1Cademy - Factoring $$81y^2

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Perfect Square Trinomials Pattern

Example

Factoring $81y^2 - 72y + 16$

Factor $81y^2 - 72y + 16$ by recognizing it as a perfect square trinomial with a negative middle term.

Step 1 — Check whether the trinomial fits the pattern $a^2 - 2ab + b^2$ :

Is the first term a perfect square? Yes: $81y^2 = (9y)^2$ , so $a = 9y$ .
Is the last term a perfect square? Yes: $16 = 4^2$ , so $b = 4$ .
Is the middle term $2ab$ ? Check: $2 \cdot 9y \cdot 4 = 72y$ . The middle term is $-72y$ , and since the middle term is negative, the pattern $a^2 - 2ab + b^2 = (a - b)^2$ applies. ✓

Step 2 — Write the square of the binomial: Since the trinomial matches the subtraction pattern with $a = 9y$ and $b = 4$ :

$81y^2 - 72y + 16 = (9y)^2 - 2(9y)(4) + 4^2 = (9y - 4)^2$

Step 3 — Check by multiplying: $(9y - 4)^2 = (9y)^2 - 2 \cdot 9y \cdot 4 + 4^2 = 81y^2 - 72y + 16$ ✓.

The factored form is $(9y - 4)^2$ . This example applies the subtraction form of the Perfect Square Trinomials Pattern. When the middle term of the trinomial is negative, the factored binomial square uses subtraction: $(a - b)^2$ . The first and last terms (the two squares) are always positive regardless of the middle term's sign.

Updated 2026-04-29

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