Simplifying (a^2 - 5a) / (a^2 - 4a - 5) A Step-by-Step Guide

In the realm of algebra, simplifying rational expressions is a fundamental skill. Rational expressions, which are essentially fractions with polynomials in the numerator and denominator, often appear complex at first glance. However, with the right techniques, they can be tamed and expressed in their simplest forms. This article delves into the process of simplifying the rational expression (a² - 5a) / (a² - 4a - 5), providing a step-by-step guide and shedding light on the underlying concepts.

Understanding Rational Expressions

Before diving into the specifics, let's establish a solid understanding of what rational expressions are. A rational expression is a fraction where both 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 x² + 3x - 2, 5y - 1, and even just the constant 7. The rational expression (a² - 5a) / (a² - 4a - 5) perfectly fits this definition, with both the numerator (a² - 5a) and the denominator (a² - 4a - 5) being polynomials.

The key to simplifying rational expressions lies in identifying common factors between the numerator and the denominator. Just like we can simplify numerical fractions by canceling out common factors (e.g., 6/8 = (2 * 3) / (2 * 4) = 3/4), we can do the same with rational expressions. To do this effectively, we often need to factor the polynomials in the numerator and denominator. Factoring is the process of breaking down a polynomial into a product of simpler polynomials. For instance, factoring x² - 4 gives us (x + 2)(x - 2). Once we have factored the numerator and denominator, we can look for common factors that can be canceled out, leading to the simplified form of the rational expression.

The process of simplifying rational expressions is crucial in various areas of mathematics, including calculus, algebra, and even real-world applications. Simplified expressions are easier to work with, making it simpler to solve equations, analyze functions, and model real-world phenomena. Therefore, mastering this skill is an essential step in your mathematical journey. In the following sections, we will systematically break down the simplification of (a² - 5a) / (a² - 4a - 5), demonstrating the power and elegance of factoring and cancellation.

Step-by-Step Simplification of (a² - 5a) / (a² - 4a - 5)

Now, let's embark on the journey of simplifying the given rational expression: (a² - 5a) / (a² - 4a - 5). We will break down the process into clear, manageable steps, ensuring that each stage is thoroughly understood.

Step 1: Factoring the Numerator

The first step in simplifying rational expressions is to factor both the numerator and the denominator. Let's begin with the numerator, which is a² - 5a. Notice that both terms in this polynomial have a common factor of 'a'. We can factor out 'a' from both terms, resulting in:

a² - 5a = a(a - 5)

This factored form is much easier to work with than the original expression. By factoring out the common factor, we have effectively broken down the numerator into a product of two simpler expressions: 'a' and '(a - 5)'. This is a crucial step because it allows us to identify potential common factors with the denominator later on.

Factoring is a fundamental skill in algebra, and it's essential to practice and master various factoring techniques. In this case, we used the simplest form of factoring, which is factoring out a common monomial. However, there are other factoring techniques, such as factoring quadratic expressions, factoring by grouping, and using special factoring patterns (e.g., the difference of squares). The more comfortable you are with these techniques, the easier it will be to simplify rational expressions.

Step 2: Factoring the Denominator

Next, we turn our attention to the denominator, which is a² - 4a - 5. This is a quadratic expression, and factoring quadratics often requires a bit more effort. We are looking for two numbers that multiply to -5 and add up to -4. These numbers are -5 and 1. Therefore, we can factor the denominator as:

a² - 4a - 5 = (a - 5)(a + 1)

Again, factoring the denominator is a crucial step in simplifying rational expressions. By breaking down the quadratic expression into a product of two linear expressions, we have unveiled the potential for common factors with the numerator. The factored form (a - 5)(a + 1) clearly shows the linear expressions that make up the denominator.

Factoring quadratic expressions is a common task in algebra, and there are several techniques you can use. One common method is trial and error, where you try different combinations of factors until you find the ones that work. Another method is to use the quadratic formula to find the roots of the quadratic, and then use those roots to construct the factors. The key is to practice and develop your factoring skills so that you can quickly and accurately factor quadratic expressions.

Step 3: Identifying and Canceling Common Factors

Now that we have factored both the numerator and the denominator, we can rewrite the original rational expression as:

(a² - 5a) / (a² - 4a - 5) = [a(a - 5)] / [(a - 5)(a + 1)]

At this point, the common factor (a - 5) becomes apparent in both the numerator and the denominator. This is the key to simplifying rational expressions. We can cancel out this common factor, as long as a ≠ 5 (because division by zero is undefined). This cancellation leaves us with:

a / (a + 1)

Canceling common factors is a fundamental step in simplifying rational expressions, and it's crucial to understand why this is allowed. When we cancel a common factor, we are essentially dividing both the numerator and the denominator by that factor. As long as the factor is not equal to zero, this operation does not change the value of the expression. However, it's important to remember to state any restrictions on the variable that arise from the cancellation, as we did with a ≠ 5.

Step 4: Stating the Restrictions

It's crucial to state the restrictions on the variable 'a' to ensure that the simplified expression is equivalent to the original expression. From the original denominator, a² - 4a - 5, we know that the expression is undefined when the denominator is equal to zero. We already factored the denominator as (a - 5)(a + 1), so the denominator is zero when a = 5 or a = -1. Therefore, the restrictions are a ≠ 5 and a ≠ -1.

These restrictions are important because they tell us the values of 'a' for which the original expression is undefined. Even though we canceled out the factor (a - 5) during the simplification process, the restriction a ≠ 5 still applies. This is because the original expression is undefined when a = 5, and the simplified expression should reflect this fact. Similarly, the restriction a ≠ -1 arises from the factor (a + 1) in the original denominator.

Stating the restrictions is an essential part of simplifying rational expressions, and it ensures that the simplified expression is mathematically equivalent to the original expression. Always remember to identify and state the restrictions whenever you simplify a rational expression.

Final Simplified Form

Therefore, the simplified form of the rational expression (a² - 5a) / (a² - 4a - 5) is:

a / (a + 1), where a ≠ 5 and a ≠ -1

This is the most concise and simplified form of the original expression, while also taking into account the restrictions on the variable 'a'. By factoring, canceling common factors, and stating the restrictions, we have successfully simplified the rational expression.

Conclusion

Simplifying rational expressions is a crucial skill in algebra, and this step-by-step guide has demonstrated the process using the example (a² - 5a) / (a² - 4a - 5). The key steps involve factoring the numerator and denominator, identifying and canceling common factors, and stating any restrictions on the variable. By mastering these techniques, you can confidently tackle more complex rational expressions and apply them in various mathematical contexts. Remember that practice is key, so work through numerous examples to solidify your understanding and develop your skills in simplifying rational expressions.

By understanding and applying these principles, you can confidently simplify a wide range of rational expressions. This skill is not just a mathematical exercise; it's a gateway to solving more complex problems and understanding deeper mathematical concepts. So, keep practicing, keep exploring, and keep simplifying!