In this talk, we shall discuss a recent result on polynomial equations in an arbitrary real composition algebra A. If A is the algebra of complex numbers, then every nonconstant polynomial over A has a root in A (the fundamental theorem of algebra). Certain polynomial equations over the quaternions H, however, are known not to have any solutions in H nor in any ring extension of H. On the other hand, by a result of Eilenberg and Niven, every nonconstant polynomial equation in H whose highest degree term is a monomial does have a solution in H. It turns out that there is a much wider class of polynomial equations with a tame tail that also have this property. The same result holds for every composition algebra of real dimension greater than one.