1Cademy - Factoring $$4y^2 + 24y + 28$$

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Factoring the Greatest Common Factor from a Polynomial

Example

Factoring $4y^2 + 24y + 28$

Factor $4y^2 + 24y + 28$ by extracting the greatest common factor from all three terms of the trinomial.

Step 1 — Find the GCF of $4y^2$ , $24y$ , and $28$ : Factor each term into primes and expanded variables: $4y^2 = 2 \cdot 2 \cdot y \cdot y$ , $24y = 2 \cdot 2 \cdot 2 \cdot 3 \cdot y$ , and $28 = 2 \cdot 2 \cdot 7$ . The factors shared by all three terms are two $2$ s, so $\text{GCF} = 2 \cdot 2 = 4$ .

Step 2 — Rewrite each term as a product of the GCF: Express $4y^2 = 4 \cdot y^2$ , $24y = 4 \cdot 6y$ , and $28 = 4 \cdot 7$ , giving $4 \cdot y^2 + 4 \cdot 6y + 4 \cdot 7$ .

Step 3 — Factor out the GCF: $4(y^2 + 6y + 7)$ .

Step 4 — Check by multiplying: $4(y^2 + 6y + 7) = 4 \cdot y^2 + 4 \cdot 6y + 4 \cdot 7 = 4y^2 + 24y + 28$ ✓.

The factored form is $4(y^2 + 6y + 7)$ . This example extends GCF factoring from binomials to trinomials. The same four-step process applies — the only difference is that the GCF must be a factor of all three terms, not just two, and each of the three terms is rewritten using the GCF before factoring.

Updated 2026-04-21

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OpenStax Elementary Algebra

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