jmc

Because the coefficients of the polynomial are rational, the radical conjugate of $5-\sqrt{2},$ which is $5+\sqrt{2},$ must also be a root of the polynomial. By Vieta's formulas, the sum of the roots of this polynomial is $0$; since $(5-\sqrt{2})+(5+\sqrt{2})=10,$ the third, integer root must be $0-10=\boxed{-10}.$