Precalculus Examples

$f (x) = \frac{4 x^{3} - 7 x^{2} - 3 x + 7}{x - 2}$

Step 1

Find where the expression $\frac{4 x^{3} - 7 x^{2} - 3 x + 7}{x - 2}$ 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 $m$ .

$n = 3$

$m = 1$

Step 4

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

No Horizontal Asymptotes

Step 5

Find the oblique asymptote using polynomial division.

Tap for more steps...

Step 5.1

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

$x$

$2$

$4 x^{3}$

$7 x^{2}$

$3 x$

$7$

Step 5.2

Divide the highest order term in the dividend $4 x^{3}$ by the highest order term in divisor $x$ .

					$4 x^{2}$
	$x$	-	$2$		$4 x^{3}$	-	$7 x^{2}$	-	$3 x$	+	$7$

Step 5.3

Multiply the new quotient term by the divisor.

				$4 x^{2}$
$x$	-	$2$		$4 x^{3}$	-	$7 x^{2}$	-	$3 x$	+	$7$
			+	$4 x^{3}$	-	$8 x^{2}$

Step 5.4

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{3} - 8 x^{2}$

				$4 x^{2}$
$x$	-	$2$		$4 x^{3}$	-	$7 x^{2}$	-	$3 x$	+	$7$
			-	$4 x^{3}$	+	$8 x^{2}$

Step 5.5

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

				$4 x^{2}$
$x$	-	$2$		$4 x^{3}$	-	$7 x^{2}$	-	$3 x$	+	$7$
			-	$4 x^{3}$	+	$8 x^{2}$
					+	$x^{2}$

Step 5.6

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

				$4 x^{2}$
$x$	-	$2$		$4 x^{3}$	-	$7 x^{2}$	-	$3 x$	+	$7$
			-	$4 x^{3}$	+	$8 x^{2}$
					+	$x^{2}$	-	$3 x$

Step 5.7

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

				$4 x^{2}$	+	$x$
$x$	-	$2$		$4 x^{3}$	-	$7 x^{2}$	-	$3 x$	+	$7$
			-	$4 x^{3}$	+	$8 x^{2}$
					+	$x^{2}$	-	$3 x$

Step 5.8

Multiply the new quotient term by the divisor.

				$4 x^{2}$	+	$x$
$x$	-	$2$		$4 x^{3}$	-	$7 x^{2}$	-	$3 x$	+	$7$
			-	$4 x^{3}$	+	$8 x^{2}$
					+	$x^{2}$	-	$3 x$
					+	$x^{2}$	-	$2 x$

Step 5.9

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

				$4 x^{2}$	+	$x$
$x$	-	$2$		$4 x^{3}$	-	$7 x^{2}$	-	$3 x$	+	$7$
			-	$4 x^{3}$	+	$8 x^{2}$
					+	$x^{2}$	-	$3 x$
					-	$x^{2}$	+	$2 x$

Step 5.10

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

				$4 x^{2}$	+	$x$
$x$	-	$2$		$4 x^{3}$	-	$7 x^{2}$	-	$3 x$	+	$7$
			-	$4 x^{3}$	+	$8 x^{2}$
					+	$x^{2}$	-	$3 x$
					-	$x^{2}$	+	$2 x$
							-	$x$

Step 5.11

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

				$4 x^{2}$	+	$x$
$x$	-	$2$		$4 x^{3}$	-	$7 x^{2}$	-	$3 x$	+	$7$
			-	$4 x^{3}$	+	$8 x^{2}$
					+	$x^{2}$	-	$3 x$
					-	$x^{2}$	+	$2 x$
							-	$x$	+	$7$

Step 5.12

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

				$4 x^{2}$	+	$x$	-	$1$
$x$	-	$2$		$4 x^{3}$	-	$7 x^{2}$	-	$3 x$	+	$7$
			-	$4 x^{3}$	+	$8 x^{2}$
					+	$x^{2}$	-	$3 x$
					-	$x^{2}$	+	$2 x$
							-	$x$	+	$7$

Step 5.13

Multiply the new quotient term by the divisor.

				$4 x^{2}$	+	$x$	-	$1$
$x$	-	$2$		$4 x^{3}$	-	$7 x^{2}$	-	$3 x$	+	$7$
			-	$4 x^{3}$	+	$8 x^{2}$
					+	$x^{2}$	-	$3 x$
					-	$x^{2}$	+	$2 x$
							-	$x$	+	$7$
							-	$x$	+	$2$

Step 5.14

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

				$4 x^{2}$	+	$x$	-	$1$
$x$	-	$2$		$4 x^{3}$	-	$7 x^{2}$	-	$3 x$	+	$7$
			-	$4 x^{3}$	+	$8 x^{2}$
					+	$x^{2}$	-	$3 x$
					-	$x^{2}$	+	$2 x$
							-	$x$	+	$7$
							+	$x$	-	$2$

Step 5.15

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

				$4 x^{2}$	+	$x$	-	$1$
$x$	-	$2$		$4 x^{3}$	-	$7 x^{2}$	-	$3 x$	+	$7$
			-	$4 x^{3}$	+	$8 x^{2}$
					+	$x^{2}$	-	$3 x$
					-	$x^{2}$	+	$2 x$
							-	$x$	+	$7$
							+	$x$	-	$2$
									+	$5$

Step 5.16

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

$4 x^{2} + x - 1 + \frac{5}{x - 2}$

Step 5.17

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

$y = 4 x^{2} + x - 1$

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = 2$

No Horizontal Asymptotes

Oblique Asymptotes: $y = 4 x^{2} + x - 1$

Step 7