x^9 - 216

We will rewrite as cubes.

==> (x^3)^3 - 6^3

Now we will factor as a difference between two cubes.

==> We know that:

(a^3 - b^3) = (a-b) ( a^2 +ab + b^2)

==> (x^3)^3 - 6^3 = (x^3-6)(x^6 + 6x^3 + 36)

Now we can not factor any further.

Then we conclude that:

**x^9 - 216 = (x^3-6)(x^6 + 6x^3 + 36) **

We have to factorize x^9 - 216

x^9 - 216

=> x^9 - 6^3

=> (x^3)^3 - 6^3

Use (a-b)(a^2+b^2+ab)

=> (x^3 - 6)(x^6 + 36 + 6x^6)

**The required result is (x^3 - 6)(x^6 + 36 + 6x^3)**

We'll re-write the given expression as a difference of two cubes:

x^9 - 216 = (x^3)^3 - (6)^3

We'll apply the formula:

a^3 - b^3 = (a-b)(a^2 + ab + b^2) (*)

We'll put a = x^3 and b = 6

We'll substitute a and b into the formula (*):

The completely factorised expression is:

**(x^3)^3 - 216 = (x^3 - 6)(x^6 + 6x^3 + 36)**