A polynomial is continuous, roughly meaning you can draw its graph without lifting your pen. So if P(x) is negative somewhere (say P(a) < 0) and positive somewhere else (say P(b) > 0), then it makes sense that P must be zero somewhere between a and b (meaning P(c) = 0 for some value of c between a and b). Corollary: Every odd degree polynomial has a real root somewhere!.