Pre-Algebra Examples

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

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

Step 1

Find where the expression

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

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

x = 0

$x = 0$

Step 2

Consider the rational function

R (x) = \frac{a x^{n}}{b x^{m}}

$R (x) = \frac{a x^{n}}{b x^{m}}$ where

n

$n$ is the degree of the numerator and

m

$m$ is the degree of the denominator.

1. If

n < m

$n < m$ , then the x-axis,

y = 0

$y = 0$ , is the horizontal asymptote.

2. If

n = m

$n = m$ , then the horizontal asymptote is the line

y = \frac{a}{b}

$y = \frac{a}{b}$ .

3. If

n > m

$n > m$ , then there is no horizontal asymptote (there is an oblique asymptote).

Step 3

Find

n

$n$ and

m

$m$ .

n = 2

$n = 2$

m = 1

$m = 1$

Step 4

Since

n > m

$n > m$ , 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

$0.1$ out of

360 + 30 x + 0.1 x^{2}

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

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Step 5.1.1

Factor

0.1

$0.1$ out of

360

$360$ .

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

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

Step 5.1.2

Factor

0.1

$0.1$ out of

30 x

$30 x$ .

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

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

Step 5.1.3

Factor

0.1

$0.1$ out of

0.1 \cdot 3600 + 0.1 (300 x)

$0.1 \cdot 3600 + 0.1 (300 x)$ .

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

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

Step 5.1.4

Factor

0.1

$0.1$ out of

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

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

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

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

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

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

Step 5.2

Expand

0.1 (3600 + 300 x + x^{2})

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

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Step 5.2.1

Apply the distributive property.

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

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

Step 5.2.2

Apply the distributive property.

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

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

Step 5.2.3

Remove parentheses.

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

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

Step 5.2.4

Multiply

0.1

$0.1$ by

3600

$3600$ .

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

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

Step 5.2.5

Multiply

0.1

$0.1$ by

300

$300$ .

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

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

Step 5.2.6

Reorder

360

$360$ and

30 x

$30 x$ .

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

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

Step 5.2.7

Move

360

$360$ .

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

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

Step 5.2.8

Reorder

30 x

$30 x$ and

0.1 x^{2}

$0.1 x^{2}$ .

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

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

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

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

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

$0$ .

x

$x$

0

$0$

0.1 x^{2}

$0.1 x^{2}$

30 x

$30 x$

360

$360$

Step 5.4

Divide the highest order term in the dividend

0.1 x^{2}

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

x

$x$ .

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

Step 5.5

Multiply the new quotient term by the divisor.

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

Step 5.6

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

0.1 x^{2} + 0

$0.1 x^{2} + 0$

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

Step 5.7

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

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

Step 5.8

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

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

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