Factoring by GCF
AlgebraFactoring 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 largest factor that divides every term, then pulling it out front. GCF stands for Greatest Common Factor.
Example
$6x^2 + 9x$: the GCF of $6$ and $9$ is $3$, and both terms have at least $x^1$. GCF $= 3x$. Factor: $3x(2x + 3)$.
Key Insight
Always look for a GCF first before trying any other factoring method. It simplifies what's left inside the parentheses.
Definition
To factor by GCF, find the greatest common factor of all coefficients (using GCF of integers) and the lowest power of each common variable. Write the polynomial as GCF times the remaining polynomial. The remaining polynomial should have no further common factor.
Example
$12x^4 - 8x^3 + 4x^2$: GCF of $12, 8, 4$ is $4$; lowest x-power is $x^2$. Factor: $4x^2(3x^2 - 2x + 1)$. Check: $4x^2 \cdot 3x^2 = 12x^4$. Correct.
Key Insight
Factoring by GCF is always the first step. Not doing it first means working with larger numbers and missing simpler forms. Many problems become trivial once the GCF is removed.
Definition
Factoring by GCF corresponds to the ring-theoretic operation of extracting the GCD of all generators of the ideal generated by the terms of the polynomial. In $\mathbb{Z}[x]$, the GCF includes both the integer GCD of coefficients (called the content) and the polynomial GCD of all terms. Gauss's Lemma states that the product of primitive polynomials (content $= 1$) is primitive, so the content is multiplicative.
Example
$15x^3y + 10x^2y^2 - 5xy^3$: integer GCF $= 5$, variable GCF $= xy$. Result: $5xy(3x^2 + 2xy - y^2)$. The remaining trinomial can be factored as $(3x - y)(x + y)$.
Key Insight
The content-primitive factorization $c(f) \cdot f_{\text{prim}}$ is the canonical decomposition of a polynomial in $\mathbb{Z}[x]$. This decomposition is essential in polynomial factorization algorithms, which first extract content, then factor the primitive part.