Simplifying Algebraic Expressions: A Step-by-Step Guide

Hey guys! Let's dive into the world of simplifying algebraic expressions. It might seem a little daunting at first, but trust me, with a systematic approach, you can totally ace this. We'll tackle a specific problem and break it down step by step, ensuring you grasp every concept. This guide will walk you through the process, making sure you understand not just how to simplify, but why we do what we do. We will be using the expression given in the question and simplify the expression. Remember, practice makes perfect, so grab a pen and paper and let's get started! This method is especially important for students studying in the field of physics, as these equations can be frequently used in their study.

Understanding the Problem and the Tools We Need

First, let's understand the problem we're trying to solve. We are given a complex algebraic expression and our goal is to simplify it to a more manageable form. This involves several key steps, including factoring, finding common denominators, and combining like terms. Let's clarify what we are working with, just to make sure we are all on the same page. The expression to simplify is: x − 2 / (x^2 + x − 2) − (x + 1) / (x^2 − 4) + (x + 2) / (x^2 − 3x + 2).

We are given a number of options, and we need to check which one is the same as our simplification result.

To succeed, we'll need a strong grasp of several algebraic concepts, including:

• Factoring: Breaking down expressions into their component parts (e.g., x^2 - 4 can be factored into (x - 2)(x + 2)).
• Finding Common Denominators: Necessary when adding or subtracting fractions.
• Combining Like Terms: Simplifying expressions by grouping terms with the same variable and exponent.
• Rules of Exponents: Understanding how to handle powers and roots.

With these tools in our toolbox, we're well-equipped to simplify any expression! We'll start by breaking down each term in the expression, factoring where possible, and then finding a common denominator to combine the fractions. Keep in mind, the goal is to transform the initial expression into its simplest, most concise form while maintaining its original value. Remember the use of physics is essential in helping solve these problems.

Step 1: Factoring the Denominators

Let's start by factoring the denominators of each fraction. Factoring allows us to identify common factors and simplify the expression. Let's factor each one:

• x^2 + x - 2: This can be factored into (x + 2)(x - 1). We need to find two numbers that multiply to -2 and add up to 1. These numbers are 2 and -1.
• x^2 - 4: This is a difference of squares and factors into (x - 2)(x + 2).
• x^2 - 3x + 2: This can be factored into (x - 2)(x - 1). We need two numbers that multiply to 2 and add up to -3. These numbers are -2 and -1.

With the factored denominators, our expression now looks like this:

(x - 2) / ((x + 2)(x - 1)) - (x + 1) / ((x - 2)(x + 2)) + (x + 2) / ((x - 2)(x - 1))

Factoring is a super important skill! It’s like the key to unlocking the simplification process. Be patient, take your time, and make sure you understand each factorization.

Step 2: Finding a Common Denominator

Next, we need to find a common denominator for all the fractions. The common denominator is the least common multiple (LCM) of all the denominators. Looking at our factored denominators: (x + 2)(x - 1), (x - 2)(x + 2), and (x - 2)(x - 1), the LCM (and thus the common denominator) is (x - 1)(x - 2)(x + 2).

Now, we'll rewrite each fraction with this common denominator. To do this, we'll multiply the numerator and the denominator of each fraction by the factors missing from its original denominator to match the common denominator.

• For the first fraction, (x - 2) / ((x + 2)(x - 1)), we need to multiply the numerator and denominator by (x - 2). This gives us ((x - 2)(x - 2)) / ((x - 1)(x - 2)(x + 2)).
• For the second fraction, (x + 1) / ((x - 2)(x + 2)), we need to multiply the numerator and denominator by (x - 1). This gives us -((x + 1)(x - 1)) / ((x - 1)(x - 2)(x + 2)).
• For the third fraction, (x + 2) / ((x - 2)(x - 1)), we need to multiply the numerator and denominator by (x + 2). This gives us ((x + 2)(x + 2)) / ((x - 1)(x - 2)(x + 2)).

With the common denominator in place, we're ready to combine the fractions. Remember, the common denominator is like the foundation upon which we're building our simplified expression! The use of the common denominator is important in physics for calculating things like resistance and charge.

Step 3: Combining the Fractions

Now that all the fractions have the same denominator, we can combine them into a single fraction. Our expression now looks like this:

((x - 2)(x - 2) - (x + 1)(x - 1) + (x + 2)(x + 2)) / ((x - 1)(x - 2)(x + 2))

Next, let's expand the numerators and simplify.

• (x - 2)(x - 2) = x^2 - 4x + 4
• (x + 1)(x - 1) = x^2 - 1
• (x + 2)(x + 2) = x^2 + 4x + 4

So our expression becomes:

(x^2 - 4x + 4 - (x^2 - 1) + x^2 + 4x + 4) / ((x - 1)(x - 2)(x + 2))

Simplify the numerator by combining like terms: x^2 - 4x + 4 - x^2 + 1 + x^2 + 4x + 4 = x^2 + 9. So the simplified expression is (x^2 + 9) / ((x - 1)(x - 2)(x + 2)).

Step 4: Identifying the Correct Answer and Discussion

After simplifying the expression, we have: (x^2 + 9) / ((x - 1)(x - 2)(x + 2)).

However, this is not one of our options. Let's go back and review the calculations. If we did the calculations right, then it's likely that there was an error in our work. After checking each step again, the equation simplifies to: (x^2 + 9) / ((x - 1)(x - 2)(x + 2)). Upon checking the question again, it can be seen that the expression is not in the provided options. This might occur when the equation itself has some kind of error or there is a typo.

Let's analyze the answers:

• A: (x^2 + 6x + 1) / ((x - 1)(x - 2)(x + 2))
• B: (x^2 - 6x + 7) / ((x - 1)(x - 2)(x + 2))
• C: (x^2 - 5x - 7) / ((x - 1)(x - 2)(x + 2))
• D: 1 / ((x - 1)(x - 2))

None of the answers match our calculated result. This is a good reminder to always double-check your work. Mistakes can happen! Always make sure to re-evaluate your answers.

Conclusion

So there you have it! We've walked through the process of simplifying a complex algebraic expression step by step. Remember that the key is to break down the problem into smaller, manageable parts. Practice consistently, and you'll become a pro at simplifying expressions in no time. The concepts we covered, like factoring and finding common denominators, are fundamental not just in algebra, but across many fields, including physics. Keep practicing, guys, and you'll become math whizzes! Understanding these concepts is the foundation for more advanced topics.