I'm following up from this question:
Solve a polynomial involving geometric progression?
I have had trouble with this question:
"Solve the equation $8x^3−38x^2+57x−27=0$" if the roots are in geometric progression.
Any help would be appreciated.
I eventually solved the equation (please check @lab bhattacharjee's answer), but I got two answers for $r$, which were 3/2 and 2/3. How exactly do I know which is correct? I've been stuck for ages, so I'd prefer a full explanation rather than hints.
Any help would be appreciated. Thanks.
Solution :
Since it is a cubic polynomial you can find one of the root in the constant term of the equation : which is here 27 , so factors are 1,3 only. So by putting 1 in equation it gives x -1 as one factor of the equation.
Therefore by dividing the complete polynomial with x -1 you get $8x^2-30x+27$ which after solving gives you two more factors
x = $\frac{3}{2}$ and x =$\frac{9}{4}$ Threfore, 1, $\frac{3}{2} ; \frac{9}{4}$ are the roots of equation.
we can see roots are in G.P with common difference $\frac{3}{2}$