Problem 38 - College Algebra and Calculus: An Applied Approach

Question

Perform the indicated operations and simplify.\(\frac{(x+5)(x-3)}{x+2} \cdot \frac{1}{(x+5)(x+2)}\)

Step-by-Step Solution

Answer

The simplified form of the given expression is \((x-3)\)

Step 1: Rearrange The Expressions

Rearrange the expressions so that the like terms are together. \((x+5)(x-3) \cdot 1 \div (x+2) \cdot (x+5)(x+2)\)

Step 2: Simplify the Common Terms

Observe that the term \((x+5)\) and \(x+2\) is common in the numerator and the denominator of the expression. By the divisional property of equality, we can divide the common terms. Hence the expression becomes \((x-3)\).

Step 3: Final Result

The final simplified expression is \((x-3)\). No additional simplification can be done, so this is the final result.

Key Concepts

Algebraic Expressions

Algebraic expressions play a crucial role in mathematics. They consist of variables, numbers, and operations such as addition, subtraction, multiplication, and division. A typical algebraic expression could be something like \(x + 5\) or \(3x^2 - 2x + 7\). In these expressions, "x" is a variable which can take on different values. The constants like 5 or 7 are known as coefficients when they multiply the variables.
Your task, when working with algebraic expressions, is often to simplify or manipulate them in order to find an unknown value. Simplifying algebraic expressions often involves combining like terms, factoring, or performing operations such as addition or subtraction between expressions. The goal is to express the equation in its simplest form.

• Combine like terms: Terms that have the same variable raised to the same power.
• Factor: Find common factors or use special formulas like \((a+b)^2\).
• Use the distributive property: \(a(b+c) = ab + ac\).

Every operation performed follows specific mathematical rules and principles, ensuring the expression retains its equality.

Rational Expressions

Rational expressions are a specific type of algebraic expression. They resemble fractions and consist of a numerator and a denominator that are both polynomials. For example, \(\frac{x+3}{x-2}\) is a rational expression because both the numerator and the denominator are polynomials.
When dealing with rational expressions, the aim is often to simplify them or solve equations that include them. Simplifying rational expressions involves the following steps:

• Factor polynomials: Break down the numerator and the denominator into products of simpler terms.
• Cancel common factors: Identify and divide out common factors between the numerator and the denominator to simplify the expression.

A key point to remember is that division by zero is undefined. Therefore, it's crucial to identify values for the variable that would make the denominator zero and exclude these from the domain of the expression.
The process of simplifying rational expressions, as shown in the exercise, involves carefully identifying common factors and then canceling them across the numerator and denominator, adhering to the rules of mathematics.

Mathematics Education

In mathematics education, understanding and simplifying expressions is a fundamental skill. This is essential because mathematics builds on itself; as students progress, the complexity of problems increases. Mastery of algebraic and rational expressions paves the way for more advanced topics such as calculus and linear algebra.
Teaching students to simplify expressions involves more than memorizing techniques. It's about helping them develop a logical approach to problem-solving. Critical thinking skills are honed as students learn to identify patterns, make deductions, and apply theoretical understanding to reach conclusions.
Effective mathematics education encourages:

• Conceptual understanding: Students grasp underlying principles, rather than just procedures.
• Problem-solving skills: Students learn to find solutions through structured methods.
• Reasoning: Students validate their processes and results through logical thinking.

Through practice and application, students gain confidence in mathematics, finding joy in solving what might initially appear as complex challenges.