What Is the Remainder Theorem? – A Brief Concept

In mathematics, the remainder is the value that is left behind after a certain number is divided by another number. Usually, the division is a procedure that is carried out to determine the possible parts in which a certain number can equally be disbursed. And when it comes to polynomial division including higher terms, they are divided by using various mathematical techniques. One such technique is the remainder theorem which is used by the remainder theorem calculator to determine fine and accurate results. 

Let us learn what is the remainder theorem and how it assists to determine a polynomial remainder!

Who Introduced the Remainder Theorem?

This particular idea of mathematical technique was proposed by the Chinese mathematician Sun Zi. But later on in the year 1247, the proper proof of the theorem was given by Qin Jiushao. And keeping in view the main idea as proposed by him, the calculator online has a programmed best remainder factor theorem calculator. 

What Is The Remainder Theorem?


“When you divide a polynomial p(x) by any other polynomial that is linear {(x-a)}, you will get a remainder as p(a)”

Digging deeper!

You must keep in mind that the degree of the dividend polynomial must be equal to 1 or greater than it. Moreover, the linear polynomial q(x) has zero at x=a and the remainder value is represented as r=p(a). The main advantage of the remainder theorem is that you are not bound to follow the long and complicated procedure of polynomial long division. Also for instance, you may better subject to the best online remainder theorem calculator that will do exactly the same procedure but in seconds.

Remainder Theorem Formula:

If p(x) is divided by a linear factor (x-a), 

Remainder = p(a)

If p(x) is divided by a linear factor (ax+b),

Remainder = p(-ba)

The online remainder theorem calculator also utilizes the same formula to compute the results.

Remainder Theorem Proof:

Well let us suppose that we have a quotient and remainder as follows:

  • Quotient = q(x)
  • Remainder = r

According to the standard division procedure, we have:

Dividend = (Divisor × Quotient) + Remainder

p(x) = (x – a) · q(x) + r

Putting the value of x = a in the above expression:

p(a) = (a – a) · q(a) + r

p(a) = (0) · q(a) + r 

p(a) = r

Which is the required remainder that can also be determined by using the remainder theorem calculator.


Now we will resolve an example that will help you in getting a firm grip on the concept. 

Let’s go!


Determine the remainder when x^3+4x+2 is divided by x-2.


First we have:

x-2=0 => x=2

Putting it in the given dividend polynomial:


= (2)^3+4(2)+2

= 8+8+2

= 18

Why Use Remainder Theorem Calculator?

The reason is quite straightforward! Who does not want to get immediate results and so do you? A special thanks to calculator-online.net which has developed this smart polynomial remainder calculator. The tool will take in a couple of polynomials (dividend and divisor) and will let you know the exact remainders you are looking for!

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Vikramjit Singh, founder of Newscoop, an esteemed UPSC educational website, helps aspirants prepare for UPSC CSE with 5 years of experience, offering a deep understanding of crucial concepts.

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