\displaystyle p\left( x\right) =\sum_{i=0}^n a_ix^i\\
p\left( x\right) =p\left( -x\right) \Longrightarrow \sum_{i=0}^n a_ix^i = \sum_{i=0}^n a_i(-x)^i \Longrightarrow 2\left( \sum_{ \left\{ i \in \mathbb{N}|\exists k\in \mathbb{N}, i=2k-1, i\leq n \right\} } a_ix^i \right) =0\\
a_i=0, \forall i\in \left\{ m \in \mathbb{N} | \exists k\in \mathbb{N}, m=2k-1, m\leq n \right\}