1Cademy - Factoring $$8x^2y

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Perfect Square Trinomial

Example

Factoring $8x^2y - 24xy + 18y$

Factor $8x^2y - 24xy + 18y$ completely by factoring out the greatest common factor and recognizing the remaining trinomial as a perfect square.

Step 1 — Factor out the GCF: The terms $8x^2y$ , $-24xy$ , and $18y$ share a common factor of $2y$ . Factor it out: $8x^2y - 24xy + 18y = 2y(4x^2 - 12x + 9)$

Step 2 — Identify the pattern: Check if the trinomial $4x^2 - 12x + 9$ fits the perfect square pattern $a^2 - 2ab + b^2$ :

The first term is a perfect square: $4x^2 = (2x)^2$ , so $a = 2x$ .
The last term is a perfect square: $9 = 3^2$ , so $b = 3$ .
The middle term is $-2ab$ : $-2(2x)(3) = -12x$ . It matches.

Step 3 — Factor the perfect square trinomial: Write the trinomial as the square of a binomial: $4x^2 - 12x + 9 = (2x - 3)^2$ Include the GCF $2y$ in the final product: $2y(2x - 3)^2$

The completely factored form is $2y(2x - 3)^2$ . Factoring out the GCF is an essential first step before the perfect square pattern can be applied.

Updated 2026-04-29

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OpenStax Intermediate Algebra

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