Precalculus Examples

$f (x) = \frac{x^{2} + x - 6}{- 4 x - 8}$

Step 1

Find where the expression $\frac{x^{2} + x - 6}{- 4 x - 8}$ is undefined.

$x = - 2$

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 $x^{2} + x - 6$ 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 $- 6$ and whose sum is $1$ .

$- 2, 3$

Step 5.1.1.2

Write the factored form using these integers.

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

Step 5.1.2

Simplify with factoring out.

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Step 5.1.2.1

Factor $4$ out of $- 4 x - 8$ .

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Step 5.1.2.1.1

Factor $4$ out of $- 4 x$ .

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

Step 5.1.2.1.2

Factor $4$ out of $- 8$ .

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

Step 5.1.2.1.3

Factor $4$ out of $4 (- x) + 4 (- 2)$ .

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

Step 5.1.2.2

Factor $- 1$ out of $- x$ .

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

Step 5.1.2.3

Rewrite $- 2$ as $- 1 (2)$ .

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

Step 5.1.2.4

Factor $- 1$ out of $- (x) - 1 (2)$ .

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

Step 5.1.2.5

Rewrite negatives.

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Step 5.1.2.5.1

Rewrite $- (x + 2)$ as $- 1 (x + 2)$ .

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

Step 5.1.2.5.2

Move the negative in front of the fraction.

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

Step 5.2

Expand $- (x - 2) (x + 3)$ .

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Step 5.2.1

Negate $x - 2$ .

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

Step 5.2.2

Apply the distributive property.

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

Step 5.2.3

Apply the distributive property.

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

Step 5.2.4

Apply the distributive property.

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

Step 5.2.5

Apply the distributive property.

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

Step 5.2.6

Move $x$ .

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

Step 5.2.7

Factor out negative.

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

Step 5.2.8

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

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

Step 5.2.9

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

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

Step 5.2.10

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

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

Step 5.2.11

Add $1$ and $1$ .

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

Step 5.2.12

Multiply $- 1$ by $3$ .

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

Step 5.2.13

Multiply $- 1$ by $- 2$ .

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

Step 5.2.14

Multiply $- 1$ by $- 2$ .

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

Step 5.2.15

Multiply $2$ by $3$ .

$\frac{- x^{2} - 3 x + 2 x + 6}{4 (x + 2)}$

Step 5.2.16

Add $- 3 x$ and $2 x$ .

$\frac{- x^{2} - x + 6}{4 (x + 2)}$

Step 5.3

Expand $4 (x + 2)$ .

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Step 5.3.1

Apply the distributive property.

$\frac{- x^{2} - x + 6}{4 x + 4 \cdot 2}$

Step 5.3.2

Multiply $4$ by $2$ .

$\frac{- x^{2} - x + 6}{4 x + 8}$

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 $0$ .

$4 x$

$8$

$x^{2}$

$x$

$6$

Step 5.5

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

				-	$\frac{x}{4}$
	$4 x$	+	$8$	-	$x^{2}$	-	$x$	+	$6$

Step 5.6

Multiply the new quotient term by the divisor.

			-	$\frac{x}{4}$
$4 x$	+	$8$	-	$x^{2}$	-	$x$	+	$6$
			-	$x^{2}$	-	$2 x$

Step 5.7

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

			-	$\frac{x}{4}$
$4 x$	+	$8$	-	$x^{2}$	-	$x$	+	$6$
			+	$x^{2}$	+	$2 x$

Step 5.8

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

			-	$\frac{x}{4}$
$4 x$	+	$8$	-	$x^{2}$	-	$x$	+	$6$
			+	$x^{2}$	+	$2 x$
					+	$x$

Step 5.9

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

			-	$\frac{x}{4}$
$4 x$	+	$8$	-	$x^{2}$	-	$x$	+	$6$
			+	$x^{2}$	+	$2 x$
					+	$x$	+	$6$

Step 5.10

Divide the highest order term in the dividend $x$ by the highest order term in divisor $4 x$ .

			-	$\frac{x}{4}$	+	$\frac{1}{4}$
$4 x$	+	$8$	-	$x^{2}$	-	$x$	+	$6$
			+	$x^{2}$	+	$2 x$
					+	$x$	+	$6$

Step 5.11

Multiply the new quotient term by the divisor.

			-	$\frac{x}{4}$	+	$\frac{1}{4}$
$4 x$	+	$8$	-	$x^{2}$	-	$x$	+	$6$
			+	$x^{2}$	+	$2 x$
					+	$x$	+	$6$
					+	$x$	+	$2$

Step 5.12

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

			-	$\frac{x}{4}$	+	$\frac{1}{4}$
$4 x$	+	$8$	-	$x^{2}$	-	$x$	+	$6$
			+	$x^{2}$	+	$2 x$
					+	$x$	+	$6$
					-	$x$	-	$2$

Step 5.13

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

			-	$\frac{x}{4}$	+	$\frac{1}{4}$
$4 x$	+	$8$	-	$x^{2}$	-	$x$	+	$6$
			+	$x^{2}$	+	$2 x$
					+	$x$	+	$6$
					-	$x$	-	$2$
							+	$4$

Step 5.14

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

$- \frac{x}{4} + \frac{1}{4} + \frac{4}{4 x + 8}$

Step 5.15

Split the solution into the polynomial portion and the remainder.

$- \frac{x}{4} + \frac{1}{4} - \frac{4}{- 4 x - 8}$

Step 5.16

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

$y = - \frac{x}{4} + \frac{1}{4}$

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = - 2$

No Horizontal Asymptotes

Oblique Asymptotes: $y = - \frac{x}{4} + \frac{1}{4}$

Step 7