# Answer to Question #10310 in Algebra for nks prasad

Question #10310

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)

Expert's answer

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.

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.

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