Remember that the dividing procedure for long division of integers finishes whenever the remainder is smaller than the divisor.

dividend equals divisor* quotient + remainder

Polynomial division follows a similar procedure. When the residue is smaller than the degree of the divisor, the division is completed.

According to the remainder theorem: If you divide a polynomial f(x) by a linear divisor (x – a), the remainder equals f (a). As a result, if the divisor is linear, the remainder may be calculated using the Remainder Theorem.

## What is Remainder Theorem?

We already know a little about the remainder theorem, it is a method to Euclidean polynomial division. According to the Remainder theorem which we are going to discuss, dividing a polynomial P(x) by a factor (x – a); that isn’t an element of the polynomial yields a smaller polynomial and a remainder. This obtained residual is essentially a value of P(x) when x = a, more especially P (a).

x -a is therefore the divisor of P(x) if and only if P(a) = 0. The remainder theorem is known to be used to factor polynomials of any degree in an easy way.

## Remainder Theorem – Definition

The Remainder Theorem starts with a polynomial, say p(x), where “p(x)” is a polynomial p with x as a variable. Then, according to the theory, divide that polynomial p(x) by some linear factor x – a, where an is just some number. A lengthy polynomial division is performed here, yielding some polynomial q(x) (the variable “q” stands for “the quotient polynomial”) and a polynomial remainder is r (x). It may be stated as follows:

p(x) divided by (x-a) equals q(x) + r(x)

## How to Find Remainder of Polynomials?

How can the remainder and factor theorem be used to discover the remainder of polynomial divisions as well as the factors of polynomial divisions?

According to the Remainder Theorem, if a polynomial f(x) is divided by (x – k), the remainder r = f (k). It can aid in the factoring of more complicated polynomial expressions.

Now according to the Factor Theorem, which says that a polynomial f(x) has a factor (x – k) if and only if f(k) = 0. It is a variant of the Remainder Theorem in which the remainder equals zero.

## Use of Remainder Theorem Formula

When a polynomial p(x) is divided by (ax + b), the remainder theorem formula is used to get the remainder. We may use the remainder theorem to determine whether (ax + b) is a factor of p(x). If the remainder is zero, (ax + b) is a factor of the polynomial p(x), else it is not. The remainder theorem formula’s principal use is the factor theorem. The remainder theorem is required to show the factor theorem.

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