Multiple choice

If $u,v$ and $w$ are the three roots of the equation $z^3 - 1 = 0$. Calculate $u.v + v.w + w.u$.

  1. $0$
  2. $1$
  3. $2$
  4. $-1$
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A Correct answer
Explanation

The roots of z^3 - 1 = 0 are 1, omega, and omega^2. The sum of roots taken two at a time (uv + vw + wu) is the coefficient of the z term in the polynomial z^3 + 0z^2 + 0z - 1 = 0. By Vieta's formulas, this sum is 0.

AI explanation

Using the relationship between roots and coefficients for the cubic equation z^3 - 1 = 0, the sum of the products of the roots taken two at a time is the coefficient of the z term. In this standard form, the coefficient of z is 0. Therefore, the value of uv + vw + wu is 0.

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