Problem
ALG-B1-M06-P012 Equal values
#12
★★★☆☆ Level 3 of 5
Let \(x+y+z=0\) and \(x^2+y^2+z^2=2\). Prove that the numbers \(x^3-x\), \(y^3-y\), \(z^3-z\) are equal.
Find \(xy+yz+zx\), then consider the polynomial with roots \(x,y,z\).
From the conditions we get \(xy+yz+zx=-1\). Hence \(x,y,z\) are roots of the polynomial \(t^3+(xy+yz+zx)t-xyz=t^3-t-xyz\).
Therefore each of \(x,y,z\) satisfies \(t^3-t=xyz\). Thus \(x^3-x=y^3-y=z^3-z=xyz\).
This is a first step toward the idea of viewing variables as roots of a polynomial.