Multiple choice

If α, β, γ are the roots of the equation x3 + x + 1 = 0, then the value of α3 + β3 + γ3 is

  1. 0

  2. 3

  3. -3

  4. -1
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C Correct answer
Explanation

For x^3 + x + 1 = 0, sum of roots alpha+beta+gamma = 0. Sum of products taken two at a time = 1. Product = -1. Newton's sums: S3 + S1 + 3 = 0. S1 = 0, so S3 + 0 + 3 = 0, S3 = -3.

AI explanation

Since alpha, beta, and gamma are roots of the equation x^3 + x + 1 = 0, each root satisfies the equation, meaning r^3 = -r - 1 for any root r. Multiplying this by r gives r^4 = -r^2 - r. Using Newton's identities or polynomial relations, we evaluate the sum of the cubes of the roots, which can be found using the algebraic identity a^3 + b^3 + c^3 = (a + b + c)^3 - 3(a + b + c)(ab + bc + ca) + 3abc. From the given cubic equation, the sum of the roots is 0, the sum of the product of the roots two at a time is 1, and the product of the roots is -1. Plugging these values into the identity yields 0^3 - 3 multiplied by 0 multiplied by 1 + 3 multiplied by -1, which equals -3.

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