Question

Dividing polynomial f(z) by zi, we get the remainder i and dividing it by z+i, we get the remainder 1+i. Find the remainder upon the division of f(z) by z2+1.

Answer 1

The remainder upon division of f(z) by z^2+1 is (i/2)z + (1+2i)/2

• Now according to remainder theorem of polynomials,  f(z) = g(z)(z-i) + i and f(z) = h(z)(z+i) + 1 + i          

• Now when z = i, then f(i) = i  and when z = -i, then f(-i) = 1 + i

• Now let assume that the remainder when f(z) is divided by z^2 + 1 is az + b as a one degree polynomial because, according to remainder theorem, the degree of the remainder should always be less than the degree of the divisor which is two here.

• So according to remainder theorem f(z) = p(z)(z^2 +1) + az + b

• Now at z = i, f(i) = ai + b but we already know that f(i) = i  so, ai + b = i
• Now at z = -i, f(-i) = - ai  + b but  we already know that f(-i) =1 + i, So - ai  + b = 1 + i
• Adding these two equations we get  b = (1+2i)/2
• From this we get a = i/2

• So finally the remainder is (i/2)z + (1+2i)/2