Precalculus Examples

$f (x) = \frac{3 x^{4} - 2 x + 1}{- x^{3} + 8}$

Step 1

Find where the expression $\frac{3 x^{4} - 2 x + 1}{- x^{3} + 8}$ is undefined.

$x = 2$

Step 2

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 3

Find $n$ and $m$ .

$n = 4$

$m = 3$

Step 4

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

No Horizontal Asymptotes

Step 5

Find the oblique asymptote using polynomial division.

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Step 5.1

Simplify the expression.

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Step 5.1.1

Simplify the denominator.

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Step 5.1.1.1

Rewrite $- x^{3}$ as ${(- x)}^{3}$ .

$\frac{3 x^{4} - 2 x + 1}{{(- x)}^{3} + 8}$

Step 5.1.1.2

Rewrite $8$ as $2^{3}$ .

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

Step 5.1.1.3

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

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

Step 5.1.1.4

Simplify.

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Step 5.1.1.4.1

Apply the product rule to $- x$ .

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

Step 5.1.1.4.2

Raise $- 1$ to the power of $2$ .

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

Step 5.1.1.4.3

Multiply $x^{2}$ by $1$ .

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

Step 5.1.1.4.4

Multiply $- - x$ .

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Step 5.1.1.4.4.1

Multiply $- 1$ by $- 1$ .

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

Step 5.1.1.4.4.2

Multiply $x$ by $1$ .

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

Step 5.1.1.4.5

Move $2$ to the left of $x$ .

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

Step 5.1.1.4.6

Raise $2$ to the power of $2$ .

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

Step 5.1.2

Simplify with factoring out.

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Step 5.1.2.1

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

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

Step 5.1.2.2

Rewrite $2$ as $- 1 (- 2)$ .

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

Step 5.1.2.3

Factor $- 1$ out of $- (x) - 1 (- 2)$ .

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

Step 5.1.2.4

Rewrite negatives.

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Step 5.1.2.4.1

Rewrite $- (x - 2)$ as $- 1 (x - 2)$ .

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

Step 5.1.2.4.2

Move the negative in front of the fraction.

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

Step 5.2

Expand $- (3 x^{4} - 2 x + 1)$ .

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Step 5.2.1

Negate $3 x^{4} - 2 x + 1$ .

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

Step 5.2.2

Apply the distributive property.

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

Step 5.2.3

Apply the distributive property.

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

Step 5.2.4

Remove parentheses.

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

Step 5.2.5

Remove parentheses.

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

Step 5.2.6

Multiply $- 1$ by $3$ .

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

Step 5.2.7

Multiply $- 1$ by $- 2$ .

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

Step 5.2.8

Multiply $- 1$ by $1$ .

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

Step 5.3

Expand $(x - 2) (x^{2} + 2 x + 4)$ .

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Step 5.3.1

Apply the distributive property.

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

Step 5.3.2

Apply the distributive property.

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

Step 5.3.3

Apply the distributive property.

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

Step 5.3.4

Apply the distributive property.

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

Step 5.3.5

Apply the distributive property.

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

Step 5.3.6

Remove parentheses.

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

Step 5.3.7

Reorder $x$ and $2$ .

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

Step 5.3.8

Reorder $x$ and $4$ .

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

Step 5.3.9

Remove parentheses.

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

Step 5.3.10

Multiply $2$ by $x$ .

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

Step 5.3.11

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

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

Step 5.3.12

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

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

Step 5.3.13

Add $1$ and $2$ .

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

Step 5.3.14

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

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

Step 5.3.15

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

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

Step 5.3.16

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

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

Step 5.3.17

Add $1$ and $1$ .

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

Step 5.3.18

Multiply $- 2$ by $2$ .

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

Step 5.3.19

Multiply $- 2$ by $4$ .

$\frac{- 3 x^{4} + 2 x - 1}{x^{3} + 2 x^{2} + 4 x - 2 x^{2} - 4 x - 8}$

Step 5.3.20

Move $4 x$ .

$\frac{- 3 x^{4} + 2 x - 1}{x^{3} + 2 x^{2} - 2 x^{2} + 4 x - 4 x - 8}$

Step 5.3.21

Subtract $2 x^{2}$ from $2 x^{2}$ .

$\frac{- 3 x^{4} + 2 x - 1}{x^{3} + 0 + 4 x - 4 x - 8}$

Step 5.3.22

Add $x^{3}$ and $0$ .

$\frac{- 3 x^{4} + 2 x - 1}{x^{3} + 4 x - 4 x - 8}$

Step 5.3.23

Subtract $4 x$ from $4 x$ .

$\frac{- 3 x^{4} + 2 x - 1}{x^{3} + 0 - 8}$

Step 5.3.24

Add $x^{3}$ and $0$ .

$\frac{- 3 x^{4} + 2 x - 1}{x^{3} - 8}$

Step 5.4

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

$x^{3}$

$0 x^{2}$

$0 x$

$8$

$3 x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$1$

Step 5.5

Divide the highest order term in the dividend $- 3 x^{4}$ by the highest order term in divisor $x^{3}$ .

$3 x$

$x^{3}$

$0 x^{2}$

$0 x$

$8$

$3 x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$1$

Step 5.6

Multiply the new quotient term by the divisor.

$3 x$

$x^{3}$

$0 x^{2}$

$0 x$

$8$

$3 x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$1$

$3 x^{4}$

$0$

$24 x$

Step 5.7

The expression needs to be subtracted from the dividend, so change all the signs in $- 3 x^{4} + 0 + 0 + 24 x$

							-	$3 x$
$x^{3}$	+	$0 x^{2}$	+	$0 x$	-	$8$	-	$3 x^{4}$	+	$0 x^{3}$	+	$0 x^{2}$	+	$2 x$	-	$1$
							+	$3 x^{4}$	-	$0$	-	$0$	-	$24 x$

Step 5.8

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

							-	$3 x$
$x^{3}$	+	$0 x^{2}$	+	$0 x$	-	$8$	-	$3 x^{4}$	+	$0 x^{3}$	+	$0 x^{2}$	+	$2 x$	-	$1$
							+	$3 x^{4}$	-	$0$	-	$0$	-	$24 x$
													-	$22 x$

Step 5.9

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

							-	$3 x$
$x^{3}$	+	$0 x^{2}$	+	$0 x$	-	$8$	-	$3 x^{4}$	+	$0 x^{3}$	+	$0 x^{2}$	+	$2 x$	-	$1$
							+	$3 x^{4}$	-	$0$	-	$0$	-	$24 x$
													-	$22 x$	-	$1$

Step 5.10

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

$- 3 x + \frac{- 22 x - 1}{x^{3} - 8}$

Step 5.11

Split the solution into the polynomial portion and the remainder.

$- 3 x + \frac{22 x + 1}{- x^{3} + 8}$

Step 5.12

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

$y = - 3 x$

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = 2$

No Horizontal Asymptotes

Oblique Asymptotes: $y = - 3 x$

Step 7