Find the Roots (Zeros) f(x)=-x(3x-2)^2(x+9)^5

$f (x) = - x {(3 x - 2)}^{2} {(x + 9)}^{5}$

Set $- x {(3 x - 2)}^{2} {(x + 9)}^{5}$ equal to $0$ .

$- x {(3 x - 2)}^{2} {(x + 9)}^{5} = 0$

Solve for $x$ .

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

$x = 0$

${(3 x - 2)}^{2} = 0$

${(x + 9)}^{5} = 0$

Set $x$ equal to $0$ .

$x = 0$

Set ${(3 x - 2)}^{2}$ equal to $0$ and solve for $x$ .

Set ${(3 x - 2)}^{2}$ equal to $0$ .

${(3 x - 2)}^{2} = 0$

Solve ${(3 x - 2)}^{2} = 0$ for $x$ .

Set the $3 x - 2$ equal to $0$ .

$3 x - 2 = 0$

Solve for $x$ .

Add $2$ to both sides of the equation.

$3 x = 2$

Divide each term in $3 x = 2$ by $3$ and simplify.

Divide each term in $3 x = 2$ by $3$ .

$\frac{3 x}{3} = \frac{2}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{2}{3}$

Divide $x$ by $1$ .

$x = \frac{2}{3}$

Set ${(x + 9)}^{5}$ equal to $0$ and solve for $x$ .

Set ${(x + 9)}^{5}$ equal to $0$ .

${(x + 9)}^{5} = 0$

Solve ${(x + 9)}^{5} = 0$ for $x$ .

Set the $x + 9$ equal to $0$ .

$x + 9 = 0$

Subtract $9$ from both sides of the equation.

$x = - 9$

The final solution is all the values that make $- x {(3 x - 2)}^{2} {(x + 9)}^{5} = 0$ true.

$x = 0, \frac{2}{3}, - 9$