Problem
ALG-B1-M03-P023 Three Symmetric Sums
#23
★★★★★ Level 5 of 5
Find all real triples \(x,y,z\) such that \(x+y+z=3\), \(xy+yz+zx=3\), \(xyz=1\).
Treat \(x,y,z\) as roots of a cubic polynomial.
The numbers \(x,y,z\) are roots of \(t^3-3t^2+3t-1\).
But \(t^3-3t^2+3t-1=(t-1)^3\). Therefore all three roots equal \(1\), so \(x=y=z=1\).
This is a first serious step toward symmetric polynomials.