Question

Find a thrid degree polynomial equation with rational coeffcients that has roots -4 and 6 +i

Answer 1

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.