Find the Asymptotes (-x^2-3x+3)/(x+1)

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

Step 1

Find where the expression $\frac{- x^{2} - 3 x + 3}{x + 1}$ is undefined.

$x = - 1$

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 $m$ .

$n = 2$

$m = 1$

Step 4

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

No Horizontal Asymptotes

Step 5

Find the oblique asymptote using polynomial division.

Simplify the expression.

Factor $- 1$ out of $- x^{2}$ .

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

Factor $- 1$ out of $- 3 x$ .

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

Factor $- 1$ out of $- (x^{2}) - (3 x)$ .

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

Rewrite $3$ as $- 1 (- 3)$ .

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

Factor $- 1$ out of $- (x^{2} + 3 x) - 1 (- 3)$ .

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

Simplify the expression.

Rewrite $- (x^{2} + 3 x - 3)$ as $- 1 (x^{2} + 3 x - 3)$ .

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

Move the negative in front of the fraction.

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

Expand $- (x^{2} + 3 x - 3)$ .

Negate $x^{2} + 3 x - 3$ .

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

Apply the distributive property.

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

Apply the distributive property.

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

Remove parentheses.

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

Multiply $- 1$ by $3$ .

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

Multiply $- 1$ by $- 3$ .

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

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

$x$

$1$

$x^{2}$

$3 x$

$3$

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

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

Multiply the new quotient term by the divisor.

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

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

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

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

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

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

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

Divide the highest order term in the dividend $- 2 x$ by the highest order term in divisor $x$ .

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

Multiply the new quotient term by the divisor.

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

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x - 2$

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

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

			-	$x$	-	$2$
$x$	+	$1$	-	$x^{2}$	-	$3 x$	+	$3$
			+	$x^{2}$	+	$x$
					-	$2 x$	+	$3$
					+	$2 x$	+	$2$
							+	$5$

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

$- x - 2 + \frac{5}{x + 1}$

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

$y = - x - 2$

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = - 1$

No Horizontal Asymptotes

Oblique Asymptotes: $y = - x - 2$

Step 7

Find the Asymptotes (-x^2-3x+3)/(x+1)

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