Unformatted text preview: d. The
authors are fully aware that the full impact and profound nature of the Fundamental Theorem
of Algebra is lost on most students this level, and that’s ﬁne. It took mathematicians literally
hundreds of years to prove the theorem in its full generality, and some of that history is recorded
here. Note that the Fundamental Theorem of Algebra applies to polynomial functions with not
only real coeﬃcients, but, those with complex number coeﬃcients as well.
Suppose f is a polynomial of degree n with n ≥ 1. The Fundamental Theorem of Algebra guarantees
us at least one complex zero, z1 , and, as such, the Factor Theorem guarantees that f (x) factors
as f (x) = (x − z1 ) q1 (x) for a polynomial function q1 , of degree exactly n − 1. If n − 1 ≥ 1, then
the Fundamental Theorem of Algebra guarantees a complex zero of q1 as well, say z2 , and so the
Factor Theorem gives us q1 (x) = (x − z2 ) q2 (x), and hence f (x) = (x − z1 ) (x − z2 ) q2 (x). We can
continue this process exactly n times, at which point...
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