Question

Factorise the following:

x⁶ – 63x³ – 64

Answer 1

Given expression: x⁶ – 63x³ – 64

Step-wise calculation:

1. Let y = x³.

Then the expression becomes y² – 63y – 64.

2. Factorising the quadratic in y: 

We look for two numbers that multiply to (–64) and add to (–63).

These numbers are (–64) and (1).

So, y² – 63y – 64 = (y – 64)(y + 1).

3. Substitute back y = x³: (x³ – 64)(x³ + 1).

4. Factorise each as a difference and sum of cubes:

x³ – 64 = x³ – 4³

x³ – 64 = (x – 4)(x² + 4x + 16)

x³ + 1 = x³ + 1³

x³ + 1 = (x + 1)(x² – x + 1)