Problem 64 - Precalculus

Question

Explain the mistake that is made. Find all complex solutions to \(x^{5}-1=0\) Solution: Add 1 to both sides. Raise both sides to the fifth power. \(x=1^{1 / 5}\) Simplify. \(x=1\) This is incorrect. What mistake was made?

Step-by-Step Solution

Answer

The mistake was assuming there is only one solution, while there are five complex solutions (fifth roots of unity).

Step 1: Recall the equation and its nature

The given equation is \( x^{5} - 1 = 0 \). This is a polynomial equation that can potentially have multiple roots, corresponding to the complex solutions of a fifth-degree polynomial.

Step 2: Identify the correct process for solving

The equation \( x^{5} = 1 \) suggests that we are looking for all complex numbers \( x \) such that raising \( x \) to the fifth power results in 1. This requires considering all possible complex solutions, not just one.

Step 3: Recognize the multiple roots of unity

The equation \( x^{5} = 1 \) leads us to the concept of complex numbers known as the fifth roots of unity. These roots are the solutions of this equation and can be expressed as \( x = e^{2\pi i k/5} \) where \( k \) is an integer from 0 to 4.

Step 4: Calculate each root

For each \( k \) from 0 to 4, compute the distinct roots: 1. \( x = e^{2 \pi i \cdot 0/5} = 1 \)2. \( x = e^{2 \pi i \cdot 1/5} \)3. \( x = e^{2 \pi i \cdot 2/5} \)4. \( x = e^{2 \pi i \cdot 3/5} \)5. \( x = e^{2 \pi i \cdot 4/5} \).

Step 5: Error explanation

The mistake made in the original solution is assuming the only solution is \( x = 1 \), which represents only one of the five complex solutions. The process ignored the possibility of other roots by only finding the principal root.

Key Concepts

Fifth-degree polynomial

A fifth-degree polynomial is an equation where the highest power of the variable, usually denoted as \(x\), is five. This means the polynomial will have a term like \(x^5\), making it a quintic equation. Such polynomials can be complicated due to the high degree, which means they can potentially have up to five distinct solutions. These solutions, or roots, could be real or complex numbers.

In the context of our exercise, the polynomial equation \(x^5 - 1 = 0\) is a fifth-degree polynomial. When setting it equal to zero, we are looking for numbers that can replace \(x\) to make the equation true. Understanding how to handle high-degree polynomials involves breaking down the equation into simpler parts and recognizing the potential for multiple solutions, especially when dealing with complex numbers.

Complex numbers

Complex numbers are numbers that include a real part and an imaginary part. Usually expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property that \(i^2 = -1\).

These numbers expand the number system to include solutions to equations that do not have real roots. Complex numbers can be plotted on the complex plane, where the x-axis represents the real part and the y-axis corresponds to the imaginary part.

• Real numbers are when \(b = 0\).
• Purely imaginary numbers when \(a = 0\).
• Complex conjugates are formed by changing the sign of the imaginary part.

In polynomial equations like \(x^5 - 1 = 0\), complex numbers allow us to find all possible roots, especially when the equation involves roots of unity.

Complex solutions

Complex solutions occur when the roots of an equation are not purely real numbers. In the case of polynomial equations, when a solution involves the square root of a negative number, it results in a complex root.

For the equation \(x^5 = 1\), although \(x = 1\) is a real number solution, there are other complex solutions called the fifth roots of unity. These solutions are obtained by using Euler's formula: \(e^{i\theta} = \cos(\theta) + i\sin(\theta)\), which represents complex numbers on the unit circle in the complex plane.

• The principal root of unity is always 1.
• Additional roots are found by rotating 72 degrees (\(2\pi/5\) radians) around the unit circle.
• Each complex solution is uniquely expressed as \(x = e^{2\pi i k/5}\) for \(k = 0, 1, 2, 3, 4\).

Recognizing complex solutions is crucial as they provide all possible roots for higher-degree polynomials.

Polynomial equation

A polynomial equation is formed by setting a polynomial equal to a value, typically zero. It represents an expression consisting of variables raised to whole number exponents and coefficients. Solving a polynomial equation involves finding the roots or values of the variable that satisfy the equation.

For example, the polynomial equation \(x^5 - 1 = 0\) is formed by subtracting 1 from \(x^5\). Solving this equation requires finding all values for \(x\) such that the equation holds true.

• Each root found is a solution to the equation.
• Unlike simple linear or quadratic equations, higher-degree polynomials like fifth-degree require more comprehensive methods.

Techniques such as factoring, substitution, and recognizing roots of unity are essential in solving these equations efficiently. Understanding polynomial equations helps with solving various problems across mathematics and engineering fields.