Precalculus Examples

$\frac{x^{2} + x}{- 5 x - 5}$

Step 1

Find where the expression $\frac{x^{2} + x}{- 5 x - 5}$ is undefined.

$x = - 1$

Step 2

The vertical asymptotes occur at areas of infinite discontinuity.

No Vertical Asymptotes

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 $m$ .

$n = 2$

$m = 1$

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

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

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Step 6.1.1.1

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

$\frac{x \cdot x + x}{- 5 x - 5}$

Step 6.1.1.2

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

$\frac{x \cdot x + x^{1}}{- 5 x - 5}$

Step 6.1.1.3

Factor $x$ out of $x^{1}$ .

$\frac{x \cdot x + x \cdot 1}{- 5 x - 5}$

Step 6.1.1.4

Factor $x$ out of $x \cdot x + x \cdot 1$ .

$\frac{x (x + 1)}{- 5 x - 5}$

Step 6.1.2

Factor $5$ out of $- 5 x - 5$ .

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Step 6.1.2.1

Factor $5$ out of $- 5 x$ .

$\frac{x (x + 1)}{5 (- x) - 5}$

Step 6.1.2.2

Factor $5$ out of $- 5$ .

$\frac{x (x + 1)}{5 (- x) + 5 (- 1)}$

Step 6.1.2.3

Factor $5$ out of $5 (- x) + 5 (- 1)$ .

$\frac{x (x + 1)}{5 (- x - 1)}$

Step 6.1.3

Cancel the common factor of $x + 1$ and $- x - 1$ .

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Step 6.1.3.1

Factor $- 1$ out of $x$ .

$\frac{x (- 1 (- x) + 1)}{5 (- x - 1)}$

Step 6.1.3.2

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

$\frac{x (- 1 (- x) - 1 (- 1))}{5 (- x - 1)}$

Step 6.1.3.3

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

$\frac{x (- 1 (- x - 1))}{5 (- x - 1)}$

Step 6.1.3.4

Cancel the common factor.

$\frac{x (- 1 (- x - 1))}{5 (- x - 1)}$

Step 6.1.3.5

Rewrite the expression.

$\frac{x \cdot (- 1)}{5}$

Step 6.1.4

Simplify the expression.

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Step 6.1.4.1

Move $- 1$ to the left of $x$ .

$\frac{- 1 \cdot x}{5}$

Step 6.1.4.2

Move the negative in front of the fraction.

$- \frac{x}{5}$

Step 6.2

Negate $x$ .

$\frac{- x}{5}$

Step 6.3

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

$5$

$x$

$0$

Step 6.4

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

		-	$\frac{x}{5}$
	$5$	-	$x$	+	$0$

Step 6.5

Multiply the new quotient term by the divisor.

	-	$\frac{x}{5}$
$5$	-	$x$	+	$0$
	-	$x$

Step 6.6

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

	-	$\frac{x}{5}$
$5$	-	$x$	+	$0$
	+	$x$

Step 6.7

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

	-	$\frac{x}{5}$
$5$	-	$x$	+	$0$
	+	$x$
		$0$

Step 6.8

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

	-	$\frac{x}{5}$
$5$	-	$x$	+	$0$
	+	$x$
		$0$	+	$0$

Step 6.9

Since the remander is $0$ , the final answer is the quotient.

$- \frac{x}{5}$

Step 6.10

Since there is no polynomial portion from the polynomial division, there are no oblique asymptotes.

No Oblique Asymptotes

Step 7

This is the set of all asymptotes.

No Vertical Asymptotes

No Horizontal Asymptotes

No Oblique Asymptotes

Step 8