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.

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 deposits made at different years and the interest accumulated over the 8-year period.

The model polynomial can be written as:
C(x) = (500 * x^0) + (800 * x^3) + (370 * x^7)

Let's break this down:

- (500 * x^0): $500 was deposited at the beginning of the first year, which is equivalent to x^0 since anything raised to the power of 0 is 1.

- (800 * x^3): $800 was deposited at the beginning of the fourth year, which is equivalent to x^3 since it has been three years since the first deposit (1 + r)^3.

- (370 * x^7): $370 was deposited at the beginning of the eighth year, which is equivalent to x^7 since it has been seven years since the first deposit (1 + r)^7.

The polynomial C(x) represents the final amount of the investment after 8 years, considering the deposits made at different points in time and the interest accumulated over the years.

Answer 2

To write the model polynomial C(x) that represents the final amount of the investment, we need to consider the deposits made at different times.

Let's break down the calculation step-by-step:

Step 1: Calculate the amount after 8 years for the $500 deposit made at the beginning of the first year.
Since the interest rate is r and x = 1 + r, the amount after 8 years can be represented as (x^8) * 500.

Step 2: Calculate the amount after 8 years for the $800 deposit made at the beginning of the fourth year.
The amount after 8 years for this deposit will be calculated from the third year, as it has only 5 more years to accumulate interest. So we take (x^5) * 800.

Step 3: Calculate the amount after 8 years for the additional $370 deposit made at the beginning of the eighth year.
Since this deposit is made in the eighth year itself, it will only accumulate interest for that year. So we take (x^1) * 370.

Step 4: Add up the amounts from all three steps to get the final amount after 8 years.
The model polynomial C(x) can be written as:
C(x) = (x^8) * 500 + (x^5) * 800 + (x^1) * 370.

Note: In this model, we assume the interest is compounded annually and that there are no other deposits or withdrawals made during the 8-year period.

Answer 3

To write a polynomial model, C(x), that represents the final amount of an 8-year investment, we need to consider the deposits made at different years and the interest earned each year.

Let's break it down step by step:

1. First, we start with the initial deposit of $500 at the beginning of the first year. We can represent this as x^0 * 500, where x^0 is 1.

2. Then, at the beginning of the fourth year, an additional $800 is deposited. Since x = 1 + r, the fourth year corresponds to x^3. Therefore, we can represent this deposit as x^3 * 800.

3. Finally, at the beginning of the eighth year, an additional $370 is deposited. The eighth year corresponds to x^7, so we can represent this deposit as x^7 * 370.

Now, let's put it all together:

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

Simplifying this equation, we get:

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

So, the model polynomial, C(x), representing the final amount of an 8-year investment, considering the given deposits, is C(x) = 500 + 800x^3 + 370x^7.