- [Voiceover] So we have a function f of x
it's written as the product of a bunch of
first degree expressions.
We obviously could also view this as a (mumbles),
specially if we expand this all out,
we'll have a more traditional form.
But what's nice about this form,
and this is where we try to get
when we're trying to find the zeros
or the root of a (mumbles),
is that is very easy to pick out the zeros.
The zeros here, these are the x values that
make our function equal to zero.
Well that's gonna be x equals to two,
x is equal to negative four,
that would make this expression zero.
X is equal to negative five,
remember, if you make any one of these equal to zero,
then the entire function is going to be equal to zero.
Cause zero times anything is zero.
X equals one,
and x is equal to nine.
Now, as we've talked about in multiple videos,
these are also going to be,
these x values when you input into the function
give you f of two equals zero,
f of negative four is equal to zero.
Then that means that, there're also a way
to find our x intercepts.
What am I talking about?
So, let me draw,
let me draw a big axis here.
See it goes lows x equals negative five
as high as x equals 9.
So, let me draw something like this.
So, it's my y axis, this is my x axis.
X's one, two, three,
four, five, six, seven,
eight, nine.
One, two, three, four, five.
Alright.
Now let's just plot these.
So when x is equal to two,
f of two we already know it's zero.
If you input two here, then this expression's gonna be
zero, zero, times a bunch of stuff's gonna be zero.
So f of two is equal to zero.
So the point two comma zero
is going to be on the graph of y equals f of x.
So one, two, comma, zero.
So that's that point right over there.
So notice, x equals two is a zero
and makes our function equal to zero
if we input it into it.
And it's also the point two, comma, zero.
It's going to be the x coordinate
of this point right over here.
It's going to define our x,
or one of our x intercepts.
Let's do it with the other ones.
X is equal to negative four, I can do that with a
better color.
X is equal to negative four.
Negative one, negative two, negative three,
negative four.
Right over there, x is equal to negative five.
X is equal to negative five.
X is equal to one, and do that in,
I will do that in pink.
X is equal to one.
And x is equal to nine.
And x is equal to nine, that was right over there, right?
One, two, three, four, five,
six, seven, eight, nine.
Now I'm not gonna be able to exactly
draw the graph of this function.
But what's neat is, I can realize, well look,
if these are the points where I'm going to intersect
the x axis.
And these are the only points
where I'm going to intersect the x axis.
The zeros, the real zeros define the
all of the points that (mumbles) to intersect the x axis,
because, think about it, there's a point that
intersected the x axis that wasn't a zero,
well, it would be a zero
cause the function would take on the value of zero there.
So between those points, if the function will
stay positive or will stay negative.
And we can test that a little bit.
Let's think about what the function is doing
between x equals negative four
and x equals one, so, over this interval right over here.
Well let's just pick a value there,
and then think about whether the function
is gonna be positive or negative at that value.
And an easy value to think about is maybe zero.
And I just want to think about the sign,
is the function going to be positive or negative
when x is equal to zero.
So f of zero, f of zero,
I'm just going to think about the sign.
So the f of zero is gonna be zero minus two,
well that's going to be negative.
Zero plus four, that's gonna be positive.
Zero plus five, that's gonna be positive.
Zero minus one, is gonna be negative.
Zero minus nine is going to be negative.
So we can see, at x equals zero,
we're going to get a, let's see,
this is going to be a negative times a negative,
which would be a positive,
then we multiply by a negative again.
So this is going to be equal to,
this is going to get us to a negative.
So the function's gonna take on a negative value here.
And we know it's gonna take a negative value
over this entire interval,
over this entire interval.
Because if for some reason it took on a positive value
over that interval, it's a continuous function.
It would have to cross the x axis
and that would define another zero.
But we know there aren't any zeros between these two points.
So the function, I don't know what it looks like,
but it's going to stay positive
between those two values.
Now, as you go over to our next interval,
it's probably going to change signs.
And we can verify that if you look at f,
if you look at, say, f of negative of 4.5.
So we'll do the same drill.
F of negative 4.5, well that's going to be,
that's going to be a see,
negative times a negative,
times a positive.
Negative 4.5 plus five is a positive.
Times a negative, times a negative.
Well we have an even number of negatives
being multiplied together here.
So this gonna give us a positive.
So this is going to give us a positive value.
So we're going to have a positive value
between these two points.
And then we're likely to flip back to negative after that.
So then the graph is likely to flip back
flip back to negative, right over there.
You can test that, try a really negative value.
So f of negative 10.
Well that's going to be a negative
times a negative, times a negative,
let's see, times a negative, times a negative,
times a negative.
So you have five negatives being multiplied together,
you're going to have a negative.
Hopefully you see where all of this is going,
so this is, we're likely to switch signs here.
You can test 1.5 and see if you get a positive value.
Then we're likely to switch back here.
And once again, the graph isn't gonna look
exactly like this.
It might deep, it might be fairly shallow here,
and then it might deep down a lot right here.
I'm just making up the parts on the ups and downs.
And then it probably does another sign change
right over there, and we can verify that.
If f of positive 10
well that's gonna be a positive
times a positive,
times a positive, times a positive, times a positive.
Positive times a positive, so we're positive.
After that, I don't know exactly what it looks like.
But the key realization here is
that the zeros are the only places
we're gonna intercept the x axis.
And between those zeros, if you were positive,
if you're positive on the interval, you're gonna be
positive on any point.
If you're positive at one point in that interval,
you're gonna be positive at any point in that interval.
And if you're negative at any point of that interval,
then you're going to be negative
throughout the interval.
And of course, at the end of the interval,
you're going to be at zero.
So, if someone were to say,
what is the sign of f?
If someone would say sing
of f on the interval on
say negative four is less than x, is less than one.
Well this interval goes from negative four
to one, so that's the interval we actually started off with.
That's that interval right over there,
you'd say, oh well look,
f is always negative on this interval.
If you don't include the end points.
And we're not including the end points here.
We don't have a less and/or equal either point here.
So here's always negative.
If someone gave you a different interval,
if they give you an interval that includes
where zero is part of the interval.
So someone said,
if someone said the interval from
I don't know, the interval from one to nine,
well that's gonna include a zero in it.
So it's gonna be positive,
it's gonna be sometimes positive and sometimes negative
on this interval.
So the key is,
between two roots, not including the roots,
if you have an interval between two roots,
then you're always going to be positive,
or you're always going to be negative over that interval.
Whatever the sign is, at any point, the interval's
gonna be the sign throughout the interval,
not including the end points.