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Answer :
Let's factor the expression [tex]\(3x^2 - 27\)[/tex] completely. Follow these steps:
1. Factor out the Greatest Common Factor (GCF):
First, identify the greatest common factor of the terms in the expression. In this case, both terms [tex]\(3x^2\)[/tex] and [tex]\(27\)[/tex] have a common factor of 3.
So, factor out the 3:
[tex]\[
3x^2 - 27 = 3(x^2 - 9)
\][/tex]
2. Recognize and Apply the Difference of Squares:
Notice that the expression inside the parentheses, [tex]\(x^2 - 9\)[/tex], is a difference of squares. A difference of squares can be factored using the formula:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
Here, [tex]\(x^2 - 9\)[/tex] can be written as [tex]\((x)^2 - (3)^2\)[/tex], fitting the difference of squares formula.
Therefore:
[tex]\[
x^2 - 9 = (x - 3)(x + 3)
\][/tex]
3. Write the Completely Factored Form:
Substitute the factored form of [tex]\(x^2 - 9\)[/tex] back into the expression:
[tex]\[
3(x^2 - 9) = 3(x - 3)(x + 3)
\][/tex]
So, the completely factored form of [tex]\(3x^2 - 27\)[/tex] is:
[tex]\[
3(x - 3)(x + 3)
\][/tex]
1. Factor out the Greatest Common Factor (GCF):
First, identify the greatest common factor of the terms in the expression. In this case, both terms [tex]\(3x^2\)[/tex] and [tex]\(27\)[/tex] have a common factor of 3.
So, factor out the 3:
[tex]\[
3x^2 - 27 = 3(x^2 - 9)
\][/tex]
2. Recognize and Apply the Difference of Squares:
Notice that the expression inside the parentheses, [tex]\(x^2 - 9\)[/tex], is a difference of squares. A difference of squares can be factored using the formula:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
Here, [tex]\(x^2 - 9\)[/tex] can be written as [tex]\((x)^2 - (3)^2\)[/tex], fitting the difference of squares formula.
Therefore:
[tex]\[
x^2 - 9 = (x - 3)(x + 3)
\][/tex]
3. Write the Completely Factored Form:
Substitute the factored form of [tex]\(x^2 - 9\)[/tex] back into the expression:
[tex]\[
3(x^2 - 9) = 3(x - 3)(x + 3)
\][/tex]
So, the completely factored form of [tex]\(3x^2 - 27\)[/tex] is:
[tex]\[
3(x - 3)(x + 3)
\][/tex]
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