Find the Asymptotes y=(X^2+11X+18)/(2X+1)

$y = \frac{X^{2} + 11 X + 18}{2 X + 1}$

Step 1

Find where the expression $\frac{x^{2} + 11 x + 18}{2 x + 1}$ is undefined.

$x = - \frac{1}{2}$

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.

Factor $x^{2} + 11 x + 18$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $18$ and whose sum is $11$ .

$2, 9$

Write the factored form using these integers.

$\frac{(x + 2) (x + 9)}{2 x + 1}$

Expand $(x + 2) (x + 9)$ .

Apply the distributive property.

$\frac{x (x + 9) + 2 (x + 9)}{2 x + 1}$

Apply the distributive property.

$\frac{x \cdot x + x \cdot 9 + 2 (x + 9)}{2 x + 1}$

Apply the distributive property.

$\frac{x \cdot x + x \cdot 9 + 2 x + 2 \cdot 9}{2 x + 1}$

Reorder $x$ and $9$ .

$\frac{x \cdot x + 9 x + 2 x + 2 \cdot 9}{2 x + 1}$

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

$\frac{x \cdot x + 9 x + 2 x + 2 \cdot 9}{2 x + 1}$

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

$\frac{x \cdot x + 9 x + 2 x + 2 \cdot 9}{2 x + 1}$

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

$\frac{x^{1 + 1} + 9 x + 2 x + 2 \cdot 9}{2 x + 1}$

Add $1$ and $1$ .

$\frac{x^{2} + 9 x + 2 x + 2 \cdot 9}{2 x + 1}$

Multiply $2$ by $9$ .

$\frac{x^{2} + 9 x + 2 x + 18}{2 x + 1}$

Add $9 x$ and $2 x$ .

$\frac{x^{2} + 11 x + 18}{2 x + 1}$

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

$2 x$

$1$

$x^{2}$

$11 x$

$18$

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

					$\frac{x}{2}$
	$2 x$	+	$1$		$x^{2}$	+	$11 x$	+	$18$

Multiply the new quotient term by the divisor.

				$\frac{x}{2}$
$2 x$	+	$1$		$x^{2}$	+	$11 x$	+	$18$
			+	$x^{2}$	+	$\frac{x}{2}$

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

				$\frac{x}{2}$
$2 x$	+	$1$		$x^{2}$	+	$11 x$	+	$18$
			-	$x^{2}$	-	$\frac{x}{2}$

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

				$\frac{x}{2}$
$2 x$	+	$1$		$x^{2}$	+	$11 x$	+	$18$
			-	$x^{2}$	-	$\frac{x}{2}$
					+	$\frac{21 x}{2}$

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

				$\frac{x}{2}$
$2 x$	+	$1$		$x^{2}$	+	$11 x$	+	$18$
			-	$x^{2}$	-	$\frac{x}{2}$
					+	$\frac{21 x}{2}$	+	$18$

Divide the highest order term in the dividend $\frac{21 x}{2}$ by the highest order term in divisor $2 x$ .

				$\frac{x}{2}$	+	$\frac{21}{4}$
$2 x$	+	$1$		$x^{2}$	+	$11 x$	+	$18$
			-	$x^{2}$	-	$\frac{x}{2}$
					+	$\frac{21 x}{2}$	+	$18$

Multiply the new quotient term by the divisor.

				$\frac{x}{2}$	+	$\frac{21}{4}$
$2 x$	+	$1$		$x^{2}$	+	$11 x$	+	$18$
			-	$x^{2}$	-	$\frac{x}{2}$
					+	$\frac{21 x}{2}$	+	$18$
					+	$\frac{21 x}{2}$	+	$\frac{21}{4}$

The expression needs to be subtracted from the dividend, so change all the signs in $\frac{21 x}{2} + \frac{21}{4}$

				$\frac{x}{2}$	+	$\frac{21}{4}$
$2 x$	+	$1$		$x^{2}$	+	$11 x$	+	$18$
			-	$x^{2}$	-	$\frac{x}{2}$
					+	$\frac{21 x}{2}$	+	$18$
					-	$\frac{21 x}{2}$	-	$\frac{21}{4}$

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

				$\frac{x}{2}$	+	$\frac{21}{4}$
$2 x$	+	$1$		$x^{2}$	+	$11 x$	+	$18$
			-	$x^{2}$	-	$\frac{x}{2}$
					+	$\frac{21 x}{2}$	+	$18$
					-	$\frac{21 x}{2}$	-	$\frac{21}{4}$
							+	$\frac{51}{4}$

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

$\frac{x}{2} + \frac{21}{4} + \frac{51}{4 (2 x + 1)}$

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

$y = \frac{x}{2} + \frac{21}{4}$

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = - \frac{1}{2}$

No Horizontal Asymptotes

Oblique Asymptotes: $y = \frac{x}{2} + \frac{21}{4}$

Step 7

Find the Asymptotes y=(X^2+11X+18)/(2X+1)

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