Polynomial Long Division Calculator

Long division works on polynomials the same way it works on numbers — repeatedly cancel the leading term, subtract, and bring down the next one — but it's easy to lose track of a sign partway through, especially once the divisor is more than just x − a. This calculator runs the whole process for you against a divisor of any degree and shows every subtraction step, with exact fraction arithmetic so a non-evenly-dividing leading coefficient never gets rounded.

How to use it

1. Enter the dividend's coefficients, highest degree first — for x⁴ − 4x² + 3, enter 1, 0, -4, 0, 3 (the missing x³ and x terms are zeros, and they matter: don't skip them).
2. Enter the divisor's coefficients the same way — for x² − 1, enter 1, 0, -1.
3. Read off the quotient, the remainder, and the step-by-step table.

The divisor can be any degree, not just linear — for a divisor of exactly x − a, Synthetic Division Calculator is the faster shorthand route to the same answer.

The method

At each step:

1. Divide the remainder's leading term by the divisor's leading term to get the next quotient term.
2. Multiply the divisor by that term and subtract the result from the remainder.
3. Repeat until the remainder's degree drops below the divisor's degree.

FAQ

What if the coefficients don't divide evenly?

The quotient term is still computed exactly as a fraction (e.g. a leading coefficient of 2 divided by 3 becomes 2/3, not a rounded decimal) — the arithmetic never loses precision, even when the final remainder isn't zero.

What does a non-zero remainder mean?

The divisor isn't a factor of the dividend. dividend = divisor × quotient + remainder always holds — a non-zero remainder is the leftover after taking out as many full copies of the divisor as possible.

Do I need to write zero coefficients for missing terms?

Yes — the coefficient list is positional (each entry is one degree, highest to lowest), so a missing x-term still needs a 0 placeholder or every following term shifts to the wrong power.

Related tools

For the fast shorthand version when the divisor is exactly x − a, see the Synthetic Division Calculator. To multiply factors back out instead of dividing, see the Expand Polynomials Calculator.

Privacy

All calculations run entirely in your browser — the coefficients you enter are never sent to a server.