Find the Asymptotes f(x)=(x^3-x^2-6x)/(-3x^2-3x+18)

$f (x) = \frac{x^{3} - x^{2} - 6 x}{- 3 x^{2} - 3 x + 18}$

Step 1

Find where the expression $\frac{x^{3} - x^{2} - 6 x}{- 3 x^{2} - 3 x + 18}$ is undefined.

$x = - 3, x = 2$

Step 2

Since $\frac{x^{3} - x^{2} - 6 x}{- 3 x^{2} - 3 x + 18}$ $\to$ $\infty$ as $x$ $\to$ $- 3$ from the left and $\frac{x^{3} - x^{2} - 6 x}{- 3 x^{2} - 3 x + 18}$ $\to$ $- \infty$ as $x$ $\to$ $- 3$ from the right, then $x = - 3$ is a vertical asymptote.

$x = - 3$

Step 3

Since $\frac{x^{3} - x^{2} - 6 x}{- 3 x^{2} - 3 x + 18}$ $\to$ $- \infty$ as $x$ $\to$ $2$ from the left and $\frac{x^{3} - x^{2} - 6 x}{- 3 x^{2} - 3 x + 18}$ $\to$ $\infty$ as $x$ $\to$ $2$ from the right, then $x = 2$ is a vertical asymptote.

$x = 2$

Step 4

List all of the vertical asymptotes:

$x = - 3, 2$

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 $x^{3} - x^{2} - 6 x$ .

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

$\frac{x \cdot x^{2} - x^{2} - 6 x}{- 3 x^{2} - 3 x + 18}$

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

$\frac{x \cdot x^{2} + x (- x) - 6 x}{- 3 x^{2} - 3 x + 18}$

Factor $x$ out of $- 6 x$ .

$\frac{x \cdot x^{2} + x (- x) + x \cdot - 6}{- 3 x^{2} - 3 x + 18}$

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

$\frac{x (x^{2} - x) + x \cdot - 6}{- 3 x^{2} - 3 x + 18}$

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

$\frac{x (x^{2} - x - 6)}{- 3 x^{2} - 3 x + 18}$

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.

$\frac{x ((x - 3) (x + 2))}{- 3 x^{2} - 3 x + 18}$

$\frac{x (x - 3) (x + 2)}{- 3 x^{2} - 3 x + 18}$

Simplify the denominator.

Factor $3$ out of $- 3 x^{2} - 3 x + 18$ .

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

$\frac{x (x - 3) (x + 2)}{3 (- x^{2}) - 3 x + 18}$

Factor $3$ out of $- 3 x$ .

$\frac{x (x - 3) (x + 2)}{3 (- x^{2}) + 3 (- x) + 18}$

Factor $3$ out of $18$ .

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

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

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

Factor $3$ out of $3 (- x^{2} - x) + 3 (6)$ .

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

Factor by grouping.

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 $b = - 1$ .

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

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

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

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

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

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

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

Simplify with factoring out.

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

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

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

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

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

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

Rewrite negatives.

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

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

Move the negative in front of the fraction.

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

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

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

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

Negate $x$ .

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Move $x$ .

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

Move $x$ .

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

Move $x$ .

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

Move $x$ .

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

Move $x$ .

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

Factor out negative.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Factor out negative.

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

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

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

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

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

Add $2$ and $1$ .

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

Multiply $- 1$ by $2$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Multiply $- 1$ by $- 3$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Multiply $- 1$ by $- 3$ .

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

Multiply $3$ by $2$ .

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

Add $- 2 x^{2}$ and $3 x^{2}$ .

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Move $x$ .

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

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

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

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

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

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

$\frac{- x^{3} + x^{2} + 6 x}{3 x^{1 + 1} + 3 \cdot (3 x) + 3 \cdot (- 2 x) + 3 \cdot - 2 \cdot 3}$

Add $1$ and $1$ .

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

Multiply $3$ by $3$ .

$\frac{- x^{3} + x^{2} + 6 x}{3 x^{2} + 9 x + 3 \cdot (- 2 x) + 3 \cdot - 2 \cdot 3}$

Multiply $3$ by $- 2$ .

$\frac{- x^{3} + x^{2} + 6 x}{3 x^{2} + 9 x - 6 x + 3 \cdot - 2 \cdot 3}$

Multiply $3$ by $- 2$ .

$\frac{- x^{3} + x^{2} + 6 x}{3 x^{2} + 9 x - 6 x - 6 \cdot 3}$

Multiply $- 6$ by $3$ .

$\frac{- x^{3} + x^{2} + 6 x}{3 x^{2} + 9 x - 6 x - 18}$

Subtract $6 x$ from $9 x$ .

$\frac{- x^{3} + x^{2} + 6 x}{3 x^{2} + 3 x - 18}$

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

$3 x^{2}$

$3 x$

$18$

$x^{3}$

$x^{2}$

$6 x$

$0$

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

$\frac{x}{3}$

$3 x^{2}$

$3 x$

$18$

$x^{3}$

$x^{2}$

$6 x$

$0$

Multiply the new quotient term by the divisor.

$\frac{x}{3}$

$3 x^{2}$

$3 x$

$18$

$x^{3}$

$x^{2}$

$6 x$

$0$

$x^{3}$

$x^{2}$

$6 x$

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

$\frac{x}{3}$

$3 x^{2}$

$3 x$

$18$

$x^{3}$

$x^{2}$

$6 x$

$0$

$x^{3}$

$x^{2}$

$6 x$

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

$\frac{x}{3}$

$3 x^{2}$

$3 x$

$18$

$x^{3}$

$x^{2}$

$6 x$

$0$

$x^{3}$

$x^{2}$

$6 x$

$2 x^{2}$

$0$

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

$\frac{x}{3}$

$3 x^{2}$

$3 x$

$18$

$x^{3}$

$x^{2}$

$6 x$

$0$

$x^{3}$

$x^{2}$

$6 x$

$2 x^{2}$

$0$

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

$\frac{x}{3}$

$\frac{2}{3}$

$3 x^{2}$

$3 x$

$18$

$x^{3}$

$x^{2}$

$6 x$

$0$

$x^{3}$

$x^{2}$

$6 x$

$2 x^{2}$

$0$

Multiply the new quotient term by the divisor.

$\frac{x}{3}$

$\frac{2}{3}$

$3 x^{2}$

$3 x$

$18$

$x^{3}$

$x^{2}$

$6 x$

$0$

$x^{3}$

$x^{2}$

$6 x$

$2 x^{2}$

$0$

$2 x^{2}$

$2 x$

$12$

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{2} + 2 x - 12$

$\frac{x}{3}$

$\frac{2}{3}$

$3 x^{2}$

$3 x$

$18$

$x^{3}$

$x^{2}$

$6 x$

$0$

$x^{3}$

$x^{2}$

$6 x$

$2 x^{2}$

$0$

$2 x^{2}$

$2 x$

$12$

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

$\frac{x}{3}$

$\frac{2}{3}$

$3 x^{2}$

$3 x$

$18$

$x^{3}$

$x^{2}$

$6 x$

$0$

$x^{3}$

$x^{2}$

$6 x$

$2 x^{2}$

$0$

$2 x^{2}$

$2 x$

$12$

$2 x$

$12$

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

$- \frac{x}{3} + \frac{2}{3} + \frac{- 2 x + 12}{3 x^{2} + 3 x - 18}$

Split the solution into the polynomial portion and the remainder.

$- \frac{x}{3} + \frac{2}{3} + \frac{2 x - 12}{- 3 x^{2} - 3 x + 18}$

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

$y = - \frac{x}{3} + \frac{2}{3}$

Step 9

This is the set of all asymptotes.

Vertical Asymptotes: $x = - 3, 2$

No Horizontal Asymptotes

Oblique Asymptotes: $y = - \frac{x}{3} + \frac{2}{3}$

Step 10

Find the Asymptotes f(x)=(x^3-x^2-6x)/(-3x^2-3x+18)

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