Factoring by GCF
Factoring by GCF means pulling out the greatest common factor from all terms of a polynomial as the first step in factoring.
Definition
Factoring by GCF means finding the greatest common factor, the largest factor dividing every term (using the GCF of the coefficients and the lowest shared power of each variable), and pulling it out front, leaving a remaining polynomial with no further common factor. This step should always come first, before trying any other factoring method, since it simplifies everything that follows. Ring-theoretically, this is extracting the GCD of all generators of the ideal formed by the polynomial's terms; in $\mathbb{Z}[x]$, the GCF includes both the integer GCD of coefficients (the content) and the polynomial GCD of the terms, and Gauss's Lemma states that the product of primitive polynomials (content $= 1$) is itself primitive, so content is multiplicative.
Example
$6x^2 + 9x$: GCF of $6$ and $9$ is $3$, and both terms have at least $x^1$, so GCF $= 3x$, giving $3x(2x + 3)$. $12x^4 - 8x^3 + 4x^2$: GCF is $4x^2$, giving $4x^2(3x^2 - 2x + 1)$. $15x^3y + 10x^2y^2 - 5xy^3$: GCF is $5xy$, giving $5xy(3x^2 + 2xy - y^2)$, and the remaining trinomial further factors as $(3x - y)(x + y)$.
Key Insight
Not pulling out the GCF first means working with larger numbers and missing simpler forms, many problems become trivial once it is removed. The content-primitive factorization $c(f) \cdot f_{\text{prim}}$ is the canonical decomposition of a polynomial in $\mathbb{Z}[x]$, and factorization algorithms rely on this same two-step process: extract the content, then factor the primitive part.