Problem
ALG-B1-M11-P020 Symmetric sums
#20
★★★★★ Level 5 of 5
Find all real triples \((a,b,c)\) such that \(a+b+c=0\), \(a^2+b^2+c^2=14\), \(a^3+b^3+c^3=18\).
Find \(ab+bc+ca\) and \(abc\), then form the polynomial.
From the first two conditions, \(ab+bc+ca=-7\). By the zero-sum identity, \(3abc=18\), so \(abc=6\). Thus \(a,b,c\) are roots of \(t^3-7t-6=0=(t-3)(t+1)(t+2)\). Answer: all permutations of \((3,-1,-2)\).
Strategy: symmetry turns the triple into roots of a polynomial.