Precalculus Examples

$\frac{a^{2} + a x + x^{2}}{a x + 2 a y} \div \frac{a^{3} - x^{3}}{b x + 2 b y}$

Step 1

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

$a x + 2 a y = 0$

Step 2

Solve for $a$ .

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Step 2.1

Factor $a$ out of $a x + 2 a y$ .

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Step 2.1.1

Factor $a$ out of $a x$ .

$a (x) + 2 a y = 0$

Step 2.1.2

Factor $a$ out of $2 a y$ .

$a (x) + a (2 y) = 0$

Step 2.1.3

Factor $a$ out of $a (x) + a (2 y)$ .

$a (x + 2 y) = 0$

Step 2.2

Divide each term in $a (x + 2 y) = 0$ by $x + 2 y$ and simplify.

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Step 2.2.1

Divide each term in $a (x + 2 y) = 0$ by $x + 2 y$ .

$\frac{a (x + 2 y)}{x + 2 y} = \frac{0}{x + 2 y}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $x + 2 y$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{a (x + 2 y)}{x + 2 y} = \frac{0}{x + 2 y}$

Step 2.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{0}{x + 2 y}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $0$ by $x + 2 y$ .

$a = 0$

Step 3

Set the denominator in $\frac{a^{3} - x^{3}}{b x + 2 b y}$ equal to $0$ to find where the expression is undefined.

$b x + 2 b y = 0$

Step 4

Solve for $a$ .

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Step 4.1

Factor $b$ out of $b x + 2 b y$ .

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Step 4.1.1

Factor $b$ out of $b x$ .

$b (x) + 2 b y = 0$

Step 4.1.2

Factor $b$ out of $2 b y$ .

$b (x) + b (2 y) = 0$

Step 4.1.3

Factor $b$ out of $b (x) + b (2 y)$ .

$b (x + 2 y) = 0$

Step 4.2

Divide each term in $b (x + 2 y) = 0$ by $x + 2 y$ and simplify.

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Step 4.2.1

Divide each term in $b (x + 2 y) = 0$ by $x + 2 y$ .

$\frac{b (x + 2 y)}{x + 2 y} = \frac{0}{x + 2 y}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $x + 2 y$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{b (x + 2 y)}{x + 2 y} = \frac{0}{x + 2 y}$

Step 4.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{0}{x + 2 y}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $0$ by $x + 2 y$ .

$b = 0$

Step 5

Set the denominator in $\frac{a^{2} + a x + x^{2}}{a x + 2 a y} \div \frac{a^{3} - x^{3}}{b x + 2 b y}$ equal to $0$ to find where the expression is undefined.

$\frac{a^{3} - x^{3}}{b x + 2 b y} = 0$

Step 6

Solve for $a$ .

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Step 6.1

Set the numerator equal to zero.

$a^{3} - x^{3} = 0$

Step 6.2

Solve the equation for $a$ .

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Step 6.2.1

Add $x^{3}$ to both sides of the equation.

$a^{3} = x^{3}$

Step 6.2.2

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$a = x$

Step 7

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

Interval Notation:

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

Set-Builder Notation:

${a | a \neq 0}$