find the constant c such that the denominator will divide evenly into the numerator x^4+10x^2+c/x+i

Answer 1

assuming c is real, the we also need to have x-i as a root. That means that x^2+1 must divide it.

A little synthetic division shows that the remainder is c-9. The remainder must be zero, so c = 9.

x^4+10x^2+9 = (x^2+1)((x^2+9)