# Polynomial factorization tutor - Human page: https://happytails.ai/en/calculators/polynomial-factorization-tutor/ - Markdown: https://happytails.ai/en/calculators/polynomial-factorization-tutor.md - Structured JSON: https://happytails.ai/agents/en/calculators/polynomial-factorization-tutor/index.json - Visibility: public ## Tool capabilities | Property | Value | | --- | --- | | Tool ID | `polynomial-factorization-tutor` | | Category | calculators | | Implementation | Registered implementation. Registration alone is not a production-quality guarantee. | | Browser execution | Use the visible action or interactive controls. | | Browser processing | Browser; see tool notes for network behavior. | | HTTP API | Not callable through HTTP. | | Browser-agent interface | run_current_tool | ### Implementation notes Factors exactly over the rational numbers on your device and proves each factor irreducible, for one variable up to degree 24 and 1,000 characters. ### Inputs and settings Text input: Polynomial. Example or placeholder shown in the interface (not a validated fixture): ```text 3x^5 - 3x^4 - 12x + 12 ``` One variable, whole-number powers and exact numbers such as 1/2. Up to 1,000 characters. Review the settings before running. No additional settings are declared for this tool. ### Browser-agent workflow 1. Open the human page in a browser that supports WebMCP. 2. Discover the interface exposed by that page; use its actual schema. 3. Select files on the page first when the tool requires files. 4. Call `run_current_tool` with valid input and settings. 5. Inspect the returned result or error and the displayed output. Retrieve files from the displayed download links. Calls do not automatically copy or download results. Browser permissions still apply. Interface availability alone is not proof of successful execution. ### Planning specification 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. This describes intended requirements. Use the implementation notes, interface schema and observed output to determine current support. ### Complete machine-readable capability record ```json { "id": "polynomial-factorization-tutor", "category": "calculators", "built": true, "requirements": "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.", "implementation": { "fields": [], "presentation": "metrics", "inputCodeDirection": "ltr", "registrationFamily": "exact-math-functions", "inputMaxLength": 1000, "inputErrorCode": "EXACT_MATH_FUNCTIONS_EXPRESSION_TOO_LONG", "note": "Factors exactly over the rational numbers on your device and proves each factor irreducible, for one variable up to degree 24 and 1,000 characters.", "experience": { "automatic": false, "single": true, "inputMaxLength": 1000, "inputMode": "text", "label": "Polynomial", "placeholder": "3x^5 - 3x^4 - 12x + 12", "help": "One variable, whole-number powers and exact numbers such as 1/2. Up to 1,000 characters. Review the settings before running.", "helpParts": [ "One variable, whole-number powers and exact numbers such as 1/2.", "Up to 1,000 characters.", "Review the settings before running." ], "action": null }, "mode": "explicit", "interactive": false }, "execution": "Browser interface", "api": null, "browser_agent": { "supported": true, "name": "run_current_tool", "discovery": "WebMCP on the human page in a supporting browser", "verification": "See audit/API-AUDIT.md; availability is not verification" } } ``` ## Page guide 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. ### How to use Polynomial factorization tutor 1. Type a polynomial in one variable, such as 3x^5 - 3x^4 - 12x + 12. 2. Run the tutor to take out the greatest common factor and try the special patterns. 3. Read the factored form, the method steps and the irreducibility proof for each factor. 4. Download the JSON report or the CSV table of factors. How can we improve this tool? Report a problem or share an idea. Please leave out private information. Your message and the tool name go to our team via PostHog. ## Useful next steps - [Quadratic formula calculator](https://happytails.ai/en/calculators/quadratic-formula/) ([Markdown](https://happytails.ai/en/calculators/quadratic-formula.md)) - [GCF calculator](https://happytails.ai/en/calculators/gcf-calculator/) ([Markdown](https://happytails.ai/en/calculators/gcf-calculator.md)) - [Prime factorization calculator](https://happytails.ai/en/calculators/prime-factorization/) ([Markdown](https://happytails.ai/en/calculators/prime-factorization.md)) 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).