Polynomial factorization over the rationals

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

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 — 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.