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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+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)=5,000x+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

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To represent the final amount of the investments using a polynomial, we take into account the different deposits made in different years and how they grow over time due to the interest rate.

  1. The first deposit of $5,000 is made at the beginning of the first year and will earn interest for 7 years. The value of this deposit at the end of 7 years can be expressed as: \[ 5,000(1 + r)^7 = 5,000x^7 \] (where \( x = 1 + r \)).

  2. The second deposit of $2,000 is made at the beginning of the third year and will earn interest for 5 years (since it will be invested for the 3rd, 4th, 5th, 6th, and 7th years). The value of this deposit at the end of 7 years can be expressed as: \[ 2,000(1 + r)^5 = 2,000x^5 \]

Combining both contributions, the model polynomial representing the total future value of the investment after 7 years is: \[ C(x) = 5,000x^7 + 2,000x^5 \]

Among the available choices, the correct polynomial is: \[ C(x) = 5,000x^7 + 2,000x^5 \]

So the best response is:

  • C(x)=5,000x^7+2,000x^5.