1Cademy - Factoring $$-6a^2 + 36a$$

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Factoring the Greatest Common Factor from a Polynomial

Example

Factoring $-6a^2 + 36a$

Factor $-6a^2 + 36a$ by extracting the greatest common factor, which includes both a negative coefficient and a variable.

Step 1 — Find the GCF: Ignoring the signs, determine the GCF of $6a^2$ and $36a$ . Factor each: $6a^2 = 2 \cdot 3 \cdot a \cdot a$ and $36a = 2 \cdot 2 \cdot 3 \cdot 3 \cdot a$ . The shared factors are one $2$ , one $3$ , and one $a$ , so the GCF is $6a$ . Because the leading coefficient is negative, use $-6a$ as the GCF.

Step 2 — Rewrite each term using the GCF: Express $-6a^2 = (-6a) \cdot a$ and $36a = (-6a) \cdot (-6)$ , giving $(-6a) \cdot a + (-6a)(-6)$ .

Step 3 — Factor out the GCF: $-6a(a - 6)$ .

Step 4 — Check by multiplying: $-6a(a - 6) = -6a \cdot a + (-6a)(-6) = -6a^2 + 36a$ ✓.

The factored form is $-6a(a - 6)$ . This example combines two features: the GCF includes a variable component ( $a$ , the lowest power of $a$ present in both terms) and is made negative because of the negative leading coefficient. The negative GCF causes the sign of the second term to flip — the original $+36a$ becomes $-6$ inside the parentheses.

Updated 2026-04-21

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OpenStax Elementary Algebra

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