https:/.../Polynomial+from+linear+factors.mp4

These are the five linear
factors that the problem gave us

for this polynomial.

It wanted us to figure out
a few things.

It wanted us to figure out
what the factored form

of the polynomial would be
and it wanted us to figure out

what the x-intercepts of the
polynomial would be.

So if we look at these
linear factors,

we have y = x+1,

y = -x-2,

y = 2x-5,

y = x+1/2-√3,

and y = x+1/2+√3.

Those were given to us
in the original problem.

And are graphed on this graph here.

So how do we find the x-intercepts
of these linear graphs?

Well we simply do it,

by setting each one equal
to zero.

Now, you might notice
that on desmos it will give

you the decimal equivalent,

but you can see over here
to the left,

it's -1/2+√3,

and then -1/2-√3.

So it's shown over here
in its exact form,

and then it's shown on
here in this decimal equivalency.

So we have these five
x-intercepts now.

Here's the deal,
remember we just learned

that the x-intercepts
of the linear factors

are the same x-intercepts
as the actual graph.

So if we take these five
linear factors and we multiply

them altogether to get our function,

which is so big I have to even
make this wider for a sec so

we can look at it.

So we took the first linear function
times the second linear function

times the third linear function
times the fourth linear function,

times the fifth linear function.

So it's the product of all those
that were given.

Let me make this smaller
again so you can see the graph.

That's what I want you to
be able to see.

If that is the factored form
of the polynomial expression.

So that was one of the questions
that was asked.

I'll show it to you again
in just a second but we want

to test it on here and make sure
that our x-intercept

of our linear is the same
x-intercepts as our polynomial.

And you can see that it is.

I'm going to zoom out a little
bit because--

I'll zoom back in so you can
get closer,

but you can see the um--
there we go,

the local, there we go,

the local maximum and
the local minimum.

So that you can see how that
all plays in there.

It's much um...more increasing
and decreasing as in our

last example.

So lets go back in again
so you can see the closer.

So you can see exactly the
same spots that was the

x-intercepts of the linear
are now the x-intercepts

of our polynomial.

And one more time,

allow me to put the
equation on the screen

so that you can see the
factored form of the polynomial.