Find the Roots (Zeros) f(x)=x^3+6x^2-4x-24

$f (x) = x^{3} + 6 x^{2} - 4 x - 24$

Set $x^{3} + 6 x^{2} - 4 x - 24$ equal to $0$ .

$x^{3} + 6 x^{2} - 4 x - 24 = 0$

Solve for $x$ .

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.

$(x^{3} + 6 x^{2}) - 4 x - 24 = 0$

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

$x^{2} (x + 6) - 4 (x + 6) = 0$

Factor the polynomial by factoring out the greatest common factor, $x + 6$ .

$(x + 6) (x^{2} - 4) = 0$

Rewrite $4$ as $2^{2}$ .

$(x + 6) (x^{2} - 2^{2}) = 0$

Factor.

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

$(x + 6) ((x + 2) (x - 2)) = 0$

Remove unnecessary parentheses.

$(x + 6) (x + 2) (x - 2) = 0$

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

$x + 6 = 0$

$x + 2 = 0$

$x - 2 = 0$

Set $x + 6$ equal to $0$ and solve for $x$ .

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

Subtract $6$ from both sides of the equation.

$x = - 6$

Set $x + 2$ equal to $0$ and solve for $x$ .

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

The final solution is all the values that make $(x + 6) (x + 2) (x - 2) = 0$ true.

$x = - 6, - 2, 2$