Question

If $a,b,c$ are the roots of $x^3+qx+r=0$, form the equation whose roots are
$bc+\frac{1}{a},ca+\frac{1}{b},ab+\frac{1}{c}$.

Answer 1

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

$bc+\frac{1}{a}=\frac{1+abc}{a}=\frac{1-r}{a}$

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

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

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

$⟹r{y}^3+q(1-r)y^2+(1-r{)}^3=0$