Algebra Examples

$x^{6} - x^{4} - x^{2} + 1 = 0$

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.

$(x^{6} - x^{4}) - x^{2} + 1 = 0$

Step 1.1.2

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

$x^{4} (x^{2} - 1) - (x^{2} - 1) = 0$

Step 1.2

Factor the polynomial by factoring out the greatest common factor, $x^{2} - 1$ .

$(x^{2} - 1) (x^{4} - 1) = 0$

Step 1.3

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

$(x^{2} - 1^{2}) (x^{4} - 1) = 0$

Step 1.4

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 $b = 1$ .

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

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

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 $b = 1$ .

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

Step 1.8

Factor.

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Step 1.8.1

Simplify.

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Step 1.8.1.1

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

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

Step 1.8.1.2

Factor.

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Step 1.8.1.2.1

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 $b = 1$ .

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

Step 1.8.1.2.2

Remove unnecessary parentheses.

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

Step 1.8.2

Remove unnecessary parentheses.

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

Step 1.9

Combine exponents.

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Step 1.9.1

Raise $x + 1$ to the power of $1$ .

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

Step 1.9.2

Raise $x + 1$ to the power of $1$ .

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

Step 1.9.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.9.4

Add $1$ and $1$ .

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

Step 1.9.5

Raise $x - 1$ to the power of $1$ .

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

Step 1.9.6

Raise $x - 1$ to the power of $1$ .

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

Step 1.9.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.9.8

Add $1$ and $1$ .

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

Step 2

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

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

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

$x^{2} + 1 = 0$

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

$x + 1 = 0$

Step 3.2.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 4

Set ${(x - 1)}^{2}$ equal to $0$ and solve for $x$ .

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Step 4.1

Set ${(x - 1)}^{2}$ equal to $0$ .

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

Step 4.2

Solve ${(x - 1)}^{2} = 0$ for $x$ .

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Step 4.2.1

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

$x - 1 = 0$

Step 4.2.2

Add $1$ to 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 ${(x + 1)}^{2} {(x - 1)}^{2} (x^{2} + 1) = 0$ true.

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

Step 7