Algebra Examples

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

Step 1

Regroup terms.

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

Step 2

Factor $- 1$ out of $- x^{4} + 2 x^{2} - 1$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 2.4

Factor $- 1$ out of $- (x^{4}) - (- 2 x^{2})$ .

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

Step 2.5

Factor $- 1$ out of $- (x^{4} - 2 x^{2}) - 1 (1)$ .

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

Step 3

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

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

Step 4

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

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

Step 5

Factor using the perfect square rule.

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Step 5.1

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

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

Step 5.2

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

$2 u = 2 \cdot u \cdot 1$

Step 5.3

Rewrite the polynomial.

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

Step 5.4

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

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

Step 6

Replace all occurrences of $u$ with $x^{2}$ .

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

Step 7

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

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

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 $b = 1$ .

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

Step 9

Factor.

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Step 9.1

Apply the product rule to $(x + 1) (x - 1)$ .

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

Step 9.2

Remove unnecessary parentheses.

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

Step 10

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

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

Step 11

Rewrite ${(x + 1)}^{2} {(x - 1)}^{2}$ as ${((x + 1) (x - 1))}^{2}$ .

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

Step 12

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

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

Step 13

Simplify.

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Step 13.1

Expand $(x + 1) (x - 1)$ using the FOIL Method.

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Step 13.1.1

Apply the distributive property.

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

Step 13.1.2

Apply the distributive property.

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

Step 13.1.3

Apply the distributive property.

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

Step 13.2

Simplify and combine like terms.

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Step 13.2.1

Simplify each term.

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Step 13.2.1.1

Multiply $x$ by $x$ .

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

Step 13.2.1.2

Move $- 1$ to the left of $x$ .

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

Step 13.2.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 13.2.1.4

Multiply $x$ by $1$ .

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

Step 13.2.1.5

Multiply $- 1$ by $1$ .

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

Step 13.2.2

Add $- x$ and $x$ .

$(x^{3} + x^{2} + 0 - 1) (x^{3} - ((x + 1) (x - 1)))$

Step 13.2.3

Add $x^{2}$ and $0$ .

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

Step 13.3

Expand $(x + 1) (x - 1)$ using the FOIL Method.

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Step 13.3.1

Apply the distributive property.

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

Step 13.3.2

Apply the distributive property.

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

Step 13.3.3

Apply the distributive property.

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

Step 13.4

Simplify and combine like terms.

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Step 13.4.1

Simplify each term.

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Step 13.4.1.1

Multiply $x$ by $x$ .

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

Step 13.4.1.2

Move $- 1$ to the left of $x$ .

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

Step 13.4.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 13.4.1.4

Multiply $x$ by $1$ .

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

Step 13.4.1.5

Multiply $- 1$ by $1$ .

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

Step 13.4.2

Add $- x$ and $x$ .

$(x^{3} + x^{2} - 1) (x^{3} - (x^{2} + 0 - 1))$

Step 13.4.3

Add $x^{2}$ and $0$ .

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

Step 13.5

Apply the distributive property.

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

Step 13.6

Multiply $- 1$ by $- 1$ .

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