Precalculus Examples

$R (x) = \frac{x^{3} - 8}{x^{2} - x - 2}$

Step 1

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

$x = - 1, x = 2$

Step 2

Since $\frac{x^{3} - 8}{x^{2} - x - 2}$ $\to$ $- \infty$ as $x$ $\to$ $- 1$ from the left and $\frac{x^{3} - 8}{x^{2} - x - 2}$ $\to$ $\infty$ 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

Rewrite $8$ as $2^{3}$ .

$\frac{x^{3} - 2^{3}}{x^{2} - x - 2}$

Step 6.1.1.2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 2$ .

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

Step 6.1.1.3

Simplify.

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Step 6.1.1.3.1

Move $2$ to the left of $x$ .

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

Step 6.1.1.3.2

Raise $2$ to the power of $2$ .

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

Step 6.1.2

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

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Step 6.1.2.1

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

$- 2, 1$

Step 6.1.2.2

Write the factored form using these integers.

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

Step 6.1.3

Cancel the common factor of $x - 2$ .

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Step 6.1.3.1

Cancel the common factor.

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

Step 6.1.3.2

Rewrite the expression.

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

Step 6.2

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

$x$

$1$

$x^{2}$

$2 x$

$4$

Step 6.3

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

					$x$
	$x$	+	$1$		$x^{2}$	+	$2 x$	+	$4$

Step 6.4

Multiply the new quotient term by the divisor.

				$x$
$x$	+	$1$		$x^{2}$	+	$2 x$	+	$4$
			+	$x^{2}$	+	$x$

Step 6.5

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

				$x$
$x$	+	$1$		$x^{2}$	+	$2 x$	+	$4$
			-	$x^{2}$	-	$x$

Step 6.6

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

				$x$
$x$	+	$1$		$x^{2}$	+	$2 x$	+	$4$
			-	$x^{2}$	-	$x$
					+	$x$

Step 6.7

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

				$x$
$x$	+	$1$		$x^{2}$	+	$2 x$	+	$4$
			-	$x^{2}$	-	$x$
					+	$x$	+	$4$

Step 6.8

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

				$x$	+	$1$
$x$	+	$1$		$x^{2}$	+	$2 x$	+	$4$
			-	$x^{2}$	-	$x$
					+	$x$	+	$4$

Step 6.9

Multiply the new quotient term by the divisor.

				$x$	+	$1$
$x$	+	$1$		$x^{2}$	+	$2 x$	+	$4$
			-	$x^{2}$	-	$x$
					+	$x$	+	$4$
					+	$x$	+	$1$

Step 6.10

The expression needs to be subtracted from the dividend, so change all the signs in $x + 1$

				$x$	+	$1$
$x$	+	$1$		$x^{2}$	+	$2 x$	+	$4$
			-	$x^{2}$	-	$x$
					+	$x$	+	$4$
					-	$x$	-	$1$

Step 6.11

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

				$x$	+	$1$
$x$	+	$1$		$x^{2}$	+	$2 x$	+	$4$
			-	$x^{2}$	-	$x$
					+	$x$	+	$4$
					-	$x$	-	$1$
							+	$3$

Step 6.12

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

$x + 1 + \frac{3}{x + 1}$

Step 6.13

Split the solution into the polynomial portion and the remainder.

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

Step 6.14

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

$y = x + 1$

Step 7

This is the set of all asymptotes.

Vertical Asymptotes: $x = - 1$

No Horizontal Asymptotes

Oblique Asymptotes: $y = x + 1$

Step 8