Remainder Theorem

In Mathematics, the Remainder Theorem is a way of addressing Euclidean’s division of polynomials. The other name for the Remainder Theorem is Bezout’s theorem of approaching polynomials of Euclidean’s division. The remainder theorem definition states that when a polynomial f(x) is divided by the factor (x -a) when the factor is not necessarily an element of the polynomial, then you will find a smaller polynomial along with a remainder. The resultant obtained is the value of the polynomial f(x) where x = a and this is possible only if f(a) = 0. In order to factorize polynomials easily, the remainder theorem is applied.


Example: If p(x) = x3 - 12x2 - 42  is divided by x - 3. The quotient is x2 - 9x - 27  and the remainder is - 123.


Assuming, x - 3 = 0.


x = - 2.


Substituting x’s value, we get: 


p (x) = -123


Therefore, this proves and satisfies the remainder theorem. 


Remainder Theorem Definition

The Remainder Theorem Definition states that when a polynomial is p ( a ) is divided by another binomial ( a - x ), then the remainder of the end result that is obtained is p ( x ).


Example: 2a2 - 5a - 1 is divided by a - 3


Solution: Here p (a) = 2a2 - 5a - 1 and the divider is ( a - 3 )


Remainder Theorem Formula

Consider a polynomial f ( a ) where f is the polynomial and a is the variable. Now, as per the theorem, the polynomial f ( a ) is now divided by a binomial ( a - x ) where x is a random number. Here, the polynomial is divided by ( a - x ) and the remainder that is produced is r ( a ). The above definition can be expressed as:


f ( a ) / ( a - x ) = q ( x ) + r ( x )


Factor Theorem

To find the roots of a polynomial equation, the factor theorem is applied to factorize the equation. With the help of synthetic division, you can solve problems and also you can check for a 0 remainder.


When f ( a ) = 0, then y - a can be considered as the factor of the polynomial f ( a ). When the y - a is a factor of the polynomial f (  a ), then the polynomial f ( a ) = 0.


Remainder Theorem Proof

The remainder theorem is applicable only when the polynomial can be divided entirely at least one time by the binomial factor to reduce the bigger polynomial to a smaller polynomial a, and the remainder to be 0. This is one of the ways which are used to find out the value of a and root of the given polynomial f ( a ).


Proof: 


When f ( a ) is divided by  ( a - x ), then: 


f ( a ) = ( a - x ) . q ( a ) + r


Consider x = a;


Then,


f ( a ) = ( a - a ) . q ( a ) + r


f ( a ) = r


Therefore, the above proves the remainder theorem. 


The Steps Involved in Dividing a Polynomial by a Non Zero Polynomial

Step 1: The polynomial ( the dividend and the divisor) is arranged in the decreasing order of its degree. 

Step 2: With the first term of the divisor, divide the first term of the dividend in order to find out the first term of the quotient. 

Step 3: Now multiply the first term of the quotient with the first term of the divisor and with the obtained result, subtract the result from the divided to find out the remainder. 

Step 4: Next, divide the remainder with the division. 

Step 5: Repeat Step 4 until you cannot divide the remainder any more.


Remainder Theorem Examples

Question 1: Find the root of the polynomial a2 - 3a - 4


Solution:  Consider the value of a to be 4. 


Substituting the value of a = 4 in the polynomial, we get:


f ( 4 ) = 42 - 3 ( 4 ) - 4


f ( 4 ) = 16 - 12 - 4


Therefore, f ( 4 ) = 0.


Question 2: Find the r ( d ) of the polynomial d4 - 2d3 + 4d2 - 5 if it is divided by d - 2


Solution: 


d - 2  = 0


d = 2


Substituting d value in polynomial we get:


r ( 2 ) = 24 - 2 ( 2 )3 + 4 ( 2 ) - 5

r ( 2 ) =  16 - 16 + 8 - 5

r ( 2 ) = 3


Question 3: Find the r ( d ) of the polynomial 4d2 - d + 9 if it is divided by d - 1


Solution: 


d - 1  = 0


d = 1


Substituting d value in polynomial we get:


r ( 1 ) = 4 ( 12 ) - 1 + 9

r ( 1 ) =  4 - 1 + 9

r ( 1 ) = 12

FAQ (Frequently Asked Questions)

1) State Remainder Theorem and Derive its Proof.

The Remainder Theorem states that when a polynomial is f ( a ) is divided by another binomial ( a - x ), then the remainder of the end result that is obtained is f ( a ). The remainder theorem is applicable only when the polynomial can be divided entirely at least one time by the binomial factor to reduce the bigger polynomial to a smaller polynomial a, and the remainder to be 0. This is one of the ways which are used to find out the value of a and root of the given polynomial f ( a ).


Proof: 


When f ( a ) is divided by  ( a - x ), then: 


f ( a ) = ( a - x ) . q ( a ) + r


Consider x = a;


Then,


f ( a ) = ( a - a ) . q ( a ) + r


f ( a ) = r

2) What are the Steps Involved in the  Remainder Theorem?

Step 1: The polynomial ( the dividend and the divisor) is arranged in the decreasing order of its degree. 

Step 2: With the first term of the divisor, divide the first term of the dividend in order to find out the first term of the quotient. 

Step 3: Now multiply the first term of the quotient with the first term of the divisor and with the obtained result, subtract the result from the divided to find out the remainder. 

Step 4: Next, divide the remainder with the division. 

Step 5: Repeat Step 4 until you cannot divide the remainder any more.