Algebra Examples

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

Step 1

Expand $(x - 1 + x^{2}) (x - 1 - x^{2})$ by multiplying each term in the first expression by each term in the second expression.

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

Step 2

Simplify terms.

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Step 2.1

Combine the opposite terms in $x \cdot x + x \cdot - 1 + x (- x^{2}) - 1 x - 1 \cdot - 1 - 1 (- x^{2}) + x^{2} x + x^{2} \cdot - 1 + x^{2} (- x^{2})$ .

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Step 2.1.1

Reorder the factors in the terms $x (- x^{2})$ and $x^{2} x$ .

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

Step 2.1.2

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

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

Step 2.1.3

Add $x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1 - 1 (- x^{2})$ and $0$ .

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

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply $x$ by $x$ .

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Multiply $- 1$ by $- 1$ .

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

Step 2.2.6

Multiply $- 1 (- x^{2})$ .

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Step 2.2.6.1

Multiply $- 1$ by $- 1$ .

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

Step 2.2.6.2

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

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

Step 2.2.7

Move $- 1$ to the left of $x^{2}$ .

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

Step 2.2.8

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

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

Step 2.2.9

Rewrite using the commutative property of multiplication.

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

Step 2.2.10

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

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Step 2.2.10.1

Move $x^{2}$ .

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

Step 2.2.10.2

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

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

Step 2.2.10.3

Add $2$ and $2$ .

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

Step 2.3

Simplify by adding terms.

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Step 2.3.1

Combine the opposite terms in $x^{2} - x - x + 1 + x^{2} - x^{2} - x^{4}$ .

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Step 2.3.1.1

Subtract $x^{2}$ from $x^{2}$ .

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

Step 2.3.1.2

Add $- x - x + 1 + x^{2}$ and $0$ .

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

Step 2.3.2

Subtract $x$ from $- x$ .

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

Step 2.3.3

Simplify the expression.

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Step 2.3.3.1

Move $1$ .

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

Step 2.3.3.2

Move $- 2 x$ .

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

Step 2.3.3.3

Reorder $x^{2}$ and $- x^{4}$ .

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