x^3+(1/x^3) as part of the expression factorises into (x+(1/x))((x^2)-1+(1/x^2)), but 52 factorises into {26*2} or {13*4} only. The whole expression can be written (x^6+1-52x^3)/(x^3). If we let y=x^3, this becomes (y^2-52y+1)/y. The quadratic expression for y in the numerator doesn't factorise into rational terms. It can be written (y-26+15sqrt(3))(y-26-15sqrt(3))/y. Then we can put x^3 back in place of y. This makes the factors x-2-sqrt(3) and x-2+sqrt(3).