How to use the synthetic division calculator
Synthetic division is a shortcut that works only when the divisor is linear, x − a. You write down the coefficients, then repeatedly multiply by a and add. The quotient coefficients and the remainder appear in one row. Enter the polynomial and a, and this tool prints the whole table: the coefficient row, the carried-product row and the sum row.
The last sum is the remainder and the values before it are the coefficients of the quotient. Dividing x³ − 2x² + 3x − 4 by x − 2, for example, gives the sums 1, 0, 3, 2, so the quotient is x² + 3 and the remainder is 2. The result also prints the identity f(x) = Q(x)(x − a) + R so you can multiply back and confirm.
By the remainder theorem, the remainder of f(x) divided by x − a equals f(a). The calculator evaluates f(a) separately and compares it with the remainder from the table. The two count as equal when they differ by less than 1e-9, because decimal coefficients introduce floating point error and an exact equality test would be unreliable.
To divide by x + 1, read it as x − (−1) and enter −1 for a. A linear divisor whose leading coefficient is not 1, such as 2x − 1, cannot be used directly, so it is not supported, and neither is long division by a quadratic or higher divisor. Coefficients may be decimals and results print to 6 places.
Frequently asked questions
x + 3 is x − (−3), so enter −3. Forgetting to flip the sign is the most common mistake in synthetic division.
No. Synthetic division as shown here applies only to a monic linear divisor of the form x − a.
It means f(a) = 0, so x − a is a factor of f(x). By the factor theorem, a is a root of f(x) = 0.