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Question: theorem 109 suppose that fx is a polynomial of positive...

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Theorem 10.9. Suppose that f(x) is a polynomial of positive degree饥Ral and that r is a root of f(x) in C 1. The conjugate T of r is also a root of f(x) 2. The polynomial (r- r)(r - r) lies in Rr 3. If r is not real, then (a -r)(z -) is a divisor of f(x) in RTheorem 10.7. Every nonconstant polynomial in Cz has a complex root. Equivalently, the only irreducible polymomials in Clx] are the polynomials of degree one, and every polmomial in Cz] of positive degree factors as a product of degree-one polynomials It is not hard to prove Theorem 10.7 as a consequence of Theorem 10.5. In order to do so, we must study conjugation of polynomials in Clr]. First, let us obtain a formula for the coefficients of the product of two polynomials. We have been assuming that polynomials have coefficients in a field, but a ring will do just as well. For instance, we can consider the set Z[x] of polynomials with integer coefficients. We wl work in this level of generality.Theorem 10.5 (Gauss). Every irreducible polymomial in R[r] has degree ei- ther one or two. There are many proofs of the fundamental theorem of algebra. The first correct proof was given in 1799 by Gauss in his doctoral thesis. Every proof requires the use of some results from mathematical analysis (calculus). The amount of analysis required can be reduced to just some elementary calculus, but the weaker the results of analysis that a proof uses, the more intricate the proof tends to be. No proof will be provided here. However, given the profound importance of the fundamental theorem of algebra, one should read through a proof of it at least one time in ones life, if only to get a sense of what is involved in proving it. Even though we will not prove the fundamental theorem, we will illustrate its power by deducing from it the following result.Exercise 10.40. Prove Theorem 10.9. I. For the first part, show that f(x) f(x). 2. For the second part, check that the coefficients of (x - r)(a F) are real, 3. Deduce that if r is a nonreal complex number and - r divides f(x) in 4. Observe that to prove that (x-r) (a -T) divides f(x) in R[r], it suffices to so that (-r( lies in R[r. C[], then (x -r)(x - F) divides f(x) in C] prove the following statement: Suppose f (x) and g(x) are nonzero polyno- mials in R[] and h() is a polynomial in Cr] such that f(x)gx)h(x) Then h() lies in Ra] 5. Prove this last statement Finally, let us deduce Theorem 10.7 from Theorem 10.5.

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