Question

two boys are given identical monetary gifts which they invest in the same bank at the same time. at the end of the investment period, tonys investment has increased to the second power and he deposits an amount that is 6 times his original investment, plus $100. thamuds investment has also increased to the second power. he withdraws $50 but then deposits 4 times his 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 boys accounts?

Explanation:

Step1: Find Tony's current amount

Tony's original investment is $x$. His investment increased to the second power, so that's $x^2$. Then he deposits 6 times his original investment plus $100$, so we add $6x + 100$. So Tony's current amount is $x^2 + 6x + 100$.

Step2: Find Thamud's current amount

Thamud's original investment is also $x$. His investment increased to the second power, so $x^2$. He withdraws $50$, so we subtract $50$, then deposits 4 times his original investment, so we add $4x$. So Thamud's current amount is $x^2 - 50 + 4x = x^2 + 4x - 50$.

Step3: Find the product of their current amounts

We need to multiply $(x^2 + 6x + 100)$ and $(x^2 + 4x - 50)$.
First, use the distributive property (FOIL for polynomials):
\[

$$\begin{align*} &(x^2 + 6x + 100)(x^2 + 4x - 50)\\ =&x^2(x^2 + 4x - 50) + 6x(x^2 + 4x - 50) + 100(x^2 + 4x - 50)\\ =&x^4 + 4x^3 - 50x^2 + 6x^3 + 24x^2 - 300x + 100x^2 + 400x - 5000\\ \end{align*}$$

\]
Now, combine like terms:

• $x^4$: $1x^4$
• $x^3$: $4x^3 + 6x^3 = 10x^3$
• $x^2$: $-50x^2 + 24x^2 + 100x^2 = 74x^2$
• $x$: $-300x + 400x = 100x$
• Constants: $-5000$

So the simplified expression is $x^4 + 10x^3 + 74x^2 + 100x - 5000$.

Answer:

The simplified expression for the product of the current amount in the boys' accounts is $\boldsymbol{x^4 + 10x^3 + 74x^2 + 100x - 5000}$.