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Polynomial Long Division Calculator

This polynomial long division calculator divides two polynomials step by step and shows quotient, remainder and check. Example: (x³ − 6x² + 11x − 6) : (x − 1) = x² − 5x + 6. A second mode finds all roots.

Source: Wolfram MathWorld – Polynomial division. Updated: .

e.g. x^3 - 6x^2 + 11x - 6; exponents with ^ or ², fractions like 1/2x or 0.5x
e.g. x - 1 or x^2 + 1
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Result

Result (quotient or roots)
x² − 5x + 6
Remainder
0
Equation
(x³ − 6x² + 11x − 6) : (x − 1) = x² − 5x + 6
Check
(x² − 5x + 6) · (x − 1) + 0 = x³ − 6x² + 11x − 6
Steps
  • + x² → x² · (x − 1) = x³ − x² → −5x² + 11x − 6
  • + −5x → −5x · (x − 1) = −5x² + 5x → 6x − 6
  • + 6 → 6 · (x − 1) = 6x − 6 → 0
Synthetic division (for divisor x − a)
x³x²x1
Coefficients1−611−6
Product (a = 1)1−56
Sum1−560

How it is calculated

Polynomial long division divides one polynomial by another, much like long division with numbers. It gives a quotient and a remainder whose degree is smaller than the divisor's. If the division leaves no remainder, the divisor is a factor of the polynomial.

How to do it

  1. Divide the leading term of the dividend by the leading term of the divisor: x³ : x = x².
  2. Multiply that result by the whole divisor and subtract it from the dividend.
  3. Repeat with the remainder until its degree is smaller than the divisor's.
  4. Check: quotient · divisor + remainder = dividend.

Example

(x³ − 6x² + 11x − 6) : (x − 1): x³ : x = x²; x² · (x − 1) = x³ − x²; remainder −5x² + 11x − 6. Then −5x, then 6. Result x² − 5x + 6, remainder 0. Because there is no remainder, x = 1 is a root.

Finding roots with polynomial division

If one root is known, divide by the linear factor (x − root) to get a polynomial of lower degree. Integer candidates are divisors of the constant term; for fractions the rational root theorem helps. The calculator tests all candidates, splits off the roots one after another and solves the rest, using the quadratic formula for degree 2.

Synthetic (Horner) division

For a divisor of the form (x − a) it is quicker with the Horner scheme: write the coefficients in a row, multiply the running value by a and add it to the next coefficient. The last value is the remainder and also p(a).

Frequently asked questions

How does polynomial long division work?

Divide the leading term of the dividend by the leading term of the divisor, multiply the result by the divisor, subtract, and repeat with the remainder until its degree is smaller than the divisor’s.

What does remainder 0 mean in polynomial division?

The divisor is then a factor of the polynomial. If the divisor is (x − a), then a is a root of the polynomial.

How do I find the first root for the division?

By trying the divisors of the constant term (for integer coefficients): substitute ±1, ±2, ±3 … until the value is 0. The calculator does this search automatically, including fractions.

What is synthetic (Horner) division?

A short form of dividing by (x − a): write down the coefficients, multiply by a as you go and add. The sums are the coefficients of the quotient and the last value is the remainder.

Can the calculator handle fractions and decimals?

Yes. Inputs like 0.5x^2 − 1/3 are processed exactly as fractions, so quotient and remainder appear without rounding errors.

Sources and legal basis

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