1Cademy - Factoring $$-4a^3 + 36a^2 - 8a$$

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Factoring with a Negative Leading Coefficient

Example

Factoring $-4a^3 + 36a^2 - 8a$

Factor $-4a^3 + 36a^2 - 8a$ by extracting the greatest common factor, keeping in mind that the leading coefficient is negative.

Step 1 — Find the GCF: Because the leading coefficient is negative, the GCF will be negative. The numerical GCF of $4$ , $36$ , and $8$ is $4$ . The only shared variable is $a$ , and its lowest power is $a^1$ (or $a$ ). Therefore, the GCF is $-4a$ .

Step 2 — Rewrite each term using the GCF: $(-4a) \cdot a^2 - (-4a) \cdot 9a + (-4a) \cdot 2$

Step 3 — Factor out the GCF: $-4a(a^2 - 9a + 2)$

Step 4 — Check by multiplying: $-4a(a^2 - 9a + 2) = -4a \cdot a^2 - (-4a) \cdot 9a + (-4a) \cdot 2 = -4a^3 + 36a^2 - 8a$ ✓

The factored form is $-4a(a^2 - 9a + 2)$ . Factoring out a negative GCF reverses the signs of all the terms inside the parentheses.

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Updated 2026-04-29

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References

OpenStax Intermediate Algebra

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Intermediate Algebra @ OpenStax

Ch.6 Factoring - Intermediate Algebra @ OpenStax

Algebra

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