<s>[INST] Value of $a^3+b^3+c^3$ when values of $a+b+c$, $abc$ and $ab+bc+ca$ are known. Is there a way to to find out what $a^3+b^3+c^3$ evaluates to, when the values of $abc$, $ab+bc+ca$ and $a+b+c$ are given?
Alternatively, is there a way to express $a^3+b^3+c^3$ in terms of the aforementioned expressions?
 [/INST] Use Newton's identities:
\begin{align}
  p_1 &= e_1\\
  p_2 &= e_1p_1-2e_2\\
  p_3 &= e_1p_2 - e_2p_1 + 3e_3 \\
\end{align}
where
\begin{align}
  e_1 &= a+b+c    &\qquad p_1 &= a^1+b^1+c^1\\
  e_2 &= ab+bc+ca &\qquad p_2 &= a^2+b^2+c^2\\
  e_3 &=abc       &\qquad p_3 &= a^3+b^3+c^3\\
\end{align}
Newton's identities give
$$
p_3 = e_1^3-3e_1e_2+3e_3
$$
that is
$$
a^3+b^3+c^3=(a+b+c)^3-3(a+b+c)(ab+bc+ca)+3abc
$$
</s>