Find the Asymptotes y=(x^2-4)/(x+5)

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

Step 1

Find where the expression $\frac{x^{2} - 4}{x + 5}$ is undefined.

$x = - 5$

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.

Simplify the numerator.

Rewrite $4$ as $2^{2}$ .

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

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

Expand $(x + 2) (x - 2)$ .

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Reorder $x$ and $- 2$ .

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

Raise $x$ to the power of $1$ .

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

Raise $x$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{1 + 1} - 2 x + 2 x + 2 \cdot - 2}{x + 5}$

Add $1$ and $1$ .

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

Multiply $2$ by $- 2$ .

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

Add $- 2 x$ and $2 x$ .

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

Subtract $4$ from $0$ .

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

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

$x$

$5$

$x^{2}$

$0 x$

$4$

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

					$x$
	$x$	+	$5$		$x^{2}$	+	$0 x$	-	$4$

Multiply the new quotient term by the divisor.

				$x$
$x$	+	$5$		$x^{2}$	+	$0 x$	-	$4$
			+	$x^{2}$	+	$5 x$

The expression needs to be subtracted from the dividend, so change all the signs in $x^{2} + 5 x$

				$x$
$x$	+	$5$		$x^{2}$	+	$0 x$	-	$4$
			-	$x^{2}$	-	$5 x$

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

				$x$
$x$	+	$5$		$x^{2}$	+	$0 x$	-	$4$
			-	$x^{2}$	-	$5 x$
					-	$5 x$

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

				$x$
$x$	+	$5$		$x^{2}$	+	$0 x$	-	$4$
			-	$x^{2}$	-	$5 x$
					-	$5 x$	-	$4$

Divide the highest order term in the dividend $- 5 x$ by the highest order term in divisor $x$ .

				$x$	-	$5$
$x$	+	$5$		$x^{2}$	+	$0 x$	-	$4$
			-	$x^{2}$	-	$5 x$
					-	$5 x$	-	$4$

Multiply the new quotient term by the divisor.

				$x$	-	$5$
$x$	+	$5$		$x^{2}$	+	$0 x$	-	$4$
			-	$x^{2}$	-	$5 x$
					-	$5 x$	-	$4$
					-	$5 x$	-	$25$

The expression needs to be subtracted from the dividend, so change all the signs in $- 5 x - 25$

				$x$	-	$5$
$x$	+	$5$		$x^{2}$	+	$0 x$	-	$4$
			-	$x^{2}$	-	$5 x$
					-	$5 x$	-	$4$
					+	$5 x$	+	$25$

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

				$x$	-	$5$
$x$	+	$5$		$x^{2}$	+	$0 x$	-	$4$
			-	$x^{2}$	-	$5 x$
					-	$5 x$	-	$4$
					+	$5 x$	+	$25$
							+	$21$

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

$x - 5 + \frac{21}{x + 5}$

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

$y = x - 5$

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = - 5$

No Horizontal Asymptotes

Oblique Asymptotes: $y = x - 5$

Step 7

Find the Asymptotes y=(x^2-4)/(x+5)

Related Questions

Support LCMGCF.com by disabling your adblocker.