The “fundamental theorem of algebra” for quaternions

The continuity of ƒ at oo follows from the fact that \f(x)\ increases without limit as \x\ increases without limit, a fact which is obvious from the definition of/. It should be noted that this argument is not valid for polynomials of degree n with more than one term of degree n. Theorem 1 is then a consequence of the following theorem which asserts that "essentially" the equation/(x) =0 has exactly n solutions.