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Answer :
To factor the expression [tex]\(121x^4 - 9y^4\)[/tex] completely, we can follow these steps:
1. Recognize the expression as a difference of squares:
The given expression is [tex]\(121x^4 - 9y^4\)[/tex].
We notice that [tex]\(121x^4 = (11x^2)^2\)[/tex] and [tex]\(9y^4 = (3y^2)^2\)[/tex].
Therefore, the expression is a difference of squares, which can be expressed as:
[tex]\((a^2 - b^2) = (a + b)(a - b)\)[/tex].
2. Apply the difference of squares formula:
Let [tex]\(a = 11x^2\)[/tex] and [tex]\(b = 3y^2\)[/tex].
Using the formula, we can factor the expression:
[tex]\((11x^2)^2 - (3y^2)^2 = (11x^2 + 3y^2)(11x^2 - 3y^2)\)[/tex].
3. Further factor each quadratic expression if possible:
- Now, let's address each factor:
- [tex]\(11x^2 + 3y^2\)[/tex] is a sum of squares and is not further factorable over the real numbers.
- [tex]\(11x^2 - 3y^2\)[/tex] is still a difference of squares, so it can be factored further:
[tex]\((11x^2 - 3y^2) = (\sqrt{11}x + \sqrt{3}y)(\sqrt{11}x - \sqrt{3}y)\)[/tex].
4. Write the complete factorization:
The complete factorization of the expression [tex]\(121x^4 - 9y^4\)[/tex] is:
[tex]\((11x^2 + 3y^2)(\sqrt{11}x + \sqrt{3}y)(\sqrt{11}x - \sqrt{3}y)\)[/tex].
Thus, the expression [tex]\(121x^4 - 9y^4\)[/tex] factors completely into [tex]\((\sqrt{11}x + \sqrt{3}y)(\sqrt{11}x - \sqrt{3}y)(\sqrt{11}x + \sqrt{3}y)(\sqrt{11}x - \sqrt{3}y)\)[/tex].
1. Recognize the expression as a difference of squares:
The given expression is [tex]\(121x^4 - 9y^4\)[/tex].
We notice that [tex]\(121x^4 = (11x^2)^2\)[/tex] and [tex]\(9y^4 = (3y^2)^2\)[/tex].
Therefore, the expression is a difference of squares, which can be expressed as:
[tex]\((a^2 - b^2) = (a + b)(a - b)\)[/tex].
2. Apply the difference of squares formula:
Let [tex]\(a = 11x^2\)[/tex] and [tex]\(b = 3y^2\)[/tex].
Using the formula, we can factor the expression:
[tex]\((11x^2)^2 - (3y^2)^2 = (11x^2 + 3y^2)(11x^2 - 3y^2)\)[/tex].
3. Further factor each quadratic expression if possible:
- Now, let's address each factor:
- [tex]\(11x^2 + 3y^2\)[/tex] is a sum of squares and is not further factorable over the real numbers.
- [tex]\(11x^2 - 3y^2\)[/tex] is still a difference of squares, so it can be factored further:
[tex]\((11x^2 - 3y^2) = (\sqrt{11}x + \sqrt{3}y)(\sqrt{11}x - \sqrt{3}y)\)[/tex].
4. Write the complete factorization:
The complete factorization of the expression [tex]\(121x^4 - 9y^4\)[/tex] is:
[tex]\((11x^2 + 3y^2)(\sqrt{11}x + \sqrt{3}y)(\sqrt{11}x - \sqrt{3}y)\)[/tex].
Thus, the expression [tex]\(121x^4 - 9y^4\)[/tex] factors completely into [tex]\((\sqrt{11}x + \sqrt{3}y)(\sqrt{11}x - \sqrt{3}y)(\sqrt{11}x + \sqrt{3}y)(\sqrt{11}x - \sqrt{3}y)\)[/tex].
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