Question

HELP PLEASE WILL FAN AND MEDAL!!!
A company estimates its total profit (profit = total revenue minus total cost) as
P(x) = 2x^5 − 3x^4 − 5x^2 − 2, where P is in thousands of dollars and x is the number of years elapsed since the company was founded. How many times can the total profit become exactly zero? Hint: Use Descartes's rule of signs.

Answer 1

@rishavraj

Answer 2

@mathmate

Answer 3

That's a very clever question to apply the rule of Descarts by ignoring the negative zeroes.

Since the formula applies only AFTER the company was founded, so any value of x<0 will not count, since they apply to time before the company was founded.

Descartes rule says that the \(possible\) number of POSITIVE zeroes is the number of change of signs of the coefficients. The reason it says "possible" is because some roots could be complex, and does not give a real zero. However, if the number of sign changes is odd, then there is at least one real, positive zero, because complex roots come in conjugate pairs.

Is that enough information for you to solve the problem?
Start by applying Descartes rule and find the possible number of POSITIVE zeroes.