Zeros of polynomials (with factoring): common factor | Polynomial graphs | Algebra 2 | Khan Academy

- So we're given a p of x,
it's a third degree polynomial,

and they say, plot all the
zeroes or the x-intercepts

of the polynomial in
the interactive graph.

And the reason why they
say interactive graph,

this is a screen shot from
the exercise on Kahn Academy,

where you could click
and place the zeroes.

But the key here is, lets
figure out what x values make

p of x equal to zero,

those are the zeroes.

And then we can plot them.

So pause this video,

and see if you can figure that out.

So the key here is to try
to factor this expression

right over here, this
third degree expression,

because really we're
trying to solve the X's

for which five x to
third plus five x squared

minus 30 x is equal to zero.

And the way we do that is

by factoring this left-hand expression.

So the first thing I always look for

is a common factor
across all of the terms.

It looks like all of the
terms are divisible by five x.

So let's factor out a five x.

So this is going to be five x times,

if we take a five x out
of five x to the third,

we're left with an x squared.

If we take out a five x
out of five x squared,

we're left with an x, so plus x.

And if we take out a
five x of negative 30 x,

we're left with a negative
six is equal to zero.

And now, we have five x
times this second degree,

the second degree expression
and to factor that,

let's see, what two numbers add up to one?

You could use as a one x here.

And their product is
equal to negative six.

And let's see, positive
three and negative two

would do the trick.

So I can rewrite this as five x times,

so x plus three, x plus three,

times x minus two, and if
what I did looks unfamiliar,

I encourage you to review
factoring quadratics

on Kahn Academy,

and that is all going to be equal to zero.

And so if I try to
figure out what x values

are going to make this
whole expression zero,

it could be the x values

or the x value that
makes five x equal zero.

Because if five x zero,

zero times anything else
is going to be zero.

So what makes five x equal zero?

Well if we divide five, if
you divide both sides by five,

you're going to get x is equal to zero.

And it is the case.

If x equals zero, this becomes zero,

and then doesn't matter what these are,

zero times anything is zero.

The other possible x value
that would make everything zero

is the x value that makes
x plus three equal to zero.

Subtract three from both sides

you get x is equal to negative three.

And then the other x value
is the x value that makes

x minus two equal to zero.

Add two to both sides,
that's gonna be x equals two.

So there you have it.

We have identified three x
values that make our polynomial

equal to zero and those
are going to be the zeros

and the x intercepts.

So we have one at x equals zero.

We have one at x equals negative three.

We have one at x equals, at x equals two.

And the reason why it's,

we're done now with this exercise,

if you're doing this on Kahn Academy

or just clicked in these three places,

but the reason why folks
find this to be useful is

it helps us start to think
about what the graph could be.

Because the graph has to intercept

the x axis at these points.

So the graph might look
something like that,

it might look something like that.

And to figure out what it
actually does look like

we'd probably want to try
out a few more x values

in between these x intercepts

to get the general sense of the graph.