Question

Given that 3 + i is a root of the equation z^3 − 3z^2 − 8z + 30= 0, find the remaining roots.

Answer 1

$z^3 − 3z^2 − 8z + 30= 0$

$(z+3)(z^2 − 6z + 10)= 0$

$z+3=0$ $z^2-6z+10=0$

$z^2-6z+10=0$

$D= (-6)^2 - 4*1*10=-4$

$z_1=\frac{6+\sqrt{D}}{2*1}=\frac{6+2i}{2}=3+i$

$z_2=\frac{6-\sqrt{D}}{2*1}=\frac{6-2i}{2}=3-i$

$z+3=0$

$z_3=-3$

Answer:

$z_1=3+i, z_2=3-i, z_3=-3$