Question

Factorise the following:
$x^6-26x^3-27$

Answer 1

Let $x^3=a$, then the equation $x^6-26x^3-27$ becomes $a^2-26a-27$

Consider the equation $a^2-26a-27$ and factorise it as follows:

$a^2-26a-27=a^2-27a+a-27=a(a-27)+1(a-27)=(a-27)(a+1)$

Now, substitute the value of $a$ as $a=x^3$:

$a^2-26a-27=a^2-27a+a-27=a(a-27)+1(a-27)=(a-27)(a+1)=(x^3-27)(x^3+1)$

$=(x^3-3^3)(x^3+1^3)=(x-3)(x^2+3x+9)(x+1)(x^2-x+1)$

(Using identities $a^3+b^3=(a+b)(a^2+b^2-ab)$ and $a^3-b^3=(a-b)(a^2+b^2+ab)$)

Hence, $x^6-26x^3-27=(x-3)(x^2+3x+9)(x+1)(x^2-x+1)$