Multiple choice

If $\alpha, \beta $ are the roots of equation $x^2-4x-1=0$, then the value of $\left ( \alpha ^{{1}/{3}}+\beta ^{{1}/{3}} \right)$ is

  1. $1$
  2. $2$
  3. $3$
  4. $4$
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A Correct answer
Explanation

Let S = alpha^(1/3) + beta^(1/3). S^3 = alpha + beta + 3 * (alpha*beta)^(1/3) * (alpha^(1/3) + beta^(1/3)). From x^2 - 4x - 1 = 0, alpha+beta = 4 and alpha*beta = -1. S^3 = 4 + 3 * (-1)^(1/3) * S. This leads to S^3 + 3S - 4 = 0. Testing S=1: 1 + 3 - 4 = 0. Thus, S=1 is a root.

AI explanation

From the equation x^2 - 4x - 1 = 0, the sum and product of the roots are alpha + beta = 4 and alpha*beta = -1. Let y be the sum of their cube roots, so y = a + b where a^3 = alpha and b^3 = beta. Cubing both sides gives y^3 = alpha + beta + 3ab(a + b), which simplifies to y^3 = 4 + 3*(-1)*y, resulting in the equation y^3 + 3y - 4 = 0. Testing for rational roots, y = 1 satisfies the equation perfectly because 1 + 3 - 4 equals 0.

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