Pre-Algebra Examples

$C (x) = \frac{360 + 30 x + 0.1 x^{2}}{x}$

Step 1

Find where the expression $\frac{0.1 (3600 + 300 x + x^{2})}{x}$ 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

Factor $0.1$ out of $360 + 30 x + 0.1 x^{2}$ .

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Step 5.1.1

Factor $0.1$ out of $360$ .

$\frac{0.1 \cdot 3600 + 30 x + 0.1 x^{2}}{x}$

Step 5.1.2

Factor $0.1$ out of $30 x$ .

$\frac{0.1 \cdot 3600 + 0.1 (300 x) + 0.1 x^{2}}{x}$

Step 5.1.3

Factor $0.1$ out of $0.1 \cdot 3600 + 0.1 (300 x)$ .

$\frac{0.1 \cdot (3600 + 300 x) + 0.1 x^{2}}{x}$

Step 5.1.4

Factor $0.1$ out of $0.1 \cdot (3600 + 300 x) + 0.1 x^{2}$ .

$\frac{0.1 (3600 + 300 x + x^{2})}{x}$

Step 5.2

Expand $0.1 (3600 + 300 x + x^{2})$ .

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Step 5.2.1

Apply the distributive property.

$\frac{0.1 (3600 + 300 x) + 0.1 x^{2}}{x}$

Step 5.2.2

Apply the distributive property.

$\frac{0.1 \cdot 3600 + 0.1 (300 x) + 0.1 x^{2}}{x}$

Step 5.2.3

Remove parentheses.

$\frac{0.1 \cdot 3600 + 0.1 \cdot (300 x) + 0.1 x^{2}}{x}$

Step 5.2.4

Multiply $0.1$ by $3600$ .

$\frac{360 + 0.1 \cdot (300 x) + 0.1 x^{2}}{x}$

Step 5.2.5

Multiply $0.1$ by $300$ .

$\frac{360 + 30 x + 0.1 x^{2}}{x}$

Step 5.2.6

Reorder $360$ and $30 x$ .

$\frac{30 x + 360 + 0.1 x^{2}}{x}$

Step 5.2.7

Move $360$ .

$\frac{30 x + 0.1 x^{2} + 360}{x}$

Step 5.2.8

Reorder $30 x$ and $0.1 x^{2}$ .

$\frac{0.1 x^{2} + 30 x + 360}{x}$

Step 5.3

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

$x$

$0$

$0.1 x^{2}$

$30 x$

$360$

Step 5.4

Divide the highest order term in the dividend $0.1 x^{2}$ by the highest order term in divisor $x$ .

					$0.1 x$
	$x$	+	$0$		$0.1 x^{2}$	+	$30 x$	+	$360$

Step 5.5

Multiply the new quotient term by the divisor.

				$0.1 x$
$x$	+	$0$		$0.1 x^{2}$	+	$30 x$	+	$360$
			+	$0.1 x^{2}$	+	$0$

Step 5.6

The expression needs to be subtracted from the dividend, so change all the signs in $0.1 x^{2} + 0$

				$0.1 x$
$x$	+	$0$		$0.1 x^{2}$	+	$30 x$	+	$360$
			-	$0.1 x^{2}$	-	$0$

Step 5.7

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

				$0.1 x$
$x$	+	$0$		$0.1 x^{2}$	+	$30 x$	+	$360$
			-	$0.1 x^{2}$	-	$0$
					+	$30 x$

Step 5.8

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

				$0.1 x$
$x$	+	$0$		$0.1 x^{2}$	+	$30 x$	+	$360$
			-	$0.1 x^{2}$	-	$0$
					+	$30 x$	+	$360$

Step 5.9

Divide the highest order term in the dividend $30 x$ by the highest order term in divisor $x$ .

				$0.1 x$	+	$30$
$x$	+	$0$		$0.1 x^{2}$	+	$30 x$	+	$360$
			-	$0.1 x^{2}$	-	$0$
					+	$30 x$	+	$360$

Step 5.10

Multiply the new quotient term by the divisor.

				$0.1 x$	+	$30$
$x$	+	$0$		$0.1 x^{2}$	+	$30 x$	+	$360$
			-	$0.1 x^{2}$	-	$0$
					+	$30 x$	+	$360$
					+	$30 x$	+	$0$

Step 5.11

The expression needs to be subtracted from the dividend, so change all the signs in $30 x + 0$

				$0.1 x$	+	$30$
$x$	+	$0$		$0.1 x^{2}$	+	$30 x$	+	$360$
			-	$0.1 x^{2}$	-	$0$
					+	$30 x$	+	$360$
					-	$30 x$	-	$0$

Step 5.12

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

				$0.1 x$	+	$30$
$x$	+	$0$		$0.1 x^{2}$	+	$30 x$	+	$360$
			-	$0.1 x^{2}$	-	$0$
					+	$30 x$	+	$360$
					-	$30 x$	-	$0$
							+	$360$

Step 5.13

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

$0.1 x + 30 + \frac{360}{x}$

Step 5.14

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

$y = 0.1 x + 30$

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = 0$

No Horizontal Asymptotes

Oblique Asymptotes: $y = 0.1 x + 30$

Step 7