Problem 63 - Algebra for College Students

Question

Find all real number solutions for each equation. $$3 x^{3}=3 x$$

Step-by-Step Solution

Answer

The solutions are \(x = 0\), \(x = -1\), and \(x = 1\).

Step 1: Simplify the Equation

Start by simplifying the given equation. The equation is \(3x^3 = 3x\). Divide both sides of the equation by 3 to make the equation simpler: \(x^3 = x\).

Step 2: Rearrange the Equation

To find the solutions, rearrange the equation so that it equals zero. Subtract \(x\) from both sides: \(x^3 - x = 0\).

Step 3: Factor the Equation

Factor the equation \(x^3 - x = 0\). First, factor out the common factor \(x\) to get \(x(x^2 - 1) = 0\).

Step 4: Factor Further

Continue factoring the equation. Notice that \(x^2 - 1\) is a difference of squares. It can be factored as \((x + 1)(x - 1)\). Thus, the equation becomes \(x(x + 1)(x - 1) = 0\).

Step 5: Solve for the Roots

Use the zero product property to find the solutions. Set each factor equal to zero: 1) \(x = 0\), 2) \(x + 1 = 0\), 3) \(x - 1 = 0\). This gives the solutions \(x = 0\), \(x = -1\), and \(x = 1\).

Key Concepts

Real Number Solutions

Real number solutions refer to the values of the variable that satisfy the equation and are real numbers. Real numbers include all the numbers on the number line, like integers, fractions, and irrational numbers.

In our exercise, we seek to find these real solutions for the equation given: \(3x^3 = 3x\). By simplifying and factoring the equation, we eventually find values of \(x\) that make the equation true.

In this instance, our solutions are \(x = 0\), \(x = -1\), and \(x = 1\). This means if you substitute any of these values back into the original equation, the equation will hold true, showing that these are the valid real number solutions.

Factoring Equations

Factoring is a process where we rewrite an equation as a product of its simpler terms or factors. This method is crucial for solving cubic equations because it breaks them down into manageable parts.

In our example, after simplifying the equation \(x^3 = x\), we rearrange it to \(x^3 - x = 0\). This step is essential as it prepares the equation for factoring.

The first step of factoring involves taking out the common factor, which in this case is \(x\). We factor it out to get \(x(x^2 - 1) = 0\).

Recognize that the term \(x^2 - 1\) is a difference of squares \((x^2 - 1 = (x + 1)(x - 1))\). Applying the difference of squares formula allows us to completely factor the equation into \(x(x + 1)(x - 1) = 0\). This shows us the simpler expressions that multiply to form the original equation.

Zero Product Property

The zero product property states that if the product of two or more factors is zero, then at least one of the factors must be zero. This property is especially useful once we have factored our equation into a product of terms.

For our equation \(x(x + 1)(x - 1) = 0\), we apply the zero product property by setting each factor equal to zero:

• When \(x = 0\)
• When \(x + 1 = 0\), solving gives \(x = -1\)
• When \(x - 1 = 0\), solving gives \(x = 1\)

Applying this property is often the final step in solving the equation. Each equation set up from the individual factors provides us with the real solutions for the original cubic equation. This step helps confirm that each found solution actually results in the original equation equaling zero.