Math
Synthetic Division Calculator
Divide a polynomial by (x - c) using synthetic division. Get the quotient, remainder, full step table, and verify with the Remainder Theorem.
P(x) divided by (x - c)
Polynomial coefficients (highest power first)
Use 0 for missing terms so the powers line up.
To divide by (x + 3), enter c = -3. To divide by x, enter c = 0.
Quotient and remainder
(2x^3 - 5x^2 + 3x - 7) / (x - 2)
2x^2 - x + 1remainder -5
| Row | c | a3 | a2 | a1 | a0 |
|---|---|---|---|---|---|
| Coefficients | 2 | 2 | -5 | 3 | -7 |
| Carry | 4 | -2 | 2 | ||
| Result | 2 | -1 | 1 | -5 |
The last entry in the result row is the remainder. Earlier entries are the quotient coefficients.
Quotient coefficients
2, -1, 1
P(2) by Remainder Theorem
-5
Frequently Asked Questions about the Synthetic Division Calculator
What is synthetic division?
Synthetic division is a shortcut for dividing a polynomial P(x) by a linear factor of the form (x - c). You write only the coefficients, bring down the first one, multiply by c, add to the next coefficient, and repeat. The last value is the remainder, and the earlier values form the coefficients of the quotient, which has a degree one lower than P(x).
How do I divide by (x + 3) using this calculator?
Synthetic division assumes the divisor has the form (x - c). To divide by (x + 3), rewrite it as (x - (-3)) and enter c = -3 in the divisor field. The algorithm runs the same way, and the quotient and remainder come out correctly.
What do I do about missing terms in the polynomial?
Every power between the highest and the constant term needs a coefficient, even if it is zero. For x^4 - 1, enter the coefficients 1, 0, 0, 0, -1. The zeros keep the columns aligned so the carry values add to the right place.
How does the Remainder Theorem connect to synthetic division?
The Remainder Theorem says that the remainder when P(x) is divided by (x - c) equals P(c). So synthetic division also evaluates the polynomial at x = c for free. If the remainder is 0, then c is a root of P(x) and (x - c) is a factor.
When can I not use synthetic division?
Synthetic division only works when the divisor is linear and of the form (x - c) with a leading coefficient of 1. For higher-degree divisors like (x^2 + 1) or scaled divisors like (2x - 5), use long polynomial division instead, or factor out the leading coefficient first.