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?

Yes, indeed,
$$(a+b+c)^3$$
$$=a^3+b^3+c^3+3a^2b+3a^2c+3b^2a+3b^2c+3c^2a+3c^2b+6abc$$
$$=a^3+b^3+c^3+3a^2b+3a^2c+3abc+3b^2a+3b^2c+3abc+3c^2a+3c^2b+3abc-3abc$$
$$=a^3+b^3+c^3+3a(ac+ab+bc)+3b(ac+ab+bc)+3c(ac+ab+bc)-3abc$$
$$=a^3+b^3+c^3+3(a+b+c)(ac+ab+bc)-3abc$$
So
$$\color{red}{a^3+b^3+c^3=(a+b+c)^3+3abc-3(a+b+c)(ac+ab+bc)}$$
