How to find a polynomial identity to evaluate sum of fifth powers?
I saw the following problem on a book of mine:
Let $a,b,c\in \mathbb C$ satisfies $a+b+c=3,a^2+b^2+c^2=5, a^3+b^3+c^3=7$. Compute $a^5+b^5+c^5$.
How one can do it by Sage? I found a mathematician who was able to find the identity that solves the problem.
sage: a = var('a') sage: b = var('b') sage: c = var('c') sage: x = a+b+c sage: y = a^2+b^2+c^2 sage: z = a^3+b^3+c^3 sage: expand((x^5-5*x^3*y+5*x^2*z+5*y*z)/6) a^5 + b^5 + c^5 sage: expand((3^5-5*3^3*5+5*3^2*7+5*5*7)/6) 29/3
But can I use Sage to find the correct identity?