Simplifying Rational Expressions A Step-by-Step Solution

In the realm of algebra, simplifying complex expressions is a fundamental skill. This article delves into the process of simplifying a product of rational expressions, providing a clear, step-by-step approach to arrive at the correct equivalent expression. We'll dissect the given problem, employing factoring techniques and strategic cancellations to unveil the solution. So, let's embark on this algebraic journey and master the art of simplifying rational expressions.

The Challenge: Decoding the Expression

At the heart of our exploration lies the following expression:

$$\frac{2 x^2+7 x-15}{3 x^2+16 x+5} \cdot \frac{3 x^2-2 x-1}{2 x^2-x-3}$$

Our mission is to navigate through this intricate expression and identify its equivalent form from the given options: A. -1, B. 1, C. $\frac{x+1}{x-1}$, and D. $\frac{x-1}{x+1}$. The key to conquering this challenge lies in the strategic application of factoring and simplification techniques.

Step 1: Factoring the Quadratic Expressions

The cornerstone of simplifying rational expressions is factoring. We begin by meticulously factoring each quadratic expression present in the numerator and denominator of the given fractions. This process involves breaking down the quadratic expressions into their constituent linear factors. Let's embark on this factoring journey step by step.

Factoring $2x^2 + 7x - 15$

To factor this quadratic expression, we seek two numbers that multiply to -30 (2 * -15) and add up to 7. These numbers are 10 and -3. Thus, we can rewrite the middle term and factor by grouping:

$2x^2 + 7x - 15 = 2x^2 + 10x - 3x - 15$

$= 2x(x + 5) - 3(x + 5)$

$= (2x - 3)(x + 5)$

Factoring $3x^2 + 16x + 5$

Similarly, for this quadratic expression, we need two numbers that multiply to 15 (3 * 5) and add up to 16. These numbers are 15 and 1. Rewriting the middle term and factoring by grouping yields:

$3x^2 + 16x + 5 = 3x^2 + 15x + x + 5$

$= 3x(x + 5) + 1(x + 5)$

$= (3x + 1)(x + 5)$

Factoring $3x^2 - 2x - 1$

For this expression, we seek two numbers that multiply to -3 (3 * -1) and add up to -2. These numbers are -3 and 1. Factoring by grouping, we get:

$3x^2 - 2x - 1 = 3x^2 - 3x + x - 1$

$= 3x(x - 1) + 1(x - 1)$

$= (3x + 1)(x - 1)$

Factoring $2x^2 - x - 3$

Finally, for this quadratic expression, we need two numbers that multiply to -6 (2 * -3) and add up to -1. These numbers are -3 and 2. Rewriting the middle term and factoring by grouping, we obtain:

$2x^2 - x - 3 = 2x^2 - 3x + 2x - 3$

$= x(2x - 3) + 1(2x - 3)$

$= (x + 1)(2x - 3)$

Step 2: Rewriting the Expression with Factored Forms

Now that we have successfully factored each quadratic expression, we can rewrite the original expression in its factored form. This crucial step sets the stage for simplification through cancellation of common factors.

Substituting the factored forms, our expression transforms into:

$$\frac{(2x - 3)(x + 5)}{(3x + 1)(x + 5)} \cdot \frac{(3x + 1)(x - 1)}{(x + 1)(2x - 3)}$$

Step 3: The Art of Cancellation

With the expression in its factored form, we can now identify and cancel common factors that appear in both the numerator and the denominator. This process of cancellation is akin to simplifying fractions by dividing both the numerator and denominator by their greatest common divisor. Let's strategically cancel the common factors to reveal the simplified expression.

Observing the expression, we can identify the following common factors:

• $(2x - 3)$ appears in both the numerator and denominator.
• $(x + 5)$ appears in both the numerator and denominator.
• $(3x + 1)$ appears in both the numerator and denominator.

Cancelling these common factors, we are left with:

$$\frac{\cancel{(2x - 3)}\cancel{(x + 5)}}{\cancel{(3x + 1)}\cancel{(x + 5)}} \cdot \frac{\cancel{(3x + 1)}(x - 1)}{(x + 1)\cancel{(2x - 3)}} = \frac{x - 1}{x + 1}$$

The Solution: Unveiling the Equivalent Expression

Through the meticulous process of factoring and cancellation, we have successfully simplified the given expression. The equivalent expression we have arrived at is:

$$\frac{x - 1}{x + 1}$$

Therefore, the correct answer is D. $\frac{x-1}{x+1}$.

Mastering Rational Expressions A Recap

In this exploration, we have navigated the intricacies of simplifying rational expressions. We've witnessed the power of factoring quadratic expressions and the elegance of canceling common factors. By mastering these techniques, you can confidently tackle a wide array of algebraic challenges.

Remember, the key to simplifying rational expressions lies in:

1. Factoring: Decompose quadratic expressions into their linear factors.
2. Rewriting: Substitute the factored forms into the original expression.
3. Cancelling: Identify and cancel common factors in the numerator and denominator.

With these steps in your arsenal, you're well-equipped to conquer the world of rational expressions.

Practice Makes Perfect Sharpening Your Skills

To solidify your understanding and enhance your skills in simplifying rational expressions, it's essential to practice diligently. Seek out additional problems and exercises that challenge you to apply the techniques we've discussed. The more you practice, the more confident and proficient you'll become in navigating the realm of algebraic expressions.

Consider exploring resources such as textbooks, online tutorials, and practice worksheets to further expand your knowledge and hone your skills. Remember, consistent effort and dedication are the keys to mastering any mathematical concept.

Conclusion: The Beauty of Simplification

Simplifying expressions is not merely a mechanical process; it's an art form that reveals the underlying structure and elegance of mathematical relationships. By mastering the techniques of factoring and cancellation, we gain a deeper appreciation for the interconnectedness of algebraic concepts.

As you continue your mathematical journey, embrace the challenges, explore the nuances, and revel in the beauty of simplification. With perseverance and a passion for learning, you'll unlock new levels of mathematical understanding and problem-solving prowess.