Multiple choice

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3 + 4x + 2 = 0$, then ${\alpha}^3 + {\beta}^3 + {\gamma}^3 =$ ?

  1. $2$
  2. $6$
  3. $-2$
  4. $-6$
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D Correct answer
Explanation

For x^3 + 4x + 2 = 0, the sum of roots alpha+beta+gamma = 0, sum of products taken two at a time = 4, product = -2. Using Newton's sums: S1 = 0. S2 = (S1)(0) - 2(4) = -8. S3 = (S2)(0) - (S1)(4) - 3(2) = -6.

AI explanation

From the equation x^3 + 4x + 2 = 0, we find the sum of the roots alpha + beta + gamma = 0, the sum of products taken two at a time is 4, and the product of the roots is -2. Using the identity for the sum of cubes, alpha^3 + beta^3 + gamma^3 - 3(alpha)(beta)(gamma) equals (alpha + beta + gamma)(alpha^2 + beta^2 + gamma^2 - alpha*beta - beta*gamma - gamma*alpha). Since the sum of the roots is 0, the entire right side becomes 0. This gives alpha^3 + beta^3 + gamma^3 = 3(alpha)(beta)(gamma). Substituting the product of the roots, we get 3 multiplied by -2, which is -6.

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