Question

User x=1+r
, where r
is the interest rate paid each year. Write a model polynomial, C(x)
. Represent the final amount of a 7-year investment if $5,000
was deposited at the beginning of the first year, and $2,000
was deposited at the beginning of the third year. (1 point)
Responses

C(x)=5,000x7+7,000x4
cap c times x is equal to 5 comma 000 x to the 7th power plus 7 comma 000 x to the 4th power

C(x)=5,000x7+2,000x3
cap c times x is equal to 5 comma 000 x to the 7th power plus 2 comma 000 x cubed

C(x)=5,000x7+2,000x5
cap c times x is equal to 5 comma 000 x to the 7th power plus 2 comma 000 x to the 5th power

C(x)=5,000x+2,000x3

Answer 1

To derive the polynomial \( C(x) \) that represents the final amount of the investment after 7 years, we need to consider both deposits and how long each amount will earn interest based on the interest rate \( r \).

1. **First deposit of $5,000** at the beginning of the first year:
- This amount will earn interest for 7 years. Therefore, its contribution to the final amount is:
\[
5,000 \cdot x^7
\]

2. **Second deposit of $2,000** at the beginning of the third year:
- This amount will earn interest for 5 years (since it's deposited at the beginning of year 3 and we are looking at the end of year 7). Thus, its contribution is:
\[
2,000 \cdot x^5
\]

Putting these contributions together, the polynomial \( C(x) \) is:
\[
C(x) = 5,000x^7 + 2,000x^5
\]

So, the correct response is:
\[
C(x) = 5,000x^7 + 2,000x^5
\]