Question

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

Answer 1

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

$\frac{b}{c}+\frac{c}{b}=\frac{b^2+c^2}{bc}=a\frac{a^2+b^2+c^2-a^2}{abc}=a\frac{(\sum a{)}^2-2\sum ab-a^2}{abc}=a\left(\frac{a^2+2q}{r}\right)$

$⟹y=x\left(\frac{x^2+2q}{r}\right)$

$⟹x^3+2qx–ry=0\dots \dots \dots \dots \dots ..\left(2\right)$

Subtracting (1) from (2), we have

$qx–ry-r=0⟹x=\left(\frac{r}{q}\right)(1+y)\dots \dots \dots \dots \dots ..\left(3\right)$

Substituting value of $x$ from (3) in (1), we get

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

$⟹r^2{y}^3+3r^2{y}^2+(2r^2+q^3)y+(r^2+2q^3)=0$