From the text: Every polynomial of degree $n$ can be expressed as a product of factors. It is possible to find identities relating the sums of the powers of the roots defined by $s_i=\sum_j (α_j)^i i=1,2,3,\dots$, to the coefficients, $ω_j$, of the monic polynomial, without actually finding the roots $α_k$ explicitly. It is straightforward to find recursion relations linking $ω_i$ and $s_i$. We can then use the recursion relations to obtain $s_i$ as functions of $ω_1$, $ω_2$,\dots, $ω_i$ as functions of $s_1$, $s_2$,\dots, $s_i$. The process is laborious. We now describe a more economical way to obtain these expressions for $ω_i$.