Precalculus Examples

$f (x) = \frac{1}{4} \cdot (x^{3} (x^{2} - 9))$

Step 1

Set $\frac{1}{4} \cdot (x^{3} (x^{2} - 9))$ equal to $0$ .

$\frac{1}{4} \cdot (x^{3} (x^{2} - 9)) = 0$

Step 2

Solve for $x$ .

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Step 2.1

Multiply both sides of the equation by $4$ .

$4 (\frac{1}{4} \cdot (x^{3} (x^{2} - 9))) = 4 \cdot 0$

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 $4 (\frac{1}{4} \cdot (x^{3} (x^{2} - 9)))$ .

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Step 2.2.1.1.1

Apply the distributive property.

$4 (\frac{1}{4} \cdot (x^{3} x^{2} + x^{3} \cdot - 9)) = 4 \cdot 0$

Step 2.2.1.1.2

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 2.2.1.1.2.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$4 (\frac{1}{4} \cdot (x^{3 + 2} + x^{3} \cdot - 9)) = 4 \cdot 0$

Step 2.2.1.1.2.2

Add $3$ and $2$ .

$4 (\frac{1}{4} \cdot (x^{5} + x^{3} \cdot - 9)) = 4 \cdot 0$

Step 2.2.1.1.3

Move $- 9$ to the left of $x^{3}$ .

$4 (\frac{1}{4} \cdot (x^{5} - 9 \cdot x^{3})) = 4 \cdot 0$

Step 2.2.1.1.4

Apply the distributive property.

$4 (\frac{1}{4} x^{5} + \frac{1}{4} (- 9 x^{3})) = 4 \cdot 0$

Step 2.2.1.1.5

Combine $\frac{1}{4}$ and $x^{5}$ .

$4 (\frac{x^{5}}{4} + \frac{1}{4} (- 9 x^{3})) = 4 \cdot 0$

Step 2.2.1.1.6

Multiply $\frac{1}{4} (- 9 x^{3})$ .

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Step 2.2.1.1.6.1

Combine $- 9$ and $\frac{1}{4}$ .

$4 (\frac{x^{5}}{4} + \frac{- 9}{4} x^{3}) = 4 \cdot 0$

Step 2.2.1.1.6.2

Combine $\frac{- 9}{4}$ and $x^{3}$ .

$4 (\frac{x^{5}}{4} + \frac{- 9 x^{3}}{4}) = 4 \cdot 0$

Step 2.2.1.1.7

Move the negative in front of the fraction.

$4 (\frac{x^{5}}{4} - \frac{9 x^{3}}{4}) = 4 \cdot 0$

Step 2.2.1.1.8

Apply the distributive property.

$4 \frac{x^{5}}{4} + 4 (- \frac{9 x^{3}}{4}) = 4 \cdot 0$

Step 2.2.1.1.9

Cancel the common factor of $4$ .

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Step 2.2.1.1.9.1

Cancel the common factor.

$4 \frac{x^{5}}{4} + 4 (- \frac{9 x^{3}}{4}) = 4 \cdot 0$

Step 2.2.1.1.9.2

Rewrite the expression.

$x^{5} + 4 (- \frac{9 x^{3}}{4}) = 4 \cdot 0$

Step 2.2.1.1.10

Cancel the common factor of $4$ .

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Step 2.2.1.1.10.1

Move the leading negative in $- \frac{9 x^{3}}{4}$ into the numerator.

$x^{5} + 4 \frac{- 9 x^{3}}{4} = 4 \cdot 0$

Step 2.2.1.1.10.2

Cancel the common factor.

$x^{5} + 4 \frac{- 9 x^{3}}{4} = 4 \cdot 0$

Step 2.2.1.1.10.3

Rewrite the expression.

$x^{5} - 9 x^{3} = 4 \cdot 0$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Multiply $4$ by $0$ .

$x^{5} - 9 x^{3} = 0$

Step 2.3

Factor the left side of the equation.

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Step 2.3.1

Factor $x^{3}$ out of $x^{5} - 9 x^{3}$ .

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Step 2.3.1.1

Factor $x^{3}$ out of $x^{5}$ .

$x^{3} x^{2} - 9 x^{3} = 0$

Step 2.3.1.2

Factor $x^{3}$ out of $- 9 x^{3}$ .

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

Step 2.3.1.3

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

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

Step 2.3.2

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

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

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 $b = 3$ .

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

Step 2.3.3.2

Remove unnecessary parentheses.

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

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 $0$ .

$x^{3} = 0$

$x + 3 = 0$

$x - 3 = 0$

Step 2.5

Set $x^{3}$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x^{3}$ equal to $0$ .

$x^{3} = 0$

Step 2.5.2

Solve $x^{3} = 0$ for $x$ .

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Step 2.5.2.1

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

$x = \sqrt[3]{0}$

Step 2.5.2.2

Simplify $\sqrt[3]{0}$ .

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Step 2.5.2.2.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 2.5.2.2.2

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

$x = 0$

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

$x = 0, - 3, 3$

Step 3