Pre-Algebra Examples

$\frac{6 z^{3} + 13 z^{2} - 4 z + 4}{z + \frac{1}{6}}$

Step 1

Find where the expression $\frac{6 (6 x^{3} + 13 x^{2} - 4 x + 4)}{6 x + 1}$ is undefined.

$x = - \frac{1}{6}$

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 $m$ .

$n = 3$

$m = 1$

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 $6$ out of $36 x^{3} + 78 x^{2} - 24 x + 24$ .

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Step 5.1.1

Factor $6$ out of $36 x^{3}$ .

$\frac{6 (6 x^{3}) + 78 x^{2} - 24 x + 24}{6 x + 1}$

Step 5.1.2

Factor $6$ out of $78 x^{2}$ .

$\frac{6 (6 x^{3}) + 6 (13 x^{2}) - 24 x + 24}{6 x + 1}$

Step 5.1.3

Factor $6$ out of $- 24 x$ .

$\frac{6 (6 x^{3}) + 6 (13 x^{2}) + 6 (- 4 x) + 24}{6 x + 1}$

Step 5.1.4

Factor $6$ out of $24$ .

$\frac{6 (6 x^{3}) + 6 (13 x^{2}) + 6 (- 4 x) + 6 (4)}{6 x + 1}$

Step 5.1.5

Factor $6$ out of $6 (6 x^{3}) + 6 (13 x^{2})$ .

$\frac{6 (6 x^{3} + 13 x^{2}) + 6 (- 4 x) + 6 (4)}{6 x + 1}$

Step 5.1.6

Factor $6$ out of $6 (6 x^{3} + 13 x^{2}) + 6 (- 4 x)$ .

$\frac{6 (6 x^{3} + 13 x^{2} - 4 x) + 6 (4)}{6 x + 1}$

Step 5.1.7

Factor $6$ out of $6 (6 x^{3} + 13 x^{2} - 4 x) + 6 (4)$ .

$\frac{6 (6 x^{3} + 13 x^{2} - 4 x + 4)}{6 x + 1}$

Step 5.2

Expand $6 (6 x^{3} + 13 x^{2} - 4 x + 4)$ .

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Step 5.2.1

Apply the distributive property.

$\frac{6 (6 x^{3} + 13 x^{2} - 4 x) + 6 \cdot 4}{6 x + 1}$

Step 5.2.2

Apply the distributive property.

$\frac{6 (6 x^{3} + 13 x^{2}) + 6 (- 4 x) + 6 \cdot 4}{6 x + 1}$

Step 5.2.3

Apply the distributive property.

$\frac{6 (6 x^{3}) + 6 (13 x^{2}) + 6 (- 4 x) + 6 \cdot 4}{6 x + 1}$

Step 5.2.4

Remove parentheses.

$\frac{6 \cdot (6 x^{3}) + 6 (13 x^{2}) + 6 (- 4 x) + 6 \cdot 4}{6 x + 1}$

Step 5.2.5

Remove parentheses.

$\frac{6 \cdot (6 x^{3}) + 6 \cdot (13 x^{2}) + 6 (- 4 x) + 6 \cdot 4}{6 x + 1}$

Step 5.2.6

Remove parentheses.

$\frac{6 \cdot (6 x^{3}) + 6 \cdot (13 x^{2}) + 6 \cdot (- 4 x) + 6 \cdot 4}{6 x + 1}$

Step 5.2.7

Multiply $6$ by $6$ .

$\frac{36 x^{3} + 6 \cdot (13 x^{2}) + 6 \cdot (- 4 x) + 6 \cdot 4}{6 x + 1}$

Step 5.2.8

Multiply $6$ by $13$ .

$\frac{36 x^{3} + 78 x^{2} + 6 \cdot (- 4 x) + 6 \cdot 4}{6 x + 1}$

Step 5.2.9

Multiply $6$ by $- 4$ .

$\frac{36 x^{3} + 78 x^{2} - 24 x + 6 \cdot 4}{6 x + 1}$

Step 5.2.10

Multiply $6$ by $4$ .

$\frac{36 x^{3} + 78 x^{2} - 24 x + 24}{6 x + 1}$

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 $0$ .

$6 x$

$1$

$36 x^{3}$

$78 x^{2}$

$24 x$

$24$

Step 5.4

Divide the highest order term in the dividend $36 x^{3}$ by the highest order term in divisor $6 x$ .

					$6 x^{2}$
	$6 x$	+	$1$		$36 x^{3}$	+	$78 x^{2}$	-	$24 x$	+	$24$

Step 5.5

Multiply the new quotient term by the divisor.

				$6 x^{2}$
$6 x$	+	$1$		$36 x^{3}$	+	$78 x^{2}$	-	$24 x$	+	$24$
			+	$36 x^{3}$	+	$6 x^{2}$

Step 5.6

The expression needs to be subtracted from the dividend, so change all the signs in $36 x^{3} + 6 x^{2}$

				$6 x^{2}$
$6 x$	+	$1$		$36 x^{3}$	+	$78 x^{2}$	-	$24 x$	+	$24$
			-	$36 x^{3}$	-	$6 x^{2}$

Step 5.7

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

				$6 x^{2}$
$6 x$	+	$1$		$36 x^{3}$	+	$78 x^{2}$	-	$24 x$	+	$24$
			-	$36 x^{3}$	-	$6 x^{2}$
					+	$72 x^{2}$

Step 5.8

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

				$6 x^{2}$
$6 x$	+	$1$		$36 x^{3}$	+	$78 x^{2}$	-	$24 x$	+	$24$
			-	$36 x^{3}$	-	$6 x^{2}$
					+	$72 x^{2}$	-	$24 x$

Step 5.9

Divide the highest order term in the dividend $72 x^{2}$ by the highest order term in divisor $6 x$ .

				$6 x^{2}$	+	$12 x$
$6 x$	+	$1$		$36 x^{3}$	+	$78 x^{2}$	-	$24 x$	+	$24$
			-	$36 x^{3}$	-	$6 x^{2}$
					+	$72 x^{2}$	-	$24 x$

Step 5.10

Multiply the new quotient term by the divisor.

				$6 x^{2}$	+	$12 x$
$6 x$	+	$1$		$36 x^{3}$	+	$78 x^{2}$	-	$24 x$	+	$24$
			-	$36 x^{3}$	-	$6 x^{2}$
					+	$72 x^{2}$	-	$24 x$
					+	$72 x^{2}$	+	$12 x$

Step 5.11

The expression needs to be subtracted from the dividend, so change all the signs in $72 x^{2} + 12 x$

				$6 x^{2}$	+	$12 x$
$6 x$	+	$1$		$36 x^{3}$	+	$78 x^{2}$	-	$24 x$	+	$24$
			-	$36 x^{3}$	-	$6 x^{2}$
					+	$72 x^{2}$	-	$24 x$
					-	$72 x^{2}$	-	$12 x$

Step 5.12

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

				$6 x^{2}$	+	$12 x$
$6 x$	+	$1$		$36 x^{3}$	+	$78 x^{2}$	-	$24 x$	+	$24$
			-	$36 x^{3}$	-	$6 x^{2}$
					+	$72 x^{2}$	-	$24 x$
					-	$72 x^{2}$	-	$12 x$
							-	$36 x$

Step 5.13

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

				$6 x^{2}$	+	$12 x$
$6 x$	+	$1$		$36 x^{3}$	+	$78 x^{2}$	-	$24 x$	+	$24$
			-	$36 x^{3}$	-	$6 x^{2}$
					+	$72 x^{2}$	-	$24 x$
					-	$72 x^{2}$	-	$12 x$
							-	$36 x$	+	$24$

Step 5.14

Divide the highest order term in the dividend $- 36 x$ by the highest order term in divisor $6 x$ .

				$6 x^{2}$	+	$12 x$	-	$6$
$6 x$	+	$1$		$36 x^{3}$	+	$78 x^{2}$	-	$24 x$	+	$24$
			-	$36 x^{3}$	-	$6 x^{2}$
					+	$72 x^{2}$	-	$24 x$
					-	$72 x^{2}$	-	$12 x$
							-	$36 x$	+	$24$

Step 5.15

Multiply the new quotient term by the divisor.

				$6 x^{2}$	+	$12 x$	-	$6$
$6 x$	+	$1$		$36 x^{3}$	+	$78 x^{2}$	-	$24 x$	+	$24$
			-	$36 x^{3}$	-	$6 x^{2}$
					+	$72 x^{2}$	-	$24 x$
					-	$72 x^{2}$	-	$12 x$
							-	$36 x$	+	$24$
							-	$36 x$	-	$6$

Step 5.16

The expression needs to be subtracted from the dividend, so change all the signs in $- 36 x - 6$

				$6 x^{2}$	+	$12 x$	-	$6$
$6 x$	+	$1$		$36 x^{3}$	+	$78 x^{2}$	-	$24 x$	+	$24$
			-	$36 x^{3}$	-	$6 x^{2}$
					+	$72 x^{2}$	-	$24 x$
					-	$72 x^{2}$	-	$12 x$
							-	$36 x$	+	$24$
							+	$36 x$	+	$6$

Step 5.17

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

				$6 x^{2}$	+	$12 x$	-	$6$
$6 x$	+	$1$		$36 x^{3}$	+	$78 x^{2}$	-	$24 x$	+	$24$
			-	$36 x^{3}$	-	$6 x^{2}$
					+	$72 x^{2}$	-	$24 x$
					-	$72 x^{2}$	-	$12 x$
							-	$36 x$	+	$24$
							+	$36 x$	+	$6$
									+	$30$

Step 5.18

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

$6 x^{2} + 12 x - 6 + \frac{30}{6 x + 1}$

Step 5.19

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

$y = 6 x^{2} + 12 x - 6$

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = - \frac{1}{6}$

No Horizontal Asymptotes

Oblique Asymptotes: $y = 6 x^{2} + 12 x - 6$

Step 7