Multiple choice

If $\displaystyle \alpha $ and $\displaystyle \beta $ are roots of the quadratic equation $\displaystyle { 3x }^{ 2 }+3x+2=0$ then $\displaystyle { \alpha }^{ 3 }+{ \beta }^{ 3 }$=.......

  1. $\displaystyle -\frac { 1 }{ 3 } $
  2. $\displaystyle -1$

  3. 1

  4. None of the above
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C Correct answer
Explanation

For 3x^2 + 3x + 2 = 0, alpha+beta = -1, alpha*beta = 2/3. alpha^3 + beta^3 = (alpha+beta)((alpha+beta)^2 - 3*alpha*beta) = (-1)((-1)^2 - 3*(2/3)) = (-1)(1-2) = (-1)(-1) = 1.

AI explanation

Using Vieta's formulas for 3x^2 + 3x + 2 = 0, the sum of the roots alpha + beta is -3/3 = -1 and the product alpha * beta is 2/3. We use the identity for the sum of cubes, which states alpha^3 + beta^3 = (alpha + beta)^3 - 3 * alpha * beta * (alpha + beta). Substituting the known values gives (-1)^3 - 3 * (2/3) * (-1) = -1 + 2 = 1.

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