2026-09-29 9 min

Polynomial GCD Calculator

Find the GCD of two integer-coefficient polynomials via the Euclidean pseudo-remainder algorithm with monic normalization. Free online calculator.

Calculator

GCD Inputs

Polynomial P coefficients Polynomial Q coefficients

Integer coefficients, highest degree first, degree ≤ 4 each. Results update automatically.

Results update live as you type

Status:P = x^2 - 1GCD = x - 1Degree 1 shared

GCD Summary

GCD(P, Q)x - 1

P / Qx^2 - 1 | x^2 - 3x + 2

VerdictShare a nontrivial factor

Euclidean Trace

1. GCD(x^2 - 1, x^2 - 3x + 2)

2. x^2 - 1 ÷ x^2 - 3x + 2 → remainder 3x - 3

3. x^2 - 3x + 2 ÷ 3x - 3 → remainder 0

→ Monic GCD: x - 1

Results

Optimal

GCD(P, Q)

x - 1

Monic greatest common divisor

GCD Degree

Shared-degree dimension

Degree of P

Highest power in P

Degree of Q

Highest power in Q

Coprime?

No — share a factor

Whether any nontrivial factor is shared

Euclidean pseudo-remainder steps use exact-feeling integer-safe arithmetic with monic normalization, and the GCD is verified to divide both inputs.

Visualizations

Degrees: Inputs vs GCD

2deg P

2deg Q

1deg GCD

Euclid Completeness

Done100%

GCD Degree vs min(deg P, deg Q)

deg GCD1

Formula

Repeated pseudo-division shrinks the pair until one divides the other; monic normalization picks the canonical GCD.

GCD(P, Q) = monic form of last nonzero pseudo-remainder

Alternative Form

GCD(A, B) = GCD(B, rem(A, B)) until rem = 0

Variables

Symbol	Name	Unit	Description
P, Q	Input polynomials	expressions	Integer-coefficient, degree ≤ 4
rem	Pseudo-remainder	expression	Scaled remainder of A ÷ B
GCD	Result	expression	Monic highest shared divisor
deg	Degrees	count	GCD degree ≤ min input degree

Explanation

Each Euclidean round strips away the non-shared part; what survives every round is exactly the common factor content.

Derivation

Any common divisor of A and B divides every remainder combination, and the last nonzero remainder divides both — so it is greatest. Scaling keeps integer arithmetic exact.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Write both polynomials and their degrees.

deg = highest nonzero power

P:x^2 - 1

Q:x^2 - 3x + 2

degP:2

degQ:2

Result: P = x^2 - 1, Q = x^2 - 3x + 2

Step 2

Run the Euclidean pseudo-remainder loop.

GCD(A,B) = GCD(B, rem(A,B))

first:x^2 - 1 ÷ x^2 - 3x + 2 → remainder 3x - 3

Intermediate: GCD(x^2 - 1, x^2 - 3x + 2) ‖ x^2 - 1 ÷ x^2 - 3x + 2 → remainder 3x - 3 ‖ x^2 - 3x + 2 ÷ 3x - 3 → remainder 0

Result: x^2 - 3x + 2 ÷ 3x - 3 → remainder 0

Step 3

Take the last nonzero remainder.

GCD up to a constant factor

raw:see loop above

Result: Raw GCD candidate found

Step 4

Normalize to monic (leading coefficient 1).

divide by the leading coefficient

GCD:x - 1

Result: GCD(P, Q) = x - 1

Step 5

Verify by division and read the verdict.

GCD divides both P and Q

degree:1

coprime:no

Result: Share factor(s) of degree 1

Final Answer

x - 1

Worked Examples

See how the calculator works with different scenarios.

Scenario

Two quadratics sharing (x−1).

Inputs

p:1, 0, -1

q:1, -3, 2

Calculation

rem = −2x+2 → GCD (x−1)

Result

GCD = x − 1

Interpretation

Shared root x = 1 with multiplicity 1.

Understanding Your Result

Your Result: Monic GCD with degree and coprime verdict

Typical range: Degree 0 (coprime) up to the smaller input degree (full divisibility).

What This Means

The GCD is the largest polynomial piece hidden inside both inputs — their shared factor content.

Why It Matters

It cancels rational expressions, sets up partial fractions, detects shared roots, and simplifies control models.

Professional Interpretation

GCD degree counts shared roots with multiplicity; a degree-0 result certifies coprimality.

Recommendations

•Factor by hand to confirm small cases
•Cancel only proven GCD factors
•Read degree as shared-root count
•Use monic form for comparisons
•Check divisibility of the answer into both inputs

Educational Content

Introduction

This calculator finds the greatest common divisor of two integer polynomials using Euclid with pseudo-remainders.

Concept Explanation

Remainders preserve common divisors while shrinking degrees, so iterating must land on the greatest shared piece.

Applications

▸Simplifying rational expressions

▸Partial fraction setup

▸Pole-zero cancellation

▸Error-correcting codes

▸Computer algebra kernels

▸Common-root detection

Benefits

•Step-by-step Euclidean trace
•Monic canonical answer
•Coprime verdict included
•Degree-safe (≤ 4)
•Integer-friendly arithmetic

Limitations

•Integer coefficients expected
•Degrees capped at 4
•Numeric, not symbolic

Best Practices

•Enter highest-degree first
•Keep coefficients integral
•Verify GCD divides both inputs
•Compare monic forms only
•Treat constants as coprime signals

Common Misconceptions

Myth: GCD must look like one of the inputs.

Truth: It is usually a smaller new polynomial dividing both.

Myth: GCD 1 means a mistake.

Truth: Coprime inputs legitimately have GCD 1.

Myth: Any multiple of the GCD is wrong.

Truth: Multiples are valid GCDs too; monic form is just the convention.

Myth: Euclid needs factoring first.

Truth: Euclid finds the GCD without ever factoring.

Practical Tips

•List coeffs highest-first
•Prefer integer coefficients
•Follow the remainder trace
•Read monic output as canonical
•Verify by dividing both
•Coprime means GCD 1
•Degree ≤ 4 per input

Common Mistakes

Reversing coefficient order

Highest-degree coefficient comes first.

Skipping zero placeholders

Pad missing powers with 0.

Comparing non-monic GCDs

Normalize leading coefficients to 1 first.

Assuming GCD 1 is an error

It certifies coprime inputs.

Using non-integer coefficients

Scale to integers for the pseudo-remainder path.

Assumptions

This calculator makes the following assumptions to provide estimates:

Integer coefficients: Pseudo-remainder arithmetic assumes integral inputs.

Degree cap 4: Each input has degree at most 4.

Univariate: Both are polynomials in the same variable x.

Monic convention: The reported GCD has leading coefficient 1.

Exact arithmetic: Near-zero values below 1e-9 count as zero.

Limitations

Be aware of these limitations when using this calculator:

Degree cap: Inputs above degree 4 are rejected.

Numeric method: No symbolic factorization is shown.

Tolerance rounding: Tiny floating residue is snapped to zero.

Single pair: Each run handles exactly two polynomials.

Frequently Asked Questions

Answers to common questions about this calculator.

What is the GCD of two polynomials?

The monic polynomial of highest degree dividing both P and Q. It captures every common factor (and hence every shared root) with the correct multiplicity.

How does the Euclidean algorithm work for polynomials?

Divide P by Q, keep the remainder R, then repeat with (Q, R) until the remainder is zero. The last nonzero remainder — normalized to monic — is the GCD.

What is pseudo-remainder division?

A variant that pre-multiplies by powers of the leading coefficient so intermediate divisions stay exact for integer coefficients — avoiding fraction blowup during the Euclidean steps.

Why normalize the GCD to monic?

Any nonzero constant multiple of a GCD is also a GCD. Forcing the leading coefficient to 1 picks one canonical representative everyone agrees on.

What does GCD = 1 mean?

The polynomials are coprime: they share no factor of positive degree and no common roots. Their LCM is then simply their product.

How do I enter the polynomials?

Type comma-separated integer coefficients, highest degree first. For x² − 1 enter "1, 0, -1". Degrees up to 4 are supported here.

How is polynomial GCD used?

Simplifying rational expressions, partial fractions, control-system pole-zero cancellation, decoding error-correcting codes, and computer algebra systems.

What is the relation between GCD and LCM?

GCD(P,Q) × LCM(P,Q) equals P × Q up to a constant. Knowing the GCD immediately gives the LCM structure.

Can I check the GCD by hand?

Factor both polynomials if possible and multiply the shared factors. For x² − 1 and x² − 3x + 2 the shared factor is (x − 1), matching the Euclidean result.