[tex]P = X^n-(X^{n-1}+X^{n-2}+...+X+2) \\ \\ X^n- (X^{n-1}+X^{n-2}+...+X+2) = 0 \\ \\ X^n = X^{n-1}+X^{n-2}+...+X+2 \\ \\ X^n-1 = X^{n-1}+X^{n-2}+...+X+1 \Big|\cdot (X-1),\quad X\neq 1\\ \\ (X^n-1)(X-1) = X^n+X^{n-1}+...+X - (X^{n-1}+X^{n-2}+...+X+1) \\ \\ (X^n-1)(X-1) = X^n-1 \\ \\ (X^n-1)(X-2) = 0[/tex]
[tex]\Rightarrow X^n = 1\quad sau\quad X = 2[/tex]
[tex]\Rightarrow x_k^n = 1,\quad k = \overline{1,2,3,..,n-1},\quad x_k\neq 1\\ \Rightarrow x_n=2\\ \\ x_k^{n} = 1 \Big|\cdot x_k^2 \Rightarrow x_k^{n+2} = x_k^2\\ x_n = 2 \Big|^{n+2}\Rightarrow x_n^{n+2} = 2^{n+2}\\ \\ \Rightarrow \displaystyle \sum\limits_{k=1}^n x_k^{n+2} =(x_1^{n+2}+x_2^{n+2}+...+x_{n-1}^{n+2})+x_{n}^{n+2} = \\ \\ =(x_1^2+x_2^2+...+x_{n-1}^2)+(x_{n}^2-x_{n}^2)+x_{n}^{n+2} = \\ \\ =(x_1^2+x_2^2+...+x_n^2) +(-x_n^2+x_n^{n+2})= \\ \\= S_1^2-2S_2+(-2^2+2^{n+2})= \\ \\ = 1+2-4+2^{n+2} = \boxed{2^{n+2}-1}[/tex]
