Answer
$7(2k-5)(4k^2+10k+25)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To factor the given expression, $
56k^3-875
,$ factor first the $GCF.$ Then use the factoring of the sum or difference of $2$ cubes.
$\bf{\text{Solution Details:}}$
The $GCF$ of the terms in the given expression is $
7
,$ since it is the greatest expression that can divide all the terms evenly (no remainder.) Factoring the $GCF$ results to
\begin{array}{l}\require{cancel}
7(8k^3-125)
.\end{array}
The expressions $
8k^3
$ and $
125
$ are both perfect cubes (the cube root is exact). Hence, $
8k^3-125
$ 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}
7[(2k)^3-(5)^3]
\\\\=
7(2k-5)[(2k)^2+2k(5)+(5)^2]
\\\\=
7(2k-5)(4k^2+10k+25)
.\end{array}
