SOLUTION: One zero of f(x)=x^5+x^4+23x^3+23x^2−50x−50 is 5i where i=√−1. Completely factor f(x)over the set of complex numbers using this. All the values should be exact.

Since the coefficients of the polynomial are real numbers (they even are integers (!) ),
it implies that together with the imaginary root 5i, its complex conjugate -5i is also the root.


Hence, the polynimial f(x) is divisible by (x-5i)*(x+5i) = .


When you perform long division, you will get the quotient  q(x) =  = .


You can factor this quotient further using grouping/re-grouping


     =  -  =  -  = .


Therefore, the full decomposition of the given polynomial over complex number domain is


    f(x) = ,


and its roots are  5i, -5i, ,   and -1.    ANSWER