Answer
$(5x-1)(25x^2+5x+1)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To factor the given expression, $
125x^3-1
,$ use the factoring of the sum/difference of $2$ cubes.
$\bf{\text{Solution Details:}}$
The expressions $
125x^3
$ and $
1
$ are both perfect cubes (the cube root is exact). Hence, $
125x^3-1
$ is a $\text{
difference
}$ of $2$ cubes. Using the factoring of the sum or difference of $2$ cubes which is given by $a^3+b^3=(a+b)(a^2-ab+b^2)$ or by $a^3-b^3=(a-b)(a^2+ab+b^2)$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
(5x)^3-(1)^3
\\\\=
(5x-1)[(5x)^2+5x(1)+(1)^2]
\\\\=
(5x-1)(25x^2+5x+1)
.\end{array}
