Question

Factorise:

x⁶ – 124x³ – 125

Answer 1

To factorise x⁶ – 124x³ – 125, let’s first treat this expression as a quadratic in terms of x³.

Let y = x³.

The expression becomes:

y² – 124y – 125

Now, factor the quadratic expression:

y² – 124y – 125 = (y – 125) (y + 1)

Substitute y = x³ back into the factored expression:

(x³ – 125) (x³ + 1)

Both of these terms are differences and sums of cubes:

x³ – 125 = (x – 5) (x² + 5x + 25)

x³ + 1 = (x + 1) (x² – x + 1)

Thus, the fully factorised form of x⁶ – 124x³ – 125 is (x – 5) (x + 1)(x² + 5x + 25) (x² – x + 1)