Polynomial GCD Calculator

Exact polynomial GCD over rational coefficients from bounded integer input, normalized monic; zero result is ["0"], rational coefficients are exact decimal numerator/denominator strings.

Description

A polynomial GCD describes shared algebraic factors.

A polynomial GCD describes shared algebraic factors. Monic normalization removes an arbitrary nonzero constant multiplier and can introduce rational coefficients.^[1]

Exact polynomial GCD over rational coefficients from bounded integer input, normalized monic; zero result is ["0"], rational coefficients are exact decimal numerator/denominator strings.

When to use Polynomial GCD Calculator

Use this tool to exact polynomial GCD over rational coefficients from bounded integer input, normalized monic; zero result is ["0"], rational coefficients are exact decimal numerator/denominator strings.

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.

How to use Polynomial GCD Calculator and understand its results

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
Enter leftPolynomial, rightPolynomial using the field table and calculation rules, then run the tool.
Read monicGcdCoefficients. The explanation below defines the units, signs, ordering and any flags or placeholder values.
If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.

Input field	Meaning	Value type; see rules below	Requirement
leftPolynomial	Left polynomial	list of number	Required
rightPolynomial	Right polynomial	list of number	Required

Supply 1–13 constant-first signed integer coefficients per polynomial, each magnitude ≤one million. Trim high-degree zeros, use exact rational Euclidean division and return constant-first reduced fraction strings. Two zero polynomials return ["0"].

GCD = monic last nonzero Euclidean polynomial remainder

Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.

Output field	Meaning	Value type
monicGcdCoefficients	Monic gcd coefficients	list of string

json

{
  "input": {
    "leftPolynomial": [
      -1,
      0,
      1
    ],
    "rightPolynomial": [
      -2,
      1,
      1
    ]
  },
  "output": {
    "monicGcdCoefficients": [
      "-1",
      "1"
    ]
  }
}

Limitations and assumptions

Exact rational numerator and denominator components have a 4096-digit budget. Output fractions should not be silently rounded.

Numeric inputs and returned numeric values must be finite; an unrepresentable returned value produces an error. Numeric lists are limited to 100,000 entries; the rules above may require smaller limits. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.

Alternative or Complementary methods

Use Polynomial LCM Calculator: Exact polynomial LCM over rational coefficients from bounded integer input, normalized monic; zero result is ["0"], rational coefficients are exact decimal numerator/denominator strings. For a complementary task, use Extended GCD Polynomial Calculator: Compute exact rational polynomial Bézout witnesses u*f+v*g=monic gcd for degree≤12 integer input; coefficient strings preserve fractions and zero/zero returns all zeros.

References

Polynomial GCD Calculator — tool.io

Similar or alternative tools

Polynomial LCM Calculator
Exact polynomial LCM over rational coefficients from bounded integer input, normalized monic; zero result is ["0"], rational coefficients are exact decimal numerator/denominator strings.
Extended GCD Polynomial Calculator
Compute exact rational polynomial Bézout witnesses u*f+v*g=monic gcd for degree≤12 integer input; coefficient strings preserve fractions and zero/zero returns all zeros.