Precalculus Examples

$f (x) = \frac{5 x^{4} - 7}{2 x + 3}$

Step 1

Find where the expression $\frac{5 x^{4} - 7}{2 x + 3}$ is undefined.

$x = - \frac{3}{2}$

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 $m$ .

$n = 4$

$m = 1$

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 $0$ .

$2 x$

$3$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$7$

Step 5.2

Divide the highest order term in the dividend $5 x^{4}$ by the highest order term in divisor $2 x$ .

$\frac{5 x^{3}}{2}$

$2 x$

$3$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$7$

Step 5.3

Multiply the new quotient term by the divisor.

$\frac{5 x^{3}}{2}$

$2 x$

$3$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$7$

$5 x^{4}$

$\frac{15 x^{3}}{2}$

Step 5.4

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{4} + \frac{15 x^{3}}{2}$

$\frac{5 x^{3}}{2}$

$2 x$

$3$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$7$

$5 x^{4}$

$\frac{15 x^{3}}{2}$

Step 5.5

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

$\frac{5 x^{3}}{2}$

$2 x$

$3$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$7$

$5 x^{4}$

$\frac{15 x^{3}}{2}$

Step 5.6

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

$\frac{5 x^{3}}{2}$

$2 x$

$3$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$7$

$5 x^{4}$

$\frac{15 x^{3}}{2}$

$0 x^{2}$

Step 5.7

Divide the highest order term in the dividend $- \frac{15 x^{3}}{2}$ by the highest order term in divisor $2 x$ .

$\frac{5 x^{3}}{2}$

$\frac{15 x^{2}}{4}$

$2 x$

$3$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$7$

$5 x^{4}$

$\frac{15 x^{3}}{2}$

$0 x^{2}$

Step 5.8

Multiply the new quotient term by the divisor.

$\frac{5 x^{3}}{2}$

$\frac{15 x^{2}}{4}$

$2 x$

$3$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$7$

$5 x^{4}$

$\frac{15 x^{3}}{2}$

$0 x^{2}$

$\frac{15 x^{3}}{2}$

$\frac{45 x^{2}}{4}$

Step 5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- \frac{15 x^{3}}{2} - \frac{45 x^{2}}{4}$

$\frac{5 x^{3}}{2}$

$\frac{15 x^{2}}{4}$

$2 x$

$3$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$7$

$5 x^{4}$

$\frac{15 x^{3}}{2}$

$0 x^{2}$

$\frac{15 x^{3}}{2}$

$\frac{45 x^{2}}{4}$

Step 5.10

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

$\frac{5 x^{3}}{2}$

$\frac{15 x^{2}}{4}$

$2 x$

$3$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$7$

$5 x^{4}$

$\frac{15 x^{3}}{2}$

$0 x^{2}$

$\frac{15 x^{3}}{2}$

$\frac{45 x^{2}}{4}$

Step 5.11

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

$\frac{5 x^{3}}{2}$

$\frac{15 x^{2}}{4}$

$2 x$

$3$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$7$

$5 x^{4}$

$\frac{15 x^{3}}{2}$

$0 x^{2}$

$\frac{15 x^{3}}{2}$

$\frac{45 x^{2}}{4}$

$0 x$

Step 5.12

Divide the highest order term in the dividend $\frac{45 x^{2}}{4}$ by the highest order term in divisor $2 x$ .

$\frac{5 x^{3}}{2}$

$\frac{15 x^{2}}{4}$

$\frac{45 x}{8}$

$2 x$

$3$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$7$

$5 x^{4}$

$\frac{15 x^{3}}{2}$

$0 x^{2}$

$\frac{15 x^{3}}{2}$

$\frac{45 x^{2}}{4}$

$0 x$

Step 5.13

Multiply the new quotient term by the divisor.

$\frac{5 x^{3}}{2}$

$\frac{15 x^{2}}{4}$

$\frac{45 x}{8}$

$2 x$

$3$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$7$

$5 x^{4}$

$\frac{15 x^{3}}{2}$

$0 x^{2}$

$\frac{15 x^{3}}{2}$

$\frac{45 x^{2}}{4}$

$0 x$

$\frac{45 x^{2}}{4}$

$\frac{135 x}{8}$

Step 5.14

The expression needs to be subtracted from the dividend, so change all the signs in $\frac{45 x^{2}}{4} + \frac{135 x}{8}$

$\frac{5 x^{3}}{2}$

$\frac{15 x^{2}}{4}$

$\frac{45 x}{8}$

$2 x$

$3$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$7$

$5 x^{4}$

$\frac{15 x^{3}}{2}$

$0 x^{2}$

$\frac{15 x^{3}}{2}$

$\frac{45 x^{2}}{4}$

$0 x$

$\frac{45 x^{2}}{4}$

$\frac{135 x}{8}$

Step 5.15

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

$\frac{5 x^{3}}{2}$

$\frac{15 x^{2}}{4}$

$\frac{45 x}{8}$

$2 x$

$3$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$7$

$5 x^{4}$

$\frac{15 x^{3}}{2}$

$0 x^{2}$

$\frac{15 x^{3}}{2}$

$\frac{45 x^{2}}{4}$

$0 x$

$\frac{45 x^{2}}{4}$

$\frac{135 x}{8}$

Step 5.16

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

$\frac{5 x^{3}}{2}$

$\frac{15 x^{2}}{4}$

$\frac{45 x}{8}$

$2 x$

$3$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$7$

$5 x^{4}$

$\frac{15 x^{3}}{2}$

$0 x^{2}$

$\frac{15 x^{3}}{2}$

$\frac{45 x^{2}}{4}$

$0 x$

$\frac{45 x^{2}}{4}$

$\frac{135 x}{8}$

$7$

Step 5.17

Divide the highest order term in the dividend $- \frac{135 x}{8}$ by the highest order term in divisor $2 x$ .

$\frac{5 x^{3}}{2}$

$\frac{15 x^{2}}{4}$

$\frac{45 x}{8}$

$\frac{135}{16}$

$2 x$

$3$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$7$

$5 x^{4}$

$\frac{15 x^{3}}{2}$

$0 x^{2}$

$\frac{15 x^{3}}{2}$

$\frac{45 x^{2}}{4}$

$0 x$

$\frac{45 x^{2}}{4}$

$\frac{135 x}{8}$

$7$

Step 5.18

Multiply the new quotient term by the divisor.

$\frac{5 x^{3}}{2}$

$\frac{15 x^{2}}{4}$

$\frac{45 x}{8}$

$\frac{135}{16}$

$2 x$

$3$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$7$

$5 x^{4}$

$\frac{15 x^{3}}{2}$

$0 x^{2}$

$\frac{15 x^{3}}{2}$

$\frac{45 x^{2}}{4}$

$0 x$

$\frac{45 x^{2}}{4}$

$\frac{135 x}{8}$

$7$

$\frac{135 x}{8}$

$\frac{405}{16}$

Step 5.19

The expression needs to be subtracted from the dividend, so change all the signs in $- \frac{135 x}{8} - \frac{405}{16}$

$\frac{5 x^{3}}{2}$

$\frac{15 x^{2}}{4}$

$\frac{45 x}{8}$

$\frac{135}{16}$

$2 x$

$3$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$7$

$5 x^{4}$

$\frac{15 x^{3}}{2}$

$0 x^{2}$

$\frac{15 x^{3}}{2}$

$\frac{45 x^{2}}{4}$

$0 x$

$\frac{45 x^{2}}{4}$

$\frac{135 x}{8}$

$7$

$\frac{135 x}{8}$

$\frac{405}{16}$

Step 5.20

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

$\frac{5 x^{3}}{2}$

$\frac{15 x^{2}}{4}$

$\frac{45 x}{8}$

$\frac{135}{16}$

$2 x$

$3$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$7$

$5 x^{4}$

$\frac{15 x^{3}}{2}$

$0 x^{2}$

$\frac{15 x^{3}}{2}$

$\frac{45 x^{2}}{4}$

$0 x$

$\frac{45 x^{2}}{4}$

$\frac{135 x}{8}$

$7$

$\frac{135 x}{8}$

$\frac{405}{16}$

$\frac{293}{16}$

Step 5.21

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

$\frac{5 x^{3}}{2} - \frac{15 x^{2}}{4} + \frac{45 x}{8} - \frac{135}{16} + \frac{293}{16 (2 x + 3)}$

Step 5.22

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

$y = \frac{5 x^{3}}{2} - \frac{15 x^{2}}{4} + \frac{45 x}{8} - \frac{135}{16}$

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = - \frac{3}{2}$

No Horizontal Asymptotes

Oblique Asymptotes: $y = \frac{5 x^{3}}{2} - \frac{15 x^{2}}{4} + \frac{45 x}{8} - \frac{135}{16}$

Step 7