Identify the Zeros and Their Multiplicities f(x)=-0.002(x+12)(x+5)^2(x-9)^3

$f (x) = - 0.002 (x + 12) {(x + 5)}^{2} {(x - 9)}^{3}$

Step 1

Set $- 0.002 (x + 12) {(x + 5)}^{2} {(x - 9)}^{3}$ equal to $0$ .

$- 0.002 (x + 12) {(x + 5)}^{2} {(x - 9)}^{3} = 0$

Step 2

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 $0$ .

$x + 12 = 0$

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

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

Set $x + 12$ equal to $0$ and solve for $x$ .

Set $x + 12$ equal to $0$ .

$x + 12 = 0$

Subtract $12$ from both sides of the equation.

$x = - 12$

Set ${(x + 5)}^{2}$ equal to $0$ and solve for $x$ .

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

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

Solve ${(x + 5)}^{2} = 0$ for $x$ .

Set the $x + 5$ equal to $0$ .

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

Set ${(x - 9)}^{3}$ equal to $0$ and solve for $x$ .

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

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

Solve ${(x - 9)}^{3} = 0$ for $x$ .

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

$x - 9 = 0$

Add $9$ to both sides of the equation.

$x = 9$

The final solution is all the values that make $- 0.002 (x + 12) {(x + 5)}^{2} {(x - 9)}^{3} = 0$ true. The multiplicity of a root is the number of times the root appears.

$x = - 12$ (Multiplicity of $1$ )

$x = - 5$ (Multiplicity of $2$ )

$x = 9$ (Multiplicity of $3$ )

$x = - 12$ (Multiplicity of $1$ )

$x = - 5$ (Multiplicity of $2$ )

$x = 9$ (Multiplicity of $3$ )

Step 3

Identify the Zeros and Their Multiplicities f(x)=-0.002(x+12)(x+5)^2(x-9)^3

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