<s>[INST] Manipulating roots of a cubic Given that $A,B,C$ are the roots of the equation $x^3-5x^2+x+1$, how do I find the value of $$\dfrac{A}{B+C}+\dfrac{B}{A+C}+\dfrac{C}{A+B}$$ I know the Vieta's formulas but I am not able to manipulate the above expression into something known. And taking the LCM doesn't help. Please help me out. Thank you.
 [/INST] Hint: $(A+B)(B+C)(C+A) = (AB+AC+B^2+BC)(C+A) = ABC+A^2B+AC^2+A^2C+B^2C+B^2A+BC^2+BCA = 2ABC + A^2(B+C) + B^2(C+A) + C^2(A+B) = 2\cdot 1 + A^2(5-A) + B^2(5-B) + C^2(5-C) = 2 + 5(A^2+B^2+C^2) - (A^3+B^3+C^3) = 2 + (A+B+C) + 3 = 2+5+3=10$
</s>