Divide one polynomial by another and see the classic long division layout: each quotient term, the product you subtract and the new remainder. Works with any divisor, from x − 3 to x² − x + 2, and keeps every coefficient as an exact fraction.
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Examples:
Type polynomials in x, e.g. 3x^4 - 2x + 1 or (x+1)(x-2). Fractions and decimals work (x/2, 0.5x). A comma list like 1, 0, -2, 1 is read as coefficients from the highest power.
Result
Quotient q(x)
Remainder r(x)
Long division layout
Step by step
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How to use the Polynomial Long Division Calculator
Enter the dividend p(x), for example 2x^4 + 3x^3 - x + 5.
Enter the divisor d(x), for example x^2 - x + 2.
Press Divide (results also update as you type).
Read the layout: quotient terms sit above the bar, each subtraction is underlined.
Follow the steps and the check line, then copy the result.
The four moves of polynomial long division
Every round of the algorithm repeats the same four moves until the remainder is too small to divide again:
Divide the leading term of the current remainder by the leading term of the divisor. The result is the next quotient term.
Multiply that term by the entire divisor and write the product under the matching columns.
Subtract the product. The leading terms cancel by design, so the degree drops by at least one.
Bring down the next term of the dividend and repeat while the degree is still at least the divisor's degree.
The process always ends with p(x) = d(x)·q(x) + r(x), where deg r < deg d. This is the division algorithm for polynomials, and it guarantees that the quotient and remainder are unique.
Worked example: (x³ − 2x² − 4) ÷ (x − 3)
Round
Divide
Multiply
Subtract to get
1
x³ ÷ x = x²
x²(x − 3) = x³ − 3x²
x² + 0x − 4
2
x² ÷ x = x
x(x − 3) = x² − 3x
3x − 4
3
3x ÷ x = 3
3(x − 3) = 3x − 9
5
The quotient is x² + x + 3 and the remainder is 5, so (x³ − 2x² − 4)/(x − 3) = x² + x + 3 + 5/(x − 3). By the remainder theorem the remainder also equals p(3) = 27 − 18 − 4 = 5.
Tips to avoid common mistakes
Distribute the minus sign to every term of the product. Subtracting x² − 3x means adding −x² + 3x.
Write missing powers as 0, both in the dividend and in the divisor, so columns stay aligned.
Order terms by descending power before you start; x + x³ − 1 should become x³ + 0x² + x − 1.
Check the degree: the quotient's degree is the dividend's degree minus the divisor's degree.
Use the result: a zero remainder means you found a factor, and q(x) + r(x)/d(x) is the form you need for slant asymptotes and for integrating improper rational functions.
Frequently asked questions
How do you do polynomial long division?
Write both polynomials in descending powers, filling missing powers with 0. Divide the leading term of the dividend by the leading term of the divisor to get the first quotient term, multiply the whole divisor by it, subtract, and repeat with the new remainder. Stop when the remainder's degree is lower than the divisor's degree.
Why do I need 0 placeholders?
Placeholders such as 0x² keep like terms in the same column, so each subtraction lines up. Without them it is easy to subtract an x term from an x² term by mistake. You can hide them with the checkbox once you are comfortable.
When does polynomial long division stop?
It stops as soon as the remainder is 0 or has a lower degree than the divisor. For a linear divisor the remainder is always a constant; for a quadratic divisor it has degree 1 or less.
How do I write the final answer?
Use p(x)/d(x) = q(x) + r(x)/d(x), or equivalently p(x) = d(x)·q(x) + r(x). If the remainder is 0, d(x) is a factor of p(x) and p(x)/d(x) simply equals q(x).
Is long division the same as synthetic division?
They give the same quotient and remainder. Synthetic division is a shortcut that only works for linear divisors like x − c, while long division works for any divisor, including quadratics such as x² + 1.
Can the coefficients be fractions?
Yes. Dividing by a divisor whose leading coefficient is not 1, such as 2x² − 3, often produces fractional coefficients. This calculator keeps them as exact fractions like 3/2 rather than rounded decimals.
How can I check my answer?
Multiply the quotient by the divisor and add the remainder. The result must equal the original dividend exactly. The calculator does this check automatically below the result.
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