If the polynomial equation $a_n{x}^n+a_{n-1}{x}^{n-1}+\mathrm{.........}+a_2{x}^2+a_{1}x+a_0=0$, n positive integer, has two different real roots $\alpha$ and $\beta$, then between  $\alpha$ and $\beta$, the equation $n{a}_n{x}^{n-1}+(n-1)a_{n-1}{x}^{n-2}+\mathrm{.......}+a_1=0$ has

Let $f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\mathrm{.........}+a_2{x}^2+a_{1}x+a_0=0$, which is a polynomial function in x of degree n of degree n. Hence $f\left(x\right)$ is continuous and differentiable for all x

Let $\alpha <\beta$ we are given, $f\left(\alpha \right)=0=f\left(\beta \right)$

By Rolle's theorem, $f\text{'}\left(c\right)=0$ for some value c, $\alpha <c<\beta$

Hence the equation $f\text{'}\left(x\right)=n{a}_n{x}^{n-1}+(n-1)a_{n-1}{x}^{n-2}+\mathrm{......}+a_1=0$

has atleast one root between $\alpha$ and $\beta$