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Answer :
To factor the expression
[tex]$$3x^2 - 27y^2,$$[/tex]
follow these steps:
1. Factor out the common factor:
Both terms have a common factor of [tex]$3$[/tex]. Factor it out:
[tex]$$
3x^2 - 27y^2 = 3(x^2 - 9y^2).
$$[/tex]
2. Recognize the difference of squares:
Inside the parentheses, we have a difference of squares. Notice that:
[tex]$$
x^2 - 9y^2 = x^2 - (3y)^2.
$$[/tex]
A difference of squares factors as:
[tex]$$
a^2 - b^2 = (a - b)(a + b),
$$[/tex]
where in this case [tex]$a = x$[/tex] and [tex]$b = 3y$[/tex]. Thus,
[tex]$$
x^2 - 9y^2 = (x - 3y)(x + 3y).
$$[/tex]
3. Write the fully factored form:
Substitute the factored form back into the expression:
[tex]$$
3(x^2 - 9y^2) = 3(x - 3y)(x + 3y).
$$[/tex]
Thus, the completely factored form of the expression is:
[tex]$$
3x^2 - 27y^2 = 3(x - 3y)(x + 3y).
$$[/tex]
[tex]$$3x^2 - 27y^2,$$[/tex]
follow these steps:
1. Factor out the common factor:
Both terms have a common factor of [tex]$3$[/tex]. Factor it out:
[tex]$$
3x^2 - 27y^2 = 3(x^2 - 9y^2).
$$[/tex]
2. Recognize the difference of squares:
Inside the parentheses, we have a difference of squares. Notice that:
[tex]$$
x^2 - 9y^2 = x^2 - (3y)^2.
$$[/tex]
A difference of squares factors as:
[tex]$$
a^2 - b^2 = (a - b)(a + b),
$$[/tex]
where in this case [tex]$a = x$[/tex] and [tex]$b = 3y$[/tex]. Thus,
[tex]$$
x^2 - 9y^2 = (x - 3y)(x + 3y).
$$[/tex]
3. Write the fully factored form:
Substitute the factored form back into the expression:
[tex]$$
3(x^2 - 9y^2) = 3(x - 3y)(x + 3y).
$$[/tex]
Thus, the completely factored form of the expression is:
[tex]$$
3x^2 - 27y^2 = 3(x - 3y)(x + 3y).
$$[/tex]
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