SOLUTION: Can someone please help me with this question - Thanks Let P(z)= {{{ z^3-kz^2+22z-20 }}} , k is a rational number. 3+i is a root of the equation P(z)=0 Find value of k. Find t

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: Can someone please help me with this question - Thanks Let P(z)= {{{ z^3-kz^2+22z-20 }}} , k is a rational number. 3+i is a root of the equation P(z)=0 Find value of k. Find t      Log On


   

Question 1099340: Can someone please help me with this question - Thanks
Let P(z)= +z%5E3-kz%5E2%2B22z-20+ , k is a rational number.
3+i is a root of the equation P(z)=0
Find value of k.
Find the other root of the equation P(z)=0
I got two roots and multiplied them together and got +z%5E2-6z%2B10+, I did this by z=3+i and z=3-i, (z-3-i)(z-3+i).
I originally thought I had to divide +z%5E3-kz%5E2%2B22z-20+ by +z%5E2-6z%2B10+ but I don't think I can divide with the k in the way.
Answer by KMST(5397) About Me  (Show Source):
You can put this solution on YOUR website!
P%28z%29=%28z%5E2-6z%2B10%29%28z-r%29 where r is the other root.
The independent term is -10r=-20 ,
so r=2 or we were deceived about 3%2Bi being a root.
Let's multiply.

If that polynomial is the same function as P%28z%29=z%5E3-kz%5E2%2B22z-20+
(same values for every complex value of z ),
then k=8 .