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Remainder Theorem Calculator

Remainder Theorem Calculator

Enter the Numerator Polynomial:

Enter the Denominator Polynomial:

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An online remainder theorem calculator allows you to determine the remainder of given polynomial expressions by remainder theorem. The factor theorem calculator provides step-wise calculations of the factor of division. Here you can understand how to find the remainder of a polynomial using the formula.

What is the Remainder Theorem?

In algebra, the remainder theorem or little Bezout’s theorem is an application of Euclidean division of different expressions, which is discovered by Etienne Bezout. It states when an expression is divided by a factor x-j, then the remainder of the division is equal to f(j).

The polynomial remainder theorem formula that is:

Dividend = (Quotient * Divisor) + Remainder

If f(x) is a dividend, (x-j) is divisor, m(x) is a remainder, and a(x) is a quotient then remainder theorem formula which is used by remainder theorem calculator can be written as:

$$f(x) = (x-j) * a(x) + m(x)$$

However, an Online Quotient & Remainder Calculator helps you to divide two numbers, a divided and a divisor to find the quotient with a remainder.

How to Calculate Remainder with Remainder Theorem?

When the polynomial f(x) is divisible by a linear factor of the form x-j, the theorem will be used by the remainder theorem calculator. If you want to do these calculations by hand, then follow the instructions below and use them to solve the rest of the polynomial expression in a couple of minutes.

  • The polynomial f(x) is used as the dividend, and the linear expression is used as the divisor.
  • The form of the linear expression must be x-j.
  • Then, the remaining value of the polynomial becomes m(x).
  • Therefore, insert the value of c into the polynomial and evaluate it to obtain the remainder value.


Solve (x^4 + 12x^3 + 18x^2 – 9x + 22) with denominator (x – 4) using remainder theorem?


Given values are

$$f(x) = x^4 + 12x^3 + 18x^2 – 9x + 22$$

x – 4 is in the form of x – (4).

Then c = 4

$$f(4) = (4)^4 + 12(4)^3 + 18(4)^2 – 9(4) + 22$$

$$= 256 + 768 + 288 – 36 + 22$$

$$= 1298$$

The remainder of given expression is 1298.

Moreover, you can use online Factoring Calculatorallows you to factor any expression (polynomial, binomial, trinomial).

How Remainder Theorem Calculator Works?

This factor theorem calculator helps you to determines the remainder of the given polynomial factors by following these instructions:


  • First, enter the numerator polynomial.
  • Then, substitute the denominator polynomial.
  • Hit the “Calculate” button to see the remainder of the given expression.


The remainder calculator calculates:

  • The remainder theorem calculator displays standard input and the outcomes.
  • It provides all steps of the remainder theorem and substitutes the denominator polynomial in the given expression.
  • You can find the remainder many times by clicking on the “Recalculate” button.


Why some people use the Chinese remainder theorem?

The Chinese remainder theorem is usually used for large integer calculations because it allows you to replace a calculation for which you know the size limit of the result with several similar small integer calculations.

What is Euclidean Division?

In arithmetic, Euclidean division is the process of dividing one whole number (the dividend) by another (divisor) so that the quotient and remainder are less than the divisor. The remainder exists and is unique under certain conditions. Because of this uniqueness, Euclidean division is usually considered without involving the calculation method, and the quotient and remainder are not explicitly calculated.


Use this online remainder theorem calculator which is closely related to Bezout’s identity and the Euclidean algorithm, which are compulsory for proving the remainder theorem calculator. The remainder estimation theorem calculator is useful for CK 12 students, mathematicians, and teachers, also it increases the remainder theorem which related to mathematical skills by providing step-wise calculations quickly without any cost.


From the source of Wikipedia: Polynomial remainder theorem, little Bézout’s theorem, factor theorem.

From the source of Purple Math: The Remainder Theorem, Synthetic division, Division Algorithm for Polynomials.

From the source of Varsity Tutors: Remainder Theorem, Polynomial function, Long Division, Synthetic Division.