Precalculus Examples

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

Step 1

Find where the expression $\frac{(x + 7) (x - 3) (x + 1)}{(x - 3) (x - 5)}$ is undefined.

$x = 3, x = 5$

Step 2

Since $\frac{(x + 7) (x - 3) (x + 1)}{(x - 3) (x - 5)}$ $\to$ $- \infty$ as $x$ $\to$ $5$ from the left and $\frac{(x + 7) (x - 3) (x + 1)}{(x - 3) (x - 5)}$ $\to$ $\infty$ as $x$ $\to$ $5$ from the right, then $x = 5$ is a vertical asymptote.

$x = 5$

Step 3

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 4

Find $n$ and $m$ .

$n = 3$

$m = 2$

Step 5

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

No Horizontal Asymptotes

Step 6

Find the oblique asymptote using polynomial division.

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Step 6.1

Factor $x^{2} - 8 x + 15$ using the AC method.

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Step 6.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $15$ and whose sum is $- 8$ .

$- 5, - 3$

Step 6.1.2

Write the factored form using these integers.

$\frac{x^{3} + 5 x^{2} - 17 x - 21}{(x - 5) (x - 3)}$

Step 6.2

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

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Step 6.2.1

Apply the distributive property.

$\frac{x^{3} + 5 x^{2} - 17 x - 21}{x (x - 3) - 5 (x - 3)}$

Step 6.2.2

Apply the distributive property.

$\frac{x^{3} + 5 x^{2} - 17 x - 21}{x \cdot x + x \cdot - 3 - 5 (x - 3)}$

Step 6.2.3

Apply the distributive property.

$\frac{x^{3} + 5 x^{2} - 17 x - 21}{x \cdot x + x \cdot - 3 - 5 x - 5 \cdot - 3}$

Step 6.2.4

Reorder $x$ and $- 3$ .

$\frac{x^{3} + 5 x^{2} - 17 x - 21}{x \cdot x - 3 x - 5 x - 5 \cdot - 3}$

Step 6.2.5

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

$\frac{x^{3} + 5 x^{2} - 17 x - 21}{x \cdot x - 3 x - 5 x - 5 \cdot - 3}$

Step 6.2.6

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

$\frac{x^{3} + 5 x^{2} - 17 x - 21}{x \cdot x - 3 x - 5 x - 5 \cdot - 3}$

Step 6.2.7

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

$\frac{x^{3} + 5 x^{2} - 17 x - 21}{x^{1 + 1} - 3 x - 5 x - 5 \cdot - 3}$

Step 6.2.8

Add $1$ and $1$ .

$\frac{x^{3} + 5 x^{2} - 17 x - 21}{x^{2} - 3 x - 5 x - 5 \cdot - 3}$

Step 6.2.9

Multiply $- 5$ by $- 3$ .

$\frac{x^{3} + 5 x^{2} - 17 x - 21}{x^{2} - 3 x - 5 x + 15}$

Step 6.2.10

Subtract $5 x$ from $- 3 x$ .

$\frac{x^{3} + 5 x^{2} - 17 x - 21}{x^{2} - 8 x + 15}$

Step 6.3

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

$x^{2}$

$8 x$

$15$

$x^{3}$

$5 x^{2}$

$17 x$

$21$

Step 6.4

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

$x$

$x^{2}$

$8 x$

$15$

$x^{3}$

$5 x^{2}$

$17 x$

$21$

Step 6.5

Multiply the new quotient term by the divisor.

$x$

$x^{2}$

$8 x$

$15$

$x^{3}$

$5 x^{2}$

$17 x$

$21$

$x^{3}$

$8 x^{2}$

$15 x$

Step 6.6

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - 8 x^{2} + 15 x$

$x$

$x^{2}$

$8 x$

$15$

$x^{3}$

$5 x^{2}$

$17 x$

$21$

$x^{3}$

$8 x^{2}$

$15 x$

Step 6.7

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

$x$

$x^{2}$

$8 x$

$15$

$x^{3}$

$5 x^{2}$

$17 x$

$21$

$x^{3}$

$8 x^{2}$

$15 x$

$13 x^{2}$

$32 x$

Step 6.8

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

						$x$
$x^{2}$	-	$8 x$	+	$15$		$x^{3}$	+	$5 x^{2}$	-	$17 x$	-	$21$
					-	$x^{3}$	+	$8 x^{2}$	-	$15 x$
							+	$13 x^{2}$	-	$32 x$	-	$21$

Step 6.9

Divide the highest order term in the dividend $13 x^{2}$ by the highest order term in divisor $x^{2}$ .

						$x$	+	$13$
$x^{2}$	-	$8 x$	+	$15$		$x^{3}$	+	$5 x^{2}$	-	$17 x$	-	$21$
					-	$x^{3}$	+	$8 x^{2}$	-	$15 x$
							+	$13 x^{2}$	-	$32 x$	-	$21$

Step 6.10

Multiply the new quotient term by the divisor.

$x$

$13$

$x^{2}$

$8 x$

$15$

$x^{3}$

$5 x^{2}$

$17 x$

$21$

$x^{3}$

$8 x^{2}$

$15 x$

$13 x^{2}$

$32 x$

$21$

$13 x^{2}$

$104 x$

$195$

Step 6.11

The expression needs to be subtracted from the dividend, so change all the signs in $13 x^{2} - 104 x + 195$

						$x$	+	$13$
$x^{2}$	-	$8 x$	+	$15$		$x^{3}$	+	$5 x^{2}$	-	$17 x$	-	$21$
					-	$x^{3}$	+	$8 x^{2}$	-	$15 x$
							+	$13 x^{2}$	-	$32 x$	-	$21$
							-	$13 x^{2}$	+	$104 x$	-	$195$

Step 6.12

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

						$x$	+	$13$
$x^{2}$	-	$8 x$	+	$15$		$x^{3}$	+	$5 x^{2}$	-	$17 x$	-	$21$
					-	$x^{3}$	+	$8 x^{2}$	-	$15 x$
							+	$13 x^{2}$	-	$32 x$	-	$21$
							-	$13 x^{2}$	+	$104 x$	-	$195$
									+	$72 x$	-	$216$

Step 6.13

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

$x + 13 + \frac{72 x - 216}{x^{2} - 8 x + 15}$

Step 6.14

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

$y = x + 13$

Step 7

This is the set of all asymptotes.

Vertical Asymptotes: $x = 5$

No Horizontal Asymptotes

Oblique Asymptotes: $y = x + 13$

Step 8