When can synthetic division be used?

Standard synthetic division applies when the divisor is a linear monic expression x minus c.

What does the synthetic-division remainder mean?

By the remainder theorem, the remainder after dividing f(x) by x minus c equals f(c).

Polynomial Synthetic Division Calculator

Divide a polynomial by x − c. Enter every coefficient, including zero placeholders, and see the quotient coefficients, remainder and each multiply-and-add step.

Checked by TheToolNet Editorial Team · Last reviewed: September 27, 2026

Remainder theorem

f(x) = (x − c)q(x) + f(c)

Calculate Synthetic Division

How Synthetic Division Works

Bring down the leading coefficient.
Multiply it by c and write the product under the next coefficient.
Add the column.
Repeat the multiply-and-add cycle through the final coefficient.

The last value is the remainder. All earlier values are the coefficients of the quotient, whose degree is one less than the original polynomial.

Worked Example

Divide x³ − 6x² + 11x − 6 by x − 2. Bring down 1. Multiply by 2 and add to −6 to get −4. Multiply −4 by 2 and add to 11 to get 3. Multiply 3 by 2 and add to −6 to get 0. The quotient is x² − 4x + 3 and the remainder is 0.

Limits and Zero Coefficients

This shortcut is for divisors in the form x − c. For 2x − 3 or a quadratic divisor, use polynomial long division. Do not omit missing powers: x³ + 2x − 1 must be entered as 1, 0, 2, −1.

Related Polynomial Tools

Use the polynomial long division calculator for general divisors or the polynomial root finder to test potential zeros.