Graph f(x)=(x^3-16x)/(-4x^2+4x+24)

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

Step 1

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

$x = - 2, x = 3$

Step 2

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

$x = - 2$

Step 3

Since $- \frac{x (x + 4) (x - 4)}{4 (x + 2) (x - 3)}$ $\to$ $- \infty$ as $x$ $\to$ $3$ from the left and $- \frac{x (x + 4) (x - 4)}{4 (x + 2) (x - 3)}$ $\to$ $\infty$ 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.

Simplify the expression.

Simplify the numerator.

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

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

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

Factor $x$ out of $16 x$ .

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

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

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

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

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

Reorder $- x^{2}$ and $4^{2}$ .

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4$ and $b = x$ .

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

Simplify the denominator.

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

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

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

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

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

Factor $4$ out of $- 24$ .

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

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

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

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

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

Factor $x^{2} - x - 6$ using the AC method.

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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Write the factored form using these integers.

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Reorder $x$ and $4$ .

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

Move $x$ .

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

Reorder $x$ and $4$ .

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

Remove parentheses.

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

Move $x$ .

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

Move $x$ .

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

Reorder $x$ and $4$ .

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

Remove parentheses.

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

Move $x$ .

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

Reorder $x$ and $- 1$ .

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

Multiply $4$ by $x$ .

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

Multiply $- 1$ by $x$ .

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

Multiply $4$ by $4$ .

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

Multiply $4$ by $- 1$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Factor out negative.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Factor out negative.

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

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

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

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

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

Add $2$ and $1$ .

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

Add $- 4 x^{2}$ and $4 x^{2}$ .

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

Subtract $x^{3}$ from $0$ .

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

Reorder $16 x$ and $- x^{3}$ .

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Move $x$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Multiply $4$ by $2$ .

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

Multiply $4$ by $- 3$ .

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

Multiply $4$ by $- 3$ .

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

Multiply $- 12$ by $2$ .

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

Subtract $12 x$ from $8 x$ .

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

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

$4 x^{2}$

$4 x$

$24$

$x^{3}$

$0 x^{2}$

$16 x$

$0$

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

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

Multiply the new quotient term by the divisor.

$\frac{x}{4}$

$4 x^{2}$

$4 x$

$24$

$x^{3}$

$0 x^{2}$

$16 x$

$0$

$x^{3}$

$x^{2}$

$6 x$

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

$\frac{x}{4}$

$4 x^{2}$

$4 x$

$24$

$x^{3}$

$0 x^{2}$

$16 x$

$0$

$x^{3}$

$x^{2}$

$6 x$

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

$\frac{x}{4}$

$4 x^{2}$

$4 x$

$24$

$x^{3}$

$0 x^{2}$

$16 x$

$0$

$x^{3}$

$x^{2}$

$6 x$

$x^{2}$

$10 x$

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

$\frac{x}{4}$

$4 x^{2}$

$4 x$

$24$

$x^{3}$

$0 x^{2}$

$16 x$

$0$

$x^{3}$

$x^{2}$

$6 x$

$x^{2}$

$10 x$

$0$

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

$\frac{x}{4}$

$\frac{1}{4}$

$4 x^{2}$

$4 x$

$24$

$x^{3}$

$0 x^{2}$

$16 x$

$0$

$x^{3}$

$x^{2}$

$6 x$

$x^{2}$

$10 x$

$0$

Multiply the new quotient term by the divisor.

$\frac{x}{4}$

$\frac{1}{4}$

$4 x^{2}$

$4 x$

$24$

$x^{3}$

$0 x^{2}$

$16 x$

$0$

$x^{3}$

$x^{2}$

$6 x$

$x^{2}$

$10 x$

$0$

$x^{2}$

$x$

$6$

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

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

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

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

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

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

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

Graph f(x)=(x^3-16x)/(-4x^2+4x+24)

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