Algebra Examples

$p (x) = \frac{6 x - 12 x^{9}}{3 x^{3} + 7}$

Step 1

Find where the expression $\frac{6 x - 12 x^{9}}{3 x^{3} + 7}$ is undefined.

$x = - \frac{\sqrt[3]{63}}{3}$

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 $m$ .

$n = 9$

$m = 3$

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

Factor $6 x$ out of $6 x - 12 x^{9}$ .

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Step 5.1.1

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

$\frac{6 x (1) - 12 x^{9}}{3 x^{3} + 7}$

Step 5.1.2

Factor $6 x$ out of $- 12 x^{9}$ .

$\frac{6 x (1) + 6 x (- 2 x^{8})}{3 x^{3} + 7}$

Step 5.1.3

Factor $6 x$ out of $6 x (1) + 6 x (- 2 x^{8})$ .

$\frac{6 x (1 - 2 x^{8})}{3 x^{3} + 7}$

Step 5.2

Expand $6 x (1 - 2 x^{8})$ .

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Step 5.2.1

Apply the distributive property.

$\frac{6 x \cdot 1 + 6 x (- 2 x^{8})}{3 x^{3} + 7}$

Step 5.2.2

Move $x$ .

$\frac{6 \cdot (1 x) + 6 x (- 2 x^{8})}{3 x^{3} + 7}$

Step 5.2.3

Remove parentheses.

$\frac{6 \cdot (1 x) + 6 x \cdot (- 2 x^{8})}{3 x^{3} + 7}$

Step 5.2.4

Move $x$ .

$\frac{6 \cdot (1 x) + 6 \cdot (- 2 x \cdot x^{8})}{3 x^{3} + 7}$

Step 5.2.5

Multiply $6$ by $1$ .

$\frac{6 x + 6 \cdot (- 2 x \cdot x^{8})}{3 x^{3} + 7}$

Step 5.2.6

Multiply $6$ by $- 2$ .

$\frac{6 x - 12 x \cdot x^{8}}{3 x^{3} + 7}$

Step 5.2.7

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

$\frac{6 x - 12 (x \cdot x^{8})}{3 x^{3} + 7}$

Step 5.2.8

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

$\frac{6 x - 12 x^{1 + 8}}{3 x^{3} + 7}$

Step 5.2.9

Add $1$ and $8$ .

$\frac{6 x - 12 x^{9}}{3 x^{3} + 7}$

Step 5.2.10

Reorder $6 x$ and $- 12 x^{9}$ .

$\frac{- 12 x^{9} + 6 x}{3 x^{3} + 7}$

Step 5.3

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

$3 x^{3}$

$0 x^{2}$

$0 x$

$7$

$12 x^{9}$

$0 x^{8}$

$0 x^{7}$

$0 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$6 x$

$0$

Step 5.4

Divide the highest order term in the dividend $- 12 x^{9}$ by the highest order term in divisor $3 x^{3}$ .

$4 x^{6}$

$3 x^{3}$

$0 x^{2}$

$0 x$

$7$

$12 x^{9}$

$0 x^{8}$

$0 x^{7}$

$0 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$6 x$

$0$

Step 5.5

Multiply the new quotient term by the divisor.

$4 x^{6}$

$3 x^{3}$

$0 x^{2}$

$0 x$

$7$

$12 x^{9}$

$0 x^{8}$

$0 x^{7}$

$0 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$6 x$

$0$

$12 x^{9}$

$0$

$28 x^{6}$

Step 5.6

The expression needs to be subtracted from the dividend, so change all the signs in $- 12 x^{9} + 0 + 0 - 28 x^{6}$

$4 x^{6}$

$3 x^{3}$

$0 x^{2}$

$0 x$

$7$

$12 x^{9}$

$0 x^{8}$

$0 x^{7}$

$0 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$6 x$

$0$

$12 x^{9}$

$0$

$28 x^{6}$

Step 5.7

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

$4 x^{6}$

$3 x^{3}$

$0 x^{2}$

$0 x$

$7$

$12 x^{9}$

$0 x^{8}$

$0 x^{7}$

$0 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$6 x$

$0$

$12 x^{9}$

$0$

$28 x^{6}$

Step 5.8

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

$4 x^{6}$

$3 x^{3}$

$0 x^{2}$

$0 x$

$7$

$12 x^{9}$

$0 x^{8}$

$0 x^{7}$

$0 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$6 x$

$0$

$12 x^{9}$

$0$

$28 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

Step 5.9

Divide the highest order term in the dividend $28 x^{6}$ by the highest order term in divisor $3 x^{3}$ .

$4 x^{6}$

$0 x^{5}$

$0 x^{4}$

$\frac{28 x^{3}}{3}$

$3 x^{3}$

$0 x^{2}$

$0 x$

$7$

$12 x^{9}$

$0 x^{8}$

$0 x^{7}$

$0 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$6 x$

$0$

$12 x^{9}$

$0$

$28 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

Step 5.10

Multiply the new quotient term by the divisor.

$4 x^{6}$

$0 x^{5}$

$0 x^{4}$

$\frac{28 x^{3}}{3}$

$3 x^{3}$

$0 x^{2}$

$0 x$

$7$

$12 x^{9}$

$0 x^{8}$

$0 x^{7}$

$0 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$6 x$

$0$

$12 x^{9}$

$0$

$28 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$28 x^{6}$

$0$

$\frac{196 x^{3}}{3}$

Step 5.11

The expression needs to be subtracted from the dividend, so change all the signs in $28 x^{6} + 0 + 0 + \frac{196 x^{3}}{3}$

$4 x^{6}$

$0 x^{5}$

$0 x^{4}$

$\frac{28 x^{3}}{3}$

$3 x^{3}$

$0 x^{2}$

$0 x$

$7$

$12 x^{9}$

$0 x^{8}$

$0 x^{7}$

$0 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$6 x$

$0$

$12 x^{9}$

$0$

$28 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$28 x^{6}$

$0$

$\frac{196 x^{3}}{3}$

Step 5.12

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

$4 x^{6}$

$0 x^{5}$

$0 x^{4}$

$\frac{28 x^{3}}{3}$

$3 x^{3}$

$0 x^{2}$

$0 x$

$7$

$12 x^{9}$

$0 x^{8}$

$0 x^{7}$

$0 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$6 x$

$0$

$12 x^{9}$

$0$

$28 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$28 x^{6}$

$0$

$\frac{196 x^{3}}{3}$

Step 5.13

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

$4 x^{6}$

$0 x^{5}$

$0 x^{4}$

$\frac{28 x^{3}}{3}$

$3 x^{3}$

$0 x^{2}$

$0 x$

$7$

$12 x^{9}$

$0 x^{8}$

$0 x^{7}$

$0 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$6 x$

$0$

$12 x^{9}$

$0$

$28 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$28 x^{6}$

$0$

$\frac{196 x^{3}}{3}$

$0 x^{2}$

$6 x$

$0$

Step 5.14

Divide the highest order term in the dividend $- \frac{196 x^{3}}{3}$ by the highest order term in divisor $3 x^{3}$ .

$4 x^{6}$

$0 x^{5}$

$0 x^{4}$

$\frac{28 x^{3}}{3}$

$0 x^{2}$

$0 x$

$\frac{196}{9}$

$3 x^{3}$

$0 x^{2}$

$0 x$

$7$

$12 x^{9}$

$0 x^{8}$

$0 x^{7}$

$0 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$6 x$

$0$

$12 x^{9}$

$0$

$28 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$28 x^{6}$

$0$

$\frac{196 x^{3}}{3}$

$0 x^{2}$

$6 x$

$0$

Step 5.15

Multiply the new quotient term by the divisor.

$4 x^{6}$

$0 x^{5}$

$0 x^{4}$

$\frac{28 x^{3}}{3}$

$0 x^{2}$

$0 x$

$\frac{196}{9}$

$3 x^{3}$

$0 x^{2}$

$0 x$

$7$

$12 x^{9}$

$0 x^{8}$

$0 x^{7}$

$0 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$6 x$

$0$

$12 x^{9}$

$0$

$28 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$28 x^{6}$

$0$

$\frac{196 x^{3}}{3}$

$0 x^{2}$

$6 x$

$0$

$\frac{196 x^{3}}{3}$

$0$

$\frac{1372}{9}$

Step 5.16

The expression needs to be subtracted from the dividend, so change all the signs in $- \frac{196 x^{3}}{3} + 0 + 0 - \frac{1372}{9}$

$4 x^{6}$

$0 x^{5}$

$0 x^{4}$

$\frac{28 x^{3}}{3}$

$0 x^{2}$

$0 x$

$\frac{196}{9}$

$3 x^{3}$

$0 x^{2}$

$0 x$

$7$

$12 x^{9}$

$0 x^{8}$

$0 x^{7}$

$0 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$6 x$

$0$

$12 x^{9}$

$0$

$28 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$28 x^{6}$

$0$

$\frac{196 x^{3}}{3}$

$0 x^{2}$

$6 x$

$0$

$\frac{196 x^{3}}{3}$

$0$

$\frac{1372}{9}$

Step 5.17

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

$4 x^{6}$

$0 x^{5}$

$0 x^{4}$

$\frac{28 x^{3}}{3}$

$0 x^{2}$

$0 x$

$\frac{196}{9}$

$3 x^{3}$

$0 x^{2}$

$0 x$

$7$

$12 x^{9}$

$0 x^{8}$

$0 x^{7}$

$0 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$6 x$

$0$

$12 x^{9}$

$0$

$28 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$28 x^{6}$

$0$

$\frac{196 x^{3}}{3}$

$0 x^{2}$

$6 x$

$0$

$\frac{196 x^{3}}{3}$

$0$

$\frac{1372}{9}$

$6 x$

$\frac{1372}{9}$

Step 5.18

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

$- 4 x^{6} + \frac{28 x^{3}}{3} - \frac{196}{9} + \frac{6 x + \frac{1372}{9}}{3 x^{3} + 7}$

Step 5.19

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

$y = - 4 x^{6} + \frac{28 x^{3}}{3} - \frac{196}{9}$

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = - \frac{\sqrt[3]{63}}{3}$

No Horizontal Asymptotes

Oblique Asymptotes: $y = - 4 x^{6} + \frac{28 x^{3}}{3} - \frac{196}{9}$

Step 7