Multiplying Rational Expressions Simplifying (5u/(u^2+23u+132)) * ((u^2+18u+72)/(3u))

In the realm of algebra, multiplying rational expressions is a fundamental skill that builds upon the foundation of fraction manipulation and polynomial factorization. This article delves into the process of multiplying rational expressions, providing a step-by-step guide with a focus on simplifying the result. We will explore the key concepts, techniques, and potential pitfalls involved in this operation. Our focus will be on the specific example of multiplying (5u/(u^2+23u+132)) * ((u^2+18u+72)/(3u)), but the principles discussed here are applicable to a wide range of similar problems.

Before diving into the multiplication process, it's crucial to understand what rational expressions are. A rational expression is simply a fraction where the numerator and denominator are polynomials. Polynomials, in turn, are algebraic expressions consisting of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. Examples of polynomials include x^2 + 3x - 2, 5u, and u^2 + 23u + 132. Therefore, rational expressions can take forms like (x^2 + 3x - 2)/(5u) or (5u)/(u^2 + 23u + 132).

The ability to manipulate rational expressions is essential in various areas of mathematics, including calculus, precalculus, and algebra. They often appear in equations, functions, and mathematical models used to describe real-world phenomena. Mastering the techniques for working with rational expressions, such as multiplication, division, addition, subtraction, and simplification, is a cornerstone of algebraic proficiency.

The first and most crucial step in multiplying rational expressions is to factor the polynomials in both the numerators and denominators. Factoring involves breaking down a polynomial into a product of simpler expressions, usually binomials. This step is critical because it allows us to identify common factors that can be canceled out later, leading to a simplified result. In our example, we have the expression (5u/(u^2+23u+132)) * ((u^2+18u+72)/(3u)). We need to factor the quadratic expressions u^2 + 23u + 132 and u^2 + 18u + 72.

To factor u^2 + 23u + 132, we look for two numbers that multiply to 132 and add up to 23. These numbers are 11 and 12. Therefore, u^2 + 23u + 132 can be factored as (u + 11)(u + 12). Similarly, to factor u^2 + 18u + 72, we seek two numbers that multiply to 72 and add up to 18. These numbers are 6 and 12. So, u^2 + 18u + 72 factors into (u + 6)(u + 12). Now, our expression looks like this: (5u/((u + 11)(u + 12))) * (((u + 6)(u + 12))/(3u)). The factorization process is vital, as it transforms the complex polynomials into a form where common factors become readily apparent.

After factoring, the next step is to multiply the numerators together and the denominators together. This is analogous to multiplying ordinary fractions, where you multiply the tops and the bottoms separately. In our case, we have (5u/((u + 11)(u + 12))) * (((u + 6)(u + 12))/(3u)). Multiplying the numerators, we get 5u * (u + 6)(u + 12). Multiplying the denominators, we get (u + 11)(u + 12) * 3u. Now, our expression looks like this: (5u(u + 6)(u + 12)) / (3u(u + 11)(u + 12)).

This step essentially combines the two rational expressions into a single fraction. The numerator of the resulting fraction is the product of the numerators of the original fractions, and the denominator is the product of the original denominators. While the expression might appear more complex at this stage, it sets the stage for the crucial simplification process that follows. By multiplying the numerators and denominators, we create a single fraction that encapsulates the entire multiplication operation, making it easier to identify and cancel common factors in the next step.

The most important step in multiplying rational expressions is simplification. This involves canceling out common factors that appear in both the numerator and the denominator. This process is based on the fundamental principle that dividing both the numerator and denominator of a fraction by the same non-zero value does not change the value of the fraction. In our example, we have (5u(u + 6)(u + 12)) / (3u(u + 11)(u + 12)). We can see that both the numerator and the denominator have the factors u and (u + 12).

We can cancel out the common factor 'u' from the numerator and denominator. We can also cancel out the common factor (u + 12) from the numerator and denominator. After canceling these common factors, our expression simplifies to (5(u + 6)) / (3(u + 11)). This simplification step is crucial because it reduces the expression to its simplest form, making it easier to work with and understand. By canceling common factors, we eliminate redundant terms and reveal the underlying structure of the rational expression.

After canceling common factors, the final step is to write the simplified expression. This usually involves distributing any remaining factors and presenting the result in its most concise form. In our case, we have (5(u + 6)) / (3(u + 11)). We can distribute the 5 in the numerator to get 5u + 30. Similarly, we can distribute the 3 in the denominator to get 3u + 33. Therefore, the simplified expression is (5u + 30) / (3u + 33). Alternatively, we can leave the expression in its factored form, which is (5(u + 6)) / (3(u + 11)), as it might be more convenient for certain applications.

The simplified expression represents the final result of the multiplication. It is equivalent to the original expression, but it is in its most reduced form. This simplified form is often preferred because it is easier to interpret, analyze, and use in further calculations. The ability to simplify rational expressions is a valuable skill in algebra, as it allows us to work with expressions in their most manageable form.

While multiplying rational expressions is a straightforward process when done correctly, there are several potential pitfalls and common mistakes to be aware of. One of the most frequent errors is failing to factor the polynomials completely before canceling common factors. It's essential to factor every polynomial as much as possible to ensure that all common factors are identified and canceled. Another common mistake is canceling terms that are not factors. Only factors, which are expressions multiplied together, can be canceled. Terms that are added or subtracted cannot be canceled directly.

For example, in the expression (5u + 30) / (3u + 33), we cannot cancel the 'u' terms or the constant terms because they are not factors. We can only cancel factors that are multiplied by the entire numerator or denominator. Another potential pitfall is forgetting to distribute after canceling factors. If there are remaining factors outside parentheses, it's important to distribute them to obtain the final simplified expression. Avoiding these common mistakes requires careful attention to detail and a thorough understanding of the rules of algebra.

Multiplying rational expressions is a fundamental algebraic skill that involves factoring, multiplying numerators and denominators, simplifying by canceling common factors, and writing the simplified expression. By following these steps carefully, we can successfully multiply rational expressions and obtain the result in its simplest form. This process is essential for various mathematical applications, including solving equations, simplifying expressions, and working with functions. Our focus on the example of multiplying (5u/(u^2+23u+132)) * ((u^2+18u+72)/(3u)) has provided a concrete illustration of these steps.

Mastering the techniques for multiplying rational expressions requires practice and attention to detail. It's crucial to understand the underlying principles of factoring, simplifying, and working with algebraic fractions. By avoiding common mistakes and following a systematic approach, you can confidently tackle a wide range of problems involving rational expressions. The ability to manipulate rational expressions is a valuable asset in your mathematical toolkit, enabling you to solve complex problems and gain a deeper understanding of algebraic concepts. Remember to always factor completely, cancel common factors correctly, and simplify the result to its most concise form.