Precalculus Examples

$f (x) = \frac{3 (2 x + 7) (x + 8)}{x + 1}$

Step 1

Find where the expression $\frac{3 (2 x + 7) (x + 8)}{x + 1}$ is undefined.

$x = - 1$

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 numerator.

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Step 5.1.1

Factor $3$ out of $6 x^{2} + 69 x + 168$ .

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Step 5.1.1.1

Factor $3$ out of $6 x^{2}$ .

$\frac{3 (2 x^{2}) + 69 x + 168}{x + 1}$

Step 5.1.1.2

Factor $3$ out of $69 x$ .

$\frac{3 (2 x^{2}) + 3 (23 x) + 168}{x + 1}$

Step 5.1.1.3

Factor $3$ out of $168$ .

$\frac{3 (2 x^{2}) + 3 (23 x) + 3 (56)}{x + 1}$

Step 5.1.1.4

Factor $3$ out of $3 (2 x^{2}) + 3 (23 x)$ .

$\frac{3 (2 x^{2} + 23 x) + 3 (56)}{x + 1}$

Step 5.1.1.5

Factor $3$ out of $3 (2 x^{2} + 23 x) + 3 (56)$ .

$\frac{3 (2 x^{2} + 23 x + 56)}{x + 1}$

Step 5.1.2

Factor by grouping.

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Step 5.1.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot 56 = 112$ and whose sum is $b = 23$ .

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Step 5.1.2.1.1

Factor $23$ out of $23 x$ .

$\frac{3 (2 x^{2} + 23 (x) + 56)}{x + 1}$

Step 5.1.2.1.2

Rewrite $23$ as $7$ plus $16$

$\frac{3 (2 x^{2} + (7 + 16) x + 56)}{x + 1}$

Step 5.1.2.1.3

Apply the distributive property.

$\frac{3 (2 x^{2} + 7 x + 16 x + 56)}{x + 1}$

Step 5.1.2.2

Factor out the greatest common factor from each group.

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Step 5.1.2.2.1

Group the first two terms and the last two terms.

$\frac{3 ((2 x^{2} + 7 x) + 16 x + 56)}{x + 1}$

Step 5.1.2.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{3 (x (2 x + 7) + 8 (2 x + 7))}{x + 1}$

Step 5.1.2.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 7$ .

$\frac{3 ((2 x + 7) (x + 8))}{x + 1}$

$\frac{3 (2 x + 7) (x + 8)}{x + 1}$

Step 5.2

Expand $3 (2 x + 7) (x + 8)$ .

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Step 5.2.1

Apply the distributive property.

$\frac{(3 (2 x) + 3 \cdot 7) (x + 8)}{x + 1}$

Step 5.2.2

Apply the distributive property.

$\frac{3 (2 x) (x + 8) + 3 \cdot (7 (x + 8))}{x + 1}$

Step 5.2.3

Apply the distributive property.

$\frac{3 (2 x) x + 3 (2 x) \cdot 8 + 3 \cdot (7 (x + 8))}{x + 1}$

Step 5.2.4

Apply the distributive property.

$\frac{3 (2 x) x + 3 (2 x) \cdot 8 + 3 \cdot (7 x) + 3 \cdot 7 \cdot 8}{x + 1}$

Step 5.2.5

Remove parentheses.

$\frac{3 \cdot (2 x \cdot x) + 3 (2 x) \cdot 8 + 3 \cdot (7 x) + 3 \cdot 7 \cdot 8}{x + 1}$

Step 5.2.6

Remove parentheses.

$\frac{3 \cdot (2 x \cdot x) + 3 \cdot (2 x) \cdot 8 + 3 \cdot (7 x) + 3 \cdot 7 \cdot 8}{x + 1}$

Step 5.2.7

Move $x$ .

$\frac{3 \cdot (2 x \cdot x) + 3 \cdot 2 \cdot (8 x) + 3 \cdot (7 x) + 3 \cdot 7 \cdot 8}{x + 1}$

Step 5.2.8

Multiply $3$ by $2$ .

$\frac{6 x \cdot x + 3 \cdot 2 \cdot (8 x) + 3 \cdot (7 x) + 3 \cdot 7 \cdot 8}{x + 1}$

Step 5.2.9

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

$\frac{6 (x \cdot x) + 3 \cdot 2 \cdot (8 x) + 3 \cdot (7 x) + 3 \cdot 7 \cdot 8}{x + 1}$

Step 5.2.10

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

$\frac{6 (x \cdot x) + 3 \cdot 2 \cdot (8 x) + 3 \cdot (7 x) + 3 \cdot 7 \cdot 8}{x + 1}$

Step 5.2.11

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

$\frac{6 x^{1 + 1} + 3 \cdot 2 \cdot (8 x) + 3 \cdot (7 x) + 3 \cdot 7 \cdot 8}{x + 1}$

Step 5.2.12

Add $1$ and $1$ .

$\frac{6 x^{2} + 3 \cdot 2 \cdot (8 x) + 3 \cdot (7 x) + 3 \cdot 7 \cdot 8}{x + 1}$

Step 5.2.13

Multiply $3$ by $2$ .

$\frac{6 x^{2} + 6 \cdot (8 x) + 3 \cdot (7 x) + 3 \cdot 7 \cdot 8}{x + 1}$

Step 5.2.14

Multiply $6$ by $8$ .

$\frac{6 x^{2} + 48 x + 3 \cdot (7 x) + 3 \cdot 7 \cdot 8}{x + 1}$

Step 5.2.15

Multiply $3$ by $7$ .

$\frac{6 x^{2} + 48 x + 21 x + 3 \cdot 7 \cdot 8}{x + 1}$

Step 5.2.16

Multiply $3$ by $7$ .

$\frac{6 x^{2} + 48 x + 21 x + 21 \cdot 8}{x + 1}$

Step 5.2.17

Multiply $21$ by $8$ .

$\frac{6 x^{2} + 48 x + 21 x + 168}{x + 1}$

Step 5.2.18

Add $48 x$ and $21 x$ .

$\frac{6 x^{2} + 69 x + 168}{x + 1}$

Step 5.3

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

$x$

$1$

$6 x^{2}$

$69 x$

$168$

Step 5.4

Divide the highest order term in the dividend $6 x^{2}$ by the highest order term in divisor $x$ .

					$6 x$
	$x$	+	$1$		$6 x^{2}$	+	$69 x$	+	$168$

Step 5.5

Multiply the new quotient term by the divisor.

				$6 x$
$x$	+	$1$		$6 x^{2}$	+	$69 x$	+	$168$
			+	$6 x^{2}$	+	$6 x$

Step 5.6

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{2} + 6 x$

				$6 x$
$x$	+	$1$		$6 x^{2}$	+	$69 x$	+	$168$
			-	$6 x^{2}$	-	$6 x$

Step 5.7

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

				$6 x$
$x$	+	$1$		$6 x^{2}$	+	$69 x$	+	$168$
			-	$6 x^{2}$	-	$6 x$
					+	$63 x$

Step 5.8

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

				$6 x$
$x$	+	$1$		$6 x^{2}$	+	$69 x$	+	$168$
			-	$6 x^{2}$	-	$6 x$
					+	$63 x$	+	$168$

Step 5.9

Divide the highest order term in the dividend $63 x$ by the highest order term in divisor $x$ .

				$6 x$	+	$63$
$x$	+	$1$		$6 x^{2}$	+	$69 x$	+	$168$
			-	$6 x^{2}$	-	$6 x$
					+	$63 x$	+	$168$

Step 5.10

Multiply the new quotient term by the divisor.

				$6 x$	+	$63$
$x$	+	$1$		$6 x^{2}$	+	$69 x$	+	$168$
			-	$6 x^{2}$	-	$6 x$
					+	$63 x$	+	$168$
					+	$63 x$	+	$63$

Step 5.11

The expression needs to be subtracted from the dividend, so change all the signs in $63 x + 63$

				$6 x$	+	$63$
$x$	+	$1$		$6 x^{2}$	+	$69 x$	+	$168$
			-	$6 x^{2}$	-	$6 x$
					+	$63 x$	+	$168$
					-	$63 x$	-	$63$

Step 5.12

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

				$6 x$	+	$63$
$x$	+	$1$		$6 x^{2}$	+	$69 x$	+	$168$
			-	$6 x^{2}$	-	$6 x$
					+	$63 x$	+	$168$
					-	$63 x$	-	$63$
							+	$105$

Step 5.13

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

$6 x + 63 + \frac{105}{x + 1}$

Step 5.14

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

$y = 6 x + 63$

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = - 1$

No Horizontal Asymptotes

Oblique Asymptotes: $y = 6 x + 63$

Step 7