Edward A. answered • 02/04/20

High School Math Whiz grown up--I've even tutored my grandchildren

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

(x-1)(x-(5-sqrt(2))

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:

x^{2}-x(5+sqrt(2)) -x(5-sqrt(2)) + (25-2)

= x^{2} -x(10) + 23

= x^{2} -10x + 23

now multiply by the simple root (x-1)

(x-1)(x^{2} -10x + 23)

= x^{3} -11x^{2} + 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.