Find a fourth degree polynomial equation with integer coefficients that has the given number of roots: 2i and 4-i

Question

This was on my homework and I'm completely lost on it, I don't even know where to start. It seems like I have to go backwards, building an equation from the roots(instead of simplifying an equation down to its roots), but I have no idea how to do that. Any help is appreciated, thanks!

Answer 1

The complex conjugate of a complex root must also be a root, so you will have 4 roots:

2i

-2i

4 - i

4 + i

f(x) = A (x - 2i) (x + 2i) [x - (4 - i)] [x - (4 + i)],

where A is a real integer ≠ 0. If you expand this, all the coefficients will come out real and integer.