Factor the greatest common factor before anything fancy
The GCF is the largest expression dividing every term. Removing it first exposes the smaller polynomial that actually needs attention.
How to recognize the method, run it, and know when it is the wrong choice.
Updated July 13, 2026Before looking for trinomial or special patterns, ask what every term shares.
Find the numerical and variable parts separately
The numerical GCF is the greatest common divisor of the coefficients. For each variable, use the smallest exponent appearing in every term.
A term missing a variable has exponent zero, so that variable cannot be part of the common factor.
Divide each term to build the inside
After pulling out the GCF, divide every original term by it. The quotients form the polynomial inside parentheses.
Expanding the result should reproduce the original expression exactly. That one-line check catches most omissions.
A negative GCF can improve the form
When the leading term is negative, factoring out a negative common factor leaves a positive leading coefficient inside.
This is optional but often makes later factoring and interpretation cleaner.
Factor 18x³y − 24x²y² + 30x²y.
Find the common numerical factor.
Use the smallest shared exponents.
Divide each term by the GCF.
Common mistakes
- Using the largest exponent instead of the smallest shared exponent.
- Leaving one term undivided inside the parentheses.
- Skipping the GCF before another factoring pattern.
Three takeaways
- Factor numbers and variables separately.
- Use minimum shared exponents.
- Expand once to verify.