Precalculus Examples

$f (x) = \frac{x^{3} - 2 x^{2} - 8 x}{3 x^{2} - 27}$

Step 1

Find where the expression $\frac{x^{3} - 2 x^{2} - 8 x}{3 x^{2} - 27}$ is undefined.

$x = - 3, x = 3$

Step 2

Since $\frac{x^{3} - 2 x^{2} - 8 x}{3 x^{2} - 27}$ $\to$ $- \infty$ as $x$ $\to$ $- 3$ from the left and $\frac{x^{3} - 2 x^{2} - 8 x}{3 x^{2} - 27}$ $\to$ $\infty$ as $x$ $\to$ $- 3$ from the right, then $x = - 3$ is a vertical asymptote.

$x = - 3$

Step 3

Since $\frac{x^{3} - 2 x^{2} - 8 x}{3 x^{2} - 27}$ $\to$ $\infty$ as $x$ $\to$ $3$ from the left and $\frac{x^{3} - 2 x^{2} - 8 x}{3 x^{2} - 27}$ $\to$ $- \infty$ as $x$ $\to$ $3$ from the right, then $x = 3$ is a vertical asymptote.

$x = 3$

Step 4

List all of the vertical asymptotes:

$x = - 3, 3$

Step 5

Consider the rational function $R (x) = \frac{a x^{n}}{b x^{m}}$ where $n$ is the degree of the numerator and $m$ is the degree of the denominator.

1. If $n < m$ , then the x-axis, $y = 0$ , is the horizontal asymptote.

2. If $n = m$ , then the horizontal asymptote is the line $y = \frac{a}{b}$ .

3. If $n > m$ , then there is no horizontal asymptote (there is an oblique asymptote).

Step 6

Find $n$ and $m$ .

$n = 3$

$m = 2$

Step 7

Since $n > m$ , there is no horizontal asymptote.

No Horizontal Asymptotes

Step 8

Find the oblique asymptote using polynomial division.

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Step 8.1

Simplify the expression.

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Step 8.1.1

Simplify the numerator.

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Step 8.1.1.1

Factor $x$ out of $x^{3} - 2 x^{2} - 8 x$ .

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Step 8.1.1.1.1

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

$\frac{x \cdot x^{2} - 2 x^{2} - 8 x}{3 x^{2} - 27}$

Step 8.1.1.1.2

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

$\frac{x \cdot x^{2} + x (- 2 x) - 8 x}{3 x^{2} - 27}$

Step 8.1.1.1.3

Factor $x$ out of $- 8 x$ .

$\frac{x \cdot x^{2} + x (- 2 x) + x \cdot - 8}{3 x^{2} - 27}$

Step 8.1.1.1.4

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

$\frac{x (x^{2} - 2 x) + x \cdot - 8}{3 x^{2} - 27}$

Step 8.1.1.1.5

Factor $x$ out of $x (x^{2} - 2 x) + x \cdot - 8$ .

$\frac{x (x^{2} - 2 x - 8)}{3 x^{2} - 27}$

Step 8.1.1.2

Factor $x^{2} - 2 x - 8$ using the AC method.

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Step 8.1.1.2.1

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 $- 8$ and whose sum is $- 2$ .

$- 4, 2$

Step 8.1.1.2.2

Write the factored form using these integers.

$\frac{x ((x - 4) (x + 2))}{3 x^{2} - 27}$

$\frac{x (x - 4) (x + 2)}{3 x^{2} - 27}$

Step 8.1.2

Simplify the denominator.

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Step 8.1.2.1

Factor $3$ out of $3 x^{2} - 27$ .

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Step 8.1.2.1.1

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

$\frac{x (x - 4) (x + 2)}{3 (x^{2}) - 27}$

Step 8.1.2.1.2

Factor $3$ out of $- 27$ .

$\frac{x (x - 4) (x + 2)}{3 x^{2} + 3 \cdot - 9}$

Step 8.1.2.1.3

Factor $3$ out of $3 x^{2} + 3 \cdot - 9$ .

$\frac{x (x - 4) (x + 2)}{3 (x^{2} - 9)}$

Step 8.1.2.2

Rewrite $9$ as $3^{2}$ .

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

Step 8.1.2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

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

Step 8.2

Expand $x (x - 4) (x + 2)$ .

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Step 8.2.1

Apply the distributive property.

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

Step 8.2.2

Apply the distributive property.

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

Step 8.2.3

Apply the distributive property.

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

Step 8.2.4

Apply the distributive property.

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

Step 8.2.5

Move $x$ .

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

Step 8.2.6

Reorder $x$ and $2$ .

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

Step 8.2.7

Reorder $x$ and $- 4$ .

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

Step 8.2.8

Reorder $x$ and $- 4$ .

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

Step 8.2.9

Move $x$ .

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

Step 8.2.10

Multiply $2$ by $x$ .

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

Step 8.2.11

Multiply $- 4$ by $x$ .

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

Step 8.2.12

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

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

Step 8.2.13

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

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

Step 8.2.14

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

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

Step 8.2.15

Add $1$ and $1$ .

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

Step 8.2.16

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

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

Step 8.2.17

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

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

Step 8.2.18

Add $2$ and $1$ .

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

Step 8.2.19

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

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

Step 8.2.20

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

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

Step 8.2.21

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

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

Step 8.2.22

Add $1$ and $1$ .

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

Step 8.2.23

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

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

Step 8.2.24

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

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

Step 8.2.25

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

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

Step 8.2.26

Add $1$ and $1$ .

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

Step 8.2.27

Multiply $- 4$ by $2$ .

$\frac{x^{3} + 2 x^{2} - 4 x^{2} - 8 x}{3 (x + 3) (x - 3)}$

Step 8.2.28

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

$\frac{x^{3} - 2 x^{2} - 8 x}{3 (x + 3) (x - 3)}$

Step 8.3

Expand $3 (x + 3) (x - 3)$ .

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Step 8.3.1

Apply the distributive property.

$\frac{x^{3} - 2 x^{2} - 8 x}{(3 x + 3 \cdot 3) (x - 3)}$

Step 8.3.2

Apply the distributive property.

$\frac{x^{3} - 2 x^{2} - 8 x}{3 x (x - 3) + 3 \cdot (3 (x - 3))}$

Step 8.3.3

Apply the distributive property.

$\frac{x^{3} - 2 x^{2} - 8 x}{3 x \cdot x + 3 x \cdot - 3 + 3 \cdot (3 (x - 3))}$

Step 8.3.4

Apply the distributive property.

$\frac{x^{3} - 2 x^{2} - 8 x}{3 x \cdot x + 3 x \cdot - 3 + 3 \cdot (3 x) + 3 \cdot 3 \cdot - 3}$

Step 8.3.5

Move $x$ .

$\frac{x^{3} - 2 x^{2} - 8 x}{3 x \cdot x + 3 \cdot (- 3 x) + 3 \cdot (3 x) + 3 \cdot 3 \cdot - 3}$

Step 8.3.6

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

$\frac{x^{3} - 2 x^{2} - 8 x}{3 (x \cdot x) + 3 \cdot (- 3 x) + 3 \cdot (3 x) + 3 \cdot 3 \cdot - 3}$

Step 8.3.7

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

$\frac{x^{3} - 2 x^{2} - 8 x}{3 (x \cdot x) + 3 \cdot (- 3 x) + 3 \cdot (3 x) + 3 \cdot 3 \cdot - 3}$

Step 8.3.8

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

$\frac{x^{3} - 2 x^{2} - 8 x}{3 x^{1 + 1} + 3 \cdot (- 3 x) + 3 \cdot (3 x) + 3 \cdot 3 \cdot - 3}$

Step 8.3.9

Add $1$ and $1$ .

$\frac{x^{3} - 2 x^{2} - 8 x}{3 x^{2} + 3 \cdot (- 3 x) + 3 \cdot (3 x) + 3 \cdot 3 \cdot - 3}$

Step 8.3.10

Multiply $3$ by $- 3$ .

$\frac{x^{3} - 2 x^{2} - 8 x}{3 x^{2} - 9 x + 3 \cdot (3 x) + 3 \cdot 3 \cdot - 3}$

Step 8.3.11

Multiply $3$ by $3$ .

$\frac{x^{3} - 2 x^{2} - 8 x}{3 x^{2} - 9 x + 9 x + 3 \cdot 3 \cdot - 3}$

Step 8.3.12

Multiply $3$ by $3$ .

$\frac{x^{3} - 2 x^{2} - 8 x}{3 x^{2} - 9 x + 9 x + 9 \cdot - 3}$

Step 8.3.13

Multiply $9$ by $- 3$ .

$\frac{x^{3} - 2 x^{2} - 8 x}{3 x^{2} - 9 x + 9 x - 27}$

Step 8.3.14

Add $- 9 x$ and $9 x$ .

$\frac{x^{3} - 2 x^{2} - 8 x}{3 x^{2} + 0 - 27}$

Step 8.3.15

Subtract $27$ from $0$ .

$\frac{x^{3} - 2 x^{2} - 8 x}{3 x^{2} - 27}$

Step 8.4

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$3 x^{2}$

$0 x$

$27$

$x^{3}$

$2 x^{2}$

$8 x$

$0$

Step 8.5

Divide the highest order term in the dividend $x^{3}$ by the highest order term in divisor $3 x^{2}$ .

$\frac{x}{3}$

$3 x^{2}$

$0 x$

$27$

$x^{3}$

$2 x^{2}$

$8 x$

$0$

Step 8.6

Multiply the new quotient term by the divisor.

$\frac{x}{3}$

$3 x^{2}$

$0 x$

$27$

$x^{3}$

$2 x^{2}$

$8 x$

$0$

$x^{3}$

$0$

$9 x$

Step 8.7

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} + 0 - 9 x$

						$\frac{x}{3}$
$3 x^{2}$	+	$0 x$	-	$27$		$x^{3}$	-	$2 x^{2}$	-	$8 x$	+	$0$
					-	$x^{3}$	-	$0$	+	$9 x$

Step 8.8

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$\frac{x}{3}$

$3 x^{2}$

$0 x$

$27$

$x^{3}$

$2 x^{2}$

$8 x$

$0$

$x^{3}$

$0$

$9 x$

$2 x^{2}$

$x$

Step 8.9

Pull the next terms from the original dividend down into the current dividend.

						$\frac{x}{3}$
$3 x^{2}$	+	$0 x$	-	$27$		$x^{3}$	-	$2 x^{2}$	-	$8 x$	+	$0$
					-	$x^{3}$	-	$0$	+	$9 x$
							-	$2 x^{2}$	+	$x$	+	$0$

Step 8.10

Divide the highest order term in the dividend $- 2 x^{2}$ by the highest order term in divisor $3 x^{2}$ .

						$\frac{x}{3}$	-	$\frac{2}{3}$
$3 x^{2}$	+	$0 x$	-	$27$		$x^{3}$	-	$2 x^{2}$	-	$8 x$	+	$0$
					-	$x^{3}$	-	$0$	+	$9 x$
							-	$2 x^{2}$	+	$x$	+	$0$

Step 8.11

Multiply the new quotient term by the divisor.

						$\frac{x}{3}$	-	$\frac{2}{3}$
$3 x^{2}$	+	$0 x$	-	$27$		$x^{3}$	-	$2 x^{2}$	-	$8 x$	+	$0$
					-	$x^{3}$	-	$0$	+	$9 x$
							-	$2 x^{2}$	+	$x$	+	$0$
							-	$2 x^{2}$	+	$0$	+	$18$

Step 8.12

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x^{2} + 0 + 18$

						$\frac{x}{3}$	-	$\frac{2}{3}$
$3 x^{2}$	+	$0 x$	-	$27$		$x^{3}$	-	$2 x^{2}$	-	$8 x$	+	$0$
					-	$x^{3}$	-	$0$	+	$9 x$
							-	$2 x^{2}$	+	$x$	+	$0$
							+	$2 x^{2}$	-	$0$	-	$18$

Step 8.13

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

						$\frac{x}{3}$	-	$\frac{2}{3}$
$3 x^{2}$	+	$0 x$	-	$27$		$x^{3}$	-	$2 x^{2}$	-	$8 x$	+	$0$
					-	$x^{3}$	-	$0$	+	$9 x$
							-	$2 x^{2}$	+	$x$	+	$0$
							+	$2 x^{2}$	-	$0$	-	$18$
									+	$x$	-	$18$

Step 8.14

The final answer is the quotient plus the remainder over the divisor.

$\frac{x}{3} - \frac{2}{3} + \frac{x - 18}{3 x^{2} - 27}$

Step 8.15

The oblique asymptote is the polynomial portion of the long division result.

$y = \frac{x}{3} - \frac{2}{3}$

Step 9

This is the set of all asymptotes.

Vertical Asymptotes: $x = - 3, 3$

No Horizontal Asymptotes

Oblique Asymptotes: $y = \frac{x}{3} - \frac{2}{3}$

Step 10