Multiple choice

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

  1. $1$
  2. $2$
  3. $-1$
  4. $-2$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The cubic polynomial can be factored as P(x) = (x - alpha)(x - beta)(x - gamma) = x^3 - 3x^2 + 2x - 1. To find the value of (1 - alpha)(1 - beta)(1 - gamma), we simply evaluate the polynomial at x = 1, which gives 1^3 - 3(1)^2 + 2(1) - 1 = -1.

AI explanation

By Vieta's formulas for the cubic equation x^3 - 3x^2 + 2x - 1 = 0, the sum of the roots alpha + beta + gamma equals 3, the sum of the roots taken two at a time is 2, and the product alpha * beta * gamma is 1. We expand (1 - alpha)(1 - beta)(1 - gamma) to get 1 - (alpha + beta + gamma) + (alpha*beta + beta*gamma + gamma*alpha) - alpha*beta*gamma. Substituting the known values gives 1 - 3 + 2 - 1, which equals -1.

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