Question

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 500wasdepositedatthebeginningofthefirstyear, 800 was deposited at the beginning of the fourth year, and an additional $370 was deposited at the beginning of the eighth year.

Answer 1

Are you sure

Answer 2

To formulate a polynomial \( C(x) \) that represents the final amount of your investment, we need to consider the contributions made at different times and how they accumulate over the investment period, factoring in the interest rate \( r \).

Given that \( x = 1 + r \):

1. **Initial deposit of $500 at the beginning of the first year:**
This amount will earn interest for 8 years. Therefore, its future value at the end of 8 years will be:
\[
500 \cdot x^8
\]

2. **Deposit of $800 at the beginning of the fourth year:**
This amount will earn interest for 5 years (from the start of the fourth year to the end of the eighth year). Thus, its future value will be:
\[
800 \cdot x^5
\]

3. **Deposit of $370 at the beginning of the eighth year:**
This amount will only be present for the eighth year (it doesn't earn any interest). Therefore, its future value is just:
\[
370
\]

Now, we combine these contributions to form the total amount after 8 years:

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

This polynomial \( C(x) \) represents the final amount of the investment after 8 years, where \( x = 1 + r \).

Answer 3

Yes, let's carefully review the setup to ensure the investment contributions are correctly calculated.

### Breakdown of the Investment Contributions

1. **Initial deposit of $500 at the beginning of the first year:**
- **Duration:** 8 years
- **Future Value Calculation:**
\[
500 \cdot x^8
\]

2. **Deposit of $800 at the beginning of the fourth year:**
- **Duration:** 5 years (from the beginning of year 4 to year 8)
- **Future Value Calculation:**
\[
800 \cdot x^5
\]

3. **Deposit of $370 at the beginning of the eighth year:**
- **Duration:** 1 year (it will gain interest for just the eighth year)
- **Future Value Calculation:**
\[
370 \cdot x^1 = 370 \cdot x
\]

### Total Future Value

Combining these three future values together, we arrive at:

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

So, the correct polynomial that represents the final amount \( C(x) \) of the investment after 8 years, incorporating the contributions made at different times and the respective durations for which each deposit earns interest, is:

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

Thank you for prompting me to clarify! Would you like to know anything else about this model?