Solve for z z^3+3z^2-25z-75=0

$z^{3} + 3 z^{2} - 25 z - 75 = 0$

Factor the left side of the equation.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(z^{3} + 3 z^{2}) - 25 z - 75 = 0$

Factor out the greatest common factor (GCF) from each group.

$z^{2} (z + 3) - 25 (z + 3) = 0$

Factor the polynomial by factoring out the greatest common factor, $z + 3$ .

$(z + 3) (z^{2} - 25) = 0$

Rewrite $25$ as $5^{2}$ .

$(z + 3) (z^{2} - 5^{2}) = 0$

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = z$ and $b = 5$ .

$(z + 3) ((z + 5) (z - 5)) = 0$

Remove unnecessary parentheses.

$(z + 3) (z + 5) (z - 5) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$z + 3 = 0$

$z + 5 = 0$

$z - 5 = 0$

Set $z + 3$ equal to $0$ and solve for $z$ .

Set $z + 3$ equal to $0$ .

$z + 3 = 0$

Subtract $3$ from both sides of the equation.

$z = - 3$

Set $z + 5$ equal to $0$ and solve for $z$ .

Set $z + 5$ equal to $0$ .

$z + 5 = 0$

Subtract $5$ from both sides of the equation.

$z = - 5$

Set $z - 5$ equal to $0$ and solve for $z$ .

Set $z - 5$ equal to $0$ .

$z - 5 = 0$

Add $5$ to both sides of the equation.

$z = 5$

The final solution is all the values that make $(z + 3) (z + 5) (z - 5) = 0$ true.

$z = - 3, - 5, 5$