Algebra Examples

$x^{9} - x^{6} - x^{3} + 1$

Step 1

Factor out the greatest common factor from each group.

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Step 1.1

Group the first two terms and the last two terms.

$(x^{9} - x^{6}) - x^{3} + 1$

Step 1.2

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

$x^{6} (x^{3} - 1) - (x^{3} - 1)$

Step 2

Factor the polynomial by factoring out the greatest common factor, $x^{3} - 1$ .

$(x^{3} - 1) (x^{6} - 1)$

Step 3

Rewrite $1$ as $1^{3}$ .

$(x^{3} - 1^{3}) (x^{6} - 1)$

Step 4

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 1$ .

$(x - 1) (x^{2} + x \cdot 1 + 1^{2}) (x^{6} - 1)$

Step 5

Simplify.

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Step 5.1

Multiply $x$ by $1$ .

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

Step 5.2

One to any power is one.

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

Step 6

Rewrite $x^{6}$ as ${(x^{2})}^{3}$ .

$(x - 1) (x^{2} + x + 1) ({(x^{2})}^{3} - 1)$

Step 7

Rewrite $1$ as $1^{3}$ .

$(x - 1) (x^{2} + x + 1) ({(x^{2})}^{3} - 1^{3})$

Step 8

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x^{2}$ and $b = 1$ .

$(x - 1) (x^{2} + x + 1) ((x^{2} - 1) ({(x^{2})}^{2} + x^{2} \cdot 1 + 1^{2}))$

Step 9

Factor.

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Step 9.1

Simplify.

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Step 9.1.1

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

$(x - 1) (x^{2} + x + 1) ((x^{2} - 1^{2}) ({(x^{2})}^{2} + x^{2} \cdot 1 + 1^{2}))$

Step 9.1.2

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 $b = 1$ .

$(x - 1) (x^{2} + x + 1) ((x + 1) (x - 1) ({(x^{2})}^{2} + x^{2} \cdot 1 + 1^{2}))$

Step 9.1.3

Multiply $x^{2}$ by $1$ .

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

Step 9.2

Remove unnecessary parentheses.

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

Step 10

Combine exponents.

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Step 10.1

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

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

Step 10.2

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

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

Step 10.3

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

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

Step 10.4

Add $1$ and $1$ .

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

Step 11

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 11.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(x - 1)}^{2} (x^{2} + x + 1) (x + 1) (x^{2 \cdot 2} + x^{2} + 1^{2})$

Step 11.2

Multiply $2$ by $2$ .

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

Step 12

One to any power is one.

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

Step 13

Factor.

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Step 13.1

Rewrite $x^{4} + x^{2} + 1$ in a factored form.

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Step 13.1.1

Rewrite the middle term.

${(x - 1)}^{2} (x^{2} + x + 1) (x + 1) (x^{4} + 2 x^{2} \cdot 1 - x^{2} + 1)$

Step 13.1.2

Rearrange terms.

${(x - 1)}^{2} (x^{2} + x + 1) (x + 1) (x^{4} + 2 x^{2} \cdot 1 + 1 - x^{2})$

Step 13.1.3

Factor first three terms by perfect square rule.

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

Step 13.1.4

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

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

Step 13.1.5

Simplify.

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Step 13.1.5.1

Reorder terms.

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

Step 13.1.5.2

Reorder terms.

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

Step 13.2

Remove unnecessary parentheses.

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

Step 14

Combine exponents.

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Step 14.1

Raise $x^{2} + x + 1$ to the power of $1$ .

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

Step 14.2

Raise $x^{2} + x + 1$ to the power of $1$ .

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

Step 14.3

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

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

Step 14.4

Add $1$ and $1$ .

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