what is remainder theorem
Answers
Answer:
Remainder Theorem states that if polynomial ƒ(x) is divided by a linear binomial of the for (x - a) then the remainder will be ƒ(a). Factor Theorem states that if ƒ(a) = 0 in this case, then the binomial (x - a) is a factor of polynomial ƒ(x).
Step-by-step explanation:
Divide 15 by 6. What answer do you get? By using the simple division process, we find that the quotient is 2 and the remainder is 3. Hence, we write 15 = (2 x 6) + 3. Note: The remainder ‘3’ is less than the divisor ‘6’. On the other hand, when we divide 12 by 6, we get a quotient of 2 and remainder 0. In this case, we say that 6 is a factor of 12 OR 12 is a multiple of 6.
Answer:
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 the remainder of the division of a polynomial {\displaystyle f(x)} by a linear polynomial {\displaystyle x-r} is equal to {\displaystyle f(r).} In particular, {\displaystyle x-r} is a divisor of {\displaystyle f(x)} if and only if {\displaystyle f(r)=0,}[2] a property known as the factor theorem.
Step-by-step explanation:
divide by the linear factor x – 4 (so a = 4):

So we get a quotient of q(x) = x2 + 4x + 9 on top, with a remainder of r(x) = 30.
You know, from long division of regular numbers, that your remainder (if there is one) has to be smaller than whatever you divided by. In polynomial terms, since we're dividing by a linear factor (that is, a factor in which the degree on x is just an understood "1"), then the remainder must be a constant value. That is, when you divide by "x – a", your remainder will just be some number.
The Remainder Theorem then points out the connection between division and multiplication. For instance, since 12 ÷ 3 = 4, then 4 × 3 = 12. If you get a remainder, you do the multiplication and then add the remainder back in. For instance, since 13 ÷ 5 = 2 R 3, then 13 = 5 × 2 + 3. This process works the same way with polynomials. That is:
If p(x) / (x – a) = q(x) with remainder r(x),
then p(x) = (x – a) q(x) + r(x).
(Technically, this "if - then" statement is the "Division Algorithm for Polynomials". But the Algorithm is the basis for the Remainder Theorem.)
In terms of our concrete example: Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved
Since (x^3 – 7x – 6) / (x – 4) = x2 + 4x + 9 with remainder 30,
then x3 – 7x – 6 = (x – 4) (x2 + 4x + 9) + 30.
The Remainder Theorem says that we can restate the polynomial in terms of the divisor, and then evaluate the polynomial at x = a. But when x = a, the factor "x – a" is just zero! Then evaluating the polynomial at x = a gives us: