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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?

if (shown in picture) are two of the roots of a forth degree polynomial with integer-example-1

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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

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