If α and β are the roots of the quadratic equation x^2 − x + 1 = 0, then which of the following equations has roots α^3 and β^3?

Introduction / Context: This algebra problem connects the roots of a given quadratic equation with the roots of a new quadratic formed from powers of the original roots. Specifically, if α and β are roots of a given quadratic, we are asked to find the quadratic whose roots are α^3 and β^3. This is a common technique in polynomial and root transformation questions.

Given Data / Assumptions:

• α and β are roots of x^2 − x + 1 = 0.
• We must determine which equation has roots α^3 and β^3.

Concept / Approach: For a quadratic with roots r1 and r2, we know: r1 + r2 = sum of roots = coefficient of x with negative sign over coefficient of x^2. r1 r2 = product of roots = constant term over coefficient of x^2. Given the original equation x^2 − x + 1 = 0, we can find α + β and αβ. Then we compute α^3 + β^3 and α^3 β^3, and use these as the sum and product of roots for the required quadratic x^2 − (sum)x + product = 0.

Step-by-Step Solution: From x^2 − x + 1 = 0, the coefficient of x is −1 and the constant term is 1. Thus α + β = 1 and αβ = 1. We need the sum S = α^3 + β^3 and the product P = α^3 β^3. Use the identity α^3 + β^3 = (α + β)^3 − 3αβ(α + β). Substitute α + β = 1 and αβ = 1: S = 1^3 − 3*1*1 = 1 − 3 = −2. For the product, α^3 β^3 = (αβ)^3 = 1^3 = 1. Therefore, the quadratic whose roots are α^3 and β^3 is: x^2 − Sx + P = x^2 − (−2)x + 1 = x^2 + 2x + 1 = 0.

Verification / Alternative check: Note that x^2 + 2x + 1 = 0 can be written as (x + 1)^2 = 0, so both roots are −1. Indeed, if α and β are the complex cube roots associated with the original equation, their cubes both equal −1, making the derived quadratic consistent. This supports our algebraic derivation.

Why Other Options Are Wrong: x^2 − 2x − 1 = 0 and x^2 − 3x + 1 = 0 correspond to different sums and products of roots that do not match S = −2 and P = 1. x^2 + 3x − 1 = 0 and x^2 + x + 1 = 0 are similarly inconsistent with the required sum and product for α^3 and β^3.

Common Pitfalls: Common mistakes include forgetting the sign in the sum of roots formula, or misapplying the identity for α^3 + β^3. Always remember that for ax^2 + bx + c = 0, sum of roots is −b/a and product is c/a. Using these correctly and applying the cube identity systematically leads directly to the correct transformed quadratic.

Final Answer: The equation whose roots are α^3 and β^3 is x^2 + 2x + 1 = 0.