then for the second term,

00:02:18.782 --> 00:02:20.814
I just multiply times both of these.

00:02:20.814 --> 00:02:22.265
And now you might recognize,

00:02:22.265 --> 00:02:25.428
if you have something x
plus one times x minus one,

00:02:25.428 --> 00:02:28.281
that's going to be x squared minus one.

00:02:28.281 --> 00:02:30.917
So I could rewrite all
of this right over here

00:02:30.917 --> 00:02:33.782
as being equal to...

00:02:33.782 --> 00:02:36.712
as being equal to

00:02:36.712 --> 00:02:38.822
x squared minus one,

00:02:38.822 --> 00:02:41.215
and once again, that's
because this is the same thing

00:02:41.215 --> 00:02:43.663
as x squared minus one.

00:02:43.663 --> 00:02:46.506
And since I'm subtracting
an x squared minus one,

00:02:46.506 --> 00:02:49.802
actually, let me just, I don't
wanna do too much on one step

00:02:49.802 --> 00:02:51.271
so let's go to the next step.

00:02:51.271 --> 00:02:53.807
So, I could multiply this out.

00:02:53.807 --> 00:02:58.281
So I could multiply x times negative two x

00:02:58.281 --> 00:03:01.224
which would give us
negative two x squared.

00:03:01.224 --> 00:03:03.503
x times four,

00:03:03.503 --> 00:03:07.143
which is going to give us plus four x

00:03:07.143 --> 00:03:09.542
and I could multiply one
times negative two x.

00:03:09.542 --> 00:03:11.494
I'm gonna subtract two x.

00:03:11.494 --> 00:03:15.755
And then one times four,
which is gonna be plus four,

00:03:15.755 --> 00:03:18.445
and then that is going to be equal to,

00:03:18.445 --> 00:03:20.433
that is going to be equal to,

00:03:20.433 --> 00:03:24.511
we have three x minus three

00:03:24.511 --> 00:03:26.293
and then we can distribute
this negative sign.

00:03:26.293 --> 00:03:30.453
So we can say minus x squared plus one.

00:03:30.453 --> 00:03:32.926
And over here, we can
simplify it a little bit.

00:03:32.926 --> 00:03:37.497
This is going to be, that
is four x minus two x

00:03:37.497 --> 00:03:39.598
is going to be,

00:03:39.598 --> 00:03:40.960
I'm gonna try to write over, yep,

00:03:40.960 --> 00:03:43.823
four x minus two x, so
that would be two x.

00:03:43.823 --> 00:03:47.594
And so this is simplified to, let's see,

00:03:47.594 --> 00:03:49.446
well this is, we have a
negative three and a one,

00:03:49.446 --> 00:03:52.799
so those two together
are going to be equal to,

00:03:52.799 --> 00:03:54.423
subtracting at two.

00:03:54.423 --> 00:03:56.469
So we can rewrite everything as,

00:03:56.469 --> 00:03:58.434
do the neutral color now,

00:03:58.434 --> 00:04:01.170
negative two x squared

00:04:01.170 --> 00:04:04.170
plus two x plus four

00:04:04.170 --> 00:04:08.475
is equal to negative x squared

00:04:08.475 --> 00:04:12.899
plus three x minus two.

00:04:12.899 --> 00:04:15.755
Now we can try to get all of this business

00:04:15.755 --> 00:04:17.540
onto the right-hand side,

00:04:17.540 --> 00:04:19.172
so let's subtract it from both sides.

00:04:19.172 --> 00:04:20.083
So we're gonna subtract,

00:04:20.083 --> 00:04:23.605
or we'll say we'll add
x squared to both sides,

00:04:23.605 --> 00:04:25.150
add x squared,

00:04:25.150 --> 00:04:28.780
that gets rid of this
white negative x squared.

00:04:28.780 --> 00:04:31.002
We subtract three x from both sides,

00:04:31.002 --> 00:04:33.224
subtract three x from both sides,

00:04:33.224 --> 00:04:36.112
add two to both sides, add two,

00:04:36.112 --> 00:04:38.945
and we will be left with,

00:04:38.945 --> 00:04:41.591
we are going to end up with.

00:04:41.591 --> 00:04:43.503
See, negative two x squared plus x squared

00:04:43.503 --> 00:04:46.015
is negative x squared.

00:04:46.015 --> 00:04:48.608
Two x minus three x is negative x.

00:04:48.608 --> 00:04:52.095
And then four plus two is six,

00:04:52.095 --> 00:04:53.583
is going to be equal to,

00:04:53.583 --> 00:04:56.436
well that's going to
cancel that, that, that,

00:04:56.436 --> 00:04:58.138
is equal to zero.

00:04:58.138 --> 00:05:00.000
I don't like having this
negative on the x squared,

00:05:00.000 --> 00:05:02.842
so let's multiply both
sides times negative one.

00:05:02.842 --> 00:05:04.197
And so if I do that,

00:05:04.197 --> 00:05:08.087
so if I just take the
negative on both sides,

00:05:08.087 --> 00:05:10.513
so I just multiply that
times negative one.

00:05:10.513 --> 00:05:12.605
Same thing, it's taking
the negative on both sides.

00:05:12.605 --> 00:05:17.587
I'm gonna get positive x squared plus x

00:05:17.587 --> 00:05:20.713
minus six is equal to zero.

00:05:20.713 --> 00:05:22.522
And we're making some good progress here.

00:05:22.522 --> 00:05:24.023
So we can factor this.

00:05:24.023 --> 00:05:25.475
And actually, let me just
do it right over here

00:05:25.475 --> 00:05:28.585
so that we can see the original problem.

00:05:28.585 --> 00:05:31.398
So, if I were to factor
this, what two numbers,

00:05:31.398 --> 00:05:33.199
their product is negative six,

00:05:33.199 --> 00:05:34.261
they're gonna have different signs

00:05:34.261 --> 00:05:35.823
since their product is negative.

00:05:35.823 --> 00:05:37.504
And they add up to one,

00:05:37.504 --> 00:05:40.316
the coefficient on the first degree term.

00:05:40.316 --> 00:05:43.626
Well, positive three
and negative two work,

00:05:43.626 --> 00:05:47.360
so I can rewrite this as x plus three

00:05:47.360 --> 00:05:50.313
times x minus two

00:05:50.313 --> 00:05:53.493
is equal to zero.

00:05:53.493 --> 00:05:54.554
Did I do that write? Yeah.

00:05:54.554 --> 00:05:56.646
Three times negative two is negative six.

00:05:56.646 --> 00:06:00.593
Three x minus two x is positive x.

00:06:00.593 --> 00:06:03.407
All right. So, I just wrote this

00:06:03.407 --> 00:06:05.798
in this quadratic in factored form.

00:06:05.798 --> 00:06:07.410
And so the way that you
get this equaling to zero

00:06:07.410 --> 00:06:09.162
is if either one of those equals zero.

00:06:09.162 --> 00:06:12.363
x plus three equals zero

00:06:12.363 --> 00:06:15.585
or x minus two is equal to zero.

00:06:15.585 --> 00:06:16.597
Well, this is going to happen

00:06:16.597 --> 00:06:18.406
if you subtract three from both sides.

00:06:18.406 --> 00:06:20.900
That's gonna happen if x
is equal to negative three

00:06:20.900 --> 00:06:23.726
or over here, if you
add two to both sides,

00:06:23.726 --> 00:06:25.942
x is equal to two.

00:06:25.942 --> 00:06:27.444
So either one of these will satisfy,

00:06:27.444 --> 00:06:28.585
but we wanna be careful.

00:06:28.585 --> 00:06:30.900
We wanna make sure that
our original equation

00:06:30.900 --> 00:06:34.143
isn't going to be undefined
for either one of these.

00:06:34.143 --> 00:06:35.755
And negative three does not make

00:06:35.755 --> 00:06:40.359
either of the denominators
equal to zero, so that's cool.

00:06:40.359 --> 00:06:41.941
And positive two does not make

00:06:41.941 --> 00:06:44.473
either of the denominators equal to zero.

00:06:44.473 --> 00:06:45.845
So it looks like we're in good shape.

00:06:45.845 --> 00:06:47.373
There's two solutions

00:06:47.373 --> 00:06:50.540
to that equation.

00:06:50.540 --> 00:06:53.613
If one of them made any of the
denominators equal to zero,

00:06:53.613 --> 00:06:56.319
then they would have been
extraneous solutions.

00:06:56.319 --> 00:06:57.180
It would have been solutions

00:06:57.180 --> 00:06:58.902
for some of our intermediate steps,

00:06:58.902 --> 00:07:01.424
but not for the actual original equations

00:07:01.424 --> 00:07:03.066
with the expressions as they're written.

00:07:03.066 --> 00:07:05.448
But this, we can feel good
about because neither of these

00:07:05.448 --> 00:07:09.781
make any of these
denominators equal to zero.


Equations with rational expressions

WEBVTT
Kind: captions
Language: en

00:00:00.352 --> 00:00:01.921
- [Voiceover] So we have a
nice little equation here

00:00:01.921 --> 00:00:03.612
dealing with rational expressions.

00:00:03.612 --> 00:00:05.242
And I encourage you to pause the video

00:00:05.242 --> 00:00:07.988
and see if you can figure
out what values of x

00:00:07.988 --> 00:00:10.947
satisfy this equation.

00:00:10.947 --> 00:00:12.637
Alright, let's work through this together.

00:00:12.637 --> 00:00:14.690
So the first thing I'd like to do

00:00:14.690 --> 00:00:16.803
is just see if I can simplify this at all,

00:00:16.803 --> 00:00:19.278
and maybe by finding some common factors

00:00:19.278 --> 00:00:21.028
between numerators and denominators,

00:00:21.028 --> 00:00:23.956
or common factors on either
side of the equal sign.

00:00:23.956 --> 00:00:27.850
So let's factor all of the
numerators and the denominators.

00:00:27.850 --> 00:00:30.144
All the ones on the right-hand
side are already done.

00:00:30.144 --> 00:00:31.925
So this thing up here,

00:00:31.925 --> 00:00:34.370
I could rewrite this as, let's see.

00:00:34.370 --> 00:00:36.845
What's product is 21?

00:00:36.845 --> 00:00:38.958
What two numbers, when I
take their product, is 21?

00:00:38.958 --> 00:00:41.252
Positive 21, so they're
going to have the same sign.

00:00:41.252 --> 00:00:43.959
And when I add them, I get negative 10.

00:00:43.959 --> 00:00:45.438
Well, negative seven and negative three,

00:00:45.438 --> 00:00:46.434
so this could be rewritten

00:00:46.434 --> 00:00:51.022
as x minus seven times x minus three.

00:00:51.022 --> 00:00:53.005
This over here, both
are divisible by three.

00:00:53.005 --> 00:00:57.382
I could rewrite this as
three times x minus four.

00:00:57.382 --> 00:00:59.072
And these are already factored.

00:00:59.072 --> 00:01:01.085
So the one thing that
jumps out at me is I have

00:01:01.085 --> 00:01:05.060
x minus four in the denominator
on the left-hand side

00:01:05.060 --> 00:01:07.505
and on the right-hand side.

00:01:07.505 --> 00:01:11.067
And so, if I were to multiply
both sides by x minus four.

00:01:11.067 --> 00:01:12.788
So actually, let me just...

00:01:12.788 --> 00:01:16.470
Let me formally replace this with that.

00:01:16.470 --> 00:01:18.613
And up here, it's not so obvious

00:01:18.613 --> 00:01:20.002
that it's going to be valuable for me

00:01:20.002 --> 00:01:21.451
to keep this factored form.

00:01:21.451 --> 00:01:23.775
So I'm just going to keep
it in this yellow form.

00:01:23.775 --> 00:01:26.231
In the expanded out form.

00:01:26.231 --> 00:01:28.796
So let me just scratch that out for now,

00:01:28.796 --> 00:01:30.275
because once I,

00:01:30.275 --> 00:01:32.178
well, let me multiply by x minus four.

00:01:32.178 --> 00:01:35.438
So, if we multiply both
sides by x minus four,

00:01:35.438 --> 00:01:36.826
and once again, why am I doing this?

00:01:36.826 --> 00:01:39.935
Just so I get rid of the x
minus four in the denominators.

00:01:39.935 --> 00:01:44.855
x minus four, and then x minus four.

00:01:44.855 --> 00:01:47.059
That and that cancels.

00:01:47.059 --> 00:01:48.811
That and that cancels,

00:01:48.811 --> 00:01:50.773
and then we're left
with, in the numerator,

00:01:50.773 --> 00:01:55.773
we're left with our x
squared minus 10 x plus 21,

00:01:57.686 --> 00:02:00.976
and, let's see, divided by three.

00:02:00.976 --> 00:02:03.572
Divided by three

00:02:03.572 --> 00:02:07.133
is equal to x minus five.

00:02:07.133 --> 00:02:08.703
Let's see, now, what we can do,

00:02:08.703 --> 00:02:10.333
and actually I could have
done it in the last step,

00:02:10.333 --> 00:02:13.835
is I can multiply both sides by three.

00:02:13.835 --> 00:02:15.525
Multiply both sides.

00:02:15.525 --> 00:02:16.551
Do that in another color

00:02:16.551 --> 00:02:18.755
just so it sticks out a little more.

00:02:18.755 --> 00:02:22.830
So I can multiply both sides by three.

00:02:22.830 --> 00:02:26.392
So multiply both sides by three.

00:02:26.392 --> 00:02:28.957
On the left-hand side,
that and that cancels,

00:02:28.957 --> 00:02:33.957
and I'll just be left with x
squared minus 10 x plus 21,

00:02:35.538 --> 00:02:37.409
and on, and I don't have
a denominator any more.

00:02:37.409 --> 00:02:39.824
My denominator is one, so
I don't need to write it.

00:02:39.824 --> 00:02:41.757
Is going to be equal to three times,

00:02:41.757 --> 00:02:43.719
let's just distribute the three.

00:02:43.719 --> 00:02:45.803
Three times x is three x.

00:02:45.803 --> 00:02:50.089
Three times negative five is negative 15.

00:02:50.089 --> 00:02:52.223
And now I can get this into
standard quadratic form

00:02:52.223 --> 00:02:55.000
by getting all of these terms
onto the left-hand side.

00:02:55.000 --> 00:02:56.701
The best way to do that,

00:02:56.701 --> 00:02:59.026
let's subtract three x from the right,

00:02:59.026 --> 00:03:00.112
but I can't just do it from the right

00:03:00.112 --> 00:03:01.531
otherwise the equality won't held.

00:03:01.531 --> 00:03:04.217
I have to do it from both sides
if I want equality to hold.

00:03:04.217 --> 00:03:07.055
And, I want to get rid
of this negative 15,

00:03:07.055 --> 00:03:09.460
so I can add 15 to both sides.

00:03:09.460 --> 00:03:11.825
So, let's do that.

00:03:11.825 --> 00:03:14.692
And what we are left with,
scroll down a little bit,

00:03:14.692 --> 00:03:17.270
so let's have a little more space.

00:03:17.270 --> 00:03:19.232
What we are going to be left with

00:03:19.232 --> 00:03:24.083
is x squared minus 13 x

00:03:25.071 --> 00:03:26.304
and then plus,

00:03:26.304 --> 00:03:30.828
what is this, plus 36.

00:03:30.828 --> 00:03:33.603
Plus 36. Did I do that right?

00:03:33.603 --> 00:03:35.625
Yep, plus 36

00:03:35.625 --> 00:03:40.472
is equal to, is equal to zero.

00:03:42.004 --> 00:03:43.944
Alright, now let's see,
we have it as a quadratic

00:03:43.944 --> 00:03:45.211
in the standard form.

00:03:45.211 --> 00:03:46.695
How can we solve this?

00:03:46.695 --> 00:03:48.023
So first thing, can we factor this.

00:03:48.023 --> 00:03:50.196
Product of two numbers, 36.

00:03:50.196 --> 00:03:51.705
If I add them, I get negative 13.

00:03:51.705 --> 00:03:52.943
They're both going to be negative,

00:03:52.943 --> 00:03:54.041
since they have to have the same sign

00:03:54.041 --> 00:03:56.486
to get their product to be positive.

00:03:56.486 --> 00:03:59.866
And, let's see, nine and
four seem to do the trick.

00:03:59.866 --> 00:04:02.070
So, or negative nine and negative four.

00:04:02.070 --> 00:04:06.386
So, x minus four times x minus nine

00:04:06.386 --> 00:04:08.278
is equal to zero.

00:04:08.278 --> 00:04:09.274
Well, that's going to happen

00:04:09.274 --> 00:04:13.834
if either x minus four is equal to zero or

00:04:13.834 --> 00:04:17.656
x minus nine is equal to zero.

00:04:17.656 --> 00:04:19.256
We'll add four to both sides of this.

00:04:19.256 --> 00:04:21.852
This happens when x is equal to four.

00:04:21.852 --> 00:04:23.211
Add nine to both side of this.

00:04:23.211 --> 00:04:25.938
This happens when x is equal to nine.

00:04:25.938 --> 00:04:27.709
So we could say that the solutions are

00:04:27.709 --> 00:04:30.909
x equals four or x equals nine.

00:04:30.909 --> 00:04:35.909
So, x is equal to four, or x equals nine.

00:04:36.161 --> 00:04:37.821
But we need to be careful.

00:04:37.821 --> 00:04:38.968
Because we have to remember,

00:04:38.968 --> 00:04:40.538
in our original expression,

00:04:40.538 --> 00:04:43.647
x minus four was a factor
of both denominators.

00:04:43.647 --> 00:04:45.216
And so if we actually try to test

00:04:45.216 --> 00:04:47.208
x minus four in the original equation,

00:04:47.208 --> 00:04:48.506
not one of these intermediary steps,

00:04:48.506 --> 00:04:50.806
the original equation, I would end up

00:04:50.806 --> 00:04:52.805
dividing by zero right over here,

00:04:52.805 --> 00:04:56.147
and actually, I'd end up by
zero right over there as well.

00:04:56.147 --> 00:04:58.099
So the original equations, if
I tried to substitute four,

00:04:58.099 --> 00:04:59.850
they don't make sense.

00:04:59.850 --> 00:05:01.993
So this is actually an
extraneous solution.

00:05:01.993 --> 00:05:04.891
It's not going to be a solution
to the original equation.

00:05:04.891 --> 00:05:09.459
The only solution is x is equal to 9.


Least common multiple of polynomials

WEBVTT
Kind: captions
Language: en

00:00:00.504 --> 00:00:01.506
- [Voiceover] So they're asking us to find

00:00:01.506 --> 00:00:03.107
the least common multiple

00:00:03.107 --> 00:00:06.106
of these two different polynomials.

00:00:06.106 --> 00:00:07.488
So the first one's three z to the third

00:00:07.488 --> 00:00:09.850
minus six z squared minus nine z

00:00:09.850 --> 00:00:11.592
and the second is seven z to the fourth

00:00:11.592 --> 00:00:16.276
plus 21 z to the third plus 14 z squared.

00:00:16.276 --> 00:00:17.665
Now, if you're saying, "Well, what is a,"

00:00:17.665 --> 00:00:19.177
you're familiar with least common multiple

00:00:19.177 --> 00:00:21.528
of two numbers.

00:00:21.528 --> 00:00:22.466
One way to think about them

00:00:22.466 --> 00:00:25.328
is if I were to find, say,
the least common multiple

00:00:25.328 --> 00:00:29.763
between, I don't know, four and six,

00:00:29.763 --> 00:00:31.325
you literally could look
at all the multiples

00:00:31.325 --> 00:00:33.876
and see which one is least.

00:00:33.876 --> 00:00:38.876
So you go four, eight,
12, 16, so on and forth.

00:00:39.064 --> 00:00:40.586
You're gonna do the same thing for six.

00:00:40.586 --> 00:00:44.225
You'd go six, 12,

00:00:44.225 --> 00:00:48.125
18, 24, so on and so forth.

00:00:48.125 --> 00:00:49.647
And you immediately see that they do have,

00:00:49.647 --> 00:00:52.562
they'll actually have
multiple common multiples,

00:00:52.562 --> 00:00:55.018
but the least of the common multiples

00:00:55.018 --> 00:00:59.109
you immediately see is going to be 12.

00:00:59.109 --> 00:01:00.761
Now another way to think about it

00:01:00.761 --> 00:01:03.603
is to actually factor these numbers out.

00:01:03.603 --> 00:01:05.152
We could view four,

00:01:05.152 --> 00:01:08.908
you could view four as being two times two

00:01:08.908 --> 00:01:10.912
if we look at its prime factorization.

00:01:10.912 --> 00:01:15.912
And six is two times three.

00:01:16.226 --> 00:01:18.545
So you could say that the
least common multiple,

00:01:18.545 --> 00:01:22.535
the LCM, of four and six

00:01:22.535 --> 00:01:24.545
is going to be,

00:01:24.545 --> 00:01:26.357
well, it's gonna be equal to,

00:01:26.357 --> 00:01:29.191
we're gonna have to have
the factors of both of them.

00:01:29.191 --> 00:01:32.665
So it's gonna have to have
two fours, two times two.

00:01:32.665 --> 00:01:34.488
It's gonna have to have a two and a three.

00:01:34.488 --> 00:01:35.705
Well we already have a two.

00:01:35.705 --> 00:01:36.896
In fact, we have to them.

00:01:36.896 --> 00:01:39.432
So in order to be divisible
by three, we have to be,

00:01:39.432 --> 00:01:41.461
in order to be divisible by six,

00:01:41.461 --> 00:01:44.277
we have to have three
as one of the factors.

00:01:44.277 --> 00:01:45.319
And so when you look at it that way,

00:01:45.319 --> 00:01:47.241
you say, "Hey look, we have to contain

00:01:47.241 --> 00:01:48.672
"all of the factors of each,"

00:01:48.672 --> 00:01:50.684
we have to have at least two twos

00:01:50.684 --> 00:01:52.993
and we have to have at least one three

00:01:52.993 --> 00:01:54.685
because the two twos
take care of this one too

00:01:54.685 --> 00:01:55.676
right over there.

00:01:55.676 --> 00:01:58.888
And you see that this is
also going to be equal to 12.

00:01:58.888 --> 00:02:00.393
Now when we think about
it for polynomials,

00:02:00.393 --> 00:02:02.235
we're gonna think about
it a little bit more.

00:02:02.235 --> 00:02:04.324
It's essentially the same idea,

00:02:04.324 --> 00:02:05.596
but we're gonna think
about it a little bit more

00:02:05.596 --> 00:02:06.663
with the second lens.

00:02:06.663 --> 00:02:08.325
We're gonna think about the factors

00:02:08.325 --> 00:02:10.007
and say, well the least common multiple

00:02:10.007 --> 00:02:12.159
needs to contain the factors of both

00:02:12.159 --> 00:02:14.071
but it shouldn't contain more than that.

00:02:14.071 --> 00:02:16.538
You can always find a
multiple of two polynomials

00:02:16.538 --> 00:02:18.029
by just multiplying them,

00:02:18.029 --> 00:02:19.581
but we don't wanna find just any multiple.

00:02:19.581 --> 00:02:21.773
We wanna find the least common multiple.

00:02:21.773 --> 00:02:24.004
So let's factor each of them.

00:02:24.004 --> 00:02:27.546
So this first one, three z to the third

00:02:27.546 --> 00:02:30.678
minus six z squared minus nine z.

00:02:30.678 --> 00:02:32.180
Let's see. Immediately, let's see,

00:02:32.180 --> 00:02:34.568
all of these terms are
divisible by three z,

00:02:34.568 --> 00:02:36.459
so let's factor out of three z.

00:02:36.459 --> 00:02:41.361
So it's three z times z squared.

00:02:41.361 --> 00:02:43.363
We factor out of three z out of that.

00:02:43.363 --> 00:02:46.299
See, it's gonna be minus two z

00:02:46.299 --> 00:02:48.425
if we factor out of three z out of that.

00:02:48.425 --> 00:02:51.571
And then minus three.

00:02:51.571 --> 00:02:53.420
And notice, if you were to
distribute this three z back,

00:02:53.420 --> 00:02:56.843
you would get exactly
what we have up here.

00:02:56.843 --> 00:02:58.945
And so let's see, can
we factor this further?

00:02:58.945 --> 00:03:00.287
Can we think of two numbers

00:03:00.287 --> 00:03:02.409
that if we multiply, we get negative three

00:03:02.409 --> 00:03:04.831
and if we add them, we get negative two?

00:03:04.831 --> 00:03:06.636
And one's gonna be positive,
one's gonna be negative

00:03:06.636 --> 00:03:08.368
since the product is negative.

00:03:08.368 --> 00:03:10.964
So let's see. It sounds
like negative three

00:03:10.964 --> 00:03:12.425
and positive one.

00:03:12.425 --> 00:03:15.181
So we can rewrite this as three z

00:03:15.181 --> 00:03:19.058
times z plus one

00:03:19.058 --> 00:03:22.192
times z minus three.

00:03:22.192 --> 00:03:25.525
I think I have factored
this first polynomial

00:03:25.525 --> 00:03:27.704
about as much as I can.

00:03:27.704 --> 00:03:30.090
One times negative
three is negative three.

00:03:30.090 --> 00:03:33.480
One z minus three z is negative two z,

00:03:33.480 --> 00:03:34.644
so that looks good.

00:03:34.644 --> 00:03:36.386
So now let's factor,

00:03:36.386 --> 00:03:38.385
now let's factor this
other character over here,

00:03:38.385 --> 00:03:41.201
this fourth degree polynomial.

00:03:41.201 --> 00:03:43.096
So, every one of those terms

00:03:43.096 --> 00:03:46.292
look like they're visible
by seven z squared,

00:03:46.292 --> 00:03:48.975
so I could write this as seven z squared

00:03:48.975 --> 00:03:51.594
times z squared.

00:03:51.594 --> 00:03:53.233
If we factor out of seven z squared here,

00:03:53.233 --> 00:03:55.295
you're just left with z squared.

00:03:55.295 --> 00:03:57.784
And then plus,

00:03:57.784 --> 00:03:59.876
21 divided by seven is three,

00:03:59.876 --> 00:04:03.439
z to the third divided by z squared is z.

00:04:03.439 --> 00:04:06.016
And then plus, 14 divided by seven is two.

00:04:06.016 --> 00:04:08.258
z squared divided by z squared is one.

00:04:08.258 --> 00:04:10.140
So it's just gonna be a two there.

00:04:10.140 --> 00:04:14.256
And so this is gonna be the
same thing as seven z squared.

00:04:14.256 --> 00:04:16.699
And this can be factored into, let's see,

00:04:16.699 --> 00:04:19.204
two times one is two,
two plus one is three.

00:04:19.204 --> 00:04:22.585
So, z plus one

00:04:22.585 --> 00:04:25.845
times z plus two.

00:04:25.845 --> 00:04:27.757
Now let's think about the
least common multiple.

00:04:27.757 --> 00:04:30.303
We've factored each of
these just the way that we,

00:04:30.303 --> 00:04:32.342
when we did the prime
factorization for regular numbers.

00:04:32.342 --> 00:04:33.743
Now we have factored this down

00:04:33.743 --> 00:04:38.743
to simpler expressions
as we will find useful.

00:04:38.784 --> 00:04:41.925
And so, the least common
multiple of these two things

00:04:41.925 --> 00:04:45.518
has to contain each of these factors.

00:04:45.518 --> 00:04:49.928
So the least common multiples
got to contain a three z.

00:04:49.928 --> 00:04:52.492
It's got to contain,

00:04:52.492 --> 00:04:54.064
and let me expand it out a little bit.

00:04:54.064 --> 00:04:55.565
It's got to contain a three.

00:04:55.565 --> 00:04:57.766
It's got to contain a z.

00:04:57.766 --> 00:05:01.508
It's gotta contain a z plus one,

00:05:01.508 --> 00:05:04.048
z plus one.

00:05:04.048 --> 00:05:06.320
I don't have to write
the little dot there.

00:05:06.320 --> 00:05:09.342
It's got to contain a z plus one.

00:05:09.342 --> 00:05:13.084
It's gotta contain a z minus three.

00:05:13.084 --> 00:05:15.155
Let's see. It's gotta contain a seven.

00:05:15.155 --> 00:05:17.487
We do not have a seven here yet,

00:05:17.487 --> 00:05:21.240
so we have to include a seven,

00:05:21.240 --> 00:05:23.983
so I'll put the seven out
front with the numbers.

00:05:23.983 --> 00:05:25.474
It's gotta include a seven.

00:05:25.474 --> 00:05:28.618
It's gotta include a z squared.

00:05:28.618 --> 00:05:30.100
Well we only have a z right now,

00:05:30.100 --> 00:05:32.131
so let's throw in another z.

00:05:32.131 --> 00:05:34.347
So I could throw in
another, I could write,

00:05:34.347 --> 00:05:35.405
I could put a z out front

00:05:35.405 --> 00:05:37.177
or I could just make this a squared.

00:05:37.177 --> 00:05:39.849
It still contains that z,
but now we contain another z,

00:05:39.849 --> 00:05:42.796
or multiplying by another
z to have z squared.

00:05:42.796 --> 00:05:45.184
See, we already have a z plus one in here.

00:05:45.184 --> 00:05:47.346
We need a z plus two as well,

00:05:47.346 --> 00:05:49.789
z plus two as well.

00:05:49.789 --> 00:05:50.680
And there you have it.

00:05:50.680 --> 00:05:51.892
This is the least common multiple.

00:05:51.892 --> 00:05:53.614
If I were to write it all
out in a neutral color,

00:05:53.614 --> 00:05:57.356
it's gonna be 21 z squared

00:05:57.356 --> 00:05:59.565
times z plus one

00:05:59.565 --> 00:06:02.334
times z minus three

00:06:02.334 --> 00:06:04.635
times z plus two

00:06:04.635 --> 00:06:06.787
(chuckle) I say two and I write six.

00:06:06.787 --> 00:06:08.501
z plus two.

00:06:08.501 --> 00:06:10.496
And we are all done.

00:06:10.496 --> 00:06:11.348
And I really want you to appreciate.

00:06:11.348 --> 00:06:12.880
This is the exact same thing we're doing

00:06:12.880 --> 00:06:14.011
when we're doing,

00:06:14.011 --> 00:06:15.443
a very similar thing than what we're doing

00:06:15.443 --> 00:06:17.005
when we're finding these common multiples

00:06:17.005 --> 00:06:19.936
of regular numbers,
looking at their factors.

00:06:19.936 --> 00:06:22.039
In the case of numbers, prime factors.

00:06:22.039 --> 00:06:25.531
And then we say, "Okay.
The least common multiple

00:06:25.531 --> 00:06:27.790
"has to be a superset, has
to contain all of these,

00:06:27.790 --> 00:06:29.078
"but we don't wanna contain,"

00:06:29.078 --> 00:06:32.545
you know, I could multiply
this times a hundred,

00:06:32.545 --> 00:06:34.574
it's still gonna be a common
multiple of these two,

00:06:34.574 --> 00:06:36.376
but it's no longer the
least common multiple.

00:06:36.376 --> 00:06:39.716
Likewise, 12 is the least
common multiple of four and six.

00:06:39.716 --> 00:06:40.658
If I just wanted a common multiple,

00:06:40.658 --> 00:06:41.920
I can multiply that times a hundred,

00:06:41.920 --> 00:06:45.042
1200 would also be a
multiple of four and six,

00:06:45.042 --> 00:06:46.824
but it wouldn't be the
least common multiple,

00:06:46.824 --> 00:06:49.308
so we don't wanna do that.

00:06:49.308 --> 00:06:51.086
Hopefully you found that interesting.


Subtracting rational expressions: factored denominators

WEBVTT
Kind: captions
Language: en

00:00:00.245 --> 00:00:01.059
- [Voiceover] Pause this video

00:00:01.059 --> 00:00:03.101
and see if you can subtract this magenta

00:00:03.101 --> 00:00:06.502
rational expression from this yellow one.

00:00:06.502 --> 00:00:08.406
Alright, now let's do this together.

00:00:08.406 --> 00:00:09.928
And the first thing that jumps out at you

00:00:09.928 --> 00:00:11.101
is that you realize these don't

00:00:11.101 --> 00:00:12.563
have the same denominator and you would

00:00:12.563 --> 00:00:14.118
like them to have the same denominator.

00:00:14.118 --> 00:00:16.148
And so you might say,
well, let me rewrite them

00:00:16.148 --> 00:00:18.584
so that they have a common denominator.

00:00:18.584 --> 00:00:20.710
And a common denominator that will work

00:00:20.710 --> 00:00:25.710
will be one that is divisible
by each of these denominators.

00:00:26.142 --> 00:00:28.163
So it has all the factors of each

00:00:28.163 --> 00:00:29.799
of these denominators and lucky for us,

00:00:29.799 --> 00:00:32.260
each of these denominators
are already factored.

00:00:32.260 --> 00:00:34.245
So let me just write
the common denominator,

00:00:34.245 --> 00:00:37.462
I'll start rewriting
the yellow expression.

00:00:37.462 --> 00:00:39.237
So, you have the yellow expression,

00:00:39.237 --> 00:00:40.200
actually, let me just make it clear,

00:00:40.200 --> 00:00:42.186
I'm going to write both, the yellow one,

00:00:42.186 --> 00:00:45.600
and then you're going to
subtract the magenta one.

00:00:45.600 --> 00:00:49.743
Whoops. I'm saying yellow
but drawing in magenta.

00:00:49.743 --> 00:00:52.901
So you have the yellow expression

00:00:52.901 --> 00:00:54.202
which I'm about to rewrite, actually,

00:00:54.202 --> 00:00:55.263
I'm going to make a longer line,

00:00:55.263 --> 00:01:00.263
so the yellow expression
minus the magenta one,

00:01:02.391 --> 00:01:06.013
minus the magenta one, right over there.

00:01:06.013 --> 00:01:07.673
Now, as I mentioned, we
want to have a denominator

00:01:07.673 --> 00:01:09.716
that has all, the common
denominator has to have,

00:01:09.716 --> 00:01:12.537
be divisible by both,
this yellow denominator

00:01:12.537 --> 00:01:14.313
and this magenta one.

00:01:14.313 --> 00:01:19.270
So it's got to have
the Z plus eight in it.

00:01:19.270 --> 00:01:23.635
It's got to have the 9z minus five in it.

00:01:23.635 --> 00:01:25.912
And it's also got to have both of these.

00:01:25.912 --> 00:01:28.813
Well, I already, we already
accounted for the 9z minus five.

00:01:28.813 --> 00:01:33.813
So it has to have, be divisible
by Z plus six. Z plus six.

00:01:35.234 --> 00:01:36.928
Notice just by multiplying the denominator

00:01:36.928 --> 00:01:40.387
by Z plus six, we're not
divisible by both of these factors

00:01:40.387 --> 00:01:43.929
AND both of these factors
because 9z minus five

00:01:43.929 --> 00:01:46.831
was the factor common to both of them.

00:01:46.831 --> 00:01:47.887
And if you were just dealing with numbers

00:01:47.887 --> 00:01:49.628
when you were just adding
or subtracting fractions,

00:01:49.628 --> 00:01:52.530
it works the exact same way.

00:01:52.530 --> 00:01:54.667
Alright, so what will
the numerator become?

00:01:54.667 --> 00:01:58.312
Well, we multiply the denominator times

00:01:58.312 --> 00:01:59.729
Z plus six, so we have to do the

00:01:59.729 --> 00:02:01.470
same thing to the numerator.

00:02:01.470 --> 00:02:05.672
It's going to be negative Z
to the third times Z plus six.

00:02:05.672 --> 00:02:07.436
Now let's focus over here.

00:02:07.436 --> 00:02:09.932
We had, well, we want
the same denominator,

00:02:09.932 --> 00:02:13.392
so we can write this as Z plus eight

00:02:13.392 --> 00:02:18.392
Z plus eight times Z plus six,

00:02:19.499 --> 00:02:24.499
times Z plus six times 9z minus five.

00:02:29.935 --> 00:02:31.514
And these are equivalent.

00:02:31.514 --> 00:02:33.371
I've just changed the order
that we multiply in it,

00:02:33.371 --> 00:02:35.077
that doesn't change their value.

00:02:35.077 --> 00:02:39.639
And if we multiplied the, so
we had a three on top before

00:02:39.639 --> 00:02:42.135
and if we multiply the
denominator times Z plus eight,

00:02:42.135 --> 00:02:47.135
we also have to multiply the
numerator times Z plus eight.

00:02:48.181 --> 00:02:49.364
So there you go.

00:02:49.364 --> 00:02:52.047
And so, this is going to be equal to,

00:02:52.047 --> 00:02:54.287
this is going to be equal
to, actually, I'll just make

00:02:54.287 --> 00:02:58.188
a big line right over here.

00:02:58.188 --> 00:03:00.545
This is all going to be equal to.

00:03:00.545 --> 00:03:02.947
We have our, probably
don't need that much space,

00:03:02.947 --> 00:03:06.046
let me see, maybe that,
maybe about that much.

00:03:06.046 --> 00:03:08.102
So I'm going to have the same denominator

00:03:08.102 --> 00:03:10.343
and I'll just write it
in a neutral color now.

00:03:10.343 --> 00:03:15.343
Z plus eight times 9z minus
five times Z plus six.

00:03:20.035 --> 00:03:23.496
So over here, just in this blue color,

00:03:23.496 --> 00:03:26.339
we want to distribute this
negative Z to the third.

00:03:26.339 --> 00:03:30.924
Negative Z to the third times
Z is negative Z to the fourth.

00:03:30.924 --> 00:03:35.924
Negative Z to the third times
six is minus 6z to the third.

00:03:37.238 --> 00:03:39.652
And now this negative
sign, right over here,

00:03:39.652 --> 00:03:42.938
actually, instead of saying negative Z,

00:03:42.938 --> 00:03:44.413
negative of this entire thing,

00:03:44.413 --> 00:03:48.313
we could just say plus
the negative of this.

00:03:48.313 --> 00:03:49.649
Or another of thinking about it,

00:03:49.649 --> 00:03:54.129
you could view this as negative
three times Z plus eight.

00:03:54.129 --> 00:03:55.859
So we could just distribute that.

00:03:55.859 --> 00:03:57.298
So let's do that.

00:03:57.298 --> 00:04:02.298
So negative three times Z is negative 3z

00:04:02.975 --> 00:04:07.975
and negative three times
eight is negative 24.

00:04:08.037 --> 00:04:09.116
And there you go.

00:04:09.116 --> 00:04:11.752
We are, we are done.

00:04:11.752 --> 00:04:13.295
We found a common denominator.

00:04:13.295 --> 00:04:14.353
And once you have a common denominator,

00:04:14.353 --> 00:04:17.056
you could just subtract
or add the numerators,

00:04:17.056 --> 00:04:20.388
and instead of doing this
as minus this entire thing,

00:04:20.388 --> 00:04:23.187
I viewed it as adding and then having

00:04:23.187 --> 00:04:24.886
a negative three in the numerator,

00:04:24.886 --> 00:04:26.721
distributing that and then these,

00:04:26.721 --> 00:04:28.021
I can't simplify it any further.

00:04:28.021 --> 00:04:29.227
Sometimes you'll do one of these types

00:04:29.227 --> 00:04:30.469
of exercises and you might have

00:04:30.469 --> 00:04:33.929
two second-degree terms
or two first-degree terms

00:04:33.929 --> 00:04:35.601
or two constants or something like that

00:04:35.601 --> 00:04:37.121
and then you might want
to add or subtract them

00:04:37.121 --> 00:04:40.128
to simplify it but here, these
all have different degrees

00:04:40.128 --> 00:04:45.128
so I can't simplify it any
further and so we are all done.


Simplifying rational expressions: common monomial factors

WEBVTT
Kind: captions
Language: en

00:00:00.196 --> 00:00:02.018
- [Voiceover] So I have a
rational expression here

00:00:02.018 --> 00:00:05.282
and my goal is to simplify
it, but while I simplify it,

00:00:05.282 --> 00:00:07.657
I wanna make the simplified expression

00:00:07.657 --> 00:00:09.406
be algebraically equivalent.

00:00:09.406 --> 00:00:10.778
So, if there are certain x values

00:00:10.778 --> 00:00:12.570
that would make this thing undefined,

00:00:12.570 --> 00:00:14.982
that I have to restrict
my simplified expression

00:00:14.982 --> 00:00:16.223
by those x values.

00:00:16.223 --> 00:00:18.438
So, you could pause this
video and take a go at it

00:00:18.438 --> 00:00:20.604
and then we'll do it together.

00:00:20.604 --> 00:00:22.486
All right. So let's just think real quick.

00:00:22.486 --> 00:00:26.363
What x values would make
this expression undefined?

00:00:26.363 --> 00:00:28.298
Well, it's undefined if
we try to divide by zero.

00:00:28.298 --> 00:00:30.830
So if x is zero, 14 times zero is zero,

00:00:30.830 --> 00:00:32.198
it's going to be undefined.

00:00:32.198 --> 00:00:36.087
And so, we could say this is,

00:00:36.087 --> 00:00:38.798
we could say x does not,
cannot be equal to zero.

00:00:38.798 --> 00:00:41.821
For any other x, we can
evaluate this expression.

00:00:41.821 --> 00:00:43.753
Now let's actually try to simplify it.

00:00:43.753 --> 00:00:45.805
So, when you look at the
numerator and the denominator,

00:00:45.805 --> 00:00:49.125
every term we see is divisible by x

00:00:49.125 --> 00:00:51.725
and every term is divisible by seven.

00:00:51.725 --> 00:00:54.517
So it looks like we can factor
seven x out of the numerator

00:00:54.517 --> 00:00:56.749
and seven x out of the denominator.

00:00:56.749 --> 00:00:58.639
So the numerator, we can rewrite

00:00:58.639 --> 00:01:03.009
as seven x times,

00:01:03.009 --> 00:01:04.931
we factor seven x out of 14 x squared,

00:01:04.931 --> 00:01:07.203
we're gonna be left with two x

00:01:07.203 --> 00:01:09.015
and then you factor
seven x out of seven x,

00:01:09.015 --> 00:01:11.237
you're gonna be left with a one.

00:01:11.237 --> 00:01:12.105
And one way to think about it

00:01:12.105 --> 00:01:15.061
is we did the distributive
property in reverse,

00:01:15.061 --> 00:01:15.872
if we were to do it again,

00:01:15.872 --> 00:01:18.545
seven x times two x is 14 x squared,

00:01:18.545 --> 00:01:22.148
seven x times one is seven x.

00:01:22.148 --> 00:01:24.141
All right. And now let's factor seven x

00:01:24.141 --> 00:01:26.182
out of the denominator.

00:01:26.182 --> 00:01:28.644
So, 14 x could be rewritten

00:01:28.644 --> 00:01:32.344
as seven x times two,

00:01:32.344 --> 00:01:34.055
times two,

00:01:34.055 --> 00:01:36.834
and remember, I wanna keep
this algebraically equivalent,

00:01:36.834 --> 00:01:38.410
so I wanna keep the constraint

00:01:38.410 --> 00:01:41.902
that x cannot be equal to zero.

00:01:41.902 --> 00:01:43.741
And so we divide the numerator

00:01:43.741 --> 00:01:46.599
and the denominator by seven x
or one way to think about it,

00:01:46.599 --> 00:01:49.310
you can divide seven x by
seven x and just get one

00:01:49.310 --> 00:01:51.378
and we are left with

00:01:51.378 --> 00:01:55.328
two x plus one over two.

00:01:55.328 --> 00:01:57.063
Now this was the original expression,

00:01:57.063 --> 00:01:58.961
x could take on any value,

00:01:58.961 --> 00:02:01.921
but if we want it to be
algebraically equivalent

00:02:01.921 --> 00:02:03.776
to our original expression,

00:02:03.776 --> 00:02:06.146
it has to have the same constraints on.

00:02:06.146 --> 00:02:08.495
So we're going to have to say x,

00:02:08.495 --> 00:02:10.497
x does not equal zero,

00:02:10.497 --> 00:02:13.392
and it's a very subtle but
really important thing.

00:02:13.392 --> 00:02:15.345
For example, if you defined a function

00:02:15.345 --> 00:02:16.810
by this right over here,

00:02:16.810 --> 00:02:19.753
the domain of the function
could not include zero.

00:02:19.753 --> 00:02:23.154
And so, if you simplified how you defined

00:02:23.154 --> 00:02:24.445
that function to this,

00:02:24.445 --> 00:02:26.297
if you want that function to be the same,

00:02:26.297 --> 00:02:27.969
it needs to have the same domain.

00:02:27.969 --> 00:02:30.231
It has to be defined for the same inputs.

00:02:30.231 --> 00:02:31.352
And so that's why we're putting

00:02:31.352 --> 00:02:34.544
the exact same constraints
for them to be equivalent.

00:02:34.544 --> 00:02:36.806
If you got rid of this constraint,

00:02:36.806 --> 00:02:38.978
these two would be equivalent everywhere

00:02:38.978 --> 00:02:40.749
except for x equal zero.

00:02:40.749 --> 00:02:42.591
This one would have been
defined for x equal zero,

00:02:42.591 --> 00:02:43.802
this one would not have,

00:02:43.802 --> 00:02:46.194
and so they wouldn't have
been algebraically equivalent.

00:02:46.194 --> 00:02:48.576
This makes them algebraically equivalent.

00:02:48.576 --> 00:02:50.754
And of course you can write
this in different ways.

00:02:50.754 --> 00:02:53.756
You can divide each of these
terms by two, if you like.

00:02:53.756 --> 00:02:56.868
So you could divide two x by two and get x

00:02:56.868 --> 00:03:00.590
and then divide one by two and get 1/2.

00:03:00.590 --> 00:03:02.521
Once again, we would wanna keep

00:03:02.521 --> 00:03:06.292
the x cannot be equal to zero.

00:03:06.292 --> 00:03:08.783
Let's do another one of these.

00:03:08.783 --> 00:03:10.335
So it's a slightly hairier expression,

00:03:10.335 --> 00:03:11.216
but let's do the same drill.

00:03:11.216 --> 00:03:12.617
See if we could simplify it.

00:03:12.617 --> 00:03:16.059
But as we simplify it, we
really wanna be conscientious

00:03:16.059 --> 00:03:19.020
of restricting the Zs here

00:03:19.020 --> 00:03:22.622
so that we get an algebraically
equivalent expression.

00:03:22.622 --> 00:03:24.784
So, let's think about
where this is undefined.

00:03:24.784 --> 00:03:26.076
We can think about where it's undefined

00:03:26.076 --> 00:03:29.708
by factoring the denominator here.

00:03:29.708 --> 00:03:31.009
Actually, let me just...

00:03:31.009 --> 00:03:33.581
So, this is going to be equal to,

00:03:33.581 --> 00:03:35.102
actually, let me just...

00:03:35.102 --> 00:03:37.184
Let me just do the first
step where I could say,

00:03:37.184 --> 00:03:38.345
"Well, what's a common factor

00:03:38.345 --> 00:03:40.274
"in the numerator and the denominator?"

00:03:40.274 --> 00:03:43.176
Every term here is divided by z squared

00:03:43.176 --> 00:03:45.812
and every term is also divided by 17,

00:03:45.812 --> 00:03:49.395
so it looks like 17 z
squared can be factored out.

00:03:49.395 --> 00:03:53.199
So, 17 z squared can be
factored out of the numerator

00:03:53.199 --> 00:03:55.824
and then we would be left with,

00:03:55.824 --> 00:03:59.295
we factor out of 17 z squared
out of 17 z to the third

00:03:59.295 --> 00:04:01.607
and we're gonna be left with just a z.

00:04:01.607 --> 00:04:04.274
17 z squared, you factor
17 z squared out of that

00:04:04.274 --> 00:04:06.569
and you're gonna be left with just a one.

00:04:06.569 --> 00:04:09.773
And once again, you can
distribute the 17 z squared,

00:04:09.773 --> 00:04:12.495
multiply it times z, you
get 17 z to the third.

00:04:12.495 --> 00:04:16.309
17 z squared times one is 17 z squared.

00:04:16.309 --> 00:04:19.626
All right. All of that
is going to be over,

00:04:19.626 --> 00:04:21.288
we wanna factor out of 17 z squared

00:04:21.288 --> 00:04:23.186
out of the denominator,

00:04:23.186 --> 00:04:27.096
17 z squared times,

00:04:27.096 --> 00:04:31.520
and so 34 z to the third
divided by 17 z squared,

00:04:31.520 --> 00:04:34.570
34 divided by 17 is two,

00:04:34.570 --> 00:04:38.570
and z to the third
divided by z squared is z

00:04:38.570 --> 00:04:43.570
and then we have minus
51 divided 17 is three

00:04:43.622 --> 00:04:47.053
and z squared divided by
z squared is just one.

00:04:47.053 --> 00:04:50.116
So we'll just leave it like that.

00:04:50.116 --> 00:04:52.058
And so over here, it
becomes a little bit clear

00:04:52.058 --> 00:04:53.930
of, well one, how we're gonna simplify it.

00:04:53.930 --> 00:04:57.163
We're just gonna divided 17
z squared by 17 z squared,

00:04:57.163 --> 00:04:59.799
but let's be careful to
restrict the domain here.

00:04:59.799 --> 00:05:02.055
So, we can tell if z is equal to zero,

00:05:02.055 --> 00:05:05.571
then this, 17 z squared
is gonna be equal to zero

00:05:05.571 --> 00:05:06.953
and we make the denominator equal to zero

00:05:06.953 --> 00:05:08.795
and we could see that even looking here.

00:05:08.795 --> 00:05:12.055
So we could say z cannot be equal to zero.

00:05:12.055 --> 00:05:14.636
Now what else? Z cannot be equal to

00:05:14.636 --> 00:05:17.790
whatever makes this two z
minus three equals zero.

00:05:17.790 --> 00:05:18.761
So let's think about what makes

00:05:18.761 --> 00:05:21.354
two z minus three equal to zero.

00:05:21.354 --> 00:05:24.657
Two z minus three is equal to zero.

00:05:24.657 --> 00:05:26.686
You can add three to both sides

00:05:26.686 --> 00:05:29.982
and you would get two z is equal to three.

00:05:29.982 --> 00:05:32.065
Divide both sides by two

00:05:32.065 --> 00:05:35.344
and you would get z is equal to 3/2.

00:05:35.344 --> 00:05:37.925
So z cannot be equal to zero

00:05:37.925 --> 00:05:42.197
and z cannot be equal to 3/2.

00:05:42.197 --> 00:05:44.393
So that's how we're gonna
restrict our domain.

00:05:44.393 --> 00:05:47.013
But now let's simplify.

00:05:47.013 --> 00:05:51.054
So, if we simplify it,
these two cancel out

00:05:51.054 --> 00:05:52.986
and we are going to be left with,

00:05:52.986 --> 00:05:56.122
we're gonna be left with z plus one

00:05:56.122 --> 00:05:59.138
over two z minus three

00:05:59.138 --> 00:06:00.700
and we wanna keep that constraint.

00:06:00.700 --> 00:06:05.077
z cannot be equal to zero.

00:06:05.077 --> 00:06:07.173
And actually, this second
constraint is redundant

00:06:07.173 --> 00:06:09.095
because we still have the
two z minus three here.

00:06:09.095 --> 00:06:10.877
If someone were to just
look at this expression,

00:06:10.877 --> 00:06:13.506
like, well, the denominator
can't be equal to zero

00:06:13.506 --> 00:06:16.215
and so, z cannot be equal to 3/2.

00:06:16.215 --> 00:06:18.314
So this is still, if we
just left it the way it is,

00:06:18.314 --> 00:06:19.936
we don't even have to rewrite this,

00:06:19.936 --> 00:06:22.352
that would be redundant since
looking at this expression

00:06:22.352 --> 00:06:26.349
is clearly not defined for z equals 3/2.

00:06:26.349 --> 00:06:28.121
So, there you go.

00:06:28.121 --> 00:06:29.155
And so I'm gonna ask you

00:06:29.155 --> 00:06:32.258
for what values does this
expression not defined,

00:06:32.258 --> 00:06:34.010
well then you would also include that.

00:06:34.010 --> 00:06:35.031
Actually, let me just write it.

00:06:35.031 --> 00:06:37.033
It doesn't hurt to be redundant.

00:06:37.033 --> 00:06:40.578
z does not equal 3/2.

00:06:40.578 --> 00:06:42.986
But this constraint right
here is really important

00:06:42.986 --> 00:06:44.844
because it's not obvious by
looking at this expression.

00:06:44.844 --> 00:06:46.483
This expression by itself

00:06:46.483 --> 00:06:48.718
would be defined for z equals zero,

00:06:48.718 --> 00:06:50.820
but if we want it to be
algebraically equivalent

00:06:50.820 --> 00:06:52.011
to this one and that one,

00:06:52.011 --> 00:06:54.618
it has to be constrained

00:06:54.618 --> 00:06:57.544
in the same way.


Simplifying rational expressions: higher degree terms

WEBVTT
Kind: captions
Language: en

00:00:00.411 --> 00:00:02.349
- [Voiceover] Let's see if we
can simplify this expression.

00:00:02.349 --> 00:00:05.230
So pause the video and have a try at it,

00:00:05.230 --> 00:00:08.200
and then we're gonna do
it together right now.

00:00:08.200 --> 00:00:10.069
All right, so when you look at this,

00:00:10.069 --> 00:00:11.741
it looks like both the
numerator and denominator,

00:00:11.741 --> 00:00:13.401
they might, you might
be able to factor them,

00:00:13.401 --> 00:00:15.143
and maybe they have some common factors

00:00:15.143 --> 00:00:18.023
that you can divide the
numerator and the denominator by

00:00:18.023 --> 00:00:19.322
to simplify it.

00:00:19.322 --> 00:00:21.400
So let's first try to
factor the numerator.

00:00:21.400 --> 00:00:24.325
X to the fourth, plus eight
x squared, plus seven.

00:00:24.325 --> 00:00:25.962
At first it might be
a little intimidating,

00:00:25.962 --> 00:00:27.739
because you have an x to the fourth here.

00:00:27.739 --> 00:00:29.004
It's not a quadratic;

00:00:29.004 --> 00:00:30.838
it's a fourth degree polynomial,

00:00:30.838 --> 00:00:34.239
but like any, if you,
like a lot of quadratics

00:00:34.239 --> 00:00:37.186
that we've seen in the past,
it does seem to have a pattern.

00:00:37.186 --> 00:00:39.682
For example, if this said x squared

00:00:39.682 --> 00:00:43.361
plus eight x plus seven,

00:00:43.361 --> 00:00:44.940
you'd say, oh, well, this
is pretty straightforward

00:00:44.940 --> 00:00:45.839
to factor.

00:00:45.839 --> 00:00:47.532
What two numbers add up to eight

00:00:47.532 --> 00:00:49.251
and when I take their product I get seven?

00:00:49.251 --> 00:00:51.677
Well, there's only two numbers
where you take their product

00:00:51.677 --> 00:00:54.138
and you get positive seven
that are going to be positive,

00:00:54.138 --> 00:00:56.936
and they need to be
positive, if they're going to

00:00:56.936 --> 00:00:59.594
add up to positive eight,
and that's one and seven.

00:00:59.594 --> 00:01:02.009
So this would be x plus seven

00:01:02.009 --> 00:01:05.387
times x plus one.

00:01:05.387 --> 00:01:07.036
Well, if you just think of,

00:01:07.036 --> 00:01:09.520
instead of thinking in
terms of x and x squared,

00:01:09.520 --> 00:01:10.901
if you just think in terms of x squared

00:01:10.901 --> 00:01:11.958
and x to the fourth, it's going to be

00:01:11.958 --> 00:01:13.339
the exact same thing.

00:01:13.339 --> 00:01:15.440
So this thing can be written as

00:01:15.440 --> 00:01:18.018
x squared plus seven

00:01:18.018 --> 00:01:21.303
times x squared plus one.

00:01:21.303 --> 00:01:23.602
If you want, you can do
some type of a substitution

00:01:23.602 --> 00:01:28.148
saying, saying that a
is equaled to x squared,

00:01:28.148 --> 00:01:32.223
in which case, so if you said
that a is equal to x squared,

00:01:32.223 --> 00:01:35.427
then this thing would become a squared

00:01:35.427 --> 00:01:38.585
plus eight a plus seven,

00:01:38.585 --> 00:01:41.464
and then you would factor this into

00:01:41.464 --> 00:01:45.364
a plus seven and a plus one,

00:01:45.364 --> 00:01:48.023
and then you would undo the substitution,

00:01:48.023 --> 00:01:49.439
and that's x squared plus seven

00:01:49.439 --> 00:01:50.584
and x squared plus one.

00:01:50.584 --> 00:01:52.105
But hopefully you see
what's going on here;

00:01:52.105 --> 00:01:55.402
this is the higher order term,

00:01:55.402 --> 00:01:57.956
and then this is half the degree of that,

00:01:57.956 --> 00:01:59.732
so it fits this mold.

00:01:59.732 --> 00:02:01.032
And so you could do a substitution,

00:02:01.032 --> 00:02:02.368
or you could just recognize, oh okay,

00:02:02.368 --> 00:02:04.689
instead of dealing with x squared,

00:02:04.689 --> 00:02:06.895
I'm dealing with x to the fourth.

00:02:06.895 --> 00:02:09.147
All right, so that's the numerator.

00:02:09.147 --> 00:02:10.749
Now let's think about,

00:02:10.749 --> 00:02:12.885
let's think about the denominator.

00:02:12.885 --> 00:02:15.149
So the denominator, both of these terms

00:02:15.149 --> 00:02:17.169
are divisible by three x.

00:02:17.169 --> 00:02:20.048
So let's factor out a three x.

00:02:20.048 --> 00:02:22.474
So it's three x times,

00:02:22.474 --> 00:02:24.089
three x times,

00:02:24.089 --> 00:02:25.667
if you factor out a three x here,

00:02:25.667 --> 00:02:26.944
three divided by three is one,

00:02:26.944 --> 00:02:30.972
x to the fifth divided
by x is x to the fourth,

00:02:30.972 --> 00:02:33.712
and then if you factor out a three x here,

00:02:33.712 --> 00:02:35.465
you're just gonna get one.

00:02:35.465 --> 00:02:38.030
And so far this doesn't seem too helpful.

00:02:38.030 --> 00:02:39.783
I don't see an x to the fourth minus one,

00:02:39.783 --> 00:02:41.567
or a three x in the numerator,

00:02:41.567 --> 00:02:43.645
but maybe I can factor this out further,

00:02:43.645 --> 00:02:45.294
x to the fourth minus one.

00:02:45.294 --> 00:02:48.254
And that's because it is
a difference of squares.

00:02:48.254 --> 00:02:50.066
And you might say, wait, I'm always used

00:02:50.066 --> 00:02:51.690
to recognizing a difference of squares

00:02:51.690 --> 00:02:53.815
as something like a squared minus one,

00:02:53.815 --> 00:02:56.450
which you could write as a plus one

00:02:56.450 --> 00:02:59.191
times a minus one.

00:02:59.191 --> 00:03:01.326
Well, this would be a squared minus one

00:03:01.326 --> 00:03:05.749
if you say that a is equal to x squared.

00:03:05.749 --> 00:03:08.349
Then this would be a squared minus one.

00:03:08.349 --> 00:03:10.404
So let's rewrite all of this.

00:03:10.404 --> 00:03:12.273
So let's rewrite.

00:03:12.273 --> 00:03:14.004
So this is all going to be equal to,

00:03:14.004 --> 00:03:15.744
same numerator,

00:03:15.744 --> 00:03:17.079
let's see, let's do it in green.

00:03:17.079 --> 00:03:18.194
Same numerator:

00:03:18.194 --> 00:03:21.990
x squared plus seven, can't
factor that out any more,

00:03:21.990 --> 00:03:25.972
times x squared plus one,
can't factor that out any more,

00:03:25.972 --> 00:03:29.421
all of that over three x,

00:03:29.421 --> 00:03:32.159
but this I can view as
a difference of squares.

00:03:32.159 --> 00:03:33.901
So this is x squared squared,

00:03:33.901 --> 00:03:35.537
and this is obviously one squared,

00:03:35.537 --> 00:03:38.672
so this is going to be x squared plus one

00:03:38.672 --> 00:03:41.794
times x squared minus,

00:03:41.794 --> 00:03:44.739
times x squared minus one.

00:03:44.739 --> 00:03:46.806
Now clearly have an x
squared in the numerator,

00:03:46.806 --> 00:03:48.133
x squared minus one in the numerator,

00:03:48.133 --> 00:03:50.054
x squared, sorry, x squared
plus one in the numerator,

00:03:50.054 --> 00:03:52.050
x squared plus one in the denominator,

00:03:52.050 --> 00:03:53.989
and so I could cancel them out,

00:03:53.989 --> 00:03:55.788
and I'm going to be left
with, in the numerator,

00:03:55.788 --> 00:03:58.749
x squared plus seven,

00:03:58.749 --> 00:04:01.135
over three x

00:04:01.135 --> 00:04:05.653
times x squared minus one.

00:04:05.653 --> 00:04:07.330
Now, this looks pretty simple,

00:04:07.330 --> 00:04:08.348
and we want to be a little careful,

00:04:08.348 --> 00:04:09.888
because whenever we do this cancelling out

00:04:09.888 --> 00:04:11.072
we don't want, we want to make sure

00:04:11.072 --> 00:04:14.206
that we restrict the xes

00:04:14.206 --> 00:04:15.774
for which the expression's defined,

00:04:15.774 --> 00:04:18.537
if we want them to be
algebraically equivalent.

00:04:18.537 --> 00:04:21.381
So this one, would this be,

00:04:21.381 --> 00:04:23.459
this would obviously be undefined,

00:04:23.459 --> 00:04:25.757
so x cannot be equal to zero,

00:04:25.757 --> 00:04:29.066
x cannot be equal to plus or minus one.

00:04:29.066 --> 00:04:32.467
Positive or negative one
would make this expression

00:04:32.467 --> 00:04:33.802
right over here equal zero,

00:04:33.802 --> 00:04:35.439
so it cannot be equal to zero,

00:04:35.439 --> 00:04:38.818
x cannot be equal to, I'll
write plus or minus one;

00:04:38.818 --> 00:04:40.721
that would make this part zero.

00:04:40.721 --> 00:04:43.777
But this right over here, this one,

00:04:43.777 --> 00:04:45.853
unless, we're assuming we're dealing

00:04:45.853 --> 00:04:49.109
only with real numbers, this
one can't ever equal zero

00:04:49.109 --> 00:04:50.514
if you're dealing with real numbers,

00:04:50.514 --> 00:04:52.685
because x squared is always
going to be non-negative

00:04:52.685 --> 00:04:54.532
and you're adding it to a positive value,

00:04:54.532 --> 00:04:58.211
and so this part, this
factor, would have never made

00:04:58.211 --> 00:04:59.650
the entire thing undefined.

00:04:59.650 --> 00:05:01.612
So we can actually just factor it out,

00:05:01.612 --> 00:05:04.607
or cancel it out, without
worrying much about it.

00:05:04.607 --> 00:05:07.104
And so this is actually
algebraically equivalent

00:05:07.104 --> 00:05:09.111
to what we had originally.

00:05:09.111 --> 00:05:11.631
Now we could write these
constraints on it, if we want.

00:05:11.631 --> 00:05:13.733
If someone were to ask me, you know,

00:05:13.733 --> 00:05:16.425
for what x is this expression not defined,

00:05:16.425 --> 00:05:18.226
well, it's clear it's not defined for x,

00:05:18.226 --> 00:05:19.722
that would make the denominator zero,

00:05:19.722 --> 00:05:21.324
dividing by zero not defined,

00:05:21.324 --> 00:05:23.264
or if x is plus or minus one,

00:05:23.264 --> 00:05:24.947
it would make the denominator equal zero.

00:05:24.947 --> 00:05:27.501
But that is, that comes
straight of this expression,

00:05:27.501 --> 00:05:29.644
so this expression and
our original expression

00:05:29.644 --> 00:05:31.323
are algebraically equivalent.

00:05:31.323 --> 00:05:33.737
Now if you wanted to, you
could expand the bottom

00:05:33.737 --> 00:05:36.674
out a little bit, you could
multiply it out if you like.

00:05:36.674 --> 00:05:38.324
That's equivalent to,

00:05:38.324 --> 00:05:40.099
so x squared plus seven

00:05:40.099 --> 00:05:43.500
over three x times x squared
is three x to the third

00:05:43.500 --> 00:05:45.276
minus three x.

00:05:45.276 --> 00:05:48.899
So these are all, these are
all equivalent expressions,

00:05:48.899 --> 00:05:51.110
and we are done.


Simplifying rational expressions: two variables

WEBVTT
Kind: captions
Language: en

00:00:00.103 --> 00:00:02.285
- [Voiceover] Let's see if we
can simplify this expression

00:00:02.285 --> 00:00:06.663
and like always, pause the
video and have a go at it.

00:00:06.663 --> 00:00:07.694
Now this one is interesting

00:00:07.694 --> 00:00:09.496
because it involves two variables,

00:00:09.496 --> 00:00:11.755
but it's really the same
ideas that we've done

00:00:11.755 --> 00:00:14.207
when we factored things with one variable.

00:00:14.207 --> 00:00:15.942
So, for example, up here in the numerator,

00:00:15.942 --> 00:00:19.029
while I never liked having
a non-one coefficient

00:00:19.029 --> 00:00:20.730
on the second degree term here,

00:00:20.730 --> 00:00:22.485
I mean, sometimes you have to,

00:00:22.485 --> 00:00:25.899
but it looks like every term
here is divisible by five,

00:00:25.899 --> 00:00:27.521
so let's factor out of five first.

00:00:27.521 --> 00:00:29.827
So the numerator, I can rewrite

00:00:29.827 --> 00:00:33.557
as five times,

00:00:33.557 --> 00:00:37.216
five times, you factor out of
five here, you get x squared.

00:00:37.216 --> 00:00:41.416
Factor out of five
here, you get plus four.

00:00:41.416 --> 00:00:44.207
Actually, I'm gonna rewrite it as four y x

00:00:44.207 --> 00:00:46.579
and you'll see in a
second why I'm doing that.

00:00:46.579 --> 00:00:47.710
Actually, I'll tell you why.

00:00:47.710 --> 00:00:49.222
I'll tell you why I'm
doing that right now,

00:00:49.222 --> 00:00:51.173
why I'm writing the y there

00:00:51.173 --> 00:00:53.304
is that this way, it seems to,

00:00:53.304 --> 00:00:55.775
it seems to hit the pattern

00:00:55.775 --> 00:00:58.577
of how we're used to seeing quadratics.

00:00:58.577 --> 00:01:00.919
So let's see. So you have
x squared plus four y x,

00:01:00.919 --> 00:01:02.871
you can view the four y's a coefficient

00:01:02.871 --> 00:01:05.182
on the first degree in x term,

00:01:05.182 --> 00:01:07.233
on the x term right over there,

00:01:07.233 --> 00:01:11.274
plus four y squared

00:01:11.274 --> 00:01:14.255
and it's gonna be over,

00:01:14.255 --> 00:01:16.423
over.

00:01:16.423 --> 00:01:19.515
Now, the denominator here,
can we factor this out?

00:01:19.515 --> 00:01:21.037
Well, let's just think about it.

00:01:21.037 --> 00:01:25.309
Do we know two numbers,
or I guess we would say,

00:01:25.309 --> 00:01:29.361
do we know two expressions
that when you multiply,

00:01:29.361 --> 00:01:31.802
you get negative six y squared

00:01:31.802 --> 00:01:33.153
and then when you add them,

00:01:33.153 --> 00:01:37.204
you get negative x y.

00:01:37.204 --> 00:01:39.136
Sorry, when you add
them, you get negative y.

00:01:39.136 --> 00:01:40.973
That's actually why I like
to write and get like this.

00:01:40.973 --> 00:01:42.219
Actually, let me rewrite this.

00:01:42.219 --> 00:01:45.196
This is the same thing as negative y x.

00:01:45.196 --> 00:01:48.670
And so you can view the
coefficient here as negative y.

00:01:48.670 --> 00:01:51.617
And so when you think of two
numbers or two expressions,

00:01:51.617 --> 00:01:56.159
a times b, that is equal
to negative six y squared

00:01:56.159 --> 00:01:59.027
and when I add them, a plus b,

00:01:59.027 --> 00:02:02.107
that is equal to negative y.

00:02:02.107 --> 00:02:03.859
And so you can imagine, both of them

00:02:03.859 --> 00:02:05.850
are going to be expressions,

00:02:05.850 --> 00:02:09.262
a and b are gonna be
expressions that involve y.

00:02:09.262 --> 00:02:12.554
And so, let's see, if this
was just a negative one

00:02:12.554 --> 00:02:15.986
and if this was just a
negative six, what we would do,

00:02:15.986 --> 00:02:20.913
we would do negative
three and positive two.

00:02:20.913 --> 00:02:23.985
Now let's see. If we did negative three y

00:02:23.985 --> 00:02:26.933
and positive two y, that indeed

00:02:26.933 --> 00:02:29.793
is going to be equal to
negative six y squared

00:02:29.793 --> 00:02:32.952
and negative three y plus two y

00:02:32.952 --> 00:02:35.024
does indeed equal negative y.

00:02:35.024 --> 00:02:36.966
So that's our a and b right there.

00:02:36.966 --> 00:02:38.248
And it seems a little mysterious.

00:02:38.248 --> 00:02:41.537
How did Sal just all of a
sudden get negative two y,

00:02:41.537 --> 00:02:44.186
or negative three y and positive two y?

00:02:44.186 --> 00:02:47.978
Let me write an analogous quadratic here

00:02:47.978 --> 00:02:49.335
that only has one variable.

00:02:49.335 --> 00:02:53.820
If I were to write x
squared minus x minus six

00:02:53.820 --> 00:02:55.412
and I were to ask you to factor that out,

00:02:55.412 --> 00:02:57.136
you say, "Oh, okay."

00:02:57.136 --> 00:02:58.258
I have negative two.

00:02:58.258 --> 00:03:00.517
I have negative three times
negative two is negative six

00:03:00.517 --> 00:03:03.270
and if I add them, well that's
going to be negative one.

00:03:03.270 --> 00:03:06.099
So you'd say, "Well that's going to be

00:03:06.099 --> 00:03:10.857
"x minus three times x plus two."

00:03:10.857 --> 00:03:13.793
And so the only difference
between this and that

00:03:13.793 --> 00:03:15.315
is instead of having just a negative one,

00:03:15.315 --> 00:03:18.054
you have a negative one y.

00:03:18.054 --> 00:03:19.416
Instead of having just
a negative six here,

00:03:19.416 --> 00:03:21.958
you have a negative six y squared.

00:03:21.958 --> 00:03:23.770
And so you could just think of this

00:03:23.770 --> 00:03:25.792
instead of just negative
three and positive two,

00:03:25.792 --> 00:03:28.715
negative three y and positive two y.

00:03:28.715 --> 00:03:30.513
Hopefully that makes
sense and if it doesn't,

00:03:30.513 --> 00:03:33.106
I encourage you to kind
of play around with this,

00:03:33.106 --> 00:03:34.358
multiply these out a little bit,

00:03:34.358 --> 00:03:37.750
get a little bit more familiar with this.

00:03:37.750 --> 00:03:40.810
But now that we know that
it can factored like this,

00:03:40.810 --> 00:03:41.801
let's rewrite this.

00:03:41.801 --> 00:03:46.801
This is going to be x minus three y

00:03:47.463 --> 00:03:51.555
times x plus two y.

00:03:51.555 --> 00:03:54.701
And nothing seems to
simplify out just yet,

00:03:54.701 --> 00:03:56.726
but it looks like what
we have in magenta here

00:03:56.726 --> 00:03:58.317
can be simplified further.

00:03:58.317 --> 00:03:59.729
We're gonna do a very similar exercise

00:03:59.729 --> 00:04:01.610
to what we did just now.

00:04:01.610 --> 00:04:03.249
What two expressions if I multiply,

00:04:03.249 --> 00:04:05.111
I'm gonna get four y squared

00:04:05.111 --> 00:04:07.953
and if I add them, I get four y?

00:04:07.953 --> 00:04:10.236
It looks like two y would do the trick.

00:04:10.236 --> 00:04:14.053
So it seems like we can
rewrite the numerator.

00:04:14.053 --> 00:04:15.722
This is going to be.

00:04:15.722 --> 00:04:18.601
So let me draw a little
line here to make it clear

00:04:18.601 --> 00:04:19.722
that this is,

00:04:19.722 --> 00:04:22.895
this is going to be equal to five

00:04:22.895 --> 00:04:25.128
times

00:04:25.128 --> 00:04:28.818
x plus two y times,

00:04:28.818 --> 00:04:30.630
I could say this is x plus two y squared

00:04:30.630 --> 00:04:35.041
or I could just say x plus
two y times x plus two y.

00:04:35.041 --> 00:04:37.470
Once again, two y times
two y is four y squared.

00:04:37.470 --> 00:04:40.956
Two y plus two y is four y.

00:04:40.956 --> 00:04:45.956
And that's all going to be over,

00:04:46.216 --> 00:04:48.964
that is all going to be over

00:04:48.964 --> 00:04:53.604
x minus three y

00:04:53.604 --> 00:04:57.236
times x plus two y.

00:04:57.236 --> 00:05:00.366
And so now, I have a
common factor, x plus two y

00:05:00.366 --> 00:05:02.925
in both the numerator and the denominator,

00:05:02.925 --> 00:05:06.806
so I can handle x plus two
y divided by x plus two y,

00:05:06.806 --> 00:05:08.895
well that's just going to be one

00:05:08.895 --> 00:05:12.782
if we assume that x plus
two y does not equal zero.

00:05:12.782 --> 00:05:14.644
And that's actually an
important constraint

00:05:14.644 --> 00:05:17.550
because once we cancel this
out, you lose that information.

00:05:17.550 --> 00:05:19.529
If you want this to be
algebraically equivalent,

00:05:19.529 --> 00:05:24.120
we could say that x plus two
y cannot be equal to zero

00:05:24.120 --> 00:05:25.064
or another way you could say it

00:05:25.064 --> 00:05:28.327
is that x cannot be equal to,

00:05:28.327 --> 00:05:30.663
cannot be equal to negative two y.

00:05:30.663 --> 00:05:33.242
I just subtracted two y
from both sides there.

00:05:33.242 --> 00:05:34.133
And so what you're left with,

00:05:34.133 --> 00:05:35.495
and we can redistribute this five

00:05:35.495 --> 00:05:38.027
if we wanna write it out in expanded form.

00:05:38.027 --> 00:05:39.128
We could rewrite it as,

00:05:39.128 --> 00:05:41.623
the numerator would be five x.

00:05:41.623 --> 00:05:43.309
Let me write it over here.

00:05:43.309 --> 00:05:47.535
Five x plus 10 y.

00:05:47.535 --> 00:05:51.748
And the denominator is x minus three y.

00:05:51.748 --> 00:05:52.919
But once again, if we want it

00:05:52.919 --> 00:05:54.284
to be algebraically equivalent,

00:05:54.284 --> 00:05:57.360
we would have to say x cannot be equal to,

00:05:57.360 --> 00:06:00.560
x cannot be equal to negative two y.

00:06:00.560 --> 00:06:04.290
And now this is algebraically equivalent

00:06:04.290 --> 00:06:05.581
to what we had up here

00:06:05.581 --> 00:06:09.571
and you can argue that
it's a little bit simpler.


Dividing rational expressions: unknown expression

WEBVTT
Kind: captions
Language: en

00:00:00.277 --> 00:00:01.669
- [Voiceover] We're told
the following equation

00:00:01.669 --> 00:00:04.931
is true for all real values of y

00:00:04.931 --> 00:00:07.727
for which the expression
on the left is defined

00:00:07.727 --> 00:00:10.740
and D is a polynomial expression.

00:00:10.740 --> 00:00:12.425
And they have this equation here.

00:00:12.425 --> 00:00:15.014
What is D?

00:00:15.014 --> 00:00:16.873
All right. So essentially,
what they're saying

00:00:16.873 --> 00:00:19.416
is they don't want us to
somehow solve this equation.

00:00:19.416 --> 00:00:20.654
They're saying D is going to be

00:00:20.654 --> 00:00:22.853
some type of a polynomial expression.

00:00:22.853 --> 00:00:24.615
They tell us that right over there:

00:00:24.615 --> 00:00:27.160
D is a polynomial expression.

00:00:27.160 --> 00:00:30.397
And if we figure out what D is,

00:00:30.397 --> 00:00:32.579
this left-hand side of the expression

00:00:32.579 --> 00:00:36.416
is going to evaluate to one
for all real values of y

00:00:36.416 --> 00:00:38.795
for which the expression is defined.

00:00:38.795 --> 00:00:40.857
So let's think about how
we would tackle that.

00:00:40.857 --> 00:00:42.339
Well the first thing
that pops into my mind,

00:00:42.339 --> 00:00:46.146
if I'm dividing by a fraction
or a rational expression,

00:00:46.146 --> 00:00:49.672
that's the same thing as
multiplying by the reciprocal.

00:00:49.672 --> 00:00:52.375
So let's just rewrite this
on the left-hand side.

00:00:52.375 --> 00:00:56.010
So this is 20 y squared

00:00:56.010 --> 00:00:59.743
minus 80 over D

00:00:59.743 --> 00:01:03.515
times four,

00:01:03.515 --> 00:01:06.076
oh, let me do the reciprocal,
let me be careful,

00:01:06.076 --> 00:01:08.104
times the reciprocal of this.

00:01:08.104 --> 00:01:09.120
If I divide by something,

00:01:09.120 --> 00:01:11.033
the same thing as multiplying
by the reciprocal,

00:01:11.033 --> 00:01:13.354
so let me just swap the
numerator and the denominator,

00:01:13.354 --> 00:01:15.646
(chuckle) numerators and denominators,

00:01:15.646 --> 00:01:17.444
numerator and denominator.

00:01:17.444 --> 00:01:22.444
All right, y to the third
plus nine y squared,

00:01:22.445 --> 00:01:27.445
all of that over four y
squared minus eight y.

00:01:27.954 --> 00:01:30.506
That's going to be equal to one.

00:01:30.506 --> 00:01:32.095
Now let's say if we can simplify

00:01:32.095 --> 00:01:35.719
all of these business on the
left-hand side a little bit.

00:01:35.719 --> 00:01:36.824
So let's see.

00:01:36.824 --> 00:01:41.224
Over here, I can divide both terms by 20.

00:01:41.224 --> 00:01:42.676
So let me factor out 20 because I think

00:01:42.676 --> 00:01:45.178
it's gonna end up being
a difference of squares.

00:01:45.178 --> 00:01:47.774
So if I factor out of 20,

00:01:47.774 --> 00:01:49.029
so this is about the same thing

00:01:49.029 --> 00:01:50.928
as 20

00:01:50.928 --> 00:01:54.768
times y squared minus four

00:01:54.768 --> 00:01:57.310
and y squared minus four,

00:01:57.310 --> 00:02:00.763
we can rewrite as y plus
two times y minus two.

00:02:00.763 --> 00:02:02.662
It is a difference of squares.

00:02:02.662 --> 00:02:04.871
So let me write y plus two

00:02:04.871 --> 00:02:07.975
times y minus two.

00:02:07.975 --> 00:02:12.121
All right. This down here,
four y squared minus eight y,

00:02:12.121 --> 00:02:14.875
well it looks like we
can factor out of four y.

00:02:14.875 --> 00:02:16.360
And so this is going to be same thing

00:02:16.360 --> 00:02:18.156
as four y

00:02:18.156 --> 00:02:22.996
times y minus two.

00:02:22.996 --> 00:02:27.353
All right. So let me cross that out.

00:02:27.353 --> 00:02:29.115
So it's the same thing as
four y times y and minus two

00:02:29.115 --> 00:02:30.447
and I already see that
this y minus two here

00:02:30.447 --> 00:02:33.039
and this y minus two here
are gonna cancel out.

00:02:33.039 --> 00:02:36.022
And let's see. Up here, both terms

00:02:36.022 --> 00:02:39.355
are divisible by y squared,
so I can rewrite this as,

00:02:39.355 --> 00:02:41.768
I don't know if this is
actually going to be helpful

00:02:41.768 --> 00:02:42.812
because it's gonna,

00:02:42.812 --> 00:02:45.064
well let me just do it, just in case.

00:02:45.064 --> 00:02:49.836
So that's the same thing
as y squared times,

00:02:49.836 --> 00:02:54.784
y squared times y plus nine.

00:02:54.784 --> 00:02:57.424
All right. And so we can
rewrite all of these things.

00:02:57.424 --> 00:02:59.546
If we were to multiply
everything together,

00:02:59.546 --> 00:03:00.647
we would end up getting,

00:03:00.647 --> 00:03:05.078
the numerator would
get 20 times y plus two

00:03:05.078 --> 00:03:07.580
times y minus two

00:03:07.580 --> 00:03:10.523
times y squared

00:03:10.523 --> 00:03:13.973
times y plus nine, I'm just
multiplying all the numerators,

00:03:13.973 --> 00:03:15.762
and that's going to be over,

00:03:15.762 --> 00:03:17.254
in the denominator, I would have

00:03:17.254 --> 00:03:19.415
whatever the expression D is,

00:03:19.415 --> 00:03:22.765
times four y

00:03:22.765 --> 00:03:25.525
times y minus two,

00:03:25.525 --> 00:03:28.608
and that's all going to be equal to one.

00:03:28.608 --> 00:03:29.449
Now let's think about it.

00:03:29.449 --> 00:03:30.810
We can divide,

00:03:30.810 --> 00:03:33.230
we have y minus two
divided by y minus two,

00:03:33.230 --> 00:03:34.981
so those cancel out.

00:03:34.981 --> 00:03:38.070
Now let's see. We have a...

00:03:38.070 --> 00:03:40.779
We can divide the numerator
and the denominator by y,

00:03:40.779 --> 00:03:42.824
so that would just become one.

00:03:42.824 --> 00:03:46.212
And then that would just
become a y to the first power.

00:03:46.212 --> 00:03:48.161
And so, what we'd be left with,

00:03:48.161 --> 00:03:49.953
what we'd be left with in the numerator

00:03:49.953 --> 00:03:54.531
is 20 times this y

00:03:54.531 --> 00:03:59.455
times y plus two times y plus nine

00:03:59.455 --> 00:04:01.598
over,

00:04:01.598 --> 00:04:06.120
over four D

00:04:06.120 --> 00:04:07.951
is equal to one.

00:04:07.951 --> 00:04:10.243
Now if we wanna solve four D,

00:04:10.243 --> 00:04:12.205
well we could just
multiply both sides by D

00:04:12.205 --> 00:04:13.964
and one times D is just gonna be D,

00:04:13.964 --> 00:04:15.826
so we're gonna have D
equals something over here

00:04:15.826 --> 00:04:16.927
and we'd be done.

00:04:16.927 --> 00:04:17.788
So let's do that.

00:04:17.788 --> 00:04:19.199
D times that

00:04:19.199 --> 00:04:20.981
and let's multiply that times D.

00:04:20.981 --> 00:04:22.753
Let me be clear what I'm doing here.

00:04:22.753 --> 00:04:25.115
Let me draw a little divider here

00:04:25.115 --> 00:04:29.085
to make it clear that that's happening

00:04:29.085 --> 00:04:30.814
in the other side of the page.

00:04:30.814 --> 00:04:34.147
All right. So D times
this, those cancel out,

00:04:34.147 --> 00:04:38.034
and we're gonna be left with 20 y times,

00:04:38.034 --> 00:04:40.493
actually, let me, I could
simplify it even more.

00:04:40.493 --> 00:04:44.524
20 divided by four is five.

00:04:44.524 --> 00:04:46.686
So, our numerator is now just one.

00:04:46.686 --> 00:04:50.653
So we have five times y

00:04:50.653 --> 00:04:53.483
times y plus two

00:04:53.483 --> 00:04:56.159
times y plus nine

00:04:56.159 --> 00:04:59.285
is equal to D, and we're done.

00:04:59.285 --> 00:05:01.707
This is D. This is the polynomial,

00:05:01.707 --> 00:05:03.903
whoops, that is...

00:05:03.903 --> 00:05:07.063
That is the polynomial expression

00:05:07.063 --> 00:05:08.434
that we are looking for.

00:05:08.434 --> 00:05:09.636
If you were to substitute this back in

00:05:09.636 --> 00:05:11.541
and then try to simplify it,

00:05:11.541 --> 00:05:13.860
well, you would end up
with all of these over here

00:05:13.860 --> 00:05:15.265
and D would be this,

00:05:15.265 --> 00:05:17.227
and so, it would all just cancel out

00:05:17.227 --> 00:05:19.685
and you would be left with one.

00:05:19.685 --> 00:05:22.168
For all real values y
for which the expression

00:05:22.168 --> 00:05:25.287
on the left is actually defined.

00:05:25.287 --> 00:05:27.003
There are some values of y

00:05:27.003 --> 00:05:29.913
for which the expression
on the left is not defined.

00:05:29.913 --> 00:05:32.195
If y is equal to zero,
this denominator is zero

00:05:32.195 --> 00:05:34.834
and you're dividing by zero,
well that's not defined.

00:05:34.834 --> 00:05:38.003
And then when you multiply
by the reciprocal,

00:05:38.003 --> 00:05:40.654
if this were to become zero,

00:05:40.654 --> 00:05:42.508
then that wouldn't be cool either.

00:05:42.508 --> 00:05:45.250
And there's multiple ways
to make this equal to zero.

00:05:45.250 --> 00:05:47.622
y could be equal to negative nine,

00:05:47.622 --> 00:05:49.324
that would also make this bottom zero.

00:05:49.324 --> 00:05:51.196
So we could think about
that if we wanted to,

00:05:51.196 --> 00:05:52.732
but they're not asking us to do that.

00:05:52.732 --> 00:05:54.211
They're saying, for all
of the real values for y

00:05:54.211 --> 00:05:55.766
for which the expression defined,

00:05:55.766 --> 00:05:59.062
find the D that makes all of
this business equal to one

00:05:59.062 --> 00:06:01.070
and we just did that.