Question

If $a,b,c$ are the roots of $x^3+qx+r=0$, form the equation whose roots are
$\frac{b+c}{a^2},\frac{c+a}{b^2},\frac{a+b}{c^2}$.

Answer 1

Given equation, $x^3+qx+r=0\dots \dots \dots \dots \dots \dots \left(1\right)$

$\frac{b+c}{a^2}=\frac{a+b+c-a}{a^2}=-\frac{1}{a}$

$⟹y=-\frac{1}{x}⟹x=-\frac{1}{y}\dots \dots \dots \dots .\left(2\right)$

Substituting value of $x$ from (2) in (1), we get

${\left(\frac{-1}{y}\right)}^3+q\left(\frac{-1}{y}\right)+r=0$

$⟹r{y}^3-q{y}^2-1=0$