Answer
$(3x^2+1 )(9x^4-3x^2+1)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To factor the given expression, $
27x^6+1
,$ use the factoring of the sum of $2$ cubes.
$\bf{\text{Solution Details:}}$
The expressions $
27x^6
$ and $
1
$ are both perfect cubes (the cube root is exact). Hence, $
27x^6+1
,$ is a sum of $2$ cubes. Using the factoring of the sum of $2$ cubes which is given by $a^3+b^3=(a+b)(a^2-ab+b^2),$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
(3x^2)^3+(1 )^3
\\\\=
(3x^2+1 )[(3x^2)^2-3x^2(1 )+(1 )^2]
\\\\=
(3x^2+1 )(9x^4-3x^2+1)
.\end{array}
