If the equation $a_n{x}^n+a_{n-1}{x}^{n-1}+\dots +a_{1}x=0$ has a positive root $\alpha ,$ then the equation $n{a}_n{x}^{n-1}+(n-1)a_{n-1}{x}^{n-2}+\dots +a_1=0$ has

Let $f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\dots +a_{1}x$. It is given that $\alpha$ is a positive root of $f\left(x\right)=0$.

 Also, we observe that $f\left(0\right)=0\text{.}$Thus, $x=0$and $x=\alpha$ are two roots of $f\left(x\right)$

By Rolle's theorem, $f^{\text{'}}\left(x\right)$ has at least one real root between 0 and $\alpha .$

$\therefore n{a}_n{x}^{n-1}+(n-1)a_{n-1}{x}^{n-2}+\dots +a_1=0$

has at least one real root between 0 and$\alpha .$