This means that \(\displaystyle x - \sqrt{2}\), \(\displaystyle x - (2 \sqrt{2} + \pi)\), \(\displaystyle x - (2 + 2 \pi)\) are all factors of this polynomial. Therefore the simplest polynomial that has all these roots is :

\(\displaystyle (x - \sqrt{2})(x - (2 \sqrt{2} + \pi))(x - (2 + 2 \pi)) = 0\)

Or, equivalently :

\(\displaystyle (x - \sqrt{2})(x - 2 \sqrt{2} - \pi)(x - 2 - 2 \pi) = 0\)

And the degree of this polynomial is, trivially, 3 (can be checked by expanding).