Q.The number of positive zeros of the polynomial $\displaystyle\sum_{r=0}^{n} {}^{n}C_r(-1)^r x^r$ is :

So $p\le s$, and $p$ has the same parity as $s$ (if $s=4$, then $p\in\{0,2,4\}$, never $1$ or $3$). Applying the same rule to $P(-x)$ bounds the number of negative roots (since a negative root of $P$ is a positive root of $P(-x)$).

Combining both bounds — a lower bound on non-real roots. For a degree-$n$ polynomial, with $m$ = sign changes of $P(x)$ and $k$ = sign changes of $P(-x)$: since the equation has exactly $n$ roots in all (Fundamental Theorem of Algebra) and at most $m+k$ of them can be real,

$\text{number of non-real roots} \ge n-(m+k).$ …