Enter polynomial coefficients and a divisor, then click "Divide".
Example: 1,3,-4,5 with divisor 2 → x² + 5x + 6, remainder 17
Use this synthetic division calculator to quickly divide a polynomial by a linear binomial of the form (x – c). Enter the coefficients (highest degree first) and the value of c, then click "Divide" for an instant quotient and remainder, along with a detailed step‑by‑step explanation.
What is Synthetic Division?
Synthetic division is a streamlined method for dividing a polynomial by a linear factor. It uses only the coefficients, making it much faster than polynomial long division. The divisor must be of the form x – c. The algorithm brings down the first coefficient, repeatedly multiplies by c and adds to the next coefficient, producing the quotient and remainder.
Examples
x³ + 3x² - 4x + 5 ÷ (x – 2)
Coefficients: 1, 3, -4, 5
2x² - 3x + 1 ÷ (x – 1)
Coefficients: 2, -3, 1
x⁴ - 2x³ + x - 7 ÷ (x – 3)
Coefficients: 1, -2, 0, 1, -7
How to Enter Data
- List coefficients from the highest power down to the constant term.
- Use commas to separate them (e.g.,
1, -3, 0, 5 for x³ – 3x² + 5). - Include zeros for missing degrees.
- The divisor is the c in (x – c). For (x + 2), enter -2.
Understanding the AlgorithmFor a polynomial aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₀ divided by (x – c):
1. Bring down aₙ (first quotient coefficient).
2. Multiply by c, add to next coefficient → next quotient coefficient.
3. Repeat until the constant term. The last sum is the remainder.
The quotient has degree one less than the original polynomial. If the remainder is zero, (x – c) is a factor.
Why Use Synthetic Division?
Synthetic division is essential for evaluating polynomials (using the Remainder Theorem), factoring higher‑degree polynomials, and simplifying rational functions. It reduces arithmetic errors and saves time, especially on exams and homework. The calculator shows each intermediate step, reinforcing the method.
The Remainder Theorem states: when a polynomial P(x) is divided by (x – c), the remainder equals P(c). Our calculator verifies this — the remainder you see is the polynomial evaluated at c.
For factoring, if the remainder is zero, then c is a root. You can then continue factoring the quotient. This tool helps you quickly test potential rational roots.