Identify the Zeros and Their Multiplicities f(t)=1/10t^2(t-4)^3(t+5)^2

$f (t) = \frac{1}{10} t^{2} {(t - 4)}^{3} {(t + 5)}^{2}$

Step 1

Set $\frac{1}{10} \cdot (t^{2} {(t - 4)}^{3} {(t + 5)}^{2})$ equal to $0$ .

$\frac{1}{10} \cdot (t^{2} {(t - 4)}^{3} {(t + 5)}^{2}) = 0$

Step 2

Solve for $x$ .

Multiply each term in $\frac{1}{10} \cdot (t^{2} {(t - 4)}^{3} {(t + 5)}^{2}) = 0$ by $10$ to eliminate the fractions.

Multiply each term in $\frac{1}{10} \cdot (t^{2} {(t - 4)}^{3} {(t + 5)}^{2}) = 0$ by $10$ .

$\frac{1}{10} \cdot (t^{2} {(t - 4)}^{3} {(t + 5)}^{2}) \cdot 10 = 0 \cdot 10$

Simplify the left side.

Multiply $\frac{1}{10} (t^{2} {(t - 4)}^{3} {(t + 5)}^{2})$ .

Combine $t^{2}$ and $\frac{1}{10}$ .

$\frac{t^{2}}{10} ({(t - 4)}^{3} {(t + 5)}^{2}) \cdot 10 = 0 \cdot 10$

Combine ${(t - 4)}^{3}$ and $\frac{t^{2}}{10}$ .

$\frac{{(t - 4)}^{3} t^{2}}{10} {(t + 5)}^{2} \cdot 10 = 0 \cdot 10$

Combine $\frac{{(t - 4)}^{3} t^{2}}{10}$ and ${(t + 5)}^{2}$ .

$\frac{{(t - 4)}^{3} t^{2} {(t + 5)}^{2}}{10} \cdot 10 = 0 \cdot 10$

Cancel the common factor of $10$ .

Cancel the common factor.

$\frac{{(t - 4)}^{3} t^{2} {(t + 5)}^{2}}{10} \cdot 10 = 0 \cdot 10$

Rewrite the expression.

${(t - 4)}^{3} t^{2} {(t + 5)}^{2} = 0 \cdot 10$

Reorder factors in ${(t - 4)}^{3} t^{2} {(t + 5)}^{2}$ .

$t^{2} {(t - 4)}^{3} {(t + 5)}^{2} = 0 \cdot 10$

Simplify the right side.

Multiply $0$ by $10$ .

$t^{2} {(t - 4)}^{3} {(t + 5)}^{2} = 0$

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

$t^{2} = 0$

${(t - 4)}^{3} = 0$

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

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

Set $t^{2}$ equal to $0$ .

$t^{2} = 0$

Solve $t^{2} = 0$ for $t$ .

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$t = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

Rewrite $0$ as $0^{2}$ .

$t = \pm \sqrt{0^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$t = \pm 0$

Plus or minus $0$ is $0$ .

$t = 0$

Set ${(t - 4)}^{3}$ equal to $0$ and solve for $t$ .

Set ${(t - 4)}^{3}$ equal to $0$ .

${(t - 4)}^{3} = 0$

Solve ${(t - 4)}^{3} = 0$ for $t$ .

Set the $t - 4$ equal to $0$ .

$t - 4 = 0$

Add $4$ to both sides of the equation.

$t = 4$

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

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

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

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

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

$t + 5 = 0$

Subtract $5$ from both sides of the equation.

$t = - 5$

The final solution is all the values that make $t^{2} {(t - 4)}^{3} {(t + 5)}^{2} = 0$ true. The multiplicity of a root is the number of times the root appears.

$t = 0$ (Multiplicity of $2$ )

$t = 4$ (Multiplicity of $3$ )

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

$t = 0$ (Multiplicity of $2$ )

$t = 4$ (Multiplicity of $3$ )

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

Step 3

Identify the Zeros and Their Multiplicities f(t)=1/10t^2(t-4)^3(t+5)^2

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