Dividing And Simplifying Rational Expressions A Step By Step Guide

Hey guys! Today, we're diving into the world of rational expressions and tackling a common challenge: division. Specifically, we'll be working through an example where we divide and simplify rational expressions to arrive at a reduced form. So, buckle up and let's get started!

Understanding Rational Expressions

Before we jump into the problem, let's quickly recap what rational expressions are. Rational expressions are essentially fractions where the numerator and denominator are polynomials. Think of them as algebraic fractions. Just like with numerical fractions, we can perform operations like addition, subtraction, multiplication, and, of course, division on rational expressions. Simplifying these expressions often involves factoring polynomials and canceling out common factors, a crucial skill for this topic.

When dividing rational expressions, the primary goal is to simplify the expressions as much as possible. This often involves several steps, including factoring polynomials, inverting the divisor, multiplying, and canceling common factors. The most important thing is to ensure each step is performed accurately. The final answer should always be presented in its simplest form, meaning there are no more common factors between the numerator and the denominator. This process might seem intimidating at first, but with practice, it becomes a straightforward and almost mechanical procedure. The ability to manipulate and simplify rational expressions is fundamental in algebra and calculus, making it a skill worth mastering. Always remember to double-check your work, especially the factoring steps, as mistakes there can propagate through the rest of the solution. Keep practicing, and you'll become a pro in no time!

Problem Statement

Our mission today is to divide and simplify the following rational expression:

$$\frac{x^2+x-56}{x^2-5 x-14} \div \frac{x+3}{x+2}$$

We want to express our answer as a reduced rational expression. This means we need to simplify it as much as possible.

Step-by-Step Solution

1. Rewrite Division as Multiplication

The golden rule for dividing fractions (and rational expressions) is to invert and multiply. This means we flip the second fraction (the divisor) and change the division operation to multiplication. So, our expression becomes:

$$\frac{x^2+x-56}{x^2-5 x-14} \times \frac{x+2}{x+3}$$

This transformation is crucial because it sets the stage for simplifying the expression through factorization and cancellation. Rewriting division as multiplication by inverting the second fraction allows us to apply the rules of fraction multiplication, which are generally more straightforward to handle. This step is not just a procedural trick; it is grounded in the fundamental properties of division. Understanding why this works can provide a deeper insight into mathematical operations. Additionally, this technique is widely applicable in various algebraic contexts, making it an essential tool in your mathematical toolkit. Make sure to always double-check that you've correctly inverted the fraction, as this is a common area for mistakes. With this transformation, the problem is now set up for the next phase, which involves factoring the polynomials.

2. Factor the Polynomials

Now, let's dive into factoring. We need to factor the quadratic expressions in the numerators and denominators. Factoring is the process of breaking down a polynomial into its constituent factors. This step is crucial for simplifying rational expressions because it allows us to identify and cancel out common factors. If you're rusty on your factoring skills, now is a good time to brush up! Factoring not only simplifies the current problem but also reinforces a fundamental algebraic skill. The ability to factor polynomials quickly and accurately is a cornerstone of algebraic manipulation. This skill extends beyond simplifying rational expressions and is vital in solving equations, analyzing functions, and tackling more advanced mathematical concepts. So, let’s get those factoring muscles warmed up and ready to go! Remember, practice makes perfect, and the more you factor, the easier it will become.

• Factoring x² + x - 56

  We are looking for two numbers that multiply to -56 and add to 1. These numbers are 8 and -7. So, we can factor this as:

  $$x^2 + x - 56 = (x + 8)(x - 7)$$

• Factoring x² - 5x - 14

  We need two numbers that multiply to -14 and add up to -5. These numbers are -7 and 2. So, we have:

  $$x^2 - 5x - 14 = (x - 7)(x + 2)$$

With our polynomials factored, our expression now looks like this:

$$\frac{(x + 8)(x - 7)}{(x - 7)(x + 2)} \times \frac{x + 2}{x + 3}$$

3. Multiply the Rational Expressions

Now that we have factored the polynomials, the next step is to multiply the rational expressions. Multiplying rational expressions is similar to multiplying numerical fractions: we multiply the numerators together and the denominators together. Multiplying rational expressions involves combining the factored forms into a single fraction, setting the stage for the crucial step of cancellation. This step is not only a mechanical process but also an opportunity to ensure that all terms are correctly accounted for. A careful and systematic approach at this stage prevents errors that could complicate the subsequent simplification. By multiplying the numerators and denominators, we consolidate the expression into a form where common factors become more apparent and easier to identify. This consolidation is key to achieving the final simplified form of the rational expression. It's like gathering all the ingredients before you start cooking – you need everything in one place to proceed efficiently. So, let's multiply those expressions and get closer to the simplified form!

Multiplying the numerators gives us:

$$(x + 8)(x - 7)(x + 2)$$

Multiplying the denominators gives us:

$$(x - 7)(x + 2)(x + 3)$$

So, our expression becomes:

$$\frac{(x + 8)(x - 7)(x + 2)}{(x - 7)(x + 2)(x + 3)}$$

4. Simplify by Canceling Common Factors

This is where the magic happens! We can now cancel out any common factors that appear in both the numerator and the denominator. Simplifying rational expressions by canceling common factors is like weeding a garden – you remove the unnecessary elements to allow the important ones to flourish. This step is the heart of the simplification process, where we reduce the expression to its most basic form. Identifying and canceling common factors requires a keen eye and a solid understanding of factoring. Each canceled factor brings us closer to the final answer, a testament to the elegance of mathematical simplification. However, it’s crucial to remember that we can only cancel factors, not terms. This distinction is a common source of errors, so it's always wise to double-check your work. The result of this cancellation is a more streamlined and manageable expression, ready for the final touch. So, let’s carefully examine our expression and clear away those common factors!

We can see that (x - 7) and (x + 2) are common factors in both the numerator and the denominator. Canceling these out, we get:

$$\frac{(x + 8)(\cancel{x - 7})(\cancel{x + 2})}{(\cancel{x - 7})(\cancel{x + 2})(x + 3)} = \frac{x + 8}{x + 3}$$

5. State the Simplified Expression

After all that work, we've arrived at our simplified rational expression! There are no more common factors to cancel, so we can confidently state our answer. The simplified expression is the final product of our efforts, representing the original complex expression in its most concise and understandable form. This step is not just about writing down the answer; it’s about recognizing the journey we've undertaken and the mathematical principles we've applied. Presenting the simplified expression clearly and accurately is the culmination of the entire process, showcasing our ability to manipulate and simplify algebraic expressions. Moreover, the simplified expression is often more useful for further calculations or analysis, highlighting the practical importance of this step. So, with a sense of accomplishment, let’s state our final answer and appreciate the elegance of simplification!

Our simplified rational expression is:

$$\frac{x + 8}{x + 3}$$

Final Answer

Therefore, the simplified form of the given rational expression is:

$$\frac{x + 8}{x + 3}$$

And there you have it! We've successfully divided and simplified the rational expression. Remember, the key is to break down the problem into manageable steps: rewrite division as multiplication, factor the polynomials, multiply the expressions, cancel common factors, and state the simplified expression. Keep practicing, and you'll become a pro at simplifying rational expressions in no time!

Key Takeaways

Let's recap the key steps we've learned today. Mastering these steps is crucial for handling similar problems and building a solid foundation in algebra. Remember, each step is a building block, and a clear understanding of each one contributes to overall proficiency. These takeaways are not just a summary; they are a roadmap for approaching rational expression problems with confidence and accuracy. By internalizing these steps, you'll be well-equipped to tackle even more complex expressions. So, let's reinforce our knowledge and ensure we've grasped the core concepts!

• Rewrite Division as Multiplication: Always start by inverting the second fraction and changing the division to multiplication. This is the foundational step that sets up the rest of the solution.
• Factor Polynomials: Factoring is the cornerstone of simplifying rational expressions. Practice your factoring skills to make this step efficient and accurate.
• Multiply Rational Expressions: Multiply the numerators and denominators separately to combine the fractions into a single expression.
• Cancel Common Factors: Identify and cancel out factors that appear in both the numerator and the denominator to simplify the expression.
• State the Simplified Expression: Present your final answer in its simplest form, ensuring no further simplification is possible.

Practice Problems

To solidify your understanding, here are a few practice problems you can try:

1. $$\frac{x^2 - 4}{x^2 + 4x + 4} \div \frac{x - 2}{x + 2}$$

2. $$\frac{2x^2 + 5x - 3}{x^2 - 9} \div \frac{2x - 1}{x + 3}$$

3. $$\frac{x^2 - 9}{x^2 + 5x + 6} \div \frac{x - 3}{x + 2}$$

Work through these problems, applying the steps we've discussed. The more you practice, the more confident you'll become in simplifying rational expressions. Remember, the goal is not just to find the answer but to understand the process. So, grab a pencil and paper, and let's get practicing!

Conclusion

Simplifying rational expressions might seem daunting at first, but with a systematic approach and plenty of practice, it becomes a manageable task. Remember the key steps, practice regularly, and don't be afraid to ask for help when you need it. You've got this! Keep up the great work, and I'll see you in the next math adventure!