Polynomial remainder theorem

In algebra, the polynomial remainder theorem or little Bézout's theorem (named after Étienne Bézout)^[1] is an application of Euclidean division of polynomials. It states that, for every number $r,$ any polynomial $f(x)$ is the sum of $f(r)$ and the product by $x-r$ of a polynomial in $x$ of degree less than the degree of $f.$ In particular, $f(r)$ is the remainder of the Euclidean division of $f(x)$ by $x-r,$ and $x-r$ is a divisor of $f(x)$ if and only if $f(r)=0,$ ^[2] a property known as the factor theorem.

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- [Voiceover] So let's introduce ourselves to the Polynomial Remainder Theorem. And as we'll see a little, you'll feel a little magical at first. But in future videos, we will prove it and we will see, well, like many things in Mathematics. When you actually think it through, maybe it's not so much magic. So what is the Polynomial Remainder Theorem? Well it tells us that if we start with some polynomial, f of x. So this right over here is a polynomial. Polynomial. And we divide it by x minus a. Then the remainder from that essentially polynomial long division is going to be f of a. It is going to be f of a. I know this might seem a little bit abstract right now. I'm talking about f of x's and x minus a's. Let's make it a little bit more concrete. So let's say that f of x is equal to, I'm just gonna make up a, let's say a second degree polynomial. This would be true for any polynomial though. So three x squared minus four x plus seven. And let's say that a is, I don't know, a is one. So we're gonna divide that by, we're going to divide by x minus one. So a, in this case, is equal to one. So let's just do the polynomial long division. I encourage you to pause the video. If you're unfamiliar with polynomial long division, I encourage you to watch that before watching this video because I will assume you know how to do a polynomial long division. So divide three x squared minus four x plus seven. Divide it by x minus one. See what you get as the remainder and see if that remainder really is f of one. So assuming you had a go at it. So let's work through it together. So let's divide x minus one into three x squared minus four x plus seven. All right, little bit of polynomial long division is never a bad way to start your morning. It's morning for me. I don't know what it is for you. All right, so I look at the x term here, the highest degree term. And then I'll start with the highest degree term here. So how many times does x going to three x squared? What was three x times? Three x times x is three x squared. So I'll write three x over here. I'll write it in the, I could say the first degree place. Three x times x is three x squared. Three x times negative one is negative three x. And now we want to subtract this thing. It's just the way that you do traditional long division. And so, what do we get? Well, three x squared minus three x squared. That's just going to be a zero. So this just add up to zero. And this negative four x, this is going to be plus three x, right? And negative of a negative. Negative four x plus three x is going to be negative x. I'm gonna do this in a new color. So it's going to be negative x. And then we can bring down seven. Complete analogy to how you first learned long division in maybe, I don't know, third or fourth grade. So all I did is I multiplied three x times this. You get three x squared minus three x and then I subtract to that from three x squared minus four x to get this right over here or you could say I subtract it from this whole polynomial and then I got negative x plus seven. So now, how many times does x minus one go to negative x plus seven? Well x goes into negative x, negative one times x is negative x. Negative one times negative one is positive one. But then we're gonna wanna subtract this thing. We're gonna wanna subtract this thing and this is going to give us our remainder. So negative x minus negative x. Just the same thing as negative x plus x. These are just going to add up to zero. And then you have seven. This is going to be seven plus one. Remember you have this negative out so if you distribute the negative, this is going to be a negative one. Seven minus one is six. So your remainder here is six. One way to think about it, you could say that, well (mumbles). I'll save that for a future video. This right over here is the remainder. And you know when you got to the remainder, this is just all review of polynomial long division, is when you get something that has a lower degree. This is, I guess you could call this a zero degree polynomial. This has a lower degree than what you are actually dividing into or than the x minus one than your divisor. So this a lower degree so this is the remainder. You can't take this into this anymore times. Now, by the Polynomial Remainder Theorem, if it's true and I just picked a random example here. This is by no means a proof but just kinda a way to make it tangible of Polynomial (laughs) Remainder Theorem is telling us. If the Polynomial Remainder Theorem is true, it's telling us that f of a, in this case, one, f of one should be equal to six. It should be equal to this remainder. Now let's verify that. This is going to be equal to three times one squared, which is going to be three, minus four times one, so that's just going to be minus four, plus seven. Three minus four is negative one plus seven is indeed, we deserve a (mumbles), is indeed equal to six. So this is just kinda, at least for this particular case, looks like okay, it seems like the Polynomial Remainder Theorem worked. But the utility of it is if someone said, "Hey, what's the remainder if I were to divide "three x squared minus four x plus seven "by x minus one if all I care about is the remainder?" They don't care about the actual quotient. All they care about is the remainder, you could, "Hey, look, I can just take that, in this case, a is one. "I can throw that in. "I can evaluate f of one and I'm gonna get six. "I don't have to do all of this business. "All I had, would have to do is this "to figure out the remainder of three x squared." Well you take three x squared minus four plus seven and divide by x minus one.

Examples

Example 1

Let $f(x)=x^{3}-12x^{2}-42$ . Polynomial division of $f(x)$ by $(x-3)$ gives the quotient $x^{2}-9x-27$ and the remainder $-123$ . Therefore, $f(3)=-123$ .

Example 2

Proof that the polynomial remainder theorem holds for an arbitrary second degree polynomial $f(x)=ax^{2}+bx+c$ by using algebraic manipulation:

{\begin{aligned}f(x)-f(r)&=ax^{2}+bx+c-(ar^{2}+br+c)\\&=a(x^{2}-r^{2})+b(x-r)\\&=a(x-r)(x+r)+b(x-r)\\&=(x-r)(ax+ar+b)\end{aligned}}

So,

f(x)=(x-r)(ax+ar+b)+f(r),

which is exactly the formula of Euclidean division.

The generalization of this proof to any degree is given below in § Direct proof.

Proofs

Using Euclidean division

The polynomial remainder theorem follows from the theorem of Euclidean division, which, given two polynomials $f (x)$ (the dividend) and $g (x)$ (the divisor), asserts the existence (and the uniqueness) of a quotient $Q (x)$ and a remainder $R (x)$ such that

f(x)=Q(x)g(x)+R(x)\quad {\text{and}}\quad R(x)=0\ {\text{ or }}\deg(R)<\deg(g).

If the divisor is $g(x)=x-r,$ where r is a constant, then either $R (x) = 0$ or its degree is zero; in both cases, $R (x)$ is a constant that is independent of $x$ ; that is

f(x)=Q(x)(x-r)+R.

Setting $x=r$ in this formula, we obtain:

f(r)=R.

Direct proof

A constructive proof—that does not involve the existence theorem of Euclidean division—uses the identity

x^{k}-r^{k}=(x-r)(x^{k-1}+x^{k-2}r+\dots +xr^{k-2}+r^{k-1}).

If $S_{k}$ denotes the large factor in the right-hand side of this identity, and

f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0},

one has

f(x)-f(r)=(x-r)(a_{n}S_{n}+\cdots +a_{2}S_{2}+a_{1}),

(since $S_{1}=1$ ).

Adding $f(r)$ to both sides of this equation, one gets simultaneously the polynomial remainder theorem and the existence part of the theorem of Euclidean division for this specific case.

Applications

The polynomial remainder theorem may be used to evaluate $f(r)$ by calculating the remainder, $R$ . Although polynomial long division is more difficult than evaluating the function itself, synthetic division is computationally easier. Thus, the function may be more "cheaply" evaluated using synthetic division and the polynomial remainder theorem.

The factor theorem is another application of the remainder theorem: if the remainder is zero, then the linear divisor is a factor. Repeated application of the factor theorem may be used to factorize the polynomial.^[3]

References

^ Piotr Rudnicki (2004). "Little Bézout Theorem (Factor Theorem)" (PDF). Formalized Mathematics. 12 (1): 49–58.
^ Larson, Ron (2014), College Algebra, Cengage Learning
^ Larson, Ron (2011), Precalculus with Limits, Cengage Learning

This page was last edited on 30 March 2024, at 04:48

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