2012/008) Find all complex roots of p(z) = z^4− 10z^3+ 38z^2− 74z + 85 given that 4 + i is a root.

4 + i is a root

so 4 - i is a root

so a factor = z - 4 = i

(z-4)^2 = - 1 or z^2-8z + 17 = 0 or z^2 - 8z + 17 is a factor

so we have z^4− 10z^3+ 38z^2− 74z + 85 = (z^2 + az + b)(z^2-8z+ 17)
= z^4 + z^3(a-8) + z^2(17 + b - 8a) + z (17a - 8b) + 17b


which gives a - 8 = - 10 => a = - 2(coefficient of z^3)
and 17 b = 85 or b = 5(constant)

17+b- 8a = 17 + 5 + 16 = 38( coefficient of z^2)
17 a - 8b = -34 - 40 = 74(coeffificent of z)

so equtions are consistant

so we get z^2-2z + 5 = 0 => (z-1)^2 + 4 = 0 or z = 1 + 2i or 1- 2i
so all roots are 1 + 2i, 1- 2i, 4+i, 4- i