Root to Poly

Congratulations to Fok Chi Kwong from Yuen Long Merchants Association Secondary School, Hong Kong on this solution.

We may find the required polynomial by starting from the expression :

$$x = 1 + \sqrt 2 + \sqrt 3$$.

Squaring both sides and simplifying, we get

\[x - 1 = \sqrt 2+ \sqrt 3 \] \[x^2 - 2x + 1 = 5 + 2\sqrt 6 \] \[ x^2 - 2x - 4 = 2\sqrt 6 \] \[(x^2 - 2x - 4)^2 = 24 \] \[x^4 - 4x^3 + 4x^2 - 8x^2 + 16x + 16 = 24 \] \[x^4 - 4x^3 - 4x^2 + 16x - 8 = 0 \]

Thus $p(x) = x^4 - 4x^3 - 4x^2 + 16x - 8$ is the required polynomial.

Tony Cardell, State College Area High School, PA, USA, also sent in a good solution.