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- [Voiceover] Let's now do a proof

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of the polynomial remainder theorem.

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Just to make the proof
a little bit tangible,

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I'm going to start with the example

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that we saw in the video that introduced

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the polynomial remainder theorem.

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We saw that if you took three x squared

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minus four x plus seven and you divided

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by x minus one, you got three x minus one

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with the remainder of six.

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When we do polynomial long division,

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

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Well when we get to an expression that has

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a lower degree than the
thing that is the divisor,

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

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So in this example we
could have re-written

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

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Let me just write it right over here.

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So we could have said three x squared

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minus four x plus seven is equal to

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x minus one times the
quotient right over here,

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or I could say the
quotient times x minus one.

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

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

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times the divisor, times x minus one.

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When you multiply these two things,

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you're not going to get exactly this.

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You still have to add the remainder.

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So plus the remainder.

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Actually, let me write
the actual remainder down.

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So plus six.

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The analogy here is exactly the analogy

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to when you did traditional division.

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If I were to say, actually, let me just

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show you the analogy.

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If I were to say 25 divided by four,

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you would say, okay,
well four goes into 25

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six times, six times four is 24.

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You would subtract, and then you would get

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a remainder, one.

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Or another way of saying
this is you could say

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that 25 is equal to six
times four plus one.

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So we just did the exact same thing here,

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but we just did it with expressions.

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

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I just wanted to make you feel comfortable

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with what I just wrote right over here.

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If I divided this expression
into this polynomial,

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and I were to get this quotient,

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that's the same thing as
saying this polynomial

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could be equal to three x minus one

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times x minus one plus six.

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Now this is true in general.

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Let's abstract a little bit.

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This is our f of x.

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So that is f of x.

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So f of x is going to be equal to

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whatever the quotient is.

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Let me call that q of x.

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So do this in a different color.

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

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This right over here is q of x.

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

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

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

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to generalize it a little bit.

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So x minus a, and then plus the remainder.

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We know that the remainder is going

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to be a constant because the remainder

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

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X minus a is a first degree.

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So in order to have a lower degree,

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this has to be zeroth degree.

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This has to be a constant.

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So this is true in general.

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This is true for any polynomial

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f of x divided by any x minus a.

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So this is just true.

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

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

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Well if f of x can be written like this,

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well we could write f of,

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

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We could write f of a would be equal to

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q of a times, I think you might see

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

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Well what's that going to be equal to?

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

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

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I don't care what q of a is,

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if you're going to multiply it by zero

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all of this is going to be zero.

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

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And so you're done.

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This is the proof of the

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polynomial remainder theorem.

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

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

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it can then be written in this way.

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If it's written in this
way and you evaluated

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

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

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whatever that remainder was.

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That is the polynomial remainder theorem.

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And we're done.

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One of the simpler proofs that exists

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for something that at first
seems somewhat magical.