Problem 68 - Algebra for College Students

Question

Find all real number solutions for each equation. $$x^{5}-x=0$$

Step-by-Step Solution

Answer

The solutions are x = 0, x = 1, and x = -1.

Step 1: Factor the Equation

We start with the equation: x^5 - x = 0.Notice that there is a common factor of \(x\) in the terms. Factoring out \(x\) gives us:\[x(x^4 - 1) = 0\]

Step 2: Apply the Zero Product Property

According to the zero product property, if the product of two factors is zero, then at least one of the factors must be zero. So we set each factor equal to zero and solve:1. \(x = 0\)2. \(x^4 - 1 = 0\)

Step 3: Solve for x from x^4 - 1 = 0

Solving the second equation \(x^4 - 1 = 0\) involves rearranging it:\[x^4 = 1\]Taking the fourth root of both sides gives us:\[x = \pm 1\]

Step 4: List all Solutions

We have found the following values of \(x\) that solve the original equation:1. From \(x = 0\), we have \(x = 0\).2. From \(x^4 = 1\), we have \(x = 1\) and \(x = -1\).These are all the real number solutions.

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial step in solving polynomial equations. It involves breaking down a complex polynomial into simpler "factors," making it easier to find solutions. Let's consider the polynomial equation given in the original exercise: \[x^5 - x = 0\] The first thing to notice here is that each term has a common factor, in this case, "\(x\)". Extracting this common factor simplifies the equation:

• The original equation \(x^5 - x = 0\) can be rewritten as \(x(x^4 - 1) = 0\).

Factoring reduces the complexity significantly, turning a higher-degree equation into a product of simpler factors. This is particularly helpful for solving since the equation splits into parts that can individually equal zero. Recognizing common factors and effectively factoring them out is a valuable skill in handling algebraic expressions. This approach not only aids in finite solutions but also represents the equation in a form more accessible to solve using other mathematical techniques.

Zero Product Property

The zero product property is a powerful mathematical statement used to solve equations where a product equals zero. The property asserts that if the product of two or more factors results in zero, at least one of the factors must individually be zero. This fundamental property helps transition from a factored polynomial equation to finding its roots.

Applying the zero product property to our factored equation:

• For \(x(x^4 - 1) = 0\), set each factor to zero: \(x = 0\) and \(x^4 - 1 = 0\).

By setting each factor equal to zero, this property essentially splits the problem into smaller, more manageable parts. Each smaller equation can then be solved separately to find the solutions of the original equation. This step is essential as it pivots the problem from factoring to solving simpler algebraic equations, paving the way to uncover all possible solutions.

Finding Roots

Finding roots of a polynomial equation is synonymous with finding the values of the variable that make the equation true. After factoring the polynomial and applying the zero product property, we have simplified the original problem into components, each equalling zero:

• For \(x = 0\), the solution is directly \(x = 0\).
• For \(x^4 - 1 = 0\), we solve \(x^4 = 1\).

Solving \(x^4 = 1\) involves taking the fourth root of both sides. Since both positive and negative numbers raised to an even power give positive results, the solutions are \(x = 1\) and \(x = -1\).

Let's recap the roots for the original polynomial \(x^5 - x = 0\):

• \(x = 0\)
• \(x = 1\)
• \(x = -1\)

These are all the real number solutions. Understanding and executing these steps not only solves the equation but also gives insights into the behavior of polynomial functions. By mastering these techniques, one can solve a wide range of polynomial equations effectively.