Algebra Examples

$x^{4} - 2 x^{3} + 4 x - 8$

Step 1

Set $x^{4} - 2 x^{3} + 4 x - 8$ equal to $0$ .

$x^{4} - 2 x^{3} + 4 x - 8 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Factor out the greatest common factor from each group.

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Step 2.1.1.1

Group the first two terms and the last two terms.

$(x^{4} - 2 x^{3}) + 4 x - 8 = 0$

Step 2.1.1.2

Factor out the greatest common factor (GCF) from each group.

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

Step 2.1.2

Factor the polynomial by factoring out the greatest common factor, $x - 2$ .

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

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 $0$ .

$x - 2 = 0$

$x^{3} + 4 = 0$

Step 2.3

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

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Step 2.3.1

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

$x - 2 = 0$

Step 2.3.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.4

Set $x^{3} + 4$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x^{3} + 4$ equal to $0$ .

$x^{3} + 4 = 0$

Step 2.4.2

Solve $x^{3} + 4 = 0$ for $x$ .

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Step 2.4.2.1

Subtract $4$ from both sides of the equation.

$x^{3} = - 4$

Step 2.4.2.2

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

$x = \sqrt[3]{- 4}$

Step 2.4.2.3

Simplify $\sqrt[3]{- 4}$ .

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Step 2.4.2.3.1

Rewrite $- 4$ as ${(- 1)}^{3} \cdot 4$ .

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Step 2.4.2.3.1.1

Rewrite $- 4$ as $- 1 (4)$ .

$x = \sqrt[3]{- 1 (4)}$

Step 2.4.2.3.1.2

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

$x = \sqrt[3]{{(- 1)}^{3} \cdot 4}$

Step 2.4.2.3.2

Pull terms out from under the radical.

$x = - 1 \sqrt[3]{4}$

Step 2.4.2.3.3

Rewrite $- 1 \sqrt[3]{4}$ as $- \sqrt[3]{4}$ .

$x = - \sqrt[3]{4}$

Step 2.5

The final solution is all the values that make $(x - 2) (x^{3} + 4) = 0$ true.

$x = 2, - \sqrt[3]{4}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = 2, - \sqrt[3]{4}$

Decimal Form:

$x = 2, - 1.58740105 \dots$

Step 4