Answer
$z=-3,z=5$ and $z=-5$
Work Step by Step
The given equation is
$\Rightarrow z^3+3z^2-25z-75=0$
Group the terms.
$\Rightarrow (z^3+3z^2)+(-25z-75)=0$
Factor each group.
$\Rightarrow z^2(z+3)-25(z+3)=0$
Factor out $(z+3)$.
$\Rightarrow (z+3)(z^2-25)=0$
$\Rightarrow (z+3)(z^2-5^2)=0$
Use difference of two square pattern
$a^2-b^2=(a+b)(a-b)$.
We have $a=z$ and $b=5$.
$\Rightarrow (z+3)(z-5)(z+5)=0$
Use zero product property.
$\Rightarrow z+3=0$ or $z-5=0$ or $z+5=0$
Solve for $n$.
$\Rightarrow z=-3$ or $z=5$ or $z=-5$
Hence, the solutions are $z=-3,z=5$ and $z=-5$.
