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Let a, b, g be the real roots of x³ + a x² + b x + c = 0, where a, b, c are real and a, b are non-zero. If the system a*u + b*v + g*w = 0, b*u + g*v + a*w = 0, g*u + a*v + b*w = 0 (in unknowns u, v, w) has a non-trivial solution, find the value of a²/b. (Here the coefficients a, b in a²/b refer to the polynomial coefficients.)

1. 5
2. 3
3. 1
4. 0

Correct answer: 3

Solution

The circulant determinant vanishes when a+b+g = 0 (i.e. -a = 0, contradicting a != 0) or when the roots are all equal; using a+b+g = -a and the second factor zero gives the symmetric relation leading to a² = 3b, so a²/b = 3.

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