This synthetic division calculator divides a polynomial by a linear factor (x − c) the fast way — coefficients only — and shows the quotient and remainder, plus whether c is a true root of the polynomial.
Long Division's Smarter Sibling
Synthetic division strips polynomial division to the coefficients: bring down the first, multiply by c, add, repeat — the last number is the remainder, the rest are the quotient's coefficients, one degree lower. It's valid exactly when the divisor is linear (x − c), which covers the most common classroom and factoring work. Part of our Math Calculators collection.
Worked Example (Verified)
- (x³ − 6x² + 11x − 6) ÷ (x − 2) (defaults): row math 1 → (×2 + (−6)) = −4 → (×2 + 11) = 3 → (×2 − 6) = 0 → quotient x² − 4x + 3, remainder 0 — and the quotient factors again to (x−1)(x−3), so the cubic's roots are 1, 2, 3
- Non-zero case: the same cubic at c = 4 gives remainder P(4) = 64 − 96 + 44 − 6 = 6 — checkable by direct substitution, which is the Remainder Theorem doing its job
- Missing terms need zeros: x³ − 8 divided by (x − 2) enters as coefficients 1, 0, 0, −8 → quotient x² + 2x + 4, remainder 0 (the difference-of-cubes factorization, discovered by the algorithm)
- Sign discipline: dividing by (x + 3) means c = −3 — the single most common synthetic-division error in the wild
When It Works, When It Doesn't
- Divisor must be (x − c): for (2x − 1), divide synthetically by c = ½ and then halve the quotient coefficients (the divisor's leading coefficient comes back out at the end).
- Higher-degree divisors need full polynomial long division — synthetic is the linear special case, by design.
- Rational-root hunting: synthetic division tests candidate roots (factors of the constant ÷ factors of the leading coefficient) at arithmetic speed — the standard pre-calculus workflow.
Related Tools
- Quadratic Formula Calculator — solve the quotients you get
- Long Division Calculator — the arithmetic original
- Math Calculators — the full collection
Frequently Asked Questions
How does synthetic division work?
Write the coefficients (zeros for missing terms), bring down the first, then repeatedly: multiply the latest result by c, add to the next coefficient. The final value is the remainder; the preceding values are the quotient's coefficients, one degree lower than the original polynomial.
What does a remainder of 0 mean?
The divisor is an exact factor and c is a root of the polynomial (the Factor Theorem). (x³ − 6x² + 11x − 6) ÷ (x − 2) leaves 0, so x = 2 is a root and (x − 2) a factor — synthetic division is the fastest root test in algebra.
Can synthetic division divide by (x + 3)?
Yes — rewrite as (x − (−3)) and use c = −3. The sign flip is the method's classic pitfall; this calculator's input asks for c directly and reminds you of the rule next to the field.
What if a term is missing, like x³ − 8?
Insert a 0 coefficient for every missing power: x³ − 8 becomes 1, 0, 0, −8. Skipping the placeholders shifts every coefficient's degree and silently wrecks the result — the zeros are structural, not optional.
Synthetic vs long division — which should I use?
Synthetic whenever the divisor is linear (faster, fewer errors); long division for anything else (quadratic divisors, non-monic cases handled wholesale). They are the same algorithm — synthetic is long division with the variables' bookkeeping removed.
Is this synthetic division calculator free?
Yes — free, instant, no sign-up.