Equal Sums of Coefficients of Odd and Even Powers

A companion test to §3.7.3: suppose the sum of the odd-power coefficients of $P(x)$ equals the sum of the even-power coefficients. If $a$ is the coefficient of an odd-degree term in $P(x)$, the coefficient of that same odd degree in $P(-x)$ is $-a$ (odd powers flip sign under $x\mapsto-x$), while every even-degree coefficient is unchanged. So "odd-sum $=$ even-sum in $P$" is exactly the condition that makes all the coefficients of $P(-x)$ sum to zero — i.e. (by §3.7.3, applied to $P(-x)$) that $1$ is a root of $P(-x)=0$, which means $-1$ is a root of $P(x)=0$. …