Polynomial With Imaginary Roots

Question

Working on question 1 here http://www.sosmath.com/cyberexam/precalc/EA2002/EA2002.html

Find a polynomial with integer coefficients that has the following zeros: -1/3, 2, 3+i

Multiplying (3x+1) (x-2) (x-3-i) produces a polynomial with i scattered throughout the terms. Not the right answer.

Then I thought maybe the root 3+i implied another root 3-i.

That didn't produce the right answer either.

What am I missing here?

Answer 1

Alpha agrees with the solution on the page you link to $ (3x+1)(x-3-i)(x-3+i)(x-2)=3 x^4-23 x^3+58 x^2-38 x-20$

Answer 2

Hint: a polynomial with integer coefficients that has the root $3+i$ must have also the root $3-i$. However

$$ (3x+1) (x-2) (x-3-i) $$

doesn't have the root $3-i$. How can you make from this a polynomial that has also the desired root?