Multiple choice

If $\alpha_{1},\alpha_{2},\ \alpha_{3},\cdots \ \alpha_{n}$ are the roots of the equation $(x-\beta_{1})(x-\beta_{2}) \cdots \ldots(x-\beta_{n})=A$ and if the equation having $\beta_{1},\ \beta_{2},\ \beta_{3},\ \beta_{n}$ as the roots is $(x-\alpha_{1})(x-\alpha_{2})\ldots\ldots\ldots(x-\alpha_{n})=k$, then $k =$

  1. $A$
  2. $-A$
  3. $\displaystyle \frac{A}{\alpha_{1}\alpha_{2}\ldots\alpha_{n}}$
  4. $\displaystyle -\frac{A}{\alpha_{1}\alpha_{2}\ldots\alpha_{n}}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given the polynomial identity (x-beta1)(x-beta2)...(x-betan) = (x-alpha1)(x-alpha2)...(x-alphan) + A, if we swap the roles of alpha and beta, we get (x-alpha1)...(x-alphan) = (x-beta1)...(x-betan) - A. Thus, k = -A.

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