Pre-Algebra Examples

$- 7 x + \frac{9}{13 x}$

Step 1

Find where the expression $- 7 x + \frac{9}{13 x}$ is undefined.

$x = 0$

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

Combine.

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Step 5.1.1

To write $- 7 x$ as a fraction with a common denominator, multiply by $\frac{13 x}{13 x}$ .

$- 7 x \cdot \frac{13 x}{13 x} + \frac{9}{13 x}$

Step 5.1.2

Simplify terms.

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Step 5.1.2.1

Combine $- 7 x$ and $\frac{13 x}{13 x}$ .

$\frac{- 7 x (13 x)}{13 x} + \frac{9}{13 x}$

Step 5.1.2.2

Combine the numerators over the common denominator.

$\frac{- 7 x (13 x) + 9}{13 x}$

Step 5.1.3

Simplify the numerator.

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Step 5.1.3.1

Rewrite using the commutative property of multiplication.

$\frac{- 7 \cdot 13 x \cdot x + 9}{13 x}$

Step 5.1.3.2

Multiply $x$ by $x$ by adding the exponents.

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Step 5.1.3.2.1

Move $x$ .

$\frac{- 7 \cdot 13 (x \cdot x) + 9}{13 x}$

Step 5.1.3.2.2

Multiply $x$ by $x$ .

$\frac{- 7 \cdot 13 x^{2} + 9}{13 x}$

Step 5.1.3.3

Multiply $- 7$ by $13$ .

$\frac{- 91 x^{2} + 9}{13 x}$

Step 5.1.4

Simplify with factoring out.

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Step 5.1.4.1

Factor $- 1$ out of $- 91 x^{2}$ .

$\frac{- (91 x^{2}) + 9}{13 x}$

Step 5.1.4.2

Rewrite $9$ as $- 1 (- 9)$ .

$\frac{- (91 x^{2}) - 1 (- 9)}{13 x}$

Step 5.1.4.3

Factor $- 1$ out of $- (91 x^{2}) - 1 (- 9)$ .

$\frac{- (91 x^{2} - 9)}{13 x}$

Step 5.1.4.4

Simplify the expression.

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Step 5.1.4.4.1

Rewrite $- (91 x^{2} - 9)$ as $- 1 (91 x^{2} - 9)$ .

$\frac{- 1 (91 x^{2} - 9)}{13 x}$

Step 5.1.4.4.2

Move the negative in front of the fraction.

$- \frac{91 x^{2} - 9}{13 x}$

Step 5.1.5

Simplify.

$\frac{- 91 x^{2} + 9}{13 x}$

Step 5.2

Simplify the expression.

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Step 5.2.1

Factor $- 1$ out of $- 91 x^{2}$ .

$\frac{- (91 x^{2}) + 9}{13 x}$

Step 5.2.2

Rewrite $9$ as $- 1 (- 9)$ .

$\frac{- (91 x^{2}) - 1 (- 9)}{13 x}$

Step 5.2.3

Factor $- 1$ out of $- (91 x^{2}) - 1 (- 9)$ .

$\frac{- (91 x^{2} - 9)}{13 x}$

Step 5.2.4

Simplify the expression.

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Step 5.2.4.1

Rewrite $- (91 x^{2} - 9)$ as $- 1 (91 x^{2} - 9)$ .

$\frac{- 1 (91 x^{2} - 9)}{13 x}$

Step 5.2.4.2

Move the negative in front of the fraction.

$- \frac{91 x^{2} - 9}{13 x}$

Step 5.3

Expand $- (91 x^{2} - 9)$ .

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Step 5.3.1

Negate $91 x^{2} - 9$ .

$\frac{- (91 x^{2} - 9)}{13 x}$

Step 5.3.2

Apply the distributive property.

$\frac{- (91 x^{2}) + 9}{13 x}$

Step 5.3.3

Remove parentheses.

$\frac{- 1 \cdot (91 x^{2}) + 9}{13 x}$

Step 5.3.4

Multiply $- 1$ by $91$ .

$\frac{- 91 x^{2} + 9}{13 x}$

Step 5.3.5

Multiply $- 1$ by $- 9$ .

$\frac{- 91 x^{2} + 9}{13 x}$

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 $0$ .

$13 x$

$0$

$91 x^{2}$

$0 x$

$9$

Step 5.5

Divide the highest order term in the dividend $- 91 x^{2}$ by the highest order term in divisor $13 x$ .

				-	$7 x$
	$13 x$	+	$0$	-	$91 x^{2}$	+	$0 x$	+	$9$

Step 5.6

Multiply the new quotient term by the divisor.

			-	$7 x$
$13 x$	+	$0$	-	$91 x^{2}$	+	$0 x$	+	$9$
			-	$91 x^{2}$	+	$0$

Step 5.7

The expression needs to be subtracted from the dividend, so change all the signs in $- 91 x^{2} + 0$

			-	$7 x$
$13 x$	+	$0$	-	$91 x^{2}$	+	$0 x$	+	$9$
			+	$91 x^{2}$	-	$0$

Step 5.8

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

			-	$7 x$
$13 x$	+	$0$	-	$91 x^{2}$	+	$0 x$	+	$9$
			+	$91 x^{2}$	-	$0$
						$0$

Step 5.9

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

			-	$7 x$
$13 x$	+	$0$	-	$91 x^{2}$	+	$0 x$	+	$9$
			+	$91 x^{2}$	-	$0$
						$0$	+	$9$

Step 5.10

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

$- 7 x + \frac{9}{13 x}$

Step 5.11

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

$y = - 7 x$

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = 0$

No Horizontal Asymptotes

Oblique Asymptotes: $y = - 7 x$

Step 7