Question

two girls are given identical monetary gifts which they invest in the same bank at the same time. at the end of the investment period, annas investment has increased to the third power and she deposits an amount that is 4 times her original investment, plus $100. amirahs investment has increased to the second power. she withdraws $100 but then deposits 5 times her original investment. if x represents the amount of the original gift, what is the simplified expression for the product of the current amount in the girls accounts?

Explanation:

Step1: Find Anna's current amount

Anna's original investment is \( x \). Her investment increased to the third power, so that's \( x^3 \). Then she deposits 4 times her original investment plus $100, so we add \( 4x + 100 \). So Anna's current amount is \( x^3 + 4x + 100 \).

Step2: Find Amirah's current amount

Amirah's original investment is \( x \). Her investment increased to the second power, so that's \( x^2 \). She withdraws $100 (so we subtract 100) and then deposits 5 times her original investment (so we add \( 5x \)). So Amirah's current amount is \( x^2 - 100 + 5x \), which can be rewritten as \( x^2 + 5x - 100 \).

Step3: Find the product of their current amounts

We need to multiply Anna's amount \( (x^3 + 4x + 100) \) and Amirah's amount \( (x^2 + 5x - 100) \).

First, use the distributive property (FOIL for polynomials):

\[

$$\begin{align*} &(x^3 + 4x + 100)(x^2 + 5x - 100)\\ =& x^3(x^2 + 5x - 100) + 4x(x^2 + 5x - 100) + 100(x^2 + 5x - 100)\\ =& x^5 + 5x^4 - 100x^3 + 4x^3 + 20x^2 - 400x + 100x^2 + 500x - 10000\\ \end{align*}$$

\]

Now, combine like terms:

• For \( x^3 \): \( -100x^3 + 4x^3 = -96x^3 \)
• For \( x^2 \): \( 20x^2 + 100x^2 = 120x^2 \)
• For \( x \): \( -400x + 500x = 100x \)

So the simplified expression is:

\[
x^5 + 5x^4 - 96x^3 + 120x^2 + 100x - 10000
\]

Answer:

\( x^5 + 5x^4 - 96x^3 + 120x^2 + 100x - 10000 \)