Polynomial GCD Calculator

Overview

This polynomial GCD calculator finds the greatest common divisor (GCD) of two polynomials entered by their coefficients. It uses the Euclidean algorithm to compute a normalized, monic GCD (leading coefficient equal to 1), and is suitable for algebra, calculus, control theory, and computer algebra applications.

You provide the coefficients of each polynomial from highest degree to constant term, optionally with negative and decimal values. The tool then computes the greatest polynomial that divides both inputs without leaving a remainder and displays the result in your chosen variable (for example, x, t, or s).

How to use the polynomial GCD calculator

1. In Coefficients of P(x), enter the coefficients of the first polynomial from highest power to constant term, separated by commas (for example, 1,0,-1 for x² − 1).
2. In Coefficients of Q(x), enter the second polynomial in the same way.
3. In Variable, type the symbol you want to use in the result (for example, x, t, or s).
4. Click Compute gcd to run the Euclidean algorithm on your inputs.
5. Read the output polynomial GCD and, if provided, any step-by-step division details. Use the Copy Result option if you want to paste the expression elsewhere.

Input format and examples

The calculator expects polynomials in coefficient form:

• Order: From highest degree term down to the constant term.
• Separator: Use commas between coefficients. Spaces are allowed but ignored.
• Signs and decimals: Negative numbers and decimal coefficients are allowed (for example, -2.5).
• Leading zeros: You may include them; the calculator will internally trim unnecessary leading zeros.

For example:

• P(x) = x² − 1 is entered as 1,0,-1.
• Q(x) = x³ − x is entered as 1,0,-1,0.
• R(x) = −2.5x² + 0.5x − 3 is entered as -2.5,0.5,-3.

What is the greatest common divisor of polynomials?

The greatest common divisor (GCD) of two nonzero polynomials P(x) and Q(x) is the polynomial of highest degree that divides both without leaving a remainder. In other words, it captures the common factor structure of the polynomials:

• If P(x) and Q(x) share one or more polynomial factors, their GCD is the product of those shared factors (up to a constant multiple).
• If they have no nonconstant factors in common, their GCD is just 1 (they are said to be coprime or relatively prime).

This is a direct analogue of the integer GCD, but in the polynomial ring over a field (typically real or rational coefficients). For example, if

P(x) = x³ − x = x(x − 1)(x + 1) and Q(x) = x² − 1 = (x − 1)(x + 1), then the common factor is (x − 1)(x + 1), so the GCD is x² − 1 (after expanding).

Euclidean algorithm for polynomial GCD

The calculator uses the polynomial version of the Euclidean algorithm. It works just like the integer Euclidean algorithm, but with polynomial long division instead of integer division:

1. Start with two polynomials P(x) and Q(x), where deg(P) ≥ deg(Q).
2. Divide P(x) by Q(x) to get a quotient S(x) and remainder R(x): with deg(R) < deg(Q).
3. Replace P(x) with Q(x) and Q(x) with R(x).
4. Repeat the division step until the remainder becomes the zero polynomial.
5. The last nonzero remainder is a GCD of the original polynomials.

Because multiplying a GCD by a nonzero constant gives another valid GCD, the calculator normalizes the result to be monic by dividing all coefficients by the leading coefficient.

Normalization and monic GCD

Over real or rational coefficients, the polynomial GCD is unique only up to multiplication by a nonzero constant. To make the result consistent across examples and implementations, this calculator returns the monic GCD: the version whose leading coefficient is 1.

For instance, if the raw GCD were computed as −2x² + 2, the monic version would be obtained by dividing by −2, giving x² − 1. Both polynomials generate the same ideal of multiples and are considered equivalent as GCDs, but the monic form is standard in algebra.

Worked examples

Example 1: Exact integer coefficients

Take

• P(x) = x³ − x
• Q(x) = x² − 1

Coefficients to enter:

• P(x): 1,0,-1,0
• Q(x): 1,0,-1

Factoring shows

• P(x) = x(x − 1)(x + 1)
• Q(x) = (x − 1)(x + 1)

so the common factor is (x − 1)(x + 1), and the GCD is x² − 1. When you run these coefficients through the calculator, the Euclidean algorithm steps end with the monic result

gcd(P, Q) = x² − 1.

Example 2: Negative and decimal coefficients

Now consider polynomials with decimals and negative signs. Let

• P(x) = −2.5x³ + 1.25x² − 0.75x
• Q(x) = −1.25x² + 0.75x

Enter:

• P(x): -2.5,1.25,-0.75,0
• Q(x): -1.25,0.75,0

Factoring out common numeric and polynomial parts, you find that both polynomials share x(2x − 1) up to a constant factor. After normalization, the calculator reports a monic GCD of the form

gcd(P, Q) = x(2x − 1) = 2x² − x, scaled to leading coefficient 1.

Depending on how the coefficients are normalized internally, you may see the final GCD displayed as a monic polynomial equivalent to 2x² − x. Small rounding differences can appear if the arithmetic involves many decimal places, but the polynomial structure (degree and roots) is preserved.

Interpreting the results

The main output of the calculator is a single polynomial, the monic GCD of your inputs. You can interpret it as follows:

• Degree of the GCD: A higher-degree GCD means the polynomials share more complex structure (for example, multiple linear factors or higher-degree irreducible factors).
• GCD = 1: When the output is just 1, the polynomials are coprime: they have no nonconstant common factors.
• Roots of the GCD: Any root of the GCD is a root of both input polynomials (ignoring multiplicities beyond what appears in the GCD).
• Use in simplification: Dividing each polynomial by the GCD removes common factors and simplifies rational expressions like P(x) / Q(x).

If a step-by-step log of the Euclidean algorithm is shown, you can see how each division reduces the degree until the algorithm terminates, which is helpful for learning and verification.

Comparison: integer GCD vs polynomial GCD

Limitations and assumptions

This calculator is designed for practical use in most common algebra and engineering problems, but it operates under some assumptions and has limitations you should be aware of:

• Coefficient type: Inputs are treated as real (or rational) numbers. Complex coefficients are not directly supported via a + bi notation.
• Degree and size: Very high-degree polynomials or extremely large coefficients may lead to slower performance or numerical instability, depending on your device and browser.
• Floating-point arithmetic: Decimal inputs are computed using standard floating-point arithmetic. For many decimal places, you may see tiny rounding errors (for example, a coefficient like 0.499999999 instead of 0.5).
• Exactness: If you require exact symbolic results, prefer integer or rational inputs. You can often multiply all coefficients by a common denominator to turn decimals into integers before entering them.
• Normalization: The output is normalized to a monic GCD. This is conventional, but it means the numeric scale may differ from what you expect if you are used to a different normalization.
• Zero polynomials: If one of the inputs is the zero polynomial (all coefficients zero), the GCD is typically defined to be a normalized version of the other nonzero polynomial. If both are zero, the GCD is undefined; the calculator may return a special message in this case.

When to use a polynomial GCD

Computing a polynomial GCD is useful in several contexts:

• Simplifying rational expressions: Reduce P(x)/Q(x) by dividing numerator and denominator by their GCD.
• Detecting repeated structure: If you know two polynomials share factors (for example, from modeling the same system in different ways), the GCD reveals that shared structure.
• Analyzing dynamical systems: In control theory, common factors between numerator and denominator of transfer functions can affect system zeros and poles.
• Algebraic manipulation: Polynomial GCDs are building blocks in algorithms for factoring, computing resultants, and solving polynomial equations.

By combining this calculator with tools for factoring polynomials or simplifying rational functions, you can quickly move from raw coefficients to a more interpretable, simplified representation of your problem.