Polynomial remainder theorem proof

- [Voiceover] Let's now do a proof

of the polynomial remainder theorem.

Just to make the proof
a little bit tangible,

I'm going to start with the example

that we saw in the video that introduced

the polynomial remainder theorem.

We saw that if you took three x squared

minus four x plus seven and you divided

by x minus one, you got three x minus one

with the remainder of six.

When we do polynomial long division,

how do we know when we've
got to our remainder?

Well when we get to an expression that has

a lower degree than the
thing that is the divisor,

the thing that we're dividing
into the other thing.

So in this example we
could have re-written

what we just did right
over here as our f of x.

Let me just write it right over here.

So we could have said three x squared

minus four x plus seven is equal to

x minus one times the
quotient right over here,

or I could say the
quotient times x minus one.

So it's going to be
equal to this business.

It's going to be equal
to three x minus one

times the divisor, times x minus one.

When you multiply these two things,

you're not going to get exactly this.

You still have to add the remainder.

So plus the remainder.

Actually, let me write
the actual remainder down.

So plus six.

The analogy here is exactly the analogy

to when you did traditional division.

If I were to say, actually, let me just

show you the analogy.

If I were to say 25 divided by four,

you would say, okay,
well four goes into 25

six times, six times four is 24.

You would subtract, and then you would get

a remainder, one.

Or another way of saying
this is you could say

that 25 is equal to six
times four plus one.

So we just did the exact same thing here,

but we just did it with expressions.

So once again, I haven't
started the proof yet,

I just wanted to make you feel comfortable

with what I just wrote right over here.

If I divided this expression
into this polynomial,

and I were to get this quotient,

that's the same thing as
saying this polynomial

could be equal to three x minus one

times x minus one plus six.

Now this is true in general.

Let's abstract a little bit.

This is our f of x.

So that is f of x.

So f of x is going to be equal to

whatever the quotient is.

Let me call that q of x.

So do this in a different color.

So I'm going to call that q of x.

This right over here is q of x.

So f of x is going to be
equal to the quotient,

q of x, times, this is our x minus a,

in this case a is one, but I'm just trying

to generalize it a little bit.

So x minus a, and then plus the remainder.

We know that the remainder is going

to be a constant because the remainder

is going to have a lower
degree then x minus a.

X minus a is a first degree.

So in order to have a lower degree,

this has to be zeroth degree.

This has to be a constant.

So this is true in general.

This is true for any polynomial

f of x divided by any x minus a.

So this is just true.

So this is true for any
f of x and x minus a.

Now what is going to happen
if we evaluate f of a?

Well if f of x can be written like this,

well we could write f of,

let me do a in a new
color so it sticks out.

We could write f of a would be equal to

q of a times, I think you might see

where this is going,
times a minus a plus r.

Well what's that going to be equal to?

What's all this business
going to be equal to?

Well a minus a is zero, and q of a,

I don't care what q of a is,

if you're going to multiply it by zero

all of this is going to be zero.

So f of a is going to be equal to r.

And so you're done.

This is the proof of the

polynomial remainder theorem.

Any function, if when you
divide it by x minus a

you get the quotient q
of x and the remainder r,

it can then be written in this way.

If it's written in this
way and you evaluated

at f of a and you put the a over here,

you're going to see that
f of a is going to be

whatever that remainder was.

That is the polynomial remainder theorem.

And we're done.

One of the simpler proofs that exists

for something that at first
seems somewhat magical.