Simplifying Algebraic Expressions A Step-by-Step Guide

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Introduction

In this comprehensive guide, we will walk you through the process of simplifying the algebraic expression 2b2b2−4b−12⋅b2−44b\frac{2 b^2}{b^2-4 b-12} \cdot \frac{b^2-4}{4 b}. This involves factoring, canceling common factors, and applying algebraic principles to arrive at the simplest form of the given expression. Algebraic simplification is a fundamental skill in mathematics, crucial for solving equations, understanding functions, and tackling more advanced mathematical concepts. This article aims to provide a step-by-step explanation to help students and enthusiasts grasp the methodology involved in simplifying such expressions. Mastering these techniques not only enhances problem-solving capabilities but also builds a strong foundation for advanced topics in algebra and calculus. We will break down each step, providing clear explanations and insights to ensure a thorough understanding of the simplification process. Whether you are a student looking to improve your algebra skills or someone interested in a refresher, this guide will provide you with the necessary tools and knowledge to simplify algebraic expressions effectively.

Step-by-Step Simplification

Step 1: Factoring the Polynomials

The first crucial step in simplifying the algebraic expression is to factor the polynomials present in both the numerator and the denominator. Factoring helps us identify common terms that can be canceled out, making the expression simpler. In this case, we have two polynomials to factor: b2−4b−12b^2 - 4b - 12 and b2−4b^2 - 4. Factoring these polynomials correctly is essential for simplifying the entire expression.

First, let's factor the quadratic expression b2−4b−12b^2 - 4b - 12. We are looking for two numbers that multiply to -12 and add up to -4. These numbers are -6 and 2. Therefore, we can write the quadratic expression as:

b2−4b−12=(b−6)(b+2)b^2 - 4b - 12 = (b - 6)(b + 2)

Next, we need to factor b2−4b^2 - 4. This is a difference of squares, which can be factored using the formula a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b). In this case, a=ba = b and b=2b = 2. So, we have:

b2−4=(b−2)(b+2)b^2 - 4 = (b - 2)(b + 2)

Now that we have factored both polynomials, we can substitute these factored forms back into the original expression. This step is critical because it sets the stage for canceling out common factors, which will lead us to the simplified form. Factoring accurately is paramount to avoid mistakes in the simplification process. By breaking down the polynomials into their factors, we can better see the structure of the expression and identify opportunities for simplification. This methodical approach is key to solving algebraic problems and ensuring the correct final result.

Step 2: Rewriting the Expression with Factored Forms

Once we have successfully factored the polynomials, the next step is to rewrite the algebraic expression using these factored forms. This involves replacing the original polynomials with their equivalent factored expressions in both the numerator and the denominator. This step is crucial because it visually organizes the expression, making it easier to identify and cancel out common factors. By rewriting the expression with factored forms, we transition from a complex polynomial fraction to a more manageable form that highlights the relationships between terms.

Substituting the factored forms we found in Step 1, the expression 2b2b2−4b−12⋅b2−44b\frac{2 b^2}{b^2-4 b-12} \cdot \frac{b^2-4}{4 b} becomes:

2b2(b−6)(b+2)⋅(b−2)(b+2)4b\frac{2b^2}{(b - 6)(b + 2)} \cdot \frac{(b - 2)(b + 2)}{4b}

This rewritten form clearly displays the factors in both the numerator and the denominator. We can see that (b+2)(b + 2) appears in both the numerator and the denominator, indicating a potential simplification. The process of rewriting the expression in factored form is a fundamental technique in algebra, enabling us to simplify complex expressions and solve equations more efficiently. It allows us to break down the problem into smaller, more manageable parts, making the simplification process more intuitive and less prone to errors. This step bridges the gap between the original expression and its simplified form, providing a clear pathway for further algebraic manipulation.

Step 3: Canceling Common Factors

Canceling common factors is a pivotal step in simplifying algebraic expressions. After rewriting the expression in its factored form, we look for factors that appear in both the numerator and the denominator. These common factors can be canceled out because any term divided by itself equals 1. This process significantly reduces the complexity of the expression, bringing us closer to the simplest form. Identifying and canceling these factors requires careful observation and an understanding of algebraic principles. This step transforms the expression by removing redundancies and highlighting the essential components.

In our expression, 2b2(b−6)(b+2)⋅(b−2)(b+2)4b\frac{2b^2}{(b - 6)(b + 2)} \cdot \frac{(b - 2)(b + 2)}{4b}, we can identify the following common factors:

  • (b+2)(b + 2) appears in both the numerator and the denominator.
  • bb is a factor in both 2b22b^2 and 4b4b.

Let's cancel these common factors step by step:

  1. Cancel (b+2)(b + 2) from the numerator and the denominator:

    2b2(b−6)(b+2)⋅(b−2)(b+2)4b=2b2(b−6)⋅(b−2)4b\frac{2b^2}{(b - 6)\cancel{(b + 2)}} \cdot \frac{(b - 2)\cancel{(b + 2)}}{4b} = \frac{2b^2}{(b - 6)} \cdot \frac{(b - 2)}{4b}

  2. Simplify the terms involving bb. We have 2b22b^2 in the numerator and 4b4b in the denominator. We can cancel one bb from both and simplify the numerical coefficients:

    2b2(b−6)⋅(b−2)4b=2b⋅b(b−6)⋅(b−2)4b=2b⋅b(b−6)⋅(b−2)4b=2b(b−6)⋅(b−2)4\frac{2b^2}{(b - 6)} \cdot \frac{(b - 2)}{4b} = \frac{2b \cdot b}{(b - 6)} \cdot \frac{(b - 2)}{4b} = \frac{2\cancel{b} \cdot b}{(b - 6)} \cdot \frac{(b - 2)}{4\cancel{b}} = \frac{2b}{(b - 6)} \cdot \frac{(b - 2)}{4}

  3. Further simplify the numerical coefficients. We have 22 in the numerator and 44 in the denominator:

    2b(b−6)⋅(b−2)4=2b(b−6)⋅(b−2)2⋅2=b(b−6)⋅(b−2)2\frac{2b}{(b - 6)} \cdot \frac{(b - 2)}{4} = \frac{\cancel{2}b}{(b - 6)} \cdot \frac{(b - 2)}{2 \cdot \cancel{2}} = \frac{b}{(b - 6)} \cdot \frac{(b - 2)}{2}

By meticulously canceling common factors, we have significantly simplified the expression. This step is a critical application of the principles of algebraic simplification and is essential for arriving at the most concise form of the expression. The ability to identify and cancel common factors is a fundamental skill in algebra, crucial for solving complex problems and simplifying equations efficiently.

Step 4: Combining Remaining Terms

After canceling the common factors, the next step in simplifying the algebraic expression involves combining the remaining terms. This usually means multiplying the terms left in the numerator and the denominator separately. Combining these terms helps to consolidate the expression and move closer to its simplest form. This step may also involve distributing terms or applying other algebraic operations to ensure the expression is as simplified as possible. The goal is to bring together all the remaining factors in a coherent manner, resulting in a single fraction or expression.

Following the cancellation of common factors in Step 3, our expression looks like this:

b(b−6)⋅(b−2)2\frac{b}{(b - 6)} \cdot \frac{(b - 2)}{2}

To combine the remaining terms, we multiply the numerators together and the denominators together:

b⋅(b−2)(b−6)⋅2\frac{b \cdot (b - 2)}{(b - 6) \cdot 2}

Now, we distribute bb in the numerator:

b2−2b2(b−6)\frac{b^2 - 2b}{2(b - 6)}

Next, we distribute 22 in the denominator:

b2−2b2b−12\frac{b^2 - 2b}{2b - 12}

This combined form presents the simplified expression as a single fraction. This step is essential for presenting the final answer in a clear and concise manner. Combining remaining terms ensures that the expression is in its most reduced form, making it easier to understand and work with in further mathematical operations. The ability to accurately combine terms is a key skill in algebraic manipulation, allowing for efficient simplification of complex expressions.

Step 5: Final Simplified Form

After combining the remaining terms, the final step is to present the algebraic expression in its simplified form. This involves reviewing the expression to ensure that no further simplifications are possible. The simplified form should be the most concise and straightforward representation of the original expression. It is essential to double-check the final form to ensure accuracy and completeness. This step solidifies the simplification process, providing a clear and final answer that can be easily understood and utilized.

From Step 4, we have the expression:

b2−2b2b−12\frac{b^2 - 2b}{2b - 12}

We can observe that there might be an opportunity for further simplification by factoring out common factors from both the numerator and the denominator. Let's factor out bb from the numerator:

b2−2b=b(b−2)b^2 - 2b = b(b - 2)

And factor out 22 from the denominator:

2b−12=2(b−6)2b - 12 = 2(b - 6)

So, our expression becomes:

b(b−2)2(b−6)\frac{b(b - 2)}{2(b - 6)}

Now, we review the expression to ensure that no further simplifications are possible. There are no common factors between the numerator and the denominator that can be canceled out. Therefore, this is the final simplified form of the expression.

The final simplified form of the algebraic expression 2b2b2−4b−12⋅b2−44b\frac{2 b^2}{b^2-4 b-12} \cdot \frac{b^2-4}{4 b} is:

b(b−2)2(b−6)\frac{b(b - 2)}{2(b - 6)}

This final step is crucial in ensuring that the expression is in its most simplified and understandable form. By presenting the final simplified form, we provide a clear and concise answer that can be used for further mathematical analysis or problem-solving. The ability to arrive at the final simplified form is a testament to a thorough understanding of algebraic principles and techniques.

Conclusion

In this detailed walkthrough, we have demonstrated how to simplify the given algebraic expression 2b2b2−4b−12⋅b2−44b\frac{2 b^2}{b^2-4 b-12} \cdot \frac{b^2-4}{4 b} step-by-step. The process involved factoring polynomials, rewriting the expression with factored forms, canceling common factors, combining remaining terms, and presenting the final simplified form. Each step is crucial in achieving the simplest representation of the expression. By following these steps, we arrived at the simplified form: b(b−2)2(b−6)\frac{b(b - 2)}{2(b - 6)}.

Simplifying algebraic expressions is a fundamental skill in mathematics, essential for solving equations, understanding functions, and tackling more advanced mathematical concepts. Mastering these techniques not only enhances problem-solving capabilities but also builds a strong foundation for advanced topics in algebra and calculus. The ability to break down complex expressions into simpler forms is a key skill for students and professionals alike. This skill is applicable in various fields, including engineering, computer science, and economics, where mathematical models and equations need to be simplified for analysis and problem-solving.

By understanding the step-by-step process of simplification, you can confidently approach similar problems and apply these techniques to various mathematical contexts. Consistent practice and a clear understanding of algebraic principles are key to mastering simplification. The techniques discussed in this article provide a solid foundation for further exploration of algebraic concepts and problem-solving strategies. Whether you are a student looking to improve your algebra skills or a professional needing a refresher, these methods will prove invaluable in your mathematical endeavors. Simplifying expressions is not just about finding the right answer; it's about developing a methodical approach to problem-solving and enhancing your mathematical reasoning skills.