Simplifying Rational Expressions A Step-by-Step Solution
In the realm of algebra, simplifying expressions is a fundamental skill. This article delves into the process of identifying equivalent expressions, focusing on a specific example involving rational expressions. We will meticulously dissect the given problem, providing a step-by-step solution and offering insights into the underlying concepts. Whether you're a student grappling with algebraic manipulations or a seasoned mathematician seeking a refresher, this guide will equip you with the knowledge and understanding to confidently tackle similar problems.
Deconstructing the Problem
Our primary focus is to determine which expression is equivalent to the following:
(2a + 1) / (10a - 5) ÷ (10a) / (4a² - 1)
This expression involves division of rational expressions, which are essentially fractions where the numerator and denominator are polynomials. To simplify this, we'll employ the fundamental principle of dividing fractions: multiplying by the reciprocal. This transformation will set the stage for factoring and cancellation, ultimately leading us to the equivalent expression. Our journey will involve a blend of algebraic techniques, including factoring, simplifying, and recognizing patterns. Let's embark on this exploration, unraveling the intricacies of rational expressions and discovering the path to the solution.
Step 1 Transforming Division into Multiplication
The cornerstone of simplifying the given expression lies in transforming the division operation into multiplication. This is achieved by multiplying the first fraction by the reciprocal of the second fraction. The reciprocal of a fraction is obtained by simply swapping the numerator and denominator. Applying this principle to our problem, we get:
(2a + 1) / (10a - 5) * (4a² - 1) / (10a)
This transformation is crucial because it allows us to combine the two fractions into a single expression, making it easier to identify common factors and simplify. The problem now takes the form of a product of rational expressions, paving the way for the subsequent steps in our simplification process. We've effectively converted a division problem into a multiplication problem, a strategic move that simplifies the overall manipulation.
Step 2 Factoring the Polynomials Unveiling Hidden Structures
Factoring polynomials is a pivotal step in simplifying rational expressions. It allows us to break down complex expressions into their fundamental building blocks, revealing common factors that can be canceled out. In our expression, we have two polynomials that can be factored: (10a - 5) and (4a² - 1). Let's delve into the factoring process for each of these.
Factoring (10a - 5)
This expression has a common factor of 5 in both terms. Factoring out the 5, we get:
5(2a - 1)
This simple factoring step significantly simplifies the expression, making it easier to identify common factors with other parts of the rational expression.
Factoring (4a² - 1)
This expression is a difference of squares, a special pattern that can be factored as follows:
4a² - 1 = (2a)² - (1)² = (2a + 1)(2a - 1)
The difference of squares pattern is a powerful tool in factoring, and recognizing it here allows us to express the polynomial as a product of two binomials. This factorization is crucial for the subsequent cancellation steps.
By factoring these polynomials, we've unveiled the hidden structure within the expression, setting the stage for simplification through cancellation.
Step 3 Rewriting the Expression with Factored Polynomials
Now that we've factored the polynomials, we can rewrite the expression with the factored forms. This will make it easier to identify common factors and perform cancellations. Substituting the factored forms into our expression, we get:
(2a + 1) / [5(2a - 1)] * [(2a + 1)(2a - 1)] / (10a)
This rewritten expression clearly displays the factors in both the numerator and denominator. The next step involves strategically canceling out common factors, a process that will significantly simplify the expression.
The act of rewriting the expression with factored polynomials is a crucial step in the simplification process. It transforms the expression into a form where common factors are readily apparent, paving the way for the next stage of cancellation.
Step 4 Canceling Common Factors Simplifying the Expression
The heart of simplifying rational expressions lies in canceling out common factors that appear in both the numerator and the denominator. This process is analogous to simplifying numerical fractions by dividing both the numerator and denominator by their greatest common divisor. In our expression, we can identify the following common factors:
- (2a + 1): This factor appears once in the numerator and once in the denominator.
- (2a - 1): This factor also appears once in the numerator and once in the denominator.
Canceling these common factors, we are left with:
(2a + 1) / 5 * 1 / (10a)
This cancellation step dramatically simplifies the expression, reducing it to a more manageable form. The strategic elimination of common factors is a cornerstone of simplifying rational expressions.
Step 5 Multiplying the Remaining Fractions Final Simplification
After canceling the common factors, we are left with a simplified expression involving the product of two fractions. To complete the simplification, we multiply the remaining fractions together. Multiplying the numerators and the denominators, we get:
(2a + 1) / (5 * 10a) = (2a + 1) / (50a)
This final multiplication combines the remaining terms into a single fraction, representing the most simplified form of the original expression. We have successfully navigated the simplification process, arriving at the equivalent expression.
The Solution
Therefore, the expression equivalent to (2a + 1) / (10a - 5) ÷ (10a) / (4a² - 1) is:
(2a + 1) / (50a)
This corresponds to option D in the given choices.
Conclusion
Simplifying rational expressions involves a systematic approach, encompassing factoring, canceling common factors, and performing arithmetic operations on fractions. By mastering these techniques, you can confidently tackle a wide range of algebraic problems. This article has provided a detailed walkthrough of the simplification process, highlighting the key steps and underlying principles. Remember, practice is paramount in honing your algebraic skills. So, continue to explore, experiment, and apply these techniques to various problems, solidifying your understanding and building your confidence in the world of algebra.
By following these steps, we have successfully identified the equivalent expression. The process involved transforming division into multiplication, factoring polynomials, canceling common factors, and multiplying the remaining fractions. This step-by-step approach provides a clear and concise method for simplifying rational expressions.
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