Question

The profit P(x) (in millions of dollars) for a T-short manufacturer can be modeled by P(x) = -2x^3+9x^2-x, where x is the number of T-shirts produced (in millions). Currently, the company produces 4 million T-shirts and makes a profit of $12 million. What lesser number of T-shirts could the company produce and still make the same profit?

Answer 1

You just want to find for what value of x we get
-2x^3 + 9x^2 - x = 12 , or
2x^3 - 9x^2 + x + 12 = 0
You already told me that x = 4 would be a solution, so x - 4
must be a factor.
Use either synthetic or long algebraic division to show that
(x - 4)(2x^2 - x - 3) = 2x^3 - 9x^2 + x + 12
so we get two more solutions from 2x^2 - x - 3 = 0
which factors to (x+1)(2x-3) = 0
x = -1, or x = 3/2, of course we have to reject the x = -1, (not good business
practise to have negative sales)

so when they sell 3/2 or 1.5 million shirts they make the same profit

check:
-2(1.5)^3 + 9(1.5)^2 - 1.5 = 12