Remainder Theorem of Polynomials

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 and 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 the 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 to 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.

Example 1: Find the remainder when  is divided by x-1

Solution: Here, p(x) = , and the zero of x-1 is 1.

So, p(1) =  = 2.

So, by the remainder theorem, 2 is the remainder when  divided by x-1.

Example 2: Check whether the polynomial q(t) =  is a multiple of 2t+1

Solution: As we know, q(t) will be multiple of 2t+1 only, if 2t+1 divides q(t) leaving remainder zero. Now taking 2t+1=0, we have t=.

So,   = +–-1 =  + 1 +  – 1 = 0

So the remainder obtained on dividing q(t) by 2t+1 is 0.

So, 2t+1 is a factor of the given polynomial q(t), that is q(t) is a  multiple of 2t+1.

Exercise

a. Find the remainder when  is divided by

1. x+1
2. 
3. x
4. 5+2x

b. Find the remainder when  is divided by

1. x-2
2. x+3
3. 2x-1

c. Find the remainder when  is divided by x-a.

d. Check whether 7+3x is a factor of .