(1 - x)⁴

= ((1 - x)²)²

= (1 - 2x + x²)²

= (x² + (1 - 2x))²

= (x²)² + 2x²(1 - 2x) + (1 - 2x)²

= x⁴ + 2x² - 4x³ + 1 - 2(2x) + (2x)²

= x⁴ + 2x² - 4x³ + 1 - 4x + 4x²

= x⁴ - 4x³ + 6x² - 4x + 1


(You can also use binomial theorem to expand if you want)

ii)

Notice that this is very similar to i)'s result.

We just need to replace x with (1 - x³) and work backwards.


(1 - x³)⁴ - 4(1 - x³)³ + 6(1 - x³)² - 4(1 - x³) + 1

= (1 - (1 - x³))⁴

= (x³)⁴

= x¹²