Pre-Algebra Examples

$\frac{5 y - 3}{2 y + 8}$

Step 1

Find where the expression $\frac{5 y - 3}{2 (y + 4)}$ is undefined.

$y = - 4$

Step 2

$x = \frac{5 y - 3}{2 (y + 4)}$ is an equation of a line, which means there are no horizontal asymptotes.

No Horizontal Asymptotes

Step 3

Find the oblique asymptote using polynomial division.

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Step 3.1

Factor $2$ out of $2 y + 8$ .

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Step 3.1.1

Factor $2$ out of $2 y$ .

$\frac{5 y - 3}{2 (y) + 8}$

Step 3.1.2

Factor $2$ out of $8$ .

$\frac{5 y - 3}{2 y + 2 \cdot 4}$

Step 3.1.3

Factor $2$ out of $2 y + 2 \cdot 4$ .

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

Step 3.2

Expand $2 (y + 4)$ .

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Step 3.2.1

Apply the distributive property.

$\frac{5 y - 3}{2 y + 2 \cdot 4}$

Step 3.2.2

Multiply $2$ by $4$ .

$\frac{5 y - 3}{2 y + 8}$

Step 3.3

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

$2 y$

$8$

$5 y$

$3$

Step 3.4

Divide the highest order term in the dividend $5 y$ by the highest order term in divisor $2 y$ .

					$\frac{5}{2}$
	$2 y$	+	$8$		$5 y$	-	$3$

Step 3.5

Multiply the new quotient term by the divisor.

				$\frac{5}{2}$
$2 y$	+	$8$		$5 y$	-	$3$
			+	$5 y$	+	$20$

Step 3.6

The expression needs to be subtracted from the dividend, so change all the signs in $5 y + 20$

				$\frac{5}{2}$
$2 y$	+	$8$		$5 y$	-	$3$
			-	$5 y$	-	$20$

Step 3.7

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

				$\frac{5}{2}$
$2 y$	+	$8$		$5 y$	-	$3$
			-	$5 y$	-	$20$
					-	$23$

Step 3.8

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

$\frac{5}{2} - \frac{23}{2 y + 8}$

Step 3.9

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

$y = \frac{5}{2}$

Step 4

This is the set of all asymptotes.

Vertical Asymptotes: $y = - 4$

No Horizontal Asymptotes

Oblique Asymptotes: $y = \frac{5}{2}$

Step 5