Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Expand $(2 x - 1) (x^{2} - 2)$ using the FOIL Method.

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Step 1.1.1

Apply the distributive property.

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

Step 1.1.2

Apply the distributive property.

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

Step 1.1.3

Apply the distributive property.

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

Step 1.2

Simplify each term.

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Step 1.2.1

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

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Step 1.2.1.1

Move $x^{2}$ .

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

Step 1.2.1.2

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

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Step 1.2.1.2.1

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

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

Step 1.2.1.2.2

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

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

Step 1.2.1.3

Add $2$ and $1$ .

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

Step 1.2.2

Multiply $- 2$ by $2$ .

$2 x^{3} - 4 x - 1 x^{2} - 1 \cdot - 2 - x (x^{2} - x - 2)$

Step 1.2.3

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

$2 x^{3} - 4 x - x^{2} - 1 \cdot - 2 - x (x^{2} - x - 2)$

Step 1.2.4

Multiply $- 1$ by $- 2$ .

$2 x^{3} - 4 x - x^{2} + 2 - x (x^{2} - x - 2)$

Step 1.3

Apply the distributive property.

$2 x^{3} - 4 x - x^{2} + 2 - x \cdot x^{2} - x (- x) - x \cdot - 2$

Step 1.4

Simplify.

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Step 1.4.1

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

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Step 1.4.1.1

Move $x^{2}$ .

$2 x^{3} - 4 x - x^{2} + 2 - (x^{2} x) - x (- x) - x \cdot - 2$

Step 1.4.1.2

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

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Step 1.4.1.2.1

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

$2 x^{3} - 4 x - x^{2} + 2 - (x^{2} x^{1}) - x (- x) - x \cdot - 2$

Step 1.4.1.2.2

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

$2 x^{3} - 4 x - x^{2} + 2 - x^{2 + 1} - x (- x) - x \cdot - 2$

Step 1.4.1.3

Add $2$ and $1$ .

$2 x^{3} - 4 x - x^{2} + 2 - x^{3} - x (- x) - x \cdot - 2$

Step 1.4.2

Rewrite using the commutative property of multiplication.

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

Step 1.4.3

Multiply $- 2$ by $- 1$ .

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

Step 1.5

Simplify each term.

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Step 1.5.1

Multiply $x$ by $x$ by adding the exponents.

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Step 1.5.1.1

Move $x$ .

$2 x^{3} - 4 x - x^{2} + 2 - x^{3} - 1 \cdot - 1 (x \cdot x) + 2 x$

Step 1.5.1.2

Multiply $x$ by $x$ .

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

Step 1.5.2

Multiply $- 1$ by $- 1$ .

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

Step 1.5.3

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

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

Step 2

Simplify by adding terms.

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Step 2.1

Combine the opposite terms in $2 x^{3} - 4 x - x^{2} + 2 - x^{3} + x^{2} + 2 x$ .

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Step 2.1.1

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

$2 x^{3} - 4 x + 2 - x^{3} + 0 + 2 x$

Step 2.1.2

Add $2 x^{3} - 4 x + 2 - x^{3}$ and $0$ .

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

Step 2.2

Subtract $x^{3}$ from $2 x^{3}$ .

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

Step 2.3

Add $- 4 x$ and $2 x$ .

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