Factorize: x⁸ – y⁸.

We can write x⁸ – y⁸ = (x⁴)² – (y⁴)²

We know: a² – b² = (a + b)(a – b)

x⁸ – y⁸ = (x⁴ + y⁴)(x⁴ – y⁴)

Rewriting (x⁴ – y⁴) as (x²)² – (y²)², we have

x⁸ – y⁸ = (x⁴ + y⁴)[(x²)² – (y²)²]

⇒ x⁸ – y⁸ = (x⁴ + y⁴)(x² + y²)(x² – y²)

Using the above identity to factorize (x² – y²), we have

x⁸ – y⁸ = (x⁴ + y⁴)(x² + y²)(x – y)(x + y)

∴ x⁸ – y⁸ = (x – y)(x + y)(x² + y²)(x⁴ + y⁴)

Hence, the factors of x⁸ – y⁸ are (x – y), (x + y), (x² + y²) and (x⁴ + y⁴).