Use x = 1 + r, where r is the interest rate paid each year. Write a polynomial model, C(x), that represents the final amount of an 8-year investment if $500 was deposited at the beginning of the first year, $800 was deposited at the beginning of the fourth year, and an additional $370 was deposited at the beginning of the eighth year.(1 point)

To create a polynomial model \( C(x) \) that represents the final amount of an 8-year investment with the specified deposits, we need to analyze each deposit and how long it will earn interest.

1. The first deposit of $500 is made at the beginning of the first year. This amount will earn interest for 8 years. Thus, the contribution from this deposit is \( 500x^8 \).

2. The second deposit of $800 is made at the beginning of the fourth year. This amount will earn interest for 5 years (from the start of year 4 to year 8). Thus, the contribution from this deposit is \( 800x^5 \).

3. The third deposit of $370 is made at the beginning of the eighth year. This amount will earn interest for 1 year. Thus, the contribution from this deposit is \( 370x \).

Putting all of these together, the polynomial model for the final amount \( C(x) \) after 8 years is:

\[ C(x) = 500x^8 + 800x^5 + 370x \]

So the correct response is:

C(x) = 500x^8 + 800x^5 + 370x.