A4th degree polynomial function has zeros at 3 and 5-i. can 4+i also be a zero of the function?

Sure, provided that the coefficients of the polynomial are complex, and not strictly real.

If you require that the coefficients of the polynomial be real, then the answer would be no. For such a polynomial, any complex roots occur in conjugate pairs, so if is a root, then so must be .

A degree-4 polynomial can have up to four distinct roots, so if the coefficients are all real, then the remaining fourth root must be real.