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Polynomial Long Division Calculator
Enter your polynomial coefficients above and click "Perform Long Division" to see results.Polynomial long division is the algebraic equivalent of numerical long division. It divides a dividend polynomial by a divisor polynomial of any degree, producing a quotient and a remainder. Unlike synthetic division, which only handles linear divisors, long division works with quadratic, cubic, or any-degree divisors.
The algorithm repeatedly: (1) divides the leading term of the current dividend by the leading term of the divisor, (2) multiplies the entire divisor by that result, (3) subtracts to get a new (reduced) dividend, and (4) repeats until the remainder's degree is less than the divisor's degree. The result satisfies Dividend = Quotient × Divisor + Remainder.
Long division is indispensable for partial fraction decomposition in calculus, for verifying that a polynomial is a factor, and for simplifying complex rational expressions. When dealing with cubic equations, it allows division by quadratic factors that arise from complex conjugate root pairs.
Decomposing rational functions for integration in calculus requires polynomial long division when the degree of the numerator exceeds the denominator.
In control engineering, simplifying transfer functions by dividing out known factors uses polynomial long division.
Confirm whether a suspected polynomial factor divides evenly into the original polynomial.
Each column must correspond to the same power of x. Skipping a degree without a zero placeholder causes cascading errors.
You subtract the product at each step. Forgetting to distribute the negative sign is the most common arithmetic mistake.
Stop when the remainder's degree is strictly less than the divisor's degree. Going further is impossible; stopping earlier is incomplete.
| Formula | P(x) = Q(x) · D(x) + R(x) |
| Divisor Degree | Any degree (not limited to linear) |
| Stops When | deg(R) < deg(D) |
| Advantage | Handles quadratic and higher divisors |
| Verification | Q(x)·D(x) + R(x) must equal P(x) |
Use this tool whenever your divisor has an <span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">x²</span> in it, or has a leading coefficient that isn't 1 (like <span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">3x + 2</span>).
Extremely. The tool automatically injects <span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">0x²</span> or <span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">0x</span> placeholders into the algorithm to keep polynomial columns aligned properly.
Because you must subtract entire grouped quantities. This calculator distributes the negative signs flawlessly.
Yes! Unlike synthetic division, polynomial long division handles any divisor degree, making it the go-to for dividing by quadratics or other non-linear factors.
The quotient is the result of the division (akin to how many times the divisor goes into the dividend), while the remainder is what is left over after the division is complete.
