Algebra Examples

$f (x) = - \frac{1}{10} (x + 3) (x - 3) {(x + 1)}^{3}$

Step 1

Set $- \frac{1}{10} \cdot ((x + 3) (x - 3) {(x + 1)}^{3})$ equal to $0$ .

$- \frac{1}{10} \cdot ((x + 3) (x - 3) {(x + 1)}^{3}) = 0$

Step 2

Solve for $x$ .

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Step 2.1

Multiply both sides of the equation by $- 10$ .

$- 10 (- \frac{1}{10} \cdot ((x + 3) (x - 3) {(x + 1)}^{3})) = - 10 \cdot 0$

Step 2.2

Simplify both sides of the equation.

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Simplify $- 10 (- \frac{1}{10} \cdot ((x + 3) (x - 3) {(x + 1)}^{3}))$ .

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Step 2.2.1.1.1

Expand $(x + 3) (x - 3)$ using the FOIL Method.

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Step 2.2.1.1.1.1

Apply the distributive property.

$- 10 (- \frac{1}{10} \cdot ((x (x - 3) + 3 (x - 3)) {(x + 1)}^{3})) = - 10 \cdot 0$

Step 2.2.1.1.1.2

Apply the distributive property.

$- 10 (- \frac{1}{10} \cdot ((x \cdot x + x \cdot - 3 + 3 (x - 3)) {(x + 1)}^{3})) = - 10 \cdot 0$

Step 2.2.1.1.1.3

Apply the distributive property.

$- 10 (- \frac{1}{10} \cdot ((x \cdot x + x \cdot - 3 + 3 x + 3 \cdot - 3) {(x + 1)}^{3})) = - 10 \cdot 0$

Step 2.2.1.1.2

Simplify terms.

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Step 2.2.1.1.2.1

Combine the opposite terms in $x \cdot x + x \cdot - 3 + 3 x + 3 \cdot - 3$ .

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Step 2.2.1.1.2.1.1

Reorder the factors in the terms $x \cdot - 3$ and $3 x$ .

$- 10 (- \frac{1}{10} \cdot ((x \cdot x - 3 x + 3 x + 3 \cdot - 3) {(x + 1)}^{3})) = - 10 \cdot 0$

Step 2.2.1.1.2.1.2

Add $- 3 x$ and $3 x$ .

$- 10 (- \frac{1}{10} \cdot ((x \cdot x + 0 + 3 \cdot - 3) {(x + 1)}^{3})) = - 10 \cdot 0$

Step 2.2.1.1.2.1.3

Add $x \cdot x$ and $0$ .

$- 10 (- \frac{1}{10} \cdot ((x \cdot x + 3 \cdot - 3) {(x + 1)}^{3})) = - 10 \cdot 0$

Step 2.2.1.1.2.2

Simplify each term.

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Step 2.2.1.1.2.2.1

Multiply $x$ by $x$ .

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

Step 2.2.1.1.2.2.2

Multiply $3$ by $- 3$ .

$- 10 (- \frac{1}{10} \cdot ((x^{2} - 9) {(x + 1)}^{3})) = - 10 \cdot 0$

Step 2.2.1.1.2.3

Simplify by multiplying through.

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Step 2.2.1.1.2.3.1

Apply the distributive property.

$- 10 (- \frac{1}{10} \cdot (x^{2} {(x + 1)}^{3} - 9 {(x + 1)}^{3})) = - 10 \cdot 0$

Step 2.2.1.1.2.3.2

Apply the distributive property.

$- 10 (- \frac{1}{10} (x^{2} {(x + 1)}^{3}) - \frac{1}{10} (- 9 {(x + 1)}^{3})) = - 10 \cdot 0$

Step 2.2.1.1.3

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

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Step 2.2.1.1.3.1

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

$- 10 (- \frac{x^{2}}{10} {(x + 1)}^{3} - \frac{1}{10} (- 9 {(x + 1)}^{3})) = - 10 \cdot 0$

Step 2.2.1.1.3.2

Combine ${(x + 1)}^{3}$ and $\frac{x^{2}}{10}$ .

$- 10 (- \frac{{(x + 1)}^{3} x^{2}}{10} - \frac{1}{10} (- 9 {(x + 1)}^{3})) = - 10 \cdot 0$

Step 2.2.1.1.4

Multiply $- \frac{1}{10} (- 9 {(x + 1)}^{3})$ .

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Step 2.2.1.1.4.1

Multiply $- 9$ by $- 1$ .

$- 10 (- \frac{{(x + 1)}^{3} x^{2}}{10} + 9 (\frac{1}{10}) {(x + 1)}^{3}) = - 10 \cdot 0$

Step 2.2.1.1.4.2

Combine $9$ and $\frac{1}{10}$ .

$- 10 (- \frac{{(x + 1)}^{3} x^{2}}{10} + \frac{9}{10} {(x + 1)}^{3}) = - 10 \cdot 0$

Step 2.2.1.1.4.3

Combine $\frac{9}{10}$ and ${(x + 1)}^{3}$ .

$- 10 (- \frac{{(x + 1)}^{3} x^{2}}{10} + \frac{9 {(x + 1)}^{3}}{10}) = - 10 \cdot 0$

Step 2.2.1.1.5

Combine the numerators over the common denominator.

$- 10 \frac{- {(x + 1)}^{3} x^{2} + 9 {(x + 1)}^{3}}{10} = - 10 \cdot 0$

Step 2.2.1.1.6

Reorder factors in $- {(x + 1)}^{3} x^{2} + 9 {(x + 1)}^{3}$ .

$- 10 \frac{- x^{2} {(x + 1)}^{3} + 9 {(x + 1)}^{3}}{10} = - 10 \cdot 0$

Step 2.2.1.1.7

Cancel the common factor of $10$ .

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Step 2.2.1.1.7.1

Factor $10$ out of $- 10$ .

$10 (- 1) \frac{- x^{2} {(x + 1)}^{3} + 9 {(x + 1)}^{3}}{10} = - 10 \cdot 0$

Step 2.2.1.1.7.2

Cancel the common factor.

$10 \cdot - 1 \frac{- x^{2} {(x + 1)}^{3} + 9 {(x + 1)}^{3}}{10} = - 10 \cdot 0$

Step 2.2.1.1.7.3

Rewrite the expression.

$- (- x^{2} {(x + 1)}^{3} + 9 {(x + 1)}^{3}) = - 10 \cdot 0$

Step 2.2.1.1.8

Apply the distributive property.

$- (- x^{2} {(x + 1)}^{3}) - (9 {(x + 1)}^{3}) = - 10 \cdot 0$

Step 2.2.1.1.9

Multiply $- (- x^{2} {(x + 1)}^{3})$ .

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Step 2.2.1.1.9.1

Multiply $- 1$ by $- 1$ .

$1 (x^{2} {(x + 1)}^{3}) - (9 {(x + 1)}^{3}) = - 10 \cdot 0$

Step 2.2.1.1.9.2

Multiply $x^{2}$ by $1$ .

$x^{2} {(x + 1)}^{3} - (9 {(x + 1)}^{3}) = - 10 \cdot 0$

Step 2.2.1.1.10

Multiply $9$ by $- 1$ .

$x^{2} {(x + 1)}^{3} - 9 {(x + 1)}^{3} = - 10 \cdot 0$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Multiply $- 10$ by $0$ .

$x^{2} {(x + 1)}^{3} - 9 {(x + 1)}^{3} = 0$

Step 2.3

Factor the left side of the equation.

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Step 2.3.1

Factor ${(x + 1)}^{3}$ out of $x^{2} {(x + 1)}^{3} - 9 {(x + 1)}^{3}$ .

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Step 2.3.1.1

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

${(x + 1)}^{3} x^{2} - 9 {(x + 1)}^{3} = 0$

Step 2.3.1.2

Factor ${(x + 1)}^{3}$ out of $- 9 {(x + 1)}^{3}$ .

${(x + 1)}^{3} x^{2} + {(x + 1)}^{3} \cdot - 9 = 0$

Step 2.3.1.3

Factor ${(x + 1)}^{3}$ out of ${(x + 1)}^{3} x^{2} + {(x + 1)}^{3} \cdot - 9$ .

${(x + 1)}^{3} (x^{2} - 9) = 0$

Step 2.3.2

Rewrite $9$ as $3^{2}$ .

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

Step 2.3.3

Factor.

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Step 2.3.3.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

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

Step 2.3.3.2

Remove unnecessary parentheses.

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

Step 2.4

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

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

$x + 3 = 0$

$x - 3 = 0$

Step 2.5

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

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Step 2.5.1

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

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

Step 2.5.2

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

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Step 2.5.2.1

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

$x + 1 = 0$

Step 2.5.2.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.6

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 2.6.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 2.6.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 2.7

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

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Step 2.7.1

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

$x - 3 = 0$

Step 2.7.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.8

The final solution is all the values that make ${(x + 1)}^{3} (x + 3) (x - 3) = 0$ true.

$x = - 1, - 3, 3$

Step 3