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Answer :
To factor the expression [tex]\(3x^2 - 27\)[/tex], you can follow these steps:
1. Identify and Factor Out the Greatest Common Factor (GCF):
Look at the terms of the expression [tex]\(3x^2 - 27\)[/tex]. Both terms, [tex]\(3x^2\)[/tex] and [tex]\(27\)[/tex], have a common factor of 3.
[tex]\[
3x^2 - 27 = 3(x^2 - 9)
\][/tex]
2. Recognize the Difference of Squares:
The expression inside the parentheses, [tex]\(x^2 - 9\)[/tex], is a difference of squares. The difference of squares formula is:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
Here, [tex]\(a = x\)[/tex] and [tex]\(b = 3\)[/tex], because [tex]\(x^2\)[/tex] is [tex]\(x^2\)[/tex] and [tex]\(9\)[/tex] is [tex]\(3^2\)[/tex].
3. Apply the Difference of Squares Formula:
Replace [tex]\(x^2 - 9\)[/tex] with its factored form:
[tex]\[
x^2 - 9 = (x - 3)(x + 3)
\][/tex]
4. Combine the Factored Expressions:
Now substitute the factored form back into the expression:
[tex]\[
3(x^2 - 9) = 3(x - 3)(x + 3)
\][/tex]
So, the expression [tex]\(3x^2 - 27\)[/tex] factors to [tex]\(3(x - 3)(x + 3)\)[/tex].
1. Identify and Factor Out the Greatest Common Factor (GCF):
Look at the terms of the expression [tex]\(3x^2 - 27\)[/tex]. Both terms, [tex]\(3x^2\)[/tex] and [tex]\(27\)[/tex], have a common factor of 3.
[tex]\[
3x^2 - 27 = 3(x^2 - 9)
\][/tex]
2. Recognize the Difference of Squares:
The expression inside the parentheses, [tex]\(x^2 - 9\)[/tex], is a difference of squares. The difference of squares formula is:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
Here, [tex]\(a = x\)[/tex] and [tex]\(b = 3\)[/tex], because [tex]\(x^2\)[/tex] is [tex]\(x^2\)[/tex] and [tex]\(9\)[/tex] is [tex]\(3^2\)[/tex].
3. Apply the Difference of Squares Formula:
Replace [tex]\(x^2 - 9\)[/tex] with its factored form:
[tex]\[
x^2 - 9 = (x - 3)(x + 3)
\][/tex]
4. Combine the Factored Expressions:
Now substitute the factored form back into the expression:
[tex]\[
3(x^2 - 9) = 3(x - 3)(x + 3)
\][/tex]
So, the expression [tex]\(3x^2 - 27\)[/tex] factors to [tex]\(3(x - 3)(x + 3)\)[/tex].
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