1.46 Factoring Definitions

Factoring Definitions establishes the vocabulary for rewriting a polynomial as a product of simpler polynomials, describing what qualifies as a factor of a polynomial, the overall process of finding such factors, the largest factor shared among every term of a polynomial, and the standard for judging when factoring has been carried out as fully as possible.

Polynomial Factor

A polynomial factor is any polynomial that divides another polynomial exactly, with no remainder, meaning it can be multiplied by some other polynomial to reproduce the original polynomial precisely. Both x + 2 and x + 3 are polynomial factors of x² + 5x + 6, since multiplying them together reproduces that polynomial exactly.

Polynomial Factoring

Polynomial factoring is the process of rewriting a polynomial as a product of two or more polynomial factors, reversing the process of polynomial multiplication. Factoring x² + 5x + 6 into (x + 2)(x + 3) is an example of polynomial factoring, transforming the expanded form of the polynomial into an equivalent factored form.

Polynomial GCF

The polynomial GCF, or greatest common factor, is the largest monomial factor that divides every term of a polynomial evenly, combining the greatest common numerical factor of the terms' coefficients with the lowest power of each variable shared by every term. The polynomial GCF of 6x³ + 9x² is 3x², since 3 is the greatest common factor of 6 and 9, and x² is the lowest power of x appearing in either term; factoring out this GCF produces 3x²(2x + 3).

Complete Factorization

A complete factorization is a factored form of a polynomial in which every factor present is itself no longer capable of being factored any further, meaning the polynomial has been broken down into the most basic factors possible given the number system being used. Factoring x² − 4 as (x + 2)(x − 2) produces a complete factorization, since neither x + 2 nor x − 2 can be factored any further, whereas stopping at an intermediate, partially factored form would not meet this standard.

Together, these definitions describe factoring as the deliberate reversal of polynomial multiplication, guided first by identifying the polynomial GCF shared across all terms, and continuing—through whichever techniques are applicable—until a complete factorization is reached, leaving no factor within the result still capable of being broken down further.