Final answer:
The polynomial 625x⁴ - 256y⁴ is a difference of squares and fully factors into (25x² + 16y²)(5x + 4y)(5x - 4y).
Step-by-step explanation:
The polynomial given is 625x⁴ - 256y⁴, which is a difference of squares. It can be factored using the fact that a² - b² = (a+b)(a-b). In this case, we can set a = 25x² and b = 16y² since 25² = 625 and 16² = 256, and both x² and y² are perfect squares. Following this observation, the factored form of the polynomial will be (25x² + 16y²)(25x² - 16y²).
Still, these are not fully factored since the second term is again a difference of squares. We can set a = 5x and b = 4y, since (5x)² = 25x² and (4y)² = 16y², hence the fully factored form will be:
(25x² + 16y²)(5x + 4y)(5x - 4y)