Remainder Theorem Calculator

Apply the remainder theorem step by step

The calculator will calculate $$$f(a)$$$ using the remainder (little Bézout's) theorem, with steps shown.

Enter a polynomial:

Enter the point `a`:

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

Finding the value of a polynomial at a specific point can be daunting, but our Remainder Theorem Calculator makes it much more manageable. This sophisticated online tool saves your time and delivers accurate results efficiently.

How to Use the Remainder Theorem Calculator?

  • Input

    Start by entering the polynomial in the given field. Make sure to enter it correctly, following the correct format for polynomials. Specify the value of $$$a$$$ you want to substitute in the polynomial.

  • Calculation

    Once you have inputted the required information, click the "Calculate" button.

  • Result

    The calculator will quickly perform the computations based on the Remainder Theorem and display the value of the polynomial in the specified point.

Understanding the Remainder Theorem

The Remainder Theorem is a foundational concept in algebra that provides a method for finding the remainder of a polynomial division. In more precise terms, the theorem declares that if a polynomial $$$f(x)$$$ is divided by a linear divisor of the form $$$x-a$$$, the remainder is equal to the value of the polynomial at $$$a$$$, or expressed differently, $$$f(a)$$$.

Let's make this clearer with a mathematical representation: when $$$f(x)$$$ is divided by $$$x-a$$$, the remainder is given by $$$f(a)$$$.

Suppose we take a polynomial such as $$$f(x)=x^3-2x^2+3x-1$$$, and we want to figure out the remainder when divided by $$$x-2$$$. According to the Remainder Theorem, we don't need to perform the division. Instead, we just substitute the value of $$$x=2$$$ into the polynomial and calculate the result.

So, the remainder is $$$f(2)=2^3-2\cdot2^2+3\cdot2-1=8-8+6-1=5$$$.

This theorem can help avoid using polynomial division, especially when dealing with high-degree polynomials. It is a handy tool in algebraic computation.

What Are the Limitations of the Remainder Theorem?

While the Remainder Theorem is a powerful and handy tool in algebra, it does have a few limitations:

  • Applicable to Linear Divisors Only: The Remainder Theorem only works when dividing by a linear divisor of the form $$$x-a$$$. It is not applicable when dividing by a polynomial of a higher degree.
  • Works for Polynomial Functions: The theorem only applies to polynomial functions. It can't be used to calculate the remainder for non-polynomial expressions.
  • Not Useful for Complete Division Process: If you need to know the full quotient in addition to the remainder, the Remainder Theorem won't be enough. It only gives you the remainder of the division, not the complete result.
  • Not Suitable for Complex Calculations: The Remainder Theorem simplifies calculations with real numbers. It might not simplify complex number calculations. When $$$a$$$ is a complex number, substitution can make the calculation more complicated, not less.

So while the Remainder Theorem is a valuable tool for a specific set of problems in algebra, it's essential to recognize its limitations and know when to use other methods.

Why Choose Our Remainder Theorem Calculator?

  • Accuracy

    Our calculator eliminates the risk of human error, providing accurate results every time.

  • Efficiency

    Don't get bogged down by tedious calculations. Our calculator quickly computes the remainder for you, saving you precious time.

  • User-Friendly Interface

    The tool is designed for easy navigation. You only need to input the polynomial and the point, and the calculator will handle the rest.

  • Step-by-Step Solution

    The calculator gives you the answer and provides a detailed, step-by-step solution. This can help you understand the process and learn how the Remainder Theorem works.


Can you use the Remainder Theorem if the remainder is zero?

Yes, you can. If the remainder is zero, it implies that the divisor $$$x-a$$$ is a factor of the polynomial. This is the basis of the Factor Theorem.

Are the Factor Theorem and the Remainder Theorem the same?

While the Factor and Remainder Theorems are related, they are different. The Factor Theorem is a particular case of the Remainder Theorem. If a polynomial $$$f(x)$$$ is divided by $$$x-a$$$ and the remainder is zero, then $$$x-a$$$ is a factor of the polynomial.

Does the Remainder Theorem Calculator provide step-by-step solutions?

Absolutely! Our calculator gives the final remainder and shows each step of the calculation process, making it a great learning tool.

What are the limitations of the Remainder Theorem?

The Remainder Theorem only works when dividing a polynomial by a linear divisor of the form $$$x-a$$$. It cannot be used if the divisor is a higher-degree polynomial or a non-polynomial expression.