Synthetic Division Calculator

Performs synthetic division of a polynomial by the linear factor (x − c). Enter the coefficients (descending order) and the root c to get the quotient and remainder.

Polynomial Coefficients (descending order)

Divisor (c in x − c)

Result

Please check your inputs.

Enter the coefficients of your polynomial in descending order of degree (e.g., for 2x³ - 4x² + 3x - 5, enter '2', '-4', '3', '-5'). Input the root 'c' from the linear factor (x − c). For example, if dividing by (x − 2), enter '2'. Click 'Calculate' to perform synthetic division. Review the result showing the quotient polynomial (coefficients) and the remainder. Optionally copy or export the result for your homework or project.

📖 How to Use This Tool

Enter the coefficients of your polynomial in descending order of degree (e.g., for 2x³ - 4x² + 3x - 5, enter '2', '-4', '3', '-5').

Input the root 'c' from the linear factor (x − c). For example, if dividing by (x − 2), enter '2'.

Click 'Calculate' to perform synthetic division.

Review the result showing the quotient polynomial (coefficients) and the remainder.

Optionally copy or export the result for your homework or project.

📝 What Is Synthetic Division Calculator?

Synthetic division is a fast, shorthand method for dividing a polynomial by a linear factor of the form (x − c). It simplifies the long division process by focusing only on coefficients, making it ideal for quickly finding quotients and remainders. This tool automates that calculation, reducing errors and saving time for students, teachers, and professionals who work with polynomial expressions. Understanding synthetic division is essential for factoring polynomials, solving equations, and evaluating functions using the Remainder Theorem. By using this calculator, you can verify your manual work or instantly get results for complex polynomials, freeing up mental energy for deeper analysis.

🧮 Formula

The tool uses synthetic division: start with the leading coefficient, multiply by c, add to next coefficient, repeat. For polynomial coefficients aₙ, aₙ₋₁, ..., a₀ and divisor (x − c), the process yields quotient coefficients bₙ₋₁, ..., b₀ and remainder R. Mathematically: bₙ₋₁ = aₙ, then for i from n-2 down to 0: bᵢ = aᵢ₊₁ + c × bᵢ₊₁. Final remainder R = a₀ + c × b₀. The quotient polynomial is bₙ₋₁xⁿ⁻¹ + ... + b₀.

💡 Tips for Best Results

✨🔢 Double-check coefficient order — enter them from highest degree down to constant term to avoid wrong results.

✨📝 Remember the sign: for divisor (x − c), c is the root. If dividing by (x + 3), c = -3 — enter negative accordingly.

✨🧮 Use this calculator to quickly check your manual synthetic division, especially for high-degree polynomials or when studying for exams.

✨⚡ If you have a missing term (e.g., no x² term), enter '0' as its coefficient to keep the order correct.

❓ Frequently Asked Questions

What if my polynomial has a missing degree?

Enter '0' for any missing coefficient. For example, for 4x³ + 2x − 7 (missing x² term), enter coefficients: 4, 0, 2, -7. The tool will handle it correctly.

Can I use this calculator for divisors like (2x − 3)?

No, synthetic division only works for linear divisors of the form (x − c). For (2x − 3), you first divide the polynomial by 2, then use c = 3/2, or use polynomial long division instead.

What does the remainder tell me?

The remainder is the value of the polynomial when x = c (Remainder Theorem). If the remainder is 0, then (x − c) is a factor of the polynomial — useful for factoring and finding roots.