Problem 34 - Advanced Engineering Mathematics

Question

In Problems 33 and 34, find all solutions of the given equation. $$ z^{8}-2 z^{4}+1=0 $$

Step-by-Step Solution

Answer

The solutions are \( z = 1, -1, i, -i \).

Step 1: Understanding the Equation

The given equation is a polynomial equation: \( z^8 - 2z^4 + 1 = 0 \). Our objective is to find all the complex solutions for this equation.

Step 2: Substitute with a New Variable

Introduce a substitution \( w = z^4 \). Therefore, \( w^2 = z^8 \). The equation becomes \( w^2 - 2w + 1 = 0 \).

Step 3: Recognize It's a Perfect Square

Notice that the equation \( w^2 - 2w + 1 = 0 \) can be expressed as \( (w-1)^2 = 0 \). This means that \( w = 1 \) is the only solution for this quadratic.

Step 4: Substitute Back

With \( w = z^4 \) and \( w = 1 \), substitute back to get \( z^4 = 1 \).

Step 5: Find the Fourth Roots of Unity

The equation \( z^4 = 1 \) has four solutions, which are the fourth roots of unity: \( z = 1, -1, i, -i \). These are the numbers that satisfy \( z^4 = 1 \). The solutions can be calculated using the formula for the roots of unity: \( z_k = e^{i(2krac{ au}{4})} \) with \( k = 0, 1, 2, 3 \).

Step 6: List the Solutions

Therefore, the solutions to the original equation \( z^8 - 2z^4 + 1 = 0 \) are \( z = 1, -1, i, -i \).

Key Concepts

Fourth Roots of Unity

The concept of **fourth roots of unity** emerges when solving equations like \( z^4 = 1 \). In simple terms, it refers to finding all the solutions for \( z \) such that raising \( z \) to the fourth power results in 1. These numbers are crucial in complex number theory and have significant geometrical interpretations.
To determine the fourth roots of unity, we can use the formula:

• \( z_k = e^{i(2k\frac{\pi}{4})} \)

Here, \( k \) is an integer that ranges from 0 to 3. This formula finds the angles corresponding to each of the four solutions on the unit circle in the complex plane. The unit circle has a radius of one and a center at the origin.
The four solutions are:

• \( z_0 = e^{i(0)} = 1 \)
• \( z_1 = e^{i(\frac{\pi}{2})} = i \)
• \( z_2 = e^{i(\pi)} = -1 \)
• \( z_3 = e^{i(\frac{3\pi}{2})} = -i \)

Each of these values of \( z \) represents a point on the unit circle at equal distances apart, exhibiting rotational symmetry. Understanding this symmetry can make working with complex numbers a lot more intuitive.

Quadratic Equations

Quadratic equations are polynomial equations of the form \( ax^2 + bx + c = 0 \). In this exercise, we encountered the simplified form \( w^2 - 2w + 1 = 0 \) after substituting \( w = z^4 \). Recognizing this equation as a perfect square can simplify solving. Here’s how this quadratic equation was handled:
By expressing it as \((w-1)^2 = 0\), we concluded that \( w = 1 \) is the sole solution. This technique of expressing quadratics as perfect squares helps in quickly finding solutions and is especially useful in complex polynomial equations.
When solving quadratic equations in general, various methods can be employed:

• Factoring
• Completing the square
• Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Understanding the structure of quadratic equations can aid in identifying quick solutions, especially when manipulating complex equations.

Complex Numbers

**Complex numbers** are numbers that include the imaginary unit \( i \), where \( i^2 = -1 \). They take the form \( a + bi \), where \( a \) and \( b \) are real numbers. In our exercise, complex numbers like \( i \) and \( -i \) appeared as solutions.
Complex numbers extend the idea of one-dimensional number line to the two-dimensional complex plane, with the x-axis representing the real part and the y-axis representing the imaginary part. This allows for a complete system where every polynomial equation has solutions.
Key concepts when working with complex numbers include:

• **Addition and Subtraction**: Combine like terms (real parts and imaginary parts separately).
• **Multiplication**: Use distributive property. Don't forget that \( i^2 = -1 \).
• **Conjugate**: The conjugate of \( a + bi \) is \( a - bi \), useful in division.
• **Magnitude**: The distance from the origin \( |z| = \sqrt{a^2 + b^2} \).

In polynomial equations, recognizing when a complex solution is needed is often hinted by how the equation behaves or the symmetry present within the numbers. Handling these numbers with familiarity simplifies solving even complex algebraic problems.