Question

Factorise:

x⁴ + x³ + 27x + 27

Answer 1

To factorise x⁴ + x³ + 27x + 27, let’s begin by grouping the terms in pairs:

x⁴ + x³ + 27x + 27 = (x⁴ + x³) + (27x + 27)

Now, factor out the common terms from each group:

= x³(x + 1) + 27(x + 1)

Next, we can factor out the common factor (x + 1):

= (x + 1) (x³ + 27)

We now recognise that x³ + 27 is a sum of cubes, which can be factored as:

 x³ + 27 = (x + 3) (x² – 3x + 9)

Thus, the fully factorised form of the expression is:

 x⁴ + x³ + 27x + 27 = (x + 1) (x + 3) (x² – 3x + 9)