Find the relation between $a, b, c, d$ if the roots of $ax^3+bx^2+cx+d=0$ are in geometric progression.
By considering $(\alpha+\beta)(\beta+\gamma)(\alpha+\gamma)$ show that the above cubic equation has two roots equal in size but opposite in sign if and only if $ad=bc$.
I can do the second part if I am given some hints on the first. I can use $\beta$ as the middle root and make $\alpha=\beta/r$ and $\gamma=r \times \beta$, but haven't got anywhere so far.
Best Answer
The monic polynomial with roots $\beta/r,\beta, r\beta$ is $$ x^3-\beta\left(1+r+\frac1r\right) x^2+\beta^2\left(1+r+\frac1r\right)x-\beta^3$$ We find $\beta=-\frac cb$ and $\beta^3=-\frac da$, hence $$db^3=ac^3$$ as necessary condition. Why is it also sufficient? What additional condition(s) would we need if we wanted the roots to be real?