Precalculus Examples

$\frac{a b^{3} - 9 a b}{12 a b^{2} + 12 a b - 144 a}$

Step 1

Set the denominator in $\frac{a b^{3} - 9 a b}{12 a b^{2} + 12 a b - 144 a}$ equal to $0$ to find where the expression is undefined.

$12 a b^{2} + 12 a b - 144 a = 0$

Step 2

Solve for $a$ .

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Step 2.1

Factor $12 a$ out of $12 a b^{2} + 12 a b - 144 a$ .

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Step 2.1.1

Factor $12 a$ out of $12 a b^{2}$ .

$12 a (b^{2}) + 12 a b - 144 a = 0$

Step 2.1.2

Factor $12 a$ out of $12 a b$ .

$12 a (b^{2}) + 12 a (b) - 144 a = 0$

Step 2.1.3

Factor $12 a$ out of $- 144 a$ .

$12 a (b^{2}) + 12 a (b) + 12 a (- 12) = 0$

Step 2.1.4

Factor $12 a$ out of $12 a (b^{2}) + 12 a (b)$ .

$12 a (b^{2} + b) + 12 a (- 12) = 0$

Step 2.1.5

Factor $12 a$ out of $12 a (b^{2} + b) + 12 a (- 12)$ .

$12 a (b^{2} + b - 12) = 0$

Step 2.2

Factor.

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Step 2.2.1

Factor $b^{2} + b - 12$ using the AC method.

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Step 2.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 12$ and whose sum is $1$ .

$- 3, 4$

Step 2.2.1.2

Write the factored form using these integers.

$12 a ((b - 3) (b + 4)) = 0$

Step 2.2.2

Remove unnecessary parentheses.

$12 a (b - 3) (b + 4) = 0$

Step 2.3

Divide each term in $12 a (b - 3) (b + 4) = 0$ by $12 (b - 3) (b + 4)$ and simplify.

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Step 2.3.1

Divide each term in $12 a (b - 3) (b + 4) = 0$ by $12 (b - 3) (b + 4)$ .

$\frac{12 a (b - 3) (b + 4)}{12 (b - 3) (b + 4)} = \frac{0}{12 (b - 3) (b + 4)}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $12$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{12 a (b - 3) (b + 4)}{12 (b - 3) (b + 4)} = \frac{0}{12 (b - 3) (b + 4)}$

Step 2.3.2.1.2

Rewrite the expression.

$\frac{(a (b - 3)) (b + 4)}{(b - 3) (b + 4)} = \frac{0}{12 (b - 3) (b + 4)}$

Step 2.3.2.2

Cancel the common factor of $b - 3$ .

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Step 2.3.2.2.1

Cancel the common factor.

$\frac{a (b - 3) (b + 4)}{(b - 3) (b + 4)} = \frac{0}{12 (b - 3) (b + 4)}$

Step 2.3.2.2.2

Rewrite the expression.

$\frac{(a) (b + 4)}{b + 4} = \frac{0}{12 (b - 3) (b + 4)}$

Step 2.3.2.3

Cancel the common factor of $b + 4$ .

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Step 2.3.2.3.1

Cancel the common factor.

$\frac{a (b + 4)}{b + 4} = \frac{0}{12 (b - 3) (b + 4)}$

Step 2.3.2.3.2

Divide $a$ by $1$ .

$a = \frac{0}{12 (b - 3) (b + 4)}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Cancel the common factor of $0$ and $12$ .

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Step 2.3.3.1.1

Factor $12$ out of $0$ .

$a = \frac{12 (0)}{12 (b - 3) (b + 4)}$

Step 2.3.3.1.2

Cancel the common factors.

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Step 2.3.3.1.2.1

Factor $12$ out of $12 (b - 3) (b + 4)$ .

$a = \frac{12 (0)}{12 ((b - 3) (b + 4))}$

Step 2.3.3.1.2.2

Cancel the common factor.

$a = \frac{12 \cdot 0}{12 ((b - 3) (b + 4))}$

Step 2.3.3.1.2.3

Rewrite the expression.

$a = \frac{0}{(b - 3) (b + 4)}$

Step 2.3.3.2

Divide $0$ by $(b - 3) (b + 4)$ .

$a = 0$

Step 3

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

Interval Notation:

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

Set-Builder Notation:

${a | a \neq 0}$