Problem 200 - A Complete Resource Book in Mathematics for JEE Main

Question

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) \(-1,1+2 \omega, 1+2 \omega^{2}\)

Step-by-Step Solution

Answer

The roots of the equation are as per option (D): \(-1, 1+2\omega, 1+2\omega^{2}\).

Step 1: Rewrite the Equation

First, start with the given equation \((x-1)^3 + 8 = 0\). Rewrite it in a form that allows us to solve for \(x\):\[(x-1)^3 = -8\].

Step 2: Identify Cube Roots

Recognize that the equation \((x-1)^3 + 8 = 0\) is in the form of \((x-a)^3 = b^3\) where \(a = 1\) and \(b = -2\). Thus, the solutions will involve the cube roots of \(-2\).

Step 3: Solve for Cube Roots

The cube roots of \(-8\) can be given by \(-2, -2\omega, -2\omega^2\) since the cube roots of unity are \(1, \omega, \omega^2\).

Step 4: Find the Values of x

Set \(x - 1 = -2, -2\omega, -2\omega^2\) from the cube roots we found:- For \(x - 1 = -2\), we get \(x = -1\).- For \(x - 1 = -2\omega\), we get \(x = 1 + 2\omega\).- For \(x - 1 = -2\omega^2\), we get \(x = 1 + 2\omega^2\).

Step 5: Match Solutions to Choices

The roots we found are \(-1, 1 + 2\omega, 1 + 2\omega^2\). Comparing these with the given options, we see they correspond to option (D).

Key Concepts

Roots of Unity

The roots of unity are fundamental in understanding complex numbers and their applications. In mathematics, the "cube roots of unity" specifically refers to those numbers which, when raised to the power of three, result in one. These roots are particularly crucial when dealing with polynomial equations, as they help in finding solutions that are not immediately obvious.

• The primary cube root of unity is 1.
• The other two are complex numbers: often denoted as \(\omega\) and \(\omega^2\).
• These complex numbers satisfy the equation \(\omega^3 = 1\) and are also the solutions to \(x^3 - 1 = 0\).

Knowing these roots allows us to express any cube root of a number in terms of these values, leading to solutions that incorporate the multiplicative symmetry of the roots of unity.

Cube Roots

Cube roots are numbers that, when multiplied by themselves three times, result in the original number. While the concept is straightforward in real numbers, it becomes intriguing in the complex plane.

• For example, the cube root of \(-8\) involves three distinct results because complex numbers allow for more diverse solutions beyond the real number line.
• The expressions \(-2, -2\omega, -2\omega^2\) represent the cube roots of \(-8\) when viewed through the lens of complex numbers.
• Here \(-2\) is the real root, and the other two involve the complex roots of unity, multiplying \(-2\) by \(\omega\) and \(\omega^2\).

This approach broadens our ability to solve equations involving cubics, particularly those expressed in terms of unity roots.

Polynomial Equations

Polynomial equations are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. The degree of the polynomial determines the maximum number of solutions it can have. In the case of a cubic polynomial, there can be up to three unique solutions.

• A simple equation like \((x-1)^3 + 8 = 0\) actually maps to a deeper analysis once you recognize its form as \((x-a)^3 = b^3\).
• Upon identifying \(a = 1\) and \(b = -2\), you can apply the pattern of cube roots to find your values of \(x\).

Utilizing roots of unity helps in factoring complex polynomials and finding all possible roots, whether real or complex.

Algebraic Solutions

Algebraic solutions involve using algebraic manipulations to solve equations, offering exact answers instead of approximations. These solutions can be carried out through factoring, finding roots, and employing properties like those of unity roots.

• In solving \((x-1)^3 = -8\), each potential value of \(x\) is determined by translating the expression into known algebraic entities like cube roots.
• Factorization enables us to see not just immediate, obvious answers (like \(x = -1\)), but also complex solutions such as \(x = 1 + 2\omega\) and \(x = 1 + 2\omega^2\).

By shifting your perspective using algebraic techniques, complex solutions become clear and easy to identify, demonstrating the harmony between algebra and complex number theory.