TITLE: Does there exist an even polynomial that has a term with an odd power?
QUESTION [12 upvotes]: Does there exist an even polynomial (i.e. $f(-x)=f(x)$ for all $x$) that has at least one odd power of $x$?

REPLY [2 votes]: Split $\,f(x)\,$ into its even part $\,f_0(x) = \sum f_{2k} x^{2k}$ and odd part $\,f_1(x) = \sum f_{2k+1} x^{2k+1}$
Thus $\,f(x) = f_0(x)+ f_1(x)\ $ where $\ f_0(-x) = f_0(x),\,$ and $\ f_1(-x) = -f_1(x)$
So $\ \ f(-x) = f_0(x)-f_1(x)\ $ 
So $\ \ f(-x) = f(x)\!\iff\! 2 f_1(x) = 0\!\iff\! f_1(x) = 0\ $ (assuming $\,2\,$ is cancellable).