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Proof of the Polynomial Remainder Theorem

Video transcript

- [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.