Question

Factorise the following:

(x² – x)² – 26(x² – x) + 120

Answer 1

Given: (x² – x)² – 26(x² – x) + 120

Step-wise calculation:

1. Let p = x² – x.

Then the expression becomes p² – 26p + 120.

2. Factorize the quadratic in (p): 

We look for two numbers that multiply to (120) and add to (–26).

These numbers are (–20) and (–6).

3. So, factorization in terms of (p) is: 

p² – 20p – 6p + 120 = p(p – 20) – 6(p – 20) 

p² – 20p – 6p + 120 = (p – 6)(p – 20)

4. Substitute back (p = x² – x): 

(x² – x – 6)(x² – x – 20)

5. Factor each quadratic factor further:

For (x² – x – 6), factors of (–6) that sum to (–1) are (–3) and (2):

x² – x – 6 = (x – 3)(x + 2)

For (x² – x – 20), factors of (–20) that sum to (–1) are (–5) and (4): 

x² – x – 20 = (x – 5)(x + 4)