Factoring Out the GCF

Before any other factoring technique, check for a common factor.

For 12x³ + 18x², the numbers share 6 and the variables share x², so the GCF is 6x²:
12x³ + 18x² = 6x²(2x + 3).

For the variable part, take the *lowest* power present. x³ and x² share x², not x³.

Always expand your answer mentally to check it returns the original.

Skipping this step makes everything afterwards harder. 2x² + 10x + 12 looks awkward until you pull out the 2 and get 2(x² + 5x + 6), which factors instantly into 2(x + 2)(x + 3).

Factoring out the GCF applies the distributive property in reverse: ab + ac = a(b + c). The GCF combines the numeric GCF of the coefficients with the lowest power of each shared variable.

It is the mandatory first step of any factoring problem for two reasons. First, it reduces the remaining polynomial's coefficients, often turning a difficult trinomial into a standard one. Second, several patterns are only recognisable after extraction: 2x² − 18 is not a difference of squares until the 2 comes out, yielding 2(x² − 9) = 2(x + 3)(x − 3).

When the leading coefficient is negative, factoring out −1 along with the GCF is usually worthwhile, since most factoring patterns are stated for positive leading coefficients.

Complete factorisation means every factor is irreducible over the integers, so each factor must be re-examined after extraction rather than assumed finished.