Problem 89 - A Textbook of Differential Calculus

Question

If \(f(x)=\frac{x^{4}+x^{2}+1}{x^{2}-x+1}\), then find the value of \(f\left(\omega^{n}\right)\) (where ' \(\omega\) ' is the non-real root of the equation \(z^{3}=1\) and ' \(n\) ' is a multiple of 3 ).

Step-by-Step Solution

Answer

The value of \(f(\omega^n)\) is 3.

Step 1: Understand the Properties of \(\omega\)

The number \( \omega \) is a complex root of the equation \( z^3 = 1 \). This equation has roots \( 1, \omega, \omega^2 \), where \( \omega = e^{i\frac{2\pi}{3}} \) and \( \omega^2 = e^{i\frac{4\pi}{3}} \). An important property is that \( \omega^3 = 1 \) and \( \omega^2 + \omega + 1 = 0 \).

Step 2: Analyze \(f(\omega^n)\) when \(n\) is a Multiple of 3

Since \(n\) is a multiple of 3, say \(n = 3k\) for integer \(k\), it follows that \( \omega^n = (\omega^3)^k = 1^k = 1 \). This implies that we actually need to find \(f(1)\).

Step 3: Substitute and Simplify \(f(x)\) at \(x=1\)

Substitute \(x = 1\) into the function: \( f(1) = \frac{1^4 + 1^2 + 1}{1^2 - 1 + 1} = \frac{1 + 1 + 1}{1 - 1 + 1} = \frac{3}{1} = 3 \). Thus, \(f(1) = 3\).

Key Concepts

Roots of Unity

Roots of Unity are essentially solutions to the equation \( z^n = 1 \). For any given positive integer \( n \), there are precisely \( n \) distinct \( n \)-th roots of unity. These solutions are evenly distributed on the unit circle in the complex plane. Let's explore a specific case:
1. **Cube Roots of Unity**: When \( n = 3 \), the equation \( z^3 = 1 \) has three solutions: 1, \( \omega \), and \( \omega^2 \). Here, \( \omega = e^{i\frac{2\pi}{3}} \) and \( \omega^2 = e^{i\frac{4\pi}{3}} \).
2. **Properties**: The roots of unity have some fascinating properties:

• \( \omega^3 = 1 \), illustrating that raising \( \omega \) to the power of 3 brings you back to the real number 1.
• \( \omega + \omega^2 + 1 = 0 \), meaning the arithmetic sum of these roots is zero.

The geometric representation helps in visualizing how these roots are positioned on the complex plane, particularly forming an equilateral triangle for the cube roots.
Understanding these properties is beneficial in simplifying problems involving complex numbers, like the calculation in this exercise.

Complex Number Properties

Complex numbers extend real numbers by incorporating an imaginary part. A complex number is typically expressed as \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit with the property \( i^2 = -1 \).
Some essential properties of complex numbers include:

• **Addition and Subtraction**: These operations are straightforward, involving the addition or subtraction of corresponding real and imaginary parts.
• **Multiplication**: When multiplying complex numbers, use the distributive property while applying \( i^2 = -1 \) for \( i \, \times \, i \).
• **Conjugate**: The conjugate of \( a + bi \) is \( a - bi \), and it has benefits in simplifying expressions, particularly in division.

These properties come into play for operations involving roots of unity since roots like \( \omega \) are themselves complex numbers with specific geometrical interpretations, as seen in their polar form \( re^{i\theta} \).
Working with complex numbers helps us solve polynomial equations and understand the symmetrical behavior of their roots.

Polynomial Functions

Polynomial functions are expressions of the form \( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \). These functions can have integer, real, or complex coefficients, and they are the bedrock of many mathematical models. The roots of polynomial functions are particularly significant.
Here’s why polynomials are important:

• **Degree and Roots**: The degree of a polynomial signifies the highest power of \( x \) and predicts the maximum number of roots.
• **Value Substitution**: Substituting values, such as in the computation of \( f(1) \) in this exercise, is a common way to analyze behavior or find specific values.
• **Factorization**: Breaking down polynomials into products of simpler polynomials helps in finding roots and understanding graph behavior.

The function in our problem is a rational function formed by the ratio of two polynomials. This relationship is useful in analyzing complex functions.
When dealing with expressions involving roots of unity, we explore the intricacies of polynomial functions, such as finding specific values or simplifying using well-known roots properties.