1. Arrange powers
Write both polynomials in descending powers of x. Missing powers act like zero-coefficient placeholders, even when they are not displayed.
x² + 4/(x − 2)
Remainder degree is less than divisor degree
Polynomial Long Division Calculator with Steps
Get the quotient, remainder, every divide–multiply–subtract step, and an exact check that divisor × quotient + remainder equals the dividend.
Enter the polynomials
Division result
Quotient with remainder
Quotientx²
Remainder4
Identityx³ − 2x² + 4 = (x − 2) · (x²) + 4
VerificationExact coefficient check passed
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Polynomial long division steps
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How polynomial long division works
2. Divide leading terms
Divide the leading term of the current remainder by the divisor’s leading term. Add that monomial to the quotient.
3. Multiply and subtract
Multiply every divisor term by the new quotient term. Subtract that product from the current remainder, combining like powers exactly.
4. Stop and check
Repeat until the remainder is zero or its degree is lower than the divisor’s. Then multiply back to verify every coefficient.
Polynomial division identity
Dividend = Divisor × Quotient + Remainder, with deg(Remainder) < deg(Divisor)
The degree condition applies when the remainder is nonzero. It makes the quotient and remainder unique for division by any nonzero polynomial over the rational numbers.
Worked division examples
Exact division
(x³−6x²+11x−6) ÷ (x−1) = x²−5x+6. The remainder is 0, so x−1 is a factor of the dividend.
Nonzero remainder
(x³−2x²+4) ÷ (x−2) = x² with remainder 4. Therefore x³−2x²+4=(x−2)x²+4.
Higher-degree divisor
If x²+1 is divided by x³−2, no polynomial quotient term is possible. The quotient is 0 and the remainder is x²+1.
Fractional leading term
When leading coefficients do not divide evenly as integers, the quotient uses an exact fraction. No decimal rounding is introduced.
Exact arithmetic, privacy, and limits
Parsing, expansion, long division, and verification use reduced BigInt fractions. Finite decimals are converted to exact fractions, so the algebra does not depend on floating-point rounding. All processing stays on this page.
The calculator supports one variable, x, through degree 12 and numeric literals up to 18 digits. It does not accept equations, radicals, functions, negative powers, variables other than x, or division by a polynomial inside either input. Extremely large intermediate fractions are stopped with a clear message. Last reviewed: 4 August 2026.
Polynomial long division FAQs
How do you divide polynomials using long division?
Divide the current leading term by the divisor’s leading term, place the result in the quotient, multiply the divisor by it, subtract, and repeat until the remainder has lower degree.
What does the remainder mean?
It is the polynomial left when no further polynomial quotient term is possible. The remainder must be zero or have lower degree than the divisor.
Can the quotient contain fractions?
Yes. Fractional coefficients are normal when the leading coefficients do not divide evenly. This calculator keeps them exact and reduced.
What if the divisor has higher degree than the dividend?
The quotient is 0 and the dividend itself is the remainder. The division identity still holds because dividend = divisor × 0 + dividend.
Do I need to type zero terms?
No. Missing powers are inserted internally. You can enter x³+1 directly; it is treated as x³+0x²+0x+1.
When should I use synthetic division instead?
Synthetic division is a shorter method for a linear divisor such as x−3. Long division is the general method and also works for quadratic and higher-degree divisors.