# equation on 3^(1/2)+5^(1/3)
#interface(prettyprint=0);
with(LinearAlgebra):

x:=3^(1/2)+5^(1/3);
x2:=expand(x^2);
x3:=expand(x^3);
x4:=expand(x^4);
x5:=expand(x^5);
x6:=expand(x^6);

M:=Matrix([
[1,0,0,0,0,0],
[0,1,1,0,0,0],
[3,0,0,1,2,0],
[5,3,9,0,0,3],
[9,20,5,18,12,0],
[150,9,45,5,25,30]]);

v:=Vector([52,300,225,135,54,30]);
M1:=Transpose(MatrixInverse(M));
MatrixVectorMultiply(M1,v);

p:=expand(x^6-9*x^4-10*x^3+27*x^2-90*x-2);
x:='x';
factor(x^6-9*x^4-10*x^3+27*x^2-90*x-2);

M:=Matrix([
[0,1,1,0,0,0],
[3,0,0,0,1,0],
[0,0,0,1,1,0],
[5,0,0,0,0,1],
[0,0,3,0,0,1],
[0,5,0,3,0,0]]);

p:=sort(CharacteristicPolynomial(M,z));