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Synthetic Division Calculator with Steps

Divide a polynomial by x − r with synthetic division: see the full tableau, the quotient, the remainder, the remainder theorem check and whether x − r is a factor.

Dividend coefficients

Divide by x − r, where r =

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How it works

Bring down a₀ · then repeat: multiply the last bottom number by r and add it to the next coefficient · bottom row = quotient coefficients + remainder · P(r) = remainder

Synthetic division is a shortcut for dividing a polynomial by a linear factor x − r using only its coefficients. Write the coefficients in a row (with 0 for any missing power), bring the first one down, multiply it by r, add the result to the next coefficient, and repeat. The bottom row holds the coefficients of the quotient — one degree lower than the dividend — and the last number is the remainder. By the remainder theorem that remainder equals P(r), so a remainder of 0 means x − r is a factor and r is a root.

Worked example

Divide x³ − 6x² + 11x − 6 by x − 2 (r = 2). Bring down 1. 1 × 2 = 2, −6 + 2 = −4. −4 × 2 = −8, 11 + (−8) = 3. 3 × 2 = 6, −6 + 6 = 0. The bottom row 1, −4, 3 | 0 gives the quotient x² − 4x + 3 with remainder 0, so x − 2 is a factor and x³ − 6x² + 11x − 6 = (x − 2)(x² − 4x + 3).

Sources: Synthetic division — Wikipedia · Polynomial remainder theorem — Wikipedia

Frequently asked questions

How do I divide by x + 3?

Write the divisor as x − r: x + 3 = x − (−3), so enter r = −3.

Can synthetic division divide by 2x − 1?

Not directly — it only handles a leading coefficient of 1. Divide by x − 1/2 (enter r = 0.5), then divide every coefficient of the quotient by 2. The remainder stays the same.

Why do I need zeros for missing terms?

Each position in the row stands for one power of x. For x⁴ − 5x + 2 enter 1, 0, 0, -5, 2; leaving out the zeros would shift every coefficient to the wrong power.

What does the remainder tell me?

The remainder theorem says the remainder after dividing P(x) by x − r equals P(r). So synthetic division is also a fast way to evaluate a polynomial, and a zero remainder proves r is a root.

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This calculator is for educational purposes. Double-check important results before acting on them.