Question

Care era formula cu relatiile lui viete pentru x1^3+x2^3+x3^3?

Answer 1

$\text{x}^3+2\text{x}^2+\text{x}+m \\ \\ Formam sistem: \left\{ \begin{array}{c} \text{x}_1^3+2\text{x}_1^2+\text{x}_1+m=0 \\ \\\text{x}_2^3+2\text{x}_2^2+\text{x}_2+m = 0 \\ \\\text{x}_3^3+2\text{x}_3^2+\text{x}_3+m = 0\end{array} \right |_{\Big{\text{adunam}}}\Rightarrow$

$\Rightarrow \text{x}_1^3+\text{x}_2^3+\text{x}_3^3 + 2\text{x}_1^2+2\text{x}_2^3+2\text{x}_3^3+\text{x}_1+\text{x}_2+\text{x}_3+m+m+m = 0 \Rightarrow \\ \\ \Rightarrow \text{x}_1^3+\text{x}_2^3+\text{x}_3^3+2(\text{x}_1^2+\tex\text{x}_2^2+\text{x}_3^2)+\text{x}_1+\text{x}_2+\text{x}_3+3m = 0 \Rightarrow \\ \\ \Rightarrow \boxed{\text{x}_1^3+\text{x}_2^3+\text{x}_3^3= -2(\text{x}_1^2+\tex\text{x}_2^2+\text{x}_3^2)-(\text{x}_1+\text{x}_2+\text{x}_3)-3m}$