Question

A store owner buys leather for his handmade shoes at a wholesale warehouse. He finds that when he sells each pair of shoes for $75, the average number of customers per week is 30 people. Later, he finds that for each $5 decrease in the price per pair of shoes, the number of customers increases by an average of 5 people per week. The weekly revenue from his sales is given by the polynomial below, where x is the number of $5 decreases in the price per pair of shoes over $75.

R(x)=2,250+225x-25x^2

Use the information given above to determine how to best complete the following sentences.

The constant of the polynomial represents the ( ) in the price per pair of shoes.

The binomial (30 + 5x) is a factor of polynomial R(x). Does this binomial represent revenue, number of customers, or price per pair of shoes? Does this binomial take into account the price decreases?

The binomial represents the average ( ) per week, ( ) the change due to price decreases.

Answer 1

revenue with no decrease

number of customers

including

Explanation:

Answer 2

number of customers and including and revenue with no decrease

Explanation:

The constant of a polynomial is the term with degree zero. In this case, the constant of polynomial R(x) is $2,250.

Since the degree of a constant term of a polynomial is zero, the variable will not have any effect on the term. In polynomial R(x), the variable, x, represents the number of $5 decreases in pair of shoes price over $75.

By the definition of the constant of a polynomial, the number of $5 decreases in pair of shoes price over $75, x, will not have any effect on $2,250. Therefore, it can be concluded that the constant of polynomial R(x) represents the revenue with no decrease in pair of shoes price.

Polynomial R(x) can be factored into the product of two binomials.

According to the problem, the average number of customers per week is 30 people. For every $5 decrease in the price per pair of shoes over $75, the number of customers increases by an average of 5 people. The number of $5 decreases in pair of shoes price is represented by x, so 5x represents the total increase in the average number of customers per week due to decreases in the price per pair of shoes.

Therefore, the binomial (30 + 5x) represents the average number of customers per week, including the change due to price decreases.