When we divide a number by another number, we get a quotient and a

remainder. If the remainder is zero, then we say the numerator is divisible by

the denominator. Let us assume two numbers 17 and 3. We know that when we

divide 17 by 3, we get the quotient 5 and remainder 2. It can be expressed as

17= (3*5) +2. We can see that the remainder i.e. 2 is less than the divisor 3.

Similarly, when we divide 10 by 2, we get 10= (2*5) +0. So, the remainder is 0,

and we can say that 2 are a factor of 10 and 10 is a multiple of 2. The same

concept is used in polynomials. When a polynomial is divided by another

polynomial, we get a quotient which is in polynomial and a remainder. To make

some sums possible Remainder Theorem plays a vital role in finding factor or

multiple of those polynomials. The Remainder Theorem is a useful mathematical

theorem that can be used to factorize polynomials of any degree in a neat and

fast manner.

Definition

The remainder theorem is an application of polynomial long

division. When we are dividing a polynomial with another polynomial it is being

expressed in the form: f(x)= g(x).q(x)+r(x), where f(x) is a polynomial, g(x)

is a divisor, q(x) is a quotient and r(x) would be the remainder Dividend =

(Divisor*Quotient) + Remainder.

Let us assume two polynomial p(x) and g(x) such that the degree

of polynomial p(x) would be greater than the g(x) i.e. the divisor which cannot

be zero. Then we can find the factor of polynomial by f(x) = g(x).q(x) + r(x)

where the remainder is lesser than the degree of divisor i.e. g(x). So, we can

say that f(x) divided by g(x) gives us the q(x) and r(x).

Remainder Theorem operates on the fact that a polynomial is

completely divisible by its factor to obtain a smaller polynomial then the

divisor and with the remainder having value smaller value or any real number.

Theorem

Let p(x) be any

polynomial of degree greater than or equal to one and is divided by the linear

polynomial x-a where a be any number which would be the divisor and we get the

value of x = a, then the remainder is p (a).

Proof

Let p(x) be any

polynomial of degree greater than or equal to 1. Suppose that when p(x) is

divided by x-a (where a is a divisor), the quotient is q(x) and the remainder

is r(x), i.e., p(x) = (x-a) q(x) + r(x) where Dividend = (Divisor x

quotient) + Remainder.

Since

the degree of x-a is less than the p(x) and the degree of r(x) is less than the

degree of x-a, where the degree of r(x) = 0. Which means that r(x) is constant,

say r. Therefore, p(x) = (x-a) q(x) +r.

By

putting the value in p(x) where x=a, this equation gives us p(a) = (a-a) q

(a) +r = r. Hence, this proves the theorem.