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All calculatorsMethodology

Polynomial Division Calculator

Divide two polynomials entered as expressions or coefficient lists and get the exact quotient and remainder with fractions, long-division steps and the identity.

Interactive calculator · Method POLYNOMIAL-DIVISION-1.0 · Last reviewed: 2026-11-04

Calculate your scenario

Polynomial long division: divide the leading term of the current remainder by the leading term of the divisor to get the next quotient term, subtract that term × divisor, and repeat until the remainder’s degree is below the divisor’s.

Your inputs

Try an example

Default worked example. Edit the inputs to calculate your scenario.

Default worked example

Quotient x^2 − 2x − 2
Remainder 0
Quotient coefficients 1, −2, −2
Remainder coefficients 0
Identity (x − 1)(x^2 − 2x − 2) + (0) = x^3 − 3x^2 + 2

(x^3 − 3x^2 + 2) ÷ (x − 1) = x^2 − 2x − 2 with remainder 0, so dividend = divisor × quotient + remainder.

How to use the calculator

Type each polynomial as an expression in x, or as comma-separated coefficients from the highest power down. Fractions (3/4) and decimals are allowed.

What the result means

x³ − 3x² + 2 divided by x − 1 gives x² − 2x − 2 with remainder 0, so x − 1 is a factor and x = 1 is a root.

Formula and units

Polynomial long division: divide the leading term of the current remainder by the leading term of the divisor to get the next quotient term, subtract that term × divisor, and repeat until the remainder’s degree is below the divisor’s. The result satisfies dividend = divisor × quotient + remainder. All arithmetic uses exact fractions; decimals such as 0.25 are converted to 1/4.

Worked examples

x³ − 3x² + 2 ÷ (x − 1): Dividend: expression like x^3-3x^2+2 or coefficients highest degree first like 1, -3, 0, 2: x^3 - 3x^2 + 2; Divisor: expression or coefficients, highest degree first: x - 1.
Example result: Quotient: x^2 − 2x − 2; Remainder: 0; Quotient coefficients: 1, −2, −2; Remainder coefficients: 0; Identity: (x − 1)(x^2 − 2x − 2) + (0) = x^3 − 3x^2 + 2.

2x⁴ + 3x³ − x + 5 ÷ (2x² + 1): Dividend: expression like x^3-3x^2+2 or coefficients highest degree first like 1, -3, 0, 2: 2x^4 + 3x^3 - x + 5; Divisor: expression or coefficients, highest degree first: 2, 0, 1.
Example result: Quotient: x^2 + (3/2)x − 1/2; Remainder: −(5/2)x + 11/2; Quotient coefficients: 1, 3/2, −1/2; Remainder coefficients: −5/2, 11/2; Identity: (2x^2 + 1)(x^2 + (3/2)x − 1/2) + (−(5/2)x + 11/2) = 2x^4 + 3x^3 − x + 5.

Limitations and common mistakes

One variable, x, with whole-number exponents up to 12. Expressions accept terms like 3x^2, -x, 5/2x (read as (5/2)x) and constants; parentheses, implicit products of brackets and other variables are not supported, so expand first or use a coefficient list (include zeros for missing powers).

Leaving out zero coefficients in a list: x³ + 2 is 1, 0, 0, 2, not 1, 2.

Sources and assumptions

Method POLYNOMIAL-DIVISION-1.0; last formula review 2026-11-04. Sources and their supported claims. Calculation methodology.