Answer
$3(z+3)(z-3)$.
Work Step by Step
The given polynomial is
$= 3z^2-27$
Factor out $3$.
$= 3(z^2-9)$
Write the the polynomial as $a^2-b^2$.
$= 3(z^2-3^2)$
Use difference of two square pattern
$a^2-b^2=(a+b)(a-b)$.
We have $a=z$ and $b=3$.
$= 3(z+3)(z-3)$
Hence, the factor is $3(z+3)(z-3)$.