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Use the Fundamental Theorem of Algebra and the Conjugate Root Theorem to show that any odd degree polynomial equation with real coefficients has at least one real root.

Here's the outline of a proof, you can fill in the details.

Using The Fundamental Theorem of Algebra , the number of roots of the polynomial is equal to the degree, so the number of roots is odd. List the roots , r1 , r2 , r3 ,..... Since the conjugate of a root is also a root , if there is no real root, the roots may be removed from the list in conjugate pairs, two at time , until the list is exhausted. But this means there is an even number of roots in the list, which we know is false. Thus, there must be a real root.  


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