Graph (x^3+1)/(x^2+2)

$\frac{x^{3} + 1}{x^{2} + 2}$

Step 1

Find where the expression $\frac{(x + 1) (x^{2} - x + 1)}{x^{2} + 2}$ is undefined.

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 2

The vertical asymptotes occur at areas of infinite discontinuity.

No Vertical Asymptotes

Step 3

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 4

Find $n$ and $m$ .

$n = 3$

$m = 2$

Step 5

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

No Horizontal Asymptotes

Step 6

Find the oblique asymptote using polynomial division.

Simplify the numerator.

Rewrite $1$ as $1^{3}$ .

$\frac{x^{3} + 1^{3}}{x^{2} + 2}$

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 $b = 1$ .

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

Simplify.

Multiply $- 1$ by $1$ .

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

One to any power is one.

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Remove parentheses.

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

Reorder $x$ and $- 1$ .

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

Reorder $x$ and $1$ .

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

Remove parentheses.

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

Multiply $- 1$ by $x$ .

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

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

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

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

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

Add $1$ and $2$ .

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

Factor out negative.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Multiply $x$ by $1$ .

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

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

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

Multiply $- 1$ by $1$ .

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

Multiply $1$ by $1$ .

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

Move $x$ .

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

Move $- x^{2}$ .

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

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

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

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

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

Subtract $x$ from $x$ .

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

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

$\frac{x^{3} + 1}{x^{2} + 2}$

$\frac{x^{3} + 1}{x^{2} + 2}$

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

$x^{2}$

+

$0 x$

+

$2$

$x^{3}$

+

$0 x^{2}$

+

$0 x$

+

$1$

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

$x$

$x^{2}$

+

$0 x$

+

$2$

$x^{3}$

+

$0 x^{2}$

+

$0 x$

+

$1$

Multiply the new quotient term by the divisor.

$x$

$x^{2}$

+

$0 x$

+

$2$

$x^{3}$

+

$0 x^{2}$

+

$0 x$

+

$1$

+

$x^{3}$

+

$0$

+

$2 x$

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} + 0 + 2 x$

$x$

$x^{2}$

+

$0 x$

+

$2$

$x^{3}$

+

$0 x^{2}$

+

$0 x$

+

$1$

-

$x^{3}$

-

$0$

-

$2 x$

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

$x$

$x^{2}$

+

$0 x$

+

$2$

$x^{3}$

+

$0 x^{2}$

+

$0 x$

+

$1$

-

$x^{3}$

-

$0$

-

$2 x$

-

$2 x$

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

$x$

$x^{2}$

+

$0 x$

+

$2$

$x^{3}$

+

$0 x^{2}$

+

$0 x$

+

$1$

-

$x^{3}$

-

$0$

-

$2 x$

-

$2 x$

+

$1$

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

$x + \frac{- 2 x + 1}{x^{2} + 2}$

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

$y = x$

$y = x$

Step 7

This is the set of all asymptotes.

No Vertical Asymptotes

No Horizontal Asymptotes

Oblique Asymptotes: $y = x$

Step 8

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