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Answer :
To simplify the expression [tex]\(27x^2 - 3y^2\)[/tex], you can follow these steps:
1. Identify Common Factors:
- Look for any common factors in the terms of the expression. Both terms, [tex]\(27x^2\)[/tex] and [tex]\(-3y^2\)[/tex], have a common factor of 3.
2. Factor Out the Common Factor:
- Factor 3 out of the expression:
[tex]\[
27x^2 - 3y^2 = 3(9x^2 - y^2)
\][/tex]
3. Observe the Remaining Expression:
- After factoring out the 3, the expression inside the parentheses is [tex]\(9x^2 - y^2\)[/tex]. This expression is a difference of squares.
4. Factor the Difference of Squares:
- Recall that a difference of squares [tex]\(a^2 - b^2\)[/tex] can be factored as [tex]\((a + b)(a - b)\)[/tex].
- Here, [tex]\(9x^2\)[/tex] can be written as [tex]\((3x)^2\)[/tex] and [tex]\(y^2\)[/tex] is [tex]\((y)^2\)[/tex].
- Thus, [tex]\(9x^2 - y^2\)[/tex] can be factored into [tex]\((3x + y)(3x - y)\)[/tex].
5. Write the Final Factored Form:
- Incorporate the factor of 3 that was factored out earlier:
[tex]\[
27x^2 - 3y^2 = 3(3x + y)(3x - y)
\][/tex]
This is the fully factored form of the expression [tex]\(27x^2 - 3y^2\)[/tex].
1. Identify Common Factors:
- Look for any common factors in the terms of the expression. Both terms, [tex]\(27x^2\)[/tex] and [tex]\(-3y^2\)[/tex], have a common factor of 3.
2. Factor Out the Common Factor:
- Factor 3 out of the expression:
[tex]\[
27x^2 - 3y^2 = 3(9x^2 - y^2)
\][/tex]
3. Observe the Remaining Expression:
- After factoring out the 3, the expression inside the parentheses is [tex]\(9x^2 - y^2\)[/tex]. This expression is a difference of squares.
4. Factor the Difference of Squares:
- Recall that a difference of squares [tex]\(a^2 - b^2\)[/tex] can be factored as [tex]\((a + b)(a - b)\)[/tex].
- Here, [tex]\(9x^2\)[/tex] can be written as [tex]\((3x)^2\)[/tex] and [tex]\(y^2\)[/tex] is [tex]\((y)^2\)[/tex].
- Thus, [tex]\(9x^2 - y^2\)[/tex] can be factored into [tex]\((3x + y)(3x - y)\)[/tex].
5. Write the Final Factored Form:
- Incorporate the factor of 3 that was factored out earlier:
[tex]\[
27x^2 - 3y^2 = 3(3x + y)(3x - y)
\][/tex]
This is the fully factored form of the expression [tex]\(27x^2 - 3y^2\)[/tex].
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