1Cademy - Factoring $$100x^2y

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Perfect Square Trinomial

Example

Factoring $100x^2y - 80xy + 16y$

Factor $100x^2y - 80xy + 16y$ completely by first extracting the GCF and then applying the perfect square trinomial pattern to the remaining expression.

Step 1 — Factor out the GCF: The terms $100x^2y$ , $-80xy$ , and $16y$ share a greatest common factor of $4y$ . Factor it out: $100x^2y - 80xy + 16y = 4y(25x^2 - 20x + 4)$

Step 2 — Identify the pattern: Examine the trinomial inside the parentheses, $25x^2 - 20x + 4$ , to see if it fits the perfect square pattern $a^2 - 2ab + b^2$ :

Is the first term a perfect square? Yes: $25x^2 = (5x)^2$ , so $a = 5x$ .
Is the last term a perfect square? Yes: $4 = 2^2$ , so $b = 2$ .
Is the middle term $-2ab$ ? Check: $-2(5x)(2) = -20x$ . Yes, it matches.

Step 3 — Factor the perfect square trinomial: Since the trinomial fits the subtraction pattern, write it as the square of a binomial: $25x^2 - 20x + 4 = (5x - 2)^2$ Remember to keep the GCF $4y$ in the final factored form: $4y(5x - 2)^2$

Step 4 — Check by multiplying: $4y(5x - 2)^2 = 4y[(5x)^2 - 2 \cdot 5x \cdot 2 + 2^2]$ $4y(25x^2 - 20x + 4) = 100x^2y - 80xy + 16y$ ✓

The completely factored form is $4y(5x - 2)^2$ . This example demonstrates that checking for a GCF must be the first step in factoring; doing so reveals the hidden perfect square trinomial structure.

Updated 2026-04-29

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OpenStax Intermediate Algebra

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