Algebra Examples

$x^{4} + 2 x^{3} - 2 x - 1$

Step 1

Regroup terms.

$x^{4} - 1 + 2 x^{3} - 2 x$

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

Factor.

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Step 5.2.1

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 $b = 1$ .

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

Step 5.2.2

Remove unnecessary parentheses.

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

Step 6

Factor $2 x$ out of $2 x^{3} - 2 x$ .

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Step 6.1

Factor $2 x$ out of $2 x^{3}$ .

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

Step 6.2

Factor $2 x$ out of $- 2 x$ .

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

Step 6.3

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

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

Step 7

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

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

Step 8

Factor.

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Step 8.1

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 $b = 1$ .

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

Step 8.2

Remove unnecessary parentheses.

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

Step 9

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

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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

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

Step 10

Factor using the perfect square rule.

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Step 10.1

Rearrange terms.

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

Step 10.2

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

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

Step 10.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 10.4

Rewrite the polynomial.

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

Step 10.5

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 1$ .

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

Step 11

Multiply $x + 1$ by ${(x + 1)}^{2}$ by adding the exponents.

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Step 11.1

Move ${(x + 1)}^{2}$ .

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

Step 11.2

Multiply ${(x + 1)}^{2}$ by $x + 1$ .

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Step 11.2.1

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

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

Step 11.2.2

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

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

Step 11.3

Add $2$ and $1$ .

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