Factoring linear binomials

What I want to do is start with
an expression like 4x plus 18

and see if we can rewrite
this as the product of two

expressions.

Essentially, we're going
to try to factor this.

And the key here
is to figure out

are there any common
factors to both 4x and 18?

And we can factor that
common factor out.

We're essentially
going to be reversing

the distributive property.

So for example, what
is the largest number

that is-- or I could really say
the largest expression-- that

is divisible into
both 4x and 18?

Well, 4x is divisible
by 2, because we

know that 4 is divisible by 2.

And 18 is also
divisible by 2, so we

can rewrite 4x as
being 2 times 2x.

If you multiply that side,
it's obviously going to be 4x.

And then, we can write 18 as
the same thing as 2 times 9.

And now it might
be clear that when

you apply the
distributive property,

you'll usually end
up with a step that

looks something like this.

Now we're just going to
undistribute the two right over

here.

We're going to
factor the two out.

Let me actually just draw that.

So we're going to
factor the two out,

and so this is going to
be 2 times 2x plus 9.

And if you were to-- wanted
to multiply this out,

it would be 2 times
2x plus 2 times 9.

It would be exactly
this, which you

would simplify as
this, right up here.

So there we have it.

We have written
this as the product

of two expressions,
2 times 2x plus 9.

Let's do this again.

So let's say that I
have 12 plus-- let

me think of something
interesting-- 32x.

Actually since we-- just to get
a little bit of variety here,

let's put a y here, 12 plus 32y.

Well, what's the
largest number that's

divisible into both 12 and 32?

2 is clearly divisible
into both, but so is 4.

And let's see.

It doesn't look like
anything larger than 4

is divisible into
both 12 and 32.

The greatest common
factor of 12 and 32 is 4,

and y is only divisible
into the second term,

not into this first
term right over here.

So it looks like 4 is the
greatest common factor.

So we could rewrite each
of these as a product of 4

and something else.

So for example, 12, we
can rewrite as 4 times 3.

And 32, we can
rewrite-- since it's

going to be plus-- 4 times.

Well if you divide 32y by
4, it's going to be 8y.

And now once again, we
can factor out the 4.

So this is going to
be 4 times 3 plus 8y.

And once you do more and
more examples of this,

you're going to find
that you can just

do this stuff all at once.

You can say hey, what's
the largest number that's

divisible into both of these?

Well, it's 4, so let
me factor a 4 out.

12 divided by 4 is 3.

32y divided by 4 is 8y.