Sure! Let's solve the problem step by step, starting by finding the polynomial expression that models the profit and then calculating the profit when 1,000 video game systems are sold.
### Step 1: Define the Revenue and Cost Functions
The revenue [tex]\( R(x) \)[/tex] is given by the polynomial:
[tex]\[ R(x) = 5x^2 + 2x - 80 \][/tex]
The cost [tex]\( C(x) \)[/tex] is given by the polynomial:
[tex]\[ C(x) = 5x^2 - x + 100 \][/tex]
### Step 2: Subtract the Cost from the Revenue to Find the Profit
Profit [tex]\( P(x) \)[/tex] is calculated by subtracting the cost from the revenue:
[tex]\[ P(x) = R(x) - C(x) \][/tex]
Substitute the given polynomials:
[tex]\[ P(x) = (5x^2 + 2x - 80) - (5x^2 - x + 100) \][/tex]
### Step 3: Simplify the Profit Expression
Distribute the negative sign and combine like terms:
[tex]\[ P(x) = 5x^2 + 2x - 80 - 5x^2 + x - 100 \][/tex]
Combine the [tex]\( x^2 \)[/tex] terms and the [tex]\( x \)[/tex] terms:
[tex]\[ P(x) = (5x^2 - 5x^2) + (2x + x) + (-80 - 100) \][/tex]
[tex]\[ P(x) = 0 + 3x - 180 \][/tex]
[tex]\[ P(x) = 3x - 180 \][/tex]
Therefore, the expression that represents the profit is:
[tex]\[ \boxed{3x - 180} \][/tex]
### Step 4: Calculate the Profit When 1,000 Video Game Systems are Sold
To find the profit when 1,000 video game systems are sold, substitute [tex]\( x = 1000 \)[/tex] into the profit expression:
[tex]\[ P(1000) = 3(1000) - 180 \][/tex]
[tex]\[ P(1000) = 3000 - 180 \][/tex]
[tex]\[ P(1000) = 2820 \][/tex]
Therefore, if 1,000 video game systems are sold, the company's profit is:
[tex]\[ \boxed{2820} \][/tex]
In summary:
- The profit can be modeled by the polynomial expression [tex]\( \boxed{3x - 180} \)[/tex].
- If 1,000 video game systems are sold, the company's profit is \$ \boxed{2820}.
