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

Step 1. Recognise the binomial expansion. $\displaystyle\sum_{r=0}^n {}^nC_rx^r(-1)^r={}^nC_0-{}^nC_1x+{}^nC_2x^2-\cdots+(-1)^n{}^nC_nx^n=(1-x)^n$ (the binomial expansion of $(1-x)^n$).

Step 2. Find the zero(s) of $(1-x)^n$. $(1-x)^n=0 \iff x=1$, with multiplicity $n$ (since the factor $(1-x)$ — equivalently $-(x-1)$ — appears $n$ times).

Step 3. Count positive zeros, with multiplicity. Since $x=1>0$ is repeated $n$ times, the number of positive zeros counted with multiplicity is $n$. …

Descartes' Rule of Signs — recognise the coefficients as the binomial expansion of $(1-x)^n$, whos …

• Answering '1' by counting only the DISTINCT positive zero, rather than counting with multiplicity as the question (and Descartes' bound) requi …