Let the real roots of the polynomial $P(x) = x^3-6x^2+3x+1$ be $a,b,c $ . Then , find the possible values of $a^2b+b^2c+c^2a$ .
The answer is ${{-3,24}}$ . I’ve tried transformation , and algebraic manipulation , but nothing seems to work . I also found it interesting that the quantity equalling -3 implies $\sum \frac{a}{b} = 3$ , the equality case of the rearrangement inequality . Can someone shed some light on this as well ?