Identify the Zeros and Their Multiplicities V(t)=-t(t+2)^2(t-3)

$V (t) = - t {(t + 2)}^{2} (t - 3)$

Step 1

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

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

Step 2

Solve for $x$ .

Divide each term in $- t {(t + 2)}^{2} (t - 3) = 0$ by $- 1$ and simplify.

Divide each term in $- t {(t + 2)}^{2} (t - 3) = 0$ by $- 1$ .

$\frac{- t {(t + 2)}^{2} (t - 3)}{- 1} = \frac{0}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{t {(t + 2)}^{2} (t - 3)}{1} = \frac{0}{- 1}$

Divide $t {(t + 2)}^{2} (t - 3)$ by $1$ .

$t {(t + 2)}^{2} (t - 3) = \frac{0}{- 1}$

Apply the distributive property.

$t {(t + 2)}^{2} t + t {(t + 2)}^{2} \cdot - 3 = \frac{0}{- 1}$

Multiply $t$ by $t$ by adding the exponents.

Move $t$ .

$t \cdot t {(t + 2)}^{2} + t {(t + 2)}^{2} \cdot - 3 = \frac{0}{- 1}$

Multiply $t$ by $t$ .

$t^{2} {(t + 2)}^{2} + t {(t + 2)}^{2} \cdot - 3 = \frac{0}{- 1}$

Move $- 3$ to the left of $t {(t + 2)}^{2}$ .

$t^{2} {(t + 2)}^{2} - 3 \cdot (t {(t + 2)}^{2}) = \frac{0}{- 1}$

Remove parentheses.

$t^{2} {(t + 2)}^{2} - 3 t {(t + 2)}^{2} = \frac{0}{- 1}$

Simplify the right side.

Divide $0$ by $- 1$ .

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

Factor $t {(t + 2)}^{2}$ out of $t^{2} {(t + 2)}^{2} - 3 t {(t + 2)}^{2}$ .

Factor $t {(t + 2)}^{2}$ out of $t^{2} {(t + 2)}^{2}$ .

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

Factor $t {(t + 2)}^{2}$ out of $- 3 t {(t + 2)}^{2}$ .

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

Factor $t {(t + 2)}^{2}$ out of $t {(t + 2)}^{2} (t) + t {(t + 2)}^{2} (- 3)$ .

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

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

$t = 0$

${(t + 2)}^{2} = 0$

$t - 3 = 0$

Set $t$ equal to $0$ .

$t = 0$

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

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

${(t + 2)}^{2} = 0$

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

Set the $t + 2$ equal to $0$ .

$t + 2 = 0$

Subtract $2$ from both sides of the equation.

$t = - 2$

Set $t - 3$ equal to $0$ and solve for $t$ .

Set $t - 3$ equal to $0$ .

$t - 3 = 0$

Add $3$ to both sides of the equation.

$t = 3$

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

$t = 0$ (Multiplicity of $1$ )

$t = - 2$ (Multiplicity of $2$ )

$t = 3$ (Multiplicity of $1$ )

$t = 0$ (Multiplicity of $1$ )

$t = - 2$ (Multiplicity of $2$ )

$t = 3$ (Multiplicity of $1$ )

Step 3

Identify the Zeros and Their Multiplicities V(t)=-t(t+2)^2(t-3)

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