Synthetic Division Calculator
Divide a polynomial by a linear factor (x - c) using synthetic division and read off the quotient coefficients and remainder.
Equals P(c) by the Remainder Theorem
- 1
Bring down leading coefficient
1 - 2
Iteration 1
-6 + 1 × 1 = -5 - 3
Iteration 2
11 + 1 × -5 = 6 - 4
Remainder
-6 + 1 × 6 = 0By the Remainder Theorem this equals P(c).
How does this calculator work?
Synthetic division divides a polynomial by (x - c). Bring down the leading coefficient, then repeatedly multiply by c and add the next coefficient. The last number is the remainder (equal to P(c)); the rest are the quotient coefficients. A zero remainder means c is a root.
Formula
How this is calculated
Enter the dividend coefficients highest-power first, including zeros for any missing terms (e.g. for x^3 - 6x^2 + 11x - 6 enter 1, -6, 11, -6). The divisor must be linear of the form (x - c); enter the root c as a number. To divide by (x + 2), set c = -2 because x + 2 = x - (-2).
Synthetic division works by bringing down the leading coefficient, then repeatedly multiplying the running result by c and adding the next coefficient: value = a_k + c × previous. The final value produced is the remainder, and the values before it are the coefficients of the quotient, which has degree one less than the dividend.
By the Remainder Theorem the remainder equals P(c), the polynomial evaluated at c. A remainder of zero means (x - c) divides evenly and c is a root of the polynomial. This method only applies to linear divisors of leading coefficient 1; for other divisors use long division. At least two coefficients are required.
Frequently asked questions
Rewrite (x + 3) as (x - (-3)) and use c = -3. The root is the value that makes the divisor zero.
Each coefficient slot represents a power of x. Skipping a missing term (like a missing x^2) would shift the alignment and give wrong results, so enter 0 in its place.
It means (x - c) divides the polynomial exactly, so c is a root and the quotient is a clean factor of the original polynomial.
Also known as
TG we-Calculate Editorial Team. (2026). Synthetic Division Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/synthetic-division-calculator
TG we-Calculate Editorial Team. "Synthetic Division Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/synthetic-division-calculator.
TG we-Calculate Editorial Team, "Synthetic Division Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/synthetic-division-calculator
@misc{wecalculate_synthetic_division_calculator, title = {Synthetic Division Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/synthetic-division-calculator}}, year = {2026}, note = {TG we-Calculate} }
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