1Cademy - Factoring $$4a^2

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Strategy to Factor Polynomials Completely

Example

Factoring $4a^2 - 12ab + 9b^2$

Factor $4a^2 - 12ab + 9b^2$ completely by recognizing it as a perfect square trinomial in two variables.

Step 1 — Is there a GCF? No. The three terms $4a^2$ , $12ab$ , and $9b^2$ share no common factor.

Step 2 — Classify by number of terms. It is a trinomial with a leading coefficient other than $1$ (the leading coefficient is $4$ ). However, the first term is a perfect square: $4a^2 = (2a)^2$ .

Step 3 — Check whether the trinomial fits the perfect square pattern $a^2 - 2ab + b^2$ :

Is the last term a perfect square? Yes: $9b^2 = (3b)^2$ .
Does it fit the pattern $a^2 - 2ab + b^2$ ? Check the middle term: $2 \cdot (2a) \cdot (3b) = 12ab$ . The middle term is $-12ab$ , which matches $-2ab$ in the pattern. ✓

Step 4 — Write as a square of a binomial: Since the trinomial matches the subtraction pattern with $a = 2a$ and $b = 3b$ :

$4a^2 - 12ab + 9b^2 = (2a - 3b)^2$

Step 5 — Check.

Is the expression factored completely? Yes — the binomial $(2a - 3b)$ is not a difference of squares.
Verify by multiplying: $(2a - 3b)^2 = (2a)^2 - 2(2a)(3b) + (3b)^2 = 4a^2 - 12ab + 9b^2$ ✓.

The completely factored form is $(2a - 3b)^2$ . This example applies the general factoring strategy to a two-variable trinomial. The strategy directs the solver to check for a GCF first (none here), then classify by number of terms (trinomial), and finally test whether it fits a special product pattern. Recognizing that both the first and last terms are perfect squares triggers a check against the Perfect Square Trinomials Pattern, which succeeds because the middle term equals twice the product of the square roots of the first and last terms.

Updated 2026-04-21

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OpenStax Elementary Algebra

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