Precalculus Examples

$f (x) = \frac{x^{2} - 8 x + 12}{x + 3}$

Step 1

Find where the expression $\frac{x^{2} - 8 x + 12}{x + 3}$ is undefined.

$x = - 3$

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.

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Step 5.1

Factor $x^{2} - 8 x + 12$ using the AC method.

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Step 5.1.1

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 $12$ and whose sum is $- 8$ .

$- 6, - 2$

Step 5.1.2

Write the factored form using these integers.

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

Step 5.2

Expand $(x - 6) (x - 2)$ .

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Step 5.2.1

Apply the distributive property.

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

Step 5.2.2

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 2 - 6 (x - 2)}{x + 3}$

Step 5.2.3

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 2 - 6 x - 6 \cdot - 2}{x + 3}$

Step 5.2.4

Reorder $x$ and $- 2$ .

$\frac{x \cdot x - 2 x - 6 x - 6 \cdot - 2}{x + 3}$

Step 5.2.5

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

$\frac{x \cdot x - 2 x - 6 x - 6 \cdot - 2}{x + 3}$

Step 5.2.6

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

$\frac{x \cdot x - 2 x - 6 x - 6 \cdot - 2}{x + 3}$

Step 5.2.7

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

$\frac{x^{1 + 1} - 2 x - 6 x - 6 \cdot - 2}{x + 3}$

Step 5.2.8

Add $1$ and $1$ .

$\frac{x^{2} - 2 x - 6 x - 6 \cdot - 2}{x + 3}$

Step 5.2.9

Multiply $- 6$ by $- 2$ .

$\frac{x^{2} - 2 x - 6 x + 12}{x + 3}$

Step 5.2.10

Subtract $6 x$ from $- 2 x$ .

$\frac{x^{2} - 8 x + 12}{x + 3}$

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 $0$ .

$x$

$3$

$x^{2}$

$8 x$

$12$

Step 5.4

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

					$x$
	$x$	+	$3$		$x^{2}$	-	$8 x$	+	$12$

Step 5.5

Multiply the new quotient term by the divisor.

				$x$
$x$	+	$3$		$x^{2}$	-	$8 x$	+	$12$
			+	$x^{2}$	+	$3 x$

Step 5.6

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

				$x$
$x$	+	$3$		$x^{2}$	-	$8 x$	+	$12$
			-	$x^{2}$	-	$3 x$

Step 5.7

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

				$x$
$x$	+	$3$		$x^{2}$	-	$8 x$	+	$12$
			-	$x^{2}$	-	$3 x$
					-	$11 x$

Step 5.8

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

				$x$
$x$	+	$3$		$x^{2}$	-	$8 x$	+	$12$
			-	$x^{2}$	-	$3 x$
					-	$11 x$	+	$12$

Step 5.9

Divide the highest order term in the dividend $- 11 x$ by the highest order term in divisor $x$ .

				$x$	-	$11$
$x$	+	$3$		$x^{2}$	-	$8 x$	+	$12$
			-	$x^{2}$	-	$3 x$
					-	$11 x$	+	$12$

Step 5.10

Multiply the new quotient term by the divisor.

				$x$	-	$11$
$x$	+	$3$		$x^{2}$	-	$8 x$	+	$12$
			-	$x^{2}$	-	$3 x$
					-	$11 x$	+	$12$
					-	$11 x$	-	$33$

Step 5.11

The expression needs to be subtracted from the dividend, so change all the signs in $- 11 x - 33$

				$x$	-	$11$
$x$	+	$3$		$x^{2}$	-	$8 x$	+	$12$
			-	$x^{2}$	-	$3 x$
					-	$11 x$	+	$12$
					+	$11 x$	+	$33$

Step 5.12

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

				$x$	-	$11$
$x$	+	$3$		$x^{2}$	-	$8 x$	+	$12$
			-	$x^{2}$	-	$3 x$
					-	$11 x$	+	$12$
					+	$11 x$	+	$33$
							+	$45$

Step 5.13

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

$x - 11 + \frac{45}{x + 3}$

Step 5.14

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

$y = x - 11$

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = - 3$

No Horizontal Asymptotes

Oblique Asymptotes: $y = x - 11$

Step 7