Questions: Identify the asymptotes. Give your answers in exact form. Do not round. a(x) = (x^3 + 3x^2 + 5x - 3) / (x^2 - 3) The expression (x^3 + 3x^2 + 5x - 3) / (x^2 - 3) is in lowest terms, and the denominator is zero at x = -sqrt(3) and x = sqrt(3) Thus, a has vertical asymptotes of x = -sqrt(3) and x = sqrt(3). The degree of the numerator is exactly one greater than the degree of the denominator. Therefore, a has no horizontal asymptote, but does have a slant asymptote.

Identify the asymptotes. Give your answers in exact form. Do not round.

a(x) = (x^3 + 3x^2 + 5x - 3) / (x^2 - 3)

The expression (x^3 + 3x^2 + 5x - 3) / (x^2 - 3) is in lowest terms, and the denominator is zero at

x = -sqrt(3) and x = sqrt(3)

Thus, a has vertical asymptotes of x = -sqrt(3) and x = sqrt(3).

The degree of the numerator is exactly one greater than the degree of the denominator. Therefore, a has no horizontal asymptote, but does have a slant asymptote.

Transcript text: Identify the asymptotes. Give your answers in exact form. Do not round. \[ a(x)=\frac{x^{3}+3 x^{2}+5 x-3}{x^{2}-3} \] The expression $\frac{x^{3}+3 x^{2}+5 x-3}{x^{2}-3}$ is in lowest terms, and the denominator is zero at \[ x=-\sqrt{3} \text { and } x=\sqrt{3} \] Thus, $a$ has vertical asymptotes of $x=-\sqrt{3}$ and $x=\sqrt{3}$. The degree of the numerator is exactly one greater than the degree of the denominator. Therefore, $a$ has no horizontal asymptote, but does have a slant asymptote.

Solution

Solution Steps

To identify the asymptotes of the given rational function $ a(x) = \frac{x^3 + 3x^2 + 5x - 3}{x^2 - 3} $, we need to follow these steps:

Vertical Asymptotes: These occur where the denominator is zero and the numerator is non-zero. Solve $ x^2 - 3 = 0 $ to find the vertical asymptotes.
Horizontal Asymptotes: Compare the degrees of the numerator and the denominator. If the degree of the numerator is greater than the degree of the denominator by exactly one, there is no horizontal asymptote but there is a slant asymptote.
Slant Asymptote: Perform polynomial long division of the numerator by the denominator to find the equation of the slant asymptote.

Step 1: Identify Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero and the numerator is non-zero. Solve $ x^2 - 3 = 0 $: \[ x = \pm \sqrt{3} \] Thus, the vertical asymptotes are at $ x = -\sqrt{3} $ and $ x = \sqrt{3} $.

Step 2: Determine the Degree of the Numerator and Denominator

The degree of the numerator $ x^3 + 3x^2 + 5x - 3 $ is 3, and the degree of the denominator $ x^2 - 3 $ is 2. Since the degree of the numerator is exactly one greater than the degree of the denominator, there is no horizontal asymptote, but there is a slant asymptote.

Step 3: Find the Slant Asymptote

Perform polynomial long division of the numerator by the denominator: \[ \frac{x^3 + 3x^2 + 5x - 3}{x^2 - 3} = x + 3 + \frac{8x - 3}{x^2 - 3} \] The quotient $ x + 3 $ represents the slant asymptote.