Question

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

C(x)=5,000x+2,000x3
𝐶
(
𝑥
)
=
5
,
000
𝑥
+
2
,
000
𝑥
3
cap c times x is equal to 5 comma 000 x plus 2 comma 000 x cubed

C(x)=5,000x7+2,000x3
𝐶
(
𝑥
)
=
5
,
000
𝑥
7
+
2
,
000
𝑥
3
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
𝐶
(
𝑥
)
=
5
,
000
𝑥
7
+
2
,
000
𝑥
5
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

Answer 1

To correctly model the final amount \( C(x) \) of a 7-year investment where \( x = 1 + r \) and taking into account the deposits made, we should consider the following:

1. A deposit of $5,000 is made at the beginning of the first year. This amount will earn interest for 7 years.

The amount for this deposit after 7 years will be:
\[
5,000x^7
\]
because it is invested for the full 7 years.

2. A deposit of $2,000 is made at the beginning of the third year. This amount will earn interest for 5 years (from the start of year 3 to the end of year 7).

The amount for this deposit after 5 years will be:
\[
2,000x^5
\]
because it is invested for 5 years.

Combining these two contributions, the model polynomial for the total amount after 7 years is:

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

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