Simplifying (x^2 - 15x + 36) / (5x - 60) A Step-by-Step Guide

In the realm of algebra, simplifying expressions is a fundamental skill. This process allows us to manipulate complex expressions into more manageable and understandable forms. Rational expressions, which are fractions where the numerator and denominator are polynomials, are no exception. In this comprehensive guide, we will delve into the step-by-step process of simplifying the rational expression (x² - 15x + 36) / (5x - 60). This exploration will not only provide a solution but also elucidate the underlying principles and techniques applicable to a broader range of rational expressions. Understanding simplification is crucial for various mathematical operations, including solving equations, graphing functions, and performing calculus.

Factoring the Numerator: Unveiling the Components

The first crucial step in simplifying any rational expression involves factoring both the numerator and the denominator. Factoring is the process of breaking down a polynomial into its constituent factors, which are expressions that, when multiplied together, yield the original polynomial. For the numerator, which is the quadratic expression x² - 15x + 36, we seek two binomials that, when multiplied, produce this expression. To accomplish this, we need to identify two numbers that simultaneously satisfy two conditions: they must add up to the coefficient of the x term (-15) and multiply to the constant term (36). By carefully considering the factors of 36, we can deduce that -12 and -3 fulfill these criteria, since (-12) + (-3) = -15 and (-12) * (-3) = 36. Consequently, we can factor the numerator as (x - 12)(x - 3). Mastering factoring techniques is essential for simplifying rational expressions and solving algebraic equations.

This factorization process transforms the numerator from a seemingly monolithic quadratic expression into a product of two simpler binomial factors. This decomposition is a pivotal step because it allows us to identify potential common factors with the denominator, which can then be canceled out, leading to simplification. Factoring not only simplifies expressions but also provides valuable insights into the behavior of the underlying function. For example, the roots of the numerator polynomial, which are the values of x that make the expression equal to zero, are directly revealed by the factors. In this case, the roots are x = 12 and x = 3, which are the values that make the factors (x - 12) and (x - 3) equal to zero, respectively. Understanding the relationship between factors and roots is crucial for graphing polynomials and solving equations.

The ability to factor quadratic expressions like x² - 15x + 36 efficiently and accurately is a cornerstone of algebraic manipulation. There are various techniques for factoring, including trial and error, using the quadratic formula, and completing the square. The choice of method often depends on the specific characteristics of the quadratic expression and the individual's preferences. However, regardless of the method used, the underlying principle remains the same: to decompose the quadratic expression into a product of two linear factors. Efficient factoring skills are crucial for success in algebra and beyond.

Factoring the Denominator: Extracting the Common Factor

Next, we turn our attention to the denominator, which is the linear expression 5x - 60. In this case, the factoring process involves identifying the greatest common factor (GCF) of the terms. The GCF is the largest factor that divides evenly into all the terms of the expression. In the expression 5x - 60, both terms are divisible by 5. Therefore, we can factor out 5 from the denominator, resulting in 5(x - 12). Factoring out the GCF is a fundamental technique for simplifying expressions and is particularly useful when dealing with linear expressions or polynomials with a common factor among their terms. Identifying and extracting common factors is a key skill in algebraic simplification.

Factoring the denominator not only simplifies the expression but also reveals potential common factors with the numerator. This is crucial for the subsequent step of canceling common factors, which is the essence of simplifying rational expressions. By factoring out the 5, we have transformed the denominator into a product of a constant (5) and a binomial (x - 12). This decomposition makes it readily apparent that there is a common factor of (x - 12) between the numerator and the denominator. Recognizing common factors is essential for simplifying rational expressions and performing other algebraic operations.

The process of factoring out the GCF is not limited to linear expressions. It can also be applied to polynomials of higher degrees, provided that there is a common factor among all the terms. For example, in the expression 3x³ + 6x² - 9x, the GCF is 3x, which can be factored out to yield 3x(x² + 2x - 3). This factorization simplifies the expression and makes it easier to work with. Generalizing factoring techniques to various types of expressions is crucial for developing algebraic fluency.

Canceling Common Factors: The Art of Simplification

Now that we have factored both the numerator and the denominator, we can rewrite the original rational expression as (x - 12)(x - 3) / 5(x - 12). This form clearly reveals that both the numerator and the denominator share a common factor of (x - 12). The next crucial step is to cancel out this common factor. Canceling common factors is a valid operation because it is equivalent to dividing both the numerator and the denominator by the same expression, which does not change the value of the fraction, as long as the common factor is not equal to zero. In this case, we can cancel out the (x - 12) factor, leaving us with (x - 3) / 5. The principle of canceling common factors is a cornerstone of simplifying rational expressions.

However, it is essential to note a crucial caveat: we can only cancel factors, not terms. Factors are expressions that are multiplied together, while terms are expressions that are added or subtracted. For example, in the expression (x + 2)(x - 3) / (x + 2), we can cancel the (x + 2) factors because they are multiplied. However, in the expression (x² + 2x) / (x + 2), we cannot simply cancel the 2x and the 2 because they are terms, not factors. We must first factor the numerator as x(x + 2) and then cancel the common factor of (x + 2). Distinguishing between factors and terms is crucial for avoiding common errors in simplification.

Canceling common factors is not merely a mechanical process; it is a powerful tool for revealing the underlying structure of the rational expression. By removing the common factor, we are essentially eliminating a redundancy in the expression, making it more concise and easier to analyze. This simplification can be particularly beneficial when dealing with more complex expressions or when performing further mathematical operations. Simplification as a tool for analysis is a key concept in mathematics.

The Simplified Expression and Restrictions: The Final Touches

After canceling the common factor of (x - 12), we arrive at the simplified expression (x - 3) / 5. This is the simplified form of the original rational expression (x² - 15x + 36) / (5x - 60). The simplification process has transformed a seemingly complex expression into a more manageable form, making it easier to understand and work with. Achieving a simplified form is the ultimate goal of this process.

However, it is crucial to remember that simplifying rational expressions may introduce restrictions on the variable. These restrictions arise from the original denominator, which cannot be equal to zero. In the original expression, the denominator was 5x - 60. To find the restrictions, we set this expression equal to zero and solve for x: 5x - 60 = 0. Solving this equation gives us x = 12. This means that x cannot be equal to 12, because this would make the original denominator zero, which is undefined. Therefore, we must state the restriction x ≠ 12 alongside the simplified expression. Understanding and stating restrictions is an essential part of simplifying rational expressions.

The restriction x ≠ 12 indicates that the simplified expression (x - 3) / 5 is equivalent to the original expression (x² - 15x + 36) / (5x - 60) for all values of x except for x = 12. At x = 12, the original expression is undefined, while the simplified expression is equal to (12 - 3) / 5 = 9/5. This difference highlights the importance of stating the restrictions when simplifying rational expressions. Restrictions as a key component of the solution are often overlooked but are crucial for a complete understanding.

In summary, the simplified form of the rational expression (x² - 15x + 36) / (5x - 60) is (x - 3) / 5, with the restriction x ≠ 12. This solution encapsulates the entire process of simplifying rational expressions, from factoring the numerator and denominator to canceling common factors and stating the restrictions. A complete solution includes both the simplified expression and any relevant restrictions.

Conclusion: Mastering Rational Expression Simplification

Simplifying rational expressions is a fundamental skill in algebra with wide-ranging applications. This process involves factoring, canceling common factors, and stating any restrictions on the variable. By mastering these techniques, you can effectively manipulate and simplify complex expressions, making them easier to understand and work with. The example of simplifying (x² - 15x + 36) / (5x - 60) provides a comprehensive illustration of the steps involved and the underlying principles. Mastering simplification techniques is crucial for success in algebra and beyond.

Remember, the key to simplifying rational expressions lies in factoring both the numerator and the denominator, identifying common factors, and canceling them out. Always state any restrictions on the variable that arise from the original denominator. With practice and a solid understanding of these principles, you can confidently tackle a wide variety of rational expression simplification problems. Practice and understanding are the keys to proficiency in this skill.

This guide has provided a detailed explanation of the simplification process, emphasizing the importance of each step and the underlying concepts. By following these guidelines, you can confidently simplify rational expressions and enhance your algebraic skills. Confidence in algebraic manipulation is a valuable asset in mathematics and its applications.