Polynomials are fundamental objects in algebra, but unfortunately most accounts of them skimp on giving a proper definition. Here we base polynomials on the more basic objects of polynumbers. This lecture is part of the MathFoundations series, which tries to lay out proper foundations for mathematics, and will not shy away from discussing the serious logical difficulties entwined in modern pure mathematics. The full playlist is at http://www.youtube.com/playlist?list=PL5A714C94D40392AB&feature=view_allWe introduce the particular positive polynumber alpha, and show that any polynumber can be written as a linear combination of powers of alpha. Then we define a positive polynomial to be a positive polynumber written in this standard alpha form.
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