Use x=1+r, where r is the interest rate paid each year. Write a model polynomial, 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)

Answer 1

To model the final amount of the investment after 8 years with the given deposits and interest rate, we will account for each deposit and how long each one is invested.

1. Deposit of $500 at the beginning of Year 1: This deposit will accumulate interest for 8 years.

   • Final amount from this deposit: \[ 500 \cdot x^8 \]

2. Deposit of $800 at the beginning of Year 4: This deposit will accumulate interest for 5 years (from Year 4 to Year 8).

   • Final amount from this deposit: \[ 800 \cdot x^5 \]

3. Deposit of $370 at the beginning of Year 8: This deposit will not accumulate any interest, as it is made at the end of Year 8.

   • Final amount from this deposit: \[ 370 \]

Now we combine these amounts into the total final amount polynomial \( C(x) \):

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

Thus, the model polynomial \( C(x) \) representing the final amount of the investment after 8 years is:

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