Precalculus Examples

$f (x) = \frac{x^{2} + 4 x - 5}{x - 5}$

Step 1

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

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Step 5.1

Factor $x^{2} + 4 x - 5$ 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 $- 5$ and whose sum is $4$ .

$- 1, 5$

Step 5.1.2

Write the factored form using these integers.

$\frac{(x - 1) (x + 5)}{x - 5}$

Step 5.2

Expand $(x - 1) (x + 5)$ .

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Step 5.2.1

Apply the distributive property.

$\frac{x (x + 5) - 1 (x + 5)}{x - 5}$

Step 5.2.2

Apply the distributive property.

$\frac{x \cdot x + x \cdot 5 - 1 (x + 5)}{x - 5}$

Step 5.2.3

Apply the distributive property.

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

Step 5.2.4

Reorder $x$ and $5$ .

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

Step 5.2.5

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

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

Step 5.2.6

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

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

Step 5.2.7

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

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

Step 5.2.8

Add $1$ and $1$ .

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

Step 5.2.9

Multiply $- 1$ by $5$ .

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

Step 5.2.10

Subtract $1 x$ from $5 x$ .

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

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 $0$ .

$x$

$5$

$x^{2}$

$4 x$

$5$

Step 5.4

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

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

Step 5.5

Multiply the new quotient term by the divisor.

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

Step 5.6

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

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

Step 5.7

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

				$x$
$x$	-	$5$		$x^{2}$	+	$4 x$	-	$5$
			-	$x^{2}$	+	$5 x$
					+	$9 x$

Step 5.8

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

				$x$
$x$	-	$5$		$x^{2}$	+	$4 x$	-	$5$
			-	$x^{2}$	+	$5 x$
					+	$9 x$	-	$5$

Step 5.9

Divide the highest order term in the dividend $9 x$ by the highest order term in divisor $x$ .

				$x$	+	$9$
$x$	-	$5$		$x^{2}$	+	$4 x$	-	$5$
			-	$x^{2}$	+	$5 x$
					+	$9 x$	-	$5$

Step 5.10

Multiply the new quotient term by the divisor.

				$x$	+	$9$
$x$	-	$5$		$x^{2}$	+	$4 x$	-	$5$
			-	$x^{2}$	+	$5 x$
					+	$9 x$	-	$5$
					+	$9 x$	-	$45$

Step 5.11

The expression needs to be subtracted from the dividend, so change all the signs in $9 x - 45$

				$x$	+	$9$
$x$	-	$5$		$x^{2}$	+	$4 x$	-	$5$
			-	$x^{2}$	+	$5 x$
					+	$9 x$	-	$5$
					-	$9 x$	+	$45$

Step 5.12

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

				$x$	+	$9$
$x$	-	$5$		$x^{2}$	+	$4 x$	-	$5$
			-	$x^{2}$	+	$5 x$
					+	$9 x$	-	$5$
					-	$9 x$	+	$45$
							+	$40$

Step 5.13

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

$x + 9 + \frac{40}{x - 5}$

Step 5.14

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

$y = x + 9$

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = 5$

No Horizontal Asymptotes

Oblique Asymptotes: $y = x + 9$

Step 7