Problem 72 - Algebra for College Students

Question

Set up an equation and solve each of the following problems. The cube of a number equals the square of the same number. Find the number.

Step-by-Step Solution

Answer

The number is either 0 or 1.

Step 1: Define the Problem Mathematically

Let the number be represented as \( x \). We know that the cube of this number is equal to the square of the same number. Therefore, we can write the equation as \( x^3 = x^2 \).

Step 2: Simplify the Equation

To simplify the equation \( x^3 = x^2 \), we can subtract \( x^2 \) from both sides of the equation. This gives us \( x^3 - x^2 = 0 \).

Step 3: Factor the Equation

Now, factor out \( x^2 \) from the equation \( x^3 - x^2 = 0 \). This can be rewritten as \( x^2 (x - 1) = 0 \).

Step 4: Solve for x

We solve the equation \( x^2 (x - 1) = 0 \) by setting each factor equal to zero. This gives us the equations \( x^2 = 0 \) and \( x - 1 = 0 \).

Step 5: Find the Solutions

From \( x^2 = 0 \), we find that \( x = 0 \). From \( x - 1 = 0 \), we find that \( x = 1 \). Therefore, the solutions to the equation are \( x = 0 \) and \( x = 1 \).

Key Concepts

Understanding Polynomial Equations

Polynomial equations are mathematical expressions involving a variable raised to whole number powers. They often come in the form of multiple terms added or subtracted together, such as \(x^3 - x^2 = 0\) in our given problem. These equations can be as simple as linear equations like \(x + 2 = 0\) or more complex like quartic equations (fourth degree), but they all share common characteristics.

The power or degree of the polynomial is determined by the highest exponent of the variable in the equation. For example, the equation \(x^3 = x^2\) is a polynomial of degree 3 because the term \(x^3\) carries the highest power. Polynomial equations generally appear in various branches of mathematics, including algebra, calculus, and numerical analysis, each employing these expressions to solve different types of problems.

Grasping the basics of polynomial equations allows students to delve deeper into mathematical problem-solving, enhancing their ability to analyze, simplify, and find solutions.

The Art of Factoring Equations

Factoring is a powerful algebraic technique used to simplify equations and find solutions. In the context of polynomial equations, factoring involves decomposing a complex expression into simpler multiplicative components. For the problem \(x^3 - x^2 = 0\), factoring is used to transform this equation by taking out common elements.

To factor the equation, we notice that both terms share the factor \(x^2\). By factoring \(x^2\) out, the equation becomes \(x^2 (x - 1) = 0\). This transformation breaks the equation down into parts that can be easily solved.

Factoring not only simplifies solving polynomial equations but also provides insight into the nature of the solutions. It reveals the critical points where the equation equals zero, which are the solutions we are looking for when solving these types of equations.

• Identify common factors in all terms.
• Rewrite the equation using these factors.
• Solve each part to find the values of the variable.

Mathematical Problem Solving Techniques

Mathematical problem-solving involves using logical steps to find answers to complex questions. In this case study with the equation \(x^3 = x^2\), the steps outline a structured approach to reach a solution.

First, it begins with understanding and defining the problem statement correctly, which involves translating words into mathematical expressions. This step turns the descriptive problem **the cube of a number equals the square of the same number** into the equation \(x^3 = x^2\).

Next, it involves simplifying and transforming the equation, often by moving terms around or removing like terms, as seen when \(x^2\) was subtracted from both sides to yield \(x^3 - x^2 = 0\). This simplification is crucial as it leads directly to the need for factoring, a necessary skill in algebra.

• Translate the problem into a mathematical language.
• Use simplification techniques to make the problem more manageable.
• Apply appropriate methods, such as factoring, to find solutions.

Each step in solving a problem relies on a clear understanding of the tools at one's disposal and knowing when and how to use them effectively. Problem-solving in mathematics doesn't just end with finding a solution; it's also about understanding the process and being able to apply these techniques to future problems.