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Let f(x) = a*x³ + b*x² + c*x + d be a cubic polynomial with real coefficients such that f(i) = 0 and f(1 + i) = 5, where i = sqrt(-1). Find the value of a² + b² + c² + d².

1. 10
2. 5
3. 15
4. 20

Correct answer: 10

Solution

Real coefficients force roots i and -i, so f(x) = a(x²+1)(x - r); imposing f(1+i)=5 fixes a and r, yielding a=1, b=-1, c=1, d=-1 and the sum of squares 10.

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