Factorise: 16×4 – y4

This is a difference of two squares problem, done twice. Remember the rule: a² − b² = (a − b)(a + b).

Write 16x⁴ − y⁴ as squares: 16x⁴ = (4x²)² and y⁴ = (y²)². So with a = 4x² and b = y², we get 16x⁴ − y⁴ = (4x² − y²)(4x² + y²).

The first bracket, 4x² − y², is itself a difference of two squares, since 4x² = (2x)² and y² = (y)². So 4x² − y² = (2x − y)(2x + y). The second bracket, 4x² + y², is a sum of squares, and that cannot be factorised any further with real numbers.

Putting it all together: 16x⁴ − y⁴ = (2x − y)(2x + y)(4x² + y²). You can check by expanding: (2x − y)(2x + y) = 4x² − y², and (4x² − y²)(4x² + y²) = 16x⁴ − y⁴. ✔