Algebra Examples

$x^{9} - x^{5} + x^{4} - 1 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor out the greatest common factor from each group.

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Step 1.1.1

Group the first two terms and the last two terms.

$(x^{9} - x^{5}) + x^{4} - 1 = 0$

Step 1.1.2

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

$x^{5} (x^{4} - 1) + 1 (x^{4} - 1) = 0$

Step 1.2

Factor the polynomial by factoring out the greatest common factor, $x^{4} - 1$ .

$(x^{4} - 1) (x^{5} + 1) = 0$

Step 1.3

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$({(x^{2})}^{2} - 1) (x^{5} + 1) = 0$

Step 1.4

Rewrite $1$ as $1^{2}$ .

$({(x^{2})}^{2} - 1^{2}) (x^{5} + 1) = 0$

Step 1.5

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{2}$ and $b = 1$ .

$(x^{2} + 1) (x^{2} - 1) (x^{5} + 1) = 0$

Step 1.6

Simplify.

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Step 1.6.1

Rewrite $1$ as $1^{2}$ .

$(x^{2} + 1) (x^{2} - 1^{2}) (x^{5} + 1) = 0$

Step 1.6.2

Factor.

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Step 1.6.2.1

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 $b = 1$ .

$(x^{2} + 1) ((x + 1) (x - 1)) (x^{5} + 1) = 0$

Step 1.6.2.2

Remove unnecessary parentheses.

$(x^{2} + 1) (x + 1) (x - 1) (x^{5} + 1) = 0$

Step 2

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

$x^{2} + 1 = 0$

$x + 1 = 0$

$x - 1 = 0$

$x^{5} + 1 = 0$

Step 3

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

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Step 3.1

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

$x^{2} + 1 = 0$

Step 3.2

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

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Step 3.2.1

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

Step 3.2.2

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

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

Step 3.2.3

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

$x = \pm i$

Step 3.2.4

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

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Step 3.2.4.1

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

$x = i$

Step 3.2.4.2

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

$x = - i$

Step 3.2.4.3

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

$x = i, - i$

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 $x - 1$ equal to $0$ and solve for $x$ .

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Step 5.1

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

$x - 1 = 0$

Step 5.2

Add $1$ to both sides of the equation.

$x = 1$

Step 6

Set $x^{5} + 1$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x^{5} + 1$ equal to $0$ .

$x^{5} + 1 = 0$

Step 6.2

Solve $x^{5} + 1 = 0$ for $x$ .

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Step 6.2.1

Subtract $1$ from both sides of the equation.

$x^{5} = - 1$

Step 6.2.2

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

$x = \sqrt[5]{- 1}$

Step 6.2.3

Simplify $\sqrt[5]{- 1}$ .

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Step 6.2.3.1

Rewrite $- 1$ as ${(- 1)}^{5}$ .

$x = \sqrt[5]{{(- 1)}^{5}}$

Step 6.2.3.2

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

$x = - 1$

Step 7

The final solution is all the values that make $(x^{2} + 1) (x + 1) (x - 1) (x^{5} + 1) = 0$ true.

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