{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Stitz-Zeager_College_Algebra_e-book

# When we make a table of values to study the behavior

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online