Factoring Polynomials

Factoring the GCF

Factoring out the greatest common factor from a polynomial.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Undoing Multiplication

3(2x + 5) expands to 6x + 15 by distributing. Before reading on, think about the reverse question: starting from just 6x + 15, how would you recover the factored form 3(2x + 5) — and what property of the two terms, 6x and 15, would tell you 3 was the number to pull out?

Definition — Factoring and the Greatest Common Factor

Factoring rewrites an expression as a product instead of a sum, reversing distribution. The greatest common factor (GCF) of a polynomial's terms is the largest expression that divides evenly into every one of them.

Factoring out the GCF works precisely because it undoes the distributive property — if every term already shares a common factor, pulling it out front and leaving the rest behind in parentheses reconstructs exactly what distributing that factor would have produced in the first place.

Worked Example — Factoring Out a Numeric and Variable GCF

Factor 8x³ + 12x². The GCF of 8x³ and 12x² is 4x² (4 is the largest number dividing both 8 and 12; x² is the highest power of x dividing both terms). Factor it out: 4x²(2x + 3). Check by redistributing: 4x²(2x) + 4x²(3) = 8x³ + 12x². Matches.

Worked Example — Factoring Out a Negative GCF

Factor −3x² − 9x. The GCF, including sign, is −3x. Factor it out: −3x(x + 3). Check: −3x(x) + (−3x)(3) = −3x² − 9x. Matches.

Tip

Always check a factored answer by redistributing it — this catches both an incorrect GCF and a sign error in the remaining parentheses before the mistake carries into a later step.

Common Mistakes

Factoring out a common factor that isn't the greatest one, such as pulling out 2x² from 8x³ + 12x² instead of the full 4x².
Check both the numeric coefficient (find the greatest common factor of the numbers) and the variable part (find the lowest shared power) — factoring out anything less than the true GCF leaves the expression only partially factored.
Making a sign error inside the parentheses when factoring out a negative GCF.
After factoring out a negative GCF, every term inside the parentheses flips sign compared to the original — double check by redistributing to confirm the signs match the original expression.

Key Takeaways

Factoring reverses distribution, rewriting a sum as a product.
The GCF is the largest expression — numeric and variable parts both — that divides evenly into every term.
Redistributing a factored answer is a reliable way to check it against the original expression.

Summary

Factoring out the GCF is the first and simplest factoring move, always worth checking first. The next lesson factors a more complex structure — a trinomial — by working FOIL's pattern in reverse.

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