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Course: Algebra (all content)>Unit 10

Lesson 24: Polynomial Remainder Theorem

Remainder theorem: finding remainder from equation

The Polynomial Remainder Theorem simplifies the process of finding the remainder when dividing a polynomial by \[x - a\]. Instead of long division, you just evaluate the polynomial at \[a\]. This method saves time and space, making polynomial division more manageable.

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• Where did the 8 come from?
• He was just plugging in a_ (in this case, 2) for _x.
The first term of the equation is -3_x_^3. When _x_ is plugged in, it is -3*2^3, which is also -3*8.
• Does the polynomial theorem only work for x-a and not x+a?
• If you have (x+a), then the "a" value will actually be "-a" because (x+(-a))=(x-a).
• This question was asked a few comments down but I did not find the answer very satisfying. What is the significance of the remainder? In what situation would I need the remainder of a polynomial instead of both its quotient and remainder?
• If the remainder is 0, then you know that the divisior is a factor of the dividend (they are divisble).
• Would this work if the the coefficient on the x term was greater than 1?
• I think that it could work. Another way to find the remainder is to set the x - a to term equal to 0 and then solve for x. After this, you just plug it back in to find the remainder. Correct me if I am wrong.
• How can a remainder be negative?
• With polynomial division, you can get remainders that are negative.
In numeric long division, you would not have a negative remainder.
• what do you mean when someone says f(x)?
• Function of x is basically a processing machine. You give it an input, and it gives out one output and one only. For example if f(x) = x+1, you can substitute any number for x and find the output. So f(5)= 5+1 = 6. In some ways they are similar to equations and in some ways they are different. I don't know too much about it myself, but here's the link: https://www.khanacademy.org/math/algebra/algebra-functions
• After all this process, should the remainder be expressed as -27/(x-2)??
• It depends on the context of the problem (if it asks for remainder term); but usually, and in this problem, it should be expressed as -27, according to the definition of remainder.
• when you have a negative remainder what does that mean? is negative and positive remainder mean the same thing?