College Algebra students, 
Dillon here.

We are continuing on
with Section 3.2.

Yes, it's a long one.
Let's get started.

We've learned that there's a connection 
between the x-intercepts, also called zeros,

and the factors that make up 
a particular polynomial.

We've learned that if the

polynomial has the factor, 
there's a corresponding zero.

Here's the last example we did.

We started with this 
fourth-degree polynomial.

We factored it; we factored out 
a negative x squared.

We're left with a quadratic, 
which we then factored.

We end up with four factors.

Two factors of x: 
2x plus 3 and x minus 1.

Each one of these contributed 
a zero (an x-intercept),

and we learned
that the zero—

that the factor to the second degree
has some interesting behavior.

That's the last thing 
we finish up here.

We're going to study how these 
polynomials behave around their zeros.

There are two types of behavior:

one where it actually goes through
the zero, one where it's tangent.

I think it'll be instructive to
just look at this in Desmos.

Not that my graph is not—

it shows all the important details, 
but it looks a little nicer in Desmos.

Here we go: y equals negative
2x to the fourth power.

That's our leading term, our 
leading coefficient is negative 2.

And remember, these monomial terms, 
these monomial leading terms,

they determine the 
end behavior of the polynomial.

They dominate the end behavior.

Okay, there it is, a slightly more
attractive version of the graph.

We can see that the 
two types of behavior here

that we're going to study a bit
more to finish up this section.

Where it goes— the graph—

Excuse me, the polynomial goes 
through the x-intercept (the zero),

if the degree of that factor is odd.

So, in this case the 
degree of 2x plus 3 is 1.

That's an odd number.

The degree of x minus 1 
is to the first power.

So, notice for both of those zeros, 
the graph goes through the points.

The zero that had an
even degree (second degree),

we call this multiplicity two.

The other two factors 
have multiplicity one.

This factor x has 
multiplicity two.

And you'll notice that the 
graph is tangent at that point,

so the degree of the factor, 
also called the multiplicity,

is going to determine 
how it behaves.

Okay, so just to practice this 
sketching the graph one more time,

let's do another one.

And I like this one because the 
factoring is a little different.

Something we haven't
seen yet in this section.

Okay, right off the bat,

you notice there's no greatest 
common factor to take out here.

And you have to go back to the
polynomial factoring methods

that we learned probably 
in the previous class

and realize that you
have four terms here.

You'll notice that there's 
a common term in each.

If we group these
two together,

and group these two together,

there's a common term
you can take out.

Let's take that out.

I'm going to take an x squared 
out of the first two terms.

This is called factoring 
by grouping, by the way.

Then what's left in just
the first two terms?

I'm ignoring these last two.

I pull that out and 
I get x minus 2.

Notice that there's a common 
factor we can pull out here.

When the leading coefficient is negative, 
you should always pull out a negative.

So we're going to
pull out a negative four.

What's left?
X minus 2.

And that's what 
we're looking for.

We needed a common factor, 
a common binomial factor

that we can now take out of both 
of those, so take out the x minus 2

and we're left with
x squared minus 4.

I hope you recognize 
that the second term here

is actually the difference
of two squares.

We're going to
have this x minus 2

and then the difference of two squares 
factors to x plus 2 and x minus 2.

We're going to 
rewrite it like this

because we're going to talk more 
about this idea of multiplicity.

The x minus 2 is actually
to the second power.

This—

The x-intercept 
has multiplicity two.

We just learned how
that's going to behave.

It's going to be tangent 
at the point x equals 2.

And then we have the 
second zero here: x plus 2.

Okay, Let's draw a graph,
sketch a graph,

somewhat accurate.

Then we'll do it
in Desmos, also.

Okay, so the interesting points are— 
from this factor, we get x equals 2.

We know it's going to be tangent there; 
we'll figure that out in a second.

The second factor
gives us x minus 2.

And then let's look at 
this leading term.

It’s an odd degree.

The leading coefficient
is positive.

So, you know, the end behavior
is like this, in this direction,

and like this 
in this direction.

As x goes to positive infinity, 
we know y goes to positive infinity.

As x goes to negative infinity, 
because it's odd degree,

the y value is going to 
go to negative infinity.

That's the end behavior.

Then what happens in 
between these two?

Now remember, these are 
the only two x-intercepts.

That's important.

We know the graph must
come up and go through this one

(this has odd multiplicity: 
x plus 2 to the first power),

and then it's going to come up;

we know it has to go through this point, 
but it doesn't actually go through.

It's going to come down 
here and it's tangent,

and then it takes off
in that direction.

It's going to look 
something like that.

Let's just graph it in a Desmos,

so we have a slightly 
more attractive version of it.

Let's see… we're 
going to get rid of this.

Change that to 
the third power,

so minus 2x to the 
second power minus 4x plus 8.

Our scale is a 
little bit off here.

Let's just fix this.

We’re going to fix this maximum 
y value to something a little bigger.

I don't really know where it is.

I'm going to try 50 and see.
That's too big.

How about 20?
There we go.

Okay, so that's similar to 
what we sketched here.

All right, a little
prettier in Desmos.

And we can see that the zero 
that has multiplicity two—

that's our vocabulary:

"multiplicity two" because that factor
is raised to the second power—

the polynomial is
tangent at that point.

Whereas the zero with 
multiplicity one (an odd number),

it actually goes through
the x-intercept.

Okay, we're going to do a
few more examples of that

to finish up the section here.

Here we go.

Everything I just said is 
summarized in this box.

You might want to pause the 
video and take a closer look.

If the multiplicity—

and the multiplicity is determined
by the power of the factor.

Each factor has a 
power one or greater.

If it's an odd multiplicity, 
then the x-intercept is

going to behave like
this for the polynomial.

If it's even multiplicity 
it behaves like this:

either tangent from the top 
or tangent from the bottom.

Then you might ask, “Well how do
you decide?” Well, look up here.

We knew what the end behavior
of the polynomial was, okay?

We knew it was
coming up here.

It has to go through this intercept, 
and then we're above the x-axis here.

And we know it's got to
be tangent at that point.

This is the only option, we know it 
doesn't have any other x-intercepts,

so it can't go 
through the x-axis.

You just use a 
little bit of logic.

Okay, let's graph this one.

I think these are kinda fun.

The first thing we usually do is— 
so notice this is factored for us.

That's nice, right?

We're going to list the zero and 
the multiplicity, then we'll graph it.

The first zero is x equals zero, 
multiplicity four.

And notice that that's 
an even number.

The second zero from this factor, 
we get x equals two, multiplicity three.

That's odd, so it's going
to go through that point.

Then x equals negative 1.

Again, even multiplicity, so we know
it's going to be tangent at that point.

This is a pretty funky graph.

Now, what is the leading term
going to look like here?

Well, if you multiply this out—

this is kind of a crazy polynomial, 
but we can do it.

If you multiply this out, it's going
to be a third-degree polynomial

times the fourth-degree term
is going to be seventh degree.

Multiply this out, it's 
going to be a second degree,

so seventh degree 
times second degree.

We're going to
have x to the ninth.

Yikes, that's crazy, right?

This is a third-degree polynomial 
times the fourth-degree term.

That gives us
x to the seventh.

Then if this is multiplied out, 
it's going to be squared.

So we have x to the ninth.

Notice the leading coefficient
is going to be 1,

because all of the leading 
coefficients here are 1.

And we know if this is odd degree,
it's going to behave like this,

and like this with
stuff in the middle.

Odd degree, just like x to the third, 
x to the ninth behaves like x to the third.

Okay, here we go.

This is a pretty 
fancy polynomial.

I'm not sure why I'm using
blue here all of a sudden.

Should’ve changed color.

Okay.

So here are intercepts: zero, 
positive 2 and negative 1.

Okay, and then we know
the end behavior

because we know the leading
term and the leading coefficient.

End behavior has to be like this

and like this.

Okay, so what happens 
at this leftmost zero?

Well, it's got to be tangent there
because it's even multiplicity.

So it has to come up and touch
here and then come back down.

What happens at zero?

Well, it's also even multiplicity.

It's going to come 
down and be like this.

It's got to be tangent here, 
and we're below the x-axis.

And we know it's tangent there.

So it can't go through and 
come back down and touch,

it’s gotta be something like that.

And then the zero of
multiplicity three is at 2.

Then we know this has to come
down and go back up and out.

It's going to look 
something like that.

We don't know how
big this loop is here,

but let's put it in Desmos.

Should be quite interesting, 
I would think.

If you're looking for an
idea for a tattoo, here it is.

This is a pretty
cool polynomial.

Just trying to get 
both on the screen.

Okay, so x minus 2
raised to the third power.

It's a pretty fancy polynomial.

X plus 1 is to the second power.

Ooh, all right, so…

Our graph, it's close.

Okay, let's see.

We didn't know how— we knew— 
Let's see if we can—

I'm going to change the scale a little bit
so we can see this detail a little better.

Negative 2 to 3.

That's not going to 
change it much.

Then we're going to
change the y scale.

Let's make this…

Let's see, on the negative side,

we could go to negative 6 
maybe, and let's go with 2.

Yeah, there we go.

Okay.

It's different but it 
has the same features.

It’s doing the same 
things we expected.

We come up here,
we’re tangent at negative 1;

we come back down, 
we’re tangent at zero.

If we click on the graph,

we can prove that what 
we have is accurate.

Okay, tangent at negative 1, 
tangent at zero.

We have little dip here,

we didn't know how far that went, 
but that's what it looks like.

And then we have another dip,

and then it goes up
through 2 (x equals 2).

So, our sketch was pretty accurate, 
but not as pretty, of course.

Okay, I would like you to 
pause the recording please,

and try number 29.

And then we'll do it together.

All right.

I hope you tried it.

First of all, what's the
leading term here?

Let's figure that out.

The leading term, let's see.

This is going to be a second-degree
polynomial times another first-degree

(so that’s going to be third), 
multiplied by third, so this gonna be—

The leading coefficient is 1; 
the leading term is x to the sixth.

Let me just repeat that, 
you multiply this out,

that's going to be
second-degree,

times another x is 
going to be third-degree,

times x to the third is 
going to be x to the sixth.

So, we know the end behavior
of this stuff in the middle is

going to look like that.
It's going to go up:

as x goes to positive infinity,
y is going to go to positive infinity

[and] as x goes to negative infinity, 
y is going to go to positive infinity,

because this is a six-degree term and
the leading coefficient is positive 1.

Okay, what about the zeros?

Just write those out, 
and the multiplicity.

So, the first zero is zero.

A multiplicity three, that's odd.

The next one is x equals negative 2, 
multiplicity there is one, it's odd,

so it's going to go through 
both of those points.

And then x equals 3, 
and that's even.

So, it's going to be 
tangent of x equals 3.

Let's sketch a graph.

All right, so let's see…

how about at x equals zero,
there's our point.

Negative 2, there's our point, 
and positive 3.

Then our end behavior, 
we know it's got to be

something like this and
something like this.

Then what happens here at negative—
let's take care of the leftmost one first—

negative 2 is odd multiplicity, right?

So we know it must come down
and go through this point.

What happens at zero?

It's odd multiplicity, so it's going to 
come down and then go back up.

I don't know what's going on
with my Wacom board here.

We might have to reboot.

Okay, it’s misbehaving.

Okay, so it's down and 
it's going to go up.

Why? Because this
is odd multiplicity.

Then what about this zero?

Well, there are no 
other zeros, okay?

And at 3, it has 
multiplicity two.

So, you know it 
must come down—

excuse me, it must come down to this, 
touch, and then zoom back up.

It's going to look 
something like that.

Aren't these fun?

Okay, let's try that in Desmos
to see if we're close.

So, x to the third, x plus 2—
gotta get rid of this exponent here—

and then x— gotta get rid of that
exponent— I don't know, yeah,

you shouldn't have that in there— 
x minus 3, that's to the second power.

Okay, let's see 
what we have here.

Do I have that typed in right?

X to the third times x plus 
2 times x minus 3 squared.

Okay, let's see if 
we're in the ball park.

Okay so, there appears to
be something wrong.

Oh, I just don’t— 
you can't see enough of it.

We have to change the scale so 
we can see more of the graph,

so let's fix this.

The problem is we’re 
zoomed in too close.

There we go.

Okay, look, everything interesting 
happens between negative 3 and 4,

so let's change that.

Just trying to get this so
we get a better picture.

Then we need this
y and x to be bigger.

So, how about—
again, I don't know how big—

So, let’s go to negative—

Nope, gotta go bigger
than that, negative 40.

There we go.

See how you just gotta play around with it, 
so you can see what you need to see.

Let's go 50 here.
Oh, there we go.

Look at that.
Beautiful.

Okay.

Does that look like 
what we drew?

Sure does.

Pretty close.

It comes down, 
odd multiplicity,

odd multiplicity,

and then even multiplicity, 
so it's tangent here.

Isn’t it beautiful?

Polynomials are beautiful because 
they're smooth and continuous.

Okay, so believe it or not,
there's 3.2 part F.

There's another part to this.

We're going to stop right there
and we'll have one last part.

Okay, bye, see you shortly.