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# Intro to the Polynomial Remainder Theorem

The Polynomial Remainder Theorem allows us to determine whether a linear expression is a factor of a polynomial expression easily. It tells us the remainder when a polynomial is divided by
\[x - a\] is
\[f(a)\]. This means if
\[x - a\] is a factor of the polynomial, the remainder is zero. It's a neat trick to quickly find remainders without doing long division!

## Want to join the conversation?

- What is the difference between a binomial and a polynomial?(30 votes)
- binomial-they are with two terms

polynomial-monomial,binomial,trinomial everything are considered to be a polynomial(35 votes)

- why do i feel like him saying, "Starting polynomial long division is a good way to start your morning." Was a cry for help(84 votes)
- I can't relate...it's past midnight for me...

just squeezin' in some late-night studying :))(17 votes)

- When would it be useful to just calculate the remainder but not the quotient of polynomial division? Can anyone provide an example?(18 votes)
- https://www.khanacademy.org/math/algebra2/polynomial_and_rational/polynomial-remainder-theorem-tutorial/v/polynomial-remainder-theorem-to-test-factor

and

https://www.khanacademy.org/math/algebra2/polynomial_and_rational/polynomial-remainder-theorem-tutorial/v/constructing-a-polynomial-that-has-a-certain-factor(3 votes)

- Is the remainder theorem only true when you're dividing by x-a? Or is it true for x+a as well?(14 votes)
- It is true for x + a as well. x + a is another way of writing x - (-a). This comes into play when using synthetic division. Sometimes you'll be given a polynomial and a binomial in the form x + a. If it was x + 9, you would just take the opposite of 9, which is -9. Hope that helps.(36 votes)

- Shouldn't it be f(-a): You have x-1, and then you plug in 1. No?(15 votes)
- x-a! buddy a=1 so we plug in one!!

if it were x+a than you would be right buddie(18 votes)

- Does the polynomial remainder theorem also work on equations where the denominator 'x-a' where x has a coefficient or to anything greater than the 1st degree?(6 votes)
- It would work when x has a coefficient but when you have a denominator or divisor that has a degree that's greater than one, the remainder theorem wouldn't work as the remainder for higher degree terms is not constant.. (I got this from another person's answer on this website)(11 votes)

- What is the polynomial remainder theorem then?(3 votes)
- It says that if you divide a polynomial, f(x), by a linear expression, x-A, the remainder will be the same as f(A). For example, the remainder when x^2 - 4x + 2 is divided by x-3 is (3)^2 - 4(3) + 2 or -1. It may sound weird that plugging in A into the polynomial give the same value as when you divide the polynomial by x-A, but I assure you that it works.

Sal provides a proof of the theorem in another video. Here's the link: https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:poly-div/x2ec2f6f830c9fb89:remainder-theorem/v/polynomial-remainder-theorem-proof.(11 votes)

- Can you use this Theorem when dividing a polynomial with x+a, with a being some positive constant?(5 votes)
- of course the Theorem can be used in such cases..(5 votes)

- Find remaider when p(x)=x^6 +3x^2 +10 is divided by g(x)=x^4 +1(1 vote)
- you have to do polynomial long division. Your remainder will be a polynomial of degree lower than 4.

You can review polynomial long division here:

https://www.khanacademy.org/math/algebra2/arithmetic-with-polynomials/long-division-of-polynomials/v/polynomial-division(10 votes)

- and would knowing the remainder do anything useful?(3 votes)
- Yes, knowing the remainder can be useful. If you divide by (x-c) and have a remainder, you know that (x-c) is not a factor of the polynomial. However, if the remainder is 0, then it is a factor. Knowing this is useful because if (x-c) is a factor, then x=c is a solution.(5 votes)

## Video transcript

- [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 minor drumroll, 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.