Simplifying (x^2 + 5x + 6) / (x + 2) A Step-by-Step Guide

In this article, we will delve into the process of simplifying the rational expression $\frac{x^2 + 5x + 6}{x + 2}$. Rational expressions, which are essentially fractions with polynomials in the numerator and denominator, often appear complex at first glance. However, by applying algebraic techniques such as factoring and cancellation, we can often reduce them to a simpler, more manageable form. This process is crucial in various mathematical contexts, including solving equations, graphing functions, and performing calculus operations. The ability to simplify rational expressions not only streamlines calculations but also provides deeper insights into the underlying mathematical relationships. Understanding these techniques enhances problem-solving skills and builds a stronger foundation in algebra. Our main focus will be on factoring the quadratic expression in the numerator and then identifying common factors with the denominator. Factoring is a fundamental skill in algebra, and mastering it is essential for simplifying rational expressions and solving quadratic equations. The key to successful factoring lies in recognizing patterns and applying the appropriate methods, such as looking for two numbers that multiply to the constant term and add up to the coefficient of the linear term. Once we have factored the numerator, we can look for common factors between the numerator and the denominator. These common factors can then be canceled out, leading to a simplified expression. This cancellation process is based on the fundamental principle that dividing both the numerator and the denominator of a fraction by the same non-zero quantity does not change the value of the fraction. It's important to note that cancellation can only be performed when factors are multiplied, not when terms are added or subtracted. This article aims to provide a step-by-step guide to simplifying rational expressions, with a particular emphasis on the expression $\frac{x^2 + 5x + 6}{x + 2}$. By the end of this discussion, you will have a clear understanding of the process and be able to apply it to other similar expressions. We will also highlight the importance of considering the domain of the expression and any restrictions on the variable ${ x }$ that may arise due to the original expression. This is a crucial aspect of simplifying rational expressions, as it ensures that the simplified expression is equivalent to the original expression for all valid values of ${ x }$. So, let's embark on this journey of simplifying rational expressions and discover the elegance and power of algebraic manipulation.

Factoring the Numerator

The first step in simplifying the expression $\frac{x^2 + 5x + 6}{x + 2}$ is to factor the numerator, which is the quadratic expression ${ x^2 + 5x + 6 }$. Factoring a quadratic expression involves rewriting it as a product of two binomials. To accomplish this, we need to identify two numbers that satisfy two conditions: they must multiply to the constant term (in this case, 6) and add up to the coefficient of the linear term (in this case, 5). This process is often referred to as the 'sum-product' method. Let's systematically explore the factor pairs of 6. The factors of 6 are 1 and 6, as well as 2 and 3. Among these pairs, the pair 2 and 3 stands out because their sum is 5, which is precisely the coefficient of the ${ x }$ term in the quadratic expression. Therefore, we can express the quadratic expression ${ x^2 + 5x + 6 }$ as a product of two binomials using the numbers 2 and 3. This leads us to the factored form ${ (x + 2)(x + 3) }$. To verify that our factoring is correct, we can expand the product ${ (x + 2)(x + 3) }$ using the distributive property (also known as the FOIL method). Multiplying the first terms gives ${ x \times x = x^2 }$, multiplying the outer terms gives ${ x \times 3 = 3x }$, multiplying the inner terms gives ${ 2 \times x = 2x }$, and multiplying the last terms gives ${ 2 \times 3 = 6 }$. Adding these terms together, we get ${ x^2 + 3x + 2x + 6 }$, which simplifies to ${ x^2 + 5x + 6 }$. This confirms that our factoring is indeed correct. Factoring quadratic expressions is a crucial skill in algebra, and it's essential to master this technique for simplifying rational expressions and solving quadratic equations. The ability to quickly and accurately factor quadratic expressions will significantly enhance your problem-solving skills in mathematics. The factored form of the numerator, ${ (x + 2)(x + 3) }$, now allows us to rewrite the original expression as $\frac{(x + 2)(x + 3)}{x + 2}$. This form reveals a common factor between the numerator and the denominator, which is the key to simplifying the expression further. In the next step, we will explore how to cancel out this common factor and arrive at the simplified form of the expression. This process of factoring and identifying common factors is a fundamental strategy in simplifying rational expressions, and it's a technique that can be applied to a wide range of algebraic problems.

Canceling Common Factors

Now that we have factored the numerator of the expression $\frac{x^2 + 5x + 6}{x + 2}$ as ${ (x + 2)(x + 3) }$, we can rewrite the original expression as $\frac{(x + 2)(x + 3)}{x + 2}$. The next step in simplifying this expression involves identifying and canceling any common factors between the numerator and the denominator. A common factor is a term that appears in both the numerator and the denominator. In this case, we can clearly see that the binomial ${ (x + 2) }$ is present in both the numerator and the denominator. This means that ${ (x + 2) }$ is a common factor, and we can cancel it out. Canceling common factors is a fundamental operation in simplifying fractions, whether they involve numbers or algebraic expressions. The principle behind this operation is that dividing both the numerator and the denominator by the same non-zero quantity does not change the value of the fraction. In other words, $\frac{a \times c}{b \times c} = \frac{a}{b}$ as long as ${ c \neq 0 }$. Applying this principle to our expression, we can cancel the common factor ${ (x + 2) }$ from the numerator and the denominator. This leaves us with the simplified expression ${ x + 3 }$. However, it is crucial to remember that cancellation is only valid if the factor we are canceling is not equal to zero. In this case, we canceled the factor ${ (x + 2) }$, which means that ${ x + 2 }$ cannot be equal to zero. This implies that ${ x \neq -2 }$. This condition is essential because the original expression, $\frac{x^2 + 5x + 6}{x + 2}$, is undefined when ${ x = -2 }$ due to division by zero. Therefore, the simplified expression ${ x + 3 }$ is equivalent to the original expression only for values of ${ x }$ that are not equal to -2. This restriction on the domain of the expression is an important consideration when simplifying rational expressions. Failing to acknowledge this restriction can lead to incorrect conclusions or misunderstandings, especially when dealing with functions and their graphs. The simplified expression ${ x + 3 }$ represents a linear function, while the original expression represents a rational function with a removable discontinuity at ${ x = -2 }$. The cancellation of the common factor has essentially 'removed' this discontinuity, but it's crucial to remember that it still exists in the original expression. In summary, canceling common factors is a powerful technique for simplifying rational expressions, but it's essential to be mindful of any restrictions on the variable that may arise as a result of the cancellation. Always consider the domain of the original expression and ensure that the simplified expression is equivalent for all valid values of the variable. This attention to detail will help you avoid errors and develop a deeper understanding of algebraic manipulations.

The Simplified Expression and Restrictions

After factoring the numerator and canceling the common factor, we have arrived at the simplified expression ${ x + 3 }$. This linear expression is much simpler to work with than the original rational expression $\frac{x^2 + 5x + 6}{x + 2}$. However, it is crucially important to remember the restriction on the variable ${ x }$ that we identified during the cancellation process. As we discussed earlier, we canceled the common factor ${ (x + 2) }$, which implies that ${ x + 2 }$ cannot be equal to zero. This leads to the restriction ${ x \neq -2 }$. This restriction means that the simplified expression ${ x + 3 }$ is equivalent to the original expression $\frac{x^2 + 5x + 6}{x + 2}$ for all values of ${ x }$ except ${ x = -2 }$. At ${ x = -2 }$, the original expression is undefined because it would involve division by zero. This is a critical concept in simplifying rational expressions. The simplified expression is not identical to the original expression; it is equivalent to the original expression within a specific domain. In other words, the simplified expression is a valid representation of the original expression as long as we acknowledge the restriction on the variable. To fully represent the simplified expression, we should write it as: ${ x + 3, }$ for ${ x \neq -2 }$. This notation explicitly indicates that the expression ${ x + 3 }$ is valid for all values of ${ x }$ except ${ x = -2 }$. The graph of the original expression, $\frac{x^2 + 5x + 6}{x + 2}$, would be a straight line with a hole (or removable discontinuity) at the point where ${ x = -2 }$. The graph of the simplified expression, ${ x + 3 }$, is a straight line without any holes. The hole in the graph of the original expression is a visual representation of the restriction on the variable. Understanding these restrictions is essential in various mathematical contexts, such as graphing functions, solving equations, and performing calculus operations. For example, if we were to solve an equation involving the original expression, we would need to be careful to exclude ${ x = -2 }$ as a possible solution. Similarly, when evaluating limits involving the original expression, we would need to consider the behavior of the function near ${ x = -2 }$. In summary, while simplifying rational expressions can make them easier to work with, it is crucial to remember any restrictions on the variable that may arise during the simplification process. These restrictions are an integral part of the simplified expression and must be taken into account in any subsequent calculations or interpretations. The simplified expression ${ x + 3, }$ for ${ x \neq -2 }$, provides a concise and accurate representation of the original expression, capturing both its algebraic form and its domain.

Conclusion

In conclusion, we have successfully simplified the rational expression $\frac{x^2 + 5x + 6}{x + 2}$ to ${ x + 3 }$, with the crucial restriction that ${ x \neq -2 }$. This process involved factoring the numerator, identifying and canceling the common factor ${ (x + 2) }$, and explicitly stating the restriction on the variable. This example highlights the key steps in simplifying rational expressions: factoring, canceling common factors, and identifying restrictions. Factoring allows us to rewrite the numerator and denominator as products of simpler expressions, making it easier to identify common factors. Canceling common factors simplifies the expression while preserving its value (within a specific domain). Identifying restrictions ensures that the simplified expression is equivalent to the original expression for all valid values of the variable. The ability to simplify rational expressions is a fundamental skill in algebra and is essential for success in higher-level mathematics courses. It is used in a wide range of applications, including solving equations, graphing functions, and performing calculus operations. Mastering these techniques not only streamlines calculations but also provides deeper insights into the underlying mathematical relationships. Throughout this discussion, we have emphasized the importance of understanding the domain of the expression and any restrictions on the variable that may arise during the simplification process. These restrictions are an integral part of the simplified expression and must be taken into account in any subsequent calculations or interpretations. Failing to acknowledge these restrictions can lead to incorrect conclusions or misunderstandings. The simplified expression ${ x + 3, }$ for ${ x \neq -2 }$, provides a concise and accurate representation of the original expression, capturing both its algebraic form and its domain. This example serves as a valuable illustration of the power of algebraic manipulation and the importance of paying attention to detail. By following the steps outlined in this article, you can confidently simplify a wide range of rational expressions and develop a deeper understanding of algebraic concepts. Remember to always factor, cancel common factors, and identify restrictions. With practice and attention to detail, you will master the art of simplifying rational expressions and unlock new possibilities in your mathematical journey. The skills and concepts discussed here are not only applicable to this specific example but also serve as a foundation for tackling more complex algebraic problems. So, embrace the challenge, practice diligently, and enjoy the satisfaction of simplifying complex expressions into elegant and manageable forms.