Given the situation, let's break down the various components of the problem and answer the given statements.
The amusement park has an initial ticket price of \[tex]$55 and sells 500 tickets daily. When the price is increased by \$[/tex]2, 20 fewer tickets are sold for each increase. The polynomial expression provided by the management, [tex]\( P(x) = -40x^2 - 100x + 27,500 \)[/tex], models the daily earnings where [tex]\( x \)[/tex] is the number of \[tex]$2 increases.
The polynomial \( P(x) \) represents the daily earnings.
1. The constant of the polynomial:
The constant term of the polynomial \( P(x) \) is \( 27,500 \). This represents the daily earnings when there are no price increases, that is when \( x = 0 \). If no price increases (\$[/tex]2) were made, the earnings would be constant at \[tex]$27,500.
2. The binomial factor:
The binomial factor \( (500 - 20x) \) is derived from the total number of tickets sold after \( x \) increases. Initially, the park sells 500 tickets per day. For each \( x \) increase (where the price is increased by \$[/tex]2), 20 fewer tickets are sold. So, this binomial factor represents the number of tickets sold at different price levels.
Now filling in the blanks:
The constant of the polynomial expression represents the ticket earnings in the price of a facility.
The binomial [tex]\( (500 - 20x) \)[/tex] is a factor of the polynomial expression and represents the number in tickets sold.
So, the complete sentences are:
- The constant of the polynomial expression represents the ticket earnings.
- The binomial [tex]\( (500 - 20x) \)[/tex] is a factor of the polynomial expression and represents the number of tickets sold.
