Precalculus Examples

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

Step 1

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

$x = 2$

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.

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Step 5.1

Simplify the expression.

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Step 5.1.1

Factor by grouping.

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Step 5.1.1.1

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 $a \cdot c = - 1 \cdot 3 = - 3$ and whose sum is $b = - 2$ .

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Step 5.1.1.1.1

Factor $- 2$ out of $- 2 x$ .

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

Step 5.1.1.1.2

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

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

Step 5.1.1.1.3

Apply the distributive property.

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

Step 5.1.1.1.4

Multiply $x$ by $1$ .

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

Step 5.1.1.2

Factor out the greatest common factor from each group.

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Step 5.1.1.2.1

Group the first two terms and the last two terms.

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

Step 5.1.1.2.2

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

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

Step 5.1.1.3

Factor the polynomial by factoring out the greatest common factor, $- x + 1$ .

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.1.4

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

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

Step 5.1.5

Simplify the expression.

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Step 5.1.5.1

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

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

Step 5.1.5.2

Move the negative in front of the fraction.

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

Step 5.2

Expand $- (x - 1) (x + 3)$ .

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Step 5.2.1

Negate $x - 1$ .

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

Step 5.2.2

Apply the distributive property.

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

Step 5.2.3

Apply the distributive property.

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

Step 5.2.4

Apply the distributive property.

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

Step 5.2.5

Apply the distributive property.

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

Step 5.2.6

Move $x$ .

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

Step 5.2.7

Factor out negative.

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

Step 5.2.8

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

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

Step 5.2.9

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

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

Step 5.2.10

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

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

Step 5.2.11

Add $1$ and $1$ .

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

Step 5.2.12

Multiply $- 1$ by $3$ .

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

Step 5.2.13

Multiply $- 1$ by $- 1$ .

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

Step 5.2.14

Multiply $x$ by $1$ .

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

Step 5.2.15

Multiply $- 1$ by $- 1$ .

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

Step 5.2.16

Multiply $3$ by $1$ .

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

Step 5.2.17

Add $- 3 x$ and $x$ .

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

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 $0$ .

$x$

$2$

$x^{2}$

$2 x$

$3$

Step 5.4

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

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

Step 5.5

Multiply the new quotient term by the divisor.

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

Step 5.6

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

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

Step 5.7

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

			-	$x$
$x$	-	$2$	-	$x^{2}$	-	$2 x$	+	$3$
			+	$x^{2}$	-	$2 x$
					-	$4 x$

Step 5.8

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

			-	$x$
$x$	-	$2$	-	$x^{2}$	-	$2 x$	+	$3$
			+	$x^{2}$	-	$2 x$
					-	$4 x$	+	$3$

Step 5.9

Divide the highest order term in the dividend $- 4 x$ by the highest order term in divisor $x$ .

			-	$x$	-	$4$
$x$	-	$2$	-	$x^{2}$	-	$2 x$	+	$3$
			+	$x^{2}$	-	$2 x$
					-	$4 x$	+	$3$

Step 5.10

Multiply the new quotient term by the divisor.

			-	$x$	-	$4$
$x$	-	$2$	-	$x^{2}$	-	$2 x$	+	$3$
			+	$x^{2}$	-	$2 x$
					-	$4 x$	+	$3$
					-	$4 x$	+	$8$

Step 5.11

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x + 8$

			-	$x$	-	$4$
$x$	-	$2$	-	$x^{2}$	-	$2 x$	+	$3$
			+	$x^{2}$	-	$2 x$
					-	$4 x$	+	$3$
					+	$4 x$	-	$8$

Step 5.12

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

			-	$x$	-	$4$
$x$	-	$2$	-	$x^{2}$	-	$2 x$	+	$3$
			+	$x^{2}$	-	$2 x$
					-	$4 x$	+	$3$
					+	$4 x$	-	$8$
							-	$5$

Step 5.13

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

$- x - 4 - \frac{5}{x - 2}$

Step 5.14

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

$y = - x - 4$

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = 2$

No Horizontal Asymptotes

Oblique Asymptotes: $y = - x - 4$

Step 7