### Polynomial identity

I'm pretty interested in solving the following kind of problem using Sage: ``Let R be a polynomial ring in, say, x,y,z as variables over a field k. I'd like to find field-elements a,b,c such that

a(x^2+y^2)+b(xy+zx)+c(xyz)==0, if they exist (I know they do)''

so that Sage returns (a,b,c)=(0,0,0). That seems ~~is ~~to be an easy matter if one can traduce the polynomial identity into a vector space identity. I proved to be unable to do so.

I've to say that my polynomial identities are quite more cumbersome and include up to 7 variables so working them by-hand is almost impossible in a finite ammount of time or patience.