Don't really know what I'm doing with this proof here a little help would be amazing right now. Here is the question.
f(x)=a0+a1x+...+anx^n (x ϵ ℝ)
be a polynomial function, an ≠ 0. Suppose n is odd. show that the equation f(x)=0 has at least one solution in ℝ.
I like to do it with sequences instead. It is more athestically pleasing. (However, the proof using field theory is even more nicer but this is not an algebra thread.)
Let be a polynomial in of odd degree. Define . This is a sequence. Now argue that without lose of generality if the leading coefficient it positive. And define and show . Thus, this means by definition there is a number so that and there is a number so that . Thus, there is a zero by intermediate value theorem. Q.E.D.