Precalculus Examples

$f (x) = \frac{5 x^{3} - 4}{x^{2} + 4 x - 5}$

Step 1

Find where the expression $\frac{5 x^{3} - 4}{x^{2} + 4 x - 5}$ is undefined.

$x = - 5, x = 1$

Step 2

Since $\frac{5 x^{3} - 4}{x^{2} + 4 x - 5}$ $\to$ $- \infty$ as $x$ $\to$ $- 5$ from the left and $\frac{5 x^{3} - 4}{x^{2} + 4 x - 5}$ $\to$ $\infty$ as $x$ $\to$ $- 5$ from the right, then $x = - 5$ is a vertical asymptote.

$x = - 5$

Step 3

Since $\frac{5 x^{3} - 4}{x^{2} + 4 x - 5}$ $\to$ $- \infty$ as $x$ $\to$ $1$ from the left and $\frac{5 x^{3} - 4}{x^{2} + 4 x - 5}$ $\to$ $\infty$ as $x$ $\to$ $1$ from the right, then $x = 1$ is a vertical asymptote.

$x = 1$

Step 4

List all of the vertical asymptotes:

$x = - 5, 1$

Step 5

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 6

Find $n$ and $m$ .

$n = 3$

$m = 2$

Step 7

Since $n > m$ , there is no horizontal asymptote.

No Horizontal Asymptotes

Step 8

Find the oblique asymptote using polynomial division.

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Step 8.1

Factor $x^{2} + 4 x - 5$ using the AC method.

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Step 8.1.1

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 8.1.2

Write the factored form using these integers.

$\frac{5 x^{3} - 4}{(x - 1) (x + 5)}$

Step 8.2

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

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Step 8.2.1

Apply the distributive property.

$\frac{5 x^{3} - 4}{x (x + 5) - 1 (x + 5)}$

Step 8.2.2

Apply the distributive property.

$\frac{5 x^{3} - 4}{x \cdot x + x \cdot 5 - 1 (x + 5)}$

Step 8.2.3

Apply the distributive property.

$\frac{5 x^{3} - 4}{x \cdot x + x \cdot 5 - 1 x - 1 \cdot 5}$

Step 8.2.4

Reorder $x$ and $5$ .

$\frac{5 x^{3} - 4}{x \cdot x + 5 x - 1 x - 1 \cdot 5}$

Step 8.2.5

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

$\frac{5 x^{3} - 4}{x \cdot x + 5 x - 1 x - 1 \cdot 5}$

Step 8.2.6

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

$\frac{5 x^{3} - 4}{x \cdot x + 5 x - 1 x - 1 \cdot 5}$

Step 8.2.7

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

$\frac{5 x^{3} - 4}{x^{1 + 1} + 5 x - 1 x - 1 \cdot 5}$

Step 8.2.8

Add $1$ and $1$ .

$\frac{5 x^{3} - 4}{x^{2} + 5 x - 1 x - 1 \cdot 5}$

Step 8.2.9

Multiply $- 1$ by $5$ .

$\frac{5 x^{3} - 4}{x^{2} + 5 x - 1 x - 5}$

Step 8.2.10

Subtract $1 x$ from $5 x$ .

$\frac{5 x^{3} - 4}{x^{2} + 4 x - 5}$

Step 8.3

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

$x^{2}$

$4 x$

$5$

$5 x^{3}$

$0 x^{2}$

$0 x$

$4$

Step 8.4

Divide the highest order term in the dividend $5 x^{3}$ by the highest order term in divisor $x^{2}$ .

$5 x$

$x^{2}$

$4 x$

$5$

$5 x^{3}$

$0 x^{2}$

$0 x$

$4$

Step 8.5

Multiply the new quotient term by the divisor.

$5 x$

$x^{2}$

$4 x$

$5$

$5 x^{3}$

$0 x^{2}$

$0 x$

$4$

$5 x^{3}$

$20 x^{2}$

$25 x$

Step 8.6

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{3} + 20 x^{2} - 25 x$

$5 x$

$x^{2}$

$4 x$

$5$

$5 x^{3}$

$0 x^{2}$

$0 x$

$4$

$5 x^{3}$

$20 x^{2}$

$25 x$

Step 8.7

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

$5 x$

$x^{2}$

$4 x$

$5$

$5 x^{3}$

$0 x^{2}$

$0 x$

$4$

$5 x^{3}$

$20 x^{2}$

$25 x$

$20 x^{2}$

$25 x$

Step 8.8

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

$5 x$

$x^{2}$

$4 x$

$5$

$5 x^{3}$

$0 x^{2}$

$0 x$

$4$

$5 x^{3}$

$20 x^{2}$

$25 x$

$20 x^{2}$

$25 x$

$4$

Step 8.9

Divide the highest order term in the dividend $- 20 x^{2}$ by the highest order term in divisor $x^{2}$ .

						$5 x$	-	$20$
$x^{2}$	+	$4 x$	-	$5$		$5 x^{3}$	+	$0 x^{2}$	+	$0 x$	-	$4$
					-	$5 x^{3}$	-	$20 x^{2}$	+	$25 x$
							-	$20 x^{2}$	+	$25 x$	-	$4$

Step 8.10

Multiply the new quotient term by the divisor.

						$5 x$	-	$20$
$x^{2}$	+	$4 x$	-	$5$		$5 x^{3}$	+	$0 x^{2}$	+	$0 x$	-	$4$
					-	$5 x^{3}$	-	$20 x^{2}$	+	$25 x$
							-	$20 x^{2}$	+	$25 x$	-	$4$
							-	$20 x^{2}$	-	$80 x$	+	$100$

Step 8.11

The expression needs to be subtracted from the dividend, so change all the signs in $- 20 x^{2} - 80 x + 100$

$5 x$

$20$

$x^{2}$

$4 x$

$5$

$5 x^{3}$

$0 x^{2}$

$0 x$

$4$

$5 x^{3}$

$20 x^{2}$

$25 x$

$20 x^{2}$

$25 x$

$4$

$20 x^{2}$

$80 x$

$100$

Step 8.12

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

						$5 x$	-	$20$
$x^{2}$	+	$4 x$	-	$5$		$5 x^{3}$	+	$0 x^{2}$	+	$0 x$	-	$4$
					-	$5 x^{3}$	-	$20 x^{2}$	+	$25 x$
							-	$20 x^{2}$	+	$25 x$	-	$4$
							+	$20 x^{2}$	+	$80 x$	-	$100$
									+	$105 x$	-	$104$

Step 8.13

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

$5 x - 20 + \frac{105 x - 104}{x^{2} + 4 x - 5}$

Step 8.14

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

$y = 5 x - 20$

Step 9

This is the set of all asymptotes.

Vertical Asymptotes: $x = - 5, 1$

No Horizontal Asymptotes

Oblique Asymptotes: $y = 5 x - 20$

Step 10