Precalculus Examples

$f (x) = \frac{24 x^{2} - 14 x + 1}{4 x - 3}$

Step 1

Find where the expression $\frac{24 x^{2} - 14 x + 1}{4 x - 3}$ is undefined.

$x = \frac{3}{4}$

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 $a \cdot c = 24 \cdot 1 = 24$ and whose sum is $b = - 14$ .

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Step 5.1.1.1

Factor $- 14$ out of $- 14 x$ .

$\frac{24 x^{2} - 14 (x) + 1}{4 x - 3}$

Step 5.1.1.2

Rewrite $- 14$ as $- 2$ plus $- 12$

$\frac{24 x^{2} + (- 2 - 12) x + 1}{4 x - 3}$

Step 5.1.1.3

Apply the distributive property.

$\frac{24 x^{2} - 2 x - 12 x + 1}{4 x - 3}$

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.

$\frac{(24 x^{2} - 2 x) - 12 x + 1}{4 x - 3}$

Step 5.1.2.2

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

$\frac{2 x (12 x - 1) - (12 x - 1)}{4 x - 3}$

Step 5.1.3

Factor the polynomial by factoring out the greatest common factor, $12 x - 1$ .

$\frac{(12 x - 1) (2 x - 1)}{4 x - 3}$

Step 5.2

Expand $(12 x - 1) (2 x - 1)$ .

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Step 5.2.1

Apply the distributive property.

$\frac{12 x (2 x - 1) - 1 (2 x - 1)}{4 x - 3}$

Step 5.2.2

Apply the distributive property.

$\frac{12 x (2 x) + 12 x \cdot - 1 - 1 (2 x - 1)}{4 x - 3}$

Step 5.2.3

Apply the distributive property.

$\frac{12 x (2 x) + 12 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1}{4 x - 3}$

Step 5.2.4

Remove parentheses.

$\frac{12 x \cdot (2 x) + 12 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1}{4 x - 3}$

Step 5.2.5

Move $x$ .

$\frac{12 \cdot (2 x \cdot x) + 12 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1}{4 x - 3}$

Step 5.2.6

Move $x$ .

$\frac{12 \cdot (2 x \cdot x) + 12 \cdot (- 1 x) - 1 (2 x) - 1 \cdot - 1}{4 x - 3}$

Step 5.2.7

Remove parentheses.

$\frac{12 \cdot (2 x \cdot x) + 12 \cdot (- 1 x) - 1 \cdot (2 x) - 1 \cdot - 1}{4 x - 3}$

Step 5.2.8

Multiply $12$ by $2$ .

$\frac{24 x \cdot x + 12 \cdot (- 1 x) - 1 \cdot (2 x) - 1 \cdot - 1}{4 x - 3}$

Step 5.2.9

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

$\frac{24 (x \cdot x) + 12 \cdot (- 1 x) - 1 \cdot (2 x) - 1 \cdot - 1}{4 x - 3}$

Step 5.2.10

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

$\frac{24 (x \cdot x) + 12 \cdot (- 1 x) - 1 \cdot (2 x) - 1 \cdot - 1}{4 x - 3}$

Step 5.2.11

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

$\frac{24 x^{1 + 1} + 12 \cdot (- 1 x) - 1 \cdot (2 x) - 1 \cdot - 1}{4 x - 3}$

Step 5.2.12

Add $1$ and $1$ .

$\frac{24 x^{2} + 12 \cdot (- 1 x) - 1 \cdot (2 x) - 1 \cdot - 1}{4 x - 3}$

Step 5.2.13

Multiply $12$ by $- 1$ .

$\frac{24 x^{2} - 12 x - 1 \cdot (2 x) - 1 \cdot - 1}{4 x - 3}$

Step 5.2.14

Multiply $- 1$ by $2$ .

$\frac{24 x^{2} - 12 x - 2 x - 1 \cdot - 1}{4 x - 3}$

Step 5.2.15

Multiply $- 1$ by $- 1$ .

$\frac{24 x^{2} - 12 x - 2 x + 1}{4 x - 3}$

Step 5.2.16

Subtract $2 x$ from $- 12 x$ .

$\frac{24 x^{2} - 14 x + 1}{4 x - 3}$

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 $0$ .

$4 x$

$3$

$24 x^{2}$

$14 x$

$1$

Step 5.4

Divide the highest order term in the dividend $24 x^{2}$ by the highest order term in divisor $4 x$ .

					$6 x$
	$4 x$	-	$3$		$24 x^{2}$	-	$14 x$	+	$1$

Step 5.5

Multiply the new quotient term by the divisor.

				$6 x$
$4 x$	-	$3$		$24 x^{2}$	-	$14 x$	+	$1$
			+	$24 x^{2}$	-	$18 x$

Step 5.6

The expression needs to be subtracted from the dividend, so change all the signs in $24 x^{2} - 18 x$

				$6 x$
$4 x$	-	$3$		$24 x^{2}$	-	$14 x$	+	$1$
			-	$24 x^{2}$	+	$18 x$

Step 5.7

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

				$6 x$
$4 x$	-	$3$		$24 x^{2}$	-	$14 x$	+	$1$
			-	$24 x^{2}$	+	$18 x$
					+	$4 x$

Step 5.8

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

				$6 x$
$4 x$	-	$3$		$24 x^{2}$	-	$14 x$	+	$1$
			-	$24 x^{2}$	+	$18 x$
					+	$4 x$	+	$1$

Step 5.9

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

				$6 x$	+	$1$
$4 x$	-	$3$		$24 x^{2}$	-	$14 x$	+	$1$
			-	$24 x^{2}$	+	$18 x$
					+	$4 x$	+	$1$

Step 5.10

Multiply the new quotient term by the divisor.

				$6 x$	+	$1$
$4 x$	-	$3$		$24 x^{2}$	-	$14 x$	+	$1$
			-	$24 x^{2}$	+	$18 x$
					+	$4 x$	+	$1$
					+	$4 x$	-	$3$

Step 5.11

The expression needs to be subtracted from the dividend, so change all the signs in $4 x - 3$

				$6 x$	+	$1$
$4 x$	-	$3$		$24 x^{2}$	-	$14 x$	+	$1$
			-	$24 x^{2}$	+	$18 x$
					+	$4 x$	+	$1$
					-	$4 x$	+	$3$

Step 5.12

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

				$6 x$	+	$1$
$4 x$	-	$3$		$24 x^{2}$	-	$14 x$	+	$1$
			-	$24 x^{2}$	+	$18 x$
					+	$4 x$	+	$1$
					-	$4 x$	+	$3$
							+	$4$

Step 5.13

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

$6 x + 1 + \frac{4}{4 x - 3}$

Step 5.14

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

$y = 6 x + 1$

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = \frac{3}{4}$

No Horizontal Asymptotes

Oblique Asymptotes: $y = 6 x + 1$

Step 7