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When a polynomial f(z) is divided by (z - i) the remainder is i, and when divided by (z + i) the remainder is 1 + i. Find the remainder when f(z) is divided by (z² + 1).

1. (1/2)z + (1 + i/2)
2. -(1/2)z + (1/2 + i)
3. (1/2)z + (1/2 + i)
4. z + i

Correct answer: -(1/2)z + (1/2 + i)

Solution

Setting remainder R(z) = az + b and using f(i) = ai + b = i, f(-i) = -ai + b = 1 + i yields a = -1/2 and b = 1/2 + i.

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