Let's break down the question step by step:
1. We start with the polynomial expression [tex]\( P(x) = -40x^2 - 100x + 27,500 \)[/tex] which models the daily earnings of the amusement park. In this expression:
- [tex]\( x \)[/tex] represents the number of [tex]\( \$2 \)[/tex] increases in the price of a ticket.
2. Understanding the terms in the polynomial:
- The constant term is [tex]\( 27,500 \)[/tex]. This term represents the initial daily earnings before any price increase.
- The binomial expression [tex]\( (500-20x) \)[/tex] is mentioned as a factor of the polynomial. Let's interpret this:
- [tex]\( 500 \)[/tex] stands for the average number of tickets sold when no price increase is applied.
- [tex]\( 20x \)[/tex] accounts for the decrease in the number of tickets sold when the price of a ticket increases.
Given these observations, we can fill out the sentences as follows:
1. The constant of the polynomial expression represents the initial daily earnings in the price of a ticket.
2. The binomial [tex]\((500 - 20x)\)[/tex] is a factor of the polynomial expression and represents the number of tickets sold based on the new price in the price of a ticket.
Therefore, the correct answers should be:
"The constant of the polynomial expression represents the initial daily earnings in the price of [tex]\( a \)[/tex] ticket."
"The binomial [tex]\((500 - 20x)\)[/tex] is a factor of the polynomial expression and represents the number of tickets sold based on the new price in the price of [tex]\( a \)[/tex] ticket."
