9. Essay; show all work. Bob’s Barber Shop estimates their gross revenue for the second quarter to be given by the polynomial 2x3 + 5x2 – x – 1. The shop estimates their costs for that quarter to be given by 4x2 – 9x – 8. For the second quarter, find and simplify a polynomial that will represent their profit.
in general ...
profit = revenue - cost
= 2x^3 + 5x^2 - x - 1 - (4x^2 - 9x - 8)
= 2x^3 + x^2 + 8x +7
To find the polynomial that represents the profit, we need to subtract the cost polynomial from the revenue polynomial. Let's go through the steps:
Step 1: Start with the revenue polynomial: 2x^3 + 5x^2 - x - 1.
Step 2: Subtract the cost polynomial: (2x^3 + 5x^2 - x - 1) - (4x^2 - 9x - 8).
Step 3: Distribute the negative sign to the terms inside the parentheses: 2x^3 + 5x^2 - x - 1 - 4x^2 + 9x + 8.
Step 4: Simplify by combining like terms: 2x^3 + (5x^2 - 4x^2) + (-x + 9x) + (-1 + 8).
Step 5: Continue simplifying: 2x^3 + x^2 + 8x + 7.
Therefore, the polynomial that represents Bob's Barber Shop's profit for the second quarter is 2x^3 + x^2 + 8x + 7.