Precalculus Examples

$\frac{x^{4} - 16 x^{2}}{x^{3} + 3 x^{2} - 4 x}$

Step 1

Find where the expression $\frac{x^{4} - 16 x^{2}}{x^{3} + 3 x^{2} - 4 x}$ is undefined.

$x = - 4, x = 0, x = 1$

Step 2

Since $\frac{x^{4} - 16 x^{2}}{x^{3} + 3 x^{2} - 4 x}$ $\to$ $\infty$ as $x$ $\to$ $1$ from the left and $\frac{x^{4} - 16 x^{2}}{x^{3} + 3 x^{2} - 4 x}$ $\to$ $- \infty$ as $x$ $\to$ $1$ from the right, then $x = 1$ is a vertical asymptote.

$x = 1$

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 $m$ .

$n = 4$

$m = 3$

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 $x^{4} - 16 x^{2}$ .

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Step 6.1.1.1.1

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

$\frac{x^{2} x^{2} - 16 x^{2}}{x^{3} + 3 x^{2} - 4 x}$

Step 6.1.1.1.2

Factor $x^{2}$ out of $- 16 x^{2}$ .

$\frac{x^{2} x^{2} + x^{2} \cdot - 16}{x^{3} + 3 x^{2} - 4 x}$

Step 6.1.1.1.3

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} \cdot - 16$ .

$\frac{x^{2} (x^{2} - 16)}{x^{3} + 3 x^{2} - 4 x}$

Step 6.1.1.2

Rewrite $16$ as $4^{2}$ .

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

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 $b = 4$ .

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

Step 6.1.2

Simplify the denominator.

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Step 6.1.2.1

Factor $x$ out of $x^{3} + 3 x^{2} - 4 x$ .

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Step 6.1.2.1.1

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

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

Step 6.1.2.1.2

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

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

Step 6.1.2.1.3

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

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

Step 6.1.2.1.4

Factor $x$ out of $x \cdot x^{2} + x (3 x)$ .

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

Step 6.1.2.1.5

Factor $x$ out of $x (x^{2} + 3 x) + x \cdot - 4$ .

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

Step 6.1.2.2

Factor $x^{2} + 3 x - 4$ using the AC method.

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Step 6.1.2.2.1

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 $- 4$ and whose sum is $3$ .

$- 1, 4$

Step 6.1.2.2.2

Write the factored form using these integers.

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

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

Step 6.1.3

Reduce the expression by cancelling the common factors.

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Step 6.1.3.1

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

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Step 6.1.3.1.1

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

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

Step 6.1.3.1.2

Cancel the common factors.

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Step 6.1.3.1.2.1

Factor $x$ out of $x (x - 1) (x + 4)$ .

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

Step 6.1.3.1.2.2

Cancel the common factor.

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

Step 6.1.3.1.2.3

Rewrite the expression.

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

Step 6.1.3.2

Cancel the common factor of $x + 4$ .

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Step 6.1.3.2.1

Cancel the common factor.

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

Step 6.1.3.2.2

Rewrite the expression.

$\frac{(x) (x - 4)}{x - 1}$

$\frac{x (x - 4)}{x - 1}$

Step 6.2

Expand $x (x - 4)$ .

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Step 6.2.1

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 4}{x - 1}$

Step 6.2.2

Reorder $x$ and $- 4$ .

$\frac{x \cdot x - 4 x}{x - 1}$

Step 6.2.3

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

$\frac{x \cdot x - 4 x}{x - 1}$

Step 6.2.4

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

$\frac{x \cdot x - 4 x}{x - 1}$

Step 6.2.5

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

$\frac{x^{1 + 1} - 4 x}{x - 1}$

Step 6.2.6

Add $1$ and $1$ .

$\frac{x^{2} - 4 x}{x - 1}$

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 $0$ .

$x$

$1$

$x^{2}$

$4 x$

$0$

Step 6.4

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

					$x$
	$x$	-	$1$		$x^{2}$	-	$4 x$	+	$0$

Step 6.5

Multiply the new quotient term by the divisor.

				$x$
$x$	-	$1$		$x^{2}$	-	$4 x$	+	$0$
			+	$x^{2}$	-	$x$

Step 6.6

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

				$x$
$x$	-	$1$		$x^{2}$	-	$4 x$	+	$0$
			-	$x^{2}$	+	$x$

Step 6.7

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

				$x$
$x$	-	$1$		$x^{2}$	-	$4 x$	+	$0$
			-	$x^{2}$	+	$x$
					-	$3 x$

Step 6.8

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

				$x$
$x$	-	$1$		$x^{2}$	-	$4 x$	+	$0$
			-	$x^{2}$	+	$x$
					-	$3 x$	+	$0$

Step 6.9

Divide the highest order term in the dividend $- 3 x$ by the highest order term in divisor $x$ .

				$x$	-	$3$
$x$	-	$1$		$x^{2}$	-	$4 x$	+	$0$
			-	$x^{2}$	+	$x$
					-	$3 x$	+	$0$

Step 6.10

Multiply the new quotient term by the divisor.

				$x$	-	$3$
$x$	-	$1$		$x^{2}$	-	$4 x$	+	$0$
			-	$x^{2}$	+	$x$
					-	$3 x$	+	$0$
					-	$3 x$	+	$3$

Step 6.11

The expression needs to be subtracted from the dividend, so change all the signs in $- 3 x + 3$

				$x$	-	$3$
$x$	-	$1$		$x^{2}$	-	$4 x$	+	$0$
			-	$x^{2}$	+	$x$
					-	$3 x$	+	$0$
					+	$3 x$	-	$3$

Step 6.12

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

				$x$	-	$3$
$x$	-	$1$		$x^{2}$	-	$4 x$	+	$0$
			-	$x^{2}$	+	$x$
					-	$3 x$	+	$0$
					+	$3 x$	-	$3$
							-	$3$

Step 6.13

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

$x - 3 - \frac{3}{x - 1}$

Step 6.14

Split the solution into the polynomial portion and the remainder.

$x - 3 + \frac{- 3 x^{2} - 12 x}{x^{3} + 3 x^{2} - 4 x}$

Step 6.15

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

$y = x - 3$

Step 7

This is the set of all asymptotes.

Vertical Asymptotes: $x = 1$

No Horizontal Asymptotes

Oblique Asymptotes: $y = x - 3$

Step 8