Synthetic Division Calculator

Quick Answer

Last updated: 3 November 2026 ·How we calculate this: synthetic division and the remainder theorem

How This Synthetic Division Calculator Works

Type the polynomial's coefficients from the highest power down to the constant, with 0 for any missing power, and enter c from the divisor (x − c). The calculator brings down the first coefficient, then multiplies by c and adds, column by column. The last number in the bottom row is the remainder, and the numbers before it are the coefficients of the quotient.

How to Use the Synthetic Division Calculator

1. Enter the coefficients from the highest power down, separated by commas.
2. Use 0 for any missing power.
3. Enter c from the divisor x − c.
4. Press Calculate.
5. Read the quotient, remainder and the table.

Core Rules

• Bring down: b₀ = a₀
• Each next value: bₖ = aₖ + c × bₖ₋₁
• Remainder: the last value in the bottom row
• Remainder theorem: the remainder equals P(c)
• Factor theorem: x − c is a factor exactly when P(c) = 0

Key Terms

• Polynomial: an expression like 2x³ − 3x² + 6, made of powers of x with coefficients.
• Coefficient: the number in front of a power of x.
• Quotient: the result of the division, without the remainder.
• Remainder: what is left over after dividing.
• Root: a value of x that makes the polynomial equal 0.

Worked Example

Divide x³ − 7x + 6 by x − 1.

• Coefficients: 1, 0, −7, 6 (with 0 for the missing x² term), and c = 1
• Bring down 1. Then 1 × 1 = 1, 0 + 1 = 1. Then 1 × 1 = 1, −7 + 1 = −6. Then −6 × 1 = −6, 6 − 6 = 0
• Quotient: x² + x − 6, remainder 0, so x − 1 is a factor and x³ − 7x + 6 = (x − 1)(x + 3)(x − 2)

Real-Life Examples

• Finding roots: test possible roots quickly. A remainder of 0 means you found one.
• Factoring cubics: after one root, the quotient is a quadratic you can solve with our Quadratic Formula Calculator.
• Evaluating polynomials: the remainder gives P(c) fast.
• Graphing: roots show where the graph crosses the x-axis.

Synthetic Division vs Long Division

• Synthetic division: faster, uses only numbers, but only for divisors like x − c.
• Polynomial long division: works for any divisor, such as x² + 1.
• Same answer: both give the same quotient and remainder for x − c.

Synthetic Division Examples

Tips

• Missing powers: always put 0 in their place.
• Sign of c: it is the opposite of the number in x + a.
• Fractions: you can enter c as 1/2 for x − 1/2.
• Divisor ax − b: divide by x − b/a, then divide the quotient by a.

Common Mistakes to Avoid

• Using the wrong sign for c. For x + 4, c is −4.
• Skipping missing powers. x³ + 1 is 1, 0, 0, 1.
• Multiplying the wrong number. Always multiply the last bottom-row value by c.
• Reading the degree wrong. The quotient is one degree lower than the polynomial.
• Using it for non-linear divisors. Use long division for x² + 1 and similar.

Frequently Asked Questions

References & Sources