Algebra Examples

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

Step 1

Regroup terms.

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

Step 2

Factor $x$ out of $x^{5} - x$ .

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Step 2.1

Factor $x$ out of $x^{5}$ .

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

Step 2.2

Factor $x$ out of $- x$ .

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

Step 2.3

Factor $x$ out of $x \cdot x^{4} + x \cdot - 1$ .

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

Step 3

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

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

Step 4

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

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

Step 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 $b = 1$ .

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

Step 6

Factor.

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Step 6.1

Simplify.

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Step 6.1.1

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

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

Step 6.1.2

Factor.

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Step 6.1.2.1

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 $b = 1$ .

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

Step 6.1.2.2

Remove unnecessary parentheses.

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

Step 6.2

Remove unnecessary parentheses.

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

Step 7

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

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

Step 8

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 $b = 1$ .

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

Step 9

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

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Step 9.1

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

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

Step 9.2

Factor $(x + 1) (x - 1)$ out of $(x + 1) (x - 1)$ .

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

Step 9.3

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

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

Step 10

Apply the distributive property.

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

Step 11

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 11.1

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

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Step 11.1.1

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

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

Step 11.1.2

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

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

Step 11.2

Add $1$ and $2$ .

$P (x) = (x + 1) (x - 1) (x^{3} + x \cdot 1 + 1)$

Step 12

Multiply $x$ by $1$ .

$P (x) = (x + 1) (x - 1) (x^{3} + x + 1)$