Synthetic Division Calculator

Perform synthetic division step by step

The calculator will divide the polynomial by the binomial using synthetic division, with steps shown.

Related calculator: Polynomial Long Division Calculator

Divide (dividend):

By (divisor):

A binomial (of the form `ax+b`).

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.

The Synthetic Division Calculator is an intuitive and efficient online tool designed for dividing a polynomial by a binomial. Whether you're a student trying to grasp the synthetic division method or a professional seeking quick solutions, our calculator is exactly what you need.

How to Use the Synthetic Division Calculator?

  • Input

    In the specified input field, enter the polynomial you wish to divide. Ensure you've correctly typed the coefficients and degrees. Enter the divisor into the other input field.

  • Calculation

    Once you've entered all the data, click the "Calculate" button. The tool will quickly process the input and display the result.

  • Result

    The output will present the quotient and the remainder.

What Is Synthetic Division of Polynomials?

Synthetic division is a shorthand method for dividing polynomials that simplifies the process significantly but can be used only when dividing by a linear factor. It is an alternative to the long division method, allowing for faster calculations while giving the same result.

How to Do Synthetic Division?

This method requires fewer steps compared to the traditional long division approach. The only drawback is that the divisor should be a linear binomial.

The steps of the method are the following:

  1. Set Up the Coefficients:

    • List the coefficients of the polynomial you wish to divide in descending order of their respective powers.
    • If any terms are missing, represent them using a zero coefficient.
  2. Write Down the Divisor:

    • Remember that the divisor should be in the form of $$$x-c$$$, where $$$c$$$ is a constant.
  3. Begin the Division:

    • Drop down the leading coefficient of the polynomial; this starts your division.
    • Multiply this coefficient by the constant term of the divisor with the opposite sign.
    • Write this product under the next coefficient and add them.
    • Continue multiplying the constant term of the divisor with the opposite sign by the obtained sum and add the result to the next coefficient in the line.
  4. Continue Till the End:

    • Keep doing this process, moving left to right, until you've accounted for all the coefficients.
  5. Determine the Quotient and Remainder:

    • The numbers you generate in the final row (except the last one) represent the coefficients of the quotient.
    • The very last number in this sequence is the remainder. If the remainder is zero, the divisor is a factor of the polynomial.

For example, suppose you have the polynomial $$$p(x)=x^3-4x^2+5x-2$$$ and want to divide it by $$$x-2$$$.

Using synthetic division, you'll eventually determine that the quotient is $$$x^2-2x+1$$$ and the remainder is $$$0$$$, indicating $$$x-2$$$ is a factor of $$$x^3-4x^2+5x-2$$$.

Practicing synthetic division on various examples will enhance your understanding and speed. Over time, this method becomes intuitive, allowing you to perform division quickly.

Why Choose Our Synthetic Division Calculator?

  • Accuracy & Precision

    Our calculator has been designed to provide accurate results every time, eliminating potential human errors that can occur with manual calculations.

  • User-Friendly Interface

    With its intuitive design, users of all skill levels can easily navigate and use our calculator.

  • Speed

    Our calculator delivers immediate results, saving you valuable time and effort.

  • Step-by-Step Solution

    Our tool returns the quotient and remainder and provides a detailed breakdown of each step, enhancing understanding and making it a valuable learning tool.


What is the use of synthetic division?

The synthetic division method is used for dividing polynomials by linear divisors. Additionally, it's a helpful tool for evaluating polynomials at a specific point, determining factors, and identifying their zeros. Its efficiency and simplicity make it a perfect choice for dividing a polynomial by a binomial.

Why is synthetic division important?

Synthetic division offers a quicker and simpler approach to dividing a polynomial by a first-degree binomial. It's also invaluable for evaluating polynomials and determining factors or zeros. Its efficiency can simplify the process of polynomial factoring and root-finding.

Can you always use the synthetic division method?

While the synthetic division method is highly efficient, it's used only for dividing a polynomial by a first-degree binomial, like $$$x-c$$$. For higher-degree divisors, only traditional polynomial long division can be used.

What are the types of polynomial division?

Polynomial division can be categorized mainly into two types:

  • Long Division: This method is similar to arithmetic long division and also works with polynomials. It's versatile and can handle any polynomial division.
  • Synthetic Division: This is a shorthand method suitable for dividing by a first-degree binomial.