Multiply The Following Rational Expressions: (30x + 30)/(2x + 6) * (x + 3)/(6x^2 - 6)

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In this comprehensive guide, we will delve into the process of multiplying rational expressions, using the specific example of (30x + 30)/(2x + 6) * (x + 3)/(6x^2 - 6). Multiplying rational expressions involves several key steps, including factoring, simplifying, and combining terms. By mastering these steps, you'll be able to confidently tackle more complex algebraic problems. Let's break down the process step by step.

1. Factoring the Expressions

The first critical step in multiplying rational expressions is to factor each polynomial completely. This involves identifying common factors and applying factoring techniques such as the difference of squares or quadratic factoring. Factoring simplifies the expressions and allows for cancellation of common factors later on.

Let's start by factoring the first expression, (30x + 30)/(2x + 6). In the numerator, 30x + 30, we can factor out the greatest common factor (GCF), which is 30. This gives us 30(x + 1). In the denominator, 2x + 6, the GCF is 2, so we factor out 2 to get 2(x + 3). Thus, the first expression becomes 30(x + 1) / 2(x + 3).

Now, let's factor the second expression, (x + 3)/(6x^2 - 6). The numerator, x + 3, is already in its simplest form and cannot be factored further. In the denominator, 6x^2 - 6, we can factor out the GCF, which is 6. This gives us 6(x^2 - 1). Notice that x^2 - 1 is a difference of squares, which can be factored as (x + 1)(x - 1). So, the denominator becomes 6(x + 1)(x - 1), and the second expression is (x + 3) / 6(x + 1)(x - 1).

After factoring both expressions, we have:

  • First expression: 30(x + 1) / 2(x + 3)
  • Second expression: (x + 3) / 6(x + 1)(x - 1)

Factoring is the cornerstone of simplifying rational expressions. By breaking down each polynomial into its simplest factors, we pave the way for canceling common terms and reducing the expression to its most manageable form. This meticulous approach ensures that we handle each component with precision, setting the stage for accurate simplification in the subsequent steps. Recognizing patterns such as the difference of squares or common factors becomes second nature with practice, making the process more efficient and less prone to errors. The ability to factor correctly not only simplifies the multiplication process but also enhances understanding of the underlying algebraic structures, which is invaluable in more advanced mathematical contexts.

2. Multiplying the Factored Expressions

Once we have factored each expression, the next step is to multiply the numerators together and the denominators together. This involves combining the factored forms we obtained in the previous step. It’s a straightforward process of multiplying across, but it’s essential to keep track of each factor to ensure no terms are missed.

We have the two expressions:

  • 30(x + 1) / 2(x + 3)
  • (x + 3) / 6(x + 1)(x - 1)

To multiply these, we multiply the numerators:

30(x + 1) * (x + 3) = 30(x + 1)(x + 3)

And we multiply the denominators:

2(x + 3) * 6(x + 1)(x - 1) = 12(x + 3)(x + 1)(x - 1)

So, the combined expression is:

[30(x + 1)(x + 3)] / [12(x + 3)(x + 1)(x - 1)]

This multiplication step essentially brings all the individual factors together into a single fraction. While the expression might look complex at this stage, it sets the stage for the next crucial step: simplification. Multiplying the factored expressions is like gathering all the ingredients for a recipe; now, we need to combine and cook them properly. Ensuring that each term is accounted for during multiplication is vital, as any missed factor can lead to errors in the final simplified form. This step is more than just a mechanical process; it’s a consolidation of the factored components, preparing them for the simplification process that will reveal the most concise form of the expression. Attention to detail in this phase is what distinguishes a correct solution from a flawed one, emphasizing the importance of a methodical approach.

3. Simplifying by Canceling Common Factors

The heart of multiplying rational expressions lies in simplifying the resulting fraction by canceling out common factors. This is where the factoring work we did earlier truly pays off. By identifying and canceling common factors in the numerator and denominator, we can reduce the expression to its simplest form. This not only makes the expression easier to work with but also provides a clearer understanding of the relationship between the variables.

Looking at our combined expression:

[30(x + 1)(x + 3)] / [12(x + 3)(x + 1)(x - 1)]

We can see several common factors. First, we have (x + 1) in both the numerator and the denominator, so we can cancel them out. Similarly, (x + 3) appears in both, so we can cancel those as well. We are left with:

30 / [12(x - 1)]

Now, let's look at the numerical coefficients. Both 30 and 12 are divisible by 6. Dividing 30 by 6 gives us 5, and dividing 12 by 6 gives us 2. So, we can simplify further:

5 / [2(x - 1)]

Thus, the simplified expression is 5 / 2(x - 1).

Simplification through cancellation is the key to elegance in algebraic manipulations. It transforms a potentially unwieldy expression into a concise and understandable form. The process of canceling common factors is akin to pruning a plant, removing the unnecessary parts to allow the essential elements to flourish. It’s a testament to the power of factoring, as the factored form reveals the common elements that can be eliminated. This step is not just about reducing complexity; it’s about revealing the underlying structure of the expression. Each cancellation is a step towards clarity, making the final form easier to interpret and use in further calculations. The simplified expression often holds the most critical insights, and the ability to arrive at this form efficiently is a hallmark of algebraic proficiency.

4. Final Simplified Expression

After performing the factoring, multiplication, and simplification steps, we arrive at the final simplified expression: 5 / 2(x - 1). This form represents the original expression in its most concise and understandable manner. The process of simplifying rational expressions is akin to refining a raw material into a polished product, removing the extraneous elements to reveal the core essence. The final expression is not just an answer; it’s a testament to the systematic approach we’ve taken, ensuring each step was performed with precision and care.

This simplified form allows for easier evaluation and manipulation in further algebraic operations. It highlights the relationships between the variables without the clutter of unnecessary terms. The journey from the initial complex expression to this simplified form underscores the importance of mastering the fundamental techniques of factoring and simplification. This methodical approach not only yields the correct result but also enhances understanding of the underlying algebraic structures, making it easier to tackle more complex problems in the future.

The expression 5 / 2(x - 1) is not just a conclusion; it’s a foundation. It’s the building block for further mathematical explorations, allowing for substitutions, graphical representations, and deeper analyses. The simplicity of this final form belies the intricate steps that led to its creation, each step a critical component in the process of algebraic refinement. This is the power of simplification – transforming complexity into clarity, and in doing so, unlocking new possibilities.

Summary

Multiplying rational expressions involves factoring, multiplying, and simplifying. By following these steps carefully, we can reduce complex expressions to their simplest forms. The example of (30x + 30)/(2x + 6) * (x + 3)/(6x^2 - 6) illustrates this process effectively, culminating in the simplified expression 5 / 2(x - 1). Mastering these techniques is essential for success in algebra and beyond.

In conclusion, the multiplication and simplification of rational expressions are foundational skills in algebra. The process, although methodical, rewards diligent practice with the ability to transform complex expressions into manageable forms. The journey from the initial problem to the final simplified answer is a testament to the power of systematic problem-solving, a skill that transcends mathematics and applies to various domains of life. This comprehensive guide has not only provided a step-by-step walkthrough of a specific problem but has also illuminated the broader principles that underpin algebraic manipulation. The ability to confidently handle rational expressions is a key milestone in mathematical proficiency, opening doors to more advanced topics and applications. So, embrace the challenge, practice diligently, and revel in the elegance of simplified solutions.