Precalculus Examples

$(\frac{2 a + b}{a} - \frac{a + 2 b}{b}) \cdot (\frac{a - b}{b^{2} + a b} + \frac{a + b}{b^{2} - a b})$

Step 1

Set the denominator in $\frac{2 a + b}{a}$ equal to $0$ to find where the expression is undefined.

$a = 0$

Step 2

Set the denominator in $\frac{a + 2 b}{b}$ equal to $0$ to find where the expression is undefined.

$b = 0$

Step 3

Set the denominator in $\frac{a - b}{b^{2} + a b}$ equal to $0$ to find where the expression is undefined.

$b^{2} + a b = 0$

Step 4

Solve for $a$ .

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Step 4.1

Subtract $b^{2}$ from both sides of the equation.

$a b = - b^{2}$

Step 4.2

Divide each term in $a b = - b^{2}$ by $b$ and simplify.

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Step 4.2.1

Divide each term in $a b = - b^{2}$ by $b$ .

$\frac{a b}{b} = \frac{- b^{2}}{b}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $b$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{a b}{b} = \frac{- b^{2}}{b}$

Step 4.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{- b^{2}}{b}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Cancel the common factor of $b^{2}$ and $b$ .

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Step 4.2.3.1.1

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

$a = \frac{b (- b)}{b}$

Step 4.2.3.1.2

Cancel the common factors.

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Step 4.2.3.1.2.1

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

$a = \frac{b (- b)}{b^{1}}$

Step 4.2.3.1.2.2

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

$a = \frac{b (- b)}{b \cdot 1}$

Step 4.2.3.1.2.3

Cancel the common factor.

$a = \frac{b (- b)}{b \cdot 1}$

Step 4.2.3.1.2.4

Rewrite the expression.

$a = \frac{- b}{1}$

Step 4.2.3.1.2.5

Divide $- b$ by $1$ .

$a = - b$

Step 5

Set the denominator in $\frac{a + b}{b^{2} - a b}$ equal to $0$ to find where the expression is undefined.

$b^{2} - a b = 0$

Step 6

Solve for $a$ .

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Step 6.1

Subtract $b^{2}$ from both sides of the equation.

$- a b = - b^{2}$

Step 6.2

Divide each term in $- a b = - b^{2}$ by $- b$ and simplify.

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Step 6.2.1

Divide each term in $- a b = - b^{2}$ by $- b$ .

$\frac{- a b}{- b} = \frac{- b^{2}}{- b}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Dividing two negative values results in a positive value.

$\frac{a b}{b} = \frac{- b^{2}}{- b}$

Step 6.2.2.2

Cancel the common factor of $b$ .

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Step 6.2.2.2.1

Cancel the common factor.

$\frac{a b}{b} = \frac{- b^{2}}{- b}$

Step 6.2.2.2.2

Divide $a$ by $1$ .

$a = \frac{- b^{2}}{- b}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Dividing two negative values results in a positive value.

$a = \frac{b^{2}}{b}$

Step 6.2.3.2

Cancel the common factor of $b^{2}$ and $b$ .

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Step 6.2.3.2.1

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

$a = \frac{b \cdot b}{b}$

Step 6.2.3.2.2

Cancel the common factors.

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Step 6.2.3.2.2.1

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

$a = \frac{b \cdot b}{b^{1}}$

Step 6.2.3.2.2.2

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

$a = \frac{b \cdot b}{b \cdot 1}$

Step 6.2.3.2.2.3

Cancel the common factor.

$a = \frac{b \cdot b}{b \cdot 1}$

Step 6.2.3.2.2.4

Rewrite the expression.

$a = \frac{b}{1}$

Step 6.2.3.2.2.5

Divide $b$ by $1$ .

$a = b$

Step 7

The domain is all values of $a$ that make the expression defined.

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${a | a \neq 0}$