The function f(x)=x^4−15(x^3)+81(x^2)−201x+182 has four complex roots, one of which is 3−2i. What is the sum of all real and imaginary coefficients of these roots?

Question

Details and assumptions
i is the imaginary unit, where i2=−1.

Answer 1

one other complex root must be 3+2i
So, (x-(3-2i))(x-(3+2i)) are factors of f(x)

That is, (x-3)^2+4 = (x^2-6x+13) divides f(x)

f(x) = (x^2-6x+13)(x^2-9x+14)
= (x^2-6x+13)(x-2)(x-7)

So, the roots are
2,7,3-2i,3+2i
You can add 'em up