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(C) For the quadratic equation $x^2 + ax + b = 0$,the sum of the roots is $\alpha + \beta = -a$ and the product of the roots is $\alpha\beta = b$.We u

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$- a^3 + 3ab$

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(C) For the quadratic equation $x^2 + ax + b = 0$,the sum of the roots is $\alpha + \beta = -a$ and the product of the roots is $\alpha\beta = b$.We use the algebraic identity for the sum of cubes: $\alpha^3 + \beta^3 = (\alpha + \beta)(\alpha^2 - \alpha\beta + \beta^2)$.This can be rewritten in terms of the sum and product of roots as: $\alpha^3 + \beta^3 = (\alpha + \beta)[(\alpha + \beta)^2 - 3\alpha\beta]$.Substituting the values $\alpha + \beta = -a$ and $\alpha\beta = b$:$\alpha^3 + \beta^3 = (-a)[(-a)^2 - 3(b)]$$= (-a)(a^2 - 3b)$$= -a^3 + 3ab$.

If $\alpha$ and $\beta$ are the roots of the equation $x^2 + ax + b = 0$,then the value of $\alpha^3 + \beta^3$ is equal to:

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