About the Synthetic Division Calculator
Synthetic Division Calculator divides a polynomial by a linear factor (x − r) using the quick synthetic division method. Type the coefficients from the highest power to the constant, write 0 for any missing power, and enter r. The page draws the division table, gives the quotient and the remainder, and tells you whether (x − r) is a factor. Because the remainder equals the value of the polynomial at r, it also works as a quick way to evaluate a polynomial and to find roots. It shows the polynomial in normal notation as well.
Updated October 2026
How to use the Synthetic Division Calculator
- 1Type the coefficients from highest power to lowest, separated by commas.
- 2Enter 0 for any power that is missing.
- 3Enter r for the divisor (x − r); use a negative r for (x + 2).
- 4Read the quotient, the remainder and the table.
- 5Copy the answer.
How synthetic division works
Write r on the left and the coefficients along the top. Bring the first coefficient down. Multiply it by r, write the result under the next coefficient and add. Repeat across the row. The numbers in the bottom row are the coefficients of the quotient, one degree lower than the original, and the last number is the remainder.
Worked example
Divide x³ − 6x² + 11x − 6 by (x − 1). The coefficients are 1, −6, 11, −6 and r = 1. Bring down 1; 1×1 = 1, −6 + 1 = −5; −5×1 = −5, 11 − 5 = 6; 6×1 = 6, −6 + 6 = 0. The quotient is x² − 5x + 6 and the remainder is 0, so x − 1 is a factor, and the quotient factors as (x − 2)(x − 3).
- Bring down the first coefficient
- Multiply by r and add to the next coefficient
- The last number is the remainder
When to use something else
Synthetic division works only for linear divisors of the form x − r. For divisors of higher degree, such as x² + 1, use polynomial long division. For finding roots numerically use a polynomial solver.
Related tools
Multiply brackets with the FOIL Calculator, solve quadratics with the Quadratic Equation Solver and cubics with the Cubic Equation Solver. The Completing the Square Calculator rewrites a quadratic.
Features
- Division table with each step
- Quotient and remainder
- Factor test (remainder 0)
- Evaluates the polynomial at r
- Up to degree 14
- Private: nothing is uploaded
Frequently asked questions
What is synthetic division?
A short way to divide a polynomial by (x − r) that uses only the coefficients. You bring down the first one, multiply by r, add to the next coefficient and repeat.
How do I divide by (x + 2)?
Write (x + 2) as (x − (−2)), so r = −2.
What does the remainder tell me?
By the Remainder Theorem, it equals the polynomial's value at x = r. If it is 0, then (x − r) is a factor and r is a root.
What if a power is missing?
Enter 0 for its coefficient. For x³ − 2x + 5 type 1, 0, −2, 5.
Can I divide by 2x − 4?
Synthetic division needs a divisor of the form x − r. Divide the polynomial and the divisor by 2 first, or use the root r = 2 and then divide the quotient by 2.
How do I use it to find roots?
Try candidate roots from the rational root test. Each time the remainder is 0, you have found a root and a smaller polynomial to continue with.
How many coefficients can I enter?
Up to 15, which is a polynomial of degree 14.
Is my data uploaded?
No. The division is done in your browser.