Precalculus Examples

$f (x) = \frac{x^{3} - 16 x}{- 4 x^{2} + 4 x + 24}$

Step 1

Find where the expression $\frac{x^{3} - 16 x}{- 4 x^{2} + 4 x + 24}$ is undefined.

$x = - 2, x = 3$

Step 2

Since $\frac{x^{3} - 16 x}{- 4 x^{2} + 4 x + 24}$ $\to$ $- \infty$ as $x$ $\to$ $- 2$ from the left and $\frac{x^{3} - 16 x}{- 4 x^{2} + 4 x + 24}$ $\to$ $\infty$ as $x$ $\to$ $- 2$ from the right, then $x = - 2$ is a vertical asymptote.

$x = - 2$

Step 3

Since $\frac{x^{3} - 16 x}{- 4 x^{2} + 4 x + 24}$ $\to$ $- \infty$ as $x$ $\to$ $3$ from the left and $\frac{x^{3} - 16 x}{- 4 x^{2} + 4 x + 24}$ $\to$ $\infty$ as $x$ $\to$ $3$ from the right, then $x = 3$ is a vertical asymptote.

$x = 3$

Step 4

List all of the vertical asymptotes:

$x = - 2, 3$

Step 5

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 6

Find $n$ and $m$ .

$n = 3$

$m = 2$

Step 7

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

No Horizontal Asymptotes

Step 8

Find the oblique asymptote using polynomial division.

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Step 8.1

Simplify the expression.

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Step 8.1.1

Simplify the numerator.

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Step 8.1.1.1

Factor $x$ out of $x^{3} - 16 x$ .

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Step 8.1.1.1.1

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

$\frac{x \cdot x^{2} - 16 x}{- 4 x^{2} + 4 x + 24}$

Step 8.1.1.1.2

Factor $x$ out of $- 16 x$ .

$\frac{x \cdot x^{2} + x \cdot - 16}{- 4 x^{2} + 4 x + 24}$

Step 8.1.1.1.3

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

$\frac{x (x^{2} - 16)}{- 4 x^{2} + 4 x + 24}$

Step 8.1.1.2

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

$\frac{x (x^{2} - 4^{2})}{- 4 x^{2} + 4 x + 24}$

Step 8.1.1.3

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 $b = 4$ .

$\frac{x (x + 4) (x - 4)}{- 4 x^{2} + 4 x + 24}$

Step 8.1.2

Simplify the denominator.

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Step 8.1.2.1

Factor $4$ out of $- 4 x^{2} + 4 x + 24$ .

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Step 8.1.2.1.1

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

$\frac{x (x + 4) (x - 4)}{4 (- x^{2}) + 4 x + 24}$

Step 8.1.2.1.2

Factor $4$ out of $4 x$ .

$\frac{x (x + 4) (x - 4)}{4 (- x^{2}) + 4 (x) + 24}$

Step 8.1.2.1.3

Factor $4$ out of $24$ .

$\frac{x (x + 4) (x - 4)}{4 (- x^{2}) + 4 x + 4 \cdot 6}$

Step 8.1.2.1.4

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

$\frac{x (x + 4) (x - 4)}{4 (- x^{2} + x) + 4 \cdot 6}$

Step 8.1.2.1.5

Factor $4$ out of $4 (- x^{2} + x) + 4 \cdot 6$ .

$\frac{x (x + 4) (x - 4)}{4 (- x^{2} + x + 6)}$

Step 8.1.2.2

Factor by grouping.

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Step 8.1.2.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 6 = - 6$ and whose sum is $b = 1$ .

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Step 8.1.2.2.1.1

Multiply by $1$ .

$\frac{x (x + 4) (x - 4)}{4 (- x^{2} + 1 x + 6)}$

Step 8.1.2.2.1.2

Rewrite $1$ as $- 2$ plus $3$

$\frac{x (x + 4) (x - 4)}{4 (- x^{2} + (- 2 + 3) x + 6)}$

Step 8.1.2.2.1.3

Apply the distributive property.

$\frac{x (x + 4) (x - 4)}{4 (- x^{2} - 2 x + 3 x + 6)}$

Step 8.1.2.2.2

Factor out the greatest common factor from each group.

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Step 8.1.2.2.2.1

Group the first two terms and the last two terms.

$\frac{x (x + 4) (x - 4)}{4 ((- x^{2} - 2 x) + 3 x + 6)}$

Step 8.1.2.2.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 8.1.2.2.3

Factor the polynomial by factoring out the greatest common factor, $- x - 2$ .

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

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

Step 8.1.3

Simplify with factoring out.

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Step 8.1.3.1

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

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

Step 8.1.3.2

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

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

Step 8.1.3.3

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

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

Step 8.1.3.4

Rewrite negatives.

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Step 8.1.3.4.1

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

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

Step 8.1.3.4.2

Move the negative in front of the fraction.

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

Step 8.1.3.4.3

Reorder factors in $- \frac{x (x + 4) (x - 4)}{(4 (x + 2)) (x - 3)}$ .

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

Step 8.2

Expand $- x (x + 4) (x - 4)$ .

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Step 8.2.1

Negate $x$ .

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

Step 8.2.2

Apply the distributive property.

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

Step 8.2.3

Apply the distributive property.

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

Step 8.2.4

Apply the distributive property.

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

Step 8.2.5

Apply the distributive property.

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

Step 8.2.6

Move $x$ .

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

Step 8.2.7

Move $x$ .

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

Step 8.2.8

Move $x$ .

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

Step 8.2.9

Move $x$ .

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

Step 8.2.10

Move $x$ .

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

Step 8.2.11

Factor out negative.

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

Step 8.2.12

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

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

Step 8.2.13

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

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

Step 8.2.14

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

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

Step 8.2.15

Add $1$ and $1$ .

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

Step 8.2.16

Factor out negative.

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

Step 8.2.17

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

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

Step 8.2.18

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

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

Step 8.2.19

Add $2$ and $1$ .

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

Step 8.2.20

Multiply $- 1$ by $- 4$ .

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

Step 8.2.21

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

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

Step 8.2.22

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

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

Step 8.2.23

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

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

Step 8.2.24

Add $1$ and $1$ .

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

Step 8.2.25

Multiply $- 1$ by $4$ .

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

Step 8.2.26

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

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

Step 8.2.27

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

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

Step 8.2.28

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

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

Step 8.2.29

Add $1$ and $1$ .

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

Step 8.2.30

Multiply $- 1$ by $4$ .

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

Step 8.2.31

Multiply $- 4$ by $- 4$ .

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

Step 8.2.32

Subtract $4 x^{2}$ from $4 x^{2}$ .

$\frac{- x^{3} + 0 + 16 x}{4 (x + 2) (x - 3)}$

Step 8.2.33

Add $0$ and $16 x$ .

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

Step 8.3

Expand $4 (x + 2) (x - 3)$ .

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Step 8.3.1

Apply the distributive property.

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

Step 8.3.2

Apply the distributive property.

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

Step 8.3.3

Apply the distributive property.

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

Step 8.3.4

Apply the distributive property.

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

Step 8.3.5

Move $x$ .

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

Step 8.3.6

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

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

Step 8.3.7

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

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

Step 8.3.8

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

$\frac{- x^{3} + 16 x}{4 x^{1 + 1} + 4 \cdot (- 3 x) + 4 \cdot (2 x) + 4 \cdot 2 \cdot - 3}$

Step 8.3.9

Add $1$ and $1$ .

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

Step 8.3.10

Multiply $4$ by $- 3$ .

$\frac{- x^{3} + 16 x}{4 x^{2} - 12 x + 4 \cdot (2 x) + 4 \cdot 2 \cdot - 3}$

Step 8.3.11

Multiply $4$ by $2$ .

$\frac{- x^{3} + 16 x}{4 x^{2} - 12 x + 8 x + 4 \cdot 2 \cdot - 3}$

Step 8.3.12

Multiply $4$ by $2$ .

$\frac{- x^{3} + 16 x}{4 x^{2} - 12 x + 8 x + 8 \cdot - 3}$

Step 8.3.13

Multiply $8$ by $- 3$ .

$\frac{- x^{3} + 16 x}{4 x^{2} - 12 x + 8 x - 24}$

Step 8.3.14

Add $- 12 x$ and $8 x$ .

$\frac{- x^{3} + 16 x}{4 x^{2} - 4 x - 24}$

Step 8.4

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

$4 x^{2}$

$4 x$

$24$

$x^{3}$

$0 x^{2}$

$16 x$

$0$

Step 8.5

Divide the highest order term in the dividend $- x^{3}$ by the highest order term in divisor $4 x^{2}$ .

						-	$\frac{x}{4}$
	$4 x^{2}$	-	$4 x$	-	$24$	-	$x^{3}$	+	$0 x^{2}$	+	$16 x$	+	$0$

Step 8.6

Multiply the new quotient term by the divisor.

$\frac{x}{4}$

$4 x^{2}$

$4 x$

$24$

$x^{3}$

$0 x^{2}$

$16 x$

$0$

$x^{3}$

$x^{2}$

$6 x$

Step 8.7

The expression needs to be subtracted from the dividend, so change all the signs in $- x^{3} + x^{2} + 6 x$

$\frac{x}{4}$

$4 x^{2}$

$4 x$

$24$

$x^{3}$

$0 x^{2}$

$16 x$

$0$

$x^{3}$

$x^{2}$

$6 x$

Step 8.8

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

$\frac{x}{4}$

$4 x^{2}$

$4 x$

$24$

$x^{3}$

$0 x^{2}$

$16 x$

$0$

$x^{3}$

$x^{2}$

$6 x$

$x^{2}$

$10 x$

Step 8.9

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

$\frac{x}{4}$

$4 x^{2}$

$4 x$

$24$

$x^{3}$

$0 x^{2}$

$16 x$

$0$

$x^{3}$

$x^{2}$

$6 x$

$x^{2}$

$10 x$

$0$

Step 8.10

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

$\frac{x}{4}$

$\frac{1}{4}$

$4 x^{2}$

$4 x$

$24$

$x^{3}$

$0 x^{2}$

$16 x$

$0$

$x^{3}$

$x^{2}$

$6 x$

$x^{2}$

$10 x$

$0$

Step 8.11

Multiply the new quotient term by the divisor.

$\frac{x}{4}$

$\frac{1}{4}$

$4 x^{2}$

$4 x$

$24$

$x^{3}$

$0 x^{2}$

$16 x$

$0$

$x^{3}$

$x^{2}$

$6 x$

$x^{2}$

$10 x$

$0$

$x^{2}$

$x$

$6$

Step 8.12

The expression needs to be subtracted from the dividend, so change all the signs in $- x^{2} + x + 6$

					-	$\frac{x}{4}$	-	$\frac{1}{4}$
$4 x^{2}$	-	$4 x$	-	$24$	-	$x^{3}$	+	$0 x^{2}$	+	$16 x$	+	$0$
					+	$x^{3}$	-	$x^{2}$	-	$6 x$
							-	$x^{2}$	+	$10 x$	+	$0$
							+	$x^{2}$	-	$x$	-	$6$

Step 8.13

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

					-	$\frac{x}{4}$	-	$\frac{1}{4}$
$4 x^{2}$	-	$4 x$	-	$24$	-	$x^{3}$	+	$0 x^{2}$	+	$16 x$	+	$0$
					+	$x^{3}$	-	$x^{2}$	-	$6 x$
							-	$x^{2}$	+	$10 x$	+	$0$
							+	$x^{2}$	-	$x$	-	$6$
									+	$9 x$	-	$6$

Step 8.14

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

$- \frac{x}{4} - \frac{1}{4} + \frac{9 x - 6}{4 x^{2} - 4 x - 24}$

Step 8.15

Split the solution into the polynomial portion and the remainder.

$- \frac{x}{4} - \frac{1}{4} + \frac{- 9 x + 6}{- 4 x^{2} + 4 x + 24}$

Step 8.16

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

$y = - \frac{x}{4} - \frac{1}{4}$

Step 9

This is the set of all asymptotes.

Vertical Asymptotes: $x = - 2, 3$

No Horizontal Asymptotes

Oblique Asymptotes: $y = - \frac{x}{4} - \frac{1}{4}$

Step 10