Algebra Examples

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

Step 1

Expand $(x^{2} - \frac{1}{x}) (x^{3} + 2 x)$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify each term.

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Step 2.1

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 2.1.1

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

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

Step 2.1.2

Add $2$ and $3$ .

$x^{5} + x^{2} (2 x) - \frac{1}{x} x^{3} - \frac{1}{x} (2 x)$

Step 2.2

Rewrite using the commutative property of multiplication.

$x^{5} + 2 x^{2} x - \frac{1}{x} x^{3} - \frac{1}{x} (2 x)$

Step 2.3

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

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Step 2.3.1

Move $x$ .

$x^{5} + 2 (x \cdot x^{2}) - \frac{1}{x} x^{3} - \frac{1}{x} (2 x)$

Step 2.3.2

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

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Step 2.3.2.1

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

$x^{5} + 2 (x^{1} x^{2}) - \frac{1}{x} x^{3} - \frac{1}{x} (2 x)$

Step 2.3.2.2

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

$x^{5} + 2 x^{1 + 2} - \frac{1}{x} x^{3} - \frac{1}{x} (2 x)$

Step 2.3.3

Add $1$ and $2$ .

$x^{5} + 2 x^{3} - \frac{1}{x} x^{3} - \frac{1}{x} (2 x)$

Step 2.4

Cancel the common factor of $x$ .

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Step 2.4.1

Move the leading negative in $- \frac{1}{x}$ into the numerator.

$x^{5} + 2 x^{3} + \frac{- 1}{x} x^{3} - \frac{1}{x} (2 x)$

Step 2.4.2

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

$x^{5} + 2 x^{3} + \frac{- 1}{x} (x \cdot x^{2}) - \frac{1}{x} (2 x)$

Step 2.4.3

Cancel the common factor.

$x^{5} + 2 x^{3} + \frac{- 1}{x} (x \cdot x^{2}) - \frac{1}{x} (2 x)$

Step 2.4.4

Rewrite the expression.

$x^{5} + 2 x^{3} - 1 x^{2} - \frac{1}{x} (2 x)$

Step 2.5

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

$x^{5} + 2 x^{3} - x^{2} - \frac{1}{x} (2 x)$

Step 2.6

Cancel the common factor of $x$ .

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Step 2.6.1

Move the leading negative in $- \frac{1}{x}$ into the numerator.

$x^{5} + 2 x^{3} - x^{2} + \frac{- 1}{x} (2 x)$

Step 2.6.2

Factor $x$ out of $2 x$ .

$x^{5} + 2 x^{3} - x^{2} + \frac{- 1}{x} (x \cdot 2)$

Step 2.6.3

Cancel the common factor.

$x^{5} + 2 x^{3} - x^{2} + \frac{- 1}{x} (x \cdot 2)$

Step 2.6.4

Rewrite the expression.

$x^{5} + 2 x^{3} - x^{2} - 1 \cdot 2$

Step 2.7

Multiply $- 1$ by $2$ .

$x^{5} + 2 x^{3} - x^{2} - 2$