Pre-Algebra Examples

$\frac{\frac{18 n^{2} + 81 n}{27 n^{2} - 63 n}}{\frac{12 n + 54}{15^{3} - 35 n^{2}}}$

Step 1

Find where the expression $\frac{5 (675 - 7 x^{2})}{6 (3 x - 7)}$ is undefined.

$x = \frac{7}{3}$

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

Simplify the expression.

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Step 5.1.1

Factor $5$ out of $3375 - 35 x^{2}$ .

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Step 5.1.1.1

Factor $5$ out of $3375$ .

$\frac{5 (675) - 35 x^{2}}{18 x - 42}$

Step 5.1.1.2

Factor $5$ out of $- 35 x^{2}$ .

$\frac{5 (675) + 5 (- 7 x^{2})}{18 x - 42}$

Step 5.1.1.3

Factor $5$ out of $5 (675) + 5 (- 7 x^{2})$ .

$\frac{5 (675 - 7 x^{2})}{18 x - 42}$

Step 5.1.2

Factor $6$ out of $18 x - 42$ .

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Step 5.1.2.1

Factor $6$ out of $18 x$ .

$\frac{5 (675 - 7 x^{2})}{6 (3 x) - 42}$

Step 5.1.2.2

Factor $6$ out of $- 42$ .

$\frac{5 (675 - 7 x^{2})}{6 (3 x) + 6 (- 7)}$

Step 5.1.2.3

Factor $6$ out of $6 (3 x) + 6 (- 7)$ .

$\frac{5 (675 - 7 x^{2})}{6 (3 x - 7)}$

Step 5.2

Expand $5 (675 - 7 x^{2})$ .

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Step 5.2.1

Apply the distributive property.

$\frac{5 \cdot 675 + 5 (- 7 x^{2})}{6 (3 x - 7)}$

Step 5.2.2

Remove parentheses.

$\frac{5 \cdot 675 + 5 \cdot (- 7 x^{2})}{6 (3 x - 7)}$

Step 5.2.3

Multiply $5$ by $675$ .

$\frac{3375 + 5 \cdot (- 7 x^{2})}{6 (3 x - 7)}$

Step 5.2.4

Multiply $5$ by $- 7$ .

$\frac{3375 - 35 x^{2}}{6 (3 x - 7)}$

Step 5.2.5

Reorder $3375$ and $- 35 x^{2}$ .

$\frac{- 35 x^{2} + 3375}{6 (3 x - 7)}$

Step 5.3

Expand $6 (3 x - 7)$ .

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Step 5.3.1

Apply the distributive property.

$\frac{- 35 x^{2} + 3375}{6 (3 x) + 6 \cdot - 7}$

Step 5.3.2

Remove parentheses.

$\frac{- 35 x^{2} + 3375}{6 \cdot (3 x) + 6 \cdot - 7}$

Step 5.3.3

Multiply $6$ by $3$ .

$\frac{- 35 x^{2} + 3375}{18 x + 6 \cdot - 7}$

Step 5.3.4

Multiply $6$ by $- 7$ .

$\frac{- 35 x^{2} + 3375}{18 x - 42}$

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 $0$ .

$18 x$

$42$

$35 x^{2}$

$0 x$

$3375$

Step 5.5

Divide the highest order term in the dividend $- 35 x^{2}$ by the highest order term in divisor $18 x$ .

				-	$\frac{35 x}{18}$
	$18 x$	-	$42$	-	$35 x^{2}$	+	$0 x$	+	$3375$

Step 5.6

Multiply the new quotient term by the divisor.

			-	$\frac{35 x}{18}$
$18 x$	-	$42$	-	$35 x^{2}$	+	$0 x$	+	$3375$
			-	$35 x^{2}$	+	$\frac{245 x}{3}$

Step 5.7

The expression needs to be subtracted from the dividend, so change all the signs in $- 35 x^{2} + \frac{245 x}{3}$

			-	$\frac{35 x}{18}$
$18 x$	-	$42$	-	$35 x^{2}$	+	$0 x$	+	$3375$
			+	$35 x^{2}$	-	$\frac{245 x}{3}$

Step 5.8

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

			-	$\frac{35 x}{18}$
$18 x$	-	$42$	-	$35 x^{2}$	+	$0 x$	+	$3375$
			+	$35 x^{2}$	-	$\frac{245 x}{3}$
					-	$\frac{245 x}{3}$

Step 5.9

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

			-	$\frac{35 x}{18}$
$18 x$	-	$42$	-	$35 x^{2}$	+	$0 x$	+	$3375$
			+	$35 x^{2}$	-	$\frac{245 x}{3}$
					-	$\frac{245 x}{3}$	+	$3375$

Step 5.10

Divide the highest order term in the dividend $- \frac{245 x}{3}$ by the highest order term in divisor $18 x$ .

			-	$\frac{35 x}{18}$	-	$\frac{245}{54}$
$18 x$	-	$42$	-	$35 x^{2}$	+	$0 x$	+	$3375$
			+	$35 x^{2}$	-	$\frac{245 x}{3}$
					-	$\frac{245 x}{3}$	+	$3375$

Step 5.11

Multiply the new quotient term by the divisor.

			-	$\frac{35 x}{18}$	-	$\frac{245}{54}$
$18 x$	-	$42$	-	$35 x^{2}$	+	$0 x$	+	$3375$
			+	$35 x^{2}$	-	$\frac{245 x}{3}$
					-	$\frac{245 x}{3}$	+	$3375$
					-	$\frac{245 x}{3}$	+	$\frac{1715}{9}$

Step 5.12

The expression needs to be subtracted from the dividend, so change all the signs in $- \frac{245 x}{3} + \frac{1715}{9}$

			-	$\frac{35 x}{18}$	-	$\frac{245}{54}$
$18 x$	-	$42$	-	$35 x^{2}$	+	$0 x$	+	$3375$
			+	$35 x^{2}$	-	$\frac{245 x}{3}$
					-	$\frac{245 x}{3}$	+	$3375$
					+	$\frac{245 x}{3}$	-	$\frac{1715}{9}$

Step 5.13

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

			-	$\frac{35 x}{18}$	-	$\frac{245}{54}$
$18 x$	-	$42$	-	$35 x^{2}$	+	$0 x$	+	$3375$
			+	$35 x^{2}$	-	$\frac{245 x}{3}$
					-	$\frac{245 x}{3}$	+	$3375$
					+	$\frac{245 x}{3}$	-	$\frac{1715}{9}$
							+	$\frac{28660}{9}$

Step 5.14

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

$- \frac{35 x}{18} - \frac{245}{54} + \frac{28660}{9 (18 x - 42)}$

Step 5.15

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

$y = - \frac{35 x}{18} - \frac{245}{54}$

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = \frac{7}{3}$

No Horizontal Asymptotes

Oblique Asymptotes: $y = - \frac{35 x}{18} - \frac{245}{54}$

Step 7