Key takeaway: Factoring out the GCF means finding the largest number and the highest variable power shared by every term, writing it outside parentheses, and dividing each term by it.
Grade 9 – Algebra 1A-SSE.A.2Expressions and Equations
Definition of Factoring Out the GCF
Factoring out the GCF is the first step in nearly every factoring problem. Look for the greatest common factor of all the terms, which includes the largest number that divides every coefficient and the lowest power of each variable that appears in every term.
Write that common factor in front, then open a pair of parentheses and fill it with what is left of each term after dividing by the GCF. This is the distributive property run backward. After factoring, the parentheses may hold a trinomial or binomial that can be factored further, so look again.
Formula and notation
If every term of a polynomial has the factor \(g\):
\[g\cdot a + g\cdot b + g\cdot c = g\,(a + b + c)\]
For \(12x^{3} + 18x^{2} – 30x\), the number part is \(\gcd(12, 18, 30) = 6\) and the variable part is \(x\), the smallest power present, so \(g = 6x\).
Key facts
- The number of terms inside the parentheses equals the number of terms in the original.
- Use the smallest exponent of each shared variable.
- When factoring out the GCF, divide every term by it; a term equal to the GCF leaves a 1.
- Multiply the answer out to confirm that it restores the original polynomial.
Factoring Out the GCF example
Factor out the greatest common factor: \(12x^{3} + 18x^{2} – 30x\).
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Numbers: the largest common divisor of 12, 18 and 30 is 6. Variables: the smallest power of \(x\) is \(x^{1}\). So the GCF is \(6x\).
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Divide each term: \(12x^{3} : 6x = 2x^{2}\), \(18x^{2} : 6x = 3x\), \(-30x : 6x = -5\).
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Check: \(6x \cdot 2x^{2} = 12x^{3}\), \(6x \cdot 3x = 18x^{2}\), \(6x \cdot (-5) = -30x\).
The result is \(6x(2x^{2} + 3x – 5)\).
Common mistake
Watch out
Alisha factors \(20n^{3} + 5n\) as \(5n(4n^{2})\), leaving out the second term.
Why it is wrong
Dividing \(5n\) by \(5n\) leaves 1, not nothing. The correct answer is \(5n(4n^{2} + 1)\). Multiplying her result gives only \(20n^{3}\), so the check exposes the missing \(5n\).
Related terms and lessons
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