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. Which answer is correct?

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. Which answer is correct?

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

C(x) = 500x^8 + 800x^4 + 370x

C(x) = 500x^8 + 800x^4 + 370

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

Answer 1

The correct answer is:

C(x) = 500x^8 + 800x^4 + 370x

Answer 2

To create the model polynomial, C(x), we need to consider the investments made at different points in time.

The investment of $500 at the beginning of the first year will be multiplied by x^0 since it does not accrue any interest.

The investment of $800 at the beginning of the fourth year will be multiplied by x^3 since it accrues interest for four years (3-year difference).

The investment of $370 at the beginning of the eighth year will be multiplied by x^7 since it accrues interest for eight years (7-year difference).

To calculate the total amount after 8 years, we add up these three investments:

C(x) = 500x^0 + 800x^3 + 370x^7

Simplifying the polynomial, we get:

C(x) = 500 + 800x^3 + 370x^7

Therefore, the correct answer is:

C(x) = 500 + 800x^3 + 370x^7

Answer 3

To write the model polynomial, C(x), that represents the final amount of the investment, we need to consider the amounts deposited at the beginning of the first, fourth, and eighth years.

Let's break it down:

1. At the beginning of the first year, $500 was deposited. This means the amount contributed at year 1 is 500.

2. At the beginning of the fourth year, $800 was deposited. Since we are given the investment is for 8 years, the contribution at year 4 will be compounded for the remaining 4 years. To take this into account, we multiply $800 by x^4 (since x represents the interest rate paid each year). So, the contribution at year 4 is 800x^4.

3. At the beginning of the eighth year, an additional $370 was deposited. Similar to the previous case, we need to account for the compounding over the remaining 8 - 1 = 7 years. Hence, the contribution at year 8 is 370x^7.

Now, let's sum up all the contributions to obtain the final amount:

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

Therefore, the correct answer is C(x) = 500 + 800x^4 + 370x^7.