Algebraic Long Division

Dividing one polynomial into another

I've been asked to make a video
on algebraic division or

algebraic long division.

So I'll make a video an
algebraic long division.

I'm just going to
make up a problem.

Let's say we wanted to divide--
we wanted to see how many times

does-- I'll start with a fairly
straightforward problem.

How many times does 2x plus 1
go into-- I don't know-- let's

say it's 8x to the third minus
7x squared plus 10x minus 5.

So what we do is we just take--
actually, just the exact same

way that you would do with long
division, traditional long

division of multiple digits.

In the 2x plus 1 expression
you look at, oh, what is

the highest degree term?

And that's really all we're
going to pay attention

to most of the time.

So the first step is you
say, OK, the highest

degree term is 2x.

How many times does 2x go into
the highest degree term of the

number-- not the number-- the
expression that we're

dividing into?

So you say, how many times does
2x go into 8x to the third?

Well, we could do a little
division on the side, but you

could imagine eventually this
is pretty straightforward.

So if you have 8x to the
third divided by 2x, that

is equal to 4x squared.

So 2x goes into 8x to the
third 4x squared times.

And this is the key thing.

You don't want to write
the 4x squared here.

You want to keep everything
in the correct places.

So when you're dividing numbers
you think of the ones, the

tens, the hundreds, and the
thousands place et cetera.

When you're dividing
polynomials you can kind of

think of the x to the 0 space.

The x to the 1 space
or the x space.

The x squared space.

The x to the third space.

So, when we say that 2x goes
into 8x to the third 4x squared

times let's write that
in the x squared spot.

It goes into it 4x
squared times.

Now, we take that 4x
squared and we multiply

it by our expression.

I think you're already seeing
that this is very similar

to long division.

And actually, if x was a
ten, it would be identical

to long division.

And I'll let you
think about that.

If x was 10 this would
be the thousands place.

This would be 8,000 minus--
although you would have

negative digits, which doesn't
make a bunch of sense.

But I think you get
what I'm saying.

But anyway, back to this
algebraic long division.

Although I think it is
very important to see the

parallels between this and
traditional long division.

Well, anyway, we said that
2x goes into 8x to the

third 4x squared times.

Now what we can do is we
can multiply 4x squared

times 2x plus 1.

So 4x squared times 1,
that's 4x squared.

So we can write that in
the x squared's place.

We could write it 4x squared.

And 4x squared times 2x
is 8x to the third.

This is plus here.

And now, just like we do with
traditional long division, we

can subtract this from this.

So minus 7x squared minus 4x
squared is minus 11x squared.

And then 8x to the third minus
8x to the third is 0, so we

can ignore that right there.

And if we want, we can bring
down the rest of the number,

but maybe just for fun we'll
bring down the next spot just

like we do in traditional
long division.

Actually, let me erase
this over here.

Because I think we might find
that real estate useful.

All right, I'm back.

Actually, it doesn't hurt to
bring down the whole thing.

Just so that you understand
what we're doing.

We're saying if you were to
divide 2x plus 1 to this entire

expression, and you say it
goes in 4x squared times.

Now you can kind of call
it our intermediate

remainder is what's left.

This is what's left.

You could almost imagine 4x
squared times 2x plus 1 is--

this is 8x to the third plus 4x
squared plus 0 plus 0 because

it doesn't contribute any
thing to these spots.

But then what's left over
is this expression.

If you take this minus this
whole expression, you

get what's left over.

Now we just do the
same thing over.

How many times does 2x--
we just look at the

highest order term.

How many times does 2x go
into negative 11x squared?

So let's write it here
on the side again.

Actually, let me do it here.

So if we were to take minus
11x squared divided by 2x,

that is equal to what?

That is equal to minus 11/2 x.

So 2x goes into minus 11x
squared minus 11/2 x times.

So we'll write that
in our x place.

So minus 11/2.

We could write that as 5.5.

I'll just write it
as a fraction.

Minus 11/2 x.

And now, what is minus
11/2 x times 2x plus 1?

So minus 11/2 x times
1 is minus 11/2 x.

And we'll want to write
that in the x position.

I'll switch colors just
to not be monotonous.

So minus 11/2 x times
1 is minus 11/2 x.

And then, minus 11/2
x times 2x, well, we

should know that is.

But you can multiply them out.

It'll be minus 11x squared.

I think you see
what we're doing.

After every step we're
canceling out the largest

degree of the polynomial
we're dividing into.

Fair enough?

Now let's subtract this
expression from this.

And we'll get kind of our
new intermediary remainder.

And maybe that'll be
the full remainder.

So let's see.

Minus 11x squared
minus 11x squared.

That's 0, so we don't have
to write anything there.

10x minus negative 11/2 x.

Remember, we're subtracting
this negative number from 10x.

So if you're subtracting a
negative number it's like

adding a positive number.

So you could view this
as 10 plus 11/2.

So 10 plus 11/2, that's
20/2 plus 11/2.

That's 31/2 or 15.5.

I'll just write 31/2 x.

31/2 x.

And then you could say that
there was a 0 here and

when you subtract 0 from
minus 5 you get minus 5.

And now we say, how many times
does 2x go into 31/2 x.

Let's do a little work
on the side here.

So if I have 31/2
x divided by 2x.

Well, the x's will
just cancel out.

This is equal to 31/4.

This is the same thing
as 31 over 2 times 1/2.

So it's 31/4.

So 2x goes into this expression
31/4 times and I'll

switch colors.

I'll switch to green.

And that's a positive, right?

You're dividing a positive
into a positive.

So plus 31/4 times.

And I'm writing that in the--
you could view that in the

constant space or the
x to the 0 space.

Or the 1 space even.

So it goes into it 31/4 times.

31/4 times 1 is 31/4.

And 31/4 times 2x is 31/2 x.

And now we subtract.

This is a plus here.

We subtract the green
expression from the light

blue expression and
we're left with this.

When you subtract this from
this you're left with 0, so

nothing shows up there.

And we're left with
minus 5 minus 31/4.

And we can just do a little
bit of fraction work here.

That's equal to, let's see.

Minus 5 over 4 is
minus 20 minus 31.

All of that over 4.

So that is equal to what?

Minus 20, that's
equal to minus 51.

Minus 51/4.

So, our answer is 2x plus 1
goes into 8x to the third minus

7x squared plus 10x minus 5--
it goes into it 4x squared

minus 11/2 x plus 31/4 times.

But there is a remainder,
and this is the remainder.

And so a way to visualize this
or another way to think about

this problem so it's actually
useful when we're actually

solving real problems.

And that you just don't view
this as some kind of mechanical

way to get problems right
on a test that only tests

algebraic long division.

As another way to write this
relationship you could write

that-- let me do it
in another color.

I've used many of
my colors already.

So you can write that 2x plus
1 times this-- 4x squared.

That's an x.

Minus 11/2 x plus 31/4.

Plus the remainder.

So when you multiply these two
out, and then if you were to

add the remainder-- 51/4-- that
that would equal-- and let

me draw a dividing line.

I don't want to confuse you
with all this stuff here.

That would equal this.

That would equal 8x to the
third minus 7x squared

plus 10x minus 5.

Anyway, I hope that helps.

See you in the next video.