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Answer :
To factor the polynomial expression [tex]\(16y^4 - 625x^4\)[/tex], let's follow these steps:
1. Identify as a Difference of Squares:
The expression [tex]\(16y^4 - 625x^4\)[/tex] is a difference of squares. We can recognize this because it is in the form [tex]\(a^2 - b^2\)[/tex], which factors into [tex]\((a - b)(a + b)\)[/tex].
2. Rewrite as Squares:
- [tex]\(16y^4\)[/tex] can be rewritten as [tex]\((4y^2)^2\)[/tex].
- [tex]\(625x^4\)[/tex] can be rewritten as [tex]\((25x^2)^2\)[/tex].
So, we have:
[tex]\[
16y^4 - 625x^4 = (4y^2)^2 - (25x^2)^2
\][/tex]
3. Apply the Difference of Squares Formula:
Using the formula [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex], we can factor the expression as:
[tex]\[
(4y^2 - 25x^2)(4y^2 + 25x^2)
\][/tex]
4. Further Factor the Difference of Squares:
The factor [tex]\(4y^2 - 25x^2\)[/tex] is itself a difference of squares and can be further factored:
- [tex]\(4y^2\)[/tex] is [tex]\((2y)^2\)[/tex].
- [tex]\(25x^2\)[/tex] is [tex]\((5x)^2\)[/tex].
Applying the difference of squares formula again to [tex]\(4y^2 - 25x^2\)[/tex], we get:
[tex]\[
(2y - 5x)(2y + 5x)
\][/tex]
5. Combine All Factors:
Therefore, the original polynomial [tex]\(16y^4 - 625x^4\)[/tex] can be expressed as:
[tex]\[
(2y - 5x)(2y + 5x)(4y^2 + 25x^2)
\][/tex]
And that's the fully factored form of the polynomial expression [tex]\(16y^4 - 625x^4\)[/tex].
1. Identify as a Difference of Squares:
The expression [tex]\(16y^4 - 625x^4\)[/tex] is a difference of squares. We can recognize this because it is in the form [tex]\(a^2 - b^2\)[/tex], which factors into [tex]\((a - b)(a + b)\)[/tex].
2. Rewrite as Squares:
- [tex]\(16y^4\)[/tex] can be rewritten as [tex]\((4y^2)^2\)[/tex].
- [tex]\(625x^4\)[/tex] can be rewritten as [tex]\((25x^2)^2\)[/tex].
So, we have:
[tex]\[
16y^4 - 625x^4 = (4y^2)^2 - (25x^2)^2
\][/tex]
3. Apply the Difference of Squares Formula:
Using the formula [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex], we can factor the expression as:
[tex]\[
(4y^2 - 25x^2)(4y^2 + 25x^2)
\][/tex]
4. Further Factor the Difference of Squares:
The factor [tex]\(4y^2 - 25x^2\)[/tex] is itself a difference of squares and can be further factored:
- [tex]\(4y^2\)[/tex] is [tex]\((2y)^2\)[/tex].
- [tex]\(25x^2\)[/tex] is [tex]\((5x)^2\)[/tex].
Applying the difference of squares formula again to [tex]\(4y^2 - 25x^2\)[/tex], we get:
[tex]\[
(2y - 5x)(2y + 5x)
\][/tex]
5. Combine All Factors:
Therefore, the original polynomial [tex]\(16y^4 - 625x^4\)[/tex] can be expressed as:
[tex]\[
(2y - 5x)(2y + 5x)(4y^2 + 25x^2)
\][/tex]
And that's the fully factored form of the polynomial expression [tex]\(16y^4 - 625x^4\)[/tex].
Thank you for reading the article Factor the polynomial expression tex 16y 4 625x 4 tex 16y 4 625x 4 4y 2 25x 2 4y 2 25x 2. We hope the information provided is useful and helps you understand this topic better. Feel free to explore more helpful content on our website!
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