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.