Factoring Out the GCF

GCF of a Polynomial

GCF = (GCF of coefficients) × (lowest power of each variable in all terms) ax^m + bx^n = GCF · (ax^m/GCF + bx^n/GCF)

To find the GCF of a polynomial, do two things separately:

Step A — Coefficients: Take the GCF of all the numerical coefficients (ignoring signs). Use the Euclidean algorithm or factor each coefficient.

Step B — Variables: For each variable that appears in every term, include it raised to the lowest power it appears with. If a variable is missing from any term, it is not part of the GCF.

Multiply the results from A and B together (keeping the sign of the leading term if negative) to get the GCF.

Quick check: divide each original term by your GCF — every quotient must be an integer coefficient with non-negative variable exponents.

Why Factor Out the GCF First?

It simplifies the remaining factor and makes all other factoring methods easier.
Smaller coefficients mean less to track when factoring trinomials or using grouping.
Without factoring the GCF, you can miss a fully factored form — your answer may look "done" but still have a common factor hiding inside.
Always factor the GCF before trying difference of squares, trinomials, sum/difference of cubes, or grouping.
If the leading coefficient is negative, factor out a negative GCF so the remaining polynomial leads with a positive coefficient.

Rule of thumb: the very first step in any factoring problem is "Do all terms share a common factor?"

Still have questions about factoring polynomials?