Precalculus Examples

$f (x) = \frac{x^{3} + x^{2} - 6 x}{4 x^{2} - 4 x - 8}$

Step 1

Find where the expression $\frac{x^{3} + x^{2} - 6 x}{4 x^{2} - 4 x - 8}$ is undefined.

$x = - 1, x = 2$

Step 2

Since $\frac{x^{3} + x^{2} - 6 x}{4 x^{2} - 4 x - 8}$ $\to$ $\infty$ as $x$ $\to$ $- 1$ from the left and $\frac{x^{3} + x^{2} - 6 x}{4 x^{2} - 4 x - 8}$ $\to$ $- \infty$ as $x$ $\to$ $- 1$ from the right, then $x = - 1$ is a vertical asymptote.

$x = - 1$

Step 3

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 4

Find $n$ and $m$ .

$n = 3$

$m = 2$

Step 5

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

No Horizontal Asymptotes

Step 6

Find the oblique asymptote using polynomial division.

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Step 6.1

Simplify the expression.

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Step 6.1.1

Simplify the numerator.

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Step 6.1.1.1

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

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Step 6.1.1.1.1

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

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

Step 6.1.1.1.2

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

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

Step 6.1.1.1.3

Factor $x$ out of $- 6 x$ .

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

Step 6.1.1.1.4

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

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

Step 6.1.1.1.5

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

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

Step 6.1.1.2

Factor $x^{2} + x - 6$ using the AC method.

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Step 6.1.1.2.1

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 $- 6$ and whose sum is $1$ .

$- 2, 3$

Step 6.1.1.2.2

Write the factored form using these integers.

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

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

Step 6.1.2

Simplify the denominator.

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Step 6.1.2.1

Factor $4$ out of $4 x^{2} - 4 x - 8$ .

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Step 6.1.2.1.1

Factor $4$ out of $4 x^{2}$ .

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

Step 6.1.2.1.2

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

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

Step 6.1.2.1.3

Factor $4$ out of $- 8$ .

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

Step 6.1.2.1.4

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

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

Step 6.1.2.1.5

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

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

Step 6.1.2.2

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

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Step 6.1.2.2.1

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 $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Step 6.1.2.2.2

Write the factored form using these integers.

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

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

Step 6.1.3

Cancel the common factor of $x - 2$ .

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Step 6.1.3.1

Cancel the common factor.

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

Step 6.1.3.2

Rewrite the expression.

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

Step 6.2

Expand $x (x + 3)$ .

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Step 6.2.1

Apply the distributive property.

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

Step 6.2.2

Reorder $x$ and $3$ .

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

Step 6.2.3

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

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

Step 6.2.4

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

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

Step 6.2.5

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

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

Step 6.2.6

Add $1$ and $1$ .

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

Step 6.3

Expand $4 (x + 1)$ .

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Step 6.3.1

Apply the distributive property.

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

Step 6.3.2

Multiply $4$ by $1$ .

$\frac{x^{2} + 3 x}{4 x + 4}$

Step 6.4

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

$4 x$

$4$

$x^{2}$

$3 x$

$0$

Step 6.5

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

					$\frac{x}{4}$
	$4 x$	+	$4$		$x^{2}$	+	$3 x$	+	$0$

Step 6.6

Multiply the new quotient term by the divisor.

				$\frac{x}{4}$
$4 x$	+	$4$		$x^{2}$	+	$3 x$	+	$0$
			+	$x^{2}$	+	$x$

Step 6.7

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

				$\frac{x}{4}$
$4 x$	+	$4$		$x^{2}$	+	$3 x$	+	$0$
			-	$x^{2}$	-	$x$

Step 6.8

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

				$\frac{x}{4}$
$4 x$	+	$4$		$x^{2}$	+	$3 x$	+	$0$
			-	$x^{2}$	-	$x$
					+	$2 x$

Step 6.9

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

				$\frac{x}{4}$
$4 x$	+	$4$		$x^{2}$	+	$3 x$	+	$0$
			-	$x^{2}$	-	$x$
					+	$2 x$	+	$0$

Step 6.10

Divide the highest order term in the dividend $2 x$ by the highest order term in divisor $4 x$ .

				$\frac{x}{4}$	+	$\frac{1}{2}$
$4 x$	+	$4$		$x^{2}$	+	$3 x$	+	$0$
			-	$x^{2}$	-	$x$
					+	$2 x$	+	$0$

Step 6.11

Multiply the new quotient term by the divisor.

				$\frac{x}{4}$	+	$\frac{1}{2}$
$4 x$	+	$4$		$x^{2}$	+	$3 x$	+	$0$
			-	$x^{2}$	-	$x$
					+	$2 x$	+	$0$
					+	$2 x$	+	$2$

Step 6.12

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

				$\frac{x}{4}$	+	$\frac{1}{2}$
$4 x$	+	$4$		$x^{2}$	+	$3 x$	+	$0$
			-	$x^{2}$	-	$x$
					+	$2 x$	+	$0$
					-	$2 x$	-	$2$

Step 6.13

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

				$\frac{x}{4}$	+	$\frac{1}{2}$
$4 x$	+	$4$		$x^{2}$	+	$3 x$	+	$0$
			-	$x^{2}$	-	$x$
					+	$2 x$	+	$0$
					-	$2 x$	-	$2$
							-	$2$

Step 6.14

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

$\frac{x}{4} + \frac{1}{2} - \frac{2}{4 x + 4}$

Step 6.15

Split the solution into the polynomial portion and the remainder.

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

Step 6.16

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

$y = \frac{x}{4} + \frac{1}{2}$

Step 7

This is the set of all asymptotes.

Vertical Asymptotes: $x = - 1$

No Horizontal Asymptotes

Oblique Asymptotes: $y = \frac{x}{4} + \frac{1}{2}$

Step 8