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,
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.