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 $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.

Answer 1

To find the final amount of the investment, we need to consider the deposit amounts at different years and the interest accumulated over the 8-year period.

First, let's break down the problem into different time periods based on the deposits made:

- Year 1: $500 deposited at the beginning.
- Year 2: No deposit.
- Year 3: No deposit.
- Year 4: $800 deposited at the beginning.
- Year 5: No deposit.
- Year 6: No deposit.
- Year 7: No deposit.
- Year 8: $370 deposited at the beginning.

Now let's calculate the interest accumulated for each deposit at the end of the 8-year period:

For the $500 deposit made at the beginning of Year 1:
Interest accumulated = $500 * r * (1+r)^7

For the $800 deposit made at the beginning of Year 4:
Interest accumulated = $800 * r * (1+r)^4

For the $370 deposit made at the beginning of Year 8:
Interest accumulated = $370 * r

Now, let's write the model polynomial, C(x), that represents the final amount of the investment after 8 years:

C(x) = $500 * x * (1+x)^7 + $800 * x^4 * (1+x)^4 + $370 * x^7

Or, simplified:

C(x) = $500 * x^8 * (1+x)^7 + $800 * x^4 * (1+x)^4 + $370 * x^7