HesapHanem

Polynomial GCD Calculator

Enter the coefficients of two polynomials to find their greatest common divisor.

Result

x - 1
1
Normalized to monic (leading coefficient 1).

How it is calculated

Each polynomial is typed as a comma-separated list of coefficients from the highest power down to the constant, so 1,0,-1 means x2 − 1. A missing power must be entered as 0. The tool runs the Euclidean algorithm: divide A by B, keep the remainder R, replace (A, B) by (B, R), and repeat until the remainder is zero. The last non-zero polynomial is the GCD. Because any constant multiple of a common divisor is also a common divisor, the result is scaled so its leading coefficient is 1 (monic). Coefficients are read as decimals, results are rounded to six places, and the loop stops after 100 steps. The output shows the polynomial and its degree; a GCD of degree 0 (just 1) means the two polynomials share no common factor.

Worked example

Take A = x2 − 1 (1,0,-1) and B = x2 − 3x + 2 (1,-3,2). Dividing A by B gives quotient 1 and remainder A − B = 3x − 3. Next, dividing B by 3x − 3 leaves remainder 0, so 3x − 3 is the last non-zero remainder. Scaling by its leading coefficient 3 gives the monic GCD x - 1, of degree 1. This matches the factorisations (x − 1)(x + 1) and (x − 1)(x − 2).

Try this example →

Common mistakes

  • Listing coefficients in ascending order. The first number must belong to the highest power; entering -1,0,1 describes −x2 + 1, not x2 − 1.
  • Skipping zero coefficients. For x3 − 1 you must type 1,0,0,-1; typing 1,-1 is just x − 1.
  • Typing terms such as x^2-1 instead of numbers. Only plain numeric coefficients separated by commas are accepted, and anything else returns no result.
  • Expecting the GCD to keep your scale. It is always normalised to leading coefficient 1, so a common factor of 3 or 2 does not appear in the answer.
  • Reading "1" as an error. A GCD of 1 (degree 0) is a valid answer meaning the polynomials are coprime.

About this calculator

Just as integers have a greatest common divisor, two polynomials have a polynomial GCD — the highest-degree polynomial that divides both of them with no remainder. It's computed with the same Euclidean algorithm used for integers: repeatedly take the remainder of dividing the higher-degree polynomial by the lower-degree one until the remainder is zero, then normalize the result to monic form (leading coefficient 1) for a unique answer.

Polynomial GCD is a building block in computer algebra systems (Mathematica, Maple, SymPy) for simplifying rational expressions and factoring, and it underlies decoding algorithms for Reed-Solomon error-correcting codes used in QR codes, CDs, and satellite communication.

Was this helpful?

You can pick up to 3 reactions.

Comments (1)

  • Editorial team

    The answer is always scaled so the leading coefficient is 1, so 2,-4,2 and 1,-1 both give x - 1, not 2x - 2. A result of 1 with degree 0 means the polynomials share no factor, which is the case for 1,0,-1 and 1,0,1. Use this to simplify a rational expression: divide top and bottom by the result. Which fraction are you trying to reduce?

Popular calculators

Loan Calculator

Monthly payment, total interest and full amortization schedule.

Percentage Calculator

Percent change, percent of, and ratio calculations.

Discount Calculator

Calculate final price, savings, and discount amount.

BMI Calculator

Body mass index from height and weight (metric or imperial).

Age Calculator

Compute your exact age in years, months and days from a birth date.

Compound Interest Calculator

Calculate the final amount and interest earned with compounding.

All Calculators