Let P(x) be a real polynomial of degree less than or equal to 4, such that there are at least 5 distinct solutions to P(x) = 5. Find P(5).
I've been trying to wrap my head around this supposedly basic problem, but I keep running into a blank wall. I can get at most 4 distinct solutions to P(x) = 5 for a given degree 4 polynomial, but I keep coming to the conclusion that 5 distinct solutions is impossible. For example, I used P(x) = x^4 - 2x^2 + 5.5. Given this P(x) it is indeed possible to find 4 distinct x's for which P(x) = 5. What am I missing here? Is there perhaps a complex solution, and if so, how would I find it?