Top-Notch Mathematics: The Remainder Theorem

In mathematics, the remainder theorem states that if a polynomial p(x) is divided by x-a, where ‘a’ is some specific value, then the remainder will be p(a). In other words, the theorem explains that one cannot divide a polynomial by just any number and expect to get an integral quotient. The reason for this is that there are constants (or coefficients) in the polynomial which do not become zero when we solve for them.

The remainder theorem can also be applied to other roles than the division of two roles. For example, it can be used with integration in addition: if we integrate f(x)+g(x), where g(x) = 0, then f(x) will be our integral.

The remainder theorem can also be applied to quotients instead of divisions, where we take the dividend and divide it by the divisor: in that case, the remainder is equal to the dividend minus the quotient [Q]. This works because when we divide [D], [0]/n = [Q], and when we add [A], x+y=y+x (since a number cannot change).

When dividing and taking remainders, there are two possible situations: either only one term in p(x) is not zero, or none of them is. In other words: if all coefficients in p(x) become 0 after we solve for them, then they all become 0 after we divide by x-a in addition, meaning that there is no remainder.

If some of the coefficients do not become zero when solving for them, they will also not become zero already after dividing by x-a. The proof of this can be found in the article on polynomials.


  • Remainder theorem can be used with division, integration, and quotients
  • Can only be divided by x-a, not by other values
  • When dividing and taking remainders, if all coefficients become 0 after solving for them, then they also become 0 after division; if some of the coefficients do not become zero when solving for them, then they will not become zero already after division. This is the basic character of the remainder theorem.


  • The remainder theorem can be used to simplify quotients of polynomials, which would otherwise be quite difficult.
  • The remainder theorem can also be applied to integration, with similar results as for division.
  • If one is unsure whether a number with many decimals is integral or not (which is usually the case), then it is always possible to test that with the remainder theorem. This works by taking the number, dividing it by 1, and seeing if there is a remainder – if there is none, then the number must be integral.
  • When finding remainders in the polynomial division, only one term needs to be nonzero for this method to work – so already if one term becomes zero after solving for it, the other terms may nevertheless have a remainder, and the theorem can nevertheless be applied.
  • Remainder theorem can also be used to find the original polynomial if we know that it has been divided by another polynomial in addition as finding what number was divided out. This works because dividing two polynomials produces another polynomial – so if adding unknown continued yields a number that is not zero, then it must have been divided out from both the dividend and the divisor.

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The remainder theorem tells us that we cannot divide by just any number and expect an integral quotient. We can only divide by x-a, where a is some specific value.

In other words: when dividing two polynomials, it is always possible to find what number was divided out. The remainder theorem can be used for polynomial division in addition as finding remainders in quotients of polynomials.

This remainder theorem becomes very interesting while one learns this concept from Cuemath classes. It is an excellent online platform that is being used by many students and teachers to clarify their doubts related to Mathematics and coding.

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