Question

use the dropdown menus to analyze the zeros in the denominator and determine if they correspond to holes or vertical asymptotes. answer attempt 1 out of 2 at x = 8, f has a zero in the denominator and also in the numerator. there is a hole at x = 8 because the multiplicity of the root in the numerator is greater than or equal to less than the multiplicity of the root in the denominator. at x = -2, f has a zero in the numerator.

Explanation:

Step1: Recall Hole and Vertical Asymptote Rules

For a rational function \( f(x)=\frac{N(x)}{D(x)} \), a root \( x = a \) in both numerator and denominator:

• If multiplicity of \( a \) in \( N(x) \) \( = \) multiplicity in \( D(x) \), there is a hole at \( x = a \).
• If multiplicity of \( a \) in \( N(x) \) \( < \) multiplicity in \( D(x) \), there is a vertical asymptote at \( x = a \).

Step2: Analyze \( x = 8 \)

At \( x = 8 \), \( f \) has a zero (root) in denominator \( D(x) \) and also in numerator \( N(x) \). For a hole to exist, the multiplicity of the root in numerator must be equal to the multiplicity in denominator.

Step3: Analyze \( x = -2 \)

At \( x = -2 \), \( f \) has a zero in numerator (but no mention of denominator having \( x = -2 \) as a root, or if denominator has it, the multiplicity in numerator is "less than" (since it’s a vertical asymptote or just a zero in numerator; but the dropdown is about numerator vs denominator multiplicity for \( x = 8 \)’s hole). Wait, the first blank: "At \( x = 8 \), \( f \) has a zero in the denominator and also \(\boldsymbol{\text{a zero}}\) in the numerator. There is a hole at \( x = 8 \) because the multiplicity of the root in the numerator is \(\boldsymbol{\text{equal to}}\) the multiplicity of the root in the denominator." The second part about \( x = -2 \): "At \( x = -2 \), \( f \) has a zero in the numerator (and denominator’s multiplicity for \( x = -2 \) is greater, so it’s a vertical asymptote, but the dropdown for the first part is about \( x = 8 \))".

Answer:

First blank: a zero; Second blank: equal to (for the hole condition at \( x = 8 \)).

(Note: The image’s dropdowns: first dropdown for "also in the numerator" should be "a zero", second dropdown for "multiplicity of the root in the numerator is" should be "equal to" the denominator’s multiplicity for a hole.)