Precalculus Examples

$f (x) = \frac{x^{3} - 5 x^{2} + 6 x}{- 3 x^{2} + 6 x + 9}$

Step 1

Find where the expression $\frac{x^{3} - 5 x^{2} + 6 x}{- 3 x^{2} + 6 x + 9}$ is undefined.

$x = - 1, x = 3$

Step 2

Since $\frac{x^{3} - 5 x^{2} + 6 x}{- 3 x^{2} + 6 x + 9}$ $\to$ $\infty$ as $x$ $\to$ $- 1$ from the left and $\frac{x^{3} - 5 x^{2} + 6 x}{- 3 x^{2} + 6 x + 9}$ $\to$ $- \infty$ as $x$ $\to$ $- 1$ from the right, then $x = - 1$ is a vertical asymptote.

$x = - 1$

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.

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Step 6.1

Simplify the expression.

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Step 6.1.1

Simplify the numerator.

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Step 6.1.1.1

Factor $x$ out of $x^{3} - 5 x^{2} + 6 x$ .

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Step 6.1.1.1.1

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

$\frac{x \cdot x^{2} - 5 x^{2} + 6 x}{- 3 x^{2} + 6 x + 9}$

Step 6.1.1.1.2

Factor $x$ out of $- 5 x^{2}$ .

$\frac{x \cdot x^{2} + x (- 5 x) + 6 x}{- 3 x^{2} + 6 x + 9}$

Step 6.1.1.1.3

Factor $x$ out of $6 x$ .

$\frac{x \cdot x^{2} + x (- 5 x) + x \cdot 6}{- 3 x^{2} + 6 x + 9}$

Step 6.1.1.1.4

Factor $x$ out of $x \cdot x^{2} + x (- 5 x)$ .

$\frac{x (x^{2} - 5 x) + x \cdot 6}{- 3 x^{2} + 6 x + 9}$

Step 6.1.1.1.5

Factor $x$ out of $x (x^{2} - 5 x) + x \cdot 6$ .

$\frac{x (x^{2} - 5 x + 6)}{- 3 x^{2} + 6 x + 9}$

Step 6.1.1.2

Factor $x^{2} - 5 x + 6$ using the AC method.

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Step 6.1.1.2.1

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 $- 5$ .

$- 3, - 2$

Step 6.1.1.2.2

Write the factored form using these integers.

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

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

Step 6.1.2

Simplify the denominator.

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Step 6.1.2.1

Factor $3$ out of $- 3 x^{2} + 6 x + 9$ .

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Step 6.1.2.1.1

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

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

Step 6.1.2.1.2

Factor $3$ out of $6 x$ .

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

Step 6.1.2.1.3

Factor $3$ out of $9$ .

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

Step 6.1.2.1.4

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

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

Step 6.1.2.1.5

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

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

Step 6.1.2.2

Factor by grouping.

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Step 6.1.2.2.1

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 6.1.2.2.1.1

Factor $2$ out of $2 x$ .

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

Step 6.1.2.2.1.2

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

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

Step 6.1.2.2.1.3

Apply the distributive property.

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

Step 6.1.2.2.2

Factor out the greatest common factor from each group.

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Step 6.1.2.2.2.1

Group the first two terms and the last two terms.

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

Step 6.1.2.2.2.2

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

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

Step 6.1.2.2.3

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

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

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

Step 6.1.3

Simplify terms.

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Step 6.1.3.1

Cancel the common factor of $x - 3$ .

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Step 6.1.3.1.1

Cancel the common factor.

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

Step 6.1.3.1.2

Rewrite the expression.

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

Step 6.1.3.2

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

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

Step 6.1.3.3

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

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

Step 6.1.3.4

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

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

Step 6.1.3.5

Rewrite negatives.

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Step 6.1.3.5.1

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

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

Step 6.1.3.5.2

Move the negative in front of the fraction.

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

Step 6.2

Expand $- x (x - 2)$ .

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Step 6.2.1

Negate $x$ .

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

Step 6.2.2

Apply the distributive property.

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

Step 6.2.3

Move $x$ .

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

Step 6.2.4

Factor out negative.

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

Step 6.2.5

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

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

Step 6.2.6

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

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

Step 6.2.7

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

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

Step 6.2.8

Add $1$ and $1$ .

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

Step 6.2.9

Multiply $- 1$ by $- 2$ .

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

Step 6.3

Expand $3 (x + 1)$ .

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Step 6.3.1

Apply the distributive property.

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

Step 6.3.2

Multiply $3$ by $1$ .

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

Step 6.4

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

$3 x$

$3$

$x^{2}$

$2 x$

$0$

Step 6.5

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

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

Step 6.6

Multiply the new quotient term by the divisor.

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

Step 6.7

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

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

Step 6.8

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

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

Step 6.9

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

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

Step 6.10

Divide the highest order term in the dividend $3 x$ by the highest order term in divisor $3 x$ .

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

Step 6.11

Multiply the new quotient term by the divisor.

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

Step 6.12

The expression needs to be subtracted from the dividend, so change all the signs in $3 x + 3$

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

Step 6.13

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

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

Step 6.14

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

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

Step 6.15

Split the solution into the polynomial portion and the remainder.

$- \frac{x}{3} + 1 + \frac{3 x - 9}{- 3 x^{2} + 6 x + 9}$

Step 6.16

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

$y = - \frac{x}{3} + 1$

Step 7

This is the set of all asymptotes.

Vertical Asymptotes: $x = - 1$

No Horizontal Asymptotes

Oblique Asymptotes: $y = - \frac{x}{3} + 1$

Step 8