Precalculus Examples

$f (x) = \frac{x^{3} + 4 x^{2} - 21 x}{x^{2} + 4 x - 21}$

Step 1

Find where the expression $\frac{x^{3} + 4 x^{2} - 21 x}{x^{2} + 4 x - 21}$ is undefined.

$x = - 7, x = 3$

Step 2

The vertical asymptotes occur at areas of infinite discontinuity.

No Vertical Asymptotes

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

Simplify the expression.

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Step 6.1.1

Simplify the numerator.

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Step 6.1.1.1

Factor $x$ out of $x^{3} + 4 x^{2} - 21 x$ .

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Step 6.1.1.1.1

Factor $x$ out of $x^{3}$ .

$\frac{x \cdot x^{2} + 4 x^{2} - 21 x}{x^{2} + 4 x - 21}$

Step 6.1.1.1.2

Factor $x$ out of $4 x^{2}$ .

$\frac{x \cdot x^{2} + x (4 x) - 21 x}{x^{2} + 4 x - 21}$

Step 6.1.1.1.3

Factor $x$ out of $- 21 x$ .

$\frac{x \cdot x^{2} + x (4 x) + x \cdot - 21}{x^{2} + 4 x - 21}$

Step 6.1.1.1.4

Factor $x$ out of $x \cdot x^{2} + x (4 x)$ .

$\frac{x (x^{2} + 4 x) + x \cdot - 21}{x^{2} + 4 x - 21}$

Step 6.1.1.1.5

Factor $x$ out of $x (x^{2} + 4 x) + x \cdot - 21$ .

$\frac{x (x^{2} + 4 x - 21)}{x^{2} + 4 x - 21}$

Step 6.1.1.2

Factor $x^{2} + 4 x - 21$ using the AC method.

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Step 6.1.1.2.1

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 $- 21$ and whose sum is $4$ .

$- 3, 7$

Step 6.1.1.2.2

Write the factored form using these integers.

$\frac{x ((x - 3) (x + 7))}{x^{2} + 4 x - 21}$

$\frac{x (x - 3) (x + 7)}{x^{2} + 4 x - 21}$

Step 6.1.2

Factor $x^{2} + 4 x - 21$ using the AC method.

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Step 6.1.2.1

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 $- 21$ and whose sum is $4$ .

$- 3, 7$

Step 6.1.2.2

Write the factored form using these integers.

$\frac{x (x - 3) (x + 7)}{(x - 3) (x + 7)}$

Step 6.1.3

Reduce the expression by cancelling the common factors.

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Step 6.1.3.1

Cancel the common factor of $x - 3$ .

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Step 6.1.3.1.1

Cancel the common factor.

$\frac{x (x - 3) (x + 7)}{(x - 3) (x + 7)}$

Step 6.1.3.1.2

Rewrite the expression.

$\frac{x (x + 7)}{x + 7}$

Step 6.1.3.2

Cancel the common factor of $x + 7$ .

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Step 6.1.3.2.1

Cancel the common factor.

$\frac{x (x + 7)}{x + 7}$

Step 6.1.3.2.2

Divide $x$ by $1$ .

$x$

Step 6.2

Since there is no polynomial portion from the polynomial division, there are no oblique asymptotes.

No Oblique Asymptotes

Step 7

This is the set of all asymptotes.

No Vertical Asymptotes

No Horizontal Asymptotes

No Oblique Asymptotes

Step 8