Express the power sum $\sum^{n}_{k=1}\alpha_{k}^n$ in terms of the coefficients of the polynomial $x^n+bx+c$.

Question

Given that the roots of the polynomial $x^n+bx+c$ are $\alpha_1,\alpha_2,...,\alpha_n.$

I need to express the power sum $\sum^{n}_{k=1}\alpha_{k}^n$ in terms of $b,c,n.$ Can someone help me, please? Thanks.

Answer 1

Recall: the sum of the roots $\sum \alpha_k$ is the coefficient of $x^{n-1}$, hence is $b$ (resp., $0$) if $n=2$ (resp., $n > 2$). Thus, $$ \begin{aligned}[t] \sum_{k=1}^{n} \alpha_{k}^{n} = \sum_{k=1}^{n} -(b \alpha_k + c) &= - b \sum_{k=1}^{n} \alpha_k - c \sum_{k=1}^{n} 1 \\ &= -b \left. \begin{cases} \!b &\text{if $n=2$}\\ \!0 &\text{if $n > 2$} \end{cases} \!\right\} - cn = \begin{cases} \!-b^2-cn &\text{if $n=2$}\\ \!-cn &\text{if $n > 2$.} \end{cases} \end{aligned} $$