Given the polynomial [tex]$9x^2y^6 - 25x^4y^8$[/tex], rewrite it as a product of polynomials.

A. [tex]\left(9xy^3 - 25x^2y^4\right)\left(xy^3 + x^2y^4\right)[/tex]

B. [tex]\left(9xy^3 - 25x^2y^4\right)\left(xy^3 - x^2y^4\right)[/tex]

C. [tex]\left(3xy^3 - 5x^2y^4\right)\left(3xy^3 + 5x^2y^4\right)[/tex]

D. [tex]\left(3xy^3 - 5x^2y^4\right)\left(3xy^3 - 5x^2y^4\right)[/tex]



Answer :

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]