Question

I was playing with the (beautiful) polynomial x^4 - 10x^2 + 1.
Look what happens:

 In[46]:= f[x_] := x^4 - 10x^2 + 1
          a = Sqrt[2];
          b = Sqrt[3];
          Simplify[f[ a + b]]
          Simplify[f[ a - b]]
          Simplify[f[-a + b]]
          Simplify[f[-a - b]]
 Out[49]= 0
 Out[50]= 0
 Out[51]= 0
 Out[52]= 0

 In[53]:= Solve[f[x] == 0, x]
 Out[53]= {{x->-Sqrt[5-2 Sqrt[6]]},{x->Sqrt[5-2 Sqrt[6]]},{x->-Sqrt[5+2 Sqrt[6]]},{x->Sqrt[5+2 Sqrt[6]]}}
 In[54]:= Simplify[Solve[f[x] == 0, x]]
 Out[54]= {{x->-Sqrt[5-2 Sqrt[6]]},{x->Sqrt[5-2 Sqrt[6]]},{x->-Sqrt[5+2 Sqrt[6]]},{x->Sqrt[5+2 Sqrt[6]]}}
 In[55]:= FullSimplify[Solve[f[x] == 0, x]]
 Out[55]= {{x->Sqrt[2]-Sqrt[3]},{x->Sqrt[5-2 Sqrt[6]]},{x->-Sqrt[5+2 Sqrt[6]]},{x->Sqrt[2]+Sqrt[3]}}

Sqrt[5-2 Sqrt[6]] is equal to Sqrt[3]-Sqrt[2].
However, Mathematica’s FullSimplify does not simplify Sqrt[5-2 Sqrt[6]].

Question: Should I use other more specialized functions to algebraically solve the equation? If so, which one?

Answer 1

Indeed, Solve doesn’t simplify all roots to the max:

A FullSimplify postprocessing step simplifies two roots and leaves two others untouched:

Same initially happens with Roots:

Strange enough, now FullSimplify simplifies all roots:

The reason for this is, I assume, that for the default ComplexityFunction some of the solutions written above in nested radicals are in a sense simpler than the others.

BTW FunctionExpand knows how to deal with those radicals: