Factoring undoes multiplication. You know how to multiply a monomial by a polynomial; factoring out the greatest common factor (GCF) runs that backward, the way the GCF of numbers lets you write $12 + 18 = 6(2 + 3)$.
Why it works
The distributive property reads both ways: $3x(2x + 3) = 6x^2 + 9x$, so $6x^2 + 9x = 3x(2x + 3)$. Picture a rectangle whose area is $6x^2 + 9x$: if one side is $3x$, the other side must be whatever times $3x$ gives each piece, which is $2x$ and $3$. Factoring is finding the sides of a rectangle from its area. The greatest common factor leaves nothing common inside the parentheses, so the job is finished in one step.
Key idea
Find the GCF of the coefficients and the smallest power of each variable that appears in every term. Divide every term by it, and write GCF$\,\times\,$(what is left). Check by distributing.
Worked examples
Worked example: Three terms
Factor $12x^4 - 18x^3 + 6x^2$.
- Coefficients $12$, $18$, $6$: the GCF is $6$. Every term has at least $x^2$. GCF: $6x^2$.
- Divide each term: $\frac{12x^4}{6x^2} = 2x^2$, $\frac{-18x^3}{6x^2} = -3x$, $\frac{6x^2}{6x^2} = 1$.
- Check: $6x^2(2x^2 - 3x + 1) = 12x^4 - 18x^3 + 6x^2$.
Answer: $6x^2(2x^2 - 3x + 1)$
Worked example: Two variables
Factor $10a^2b - 15ab^2$.
- Coefficients: GCF $5$. Variables: one $a$ and one $b$ are in both terms. GCF: $5ab$.
- $\frac{10a^2b}{5ab} = 2a$ and $\frac{-15ab^2}{5ab} = -3b$.
Answer: $5ab(2a - 3b)$
Common mistake
Leaving nothing behind when a whole term is factored out: writing $3x^2 + 3x = 3x(x)$. The second term, $3x$, divided by $3x$ is $1$, not nothing, so the answer is $3x(x + 1)$. Check by distributing: $3x(x) = 3x^2$ has lost a term, while $3x(x + 1)$ gives back both.