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Answer :
To factor the expression [tex]\(9y^2 - 121x^4\)[/tex] completely, we can follow these steps:
1. Recognize the Difference of Squares Pattern: The expression [tex]\(9y^2 - 121x^4\)[/tex] can be seen as a difference of squares. The difference of squares pattern is given by [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex].
2. Identify the Squares:
- The first term, [tex]\(9y^2\)[/tex], is a perfect square. It can be written as [tex]\((3y)^2\)[/tex].
- The second term, [tex]\(121x^4\)[/tex], is also a perfect square. It can be written as [tex]\((11x^2)^2\)[/tex].
3. Apply the Difference of Squares Formula:
- Using the pattern [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex], where [tex]\(a = 3y\)[/tex] and [tex]\(b = 11x^2\)[/tex], we can factor the expression as follows:
[tex]\[
9y^2 - 121x^4 = (3y)^2 - (11x^2)^2 = (3y - 11x^2)(3y + 11x^2)
\][/tex]
Therefore, the completely factored form of the expression [tex]\(9y^2 - 121x^4\)[/tex] is [tex]\(-(11x^2 - 3y)(11x^2 + 3y)\)[/tex].
1. Recognize the Difference of Squares Pattern: The expression [tex]\(9y^2 - 121x^4\)[/tex] can be seen as a difference of squares. The difference of squares pattern is given by [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex].
2. Identify the Squares:
- The first term, [tex]\(9y^2\)[/tex], is a perfect square. It can be written as [tex]\((3y)^2\)[/tex].
- The second term, [tex]\(121x^4\)[/tex], is also a perfect square. It can be written as [tex]\((11x^2)^2\)[/tex].
3. Apply the Difference of Squares Formula:
- Using the pattern [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex], where [tex]\(a = 3y\)[/tex] and [tex]\(b = 11x^2\)[/tex], we can factor the expression as follows:
[tex]\[
9y^2 - 121x^4 = (3y)^2 - (11x^2)^2 = (3y - 11x^2)(3y + 11x^2)
\][/tex]
Therefore, the completely factored form of the expression [tex]\(9y^2 - 121x^4\)[/tex] is [tex]\(-(11x^2 - 3y)(11x^2 + 3y)\)[/tex].
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