The polynomial equation \[x^3 + bx + c = 0,\]where $b$ and $c$ are rational numbers, has $3-\sqrt{7}$ as a root. It also has an integer root. What is it?

Roots come in pairs

So if one root is 3- sqtr 7 another is 3+ sqrt 7

Expand

\([x-(3-\sqrt7)]\;[x-(3+\sqrt7)]\)

Divide the question cubic polynomial by the answer.

Now think about what you have found.