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When we make a table of values to study the behavior

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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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