Question 117518
I'm assuming P(x) is of the form,
{{{P(x)=ax^3+bx^2+cx+d}}}
a) From the properties of i,
{{{i^2=-1}}}
{{{i^3=-i}}}
So if P(i)=0, then
{{{P(i)=a(-i)+b(-1)+c(i)+d}}}
{{{P(i)=(d-b)+(c-a)i}}}
For P(i)=0, both the real and complex parts must equal zero.
Real part
{{{d-b=0}}}
{{{highlight(d=b)}}}
Complex part
{{{c-a=0}}}
{{{highlight(c=a)}}}
b.) The assumption is d=b and c=a, so let's look at P(-i).
{{{P(-i)=a(-i)^3+b*(-i)^2+c(-i)+d}}}
{{{(-i)^3=(-1)^3*(i)^3=(-1)(-i)=i}}}
{{{(-i)^2=(-1)^2*(i)^2=(1)(-1)=-1}}}
{{{P(-i)=a(i)+b(-1)+c(-i)+d}}}
{{{P(-i)=(d-b)+(a-c)i}}}
{{{P(-i)=(b-b)+(a-a)i}}}
{{{P(-i)=0}}}
Yes, then P(-i)=0.