MHB Rational Root of $ax^3+bx+c=0$ is Product of 2 Rational Roots

[kaliprasad]

if for rational a,b,c $ax^3+bx+c=0$ one root is product of 2 roots then that root is rational

 

[Albert1]

if for rational a,b,c $ax^3+bx+c=0$ one root is product of 2 roots then that root is rational

my solution:

Spoiler

let 3 roots be $r,s,t$ and $r=st$
we have :$rst=r^2=\dfrac {-c}{a}---(1)$
$r+s+t=0,\rightarrow s+t=-r---(2)$
$rs+rt+st=r(1+s+t)=r(1-r)=r-r^2=\dfrac {b}{a}---(3)$
$\therefore r=\dfrac {b-c}{a}$ is rational