Precalculus Examples

$f (x) = \frac{x^{3} - 1}{x - x^{2}}$

Step 1

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

$x = 0, x = 1$

Step 2

Since $\frac{x^{3} - 1}{x - x^{2}}$ $\to$ $\infty$ as $x$ $\to$ $0$ from the left and $\frac{x^{3} - 1}{x - x^{2}}$ $\to$ $- \infty$ 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 $m$ .

$n = 3$

$m = 2$

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 $1$ as $1^{3}$ .

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

Step 6.1.1.2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 1$ .

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

Step 6.1.1.3

Simplify.

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Step 6.1.1.3.1

Multiply $x$ by $1$ .

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

Step 6.1.1.3.2

One to any power is one.

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

Step 6.1.2

Simplify terms.

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Step 6.1.2.1

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

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Step 6.1.2.1.1

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

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

Step 6.1.2.1.2

Factor $x$ out of $x^{1}$ .

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

Step 6.1.2.1.3

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

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

Step 6.1.2.1.4

Factor $x$ out of $x \cdot 1 + x (- x)$ .

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

Step 6.1.2.2

Cancel the common factor of $x - 1$ and $1 - x$ .

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Step 6.1.2.2.1

Factor $- 1$ out of $x$ .

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

Step 6.1.2.2.2

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 6.1.2.2.3

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

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

Step 6.1.2.2.4

Reorder terms.

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

Step 6.1.2.2.5

Cancel the common factor.

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

Step 6.1.2.2.6

Rewrite the expression.

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

Step 6.1.2.3

Move the negative in front of the fraction.

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

Step 6.2

Expand $- (x^{2} + x + 1)$ .

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Step 6.2.1

Negate $x^{2} + x + 1$ .

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

Step 6.2.2

Apply the distributive property.

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

Step 6.2.3

Apply the distributive property.

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

Step 6.2.4

Multiply $- 1$ by $1$ .

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

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 $0$ .

$x$

$0$

$x^{2}$

$x$

$1$

Step 6.4

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

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

Step 6.5

Multiply the new quotient term by the divisor.

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

Step 6.6

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

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

Step 6.7

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

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

Step 6.8

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

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

Step 6.9

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

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

Step 6.10

Multiply the new quotient term by the divisor.

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

Step 6.11

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

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

Step 6.12

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

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

Step 6.13

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

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

Step 6.14

Split the solution into the polynomial portion and the remainder.

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

Step 6.15

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

$y = - x - 1$

Step 7

This is the set of all asymptotes.

Vertical Asymptotes: $x = 0$

No Horizontal Asymptotes

Oblique Asymptotes: $y = - x - 1$

Step 8