Question

2. Each year a certain amount of money is deposited in an account which pays an

annual interest rate of r so that at the end of each year the balance in the account is
multiplied by a growth factor of x = 1+r. $500 is deposited at the start of the first
year, an additional $200 is deposited at the start of the next year, and $600 at the
start of the following year.
a. Write an expression for the value of the account at the end of three years in
terms of the growth factor x.
b. What is the amount (to the nearest cent) in the account at the end of three
years if the interest rate is 2%?
3. Consider the polynomial function p given by p(x) = 5x³ + 8x²-3x + 1. Evaluate the
function at x = -2.

Answer 1

2a. ((500x +200)x +600)x

2b. $1350.68

3. -1

Explanation:

2. You want to know the balance in an account at the end of 3 years with deposits at the beginning of successive years being $500, $200, and $600, and with deposits having an annual growth factor of x. You want (a) an expression for the balance in terms of x, and (b) the balance when x=1+2%.

3. You want the value of p(-2) for p(x) = 5x³ +8x² -3x + 1.

2. Account Balance

a. Expression

$500 is deposited at the beginning of the first year. The problem statement tells us that this has been multiplied by growth factor x by the end of the year, so the balance at that point is 500x.

At the beginning of the second year, $200 is added to the account, and the entire amount is multiplied by the growth factor for the year. That makes the balance at the end of the second year be ...

(500x +200)x

Then $600 is added, and again the account grows by a factor of x. At the end of the third year, the balance is ...

balance = ((500x +200)x +600)x

b. Balance for r=2%

The growth factor is given as x = 1 +r. When r = 2%, this becomes x = 1.02. Then the balance at the end of year 3 is ...

((500·1.02 +200)·1.02 +600)·1.02 = 1350.68

The amount in the account at the end of three years is $1350.68.

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3. Polynomial value

The value of p(x) = 5x³ +8x² -3x +1 for x=-2 is found by substituting -2 where x is found. Evaluation can be easier by rewriting the polynomial to Horner form:

p(x) = ((5x +8)x -3)x +1

p(-2) = ((5(-2) +8)(-2) -3)(-2) +1 = ((-2)(-2) -3)(-2) +1 = (1)(-2) +1 = -1

The value of the function for x=-2 is -1.

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Additional comment

You will note that the expression in problem 2 is also written in Horner form. If it were expanded, it would be 500x³ +200x² +600x. Evaluation takes fewer steps when Horner form is used.