Answer
$6z(2z^2-z+3)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To factor the given expression, $
12z^3-6z^2+18z
,$ factor first the $GCF.$ Then use the factoring of trinomials in the form $ax^2+bx+c.$
$\bf{\text{Solution Details:}}$
The $GCF$ of the terms in the given expression is $
6z
,$ 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}
6z(2z^2-z+3)
.\end{array}
In the trinomial expression above, $a=
2
,b=
-1
,\text{ and } c=
3
.$ There are no two numbers whose product is $ac=
2(3)=6
$ and whose sum is $b.$ Hence, the factored form is
\begin{array}{l}\require{cancel}
6z(2z^2-z+3)
.\end{array}