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Answer :
We start with the expression
[tex]$$
27y^2 - 3x^2.
$$[/tex]
Step 1: Factor Out the Greatest Common Factor (GCF).
Both terms have a common factor of [tex]$3$[/tex], so we factor it out:
[tex]$$
27y^2 - 3x^2 = 3(9y^2 - x^2).
$$[/tex]
Step 2: Factor the Difference of Two Squares.
The expression inside the parentheses, [tex]$9y^2 - x^2$[/tex], is a difference of two squares. We can write:
[tex]$$
9y^2 = (3y)^2 \quad \text{and} \quad x^2 = (x)^2.
$$[/tex]
The difference of two squares formula is given by:
[tex]$$
a^2 - b^2 = (a - b)(a + b).
$$[/tex]
Here, [tex]$a = 3y$[/tex] and [tex]$b = x$[/tex]. Therefore,
[tex]$$
9y^2 - x^2 = (3y - x)(3y + x).
$$[/tex]
Step 3: Write the Fully Factored Expression.
Substituting back into the expression, we obtain:
[tex]$$
27y^2 - 3x^2 = 3(3y - x)(3y + x).
$$[/tex]
Thus, the final answer is
[tex]$$
3(3y - x)(3y + x).
$$[/tex]
[tex]$$
27y^2 - 3x^2.
$$[/tex]
Step 1: Factor Out the Greatest Common Factor (GCF).
Both terms have a common factor of [tex]$3$[/tex], so we factor it out:
[tex]$$
27y^2 - 3x^2 = 3(9y^2 - x^2).
$$[/tex]
Step 2: Factor the Difference of Two Squares.
The expression inside the parentheses, [tex]$9y^2 - x^2$[/tex], is a difference of two squares. We can write:
[tex]$$
9y^2 = (3y)^2 \quad \text{and} \quad x^2 = (x)^2.
$$[/tex]
The difference of two squares formula is given by:
[tex]$$
a^2 - b^2 = (a - b)(a + b).
$$[/tex]
Here, [tex]$a = 3y$[/tex] and [tex]$b = x$[/tex]. Therefore,
[tex]$$
9y^2 - x^2 = (3y - x)(3y + x).
$$[/tex]
Step 3: Write the Fully Factored Expression.
Substituting back into the expression, we obtain:
[tex]$$
27y^2 - 3x^2 = 3(3y - x)(3y + x).
$$[/tex]
Thus, the final answer is
[tex]$$
3(3y - x)(3y + x).
$$[/tex]
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