Polynomial Long Division Calculator

Polynomial long division works like long division with numbers: divide the leading terms, multiply back, subtract, and repeat until what is left has a lower degree than the divisor. This solver does every step with exact fractions, then gives the quotient, the remainder and the answer written as quotient + remainder/divisor. Missing powers, such as the absent x term in x³ − 2x² − 4, are handled automatically.

How to use this tool

1. Type the dividend — the polynomial being divided.
2. Type the divisor; it can be linear like x - 3 or longer like x^2 + 1.
3. Read the quotient, the remainder and the combined answer.
4. Follow the steps table: each row shows the new quotient term, what was subtracted and what was left.

Formula

P(x) = D(x) · Q(x) + R(x), with deg R < deg D

P is the dividend, D the divisor, Q the quotient and R the remainder. At each step the next quotient term is (leading term of what is left) ÷ (leading term of D), and that term times D is subtracted. When D is linear, x − a, the remainder equals P(a) — the remainder theorem — so a remainder of 0 means x − a is a factor.

Example

(x³ − 2x² − 4) ÷ (x − 3): x³ ÷ x = x²; subtract x³ − 3x² to leave x² − 4. Then x² ÷ x = x; subtract x² − 3x to leave 3x − 4. Then 3x ÷ x = 3; subtract 3x − 9 to leave 5. Quotient x² + x + 3, remainder 5, so the answer is x² + x + 3 + 5/(x − 3). The remainder theorem agrees: P(3) = 27 − 18 − 4 = 5.