Algebra Examples

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

Step 1

Find the common denominator.

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Step 1.1

Multiply $\frac{1}{x}$ by $\frac{(x^{2} + x) (x + 3)}{(x^{2} + x) (x + 3)}$ .

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

Step 1.2

Multiply $\frac{1}{x}$ by $\frac{(x^{2} + x) (x + 3)}{(x^{2} + x) (x + 3)}$ .

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

Step 1.3

Multiply $\frac{2}{x^{2} + x}$ by $\frac{x (x + 3)}{x (x + 3)}$ .

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

Step 1.4

Multiply $\frac{2}{x^{2} + x}$ by $\frac{x (x + 3)}{x (x + 3)}$ .

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

Step 1.5

Multiply $\frac{3}{x + 3}$ by $\frac{x (x^{2} + x)}{x (x^{2} + x)}$ .

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

Step 1.6

Multiply $\frac{3}{x + 3}$ by $\frac{x (x^{2} + x)}{x (x^{2} + x)}$ .

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

Step 1.7

Reorder the factors of $x ((x^{2} + x) (x + 3))$ .

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

Step 1.8

Reorder the factors of $(x^{2} + x) (x (x + 3))$ .

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

Step 1.9

Reorder the factors of $(x + 3) (x (x^{2} + x))$ .

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

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

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

Step 2.2

Simplify each term.

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Step 2.2.1

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

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Step 2.2.1.1

Apply the distributive property.

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

Step 2.2.1.2

Apply the distributive property.

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

Step 2.2.1.3

Apply the distributive property.

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

Step 2.2.2

Simplify and combine like terms.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

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

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Step 2.2.2.1.1.1

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

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Step 2.2.2.1.1.1.1

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

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

Step 2.2.2.1.1.1.2

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

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

Step 2.2.2.1.1.2

Add $2$ and $1$ .

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

Step 2.2.2.1.2

Move $3$ to the left of $x^{2}$ .

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

Step 2.2.2.1.3

Multiply $x$ by $x$ .

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

Step 2.2.2.1.4

Move $3$ to the left of $x$ .

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

Step 2.2.2.2

Add $3 x^{2}$ and $x^{2}$ .

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

Step 2.2.3

Apply the distributive property.

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

Step 2.2.4

Multiply $x$ by $x$ .

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

Step 2.2.5

Move $3$ to the left of $x$ .

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

Step 2.2.6

Apply the distributive property.

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

Step 2.2.7

Multiply $3$ by $- 2$ .

$\frac{x^{3} + 4 x^{2} + 3 x - 2 x^{2} - 6 x - 3 (x (x^{2} + x))}{x (x + 3) (x^{2} + x)}$

Step 2.2.8

Apply the distributive property.

$\frac{x^{3} + 4 x^{2} + 3 x - 2 x^{2} - 6 x - 3 (x \cdot x^{2} + x \cdot x)}{x (x + 3) (x^{2} + x)}$

Step 2.2.9

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

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Step 2.2.9.1

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

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Step 2.2.9.1.1

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

$\frac{x^{3} + 4 x^{2} + 3 x - 2 x^{2} - 6 x - 3 (x^{1} x^{2} + x \cdot x)}{x (x + 3) (x^{2} + x)}$

Step 2.2.9.1.2

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

$\frac{x^{3} + 4 x^{2} + 3 x - 2 x^{2} - 6 x - 3 (x^{1 + 2} + x \cdot x)}{x (x + 3) (x^{2} + x)}$

Step 2.2.9.2

Add $1$ and $2$ .

$\frac{x^{3} + 4 x^{2} + 3 x - 2 x^{2} - 6 x - 3 (x^{3} + x \cdot x)}{x (x + 3) (x^{2} + x)}$

Step 2.2.10

Multiply $x$ by $x$ .

$\frac{x^{3} + 4 x^{2} + 3 x - 2 x^{2} - 6 x - 3 (x^{3} + x^{2})}{x (x + 3) (x^{2} + x)}$

Step 2.2.11

Apply the distributive property.

$\frac{x^{3} + 4 x^{2} + 3 x - 2 x^{2} - 6 x - 3 x^{3} - 3 x^{2}}{x (x + 3) (x^{2} + x)}$

Step 2.3

Simplify terms.

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Step 2.3.1

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

$\frac{- 2 x^{3} + 4 x^{2} + 3 x - 2 x^{2} - 6 x - 3 x^{2}}{x (x + 3) (x^{2} + x)}$

Step 2.3.2

Subtract $2 x^{2}$ from $4 x^{2}$ .

$\frac{- 2 x^{3} + 2 x^{2} + 3 x - 6 x - 3 x^{2}}{x (x + 3) (x^{2} + x)}$

Step 2.3.3

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

$\frac{- 2 x^{3} - x^{2} + 3 x - 6 x}{x (x + 3) (x^{2} + x)}$

Step 2.3.4

Subtract $6 x$ from $3 x$ .

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

Step 2.3.5

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

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Step 2.3.5.1

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

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

Step 2.3.5.2

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

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

Step 2.3.5.3

Factor $x$ out of $- 3 x$ .

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

Step 2.3.5.4

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

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

Step 2.3.5.5

Factor $x$ out of $x (- 2 x^{2} - x) + x \cdot - 3$ .

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

Step 3

Simplify the denominator.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.1.4

Factor $x$ out of $x \cdot x + x \cdot 1$ .

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

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

Step 3.2

Combine exponents.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

Add $1$ and $1$ .

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

Step 4

Cancel the common factors.

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Step 4.1

Factor $x$ out of $x^{2} (x + 3) (x + 1)$ .

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

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

Step 5

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

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

Step 6

Factor $- 1$ out of $- x$ .

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

Step 7

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

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

Step 8

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 9

Factor $- 1$ out of $- (2 x^{2} + x) - 1 (3)$ .

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

Step 10

Simplify the expression.

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Step 10.1

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

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

Step 10.2

Move the negative in front of the fraction.

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