If $\displaystyle \Sigma\alpha=\frac{2}{3},\Sigma\alpha\beta=\frac{4}{3}$ and $\displaystyle \alpha\beta\gamma=\frac{8}{3}$, then the equation whose roots are $\alpha,\beta,\gamma$, is

By Vieta's formulas, a monic cubic polynomial with roots alpha, beta, and gamma is given by x cubed minus (sum of the roots)x squared plus (sum of the product of roots two at a time)x minus (product of the roots) equals 0. Substituting the given values, the equation becomes x cubed minus (two thirds)x squared plus (four thirds)x minus (eight thirds) equals 0. Multiplying the entire equation by 3 to clear the denominators yields 3x cubed minus 2x squared plus 4x minus 8 equals 0.

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