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)=5,000x+2,000x3
cap c times x is equal to 5 comma 000 x 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)=5,000x7+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

Answers

GPT-4o mini
To create a model polynomial \( C(x) \) that represents the final amount of the investment after 7 years, we need to consider the two deposits and how compound interest affects them.

1. **First Deposit:** $5,000 is deposited at the beginning of the first year. The duration of this investment is 7 years. Therefore, the amount after 7 years, using the formula \( C_1 = 5,000 \cdot x^7 \), where \( x = 1 + r \) (the total factor due to interest).

2. **Second Deposit:** $2,000 is deposited at the beginning of the third year. The duration of this investment will be 5 years (from the beginning of year 3 to the end of year 7). Therefore, the amount after 7 years, using the formula \( C_2 = 2,000 \cdot x^5 \).

Combining both deposits, the total amount \( C(x) \) after 7 years can be represented as:

\[
C(x) = 5,000x^7 + 2,000x^5
\]

Thus, the correct polynomial model is:

\[
C(x) = 5,000x^7 + 2,000x^5
\]

So the correct answer is:

**C(x) = 5,000x^7 + 2,000x^5**.