Answer to Question #10310 in Algebra for nks prasad

Question

using factor theorem, show that x-y, y-z, z-x are the factors of x^2(y-z)+y^2(z-x)+z^2(x-y)

Answer 1

It is interesting property: if values of any two of three variables coincide
then polynomial becomes 0.
f(x,y,x)=f(x,x,z)=f(x,y,y)=0
As we think about
given f(x,y,z)= x^2(y-z)+y^2(z-x)+z^2(x-y) as polynomial with x - variable and
y, z are parameters then
on value x=y we have f(y,y,z)=0, so (x-y)divides
f;
on value x=z we have f(z,y,z)=0, so (x-z)divides f;
And if we consider
f as polynomial from variable y and x,z are parameters then
on value y=z we
have f(x,z,z)=0, so (y-z)divides f;

We proved that (x-y),(x-z),(y-z) are
factors of f(x,y,z) - given polynomial.