For the coefficients to be rational, any complex roots must occur in conjugate pairs. So if \(6+i\) is a root, then so must be \(6-i\).
Now such a third degree polynomial might be
\((x+4)(x-(6+i))(x-(6-i))=(x+4)(x^2-12x+37)=x^3-8x^2-11x+148\)
The only variation to this would be multiplying throughout by some non-zero constant. This doesn't change the roots.
