If the cubes roots of unity are 1, ω, ω^2 then the roots of the equation (x-1)^3+8=0, are (A) -1

step 1 Hello students in this question we have given the cube root of unity that we know it on one Omeg

If the cubes roots of unity are 1, ω, ω^2 then the roots of...

If the cubes roots of unity are $1, \omega, \omega^{2}$ then the roots of the equation $(x-1)^{3}+8=0$, are
(A) $-1,-1+2 \omega,-1-\omega^{2}$
(B) $-1,-1,-1$
(C) $-1,1-2 \omega, 1-2 \omega^{2}$
(D) $-1,1+2 \omega, 1+2 \omega^{2}$

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Key Concepts

Complex Cube Roots and Polar Form

When solving equations involving cube terms with complex numbers, it is common to express the complex number in polar form. Using De Moivre's Theorem, the cube roots of a complex number can be determined by taking the cube root of the magnitude and dividing the argument by three, then adding the appropriate multiples of 2?/3 to obtain all distinct roots. This procedure is essential for finding all solutions in the complex plane.

Cube Roots of Unity

The cube roots of unity are the three complex numbers that satisfy the equation z³ = 1. They consist of one real root (1) and two non-real complex conjugate roots, commonly denoted as ? and ?². These roots have notable properties such as ?² + ? + 1 = 0, and they are evenly distributed on the unit circle in the complex plane, which facilitates their use in expressing all cube roots of a complex number concisely.

Polynomial Equation Transformation

This concept involves rewriting a polynomial equation through substitution to simplify its structure. For example, translating an equation by letting y = x - c can center the equation at the origin, making it easier to solve by isolating the transformed variable. This method is particularly useful when dealing with shifted polynomial forms such as (x - c) raised to a power.

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