Let $x_{1}, x_{2}, \ldots, x_{n}$ be the roots of the equation $a_{n} x^{n}+\ldots+a_{1} x+a_{0}=0$. What will be the roots of the equations

a) $a_{0} x^{n}+\ldots+a_{n-1} x+a_{n}=0$

b) $a_{n} x^{2 n}+\ldots+a_{1} x^{2}+a_{0}=0$?
By the meaning of the problem $a_{0} \neq 0$, hence, all roots are non-zero.

By the substitution a) $x=t^{-1}$, b) $x^{2}=t$ the equation is reduced to the original form $a_{n} t^{n}+\ldots+a_{1} t+a_{0}=0$.

## Answer

a) $\frac{1}{x_{1}}, \ldots, \frac{1}{x_{n}}$
b) $\pm \sqrt{x_{1}}, \ldots, \pm \sqrt{x_{n}}$.

## Problem