Polynomial factorization tutor
Factor a polynomial completely over the rational numbers and see which method found each factor. Every factor of degree 2 or more comes with a proof that it cannot be split further.
Polynomials are limited to 1,000 characters, one variable, degree 24, leading and constant coefficients of 12 digits after clearing fractions and 20,000 rational root candidates; exact values may grow to 1,024 digits. Everything is computed on your device in a background worker with a 15 second limit. Downloads are a JSON report and a CSV table of factors. One variable, whole-number powers and exact numbers such as 1/2. Up to 1,000 characters. Review the settings before running.
Your result
How to use Polynomial factorization tutor
- Type a polynomial in one variable, such as 3x^5 - 3x^4 - 12x + 12.
- Run the tutor to take out the greatest common factor and try the special patterns.
- Read the factored form, the method steps and the irreducibility proof for each factor.
- Download the JSON report or the CSV table of factors.
Useful next steps
Polynomials are limited to 1,000 characters, one variable, degree 24, leading and constant coefficients of 12 digits after clearing fractions and 20,000 rational root candidates; exact values may grow to 1,024 digits. Everything is computed on your device in a background worker with a 15 second limit. Downloads are a JSON report and a CSV table of factors.
Methods it explains
The tutor takes out the greatest common factor, then looks for a difference of squares, a sum or difference of cubes, a perfect square trinomial, the ac method for quadratics and grouping for four-term cubics. It then applies the rational root test and splits repeated factors. Method: OpenStax, College Algebra 2e, section 1.5 Factoring Polynomials (2021).
How irreducibility is proven
A quadratic or cubic with no rational root cannot be factored over the rationals. For degree 4 and above the tool factors the polynomial modulo several primes: if the possible factor degrees never match, no factor exists. Otherwise it lifts the modular factors with Hensel lifting and tests every combination, which is the Zassenhaus algorithm. Method: von zur Gathen and Gerhard, Modern Computer Algebra, 3rd edition, chapters 14 and 15 (2013).
Real roots
For each factor the report counts its real roots exactly with a Sturm sequence and lists the exact roots of quadratic factors, such as sqrt(2).