Precalculus Examples

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

Step 1

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

$x = 0, x = 4$

Step 2

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

$x = 4$

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

Factor $x^{2}$ out of $x^{4} - 4 x^{2}$ .

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Step 6.1.1.1.1

Factor $x^{2}$ out of $x^{4}$ .

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

Step 6.1.1.1.2

Factor $x^{2}$ out of $- 4 x^{2}$ .

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

Step 6.1.1.1.3

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} \cdot - 4$ .

$\frac{x^{2} (x^{2} - 4)}{x^{2} - 4 x}$

Step 6.1.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{x^{2} (x^{2} - 2^{2})}{x^{2} - 4 x}$

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 $a = x$ and $b = 2$ .

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

Step 6.1.2

Simplify terms.

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Step 6.1.2.1

Factor $x$ out of $x^{2} - 4 x$ .

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Step 6.1.2.1.1

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

$\frac{x^{2} (x + 2) (x - 2)}{x \cdot x - 4 x}$

Step 6.1.2.1.2

Factor $x$ out of $- 4 x$ .

$\frac{x^{2} (x + 2) (x - 2)}{x \cdot x + x \cdot - 4}$

Step 6.1.2.1.3

Factor $x$ out of $x \cdot x + x \cdot - 4$ .

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

Step 6.1.2.2

Cancel the common factor of $x^{2}$ and $x$ .

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Step 6.1.2.2.1

Factor $x$ out of $x^{2} (x + 2) (x - 2)$ .

$\frac{x ((x (x + 2)) (x - 2))}{x (x - 4)}$

Step 6.1.2.2.2

Cancel the common factors.

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Step 6.1.2.2.2.1

Cancel the common factor.

$\frac{x ((x (x + 2)) (x - 2))}{x (x - 4)}$

Step 6.1.2.2.2.2

Rewrite the expression.

$\frac{(x (x + 2)) (x - 2)}{x - 4}$

$\frac{x (x + 2) (x - 2)}{x - 4}$

Step 6.2

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

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Step 6.2.1

Apply the distributive property.

$\frac{(x \cdot x + x \cdot 2) (x - 2)}{x - 4}$

Step 6.2.2

Apply the distributive property.

$\frac{x \cdot x (x - 2) + x \cdot (2 (x - 2))}{x - 4}$

Step 6.2.3

Apply the distributive property.

$\frac{x \cdot x \cdot x + x \cdot x \cdot - 2 + x \cdot (2 (x - 2))}{x - 4}$

Step 6.2.4

Apply the distributive property.

$\frac{x \cdot x \cdot x + x \cdot x \cdot - 2 + x \cdot (2 x) + x \cdot 2 \cdot - 2}{x - 4}$

Step 6.2.5

Move $x$ .

$\frac{x \cdot x \cdot x + x \cdot (- 2 x) + x \cdot (2 x) + x \cdot 2 \cdot - 2}{x - 4}$

Step 6.2.6

Reorder $x$ and $- 2$ .

$\frac{x \cdot x \cdot x - 2 \cdot (x \cdot x) + x \cdot (2 x) + x \cdot 2 \cdot - 2}{x - 4}$

Step 6.2.7

Reorder $x$ and $2$ .

$\frac{x \cdot x \cdot x - 2 \cdot (x \cdot x) + 2 \cdot (x \cdot x) + x \cdot 2 \cdot - 2}{x - 4}$

Step 6.2.8

Reorder $x$ and $2$ .

$\frac{x \cdot x \cdot x - 2 \cdot (x \cdot x) + 2 \cdot (x \cdot x) + 2 \cdot x \cdot - 2}{x - 4}$

Step 6.2.9

Move $x$ .

$\frac{x \cdot x \cdot x - 2 \cdot (x \cdot x) + 2 \cdot (x \cdot x) + 2 \cdot (- 2 x)}{x - 4}$

Step 6.2.10

Multiply $- 2$ by $x$ .

$\frac{x \cdot x \cdot x - 2 x \cdot x + 2 \cdot (x \cdot x) + 2 \cdot (- 2 x)}{x - 4}$

Step 6.2.11

Multiply $2$ by $x$ .

$\frac{x \cdot x \cdot x - 2 x \cdot x + 2 x \cdot x + 2 \cdot (- 2 x)}{x - 4}$

Step 6.2.12

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

$\frac{x \cdot x \cdot x - 2 x \cdot x + 2 x \cdot x + 2 \cdot (- 2 x)}{x - 4}$

Step 6.2.13

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

$\frac{x \cdot x \cdot x - 2 x \cdot x + 2 x \cdot x + 2 \cdot (- 2 x)}{x - 4}$

Step 6.2.14

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

$\frac{x^{1 + 1} x - 2 x \cdot x + 2 x \cdot x + 2 \cdot (- 2 x)}{x - 4}$

Step 6.2.15

Add $1$ and $1$ .

$\frac{x^{2} x - 2 x \cdot x + 2 x \cdot x + 2 \cdot (- 2 x)}{x - 4}$

Step 6.2.16

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

$\frac{x^{2} x - 2 x \cdot x + 2 x \cdot x + 2 \cdot (- 2 x)}{x - 4}$

Step 6.2.17

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

$\frac{x^{2 + 1} - 2 x \cdot x + 2 x \cdot x + 2 \cdot (- 2 x)}{x - 4}$

Step 6.2.18

Add $2$ and $1$ .

$\frac{x^{3} - 2 x \cdot x + 2 x \cdot x + 2 \cdot (- 2 x)}{x - 4}$

Step 6.2.19

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

$\frac{x^{3} - 2 (x \cdot x) + 2 x \cdot x + 2 \cdot (- 2 x)}{x - 4}$

Step 6.2.20

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

$\frac{x^{3} - 2 (x \cdot x) + 2 x \cdot x + 2 \cdot (- 2 x)}{x - 4}$

Step 6.2.21

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

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

Step 6.2.22

Add $1$ and $1$ .

$\frac{x^{3} - 2 x^{2} + 2 x \cdot x + 2 \cdot (- 2 x)}{x - 4}$

Step 6.2.23

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

$\frac{x^{3} - 2 x^{2} + 2 (x \cdot x) + 2 \cdot (- 2 x)}{x - 4}$

Step 6.2.24

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

$\frac{x^{3} - 2 x^{2} + 2 (x \cdot x) + 2 \cdot (- 2 x)}{x - 4}$

Step 6.2.25

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

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

Step 6.2.26

Add $1$ and $1$ .

$\frac{x^{3} - 2 x^{2} + 2 x^{2} + 2 \cdot (- 2 x)}{x - 4}$

Step 6.2.27

Multiply $2$ by $- 2$ .

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

Step 6.2.28

Add $- 2 x^{2}$ and $2 x^{2}$ .

$\frac{x^{3} + 0 - 4 x}{x - 4}$

Step 6.2.29

Subtract $4 x$ from $0$ .

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

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 $0$ .

$x$

$4$

$x^{3}$

$0 x^{2}$

$4 x$

$0$

Step 6.4

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

					$x^{2}$
	$x$	-	$4$		$x^{3}$	+	$0 x^{2}$	-	$4 x$	+	$0$

Step 6.5

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$4$		$x^{3}$	+	$0 x^{2}$	-	$4 x$	+	$0$
			+	$x^{3}$	-	$4 x^{2}$

Step 6.6

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - 4 x^{2}$

				$x^{2}$
$x$	-	$4$		$x^{3}$	+	$0 x^{2}$	-	$4 x$	+	$0$
			-	$x^{3}$	+	$4 x^{2}$

Step 6.7

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

				$x^{2}$
$x$	-	$4$		$x^{3}$	+	$0 x^{2}$	-	$4 x$	+	$0$
			-	$x^{3}$	+	$4 x^{2}$
					+	$4 x^{2}$

Step 6.8

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

				$x^{2}$
$x$	-	$4$		$x^{3}$	+	$0 x^{2}$	-	$4 x$	+	$0$
			-	$x^{3}$	+	$4 x^{2}$
					+	$4 x^{2}$	-	$4 x$

Step 6.9

Divide the highest order term in the dividend $4 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$4 x$
$x$	-	$4$		$x^{3}$	+	$0 x^{2}$	-	$4 x$	+	$0$
			-	$x^{3}$	+	$4 x^{2}$
					+	$4 x^{2}$	-	$4 x$

Step 6.10

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$4 x$
$x$	-	$4$		$x^{3}$	+	$0 x^{2}$	-	$4 x$	+	$0$
			-	$x^{3}$	+	$4 x^{2}$
					+	$4 x^{2}$	-	$4 x$
					+	$4 x^{2}$	-	$16 x$

Step 6.11

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{2} - 16 x$

				$x^{2}$	+	$4 x$
$x$	-	$4$		$x^{3}$	+	$0 x^{2}$	-	$4 x$	+	$0$
			-	$x^{3}$	+	$4 x^{2}$
					+	$4 x^{2}$	-	$4 x$
					-	$4 x^{2}$	+	$16 x$

Step 6.12

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

				$x^{2}$	+	$4 x$
$x$	-	$4$		$x^{3}$	+	$0 x^{2}$	-	$4 x$	+	$0$
			-	$x^{3}$	+	$4 x^{2}$
					+	$4 x^{2}$	-	$4 x$
					-	$4 x^{2}$	+	$16 x$
							+	$12 x$

Step 6.13

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

				$x^{2}$	+	$4 x$
$x$	-	$4$		$x^{3}$	+	$0 x^{2}$	-	$4 x$	+	$0$
			-	$x^{3}$	+	$4 x^{2}$
					+	$4 x^{2}$	-	$4 x$
					-	$4 x^{2}$	+	$16 x$
							+	$12 x$	+	$0$

Step 6.14

Divide the highest order term in the dividend $12 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$4 x$	+	$12$
$x$	-	$4$		$x^{3}$	+	$0 x^{2}$	-	$4 x$	+	$0$
			-	$x^{3}$	+	$4 x^{2}$
					+	$4 x^{2}$	-	$4 x$
					-	$4 x^{2}$	+	$16 x$
							+	$12 x$	+	$0$

Step 6.15

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$4 x$	+	$12$
$x$	-	$4$		$x^{3}$	+	$0 x^{2}$	-	$4 x$	+	$0$
			-	$x^{3}$	+	$4 x^{2}$
					+	$4 x^{2}$	-	$4 x$
					-	$4 x^{2}$	+	$16 x$
							+	$12 x$	+	$0$
							+	$12 x$	-	$48$

Step 6.16

The expression needs to be subtracted from the dividend, so change all the signs in $12 x - 48$

				$x^{2}$	+	$4 x$	+	$12$
$x$	-	$4$		$x^{3}$	+	$0 x^{2}$	-	$4 x$	+	$0$
			-	$x^{3}$	+	$4 x^{2}$
					+	$4 x^{2}$	-	$4 x$
					-	$4 x^{2}$	+	$16 x$
							+	$12 x$	+	$0$
							-	$12 x$	+	$48$

Step 6.17

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

				$x^{2}$	+	$4 x$	+	$12$
$x$	-	$4$		$x^{3}$	+	$0 x^{2}$	-	$4 x$	+	$0$
			-	$x^{3}$	+	$4 x^{2}$
					+	$4 x^{2}$	-	$4 x$
					-	$4 x^{2}$	+	$16 x$
							+	$12 x$	+	$0$
							-	$12 x$	+	$48$
									+	$48$

Step 6.18

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

$x^{2} + 4 x + 12 + \frac{48}{x - 4}$

Step 6.19

Split the solution into the polynomial portion and the remainder.

$x^{2} + 4 x + 12 + \frac{48 x}{x^{2} - 4 x}$

Step 6.20

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

$y = x^{2} + 4 x + 12$

Step 7

This is the set of all asymptotes.

Vertical Asymptotes: $x = 4$

No Horizontal Asymptotes

Oblique Asymptotes: $y = x^{2} + 4 x + 12$

Step 8