Question

find a third degree polynomial that has 3+i and 7 as zeros.

Answer 1

We know that complex roots occur as conjugate pairs: namely, if a + bi is a solution, then a-bi is a solution. A third degree polynomial has 3 complex roots, thus 3+i, 3-i, and 7 also meet this criterion.

We conclude the polynomial is
(x-(3+i))*(x-(3-i))*(x-7)

Which, after the proper expansion is
x^3 -13x^2 + 52x - 70