As the two zeros among four zeros, viz. 88, -i−i, ii and 11 have two complex numbers, which are complex conjugate of each other, it is possible to write a polynomial function of least degree (which is 44 as there are 44 zeros with no multiplicity, i.e. repeat zeros) that has real coefficients.
A function f(x)f(x) with zeros aa, bb, cc and dd is (x-a)(x-b)(x-c)(x-d)(x−a)(x−b)(x−c)(x−d) and hence the function with 88, -i−i, ii and 11as zeros, will be
(x-8)(x-(-i))(x+i)(x-1)(x−8)(x−(−i))(x+i)(x−1)
= (x-8)(x+i)(x-i)(x-1)(x−8)(x+i)(x−i)(x−1)
= (x-8)(x-1)(x^2+ix-ix+(-i^2))(x−8)(x−1)(x2+ix−ix+(−i2))
= (x-8)(x-1)(x^2+1)(x−8)(x−1)(x2+1) as i^2=-1i2=−1
= (x^2-9x+8)(x^2+1)(x2−9x+8)(x2+1)
= x^4-9x^3+8x^2+x^2-9x+8x4−9x3+8x2+x2−9x+8
= x^4-9x^3+9x^2-9x+8x4−9x3+9x2−9x+8
