 # Remainder Theorem, Definition, Formula and Examples

The remaining theorem is a formula for calculating the remainder when dividing a polynomial by a linear polynomial. The amount of items left over after dividing a specific number of things into groups with an equal number of things in each group is known as the remnant. After division, it is anything that “remains.” Let us first study the remainder theorem.

Everyone enjoys finding a shortcut, whether it is for driving directions or another sort of tedious work. Seeing a faster and more efficient method to the same endpoint makes you feel good because you’ve probably saved time, effort, and money. The remainder theorem is one of the more effective of these shortcuts in mathematics. The Chinese remainder theorem is a unique solution to simultaneous linear congruences with coprime moduli. In its most basic version, the Chinese remainder theorem will identify a number p that, when divided by certain specified divisors, leaves provided remainders. ## Remainder Theorem Definition

The following is the remaining theorem: The remainder is provided by r = a when a polynomial an (x) is divided by a linear polynomial b(x) whose zero is x = k. (k). The remainder theorem allows us to determine the remaining of dividing any polynomial by a linear polynomial without actually performing the division algorithm steps.

The Remainder Theorem is a method to Euclidean polynomial division. According to this theorem, dividing a polynomial P(x) by a factor ( x – a) that isn’t a polynomial element yields a smaller polynomial and a remainder. This obtained residual is really a value of P(x) when x = a, more particularly P (a). A is thus the divisor of P(x) if and P(a) = 0. It is used to factor polynomials of any degree in an attractive way.

## Factor theorem

The Factor Theorem is commonly used to factor and determine the roots of polynomial equations. It is the opposite of the remainder theorem. Problems are handled by using synthetic division and then checking for a zero remainder.

When p(x) = 0, y-x is a polynomial factor. Or, to put it another way, However, y-x is a polynomial factor. As a result, p(x) = 0. ## The Remainder Theorem Formula

Consider a general scenario to see how the remainder theorem works. Let an (x) represent the dividend polynomial and b(x) represent the linear divisor polynomial. We’ll call the quotient q(x) and the constant remainder r. As a result,

a(x) Equals b(x), q(x), and r

Let k represent the zero of the linear polynomial b(x). As a result, b(k) = 0. When we substitute x for k in the starred relation above, we get a(k) = b(k) q(k) + r.

This is permissible since the starred connection remains true for all values of x. It is, in reality, a polynomial identity. Because b(k)=0, we have a(k)=r. In other words, when x equals k, the remainder equals the value of an (x). That’s exactly what we discovered! The remainder theorem is exactly what it sounds like: When a polynomial an (x) is divided by a linear polynomial b(x), the remainder is given by r=a(x) (k). Consider the following example with two polynomials to show how it works in the case of polynomials:

a(x) : 6×4 – x3 + 2×2 – 7x + 2

b(x) : 2x + 3

When polynomials are divided, the quotient polynomial and the remainder are:

r = 203/4 q(x) = 3×3 – 5×2 + 17/2 x – 65/4

In this example, we estimated the remainder to be r = 203/4. Now consider what occurs when we evaluate a(x) for x equal to the zero of b(x), x = -3/2.