
Factor Theroem
Use the Factor Theorem to find the real number k such that (z – k) is a factor of z^3 + (6 + 2j) z ^2 + (12  15j) z + (9+27j)
Hence write z^3 + (6 + 2j) z ^2 + (12  15j) z + (9+27j) as the product of two factors.
Find all the solutions of the equation z^3 + (6 + 2j) z ^2 + (12  15j) z + (9+27j).

Re: Factor Theroem
Then "factor theorem" says that any polynomial of the form $\displaystyle z^3+ az^2+ bz+ c$ can be factored as $\displaystyle (z z_0)(z z_1)(z z_2)$ where $\displaystyle z_0$, $\displaystyle z_1$, $\displaystyle z_2$ are complex numbers and, of course, they make the value of the polynomial 0. One of the things that tells us is that the product of those numbers is c= 9+ 27j= 9(1+ 3j). Crossing my fingers and trying (something I strongly recommend!) I see that if I set z= 3, I get 27+ (6+2j)(9)+ (12 15j)(3)+ (9+ 27j)= (27 54+ 36 9)+ j(2(9) 15(3)+ 27)= (63 63)+ j(18 45+ 27)= 0!
Can you finish the problem now?