Question

use the dropdown menus to analyze the zeros in the denominator and determine if they correspond to holes or vertical asymptotes.
answer: to analyze f(x):
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 the multiplicity of the root in the denominator.
at x = -2, f has a zero in the denominator but not in the numerator. as a result, there is a vertical asymptote/hole at x = -2.

Explanation:

Step1: Analyze \( x = 8 \)

For a rational function, a root in the denominator that is also a root in the numerator creates a hole if the multiplicity of the root in the numerator is greater than or equal to that in the denominator. At \( x = 8 \), since \( f \) has a zero in both numerator and denominator, and the multiplicity of the root in the numerator is greater than or equal to that in the denominator, there is a hole at \( x = 8 \).

Step2: Analyze \( x = -2 \)

At \( x = -2 \), \( f \) has a zero in the denominator but not in the numerator. This means the function is undefined here and the graph approaches infinity (or negative infinity), so there is a vertical asymptote at \( x = -2 \).

Answer:

• At \( x = 8 \): hole
• At \( x = -2 \): vertical asymptote