What is synthetic division?

Synthetic division is a shorthand method for dividing a polynomial by a linear binomial (x - r). It uses only the coefficients of the polynomial, making it faster than traditional long division.

How do I enter coefficients for missing terms?

Include a 0 for any missing degree. For example, x⁴ - 1 should be entered as "1 0 0 0 -1" to account for the x³, x², and x terms that have zero coefficients.

What does the remainder mean?

By the Remainder Theorem, the remainder equals the polynomial evaluated at x = r. If the remainder is 0, then (x - r) is a factor of the polynomial and r is a root.

What format does the divisor use?

Enter only the root r of the linear factor (x - r). To divide by (x + 3), enter -3 as the root, because x + 3 = x - (-3).

Can this handle decimal or negative coefficients?

Yes. You can enter any real number for both the coefficients and the root, including decimals and negatives.

What degree of polynomial can I use?

You can enter polynomials of any degree. Just list all coefficients from the highest degree down to the constant term, separated by spaces.

Synthetic Division Calculator

Name: Synthetic Division Calculator
Availability: InStock
Author: Nham Vu

Enter a polynomial and a divisor root to perform synthetic division with full step-by-step work shown.

Enter Polynomial Data

Dividend Coefficients

Space-separated, highest degree first. Use 0 for missing terms.

Divisor Root (r)

Divide by (x − r). To divide by (x + 3), enter −3.

Quick Examples

Enter coefficients and a root, then click Calculate.

Summary

Enter a polynomial and a divisor root to perform synthetic division with full step-by-step work shown.

How it works

Enter the coefficients of your dividend polynomial, separated by spaces (e.g. "1 -6 11 -6" for x³ - 6x² + 11x - 6).
Enter the root value r of the divisor (x - r). For example, enter "2" to divide by (x - 2).
Click "Calculate" to run synthetic division.
The tool displays the full step-by-step table with each multiply-then-add operation highlighted.
The bottom row gives the coefficients of the quotient polynomial plus the final remainder.
Use the remainder to check factor status: if remainder = 0, the divisor is a factor.

Use cases

Divide polynomials quickly without long polynomial division.
Find quotient and remainder when dividing by a linear factor.
Apply the Remainder Theorem to evaluate a polynomial at a given value.
Factor polynomials by testing potential rational roots.
Check homework answers for polynomial division problems.
Simplify rational expressions in algebra or pre-calculus.
Understand step-by-step synthetic division for exam prep.
Verify that a given value is a root of a polynomial.

Frequently Asked Questions

Last updated: 2026-06-10 · Reviewed by Nham Vu