Remainder Theorem
The remainder theorem is a formula that is used to find the remainder when a polynomial is divided by a linear polynomial. When a certain number of things are divided into groups with an equal number of things in each group, the number of leftover things is known as the remainder. It is something that "remains" after division. Let us learn the concept of the remainder theorem.
1.  What Is the Remainder Theorem? 
2.  What Is Remainder Theorem Formula? 
3.  How Does Remainder Theorem Work? 
4.  FAQs on Remainder Theorem 
What Is the Remainder Theorem?
The remainder theorem is stated as follows: When a polynomial a(x) is divided by a linear polynomial b(x) whose zero is x = k, the remainder is given by r = a(k). The remainder theorem enables us to calculate the remainder of the division of any polynomial by a linear polynomial, without actually carrying out the steps of the division algorithm.
What Is Remainder Theorem Formula?
The general formula for remainder theorem is expressed as p(x) = (xc)·q(x) + r(x). Let us consider polynomials to prove the remainder theorem formula.
Proof of Remainder Theorem
You know that Dividend = (Divisor × Quotient) + Remainder.
If r(x) is the constant then, p(x) = (xc)·q(x) + r.
Let us put x=c
p(c) = (cc)·q(c) + r
p(c) = (0)·q(c) + r
p(c) = r
Hence, proved.
How Does Remainder Theorem Work?
To understand how the remainder theorem works, let us consider a general case. Let a(x) be the dividend polynomial and b(x) the linear divisor polynomial, and let q(x) be the quotient and r the constant remainder. Thus, we have
a(x) = b(x) q(x) + r
Let us denote the zero of the linear polynomial b(x) by k. This means that b(k) = 0. If we plug in x as k in the starred relation above, we have a(k) = b(k) q(k) + r
Note that doing this is allowed since the starred relation holds true for every value of x. In fact, it is a polynomial identity. Since b(k)=0 we are left with a(k)=r. In other words, the remainder is equal to the value of a(x) when x is equal to k. Precisely what we stumbled upon! This is exactly what the remainder theorem is: 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). To see how it works in the case of polynomials, let us consider the following example with two polynomials:
a(x) : 6x^{4}  x^{3} + 2x^{2}  7x + 2
b(x) : 2x + 3
On dividing polynomials, the quotient polynomial and the remainder are:
q(x) = 3x^{3}  5x^{2} + 17/2 x  65/4 r = 203/4
We calculated the remainder in this case to be r = 203/4. Now, let’s see what happens when we evaluate a(x) for x equal to the zero of b(x), which is x = 3/2. We have
Once again, this has turned out to be equal to the remainder we calculated using the division algorithm.
Important Notes
 When a polynomial a(x) is divided by a linear polynomial b(x) whose zero is x = k, the remainder is given by r = a(k)
 The remainder theorem formula is: p(x) = (xc)·q(x) + r(x).
 The basic formula to check the division is: Dividend = (Divisor × Quotient) + Remainder.
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Examples on Remainder Theorem

Example 1: Casey is solving a polynomial expression. Help her find the remainder when p(x):3x^{5}−x^{4}+x^{3}−4x^{2}+2 is divided by q(x):x−1.
Solution:
We will use the remainder theorem: we will substitute the zero of q(x) into the polynomial p(x) to find the remainder r:
r = p(1)
= 3(1)^{5}  (1)^{4} + (1)^{3}  4(1)^{2} + 2
= 3  1 + 1  4 + 2
= 1 
Example 2: Consider the following polynomial:
p(x): x^{6}−2x^{5}+x^{4}+2x^{3}+x^{2}−36
Evaluate the value of this polynomial using the remainder theorem, when x is equal to 2. What can you infer from your answer?Solution:
We have
p(2) = (2)^{6} − 2(2)^{5} + (2)^{2} + 2(2)^{3} + (2)^{2} − 36
= 64 – 64 + 16 + 16 + 4 − 36
= 0This means that if p(x) is divided by the linear polynomial q(x) : x  2 there will be no remainder! Thus, we can conclude that q(x) is a factor of p(x). In general, whenever a polynomial is divided by a linear divisor and the remainder is 0, the linear divisor must be a factor of the polynomial.

Example 3: Consider the following two polynomials:
a(x) : x^{3}−x^{2}+x−1
b(x) : 2x+1Find the remainder polynomial and the quotient when a(x) is divided by b(x) using the remainder theorem.
Solution:
We proceed as earlier:
Thus, we have
q(x): 1/2x^{2}  3/4x + 7/8
r = 15/8Let us try to find the same using the remainder theorem.
Putting \(x=1/2\) in \(x^3x^2+x1\)
we have,
=(1/2)^{3} (1/2)^{2} + (1/2) 1
=(1/8)  (1/4)  (1/2) 1
= 7/8  1
Remainder = 15/8
FAQs on Remainder Theorem
What Is the Remainder Theorem in Math?
The remainder theorem enables us to calculate the remainder of the division of any polynomial by a linear polynomial, without actually carrying out the steps of the division algorithm.
When a polynomial a(x) is divided by a linear polynomial b(x) whose zero is x = k, the remainder is given by r = a(k).
How do you Use the Remainder Theorem?
To use the remainder theorem we can concentrate on the following steps:
 Observe the given polynomial
 Arrange the polynomial in the increasing order of the power
 Either perform the long division or by using the remainder theorem that is p(x) = (xc)·q(x) + r(x) we can justify the answer.
Are the Factor Theorem and the Remainder Theorem Same?
The factor theorem says that given a polynomial a(x), if its value at x equal to some real number k is equal to 0, that is, if a(k)=0, then the linear polynomial (x  k) is a factor of a(x), and we can write a(x) = (xk) b(x) where b(x) is a polynomial of degree one less than a(x).
What Is the Remainder Theorem Formula?
The general formula for remainder therem is represented as, p(x) = (xc)·q(x) + r(x)
Who Invented the Remainder Theorem?
Chinese mathematician Sun Zi invented the remainder theorem. The complete remainder theorem was given by Qin Jiushao in 1247.
What if the Remainder Is Zero?
If the remainder is zero, then the remaining quotient and the divisor are the factors of the given expression.
What Is the Use of the Factor Theorem?
Factor theorem is used in math to find the factors of a given polynomial equation. Let say hypothetically, if f(x) is a polynomial, putting f(k) = 0, then (x – k) is a factor of f(x).
What Is the Use of Remainder Theorem Formula?
The remainder theorem formula is used to find the remainder when a polynomial p(x) is divided by (ax + b). Using the remainder theorem we can determine whether (ax + b) is a factor of p(x) or not. If the remainder is 0, then (ax + b) is a factor of a polynomial p(x), otherwise, it is not.
What Are the Applications of Remainder Theorem Formula?
The factor theorem is the main application of the remainder theorem formula. To prove the factor theorem, we need the remainder theorem. The factor theorem says if the remainder obtained by dividing p(x) by (x  r) is 0, then (x  r) is a factor of p(x).