SOLUTION: <img src="http://img177.imageshack.us/img177/9696/mathnd7.png"> I NEED HELP! :]

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: <img src="http://img177.imageshack.us/img177/9696/mathnd7.png"> I NEED HELP! :]      Log On


   

Question 117518:
I NEED HELP! :]
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
I'm assuming P(x) is of the form,
P%28x%29=ax%5E3%2Bbx%5E2%2Bcx%2Bd
a) From the properties of i,
i%5E2=-1
i%5E3=-i
So if P(i)=0, then
P%28i%29=a%28-i%29%2Bb%28-1%29%2Bc%28i%29%2Bd
P%28i%29=%28d-b%29%2B%28c-a%29i
For P(i)=0, both the real and complex parts must equal zero.
Real part
d-b=0
highlight%28d=b%29
Complex part
c-a=0
highlight%28c=a%29
b.) The assumption is d=b and c=a, so let's look at P(-i).
P%28-i%29=a%28-i%29%5E3%2Bb%2A%28-i%29%5E2%2Bc%28-i%29%2Bd
%28-i%29%5E3=%28-1%29%5E3%2A%28i%29%5E3=%28-1%29%28-i%29=i
%28-i%29%5E2=%28-1%29%5E2%2A%28i%29%5E2=%281%29%28-1%29=-1
P%28-i%29=a%28i%29%2Bb%28-1%29%2Bc%28-i%29%2Bd
P%28-i%29=%28d-b%29%2B%28a-c%29i
P%28-i%29=%28b-b%29%2B%28a-a%29i
P%28-i%29=0
Yes, then P(-i)=0.