Pre-Algebra Examples

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

Step 1

Factor the left side of the equation.

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Step 1.1

Factor $8 x$ out of $8 x^{4} + 8 x^{3} + 8 x^{2} + 8 x$ .

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Step 1.1.1

Factor $8 x$ out of $8 x^{4}$ .

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

Step 1.1.2

Factor $8 x$ out of $8 x^{3}$ .

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

Step 1.1.3

Factor $8 x$ out of $8 x^{2}$ .

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

Step 1.1.4

Factor $8 x$ out of $8 x$ .

$8 x (x^{3}) + 8 x (x^{2}) + 8 x (x) + 8 x (1) = 0$

Step 1.1.5

Factor $8 x$ out of $8 x (x^{3}) + 8 x (x^{2})$ .

$8 x (x^{3} + x^{2}) + 8 x (x) + 8 x (1) = 0$

Step 1.1.6

Factor $8 x$ out of $8 x (x^{3} + x^{2}) + 8 x (x)$ .

$8 x (x^{3} + x^{2} + x) + 8 x (1) = 0$

Step 1.1.7

Factor $8 x$ out of $8 x (x^{3} + x^{2} + x) + 8 x (1)$ .

$8 x (x^{3} + x^{2} + x + 1) = 0$

Step 1.2

Factor out the greatest common factor from each group.

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Step 1.2.1

Group the first two terms and the last two terms.

$8 x ((x^{3} + x^{2}) + x + 1) = 0$

Step 1.2.2

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

$8 x (x^{2} (x + 1) + 1 (x + 1)) = 0$

Step 1.3

Factor.

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Step 1.3.1

Factor the polynomial by factoring out the greatest common factor, $x + 1$ .

$8 x ((x + 1) (x^{2} + 1)) = 0$

Step 1.3.2

Remove unnecessary parentheses.

$8 x (x + 1) (x^{2} + 1) = 0$

Step 2

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

$x = 0$

$x + 1 = 0$

$x^{2} + 1 = 0$

Step 3

Set $x$ equal to $0$ .

$x = 0$

Step 4

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 4.1

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

$x + 1 = 0$

Step 4.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 5

Set $x^{2} + 1$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x^{2} + 1$ equal to $0$ .

$x^{2} + 1 = 0$

Step 5.2

Solve $x^{2} + 1 = 0$ for $x$ .

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Step 5.2.1

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

Step 5.2.2

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

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

Step 5.2.3

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

$x = \pm i$

Step 5.2.4

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

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Step 5.2.4.1

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

$x = i$

Step 5.2.4.2

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

$x = - i$

Step 5.2.4.3

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

$x = i, - i$

Step 6

The final solution is all the values that make $8 x (x + 1) (x^{2} + 1) = 0$ true.

$x = 0, - 1, i, - i$