Let's break down what the given polynomial [tex]\( P(x) = -40x^2 - 100x + 27,500 \)[/tex] represents in terms of the amusement park's ticket pricing and sales.
### First Sentence
The polynomial [tex]\( P(x) \)[/tex] describes the daily earnings of the amusement park, where [tex]\( x \)[/tex] is the number of \[tex]$2 increases in the price of a ticket.
The constant term of the polynomial expression, which is 27,500, represents the initial earnings in the price of a ticket. This is the earnings when no price increase (\( x = 0 \)) has been applied:
\[ P(0) = -40(0)^2 - 100(0) + 27,500 = 27,500 \]
So, we can complete the first sentence as:
The constant of the polynomial expression represents the initial earnings in the price of a ticket.
### Second Sentence
To understand the binomial \((500 - 20x)\), note that it represents the number of tickets sold after \( x \) \$[/tex]2 increases in the price of a ticket. Each \[tex]$2 increase results in 20 fewer tickets being sold.
Initially, without any increase (\( x = 0 \)), the park sells 500 tickets:
\[ \text{Initial number of tickets sold} = 500 \]
After \( x \) increases of \$[/tex]2 each, the number of tickets sold decreases:
[tex]\[ \text{Number of tickets sold} = 500 - 20x \][/tex]
Thus, the term [tex]\((500 - 20x)\)[/tex] is included in the polynomial expression as it directly influences the total revenue based on changes in tickets sold.
So, we can complete the second sentence as:
The binomial [tex]\((500 - 20x)\)[/tex] is a factor of the polynomial expression and represents the number of tickets sold in the price of a ticket.
### Summary
1. The constant of the polynomial expression represents the initial 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 in the price of a ticket.
