Question

Factorise the following:

x⁹ + y⁹

Answer 1

Given: Factorise x⁹ + y⁹

Step-wise calculation:

1. Recognize that x⁹ + y⁹ is a sum of two ninth powers.

This can be written as x⁹ + y⁹ = (x³)³ + (y³)³ i.e., the sum of cubes of x³ and y³.

2. Use the sum of cubes factorisation formula:

a³ + b³ = (a + b)(a² – ab + b²) where a = x³ and b = y³.

3. Apply this:

x⁹ + y⁹ = (x³ + y³)((x³)² – x³y³ + (y³)²) 

x⁹ + y⁹ = (x³ + y³)(x⁶ – x³y³ + y⁶)

4. Further factorise the x³ + y³ term using sum of cubes:

x³ + y³ = (x + y)(x² – xy + y²)

Therefore, the full factorisation is x⁹ + y⁹ = (x + y)(x² – xy + y²)(x⁶ – x³y³ + y⁶).