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For all those interested I have manged to do what I wanted using sympy.

 sage: var('x1,x2,x3,w1,w2,w3')
 sage: f=w1+w2+w3==2;g=(w1*x1)+(w2*x2)+(w3*x3)==0;h=(w1*(x1)^2)+(w2*(x2)^2)+(w3*(x3)^2)==(2/3);j=(w1*(x1)^3)+(w2*(x2)^3)+(w3*(x3)^3)==0;k=(w1*(x1)^4)+(w2*(x2)^4)+(w3*(x3)^4)==(2/5);l=(w1*(x1)^5)+(w2*(x2)^5)+(w3*(x3)^5)==0
 sage: import sympy
 sage sympy.solve([f,g,h,j,k,l],w1,w2,w3,x1,x2,x3)

output:

  [{w1: 5/9, w2: 5/9, w3: 8/9, x1: -sqrt(15)/5, x2: sqrt(15)/5, x3: 0},
  {w1: 5/9, w2: 5/9, w3: 8/9, x1: sqrt(15)/5, x2: -sqrt(15)/5, x3: 0},
  {w1: 5/9, w2: 8/9, w3: 5/9, x1: -sqrt(15)/5, x2: 0, x3: sqrt(15)/5},
  {w1: 5/9, w2: 8/9, w3: 5/9, x1: sqrt(15)/5, x2: 0, x3: -sqrt(15)/5},
  {w1: 8/9, w2: 5/9, w3: 5/9, x1: 0, x2: -sqrt(15)/5, x3: sqrt(15)/5},
  {w1: 8/9, w2: 5/9, w3: 5/9, x1: 0, x2: sqrt(15)/5, x3: -sqrt(15)/5}]

If anyone can enlighten me on how to use the polynomial rings to solve polynomial systems of equations I'd be very happy (I am also a noob at sagemath btw)

Thank you.