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Answer :
To completely factor the expression [tex]\(625x^4 - 81y^4\)[/tex] over the complex number system, follow these steps:
1. Recognize the Form: The expression [tex]\(625x^4 - 81y^4\)[/tex] is a 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].
2. Identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex]: In the expression [tex]\(625x^4 - 81y^4\)[/tex], observe that:
- [tex]\(a^2 = (25x^2)^2 = 625x^4\)[/tex], so [tex]\(a = 25x^2\)[/tex].
- [tex]\(b^2 = (9y^2)^2 = 81y^4\)[/tex], so [tex]\(b = 9y^2\)[/tex].
3. Apply the Difference of Squares Formula: Use the formula for difference of squares:
[tex]\[
625x^4 - 81y^4 = (25x^2)^2 - (9y^2)^2 = (25x^2 - 9y^2)(25x^2 + 9y^2)
\][/tex]
4. Further Factor [tex]\(25x^2 - 9y^2\)[/tex]: Notice that [tex]\(25x^2 - 9y^2\)[/tex] is itself a difference of squares:
- Here, [tex]\((5x)^2 = 25x^2\)[/tex] and [tex]\((3y)^2 = 9y^2\)[/tex].
- Therefore, [tex]\(25x^2 - 9y^2 = (5x - 3y)(5x + 3y)\)[/tex].
5. Combine the Results: Now, write the completely factored form as:
[tex]\[
625x^4 - 81y^4 = (5x - 3y)(5x + 3y)(25x^2 + 9y^2)
\][/tex]
This is the complete factorization of the given expression over the complex number system.
1. Recognize the Form: The expression [tex]\(625x^4 - 81y^4\)[/tex] is a 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].
2. Identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex]: In the expression [tex]\(625x^4 - 81y^4\)[/tex], observe that:
- [tex]\(a^2 = (25x^2)^2 = 625x^4\)[/tex], so [tex]\(a = 25x^2\)[/tex].
- [tex]\(b^2 = (9y^2)^2 = 81y^4\)[/tex], so [tex]\(b = 9y^2\)[/tex].
3. Apply the Difference of Squares Formula: Use the formula for difference of squares:
[tex]\[
625x^4 - 81y^4 = (25x^2)^2 - (9y^2)^2 = (25x^2 - 9y^2)(25x^2 + 9y^2)
\][/tex]
4. Further Factor [tex]\(25x^2 - 9y^2\)[/tex]: Notice that [tex]\(25x^2 - 9y^2\)[/tex] is itself a difference of squares:
- Here, [tex]\((5x)^2 = 25x^2\)[/tex] and [tex]\((3y)^2 = 9y^2\)[/tex].
- Therefore, [tex]\(25x^2 - 9y^2 = (5x - 3y)(5x + 3y)\)[/tex].
5. Combine the Results: Now, write the completely factored form as:
[tex]\[
625x^4 - 81y^4 = (5x - 3y)(5x + 3y)(25x^2 + 9y^2)
\][/tex]
This is the complete factorization of the given expression over the complex number system.
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