All right.
What we're going to do is for this problem,
we need to find the I forget--
we need to find the zeros.
That's what we need to do.
And when we're finding the zeros,
we also need to determine what is
going to be the multiplicity of the problem.
Or I'm sorry.
Multiplicity of the zeros.
And we need to know, what is the multiplicity?
Why are we figuring out what the multiplicity?
And who cares about multiplicity?
Well, let's first start out finding the zeros.
All right?
And remember guys, we're talking about the zeros.
The zero is the same thing as like the roots.
The same thing as the x intercepts.
They all mean the exact same thing.
And what we're trying to do is, we're
trying to find out when is our value of our function zero?
So we could say finding the zeros
we want to find the values of t, that make g of t zero.
0 equals t to the fifth minus 6t cubed plus 9t.
Next thing, factor out your t.
All these share a t.
You can factor it out.
There's a 6 there.
Stop!
Alert!
We have something that's very, very special.
One, we have a bigger parenthesis
than a smaller parenthesis.
The next thing that you guys see is
we have a square number at the back.
So automatically, whenever you guys see a square number,
you can look for a perfect square.
You can look for a difference of two squares.
Always look for these things when you see a square number.
And what we have is, we actually are
going to be able to create a perfect square.
So what does that mean?
Well, a square number means I'm going
to take a number multiplied by itself, right?
3 times 3 gives you 9.
So does 3 and 3 give me a 6?
Well not positive 3 and 3 but a negative 3 and 3.
So I can rewrite this as minus 3 times minus 3.
Oops.
That's like way long.
Now usually guys, you are used to this.
Like everybody is so easy.
Like, you understand this.
Oh, well OK.
You know x minus 3 times x minus 3, right?
That's easy.
Everybody's like all right.
Algebra 1, I got it.
Here, if you guys look at this, we're
not dealing with x's anymore.
We're dealing with t to the fourths and t squareds.
If you look at it, t squared times t squared,
is going to give you t to the fourth.
And the exact same thing.
These t's when added up together,
are still going to be, or t squareds
added up together, are still going to be t squareds.
So that's how I got the t squareds inside of there.
An awesome thing you need to know now, well, what do we do,
t squared minus 3 times t squared minus 3
is the same thing as saying 0 equals t times
t squared minus 3 squared.
And why is this so important?
Well because, ladies and gentlemen,
when we're looking at multiplicity of a zero,
we need to know when our zeros are written as factors.
If they are to an even power, they're
going to give us a multiplicity of 2.
And if they're to an odd power, they're
going to give us a multiplicity of 1.
Now, we'll talk about multiplicity a little bit more
in just a second, but for anyways,
next thing we need to know, is now
we have a set of linear factors.
I can say x times y equals 0.
I know that one of these numbers has to equal 0.
So we're going to set our two factors equal to zero.
t is equal to 0.
And t squared minus 3 squared equals 0.
Now one real quick thing a lot of students
make mistakes on is that just because my variable is squared,
it's a multiplicity of 2.
That is incorrect logic.
What you need to do is not the variable power,
but it's your factor.
If that's 2 raised to a power.
So since this whole factor is raised to a power,
it's a multiplicity of 2.
Just because my power is raised to 2, does not mean it's
a multiplicity of 2.
OK?
So it's the factors, not the powers.
Or I'm sorry, it's the factors, not just
your individual variable.
t squared plus 0 that's pretty easy
and that's also to a multiplicity of 1.
Here, I got to square root to get rid of this square.
And I'm left with t squared minus 3 equals 0.
Add 3 to both sides.
t squared equals 3 square root.
t equals plus or minus the square root of 3.
Now, my factors, t is equal to 0 square root
of 3 negative square root of 3.
Both of these are to a multiplicity of 2.
So what it means is the graph is going to touch.
Just touch.
Just a nice little touch.
It's going to touch at these two intercepts.
And here, since it has a multiplicity of 1,
it's going to intercept.
All right?
So just remember, when you have a multiplicity of 1,
it's intersecting.
In multiplicity of 2, it's just going to touch.
OK?
So it's just like runs out and touches it.
And that's how you determine what
the zeros are and the multiplicities for your zeros.