About the Synthetic Division Calculator
This synthetic division calculator divides a polynomial by a linear factor of the form (x − c) and shows the full working. Type the polynomial as an expression such as 2x^3 − 6x^2 + 2x − 1, or just list its coefficients from the highest power down (2, −6, 2, −1), enter c, and you get the quotient polynomial, the remainder and the complete synthetic division tableau.
It is built for algebra and precalculus students learning polynomial division, for checking homework, and for testing possible roots: by the remainder theorem the remainder equals P(c), so a remainder of zero means (x − c) is a factor and c is a root. That makes it a quick companion to the rational root test when factoring cubics and quartics.
Remember to include zeros for missing powers when you enter coefficients (x³ − 8 is 1, 0, 0, −8); when you type an expression, missing powers are filled in automatically. To divide by (x + 3), enter c = −3.
How to use the synthetic division calculator
- 1Type the polynomial, e.g. x^3 - 7x + 6, or list its coefficients with zeros for missing powers.
- 2Enter c from the divisor (x − c); for (x + 2) enter −2.
- 3Read the quotient and remainder.
- 4Follow the tableau row by row to see each multiply-and-add step.
- 5If the remainder is 0, factor the polynomial as (x − c) × quotient.
Formula and method
Synthetic division is a shortcut for dividing a polynomial by (x − c). Write the coefficients a₀ … aₙ in a row. Bring the first coefficient down; then repeatedly multiply the last number in the bottom row by c, write it under the next coefficient, and add. The bottom row gives the coefficients of the quotient, which has degree one less than the dividend, and the last number is the remainder.
The remainder theorem says that the remainder equals P(c), the value of the polynomial at x = c. So synthetic division doubles as a fast way to evaluate a polynomial and to test candidate roots: if the remainder is zero, (x − c) is a factor and P(x) = (x − c) × quotient.
- aᵢ
- Coefficients of the dividend, highest power first
- bᵢ
- Bottom-row values (quotient coefficients, then remainder)
- c
- Number in the divisor (x − c)
Worked examples
(2x³ − 6x² + 2x − 1) ÷ (x − 3)
Bring down 2. 2 × 3 = 6, −6 + 6 = 0. 0 × 3 = 0, 2 + 0 = 2. 2 × 3 = 6, −1 + 6 = 5. The quotient is 2x² + 0x + 2 = 2x² + 2 with remainder 5, and indeed P(3) = 54 − 54 + 6 − 1 = 5.
Testing the root x = 2 of x³ − 7x + 6
The coefficients are 1, 0, −7, 6 (the missing x² term is 0). The bottom row is 1, 2, −3, 0, so the remainder is 0: x − 2 is a factor and x³ − 7x + 6 = (x − 2)(x² + 2x − 3) = (x − 2)(x + 3)(x − 1).
(x³ − 8) ÷ (x + 1)
Dividing by x + 1 means c = −1. The bottom row is 1, −1, 1, −9, so the quotient is x² − x + 1 and the remainder is −9, matching P(−1) = −1 − 8 = −9.
Frequently asked questions
How do you do synthetic division?+
Write the coefficients in order, bring down the first one, multiply it by c and add to the next coefficient, and repeat to the end. The bottom row gives the quotient coefficients, and the final number is the remainder.
What do I do about missing terms?+
Every power from the highest down to x⁰ needs a coefficient, so insert 0 for any missing power. For x³ − 8 use 1, 0, 0, −8. If you type the polynomial as an expression, this calculator fills the zeros in automatically.
Can synthetic division divide by (x + c) or (2x − 1)?+
For x + c use −c, because x + c = x − (−c). For a divisor such as 2x − 1, divide by x − 1/2 (c = 0.5) and then divide every quotient coefficient by 2; the remainder stays the same.
What does a remainder of zero mean?+
A zero remainder means (x − c) divides the polynomial exactly, so c is a root (zero) of the polynomial and it factors as (x − c) × quotient. This is the factor theorem.
What is the remainder theorem?+
The remainder theorem states that dividing a polynomial P(x) by (x − c) leaves a remainder equal to P(c). Synthetic division is therefore also a quick way to evaluate a polynomial at a point.