# Factor Theroem

• May 24th 2012, 09:54 AM
user654321
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).
• May 24th 2012, 11:43 AM
HallsofIvy
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?