Problem 29 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

If \(\omega\) is a cube root of unity, then \((1+\omega-\) \(\left.\omega^{2}\right)\left(1-\omega+\omega^{2}\right)=\) (a) 1 (b) 0 (c) 2 (d) 4

Step-by-Step Solution

Answer

(b) 0

Step 1: Understanding Cube Roots of Unity

The cube roots of unity are the solutions to the equation \(x^3 = 1\). They can be represented as \(1, \omega, \omega^2\), where \(\omega\) is a primitive cube root of unity. These roots satisfy the property: \(\omega^3 = 1\), \(1 + \omega + \omega^2 = 0\), and \(\omega^2 = \omega + 1\).

Step 2: Substitution and Simplification

Given the expression \((1 + \omega - \omega^2)(1 - \omega + \omega^2)\), we substitute \(\omega^2 = \omega + 1\) in both parts of the expression.Thus, it becomes \((1 + \omega - (\omega + 1))(1 - \omega + (\omega + 1))\), which simplifies to \(0 \times 1 = 0\).

Step 3: Verification

Verify by simplifying each term separately: \(1 + \omega - \omega^2\) simplifies to \(1 + \omega - (\omega + 1) = 0\), and \(1 - \omega + \omega^2\) simplifies to \(1 - \omega + (\omega + 1) = 1\). Multiplying these results: \(0 \times 1 = 0\).

Key Concepts

Primitive Roots

Primitive roots are crucial in understanding concepts such as roots of unity in mathematics. A primitive root of unity is a number that, when raised to a certain power (the order of the root), equals one. Specifically, for cube roots of unity, this involves solving a polynomial equation where the possible solutions are complex numbers. A primitive cube root of unity is the most basic root from which all other roots of unity can be derived. For cube roots, these are numbers that satisfy the equation:

• \(x^3 = 1\)
• Where the solutions are 1, \(\omega\), and \(\omega^2\).

The primitive root in this context is \(\omega\), which is neither 1 nor another root of unity. It helps describe all roots without repetition. \(\omega\) satisfies the fundamental properties:

• \(\omega^3 = 1\)
• \(1 + \omega + \omega^2 = 0\)

Using these properties, we can express complex equations involving cube roots of unity in simplified forms. This allows for easy solving and understanding of polynomial equations linked to cube roots.

Complex Numbers

Complex numbers are an extension of the real numbers, featuring elements that are both real and imaginary components. Within the realm of complex numbers, we often encounter the concept of cube roots of unity. These appear predominantly in the study of polynomial equations. A complex number is generally expressed as \(a + bi\),

• where \(i\) is the imaginary unit \(\sqrt{-1}\).
• \(a\) is the real part, and \(b\) is the imaginary part.

Cube roots of unity are specifically useful because they evenly divide the unit circle in the complex plane into three parts. This concept is crucial when simplifying expressions such as \((1 + \omega - \omega^2)(1 - \omega + \omega^2)\), where understanding how complex numbers interact allows the simplification process to yield correct results.Notably, complex numbers make the visualization and solution of polynomial equations involving imaginary elements feasible, showcasing their importance in higher-level mathematics.

Polynomial Equations

Polynomial equations form the backbone of algebra and involve terms raised to exponent powers and their sums. The degree of the polynomial is dictated by the highest power present in the equation. For cube roots of unity, we deal with third-degree polynomial equations. The general equation for cube roots of unity is: \(x^3 - 1 = 0\). Solving this specific polynomial involves finding the three roots that satisfy the equation. These roots are fundamentally important for forming expressions such as:

• \((1 + \omega - \omega^2)(1 - \omega + \omega^2)\)

In a polynomial equation, each term multiplies and simplifies following algebraic rules. Substituting relevant equations like \(\omega^2 = \omega + 1\) into the polynomial reduces the complexity of the solution, showing:

• \(\omega^3 = 1\)
• which simplifies other expressions, verifying \(0 \times 1 = 0\).

Polynomial equations allow us to predict and derive solutions accurately by standardizing how we approach complex algebraic problems. This predictability and structure make them vital tools in mathematics.