Last updated October 7, 2026 · Report an error
Synthetic division is a shortcut for dividing a polynomial by a linear factor (x − c) that skips the tedious full long-division setup, using only the polynomial's coefficients. This calculator performs synthetic division on any polynomial and divisor root, showing the complete division table alongside the quotient and remainder.
How it works
Enter the polynomial's coefficients in descending order (comma-separated) and the divisor root, c (for dividing by x − c). The calculator brings down the first coefficient, then repeatedly multiplies by c and adds to the next coefficient across the whole row.
- Enter coefficients (comma-separated, descending order).
- Enter divisor root (dividing by x − this value).
- Click Calculate to see your results.
Examples
Dividing x⁴ − 10x³ + 35x² − 50x + 24 by (x − 1)
This divides evenly (remainder 0) into x³ − 9x² + 26x − 24, confirming that 1 is a root of the original polynomial.
Who should use it
- Testing candidate roots of a polynomial found via the rational root theorem.
- Factoring a polynomial once one root is already known.
Industry applications
- Algebra and precalculus education
- Engineering and applied mathematics coursework
Advantages
- Shows the complete division table, not just the final quotient and remainder.
- Much faster than full polynomial long division for dividing by a linear factor.
Limitations
- Only works for dividing by a linear factor (x − c), not by higher-degree divisors.
Common mistakes to avoid
- Forgetting to include a 0 coefficient for a missing term, which misaligns every subsequent step.
- Using the wrong sign for the divisor root — dividing by (x − c) uses c itself, not −c, as the synthetic division value.
Best practices
- Use a nonzero remainder as a signal to try a different candidate root (often from the rational root theorem) rather than assuming an error in the division.
Tips
- Pair this with the Polynomial root-finding process: test candidate rational roots with synthetic division until you find one with a zero remainder, then repeat on the resulting quotient to factor further.