If a polynomial P(z) of degree n>=2 is divided by z-i the remainder is A. If it is divided by z+i the

remainder is B. If the polynomial is divided by z^2 + 1 then find the remainder.

If the quotient and remainder on dividing P(z) by z^2+1 is Q(z) and az+b then
P(z)=(z^2+1)Q(z) + az+b
so P(i)=0Q(i)+ai+b and P(i)=A by the remainder theorem, so ai+b=A
Similarly P(-i)=0Q(-i) -ai+b and -ai+b=B
Solve the two equations simultaneously for a and b and you're there!