Suppose $n\ge 2$. Consider the polynomial $[Q_n(x) = 1-x^n - (1-x)^n .]$Show that the equation $Q_n(x) = 0$ has only two real roots, namely $0$ and $1$.
I have solve the given problem by showing that $g(x)=0$ for $x=0,1$. And sssuming that it true for $k\in\mathbb N$. And for $x=k+1$ I applied induction and showed that the polynomial has two roots.
But I want a solution which uses the idea of real analysis such that we assume that there is Another real number $\alpha$ in $(0,1)$ which breaks the given Sen into two sets $(0,\alpha)$ and $(\alpha ,1)$ and then the solution proceeds. I did not know and understand the complete idea to do the problem this way, so can anybody show it how to do it using the ideas of real analysis, thank you.