Let
be a polynomial. Show that P {\displaystyle {}P} is an odd function if and only if a k = 0 {\displaystyle {}a_{k}=0} for all even indices.
![{\displaystyle {}P=\sum _{k=0}^{d}a_{k}x^{k}\in \mathbb {R} [X]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6abf1d478bc7bda3e60c6a47909e90b0a912f610)
is an
odd function
if and only if
for all even indices.
