Precalculus Examples

$\frac{2 x^{3} + 4 x^{2} - 9}{- x^{2} + 3}$

Step 1

Find where the expression $\frac{2 x^{3} + 4 x^{2} - 9}{- x^{2} + 3}$ is undefined.

$x = - \sqrt{3}, x = \sqrt{3}$

Step 2

Since $\frac{2 x^{3} + 4 x^{2} - 9}{- x^{2} + 3}$ $\to$ $\infty$ as $x$ $\to$ $- \sqrt{3}$ from the left and $\frac{2 x^{3} + 4 x^{2} - 9}{- x^{2} + 3}$ $\to$ $- \infty$ as $x$ $\to$ $- \sqrt{3}$ from the right, then $x = - \sqrt{3}$ is a vertical asymptote.

$x = - \sqrt{3}$

Step 3

Since $\frac{2 x^{3} + 4 x^{2} - 9}{- x^{2} + 3}$ $\to$ $\infty$ as $x$ $\to$ $\sqrt{3}$ from the left and $\frac{2 x^{3} + 4 x^{2} - 9}{- x^{2} + 3}$ $\to$ $- \infty$ as $x$ $\to$ $\sqrt{3}$ from the right, then $x = \sqrt{3}$ is a vertical asymptote.

$x = \sqrt{3}$

Step 4

List all of the vertical asymptotes:

$x = - \sqrt{3}, \sqrt{3}$

Step 5

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 6

Find $n$ and $m$ .

$n = 3$

$m = 2$

Step 7

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

No Horizontal Asymptotes

Step 8

Find the oblique asymptote using polynomial division.

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Step 8.1

Simplify the expression.

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Step 8.1.1

Factor $- 1$ out of $- x^{2}$ .

$\frac{2 x^{3} + 4 x^{2} - 9}{- (x^{2}) + 3}$

Step 8.1.2

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{2 x^{3} + 4 x^{2} - 9}{- (x^{2}) - 1 (- 3)}$

Step 8.1.3

Factor $- 1$ out of $- (x^{2}) - 1 (- 3)$ .

$\frac{2 x^{3} + 4 x^{2} - 9}{- (x^{2} - 3)}$

Step 8.1.4

Rewrite negatives.

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Step 8.1.4.1

Rewrite $- (x^{2} - 3)$ as $- 1 (x^{2} - 3)$ .

$\frac{2 x^{3} + 4 x^{2} - 9}{- 1 (x^{2} - 3)}$

Step 8.1.4.2

Move the negative in front of the fraction.

$- \frac{2 x^{3} + 4 x^{2} - 9}{x^{2} - 3}$

Step 8.2

Expand $- (2 x^{3} + 4 x^{2} - 9)$ .

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Step 8.2.1

Negate $2 x^{3} + 4 x^{2} - 9$ .

$\frac{- (2 x^{3} + 4 x^{2} - 9)}{x^{2} - 3}$

Step 8.2.2

Apply the distributive property.

$\frac{- (2 x^{3} + 4 x^{2}) + 9}{x^{2} - 3}$

Step 8.2.3

Apply the distributive property.

$\frac{- (2 x^{3}) - (4 x^{2}) + 9}{x^{2} - 3}$

Step 8.2.4

Remove parentheses.

$\frac{- 1 \cdot (2 x^{3}) - (4 x^{2}) + 9}{x^{2} - 3}$

Step 8.2.5

Remove parentheses.

$\frac{- 1 \cdot (2 x^{3}) - 1 \cdot (4 x^{2}) + 9}{x^{2} - 3}$

Step 8.2.6

Multiply $- 1$ by $2$ .

$\frac{- 2 x^{3} - 1 \cdot (4 x^{2}) + 9}{x^{2} - 3}$

Step 8.2.7

Multiply $- 1$ by $4$ .

$\frac{- 2 x^{3} - 4 x^{2} + 9}{x^{2} - 3}$

Step 8.2.8

Multiply $- 1$ by $- 9$ .

$\frac{- 2 x^{3} - 4 x^{2} + 9}{x^{2} - 3}$

Step 8.3

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

$x^{2}$

$0 x$

$3$

$2 x^{3}$

$4 x^{2}$

$0 x$

$9$

Step 8.4

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

						-	$2 x$
	$x^{2}$	+	$0 x$	-	$3$	-	$2 x^{3}$	-	$4 x^{2}$	+	$0 x$	+	$9$

Step 8.5

Multiply the new quotient term by the divisor.

$2 x$

$x^{2}$

$0 x$

$3$

$2 x^{3}$

$4 x^{2}$

$0 x$

$9$

$2 x^{3}$

$0$

$6 x$

Step 8.6

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x^{3} + 0 + 6 x$

					-	$2 x$
$x^{2}$	+	$0 x$	-	$3$	-	$2 x^{3}$	-	$4 x^{2}$	+	$0 x$	+	$9$
					+	$2 x^{3}$	-	$0$	-	$6 x$

Step 8.7

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

					-	$2 x$
$x^{2}$	+	$0 x$	-	$3$	-	$2 x^{3}$	-	$4 x^{2}$	+	$0 x$	+	$9$
					+	$2 x^{3}$	-	$0$	-	$6 x$
							-	$4 x^{2}$	-	$6 x$

Step 8.8

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

					-	$2 x$
$x^{2}$	+	$0 x$	-	$3$	-	$2 x^{3}$	-	$4 x^{2}$	+	$0 x$	+	$9$
					+	$2 x^{3}$	-	$0$	-	$6 x$
							-	$4 x^{2}$	-	$6 x$	+	$9$

Step 8.9

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

					-	$2 x$	-	$4$
$x^{2}$	+	$0 x$	-	$3$	-	$2 x^{3}$	-	$4 x^{2}$	+	$0 x$	+	$9$
					+	$2 x^{3}$	-	$0$	-	$6 x$
							-	$4 x^{2}$	-	$6 x$	+	$9$

Step 8.10

Multiply the new quotient term by the divisor.

					-	$2 x$	-	$4$
$x^{2}$	+	$0 x$	-	$3$	-	$2 x^{3}$	-	$4 x^{2}$	+	$0 x$	+	$9$
					+	$2 x^{3}$	-	$0$	-	$6 x$
							-	$4 x^{2}$	-	$6 x$	+	$9$
							-	$4 x^{2}$	+	$0$	+	$12$

Step 8.11

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x^{2} + 0 + 12$

					-	$2 x$	-	$4$
$x^{2}$	+	$0 x$	-	$3$	-	$2 x^{3}$	-	$4 x^{2}$	+	$0 x$	+	$9$
					+	$2 x^{3}$	-	$0$	-	$6 x$
							-	$4 x^{2}$	-	$6 x$	+	$9$
							+	$4 x^{2}$	-	$0$	-	$12$

Step 8.12

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

					-	$2 x$	-	$4$
$x^{2}$	+	$0 x$	-	$3$	-	$2 x^{3}$	-	$4 x^{2}$	+	$0 x$	+	$9$
					+	$2 x^{3}$	-	$0$	-	$6 x$
							-	$4 x^{2}$	-	$6 x$	+	$9$
							+	$4 x^{2}$	-	$0$	-	$12$
									-	$6 x$	-	$3$

Step 8.13

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

$- 2 x - 4 + \frac{- 6 x - 3}{x^{2} - 3}$

Step 8.14

Split the solution into the polynomial portion and the remainder.

$- 2 x - 4 + \frac{6 x + 3}{- x^{2} + 3}$

Step 8.15

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

$y = - 2 x - 4$

Step 9

This is the set of all asymptotes.

Vertical Asymptotes: $x = - \sqrt{3}, \sqrt{3}$

No Horizontal Asymptotes

Oblique Asymptotes: $y = - 2 x - 4$

Step 10