Algebra Examples

$h (t) = 3 {(t - 5)}^{3} {(t - 3)}^{3} (t - 2)$

Step 1

Set $3 {(t - 5)}^{3} {(t - 3)}^{3} (t - 2)$ equal to $0$ .

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

Step 2

Solve for $x$ .

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Step 2.1

Divide each term in $3 {(t - 5)}^{3} {(t - 3)}^{3} (t - 2) = 0$ by $3$ and simplify.

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Step 2.1.1

Divide each term in $3 {(t - 5)}^{3} {(t - 3)}^{3} (t - 2) = 0$ by $3$ .

$\frac{3 {(t - 5)}^{3} {(t - 3)}^{3} (t - 2)}{3} = \frac{0}{3}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Simplify terms.

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Step 2.1.2.1.1

Cancel the common factor of $3$ .

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Step 2.1.2.1.1.1

Cancel the common factor.

$\frac{3 {(t - 5)}^{3} {(t - 3)}^{3} (t - 2)}{3} = \frac{0}{3}$

Step 2.1.2.1.1.2

Divide ${(t - 5)}^{3} {(t - 3)}^{3} (t - 2)$ by $1$ .

${(t - 5)}^{3} {(t - 3)}^{3} (t - 2) = \frac{0}{3}$

Step 2.1.2.1.2

Apply the distributive property.

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

Step 2.1.2.1.3

Move $- 2$ to the left of ${(t - 5)}^{3} {(t - 3)}^{3}$ .

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

Step 2.1.2.2

Remove parentheses.

${(t - 5)}^{3} {(t - 3)}^{3} t - 2 {(t - 5)}^{3} {(t - 3)}^{3} = \frac{0}{3}$

Step 2.1.2.3

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

$t {(t - 5)}^{3} {(t - 3)}^{3} - 2 {(t - 5)}^{3} {(t - 3)}^{3} = \frac{0}{3}$

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Divide $0$ by $3$ .

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

Step 2.2

Factor ${(t - 5)}^{3} {(t - 3)}^{3}$ out of $t {(t - 5)}^{3} {(t - 3)}^{3} - 2 {(t - 5)}^{3} {(t - 3)}^{3}$ .

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Step 2.2.1

Factor ${(t - 5)}^{3} {(t - 3)}^{3}$ out of $t {(t - 5)}^{3} {(t - 3)}^{3}$ .

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

Step 2.2.2

Factor ${(t - 5)}^{3} {(t - 3)}^{3}$ out of $- 2 {(t - 5)}^{3} {(t - 3)}^{3}$ .

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

Step 2.2.3

Factor ${(t - 5)}^{3} {(t - 3)}^{3}$ out of ${(t - 5)}^{3} {(t - 3)}^{3} (t) + {(t - 5)}^{3} {(t - 3)}^{3} (- 2)$ .

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

Step 2.3

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

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

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

$t - 2 = 0$

Step 2.4

Set ${(t - 5)}^{3}$ equal to $0$ and solve for $t$ .

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Step 2.4.1

Set ${(t - 5)}^{3}$ equal to $0$ .

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

Step 2.4.2

Solve ${(t - 5)}^{3} = 0$ for $t$ .

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Step 2.4.2.1

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

$t - 5 = 0$

Step 2.4.2.2

Add $5$ to both sides of the equation.

$t = 5$

Step 2.5

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

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Step 2.5.1

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

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

Step 2.5.2

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

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Step 2.5.2.1

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

$t - 3 = 0$

Step 2.5.2.2

Add $3$ to both sides of the equation.

$t = 3$

Step 2.6

Set $t - 2$ equal to $0$ and solve for $t$ .

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Step 2.6.1

Set $t - 2$ equal to $0$ .

$t - 2 = 0$

Step 2.6.2

Add $2$ to both sides of the equation.

$t = 2$

Step 2.7

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

$t = 5$ (Multiplicity of $3$ )

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

$t = 2$ (Multiplicity of $1$ )

$t = 5$ (Multiplicity of $3$ )

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

$t = 2$ (Multiplicity of $1$ )

Step 3