Precalculus Examples

$f (x) = \frac{3 x^{2} - 2 x - 5}{2 x + 7}$

Step 1

Find where the expression $\frac{3 x^{2} - 2 x - 5}{2 x + 7}$ is undefined.

$x = - \frac{7}{2}$

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

Factor by grouping.

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Step 5.1.1

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 $a \cdot c = 3 \cdot - 5 = - 15$ and whose sum is $b = - 2$ .

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Step 5.1.1.1

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

$\frac{3 x^{2} - 2 (x) - 5}{2 x + 7}$

Step 5.1.1.2

Rewrite $- 2$ as $3$ plus $- 5$

$\frac{3 x^{2} + (3 - 5) x - 5}{2 x + 7}$

Step 5.1.1.3

Apply the distributive property.

$\frac{3 x^{2} + 3 x - 5 x - 5}{2 x + 7}$

Step 5.1.2

Factor out the greatest common factor from each group.

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Step 5.1.2.1

Group the first two terms and the last two terms.

$\frac{(3 x^{2} + 3 x) - 5 x - 5}{2 x + 7}$

Step 5.1.2.2

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

$\frac{3 x (x + 1) - 5 (x + 1)}{2 x + 7}$

Step 5.1.3

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

$\frac{(x + 1) (3 x - 5)}{2 x + 7}$

Step 5.2

Expand $(x + 1) (3 x - 5)$ .

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Step 5.2.1

Apply the distributive property.

$\frac{x (3 x - 5) + 1 (3 x - 5)}{2 x + 7}$

Step 5.2.2

Apply the distributive property.

$\frac{x (3 x) + x \cdot - 5 + 1 (3 x - 5)}{2 x + 7}$

Step 5.2.3

Apply the distributive property.

$\frac{x (3 x) + x \cdot - 5 + 1 (3 x) + 1 \cdot - 5}{2 x + 7}$

Step 5.2.4

Remove parentheses.

$\frac{x \cdot (3 x) + x \cdot - 5 + 1 (3 x) + 1 \cdot - 5}{2 x + 7}$

Step 5.2.5

Reorder $x$ and $3$ .

$\frac{3 \cdot (x \cdot x) + x \cdot - 5 + 1 (3 x) + 1 \cdot - 5}{2 x + 7}$

Step 5.2.6

Reorder $x$ and $- 5$ .

$\frac{3 \cdot (x \cdot x) - 5 \cdot x + 1 (3 x) + 1 \cdot - 5}{2 x + 7}$

Step 5.2.7

Remove parentheses.

$\frac{3 \cdot (x \cdot x) - 5 \cdot x + 1 \cdot (3 x) + 1 \cdot - 5}{2 x + 7}$

Step 5.2.8

Multiply $3$ by $x$ .

$\frac{3 x \cdot x - 5 x + 1 \cdot (3 x) + 1 \cdot - 5}{2 x + 7}$

Step 5.2.9

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

$\frac{3 (x \cdot x) - 5 x + 1 \cdot (3 x) + 1 \cdot - 5}{2 x + 7}$

Step 5.2.10

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

$\frac{3 (x \cdot x) - 5 x + 1 \cdot (3 x) + 1 \cdot - 5}{2 x + 7}$

Step 5.2.11

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

$\frac{3 x^{1 + 1} - 5 x + 1 \cdot (3 x) + 1 \cdot - 5}{2 x + 7}$

Step 5.2.12

Add $1$ and $1$ .

$\frac{3 x^{2} - 5 x + 1 \cdot (3 x) + 1 \cdot - 5}{2 x + 7}$

Step 5.2.13

Multiply $3$ by $1$ .

$\frac{3 x^{2} - 5 x + 3 x + 1 \cdot - 5}{2 x + 7}$

Step 5.2.14

Multiply $- 5$ by $1$ .

$\frac{3 x^{2} - 5 x + 3 x - 5}{2 x + 7}$

Step 5.2.15

Add $- 5 x$ and $3 x$ .

$\frac{3 x^{2} - 2 x - 5}{2 x + 7}$

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 $0$ .

$2 x$

$7$

$3 x^{2}$

$2 x$

$5$

Step 5.4

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

					$\frac{3 x}{2}$
	$2 x$	+	$7$		$3 x^{2}$	-	$2 x$	-	$5$

Step 5.5

Multiply the new quotient term by the divisor.

				$\frac{3 x}{2}$
$2 x$	+	$7$		$3 x^{2}$	-	$2 x$	-	$5$
			+	$3 x^{2}$	+	$\frac{21 x}{2}$

Step 5.6

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{2} + \frac{21 x}{2}$

				$\frac{3 x}{2}$
$2 x$	+	$7$		$3 x^{2}$	-	$2 x$	-	$5$
			-	$3 x^{2}$	-	$\frac{21 x}{2}$

Step 5.7

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

				$\frac{3 x}{2}$
$2 x$	+	$7$		$3 x^{2}$	-	$2 x$	-	$5$
			-	$3 x^{2}$	-	$\frac{21 x}{2}$
					-	$\frac{25 x}{2}$

Step 5.8

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

				$\frac{3 x}{2}$
$2 x$	+	$7$		$3 x^{2}$	-	$2 x$	-	$5$
			-	$3 x^{2}$	-	$\frac{21 x}{2}$
					-	$\frac{25 x}{2}$	-	$5$

Step 5.9

Divide the highest order term in the dividend $- \frac{25 x}{2}$ by the highest order term in divisor $2 x$ .

				$\frac{3 x}{2}$	-	$\frac{25}{4}$
$2 x$	+	$7$		$3 x^{2}$	-	$2 x$	-	$5$
			-	$3 x^{2}$	-	$\frac{21 x}{2}$
					-	$\frac{25 x}{2}$	-	$5$

Step 5.10

Multiply the new quotient term by the divisor.

				$\frac{3 x}{2}$	-	$\frac{25}{4}$
$2 x$	+	$7$		$3 x^{2}$	-	$2 x$	-	$5$
			-	$3 x^{2}$	-	$\frac{21 x}{2}$
					-	$\frac{25 x}{2}$	-	$5$
					-	$\frac{25 x}{2}$	-	$\frac{175}{4}$

Step 5.11

The expression needs to be subtracted from the dividend, so change all the signs in $- \frac{25 x}{2} - \frac{175}{4}$

				$\frac{3 x}{2}$	-	$\frac{25}{4}$
$2 x$	+	$7$		$3 x^{2}$	-	$2 x$	-	$5$
			-	$3 x^{2}$	-	$\frac{21 x}{2}$
					-	$\frac{25 x}{2}$	-	$5$
					+	$\frac{25 x}{2}$	+	$\frac{175}{4}$

Step 5.12

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

				$\frac{3 x}{2}$	-	$\frac{25}{4}$
$2 x$	+	$7$		$3 x^{2}$	-	$2 x$	-	$5$
			-	$3 x^{2}$	-	$\frac{21 x}{2}$
					-	$\frac{25 x}{2}$	-	$5$
					+	$\frac{25 x}{2}$	+	$\frac{175}{4}$
							+	$\frac{155}{4}$

Step 5.13

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

$\frac{3 x}{2} - \frac{25}{4} + \frac{155}{4 (2 x + 7)}$

Step 5.14

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

$y = \frac{3 x}{2} - \frac{25}{4}$

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = - \frac{7}{2}$

No Horizontal Asymptotes

Oblique Asymptotes: $y = \frac{3 x}{2} - \frac{25}{4}$

Step 7