Question

if (shown in picture) are two of the roots of a forth degree polynomial with integer coefficients, what is the product of the other two roots?

Answer 1

The other two roots will be the radical conjugates of the given roots, it means

$\begin{gathered} 3+\sqrt[]{2} \\ 2-\sqrt[]{2} \end{gathered}$

Once we know them, we can find the product of these two roots

$\begin{gathered} (3+\sqrt[]{2})\cdot(2-\sqrt[]{2})=6-3\sqrt[]{2}+2\sqrt[]{2}-2 \\ =4-\sqrt[]{2} \end{gathered}$

The product of the roots is 4-sqrt2