Identify the Zeros and Their Multiplicities k(x)=2(x-3)^2(x+1)^3(x+2)

$k (x) = 2 {(x - 3)}^{2} {(x + 1)}^{3} (x + 2)$

Step 1

Set $2 {(x - 3)}^{2} {(x + 1)}^{3} (x + 2)$ equal to $0$ .

$2 {(x - 3)}^{2} {(x + 1)}^{3} (x + 2) = 0$

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 $0$ .

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

${(x + 1)}^{3} = 0$

$x + 2 = 0$

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

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

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

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

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

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Set ${(x + 1)}^{3}$ equal to $0$ and solve for $x$ .

Set ${(x + 1)}^{3}$ equal to $0$ .

${(x + 1)}^{3} = 0$

Solve ${(x + 1)}^{3} = 0$ for $x$ .

Set the $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

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.

$x = - 2$

The final solution is all the values that make $2 {(x - 3)}^{2} {(x + 1)}^{3} (x + 2) = 0$ true. The multiplicity of a root is the number of times the root appears.

$x = 3$ (Multiplicity of $2$ )

$x = - 1$ (Multiplicity of $3$ )

$x = - 2$ (Multiplicity of $1$ )

$x = 3$ (Multiplicity of $2$ )

$x = - 1$ (Multiplicity of $3$ )

$x = - 2$ (Multiplicity of $1$ )

Step 3

Identify the Zeros and Their Multiplicities k(x)=2(x-3)^2(x+1)^3(x+2)

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