Synthetic Division Calculator

Synthetic division is the fast, shorthand way to divide a polynomial by a linear binomial x − a — no long division, no writing out every term twice. This calculator runs the bring-down / multiply / add steps for you and shows the full table alongside the quotient, the remainder, and whether x − a is a factor of the polynomial.

How to use it

1. Enter the polynomial's coefficients, highest degree first, separated by commas or spaces — for x³ + 2x² − x − 2, enter 1, 2, -1, -2.
2. Enter a, the root of the divisor x − a (use a negative number for a divisor like x + 3, i.e. a = −3).
3. Read off the quotient, the remainder, and the step-by-step table.

The method

For coefficients c₀, c₁, …, cₙ (highest degree first) and root a:

• r₀ = c₀
• rᵢ = cᵢ + rᵢ₋₁ × a, for i = 1 to n

r₀ through rₙ₋₁ become the quotient's coefficients (one degree lower than the dividend); rₙ is the remainder.

FAQ

When can I use synthetic division instead of long division?

Only when dividing by a linear binomial of the form x − a (coefficient of x is exactly 1). For any other divisor — a quadratic, or x with a non-1 leading coefficient — use polynomial long division instead.

What does it mean if the remainder is 0?

By the factor theorem, a remainder of 0 means x − a is a factor of the polynomial and a is a root — so if you're factoring or solving p(x) = 0, a zero remainder tells you a is one of the solutions.

Related tools

To solve directly for a quadratic's roots instead of dividing it down first, see the Quadratic Formula Solver.

Privacy

All calculations run entirely in your browser — the coefficients you enter are never sent to a server.