To model the final amount \( C(x) \) after an 8-year investment with given deposits at specified times, we need to account for how long each deposit will earn interest by the end of the 8 years.
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The $500 deposit at the beginning of the first year will be invested for 8 years. Thus, this amount will be represented as: \[ 500x^8 \]
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The $800 deposit at the beginning of the fourth year will be invested for 5 years (from year 4 to year 8). Thus, this amount will be represented as: \[ 800x^5 \]
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The $370 deposit at the beginning of the eighth year will not earn any interest since it is deposited at the beginning of year 8. Thus, this amount is simply: \[ 370 \]
Putting it all together, the model polynomial \( C(x) \) is given by: \[ C(x) = 500x^8 + 800x^5 + 370 \]
Therefore, the correct response is: \[ C(x) = 500x^8 + 800x^5 + 370 \]