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Answer :
Sure! Let's work on factoring the polynomial expression [tex]\(16y^4 - 625x^4\)[/tex].
1. Identify as a Difference of Squares:
The given expression is [tex]\(16y^4 - 625x^4\)[/tex]. Notice that both [tex]\(16y^4\)[/tex] and [tex]\(625x^4\)[/tex] are perfect squares:
- [tex]\(16y^4 = (4y^2)^2\)[/tex]
- [tex]\(625x^4 = (25x^2)^2\)[/tex]
This allows us to recognize the expression as a difference of squares, which has the form [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex].
2. Factor the Initial Expression:
Applying the difference of squares formula, we get:
[tex]\[
16y^4 - 625x^4 = (4y^2 - 25x^2)(4y^2 + 25x^2)
\][/tex]
3. Further Factor the First Term:
Now, look at the first term [tex]\(4y^2 - 25x^2\)[/tex], which is itself a difference of squares:
- [tex]\(4y^2 = (2y)^2\)[/tex]
- [tex]\(25x^2 = (5x)^2\)[/tex]
Using the difference of squares again, we factor it as:
[tex]\[
4y^2 - 25x^2 = (2y - 5x)(2y + 5x)
\][/tex]
4. Combine All Factors:
Substitute this back into our expression from step 2:
[tex]\[
(16y^4 - 625x^4) = (2y - 5x)(2y + 5x)(4y^2 + 25x^2)
\][/tex]
This final expression [tex]\((2y - 5x)(2y + 5x)(4y^2 + 25x^2)\)[/tex] is the fully factored form of the original polynomial [tex]\(16y^4 - 625x^4\)[/tex].
1. Identify as a Difference of Squares:
The given expression is [tex]\(16y^4 - 625x^4\)[/tex]. Notice that both [tex]\(16y^4\)[/tex] and [tex]\(625x^4\)[/tex] are perfect squares:
- [tex]\(16y^4 = (4y^2)^2\)[/tex]
- [tex]\(625x^4 = (25x^2)^2\)[/tex]
This allows us to recognize the expression as a difference of squares, which has the form [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex].
2. Factor the Initial Expression:
Applying the difference of squares formula, we get:
[tex]\[
16y^4 - 625x^4 = (4y^2 - 25x^2)(4y^2 + 25x^2)
\][/tex]
3. Further Factor the First Term:
Now, look at the first term [tex]\(4y^2 - 25x^2\)[/tex], which is itself a difference of squares:
- [tex]\(4y^2 = (2y)^2\)[/tex]
- [tex]\(25x^2 = (5x)^2\)[/tex]
Using the difference of squares again, we factor it as:
[tex]\[
4y^2 - 25x^2 = (2y - 5x)(2y + 5x)
\][/tex]
4. Combine All Factors:
Substitute this back into our expression from step 2:
[tex]\[
(16y^4 - 625x^4) = (2y - 5x)(2y + 5x)(4y^2 + 25x^2)
\][/tex]
This final expression [tex]\((2y - 5x)(2y + 5x)(4y^2 + 25x^2)\)[/tex] is the fully factored form of the original polynomial [tex]\(16y^4 - 625x^4\)[/tex].
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