Question

Factorise the following:

x⁶ – 7x³ – 8

Answer 1

Given: x⁶ – 7x³ – 8

Step-wise calculation:

1. Let y = x³.

Then the expression becomes y² – 7y – 8.

2. Factorise the quadratic in (y):

We look for two numbers whose product is (–8) and sum is (–7).

These are (–8) and (1).

So, y² – 7y – 8 = (y – 8)(y + 1).

3. Substitute back (y = x³): (x³ – 8)(x³ + 1).

4. Recognize both as differences and sums of cubes:

x³ – 8 = x³ – 2³

x³ – 8 = (x – 2)(x² + 2x + 4)

x³ + 1 = x³ + 1³

x³ + 1 = (x + 1)(x² – x + 1)

5. Hence the fully factorised form:

x⁶ – 7x³ – 8 = (x – 2)(x² + 2x + 4)(x + 1)(x² – x + 1)