Algebra Examples

$x^{5} + 5 x^{3} + 17 x^{2} + 85$

Step 1

Set $x^{5} + 5 x^{3} + 17 x^{2} + 85$ equal to $0$ .

$x^{5} + 5 x^{3} + 17 x^{2} + 85 = 0$

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.

$(x^{5} + 5 x^{3}) + 17 x^{2} + 85 = 0$

Step 2.1.1.2

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

$x^{3} (x^{2} + 5) + 17 (x^{2} + 5) = 0$

Step 2.1.2

Factor the polynomial by factoring out the greatest common factor, $x^{2} + 5$ .

$(x^{2} + 5) (x^{3} + 17) = 0$

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 $0$ .

$x^{2} + 5 = 0$

$x^{3} + 17 = 0$

Step 2.3

Set $x^{2} + 5$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $x^{2} + 5$ equal to $0$ .

$x^{2} + 5 = 0$

Step 2.3.2

Solve $x^{2} + 5 = 0$ for $x$ .

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Step 2.3.2.1

Subtract $5$ from both sides of the equation.

$x^{2} = - 5$

Step 2.3.2.2

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

$x = \pm \sqrt{- 5}$

Step 2.3.2.3

Simplify $\pm \sqrt{- 5}$ .

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Step 2.3.2.3.1

Rewrite $- 5$ as $- 1 (5)$ .

$x = \pm \sqrt{- 1 (5)}$

Step 2.3.2.3.2

Rewrite $\sqrt{- 1 (5)}$ as $\sqrt{- 1} \cdot \sqrt{5}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{5}$

Step 2.3.2.3.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \sqrt{5}$

Step 2.3.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.3.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = i \sqrt{5}$

Step 2.3.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - i \sqrt{5}$

Step 2.3.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = i \sqrt{5}, - i \sqrt{5}$

Step 2.4

Set $x^{3} + 17$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x^{3} + 17$ equal to $0$ .

$x^{3} + 17 = 0$

Step 2.4.2

Solve $x^{3} + 17 = 0$ for $x$ .

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Step 2.4.2.1

Subtract $17$ from both sides of the equation.

$x^{3} = - 17$

Step 2.4.2.2

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

$x = \sqrt[3]{- 17}$

Step 2.4.2.3

Simplify $\sqrt[3]{- 17}$ .

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Step 2.4.2.3.1

Rewrite $- 17$ as ${(- 1)}^{3} \cdot 17$ .

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Step 2.4.2.3.1.1

Rewrite $- 17$ as $- 1 (17)$ .

$x = \sqrt[3]{- 1 (17)}$

Step 2.4.2.3.1.2

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

$x = \sqrt[3]{{(- 1)}^{3} \cdot 17}$

Step 2.4.2.3.2

Pull terms out from under the radical.

$x = - 1 \sqrt[3]{17}$

Step 2.4.2.3.3

Rewrite $- 1 \sqrt[3]{17}$ as $- \sqrt[3]{17}$ .

$x = - \sqrt[3]{17}$

Step 2.5

The final solution is all the values that make $(x^{2} + 5) (x^{3} + 17) = 0$ true.

$x = i \sqrt{5}, - i \sqrt{5}, - \sqrt[3]{17}$

Step 3