Factors of (16 − x⁶ + 2x³y³ − y⁶) are -

To find the factors of the expression 16 - x^6 + 2x^3y^3 - y^6, we can use algebraic grouping and the difference of squares identity: a^2 - b^2 = (a - b)(a + b).

Step-by-Step Solution
Rearrange the expression: Group the variable terms together and factor out a negative sign:
16 - (x^6 - 2x^3y^3 + y^6)

Identify the perfect square trinomial: The expression inside the parentheses, (x^6 - 2x^3y^3 + y^6), is a perfect square trinomial. It can be written as:
(x^3 - y^3)^2

Rewrite the expression: Now the original expression is in the form of a difference of squares:
16 - (x^3 - y^3)^2
= (4)^2 - (x^3 - y^3)^2

Apply the difference of squares identity: Using a^2 - b^2 = (a - b)(a + b), where a = 4 and b = (x^3 - y^3):
(4 - (x^3 - y^3)) * (4 + (x^3 - y^3))
= (4 - x^3 + y^3) * (4 + x^3 - y^3)

Match with the options: Rearranging the terms to match Option B:
(x^3 - y^3 + 4)(y^3 - x^3 + 4)