What is a polynomial that has the roots: 3 and 5-i and also crosses the origin with integer coefficients? question
What is a polynomial that has the roots: 3 and 5-i and also crosses the origin with integer coefficients?
My thoughts
on my first instinct, i wrote this:

$(x-(5-i))(x+(5-i))(x-3)(x)$

But then i've realized that when you factor out the complex parts, you dont get integer coefficients so I was really confused if this is even possible

If you want real coefficients, which integers implies, you need conjugate imaginary roots.  To cross the origin you need a factor $x$.  The simplest choice is then $$x(x-3)(x-5+i)(x-5-i)=x^4 - 13 x^3 + 56 x^2 - 78 x$$
You can multiply this by any polynomial with integer coefficients that you like.  You are close but did not get the conjugate right.
