Synthetic Division Calculator
Divide a polynomial by x − r (or ax + b) with synthetic division. Shows the full synthetic division table, the quotient, the remainder and whether it is a factor.
Synthetic division is a shortcut for dividing a polynomial by a linear factor such as x − 2. It uses only the coefficients, so it is faster and less error-prone than long division. Use it to factor cubics and quartics, test possible roots from the rational root theorem, or evaluate P(r) quickly with the remainder theorem.
How to use this tool
- Type the dividend, for example x^3 - 6x^2 + 11x - 6, or list its coefficients: 1, -6, 11, -6.
- Type the divisor, such as x - 2. For x + 3 the root r is -3; 2x - 1 also works.
- Read the quotient and remainder, and whether the divisor is a factor.
- Follow the three-row table to check each multiply-and-add step.
Formula
P(x) = (x − r) · Q(x) + R, with R = P(r)
Write the coefficients of P in order, including 0 for any missing power. Bring the first coefficient down, multiply it by r, add the product to the next coefficient, and repeat. The bottom row gives the coefficients of Q(x), one degree lower than P, and its last number is the remainder R. For a divisor ax + b, divide by x − (−b/a) and then divide the quotient by a.
Example
Divide x³ − 6x² + 11x − 6 by x − 2, so r = 2. Bring down 1. Multiply 1 × 2 = 2 and add to -6 to get -4. Multiply -4 × 2 = -8 and add to 11 to get 3. Multiply 3 × 2 = 6 and add to -6 to get 0. The quotient is x² − 4x + 3 with remainder 0, so x − 2 is a factor and x = 2 is a root.
Frequently asked questions
Can synthetic division divide by x² + 1 or other quadratics?
Not in its basic form. It only works for linear divisors (degree 1). Use polynomial long division for higher-degree divisors.
What do I do about missing terms?
Every power must have a place in the row, so x³ − 8 becomes 1, 0, 0, -8. When you type the polynomial with x, the calculator fills in the zeros for you.
How does the remainder theorem relate to this?
The remainder of P(x) ÷ (x − r) equals P(r). A remainder of 0 means r is a root and (x − r) is a factor.
Which r do I use for x + 5?
Rewrite it as x − (−5), so r = -5. The calculator does this automatically when you type x + 5.
Last updated: 2026-10-30