Precalculus Examples

$\frac{x^{4} - 81}{x^{2} - 3 x}$

Step 1

Find where the expression $\frac{x^{4} - 81}{x^{2} - 3 x}$ is undefined.

$x = 0, x = 3$

Step 2

Since $\frac{x^{4} - 81}{x^{2} - 3 x}$ $\to$ $- \infty$ as $x$ $\to$ $0$ from the left and $\frac{x^{4} - 81}{x^{2} - 3 x}$ $\to$ $\infty$ as $x$ $\to$ $0$ from the right, then $x = 0$ is a vertical asymptote.

$x = 0$

Step 3

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 4

Find $n$ and $m$ .

$n = 4$

$m = 2$

Step 5

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

No Horizontal Asymptotes

Step 6

Find the oblique asymptote using polynomial division.

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Step 6.1

Simplify the expression.

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Step 6.1.1

Simplify the numerator.

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Step 6.1.1.1

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$\frac{{(x^{2})}^{2} - 81}{x^{2} - 3 x}$

Step 6.1.1.2

Rewrite $81$ as $9^{2}$ .

$\frac{{(x^{2})}^{2} - 9^{2}}{x^{2} - 3 x}$

Step 6.1.1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{2}$ and $b = 9$ .

$\frac{(x^{2} + 9) (x^{2} - 9)}{x^{2} - 3 x}$

Step 6.1.1.4

Simplify.

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Step 6.1.1.4.1

Rewrite $9$ as $3^{2}$ .

$\frac{(x^{2} + 9) (x^{2} - 3^{2})}{x^{2} - 3 x}$

Step 6.1.1.4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

$\frac{(x^{2} + 9) (x + 3) (x - 3)}{x^{2} - 3 x}$

Step 6.1.2

Simplify terms.

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Step 6.1.2.1

Factor $x$ out of $x^{2} - 3 x$ .

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Step 6.1.2.1.1

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

$\frac{(x^{2} + 9) (x + 3) (x - 3)}{x \cdot x - 3 x}$

Step 6.1.2.1.2

Factor $x$ out of $- 3 x$ .

$\frac{(x^{2} + 9) (x + 3) (x - 3)}{x \cdot x + x \cdot - 3}$

Step 6.1.2.1.3

Factor $x$ out of $x \cdot x + x \cdot - 3$ .

$\frac{(x^{2} + 9) (x + 3) (x - 3)}{x (x - 3)}$

Step 6.1.2.2

Cancel the common factor of $x - 3$ .

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Step 6.1.2.2.1

Cancel the common factor.

$\frac{(x^{2} + 9) (x + 3) (x - 3)}{x (x - 3)}$

Step 6.1.2.2.2

Rewrite the expression.

$\frac{(x^{2} + 9) (x + 3)}{x}$

Step 6.2

Expand $(x^{2} + 9) (x + 3)$ .

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Step 6.2.1

Apply the distributive property.

$\frac{x^{2} (x + 3) + 9 (x + 3)}{x}$

Step 6.2.2

Apply the distributive property.

$\frac{x^{2} x + x^{2} \cdot 3 + 9 (x + 3)}{x}$

Step 6.2.3

Apply the distributive property.

$\frac{x^{2} x + x^{2} \cdot 3 + 9 x + 9 \cdot 3}{x}$

Step 6.2.4

Reorder $x^{2}$ and $3$ .

$\frac{x^{2} x + 3 x^{2} + 9 x + 9 \cdot 3}{x}$

Step 6.2.5

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

$\frac{x^{2} x + 3 x^{2} + 9 x + 9 \cdot 3}{x}$

Step 6.2.6

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

$\frac{x^{2 + 1} + 3 x^{2} + 9 x + 9 \cdot 3}{x}$

Step 6.2.7

Add $2$ and $1$ .

$\frac{x^{3} + 3 x^{2} + 9 x + 9 \cdot 3}{x}$

Step 6.2.8

Multiply $9$ by $3$ .

$\frac{x^{3} + 3 x^{2} + 9 x + 27}{x}$

Step 6.3

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

$x$

$0$

$x^{3}$

$3 x^{2}$

$9 x$

$27$

Step 6.4

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

					$x^{2}$
	$x$	+	$0$		$x^{3}$	+	$3 x^{2}$	+	$9 x$	+	$27$

Step 6.5

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$0$		$x^{3}$	+	$3 x^{2}$	+	$9 x$	+	$27$
			+	$x^{3}$	+	$0$

Step 6.6

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

				$x^{2}$
$x$	+	$0$		$x^{3}$	+	$3 x^{2}$	+	$9 x$	+	$27$
			-	$x^{3}$	-	$0$

Step 6.7

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

				$x^{2}$
$x$	+	$0$		$x^{3}$	+	$3 x^{2}$	+	$9 x$	+	$27$
			-	$x^{3}$	-	$0$
					+	$3 x^{2}$

Step 6.8

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

				$x^{2}$
$x$	+	$0$		$x^{3}$	+	$3 x^{2}$	+	$9 x$	+	$27$
			-	$x^{3}$	-	$0$
					+	$3 x^{2}$	+	$9 x$

Step 6.9

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

				$x^{2}$	+	$3 x$
$x$	+	$0$		$x^{3}$	+	$3 x^{2}$	+	$9 x$	+	$27$
			-	$x^{3}$	-	$0$
					+	$3 x^{2}$	+	$9 x$

Step 6.10

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$3 x$
$x$	+	$0$		$x^{3}$	+	$3 x^{2}$	+	$9 x$	+	$27$
			-	$x^{3}$	-	$0$
					+	$3 x^{2}$	+	$9 x$
					+	$3 x^{2}$	+	$0$

Step 6.11

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

				$x^{2}$	+	$3 x$
$x$	+	$0$		$x^{3}$	+	$3 x^{2}$	+	$9 x$	+	$27$
			-	$x^{3}$	-	$0$
					+	$3 x^{2}$	+	$9 x$
					-	$3 x^{2}$	-	$0$

Step 6.12

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

				$x^{2}$	+	$3 x$
$x$	+	$0$		$x^{3}$	+	$3 x^{2}$	+	$9 x$	+	$27$
			-	$x^{3}$	-	$0$
					+	$3 x^{2}$	+	$9 x$
					-	$3 x^{2}$	-	$0$
							+	$9 x$

Step 6.13

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

				$x^{2}$	+	$3 x$
$x$	+	$0$		$x^{3}$	+	$3 x^{2}$	+	$9 x$	+	$27$
			-	$x^{3}$	-	$0$
					+	$3 x^{2}$	+	$9 x$
					-	$3 x^{2}$	-	$0$
							+	$9 x$	+	$27$

Step 6.14

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

				$x^{2}$	+	$3 x$	+	$9$
$x$	+	$0$		$x^{3}$	+	$3 x^{2}$	+	$9 x$	+	$27$
			-	$x^{3}$	-	$0$
					+	$3 x^{2}$	+	$9 x$
					-	$3 x^{2}$	-	$0$
							+	$9 x$	+	$27$

Step 6.15

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$3 x$	+	$9$
$x$	+	$0$		$x^{3}$	+	$3 x^{2}$	+	$9 x$	+	$27$
			-	$x^{3}$	-	$0$
					+	$3 x^{2}$	+	$9 x$
					-	$3 x^{2}$	-	$0$
							+	$9 x$	+	$27$
							+	$9 x$	+	$0$

Step 6.16

The expression needs to be subtracted from the dividend, so change all the signs in $9 x + 0$

				$x^{2}$	+	$3 x$	+	$9$
$x$	+	$0$		$x^{3}$	+	$3 x^{2}$	+	$9 x$	+	$27$
			-	$x^{3}$	-	$0$
					+	$3 x^{2}$	+	$9 x$
					-	$3 x^{2}$	-	$0$
							+	$9 x$	+	$27$
							-	$9 x$	-	$0$

Step 6.17

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

				$x^{2}$	+	$3 x$	+	$9$
$x$	+	$0$		$x^{3}$	+	$3 x^{2}$	+	$9 x$	+	$27$
			-	$x^{3}$	-	$0$
					+	$3 x^{2}$	+	$9 x$
					-	$3 x^{2}$	-	$0$
							+	$9 x$	+	$27$
							-	$9 x$	-	$0$
									+	$27$

Step 6.18

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

$x^{2} + 3 x + 9 + \frac{27}{x}$

Step 6.19

Split the solution into the polynomial portion and the remainder.

$x^{2} + 3 x + 9 + \frac{27 x - 81}{x^{2} - 3 x}$

Step 6.20

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

$y = x^{2} + 3 x + 9$

Step 7

This is the set of all asymptotes.

Vertical Asymptotes: $x = 0$

No Horizontal Asymptotes

Oblique Asymptotes: $y = x^{2} + 3 x + 9$

Step 8