Precalculus Examples

$f (x) = \frac{x^{3} - 6 x^{2} + 8 x}{x^{2} - 2 x - 15}$

Step 1

Find where the expression $\frac{x^{3} - 6 x^{2} + 8 x}{x^{2} - 2 x - 15}$ is undefined.

$x = - 3, x = 5$

Step 2

Since $\frac{x^{3} - 6 x^{2} + 8 x}{x^{2} - 2 x - 15}$ $\to$ $- \infty$ as $x$ $\to$ $- 3$ from the left and $\frac{x^{3} - 6 x^{2} + 8 x}{x^{2} - 2 x - 15}$ $\to$ $\infty$ as $x$ $\to$ $- 3$ from the right, then $x = - 3$ is a vertical asymptote.

$x = - 3$

Step 3

Since $\frac{x^{3} - 6 x^{2} + 8 x}{x^{2} - 2 x - 15}$ $\to$ $- \infty$ as $x$ $\to$ $5$ from the left and $\frac{x^{3} - 6 x^{2} + 8 x}{x^{2} - 2 x - 15}$ $\to$ $\infty$ as $x$ $\to$ $5$ from the right, then $x = 5$ is a vertical asymptote.

$x = 5$

Step 4

List all of the vertical asymptotes:

$x = - 3, 5$

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 $x^{3} - 6 x^{2} + 8 x$ .

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Step 8.1.1.1.1

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

$\frac{x \cdot x^{2} - 6 x^{2} + 8 x}{x^{2} - 2 x - 15}$

Step 8.1.1.1.2

Factor $x$ out of $- 6 x^{2}$ .

$\frac{x \cdot x^{2} + x (- 6 x) + 8 x}{x^{2} - 2 x - 15}$

Step 8.1.1.1.3

Factor $x$ out of $8 x$ .

$\frac{x \cdot x^{2} + x (- 6 x) + x \cdot 8}{x^{2} - 2 x - 15}$

Step 8.1.1.1.4

Factor $x$ out of $x \cdot x^{2} + x (- 6 x)$ .

$\frac{x (x^{2} - 6 x) + x \cdot 8}{x^{2} - 2 x - 15}$

Step 8.1.1.1.5

Factor $x$ out of $x (x^{2} - 6 x) + x \cdot 8$ .

$\frac{x (x^{2} - 6 x + 8)}{x^{2} - 2 x - 15}$

Step 8.1.1.2

Factor $x^{2} - 6 x + 8$ 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 $- 6$ .

$- 4, - 2$

Step 8.1.1.2.2

Write the factored form using these integers.

$\frac{x ((x - 4) (x - 2))}{x^{2} - 2 x - 15}$

$\frac{x (x - 4) (x - 2)}{x^{2} - 2 x - 15}$

Step 8.1.2

Factor $x^{2} - 2 x - 15$ using the AC method.

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Step 8.1.2.1

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 $- 15$ and whose sum is $- 2$ .

$- 5, 3$

Step 8.1.2.2

Write the factored form using these integers.

$\frac{x (x - 4) (x - 2)}{(x - 5) (x + 3)}$

Step 8.2

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

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Step 8.2.1

Apply the distributive property.

$\frac{(x \cdot x + x \cdot - 4) (x - 2)}{(x - 5) (x + 3)}$

Step 8.2.2

Apply the distributive property.

$\frac{x \cdot x (x - 2) + x \cdot (- 4 (x - 2))}{(x - 5) (x + 3)}$

Step 8.2.3

Apply the distributive property.

$\frac{x \cdot x \cdot x + x \cdot x \cdot - 2 + x \cdot (- 4 (x - 2))}{(x - 5) (x + 3)}$

Step 8.2.4

Apply the distributive property.

$\frac{x \cdot x \cdot x + x \cdot x \cdot - 2 + x \cdot (- 4 x) + x \cdot - 4 \cdot - 2}{(x - 5) (x + 3)}$

Step 8.2.5

Move $x$ .

$\frac{x \cdot x \cdot x + x \cdot (- 2 x) + x \cdot (- 4 x) + x \cdot - 4 \cdot - 2}{(x - 5) (x + 3)}$

Step 8.2.6

Reorder $x$ and $- 2$ .

$\frac{x \cdot x \cdot x - 2 \cdot (x \cdot x) + x \cdot (- 4 x) + x \cdot - 4 \cdot - 2}{(x - 5) (x + 3)}$

Step 8.2.7

Reorder $x$ and $- 4$ .

$\frac{x \cdot x \cdot x - 2 \cdot (x \cdot x) - 4 \cdot (x \cdot x) + x \cdot - 4 \cdot - 2}{(x - 5) (x + 3)}$

Step 8.2.8

Reorder $x$ and $- 4$ .

$\frac{x \cdot x \cdot x - 2 \cdot (x \cdot x) - 4 \cdot (x \cdot x) - 4 \cdot x \cdot - 2}{(x - 5) (x + 3)}$

Step 8.2.9

Move $x$ .

$\frac{x \cdot x \cdot x - 2 \cdot (x \cdot x) - 4 \cdot (x \cdot x) - 4 \cdot (- 2 x)}{(x - 5) (x + 3)}$

Step 8.2.10

Multiply $- 2$ by $x$ .

$\frac{x \cdot x \cdot x - 2 x \cdot x - 4 \cdot (x \cdot x) - 4 \cdot (- 2 x)}{(x - 5) (x + 3)}$

Step 8.2.11

Multiply $- 4$ by $x$ .

$\frac{x \cdot x \cdot x - 2 x \cdot x - 4 x \cdot x - 4 \cdot (- 2 x)}{(x - 5) (x + 3)}$

Step 8.2.12

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

$\frac{x \cdot x \cdot x - 2 x \cdot x - 4 x \cdot x - 4 \cdot (- 2 x)}{(x - 5) (x + 3)}$

Step 8.2.13

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

$\frac{x \cdot x \cdot x - 2 x \cdot x - 4 x \cdot x - 4 \cdot (- 2 x)}{(x - 5) (x + 3)}$

Step 8.2.14

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

$\frac{x^{1 + 1} x - 2 x \cdot x - 4 x \cdot x - 4 \cdot (- 2 x)}{(x - 5) (x + 3)}$

Step 8.2.15

Add $1$ and $1$ .

$\frac{x^{2} x - 2 x \cdot x - 4 x \cdot x - 4 \cdot (- 2 x)}{(x - 5) (x + 3)}$

Step 8.2.16

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

$\frac{x^{2} x - 2 x \cdot x - 4 x \cdot x - 4 \cdot (- 2 x)}{(x - 5) (x + 3)}$

Step 8.2.17

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

$\frac{x^{2 + 1} - 2 x \cdot x - 4 x \cdot x - 4 \cdot (- 2 x)}{(x - 5) (x + 3)}$

Step 8.2.18

Add $2$ and $1$ .

$\frac{x^{3} - 2 x \cdot x - 4 x \cdot x - 4 \cdot (- 2 x)}{(x - 5) (x + 3)}$

Step 8.2.19

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

$\frac{x^{3} - 2 (x \cdot x) - 4 x \cdot x - 4 \cdot (- 2 x)}{(x - 5) (x + 3)}$

Step 8.2.20

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

$\frac{x^{3} - 2 (x \cdot x) - 4 x \cdot x - 4 \cdot (- 2 x)}{(x - 5) (x + 3)}$

Step 8.2.21

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

$\frac{x^{3} - 2 x^{1 + 1} - 4 x \cdot x - 4 \cdot (- 2 x)}{(x - 5) (x + 3)}$

Step 8.2.22

Add $1$ and $1$ .

$\frac{x^{3} - 2 x^{2} - 4 x \cdot x - 4 \cdot (- 2 x)}{(x - 5) (x + 3)}$

Step 8.2.23

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

$\frac{x^{3} - 2 x^{2} - 4 (x \cdot x) - 4 \cdot (- 2 x)}{(x - 5) (x + 3)}$

Step 8.2.24

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

$\frac{x^{3} - 2 x^{2} - 4 (x \cdot x) - 4 \cdot (- 2 x)}{(x - 5) (x + 3)}$

Step 8.2.25

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

$\frac{x^{3} - 2 x^{2} - 4 x^{1 + 1} - 4 \cdot (- 2 x)}{(x - 5) (x + 3)}$

Step 8.2.26

Add $1$ and $1$ .

$\frac{x^{3} - 2 x^{2} - 4 x^{2} - 4 \cdot (- 2 x)}{(x - 5) (x + 3)}$

Step 8.2.27

Multiply $- 4$ by $- 2$ .

$\frac{x^{3} - 2 x^{2} - 4 x^{2} + 8 x}{(x - 5) (x + 3)}$

Step 8.2.28

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

$\frac{x^{3} - 6 x^{2} + 8 x}{(x - 5) (x + 3)}$

Step 8.3

Expand $(x - 5) (x + 3)$ .

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Step 8.3.1

Apply the distributive property.

$\frac{x^{3} - 6 x^{2} + 8 x}{x (x + 3) - 5 (x + 3)}$

Step 8.3.2

Apply the distributive property.

$\frac{x^{3} - 6 x^{2} + 8 x}{x \cdot x + x \cdot 3 - 5 (x + 3)}$

Step 8.3.3

Apply the distributive property.

$\frac{x^{3} - 6 x^{2} + 8 x}{x \cdot x + x \cdot 3 - 5 x - 5 \cdot 3}$

Step 8.3.4

Reorder $x$ and $3$ .

$\frac{x^{3} - 6 x^{2} + 8 x}{x \cdot x + 3 x - 5 x - 5 \cdot 3}$

Step 8.3.5

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

$\frac{x^{3} - 6 x^{2} + 8 x}{x \cdot x + 3 x - 5 x - 5 \cdot 3}$

Step 8.3.6

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

$\frac{x^{3} - 6 x^{2} + 8 x}{x \cdot x + 3 x - 5 x - 5 \cdot 3}$

Step 8.3.7

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

$\frac{x^{3} - 6 x^{2} + 8 x}{x^{1 + 1} + 3 x - 5 x - 5 \cdot 3}$

Step 8.3.8

Add $1$ and $1$ .

$\frac{x^{3} - 6 x^{2} + 8 x}{x^{2} + 3 x - 5 x - 5 \cdot 3}$

Step 8.3.9

Multiply $- 5$ by $3$ .

$\frac{x^{3} - 6 x^{2} + 8 x}{x^{2} + 3 x - 5 x - 15}$

Step 8.3.10

Subtract $5 x$ from $3 x$ .

$\frac{x^{3} - 6 x^{2} + 8 x}{x^{2} - 2 x - 15}$

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 $0$ .

$x^{2}$

$2 x$

$15$

$x^{3}$

$6 x^{2}$

$8 x$

$0$

Step 8.5

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

$x$

$x^{2}$

$2 x$

$15$

$x^{3}$

$6 x^{2}$

$8 x$

$0$

Step 8.6

Multiply the new quotient term by the divisor.

$x$

$x^{2}$

$2 x$

$15$

$x^{3}$

$6 x^{2}$

$8 x$

$0$

$x^{3}$

$2 x^{2}$

$15 x$

Step 8.7

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - 2 x^{2} - 15 x$

$x$

$x^{2}$

$2 x$

$15$

$x^{3}$

$6 x^{2}$

$8 x$

$0$

$x^{3}$

$2 x^{2}$

$15 x$

Step 8.8

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

$x$

$x^{2}$

$2 x$

$15$

$x^{3}$

$6 x^{2}$

$8 x$

$0$

$x^{3}$

$2 x^{2}$

$15 x$

$4 x^{2}$

$23 x$

Step 8.9

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

$x$

$x^{2}$

$2 x$

$15$

$x^{3}$

$6 x^{2}$

$8 x$

$0$

$x^{3}$

$2 x^{2}$

$15 x$

$4 x^{2}$

$23 x$

$0$

Step 8.10

Divide the highest order term in the dividend $- 4 x^{2}$ by the highest order term in divisor $x^{2}$ .

$x$

$4$

$x^{2}$

$2 x$

$15$

$x^{3}$

$6 x^{2}$

$8 x$

$0$

$x^{3}$

$2 x^{2}$

$15 x$

$4 x^{2}$

$23 x$

$0$

Step 8.11

Multiply the new quotient term by the divisor.

$x$

$4$

$x^{2}$

$2 x$

$15$

$x^{3}$

$6 x^{2}$

$8 x$

$0$

$x^{3}$

$2 x^{2}$

$15 x$

$4 x^{2}$

$23 x$

$0$

$4 x^{2}$

$8 x$

$60$

Step 8.12

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x^{2} + 8 x + 60$

$x$

$4$

$x^{2}$

$2 x$

$15$

$x^{3}$

$6 x^{2}$

$8 x$

$0$

$x^{3}$

$2 x^{2}$

$15 x$

$4 x^{2}$

$23 x$

$0$

$4 x^{2}$

$8 x$

$60$

Step 8.13

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

						$x$	-	$4$
$x^{2}$	-	$2 x$	-	$15$		$x^{3}$	-	$6 x^{2}$	+	$8 x$	+	$0$
					-	$x^{3}$	+	$2 x^{2}$	+	$15 x$
							-	$4 x^{2}$	+	$23 x$	+	$0$
							+	$4 x^{2}$	-	$8 x$	-	$60$
									+	$15 x$	-	$60$

Step 8.14

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

$x - 4 + \frac{15 x - 60}{x^{2} - 2 x - 15}$

Step 8.15

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

$y = x - 4$

Step 9

This is the set of all asymptotes.

Vertical Asymptotes: $x = - 3, 5$

No Horizontal Asymptotes

Oblique Asymptotes: $y = x - 4$

Step 10