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An amusement park prices tickets at [tex] \$ 55 [/tex] and sells an average of 500 tickets daily. The management finds, over multiple increases in ticket pricing, that a [tex] \$ 2 [/tex] increase in the price of a ticket leads to an average of 20 fewer tickets being sold in a day.

Management uses the combined function [tex] P [/tex] to model the daily earnings of the amusement park, where [tex] x [/tex] is the number of [tex] \$ 2 [/tex] increases in the price of a ticket.
[tex]\[
P(x) = -40x^2 - 100x + 27,500
\][/tex]

Use the given information to complete the sentences.
The constant of the polynomial expression represents the ___.
The binomial [tex] (500 - 20x) [/tex] is a factor of the polynomial expression and represents the ___.

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Answer :

GinnyAnswer
Certainly! Let's analyze each part of the polynomial expression [tex]\( P(x) = -40x^2 - 100x + 27,500 \)[/tex] given in the problem, and match it to the provided sentences.

First, we recognize that [tex]\( P(x) \)[/tex] models the daily earnings of the amusement park based on the number of [tex]\( \$2 \)[/tex] increases in the price of a ticket, [tex]\( x \)[/tex].

### The Constant of the Polynomial Expression
In the polynomial [tex]\( P(x) \)[/tex], the constant term is [tex]\( 27,500 \)[/tex]. This term represents the baseline daily earnings when [tex]\( x = 0 \)[/tex], i.e., when there have been no [tex]\( \$2 \)[/tex] increases applied to the ticket price.

### Binomial (500 - 20x) as a Factor
When writing about the binomial [tex]\( (500 - 20x) \)[/tex], it is necessary to understand that [tex]\( 500 \)[/tex] is initially the average number of tickets sold daily. Given that each [tex]\( \$2 \)[/tex] increase leads to 20 fewer tickets sold, [tex]\( 20x \)[/tex] represents the total reduction in tickets sold as a function of [tex]\( x \)[/tex], the number of [tex]\( \$2 \)[/tex] increases.

Bringing these pieces of information into the sentences:

1. "The constant of the polynomial expression represents the ticket."

- Given that the constant [tex]\( 27,500 \)[/tex] represents the baseline daily earnings, we can infer it relates to total daily earnings when no price increase is applied.

2. "The binomial [tex]\( (500 - 20x) \)[/tex] is a factor of the polynomial expression and represents the price of a ticket."

- The binomial correctly represents the adjusted number of tickets sold, factoring in changes due to the price increase.

### Final Answer:
Based on the correct answer values for constant and binomial substitution:

1. The constant of the polynomial expression represents the total revenue. This is because [tex]\( \$27,500 \)[/tex] represents the earnings when there are no price increases.

2. The binomial [tex]\( (500 - 20x) \)[/tex] is a factor of the polynomial expression and represents the number of tickets sold. This simplifies our polynomial in terms of assessing ticket sales based on price increases.

To conclude:
- constant of the polynomial expression represents the total revenue when no price increase is applied.
- binomial [tex]\( (500 - 20x) \)[/tex] represents the number of tickets sold after [tex]\( x \)[/tex] number of $2 increases in ticket price.

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