Precalculus Examples

$\frac{(x + 6) (x + 9)}{3 (x + 10)}$

Step 1

Find where the expression $\frac{(x + 6) (x + 9)}{3 (x + 10)}$ is undefined.

$x = - 10$

Step 2

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 3

Find $n$ and $m$ .

$n = 2$

$m = 1$

Step 4

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

No Horizontal Asymptotes

Step 5

Find the oblique asymptote using polynomial division.

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Step 5.1

Simplify the expression.

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Step 5.1.1

Factor $x^{2} + 15 x + 54$ using the AC method.

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Step 5.1.1.1

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 $54$ and whose sum is $15$ .

$6, 9$

Step 5.1.1.2

Write the factored form using these integers.

$\frac{(x + 6) (x + 9)}{3 x + 30}$

Step 5.1.2

Factor $3$ out of $3 x + 30$ .

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Step 5.1.2.1

Factor $3$ out of $3 x$ .

$\frac{(x + 6) (x + 9)}{3 (x) + 30}$

Step 5.1.2.2

Factor $3$ out of $30$ .

$\frac{(x + 6) (x + 9)}{3 x + 3 \cdot 10}$

Step 5.1.2.3

Factor $3$ out of $3 x + 3 \cdot 10$ .

$\frac{(x + 6) (x + 9)}{3 (x + 10)}$

Step 5.2

Expand $(x + 6) (x + 9)$ .

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Step 5.2.1

Apply the distributive property.

$\frac{x (x + 9) + 6 (x + 9)}{3 (x + 10)}$

Step 5.2.2

Apply the distributive property.

$\frac{x \cdot x + x \cdot 9 + 6 (x + 9)}{3 (x + 10)}$

Step 5.2.3

Apply the distributive property.

$\frac{x \cdot x + x \cdot 9 + 6 x + 6 \cdot 9}{3 (x + 10)}$

Step 5.2.4

Reorder $x$ and $9$ .

$\frac{x \cdot x + 9 x + 6 x + 6 \cdot 9}{3 (x + 10)}$

Step 5.2.5

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

$\frac{x \cdot x + 9 x + 6 x + 6 \cdot 9}{3 (x + 10)}$

Step 5.2.6

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

$\frac{x \cdot x + 9 x + 6 x + 6 \cdot 9}{3 (x + 10)}$

Step 5.2.7

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

$\frac{x^{1 + 1} + 9 x + 6 x + 6 \cdot 9}{3 (x + 10)}$

Step 5.2.8

Add $1$ and $1$ .

$\frac{x^{2} + 9 x + 6 x + 6 \cdot 9}{3 (x + 10)}$

Step 5.2.9

Multiply $6$ by $9$ .

$\frac{x^{2} + 9 x + 6 x + 54}{3 (x + 10)}$

Step 5.2.10

Add $9 x$ and $6 x$ .

$\frac{x^{2} + 15 x + 54}{3 (x + 10)}$

Step 5.3

Expand $3 (x + 10)$ .

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Step 5.3.1

Apply the distributive property.

$\frac{x^{2} + 15 x + 54}{3 x + 3 \cdot 10}$

Step 5.3.2

Multiply $3$ by $10$ .

$\frac{x^{2} + 15 x + 54}{3 x + 30}$

Step 5.4

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

$3 x$

$30$

$x^{2}$

$15 x$

$54$

Step 5.5

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

					$\frac{x}{3}$
	$3 x$	+	$30$		$x^{2}$	+	$15 x$	+	$54$

Step 5.6

Multiply the new quotient term by the divisor.

				$\frac{x}{3}$
$3 x$	+	$30$		$x^{2}$	+	$15 x$	+	$54$
			+	$x^{2}$	+	$10 x$

Step 5.7

The expression needs to be subtracted from the dividend, so change all the signs in $x^{2} + 10 x$

				$\frac{x}{3}$
$3 x$	+	$30$		$x^{2}$	+	$15 x$	+	$54$
			-	$x^{2}$	-	$10 x$

Step 5.8

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

				$\frac{x}{3}$
$3 x$	+	$30$		$x^{2}$	+	$15 x$	+	$54$
			-	$x^{2}$	-	$10 x$
					+	$5 x$

Step 5.9

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

				$\frac{x}{3}$
$3 x$	+	$30$		$x^{2}$	+	$15 x$	+	$54$
			-	$x^{2}$	-	$10 x$
					+	$5 x$	+	$54$

Step 5.10

Divide the highest order term in the dividend $5 x$ by the highest order term in divisor $3 x$ .

				$\frac{x}{3}$	+	$\frac{5}{3}$
$3 x$	+	$30$		$x^{2}$	+	$15 x$	+	$54$
			-	$x^{2}$	-	$10 x$
					+	$5 x$	+	$54$

Step 5.11

Multiply the new quotient term by the divisor.

				$\frac{x}{3}$	+	$\frac{5}{3}$
$3 x$	+	$30$		$x^{2}$	+	$15 x$	+	$54$
			-	$x^{2}$	-	$10 x$
					+	$5 x$	+	$54$
					+	$5 x$	+	$50$

Step 5.12

The expression needs to be subtracted from the dividend, so change all the signs in $5 x + 50$

				$\frac{x}{3}$	+	$\frac{5}{3}$
$3 x$	+	$30$		$x^{2}$	+	$15 x$	+	$54$
			-	$x^{2}$	-	$10 x$
					+	$5 x$	+	$54$
					-	$5 x$	-	$50$

Step 5.13

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

				$\frac{x}{3}$	+	$\frac{5}{3}$
$3 x$	+	$30$		$x^{2}$	+	$15 x$	+	$54$
			-	$x^{2}$	-	$10 x$
					+	$5 x$	+	$54$
					-	$5 x$	-	$50$
							+	$4$

Step 5.14

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

$\frac{x}{3} + \frac{5}{3} + \frac{4}{3 x + 30}$

Step 5.15

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

$y = \frac{x}{3} + \frac{5}{3}$

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = - 10$

No Horizontal Asymptotes

Oblique Asymptotes: $y = \frac{x}{3} + \frac{5}{3}$

Step 7