1Cademy - Factoring $$27p^2q + 90pq + 75q$$

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Perfect Square Trinomial

Example

Factoring $27p^2q + 90pq + 75q$

Factor $27p^2q + 90pq + 75q$ completely by first extracting the GCF and then applying the perfect square trinomial pattern.

Step 1 — Factor out the GCF: The terms $27p^2q$ , $90pq$ , and $75q$ share a common factor of $3q$ . Factor it out: $27p^2q + 90pq + 75q = 3q(9p^2 + 30p + 25)$

Step 2 — Identify the pattern: Check if the trinomial $9p^2 + 30p + 25$ fits the perfect square pattern $a^2 + 2ab + b^2$ :

The first term is a perfect square: $9p^2 = (3p)^2$ , so $a = 3p$ .
The last term is a perfect square: $25 = 5^2$ , so $b = 5$ .
The middle term is $2ab$ : $2(3p)(5) = 30p$ . It matches.

Step 3 — Factor the perfect square trinomial: Write the trinomial as the square of a binomial, using addition because the middle term is positive: $9p^2 + 30p + 25 = (3p + 5)^2$ Include the GCF $3q$ in the final product: $3q(3p + 5)^2$

The completely factored form is $3q(3p + 5)^2$ . Factoring out the GCF reveals the perfect square structure that was initially obscured.

Updated 2026-04-29

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OpenStax Intermediate Algebra

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