Use 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
To represent the final amount of a 7-year investment using the given information, we can use the compound interest formula with the given polynomial substitution.
Let's start by representing the amount deposited at the beginning of each year as follows:
- At the beginning of the first year: $5,000
- At the beginning of the third year: $2,000
Using the given polynomial substitution, we have x = 1 + r, so x - 1 = r.
Since the interest rate is the same each year, we can express the amount deposited at the beginning of year 3 as the amount deposited at the beginning of year 1 multiplied by (x-1)^2, as it is 2 years after year 1.
Now, let's build the polynomial expression for the 7-year investment amount:
C(x) = 5000*(x-1) + 2000*(x-1)^2
To simplify this expression, we can expand the second term:
C(x) = 5000*(x-1) + 2000*(x^2 - 2x + 1)
Simplifying further:
C(x) = 5000x - 5000 + 2000x^2 - 4000x + 2000
Combining like terms:
C(x) = 2000x^2 + 1000x - 3000
Therefore, the model polynomial for the final amount of a 7-year investment, given the deposits of $5,000 at the beginning of the first year and $2,000 at the beginning of the third year, is C(x) = 2000x^2 + 1000x - 3000.
To find the final amount of a 7-year investment, we can use the given formula x = 1 + r, where r is the interest rate paid each year.
Let's break down the investment into two parts:
1. The initial deposit of $5,000 at the beginning of the first year (year 0).
2. The additional deposit of $2,000 at the beginning of the third year (year 2).
Now, let's create the model polynomial, C(x), to represent the final amount after 7 years.
Step 1: Calculate the formula for the first $5,000 deposit:
The initial deposit will be compounded for 7 years, so we will apply the formula x = 1 + r seven times:
C(x) = (1 + r)^7
Step 2: Calculate the formula for the additional $2,000 deposit:
The additional deposit will be compounded for 5 years (years 3 to 7), so we will apply the formula x = 1 + r five times:
C(x) = C(x) * (1 + r)^5
Step 3: Multiply the two formulas together to get the final polynomial:
Since both deposits are independent, we can multiply the formulas:
C(x) = (1 + r)^7 * (1 + r)^5
Simplifying this expression:
C(x) = (1 + r)^(7 + 5)
C(x) = (1 + r)^12
The final polynomial, C(x), to represent the final amount after 7 years with a $5,000 deposit at the beginning of the first year and a $2,000 deposit at the beginning of the third year is (1 + r)^12.