Polynomial Long Division Calculator With Steps
Enter real coefficients for a dividend and divisor polynomial. The calculator finds the quotient and remainder, shows every leading-term division and subtraction, and reconstructs the dividend to check the result.
Written and checked by TheToolNet Editorial Team · Last reviewed: August 22, 2026 · How we check math tools
For non-zero D(x), the remainder is zero or has degree lower than D(x).
Divide Polynomials Online
Choose degrees and enter coefficients from the highest power through the constant term. Include 0 for every missing power.
How to do polynomial long division
- Write both polynomials in descending powers, inserting zero-coefficient terms where powers are missing.
- Divide the leading term of the current remainder by the leading term of the divisor.
- Place that term in the quotient.
- Multiply the entire divisor by the new quotient term and subtract every coefficient.
- Repeat until the remainder is zero or its degree is lower than the divisor degree.
This is the polynomial version of ordinary long division: divide, multiply, subtract and continue.
Worked polynomial long-division example
Divide P(x) = x³ + 2x² − 5x + 3 by D(x) = x² − 1.
- x³ ÷ x² = x. Multiply: x(x² − 1) = x³ − x. Subtract to obtain 2x² − 4x + 3.
- 2x² ÷ x² = 2. Multiply: 2(x² − 1) = 2x² − 2. Subtract to obtain −4x + 5.
The quotient is x + 2 and the remainder is −4x + 5. Verification:
(x² − 1)(x + 2) + (−4x + 5) = x³ + 2x² − 5x + 3
Finding the quotient and remainder
The final answer may be written as P(x) ÷ D(x) = Q(x) + R(x)/D(x). Division stops only when deg R < deg D. A non-zero remainder is not an error; it means D(x) is not a factor of P(x).
If the divisor degree exceeds the dividend degree, no leading term can be divided. In that case Q(x) = 0 and R(x) = P(x).
Polynomial long division versus synthetic division
| Method | Works with | Advantage |
|---|---|---|
| Long division | Any non-zero polynomial divisor | General method; every multiplication and subtraction is visible. |
| Synthetic division | Usually a linear divisor x − c | Shorter coefficient-only layout. |
This calculator always uses long division so quadratic and higher-degree divisors are handled by the same documented method.
Common polynomial-division mistakes
- Skipping a missing power: enter 0 so coefficients stay aligned.
- Dividing whole polynomials at once: divide only the leading terms at each step.
- Subtracting only the first term: distribute the minus sign across the entire product.
- Stopping too early: continue while the remainder degree is at least the divisor degree.
- Forgetting the remainder: include R(x)/D(x) in the final rational expression.
Scope and numerical accuracy
The dividend may have degree 1–10 and the divisor degree 0–5. Coefficients may be real integers or decimals, but the selected leading coefficient must be non-zero. Integer-looking results can be exact; decimal calculations are numerical approximations. The reconstruction check reports the maximum coefficient error rather than claiming symbolic equality for decimal data.
Frequently asked questions
How do you find the quotient using polynomial long division?
Repeatedly divide the leading remainder term by the leading divisor term. Each result becomes the next quotient term.
How do you verify the quotient and remainder?
Compute D(x)Q(x) + R(x). It must reconstruct the original P(x), and the remainder degree must be lower than the divisor degree.
Can the divisor be quadratic?
Yes. The calculator accepts divisor degrees from 0 through 5.
Why must I enter zero for a missing power?
Coefficient positions encode powers. A zero keeps later coefficients aligned with the correct exponents.
Related polynomial tools and references
Find zeros of the dividend with the polynomial root finder. For ordinary numerical division, use the long division calculator.