Question

If the polynomial equation
$a_0{x}^n+a_{n-1}{x}^{n-1}+a_{n-2}{x}^{n-2}+...+a_2{x}^2+a_{1}x+a_0=0$
n positive integer, has two different real roots α and β, then between α and β, the equation
$n{a}_n{x}^{n-1}+\left(n-1\right)a_{n-1}{x}^{n-2}+...+a_1=0\mathrm{has}$
(a) exactly one root
(b) almost one root
(c) at least one root
(d) no root

Answer 1

(c) at least one root

We observe that, $n{a}_n{x}^{n-1}+\left(n-1\right)a_{n-1}{x}^{n-2}+...+a_1=0$ is the derivative of the
polynomial $a_n{x}^n+a_{n-1}{x}^{n-1}+a_{n-2}{x}^{n-2}+...+a_2{x}^2+a_{1}x+a_0=0$

Polynomial function is continuous every where in R and consequently derivative in R
Therefore, $a_n{x}^n+a_{n-1}{x}^{n-1}+a_{n-2}{x}^{n-2}+...+a_2{x}^2+a_{1}x+a_0$ is continuous on $\left[\alpha ,\beta \right]$ and derivative on $\left(\alpha ,\beta \right)$.
Hence, it satisfies the both the conditions of Rolle's theorem.

By algebraic interpretation of Rolle's theorem, we know that between any two roots of a function $f\left(x\right)$, there exists at least one root of its derivative.

Hence, the equation $n{a}_n{x}^{n-1}+\left(n-1\right)a_{n-1}{x}^{n-2}+...+a_1=0$ will have at least one root between $\alpha \mathrm{and}\beta$.