Thread: 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).

Then "factor theorem" says that any polynomial of the form can be factored as where , , 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?