Essay; show all work. Bob’s Barber Shop estimates their gross revenue for the second quarter to be given by the polynomial 7x3 – 9x2 – 2x + 4. The shop estimates their costs for that quarter to be given by x2 + 7x – 5. For the second quarter, find and simplify a polynomial that will represent their profit.
P=(7X^3 - 9X^2 - 2X + 4)-(X^2 + 7X -5).
P=7X^3 - 9X^2 - 2X + 4 - X^2 - 7X + 5,
Combine like-terms:
P = 7X^3 - 10X^2 - 9X + 9,
P=(7X^3 - 9X^2 - 2X + 4)-(X^2 + 7X -5).
P=7X^3 - 9X^2 - 2X + 4 - X^2 - 7X + 5,
Combine like-terms:
P = 7X^3 - 10X^2 - 9X + 9,
To find the profit, we need to subtract the costs from the revenue.
The revenue for the second quarter is given by the polynomial 7x^3 - 9x^2 - 2x + 4.
The costs for the second quarter are given by the polynomial x^2 + 7x - 5.
To find the profit, we subtract the costs from the revenue:
Profit = Revenue - Costs
Substituting the given polynomials, we have:
Profit = (7x^3 - 9x^2 - 2x + 4) - (x^2 + 7x - 5)
To simplify this expression, we'll combine like terms.
First, distribute the negative sign to the terms inside the parentheses:
Profit = 7x^3 - 9x^2 - 2x + 4 - x^2 - 7x + 5
Now, combine like terms by adding or subtracting coefficients of the same degree:
Profit = 7x^3 + (-9x^2 - x^2) + (-2x - 7x) + (4 + 5)
Simplifying the expression, we have:
Profit = 7x^3 - 10x^2 - 9x + 9
Therefore, the polynomial that represents their profit for the second quarter is 7x^3 - 10x^2 - 9x + 9.