Pre-Algebra Examples

$f (x) = \frac{2 x^{2} + 5}{6 x - 4}$

Step 1

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

$x = \frac{2}{3}$

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.

Tap for more steps...

Step 5.1

Factor $2$ out of $6 x - 4$ .

Tap for more steps...

Step 5.1.1

Factor $2$ out of $6 x$ .

$\frac{2 x^{2} + 5}{2 (3 x) - 4}$

Step 5.1.2

Factor $2$ out of $- 4$ .

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

Step 5.1.3

Factor $2$ out of $2 (3 x) + 2 (- 2)$ .

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

Step 5.2

Expand $2 (3 x - 2)$ .

Tap for more steps...

Step 5.2.1

Apply the distributive property.

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

Step 5.2.2

Remove parentheses.

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

Step 5.2.3

Multiply $2$ by $3$ .

$\frac{2 x^{2} + 5}{6 x + 2 \cdot - 2}$

Step 5.2.4

Multiply $2$ by $- 2$ .

$\frac{2 x^{2} + 5}{6 x - 4}$

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 $0$ .

$6 x$

$4$

$2 x^{2}$

$0 x$

$5$

Step 5.4

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

					$\frac{x}{3}$
	$6 x$	-	$4$		$2 x^{2}$	+	$0 x$	+	$5$

Step 5.5

Multiply the new quotient term by the divisor.

				$\frac{x}{3}$
$6 x$	-	$4$		$2 x^{2}$	+	$0 x$	+	$5$
			+	$2 x^{2}$	-	$\frac{4 x}{3}$

Step 5.6

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{2} - \frac{4 x}{3}$

				$\frac{x}{3}$
$6 x$	-	$4$		$2 x^{2}$	+	$0 x$	+	$5$
			-	$2 x^{2}$	+	$\frac{4 x}{3}$

Step 5.7

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

				$\frac{x}{3}$
$6 x$	-	$4$		$2 x^{2}$	+	$0 x$	+	$5$
			-	$2 x^{2}$	+	$\frac{4 x}{3}$
					+	$\frac{4 x}{3}$

Step 5.8

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

				$\frac{x}{3}$
$6 x$	-	$4$		$2 x^{2}$	+	$0 x$	+	$5$
			-	$2 x^{2}$	+	$\frac{4 x}{3}$
					+	$\frac{4 x}{3}$	+	$5$

Step 5.9

Divide the highest order term in the dividend $\frac{4 x}{3}$ by the highest order term in divisor $6 x$ .

				$\frac{x}{3}$	+	$\frac{2}{9}$
$6 x$	-	$4$		$2 x^{2}$	+	$0 x$	+	$5$
			-	$2 x^{2}$	+	$\frac{4 x}{3}$
					+	$\frac{4 x}{3}$	+	$5$

Step 5.10

Multiply the new quotient term by the divisor.

				$\frac{x}{3}$	+	$\frac{2}{9}$
$6 x$	-	$4$		$2 x^{2}$	+	$0 x$	+	$5$
			-	$2 x^{2}$	+	$\frac{4 x}{3}$
					+	$\frac{4 x}{3}$	+	$5$
					+	$\frac{4 x}{3}$	-	$\frac{8}{9}$

Step 5.11

The expression needs to be subtracted from the dividend, so change all the signs in $\frac{4 x}{3} - \frac{8}{9}$

				$\frac{x}{3}$	+	$\frac{2}{9}$
$6 x$	-	$4$		$2 x^{2}$	+	$0 x$	+	$5$
			-	$2 x^{2}$	+	$\frac{4 x}{3}$
					+	$\frac{4 x}{3}$	+	$5$
					-	$\frac{4 x}{3}$	+	$\frac{8}{9}$

Step 5.12

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

				$\frac{x}{3}$	+	$\frac{2}{9}$
$6 x$	-	$4$		$2 x^{2}$	+	$0 x$	+	$5$
			-	$2 x^{2}$	+	$\frac{4 x}{3}$
					+	$\frac{4 x}{3}$	+	$5$
					-	$\frac{4 x}{3}$	+	$\frac{8}{9}$
							+	$\frac{53}{9}$

Step 5.13

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

$\frac{x}{3} + \frac{2}{9} + \frac{53}{9 (6 x - 4)}$

Step 5.14

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

$y = \frac{x}{3} + \frac{2}{9}$

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = \frac{2}{3}$

No Horizontal Asymptotes

Oblique Asymptotes: $y = \frac{x}{3} + \frac{2}{9}$

Step 7