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=3

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. The polynomial f (a) is now divided by a binomial (a – x), where x is a random number, according to the theory. The polynomial is divided by (a – x) in this case, and the remainder 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.

The Factor Theorem and How to Apply It

The following are the steps to use the factor theorem to identify the factors of a polynomial:

• Step 1 : If f(-c)=0, then (x+ c) is a factor of the polynomial f(x).

• Step 2 : (cx-d) is a factor of the polynomial f(x) if p(d/c)= 0.

• Step 3 : If p(-d/c)= 0, then (cx+d) is a factor of the polynomial f(x).

• Step 4 : If p(c)=0 and p(d)=0, then (x-c) and (x-d) are polynomial factors p(x).

Rather than using the polynomial long division method to find the factors, the factor theorem and synthetic division method are the best options. This theorem is generally used to eliminate known zeros from polynomials while keeping all unknown zeros unaffected, allowing the lower degree polynomial to be readily found. The factor theorem can also be defined in another way. We usually get a reminder when a polynomial is divided by a binomial. When a polynomial is split by one of its binomial components, the quotient resulting is known as a depressed polynomial. The factor theorem is demonstrated as follows if the remainder is zero:

If f(c)= 0, the polynomial f(x) has a component (x-c), where f(x) is a polynomial of degree n, and n is larger than or equal to 1 for any real number, c.

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 anymore.

Working of Remainder Theorem

Let's look at a general scenario to see how the remainder theorem works. Let a(x) be the dividend polynomial and b(x) be the linear divisor polynomial, and q(x) and r be the quotient and constant remainder, respectively. As a result,

a(x) = b(x) q(x) + r

Let's use k to represent the zero of the linear polynomial b(x). As a result, b(k) = 0. If we insert in x as k in the stated relation above, we have a(k) = b(k) q(k) + r

It's worth noting that this is permissible since the starred connection remains true for all x values. It's a polynomial identity, in reality. We're left with a(k)=r since b(k)=0. In other words, when x equals k, the remainder equals the value of a(x). That's exactly what we found! The remainder theorem is exactly what it sounds like: When a polynomial a(x) is divided by a linear polynomial b(x) whose zero is x equal to k, the remainder is given by r=a(k).

Remainder Theorem Examples

Question 1: Find the root of the polynomia 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

Remainder Theorem is a way of addressing Euclidean's division of polynomials. It 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).