If 2 is a zero of the polynomial p(x)=x^{3}+3x^{2}-6x-a, then what is the sum of the squares of the other zeros of the polynomial?

Explanation

Since 2 is a root, p(2) = 8 + 12 - 12 - a = 0 \implies a = 8. Let the roots be 2, \beta, \gamma. Sum of roots 2 + \beta + \gamma = -3 \implies \beta + \gamma = -5. Product of roots taken two at a time: 2\beta + 2\gamma + \beta\gamma = -6 \implies 2(-5) + \beta\gamma = -6 \implies \beta\gamma = 4. The sum of squares is \beta^2 + \gamma^2 = (\beta+\gamma)^2 - 2\beta\gamma = (-5)^2 - 2(4) = 17.

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