Question

solve the equation $z^3 - z^2 - 12z = 0$. the solutions are $z = \square$, $z = \square$, and $z = \square$.

Explanation:

Step1: Factor out common term

$z^3 - z^2 - 12z = z(z^2 - z - 12)$

Step2: Factor quadratic expression

Find two numbers that multiply to $-12$ and add to $-1$: $-4$ and $3$.
$z^2 - z - 12 = (z - 4)(z + 3)$

Step3: Set factors to zero

$z = 0$, $z - 4 = 0$, $z + 3 = 0$

Step4: Solve for z

$z = 0$, $z = 4$, $z = -3$

Answer:

$z = 0$, $z = 4$, and $z = -3$