U0S2V13 Using the IVT.txt

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Let's start with this polynomial x
to the fourth minus x minus 1.
And let's say we want to know does it have any roots?
Is it ever equal to 0?
We can try to factor it, but this is a fourth degree
polynomial and I don't know how to factor that.
So let's try something else.
Since it's a polynomial, it's continuous everywhere.
So if we call this function f of x,
we can try to use the Intermediate Value Theorem.
To remind you, here's the statement of the Intermediate
Value Theorem.
If f is continuous on the closed interval a to b and M
is some value in between f of a and f of b,
then there's at least one point c between a and b such
that f of c equals M. And this conclusion of the theorem,
that's exactly what we want.
We want to know if there is some point such that f of c
equals 0, in our case.
So we're going to be using 0 for M.
And we already know that f is continuous so all that remains
is we need to find f of a and f of b such that M, which is 0
is in between f of a and f of b.
So let's just plug in some stuff.
And I'm going to graph it so that we have a good picture.
We can start with f of 1 and that's going to equal minus 1
and that's down here.
And we can try f of 3 and that's 81 minus 3 minus 1 is 77.
So that's way up here.
And then 0, the x-axis, is in between where f of 1 is
and where f of 3 is.
And therefore, by the Intermediate Value Theorem
there is at least one place where the graph of f
will intersect the x-axis.
So we definitely have at least one root.

Now we don't know where this root is exactly,
the Intermediate Value Theorem only
tells us that it's somewhere between x equals 1
and x equals 3.
If we want to narrow it down, we could try plugging in something
like x equals 2.
If we take f of 2, that's 16 minus 2 minus 1 that's 13.
So that's up here.
And applying the Intermediate Value Theorem again,
that means that for sure we'll have
at least one root between x equals 1 and x equals 2.
If you wanted to pin down the location of a root
even further, you could just repeat the process.
You could plug in x equals 1.5, for instance,
and just keep going.
However, this is a pretty slow process,
if you want to get things precise.
You'll learn faster techniques like Newton's Method
a little later in the course, but if all you care about is:
"is there a root somewhere?", then
we already have our answer.
The answer is yes.
There's at least one root between x
equals 1 and x equals 2.
Whether there are roots elsewhere,
like maybe between 2 and 3, we don't know,
but there definitely is at least one
between x equals 1 and x equals 2.
So that's basically how you use the Intermediate Value Theorem.
It's really exploiting the continuity of your function,
presuming your function is continuous to begin with.
You should always be sure to check.
Anyhow, we have some problems for you to play around with it.