$x^3 +ax^2 +bx +c = 0 $ has roots $\alpha$, $\beta$ and $\gamma$, where $\gamma=\alpha+\beta $. Show that $a^3 -4ab +8c = 0$ . I have done it through relations of roots ($\alpha$ + $\beta$ + $\gamma$ = $\frac{-b}{a} = -a$ and etc.) and substituing $\gamma$ into every single equation, then substituing them into what is required to be shown (sub in $ a, b$ and $c$). Is there a faster method to show this, maybe a smarter substitution?
Is there a better way to show $a^3 -4ab +8c = 0$ for this polynomial
cubicspolynomialsroots
Best Answer
Hint: you dont need to sub in every equation
$$\alpha+\beta+\gamma=-a$$ $$\gamma=-a/2$$
As this is a root ....substituiting in cubic $x^3+ax^2+bx+c=0$ we have $$a^3-4ab+8c=0$$