Analyze polynomials with rational root and Eisenstein checks. See graph, factors, and modular evidence quickly. Build cleaner proofs with exports, examples, and practical guidance.
| Polynomial | Main Test Triggered | Conclusion |
|---|---|---|
| x^2 + 1 | Quadratic discriminant | Irreducible over Q |
| x^2 - 5x + 6 | Square discriminant | Reducible over Q |
| x^3 - 2 | Eisenstein with p = 2 | Irreducible over Q |
| x^3 + x^2 - x - 1 | Rational root at x = 1 | Reducible over Q |
This tool studies polynomials with integer coefficients and checks whether they factor over the rational numbers.
1. Polynomial model: f(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0
2. Rational Root Test: Any rational root r = p/q must have p dividing the constant term and q dividing the leading coefficient.
3. Quadratic rule: For ax^2 + bx + c, the discriminant is D = b^2 - 4ac. If D is not a perfect square, the quadratic is irreducible over Q.
4. Cubic rule: A cubic over Q is reducible if and only if it has a rational root.
5. Eisenstein criterion: If a prime p divides every non-leading coefficient, p does not divide the leading coefficient, and p^2 does not divide the constant term, then the polynomial is irreducible over Q.
6. Modular check: If a primitive polynomial remains the same degree modulo a prime and is irreducible there, then it is irreducible over Q.
It tests whether a polynomial with integer coefficients is reducible or irreducible over the rational numbers using standard algebraic criteria and screening checks.
No. That statement is exact for cubics, but not for higher degrees. Quartics and higher polynomials can factor into non-linear pieces without any rational root.
The quadratic discriminant directly decides reducibility over Q. Higher-degree discriminants are useful, but they do not give the same simple yes-or-no factorization result.
A primitive polynomial has coefficients with greatest common divisor equal to one. Removing a shared integer factor does not change irreducibility over Q.
Reduction modulo a prime creates arithmetic in a field. That makes the irreducibility logic valid and avoids breakdowns that happen with composite moduli.
No. It proves many common cases, but some polynomials need stronger methods, shifts, substitutions, or computer algebra factorization routines.
The implemented checks may fail to certify the answer even when the polynomial truly is irreducible. Inconclusive means more theory or a stronger factoring engine is needed.
No. The graph is visual support only. It helps inspect real behavior, but irreducibility over Q must be decided by algebraic criteria, not appearance.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.