Simplify Rational Expressions A Step By Step Guide

Simplifying rational expressions is a fundamental skill in algebra, often encountered in various mathematical contexts. This article delves into the process of simplifying the expression ${ \frac{x^2-3 x-4}{x^2+4 x+3} \cdot \frac{x^2+8 x+15}{x-4} }$, providing a step-by-step guide suitable for students and enthusiasts alike. We will explore the necessary techniques, such as factoring quadratic expressions, canceling common factors, and understanding restrictions on the variable ${ x }$. By the end of this discussion, you will have a clear understanding of how to tackle similar problems and a deeper appreciation for algebraic manipulations.

Understanding Rational Expressions

Before diving into the specifics of our problem, it's essential to grasp the concept of rational expressions. A rational expression is essentially a fraction where the numerator and the denominator are polynomials. Polynomials, in turn, are expressions consisting of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. Examples of polynomials include ${ x^2 - 3x - 4 }$ and ${ x^2 + 4x + 3 }$. Rational expressions, like regular fractions, can be simplified, added, subtracted, multiplied, and divided. The key to simplifying these expressions lies in factoring and canceling common factors.

When working with rational expressions, it's crucial to be mindful of values that make the denominator zero. Division by zero is undefined in mathematics, so we must exclude any values of the variable that would lead to a zero denominator. These values are known as restrictions or excluded values, and they play a vital role in the final solution. Ignoring these restrictions can lead to incorrect or incomplete answers. Therefore, identifying and stating these restrictions is a necessary step in simplifying rational expressions.

Factoring Quadratic Expressions: A Crucial First Step

The cornerstone of simplifying rational expressions is factoring. Factoring involves breaking down a polynomial into its constituent factors, which are expressions that, when multiplied together, yield the original polynomial. In our case, we encounter quadratic expressions, which are polynomials of degree two (i.e., the highest power of the variable is two). Factoring quadratic expressions typically involves finding two binomials (expressions with two terms) that, when multiplied, produce the original quadratic. For example, factoring ${ x^2 - 3x - 4 }$ involves finding two numbers that multiply to -4 and add to -3. These numbers are -4 and 1, so we can factor the expression as ${ (x - 4)(x + 1) }$. Similarly, other quadratic expressions in our problem need to be factored to identify common factors that can be canceled.

Factoring is not just a mechanical process; it's a way of rewriting an expression in a more revealing form. By factoring, we expose the underlying structure of the expression, making it easier to identify common factors and simplify. There are several techniques for factoring quadratic expressions, including trial and error, using the quadratic formula, and recognizing special patterns like the difference of squares or perfect square trinomials. Mastering these techniques is essential for simplifying rational expressions effectively. In the following sections, we will demonstrate how to apply these techniques to the specific quadratic expressions in our problem.

Step-by-Step Simplification

Let’s break down the simplification of the given expression ${ \frac{x^2-3 x-4}{x^2+4 x+3} \cdot \frac{x^2+8 x+15}{x-4} }$ into manageable steps. This will not only provide the solution but also illustrate the general methodology for simplifying rational expressions.

1. Factoring the Quadratic Expressions

The first step in simplifying the expression is to factor each quadratic expression present. Factoring breaks down the polynomials into simpler terms, making it easier to identify and cancel common factors. This is a critical step because it transforms the expression into a form where simplification becomes much more straightforward. Let's factor each quadratic expression individually:

• Factoring ${ x^2 - 3x - 4 }$:

  We need to find two numbers that multiply to -4 and add to -3. These numbers are -4 and 1. Therefore, ${ x^2 - 3x - 4 }$ factors as ${ (x - 4)(x + 1) }$.

• Factoring ${ x^2 + 4x + 3 }$:

  We need to find two numbers that multiply to 3 and add to 4. These numbers are 3 and 1. Therefore, ${ x^2 + 4x + 3 }$ factors as ${ (x + 3)(x + 1) }$.

• Factoring ${ x^2 + 8x + 15 }$:

  We need to find two numbers that multiply to 15 and add to 8. These numbers are 5 and 3. Therefore, ${ x^2 + 8x + 15 }$ factors as ${ (x + 5)(x + 3) }$.

Now that we have factored each quadratic expression, we can rewrite the original expression with these factored forms. This sets the stage for the next crucial step: canceling common factors. The factored form allows us to clearly see which terms appear in both the numerator and the denominator, which can then be simplified. This process of factoring and rewriting is fundamental to simplifying any rational expression.

2. Rewriting the Expression

After factoring each quadratic expression, we can rewrite the original expression in its factored form. This step is crucial as it sets the stage for simplification by making common factors visible. Substituting the factored expressions, we get:

${ \frac{x^2-3 x-4}{x^2+4 x+3} \cdot \frac{x^2+8 x+15}{x-4} = \frac{(x - 4)(x + 1)}{(x + 3)(x + 1)} \cdot \frac{(x + 5)(x + 3)}{x - 4} }$

Rewriting the expression in this factored form is more than just a notational change; it's a strategic move that reveals the underlying structure of the expression. By presenting the expression as a product of factors, we can easily identify and cancel terms that appear in both the numerator and the denominator. This process of cancellation is the heart of simplifying rational expressions. The rewritten expression clearly shows the factors that can be simplified, leading us closer to the final simplified form. This step highlights the power of factoring in making complex expressions more manageable and easier to understand.

3. Canceling Common Factors

Now that the expression is in its factored form, we can proceed with canceling common factors. This involves identifying factors that appear in both the numerator and the denominator and dividing them out. Remember, canceling common factors is essentially dividing both the numerator and the denominator by the same expression, which maintains the value of the overall expression. From our rewritten expression:

${ \frac{(x - 4)(x + 1)}{(x + 3)(x + 1)} \cdot \frac{(x + 5)(x + 3)}{x - 4} }$

We can observe the following common factors:

• ${ (x - 4) }$ appears in both the numerator and the denominator.

• ${ (x + 1) }$ appears in both the numerator and the denominator.

• ${ (x + 3) }$ appears in both the numerator and the denominator.

Canceling these common factors, we get:

${ \frac{\cancel{(x - 4)}\cancel{(x + 1)}}{\cancel{(x + 3)}\cancel{(x + 1)}} \cdot \frac{(x + 5)\cancel{(x + 3)}}{\cancel{x - 4}} = x + 5 }$

The process of canceling common factors is a powerful simplification technique. It reduces the complexity of the expression by eliminating redundant terms, leaving us with a more concise form. However, it's crucial to remember that canceling factors is only valid for factors that are multiplied, not added or subtracted. This step demonstrates the elegance of factoring and cancellation in simplifying algebraic expressions. The result, ${ x + 5 }$, is significantly simpler than the original expression, making it easier to work with in further calculations or analyses.

4. Identifying Restrictions on ${ x }$

Before declaring the simplified expression as the final answer, it’s crucial to identify any restrictions on the variable ${ x }$. Restrictions are values of ${ x }$ that would make the original expression undefined. In the context of rational expressions, this typically occurs when the denominator of any fraction is equal to zero. To find these restrictions, we need to examine the original expression and identify any values of ${ x }$ that would cause a denominator to be zero.

Looking at the original expression:

${ \frac{x^2-3 x-4}{x^2+4 x+3} \cdot \frac{x^2+8 x+15}{x-4} }$

We need to consider the denominators ${ x^2 + 4x + 3 }$ and ${ x - 4 }$. Let's analyze each one:

• Denominator ${ x^2 + 4x + 3 }$:

  We already factored this as ${ (x + 3)(x + 1) }$. Setting this equal to zero gives us:

  ${ (x + 3)(x + 1) = 0 }$

  This yields two restrictions: ${ x = -3 }$ and ${ x = -1 }$.

• Denominator ${ x - 4 }$:

  Setting this equal to zero gives us:

  ${ x - 4 = 0 }$

  This yields one restriction: ${ x = 4 }$.

Therefore, the restrictions on ${ x }$ are ${ x \neq -3 }$, ${ x \neq -1 }$, and ${ x \neq 4 }$. These restrictions are essential because they define the domain over which the simplified expression is equivalent to the original expression. Ignoring these restrictions can lead to incorrect conclusions or paradoxes. Identifying and stating these restrictions is a critical part of simplifying rational expressions, ensuring the mathematical validity of the result.

Final Answer

After simplifying the expression and identifying the restrictions, we can now present the final answer. The simplified form of the expression ${ \frac{x^2-3 x-4}{x^2+4 x+3} \cdot \frac{x^2+8 x+15}{x-4} }$ is:

${ x + 5 }$

However, we must also state the restrictions on ${ x }$ to ensure the equivalence between the original and simplified expressions. The restrictions are:

${ x \neq -3, \quad x \neq -1, \quad x \neq 4 }$

Therefore, the complete answer is ${ x + 5 }$, with the conditions that ${ x }$ cannot be -3, -1, or 4. This final answer encapsulates the entire simplification process, from factoring and canceling common factors to identifying and stating restrictions. It demonstrates the importance of not only finding a simplified form but also understanding the context in which that simplification is valid. The restrictions are a crucial part of the answer, providing a complete and accurate representation of the relationship between the original and simplified expressions.

In conclusion, simplifying rational expressions involves several key steps: factoring, canceling common factors, and identifying restrictions. Mastering these techniques is essential for success in algebra and related fields. By following a systematic approach, you can confidently tackle even the most complex rational expressions.