Simplifying (x^2-x-6)/(2x^3+16) * (2x)/(x^2-3x) A Step-by-Step Guide

In the realm of algebra, simplifying complex expressions is a fundamental skill. This article delves into the process of simplifying a specific rational expression: (x^2-x-6)/(2x3+16) * (2x)/(x^2-3x). We will break down each step, providing clear explanations and justifications to help you understand the underlying principles. Mastering these techniques will empower you to tackle similar algebraic challenges with confidence. Let's embark on this mathematical journey together!

1. Introduction to Rational Expressions

Before we dive into the specifics of our expression, let's establish a solid understanding of rational expressions. Rational expressions are essentially fractions where the numerator and denominator are polynomials. Just like with numerical fractions, we can simplify rational expressions by canceling out common factors. This simplification process makes the expression easier to work with and understand.

Our target expression, (x^2-x-6)/(2x3+16) * (2x)/(x^2-3x), is a product of two rational expressions. To simplify it effectively, we will need to factor the polynomials in the numerators and denominators, identify common factors, and then cancel them out. This meticulous approach will lead us to the most simplified form of the expression. Understanding the nature of rational expressions and the rules governing their manipulation is crucial for success in algebra and beyond. The process of simplification not only makes the expression more compact but also reveals its underlying structure and behavior.

2. Factoring Polynomials: The Key to Simplification

Factoring polynomials is the cornerstone of simplifying rational expressions. It involves breaking down a polynomial into a product of simpler expressions (factors). There are several factoring techniques, and we'll employ the ones relevant to our expression. For the quadratic expressions (expressions with x^2), we'll look for two binomials that multiply to give the original quadratic. For the cubic expression (expression with x^3), we'll explore techniques like factoring out a common factor and the sum of cubes factorization.

Let's start by factoring the numerator of the first fraction, x^2 - x - 6. We need to find two numbers that multiply to -6 and add up to -1. These numbers are -3 and 2. Therefore, we can factor the quadratic as (x - 3)(x + 2). Next, consider the denominator of the first fraction, 2x^3 + 16. We can first factor out a common factor of 2, giving us 2(x^3 + 8). Now, we recognize that x^3 + 8 is a sum of cubes, which can be factored using the formula a^3 + b^3 = (a + b)(a^2 - ab + b^2). In this case, a = x and b = 2, so we have 2(x + 2)(x^2 - 2x + 4). For the second fraction, the numerator 2x is already in its simplest form. The denominator x^2 - 3x can be factored by taking out a common factor of x, resulting in x(x - 3). Factoring polynomials is not just a mechanical process; it's about recognizing patterns and applying the appropriate techniques to reveal the underlying structure of the expressions. The ability to factor efficiently is a powerful tool in algebra and beyond, enabling us to solve equations, simplify expressions, and gain deeper insights into mathematical relationships.

3. Rewriting the Expression with Factored Polynomials

Now that we've factored each polynomial, we can rewrite the original expression with its factored components. This step is crucial as it allows us to clearly see the common factors that can be canceled out. By expressing the polynomials in their factored form, we transform the complex expression into a more manageable form, setting the stage for simplification. The rewritten expression will visually highlight the common factors, making the cancellation process more intuitive and less prone to errors. This meticulous approach ensures that we accurately capture the relationships between the factors and move closer to the most simplified form of the expression.

Replacing the original polynomials with their factored forms, our expression becomes:

[(x - 3)(x + 2)] / [2(x + 2)(x^2 - 2x + 4)] * [2x] / [x(x - 3)]

This rewritten expression is equivalent to the original but presents the terms in a way that facilitates simplification. Notice how the factors are now clearly visible, allowing us to identify terms that appear in both the numerator and the denominator. This is the key to reducing the expression to its simplest form.

4. Identifying and Canceling Common Factors

With the expression rewritten in factored form, the next step is to identify and cancel out any common factors present in both the numerator and the denominator. This process is analogous to simplifying numerical fractions, where we divide both the numerator and denominator by their greatest common divisor. In the context of rational expressions, we are essentially dividing both the numerator and denominator by common polynomial factors.

Looking at our expression:

[(x - 3)(x + 2)] / [2(x + 2)(x^2 - 2x + 4)] * [2x] / [x(x - 3)]

We can observe the following common factors:

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

Now, we can cancel out these common factors. Cancelling (x - 3) from the numerator of the first fraction and the denominator of the second fraction, (x + 2) from the numerator and denominator of the first fraction, 2 from the denominator of the first fraction and the numerator of the second fraction, and x from the numerator of the second fraction and the denominator of the second fraction, we get:

[1] / [(x^2 - 2x + 4)] * [1] / [1]

This step significantly simplifies the expression, reducing it to a much more compact form. The cancellation of common factors is a crucial technique in simplifying rational expressions, allowing us to eliminate redundant terms and reveal the underlying structure of the expression. However, it's important to remember that we can only cancel factors that are multiplied, not terms that are added or subtracted.

5. The Simplified Expression

After canceling out all the common factors, we arrive at the simplified form of the expression. This simplified form is equivalent to the original expression but is presented in its most compact and easily understandable form. The process of simplification has removed the redundancy and revealed the essential mathematical relationship captured by the expression. This ability to reduce complex expressions to their simplest forms is a powerful tool in mathematics, allowing us to solve equations, analyze functions, and gain deeper insights into mathematical concepts.

Multiplying the remaining terms together, we obtain the simplified expression:

1 / (x^2 - 2x + 4)

This is the final simplified form of the original expression (x^2-x-6)/(2x3+16) * (2x)/(x^2-3x). The denominator, (x^2 - 2x + 4), is a quadratic expression that cannot be factored further using real numbers. Therefore, we have reached the most simplified form of the expression.

6. Important Considerations: Excluded Values

While we have successfully simplified the expression, it's crucial to remember an important concept: excluded values. Excluded values are values of the variable (in this case, x) that would make the original expression undefined. This typically occurs when the denominator of a rational expression equals zero. Identifying excluded values is essential to ensure that our simplified expression is equivalent to the original for all valid values of x.

To find the excluded values, we need to look at the denominators of the original expression before simplification. The original expression was:

(x^2-x-6)/(2x3+16) * (2x)/(x^2-3x)

The denominators were 2x^3 + 16 and x^2 - 3x. Let's find the values of x that make these denominators equal to zero.

For 2x^3 + 16 = 0:

1. Factor out 2: 2(x^3 + 8) = 0
2. Factor the sum of cubes: 2(x + 2)(x^2 - 2x + 4) = 0

The solutions are x = -2 (from x + 2 = 0). The quadratic factor (x^2 - 2x + 4) has no real roots (its discriminant is negative).

For x^2 - 3x = 0:

1. Factor out x: x(x - 3) = 0

The solutions are x = 0 and x = 3.

Therefore, the excluded values are x = -2, x = 0, and x = 3. These values must be excluded from the domain of the simplified expression. While the simplified expression 1 / (x^2 - 2x + 4) is defined for these values, the original expression is not. Hence, we must state that the simplified expression is equivalent to the original expression only for x ≠ -2, x ≠ 0, and x ≠ 3. The concept of excluded values highlights the importance of considering the original expression when working with rational expressions. Simplification is a powerful tool, but it's crucial to be mindful of the values that make the original expression undefined.

7. Conclusion: Mastering Rational Expression Simplification

In this article, we've walked through the process of simplifying the rational expression (x^2-x-6)/(2x3+16) * (2x)/(x^2-3x). We've covered essential techniques such as factoring polynomials, canceling common factors, and identifying excluded values. Mastering these skills is fundamental to success in algebra and calculus. By understanding the underlying principles and practicing these techniques, you can confidently tackle complex rational expressions and gain a deeper appreciation for the beauty and power of mathematics.

Simplifying rational expressions is not just a mechanical process; it's an exercise in problem-solving, logical reasoning, and attention to detail. Each step requires careful consideration and a solid understanding of the underlying mathematical concepts. By approaching these problems systematically and methodically, you can break them down into manageable steps and arrive at the correct solution. Moreover, the skills you develop in simplifying rational expressions are transferable to other areas of mathematics and beyond, equipping you with a powerful toolkit for tackling a wide range of mathematical challenges. So, keep practicing, keep exploring, and continue to deepen your understanding of the fascinating world of algebra!