➕ Polynomial Long Division: Calculator

Divide one polynomial by another to find the quotient and remainder.

QUOTIENT
—
REMAINDER
—
Dividend
—
Divisor
—
Dividend = Divisor × Quotient + Remainder

How Polynomial Long Division Works

Polynomial long division divides one polynomial (the dividend) by another (the divisor) to produce a quotient and a remainder, exactly the way ordinary long division works for numbers. This calculator uses a synthetic-division-style algorithm: it normalizes by the divisor's leading coefficient, then repeatedly eliminates the highest remaining term of the working polynomial by subtracting a scaled copy of the divisor, one degree at a time, until what's left has a lower degree than the divisor.

Dividend(x) = Divisor(x) × Quotient(x) + Remainder(x)
deg(Remainder) < deg(Divisor)

Reading the Quotient and Remainder

The quotient is the polynomial that results from the division, and the remainder is whatever is left over after the divisor no longer divides evenly — if the remainder is 0, the divisor is a factor of the dividend. For example, dividing x³ − 6x² + 11x − 6 by (x − 2) gives a quotient of x² − 4x + 3 with a remainder of 0, confirming that (x − 2) is a factor of the original cubic.

Why Polynomial Division Matters

Polynomial division is central to factoring higher-degree polynomials once you've found one root (via the Factor Theorem, if r is a root then (x − r) divides the polynomial evenly), simplifying rational expressions, and finding partial fraction decompositions used in calculus and engineering. It's also the basis of synthetic division, a faster shorthand version of the same process used specifically when dividing by a linear factor (x − r).

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❓ Frequently Asked Questions

What's the difference between polynomial long division and synthetic division?
They accomplish the same thing, but synthetic division is a streamlined shorthand that only works when dividing by a linear factor of the form (x − r). This calculator uses a general synthetic-division-style algorithm that also extends to quadratic divisors.
What does it mean if the remainder is 0?
A remainder of 0 means the divisor divides the dividend evenly — in other words, the divisor is a factor of the dividend. This is directly connected to the Factor Theorem: if (x − r) divides a polynomial with remainder 0, then r is a root of that polynomial.
Can the divisor have a higher degree than the dividend?
No meaningful division occurs in that case — the quotient is simply 0 and the remainder equals the entire dividend, since a lower-degree polynomial can't be divided evenly by a higher-degree one in the usual sense.
Why do I need to enter coefficients including zeros?
Every power of x from the highest degree down to the constant term needs a coefficient, even if it's zero (for example, x³ + 1 needs a 0 entered for both the x² and x terms), so the division algorithm correctly tracks each term's position.
Does this work for dividing by quadratic (degree-2) divisors?
Yes. Select divisor degree 2 to divide by a general quadratic like x² + bx + c, not just a linear factor. The same core algorithm handles both cases.

🏆 About This Tool — Accuracy & Trust

🔒 Data Privacy: All calculations run entirely within your browser using JavaScript. Nothing you enter here is ever transmitted to our servers, stored, sold, or shared.

📐 Accuracy Note: This tool uses standard, widely published formulas and guidelines. Results are estimates for informational purposes; for financial, medical, or engineering decisions, consult a licensed professional.

📅 Last Updated: September 2026.