Multiple choice

Let $ \alpha +i\beta :\alpha ,\beta \in R$ , be a root of the equation $ { x }^{ 3 }+qx+r=0$ ; $q, r \in R$. Find a real cubic equation, independent of $\alpha$ and $\beta$ , whose one root is $2\alpha$

  1. $ { x }^{ 3 }+rx-r=0$
  2. $ { x }^{ 3 }+rx-q=0$
  3. $ { x }^{ 3 }+qx-r=0$
  4. $ { x }^{ 3 }+2qx-r=0$
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C Correct answer
AI explanation

Let the real root of x^3 + qx + r = 0 be c, making the roots c, (a+ib), and (a-ib). The sum of the roots is c + 2a = 0, which means c = -2a, so the real root of the original equation is -2a. A real root of 2a corresponds to x = -c in the original cubic equation. Substituting -c for x gives (-c)^3 + q(-c) + r = 0, which simplifies to -c^3 - qc + r = 0, or c^3 + qc - r = 0. The required equation is x^3 + qx - r = 0.

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