A more down-to-earth way to see that every cubic polynomial has a real root (and hence a linear factor) is to notice that for large x, x, x, the lead term a x 3 ax^3 a x 3 dominates, so the sign of f (x) f(x) f (x) for large positive x x x is the sign of a, a, a, and the sign of f (x) f(x) f (x) for large negative x x x is the sign of − a.