Question

Factorise the following:

5(a – 2b)³ + 15(a – 2b)² – 25(a – 2b)

Answer 1

Given expression: 5(a – 2b)³ + 15(a – 2b)² – 25(a – 2b)

Step-wise calculation:

1. Identify the common factor in all terms.

Here, each term contains a factor of 5(a – 2b).

2. Factor 5(a – 2b) out:

5(a – 2b)³ + 15(a – 2b)² – 25(a – 2b)

= 5(a – 2b) [(a – 2b)² + 3(a – 2b) – 5]

3. Let x = (a – 2b), then the expression inside the bracket becomes x² + 3x – 5

4. Try to factorise x² + 3x – 5:

The quadratic does not factor neatly over integers, so this is the simplified factorisation.

5(a – 2b)³ + 15(a – 2b)² – 25(a – 2b)

= 5(a – 2b) [(a – 2b)² + 3(a – 2b) – 5]

This is the fully factorised form by taking out the common factor.

Alternatively, from the source, the answer given is:

5(a – 2b) (a² – 4ab + 4b² + 3a – 6b – 5)

Which matches the expanded form of (a – 2b)² + 3(a – 2b) – 5.

Hence, 5(a – 2b) (a² – 4ab + 4b² + 3a – 6b – 5).