Precalculus Examples

$\frac{3 x^{3} + x^{2} + 1}{x^{2} + 2 x}$

Step 1

Find where the expression $\frac{3 x^{3} + x^{2} + 1}{x^{2} + 2 x}$ is undefined.

$x = - 2, x = 0$

Step 2

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

$x = - 2$

Step 3

Since $\frac{3 x^{3} + x^{2} + 1}{x^{2} + 2 x}$ $\to$ $- \infty$ as $x$ $\to$ $0$ from the left and $\frac{3 x^{3} + x^{2} + 1}{x^{2} + 2 x}$ $\to$ $\infty$ as $x$ $\to$ $0$ from the right, then $x = 0$ is a vertical asymptote.

$x = 0$

Step 4

List all of the vertical asymptotes:

$x = - 2, 0$

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 $x$ out of $x^{2} + 2 x$ .

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Step 8.1.1

Factor $x$ out of $x^{2}$ .

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

Step 8.1.2

Factor $x$ out of $2 x$ .

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

Step 8.1.3

Factor $x$ out of $x \cdot x + x \cdot 2$ .

$\frac{3 x^{3} + x^{2} + 1}{x (x + 2)}$

Step 8.2

Expand $x (x + 2)$ .

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Step 8.2.1

Apply the distributive property.

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

Step 8.2.2

Reorder $x$ and $2$ .

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

Step 8.2.3

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

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

Step 8.2.4

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

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

Step 8.2.5

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

$\frac{3 x^{3} + x^{2} + 1}{x^{1 + 1} + 2 x}$

Step 8.2.6

Add $1$ and $1$ .

$\frac{3 x^{3} + x^{2} + 1}{x^{2} + 2 x}$

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 $0$ .

$x^{2}$

$2 x$

$0$

$3 x^{3}$

$x^{2}$

$0 x$

$1$

Step 8.4

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

$3 x$

$x^{2}$

$2 x$

$0$

$3 x^{3}$

$x^{2}$

$0 x$

$1$

Step 8.5

Multiply the new quotient term by the divisor.

$3 x$

$x^{2}$

$2 x$

$0$

$3 x^{3}$

$x^{2}$

$0 x$

$1$

$3 x^{3}$

$6 x^{2}$

$0$

Step 8.6

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{3} + 6 x^{2} + 0$

$3 x$

$x^{2}$

$2 x$

$0$

$3 x^{3}$

$x^{2}$

$0 x$

$1$

$3 x^{3}$

$6 x^{2}$

$0$

Step 8.7

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

$3 x$

$x^{2}$

$2 x$

$0$

$3 x^{3}$

$x^{2}$

$0 x$

$1$

$3 x^{3}$

$6 x^{2}$

$0$

$5 x^{2}$

$0$

Step 8.8

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

$3 x$

$x^{2}$

$2 x$

$0$

$3 x^{3}$

$x^{2}$

$0 x$

$1$

$3 x^{3}$

$6 x^{2}$

$0$

$5 x^{2}$

$0$

$1$

Step 8.9

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

						$3 x$	-	$5$
$x^{2}$	+	$2 x$	+	$0$		$3 x^{3}$	+	$x^{2}$	+	$0 x$	+	$1$
					-	$3 x^{3}$	-	$6 x^{2}$	-	$0$
							-	$5 x^{2}$	+	$0$	+	$1$

Step 8.10

Multiply the new quotient term by the divisor.

						$3 x$	-	$5$
$x^{2}$	+	$2 x$	+	$0$		$3 x^{3}$	+	$x^{2}$	+	$0 x$	+	$1$
					-	$3 x^{3}$	-	$6 x^{2}$	-	$0$
							-	$5 x^{2}$	+	$0$	+	$1$
							-	$5 x^{2}$	-	$10 x$	+	$0$

Step 8.11

The expression needs to be subtracted from the dividend, so change all the signs in $- 5 x^{2} - 10 x + 0$

$3 x$

$5$

$x^{2}$

$2 x$

$0$

$3 x^{3}$

$x^{2}$

$0 x$

$1$

$3 x^{3}$

$6 x^{2}$

$0$

$5 x^{2}$

$0$

$1$

$5 x^{2}$

$10 x$

$0$

Step 8.12

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

$3 x$

$5$

$x^{2}$

$2 x$

$0$

$3 x^{3}$

$x^{2}$

$0 x$

$1$

$3 x^{3}$

$6 x^{2}$

$0$

$5 x^{2}$

$0$

$1$

$5 x^{2}$

$10 x$

$0$

$10 x$

$1$

Step 8.13

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

$3 x - 5 + \frac{10 x + 1}{x^{2} + 2 x}$

Step 8.14

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

$y = 3 x - 5$

Step 9

This is the set of all asymptotes.

Vertical Asymptotes: $x = - 2, 0$

No Horizontal Asymptotes

Oblique Asymptotes: $y = 3 x - 5$

Step 10