Algebra Examples

$q (t) = - 4 t (2 - t) {(t + 1)}^{3}$

Step 1

Set $- 4 t (2 - t) {(t + 1)}^{3}$ equal to $0$ .

$- 4 t (2 - t) {(t + 1)}^{3} = 0$

Step 2

Solve for $x$ .

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Step 2.1

Divide each term in $- 4 t (2 - t) {(t + 1)}^{3} = 0$ by $- 4$ and simplify.

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Step 2.1.1

Divide each term in $- 4 t (2 - t) {(t + 1)}^{3} = 0$ by $- 4$ .

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

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 $- 4$ .

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Step 2.1.2.1.1.1

Cancel the common factor.

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

Step 2.1.2.1.1.2

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

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

Step 2.1.2.1.2

Apply the distributive property.

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

Step 2.1.2.1.3

Reorder.

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Step 2.1.2.1.3.1

Move $2$ to the left of $t$ .

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

Step 2.1.2.1.3.2

Rewrite using the commutative property of multiplication.

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

Step 2.1.2.2

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

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Step 2.1.2.2.1

Move $t$ .

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

Step 2.1.2.2.2

Multiply $t$ by $t$ .

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

Step 2.1.2.3

Apply the distributive property.

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

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Divide $0$ by $- 4$ .

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

Step 2.2

Factor $t {(t + 1)}^{3}$ out of $2 t {(t + 1)}^{3} - t^{2} {(t + 1)}^{3}$ .

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Step 2.2.1

Factor $t {(t + 1)}^{3}$ out of $2 t {(t + 1)}^{3}$ .

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

Step 2.2.2

Factor $t {(t + 1)}^{3}$ out of $- t^{2} {(t + 1)}^{3}$ .

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

Step 2.2.3

Factor $t {(t + 1)}^{3}$ out of $t {(t + 1)}^{3} (2) + t {(t + 1)}^{3} (- t)$ .

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

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 $0$ .

$t = 0$

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

$2 - t = 0$

Step 2.4

Set $t$ equal to $0$ .

$t = 0$

Step 2.5

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

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Step 2.5.1

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

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

Step 2.5.2

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

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Step 2.5.2.1

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

$t + 1 = 0$

Step 2.5.2.2

Subtract $1$ from both sides of the equation.

$t = - 1$

Step 2.6

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

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Step 2.6.1

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

$2 - t = 0$

Step 2.6.2

Solve $2 - t = 0$ for $t$ .

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Step 2.6.2.1

Subtract $2$ from both sides of the equation.

$- t = - 2$

Step 2.6.2.2

Divide each term in $- t = - 2$ by $- 1$ and simplify.

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Step 2.6.2.2.1

Divide each term in $- t = - 2$ by $- 1$ .

$\frac{- t}{- 1} = \frac{- 2}{- 1}$

Step 2.6.2.2.2

Simplify the left side.

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Step 2.6.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{t}{1} = \frac{- 2}{- 1}$

Step 2.6.2.2.2.2

Divide $t$ by $1$ .

$t = \frac{- 2}{- 1}$

Step 2.6.2.2.3

Simplify the right side.

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Step 2.6.2.2.3.1

Divide $- 2$ by $- 1$ .

$t = 2$

Step 2.7

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

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

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

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

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

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

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

Step 3