To rewrite the polynomial \( 9 x^2 y^6 - 25 x^4 y^8 \) as a product of polynomials, we'll conduct a detailed factorization.
1. Identify Common Factors:
Begin by observing if there is any common factor in both terms of the polynomial \( 9 x^2 y^6 - 25 x^4 y^8 \).
2. Factor out Common Terms:
Notice that \( x^2 y^6 \) is a common factor:
[tex]\[
9 x^2 y^6 - 25 x^4 y^8 = x^2 y^6 (9 - 25 x^2 y^2)
\][/tex]
3. Recognize the Difference of Squares:
Observe that \( 9 - 25 x^2 y^2 \) is a difference of squares which can be factored using the identity \( a^2 - b^2 = (a - b)(a + b) \). Here, we have:
[tex]\[
9 - 25 x^2 y^2 = (3)^2 - (5 x y)^2
\][/tex]
Using the difference of squares formula:
[tex]\[
9 - 25 x^2 y^2 = (3 - 5 x y)(3 + 5 x y)
\][/tex]
4. Combine the Factored Forms:
Substitute this factorization back into the expression:
[tex]\[
x^2 y^6 (3 - 5 x y)(3 + 5 x y)
\][/tex]
5. Re-arrange for a Standard Form:
This gives us:
[tex]\[
(3 x y^3 - 5 x^2 y^4)(3 x y^3 + 5 x^2 y^4)
\][/tex]
Therefore, the polynomial \( 9 x^2 y^6 - 25 x^4 y^8 \) can be rewritten as:
[tex]\[
(3 x y^3 - 5 x^2 y^4)(3 x y^3 + 5 x^2 y^4)
\][/tex]
From the given options, the correct factored form is:
[tex]\[
\left(3 x y^3 - 5 x^2 y^4\right)\left(3 x y^3 + 5 x^2 y^4\right)
\][/tex]
So, the answer is:
[tex]\[
\boxed{\left(3 x y^3 - 5 x^2 y^4\right)\left(3 x y^3 + 5 x^2 y^4\right)}
\][/tex]
