Algebra Examples

$f (x) = - 7 x - x^{4}$

Step 1

Set $- 7 x - x^{4}$ equal to $0$ .

$- 7 x - x^{4} = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor $- x$ out of $- 7 x - x^{4}$ .

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Step 2.1.1

Reorder $- 7 x$ and $- x^{4}$ .

$- x^{4} - 7 x = 0$

Step 2.1.2

Factor $- x$ out of $- x^{4}$ .

$- x \cdot x^{3} - 7 x = 0$

Step 2.1.3

Factor $- x$ out of $- 7 x$ .

$- x \cdot x^{3} - x \cdot 7 = 0$

Step 2.1.4

Factor $- x$ out of $- x (x^{3}) - x (7)$ .

$- x (x^{3} + 7) = 0$

Step 2.2

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

$x = 0$

$x^{3} + 7 = 0$

Step 2.3

Set $x$ equal to $0$ .

$x = 0$

Step 2.4

Set $x^{3} + 7$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x^{3} + 7$ equal to $0$ .

$x^{3} + 7 = 0$

Step 2.4.2

Solve $x^{3} + 7 = 0$ for $x$ .

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Step 2.4.2.1

Subtract $7$ from both sides of the equation.

$x^{3} = - 7$

Step 2.4.2.2

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

$x = \sqrt[3]{- 7}$

Step 2.4.2.3

Simplify $\sqrt[3]{- 7}$ .

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Step 2.4.2.3.1

Rewrite $- 7$ as ${(- 1)}^{3} \cdot 7$ .

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Step 2.4.2.3.1.1

Rewrite $- 7$ as $- 1 (7)$ .

$x = \sqrt[3]{- 1 (7)}$

Step 2.4.2.3.1.2

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$x = \sqrt[3]{{(- 1)}^{3} \cdot 7}$

Step 2.4.2.3.2

Pull terms out from under the radical.

$x = - 1 \sqrt[3]{7}$

Step 2.4.2.3.3

Rewrite $- 1 \sqrt[3]{7}$ as $- \sqrt[3]{7}$ .

$x = - \sqrt[3]{7}$

Step 2.5

The final solution is all the values that make $- x (x^{3} + 7) = 0$ true. The multiplicity of a root is the number of times the root appears.

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

$x = - \sqrt[3]{7}$ (Multiplicity of $1$ )

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

$x = - \sqrt[3]{7}$ (Multiplicity of $1$ )

Step 3