On this problem, it says find the 0s of the function.
So again, when we're looking at finding the 0s,
we know that we need to figure out
what are the values of x when f of x equals 0.
So again, we plug in 0 in for f of x.
And we're going to have to determine what our values of x
are.
Well, it used to be pretty easy to find the values of x,
because we had one x.
And all we had to do was just solve for that one x.
You know, isolate the variable, get everything
on the other side, and solve for x.
Here, we have a lot of stuff going on.
We have x cubed, a negative 4x squared, and a negative 9x.
So we have a lot of x's.
And we can't use our regular factoring,
because that was always when we were dealing with a trinomial--
when it has three terms-- or a quadratic.
We were always so used to doing your factoring,
setting it up as far as two linear factors.
So now we need a different kind of factoring method.
Well, the factoring method we're going to want to use--
and whenever you see four terms, you
should always look into factoring by grouping.
So what you really just want to do
is just take two different terms and see
if you can bring them together and if we can factor something
similar out of them.
So in this problem, I'm just going
to look at these first two terms and let's say, all right.
Is there something that these two have in common?
You can say, yes, I can take out an x squared out of here.
So if I factor out an x squared, I'll be left with x minus 4.
And then here, out of these two terms,
if I took out a negative 9, if I factor out a negative 9,
I'll be left with x minus 4.
Which, again, gets me to what I want when I'm looking
into factoring by grouping.
Because now, if you look at these,
treat these just as another term.
I know they're a binomial--
they have two terms within them--
but just treat these just like a regular variable.
Now both of these terms, this and this,
both share an x minus 4.
So what I'm going to do is I'm going to factor both of those
out.
So if I factor out an x minus 4, I pull it out,
I am left with an x squared.
And if here I factor out an x minus 4, I'm left with--
sorry, negative 9.
Does everybody understand that?
You can raise your hand, it's OK.
Good?
Thank you.
Why is it that we have two x minus 4's,
why do we only have--
like, why did we almost, like, drop it?
What do you mean?
Because we had two x minus 4's, and then on the next line
it was like, x minus 4.
There was only one of them.
Why is there only one?
OK, what I'm doing is I'm--
it's like this.
It's like 3x plus 3x squared, right?
What can you factor out of this?
3x.
You can factor out 3x, right?
So what I'm doing is, I'm pretty much just
saying you have these 3--
actually-- yeah.
So what I'm doing is I'm taking a 3 out of both of them, right?
Yeah.
And I'm taking an x out of both of them.
So you're left with, what's left?
Well, that's a 1 plus x.
Correct?
Yeah.
Well, that's the exact same thing I'm doing here.
These both share an x minus 4.
Just treat it like it's another variable.
So I'm taking an x minus 4 out of both of them.
So therefore that's why I only have--
when I take an x minus 4 out of both of them,
I'm just left with an x minus 4.
Oh.
So then what's left over?
Well, that's left over.
And that's left over.
So that's why I write that in its own parenthesis.
x squared minus 9.
OK.
Does that make any more sense?
Yeah, but I was just curious because when we did the work,
it was almost kind of like we were dropping
one of the ones in parentheses, so I
was just confused about that.
Yeah, you're not dropping it.
What you're doing is you're actually factoring it out.
You're taking these two x minus 4's and you're factoring out,
you're pulling it out.
And then what here was left--
Well, I'm trying to think of a different example other than
3's.
But if you can just remember that,
if you look at these, what do these two share?
They both share an x squared.
So you factored out the x squared.
What do these two, what does this term and this term share?
They both have an x minus 4.
So you can take that x minus 4 out.
And when you take that x minus 4 out,
the only thing left is that x squared and that negative 9.
So that's why we write a new equation.
Then just have to finish this off by saying, remember,
0 equals x minus 4 or x squared plus 9.
Therefore, we say either x minus 4 equals 0,
or x squared minus 9 equals 0.
Therefore, when solving for x, x equals 4 or x squared equals 9.
Make sure, now, we have to take the square root.
Whenever you're taking the square root,
remember we have a positive and a negative.
So x equals plus or minus 3.
So therefore, the 0's for this function
is going to be when x equals 4 or if x equals plus or minus 3.
Like I said, if you want to check your work,
you could plug any one of those numbers up there
and one of those might be a 0.
We'll learn how to check that a little bit later as well.
But those are all possible 0's for your function.