Euclidean Algorithm Polynomials Calculator

Calculator Form

Polynomial A coefficients

Enter highest degree to constant. Example: 1, 0, 0, 0, -1

Polynomial B coefficients

Fractions are allowed. Example: 1, 0, -1, 0

Variable symbol

Display precision

Trim epsilon

Normalize gcd to monic

Use monic gcd and scaled Bezout coefficients

Graph min x

Graph max x

Graph sample points

Example Data Table

Sample	Polynomial A Coefficients	Polynomial B Coefficients	Polynomial A	Polynomial B	Expected GCD
1	1, 0, 0, 0, -1	1, 0, -1, 0	x^4 - 1	x^3 - x	x^2 - 1
2	1, -3, 2	1, -2, 1	x^2 - 3x + 2	x^2 - 2x + 1	x - 1
3	1, 2, -1, -2	1, 1, -2	x^3 + 2x^2 - x - 2	x^2 + x - 2	x^2 + x - 2

Formula Used

The polynomial Euclidean algorithm repeats long division until the remainder becomes zero. For polynomials A(x) and B(x), the division rule is:

A(x) = Q(x)B(x) + R(x), where deg(R) < deg(B).

Then replace the pair (A(x), B(x)) with (B(x), R(x)) and repeat. The last non-zero remainder is the greatest common divisor.

This calculator also computes extended Euclidean coefficients:

S(x)A(x) + T(x)B(x) = GCD(A(x), B(x))

When monic normalization is enabled, the final gcd is divided by its leading coefficient. The Bezout coefficients are scaled the same way, so the identity still stays valid.

How to Use This Calculator

Enter polynomial coefficients from highest degree to constant term.
Use commas, spaces, or semicolons between coefficients.
Optionally enter fractions such as 1/2 or -3/4.
Choose the display precision and trimming epsilon.
Enable monic normalization when you want a leading coefficient of 1.
Set the graph range and sample count.
Press calculate to show the gcd above the form.
Review quotients, remainders, Bezout coefficients, and the plotted curves.
Use the CSV and PDF buttons to export your result set.

Frequently Asked Questions

1. How should I enter the polynomials?

Enter coefficients from highest degree to constant term. For x^4 - 1, type 1, 0, 0, 0, -1.

2. Can I use fractions in the coefficient lists?

Yes. You can enter values like 1/2, -3/4, or 5/3. The calculator converts them to decimal values internally.

3. What does monic gcd mean?

A monic gcd has leading coefficient 1. Many algebra texts prefer this normalized form because it makes answers easier to compare.

4. Why are Bezout coefficients included?

They prove that the gcd is a linear combination of the two input polynomials. That is useful in algebra, coding theory, and symbolic computation.

5. What happens if one polynomial is zero?

If one input is zero and the other is not, the non-zero polynomial becomes the gcd, after optional monic normalization.

6. Why is epsilon important?

Epsilon removes tiny floating leftovers created by division. A slightly larger value can make output cleaner when decimal coefficients are involved.

7. What does the graph show?

The graph plots Polynomial A, Polynomial B, and their gcd over the selected x-range. The second chart shows degree reduction through each Euclidean step.

8. Is the lcm always shown?

The lcm appears when the product divides exactly by the computed gcd within the chosen tolerance. Otherwise the page leaves it hidden.