Multiple choice

If $\displaystyle \alpha, : \beta, : \gamma$ are roots of the equation $\displaystyle x^{3} - ax^{2} + bx - c = 0$, then the value of the expression $\displaystyle \left ( \alpha + \beta - \gamma \right ) \left ( \beta + \gamma - \alpha \right )\left ( \gamma + \alpha - \beta \right )$ equals

  1. $\displaystyle 4ab - 8c$
  2. $\displaystyle a^{3} - 8c$
  3. $\displaystyle 4ab - a^{3} - 8c$
  4. $\displaystyle a^{3} -8c - 4ab$
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C Correct answer
Explanation

Using Vieta's formulas: alpha+beta+gamma = a, alphabeta+betagamma+gammaalpha = b, alphabetagamma = c. The expression expands to (a-2gamma)(a-2alpha)(a-2beta). This evaluates to a^3 - 2a^2(alpha+beta+gamma) + 4a(alphabeta+betagamma+gammaalpha) - 8alphabetagamma = a^3 - 2a^3 + 4ab - 8c = 4ab - a^3 - 8c.

AI explanation

From Vieta's formulas, the sum of the roots alpha, beta, and gamma is a. The given expression can be rewritten using the sum of the roots as (a minus 2 gamma) times (a minus 2 alpha) times (a minus 2 beta), which is a symmetric polynomial. Expanding this product yields a cubed minus 2 a squared times the sum of the roots plus 4 a times the sum of the products of roots taken two at a time minus 8 times the product of the roots. Substituting the sum as a, the sum of products as b, and the product as c gives a cubed minus 2 a cubed plus 4 a b minus 8 c, simplifying to 4 a b minus a cubed minus 8 c. The result is 4ab - a^3 - 8c.

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