Without any extra conditions, the answer could be 3, and the simplest polynomial with the given roots would be
(x + 5) (x - (1 + 4i )) (x + 4i )
= x ³ + 4x ² + (11 - 4i ) x + 80 - 2i
If the polynomial is supposed to have only real coefficients, then any complex roots must occur along with their complex conjugates:
(x + 5) (x - (1 + 4i )) (x - (1 - 4i )) (x + 4i ) (x - 4i )
= x ⁵ + 3x ⁴ + 23x ³ + 133x ² + 112x + 1360
and then the degree would be 5.
