# Polynomial factorization over the rationals

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Factorization writes a polynomial as a product in a declared coefficient field. Validation compares all coefficients rather than a few evaluated values.

mko-factorization · definition · v0.10.0



## Mathematical statement

In Q[x], a proposed factorization p=f·g is a polynomial identity: after expansion and collection of like powers, every coefficient agrees. This entry illustrates binomial products, common factors, and differences of squares; it does not provide a complete factorization algorithm for all polynomials.





## Representative formula (exact TeX)

x^{2}-a^{2}





## Assumptions


- Fix one formal indeterminate x and rational coefficients represented by finite coefficient data.
- Remove trailing zeros from factors and products. Multiplying factors by compensating constants or reordering them does not count as new mathematical content in this pulse.





## Symbol scope


- x — formal indeterminate
- a — rational constant in the representative difference of squares





## Intuition, definition, and misconceptions

(x+a)(x+b)=x²+(a+b)x+ab. Multiplication term by term and collection of like powers give a reproducible coefficient identity. For a∈Q, extracting a common factor, such as ax+a=a(x+1), remains in the same univariate polynomial ring.

The difference of squares x²−d²=(x−d)(x+d) is an identity. The exercises verify its coefficient expansion for integer d; agreement at one or two inputs does not establish equality of the entire polynomials.

Factorability depends on the coefficient field. For example, x²−2 cannot be expressed as a product of two nonconstant linear factors over Q, but over R the factors may use ±√2. Factorization is distinct from solving an equation, and a zero-product argument also requires a specified number system.

Common errors include applying a difference-of-squares formula to a sum of squares, or treating agreement at finitely many inputs as an identity without a justified degree bound.





## Adopted derivation guide

[Polynomial factorization over the rationals: verifying common factors, binomial products and differences of squares by comparing coefficients](https://ocme.evemisslab.com/en/derivations/derivation-mko-factorization-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Univariate polynomials over the rationals](https://ocme.evemisslab.com/en/mko/mko-polynomial/) — Coefficient comparison, multiplication by convolution, and identities are defined in the same polynomial ring.





## Verification boundary

This pulse validates identities of explicit finite coefficient products only; general irreducibility, unique factorization, and a complete algorithm have no attached formal proof.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Bind Polynomial Q multiplication, coeff_mul, the specified identities, and the conditions for changing coefficient fields.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/factorization.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-factorization/index.md)