Synthetic Division Calculator
Divide a polynomial by (x − c) using only its coefficients.
Remainder: 0. Since the remainder is 0, (x − 3) divides evenly, so it is a factor and x = 3 is a root.
| 3 | 1 | -2 | -4 | 3 |
| 3 | 3 | -3 | ||
| 1 | 1 | -1 | 0 |
Top row: coefficients. Middle row: each result × 3. Bottom row: the sums — the last one (in colour) is the remainder, the rest are the quotient.
Synthetic division divides a polynomial by a linear factor (x − c) using just its coefficients. Dividing x³ − 2x² − 4x + 3 by (x − 3) gives a quotient of x² + x − 1 with a remainder of 0. A remainder of 0 means (x − 3) is a factor and x = 3 is a root.
What synthetic division does
Synthetic division is a shorthand for dividing a polynomial by a linear divisor of the form (x − c). Instead of writing out full long division, you work with only the coefficients: bring the first one down, then repeatedly multiply by c and add. The final number is the remainder, and the numbers before it are the coefficients of the quotient — a polynomial one degree lower than the one you started with.
Worked example
Divide x³ − 2x² − 4x + 3 by (x − 3), so c = 3:
- 1 Write the coefficients in order. For x³ − 2x² − 4x + 3, list 1, −2, −4, 3 from the highest power down. Insert a 0 for any missing term. Put c = 3 to the left.
- 2 Bring down the first coefficient. Copy the leading 1 straight down to start the bottom row.
- 3 Multiply by c, then add. 1 × 3 = 3; add to −2 to get 1. Then 1 × 3 = 3; add to −4 to get −1. Then −1 × 3 = −3; add to 3 to get 0.
- 4 Read off the answer. The bottom row is 1, 1, −1, 0. The last number, 0, is the remainder; the rest are the quotient x² + x − 1.
The synthetic-division process
Work left to right across the coefficients. The single number left over at the end is the remainder.
| Position | Operation | Meaning |
|---|---|---|
| First coefficient | Bring it straight down | Starts the quotient |
| Each next column | Multiply the last result by c | Gives the number to add |
| Same column | Add it to the coefficient above | The next quotient coefficient |
| Repeat | Carry down to the last column | One column per coefficient |
| Last number | Nothing follows it | This is the remainder = P(c) |
Reading the result
It only works for a linear divisor. Synthetic division is a shortcut specifically for dividing by (x − c). To divide by something like (x² + 1) or (2x − 3) written as a general polynomial, use polynomial long division instead — although 2x − 3 can be handled by dividing by (x − 3/2) and scaling.
The remainder equals P(c). By the Remainder Theorem, dividing P(x) by (x − c) leaves a remainder of exactly P(c). In the example, P(3) = 27 − 18 − 12 + 3 = 0, which matches the remainder from the table.
A remainder of 0 means a factor. When the remainder is 0, (x − c) divides P(x) evenly, so (x − c) is a factor and x = c is a root. That is why synthetic division is a fast way to test candidate roots and to peel factors off a polynomial one at a time.