Viv, Mark’s procedure works for irrational roots as well as complex roots.
The way to construct the polynomial is simple: multiply several monomials together, one for each root, plus one for each “conjugate” (I’ll explain jn a moment).
We start with the two known roots: 1 and 5-sqrt(2).
The polynomial starts with
however, because one root is irrational (it contains an irrational (sqrt(2)) ), we need to append another root. The other, “conjugate”, root has the opposite sign for the irrational: (5 + sqrt(2)).
So the complete polynomial is:
(x-1) * (x-(5-sqrt(2))) * (x-(5 + sqrt(2)))
Multiply out the “conjugates” first:
x2-x(5+sqrt(2)) -x(5-sqrt(2)) + (25-2)
= x2 -x(10) + 23
= x2 -10x + 23
now multiply by the simple root (x-1)
(x-1)(x2 -10x + 23)
= x3 -11x2 + 33x -23
So this is the answer for the problem you posed, while Mark’s is the answer for a similar problem with complex roots.