Pre-Algebra Examples

$\frac{y^{2} + 3 y - 10}{y^{2} - 3 y - 40}$

Step 1

Find where the expression $\frac{y - 2}{y - 8}$ is undefined.

$y = 8$

Step 2

$x = \frac{y - 2}{y - 8}$ 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

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

$y$

$8$

$y$

$2$

Step 3.2

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

					$1$
	$y$	-	$8$		$y$	-	$2$

Step 3.3

Multiply the new quotient term by the divisor.

Step 3.4

The expression needs to be subtracted from the dividend, so change all the signs in $y - 8$

Step 3.5

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

Step 3.6

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

$1 + \frac{6}{y - 8}$

Step 3.7

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

$y = 1$

Step 4

This is the set of all asymptotes.

Vertical Asymptotes: $y = 8$

No Horizontal Asymptotes

Oblique Asymptotes: $y = 1$

Step 5