Simplifying Rational Expressions: A Step-by-Step Guide

Rational expressions, which are essentially fractions with polynomials, might seem daunting at first glance. However, breaking them down into smaller, manageable steps makes them much easier to tackle. In this comprehensive guide, we'll walk you through the process of simplifying a given rational expression, identifying equivalent forms, and pinpointing values that need to be excluded from the domain. Understanding these concepts is crucial for success in algebra and beyond. So, let's dive in and unravel the mysteries of rational expressions!

Understanding Rational Expressions

At its core, a rational expression is simply a fraction where the numerator and denominator are polynomials. Polynomials, in turn, are expressions consisting of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. Examples of rational expressions include (x+1)/(x-2), (3x^2 - 2x + 1)/(x^2 + 4), and the one we'll be working with today: (x^2 - 14x + 49) / (6x^2 - 42x). The key to simplifying these expressions lies in factoring and identifying common factors that can be canceled out. This process is analogous to simplifying regular numerical fractions, where we find the greatest common divisor and divide both the numerator and denominator by it. However, with rational expressions, we're dealing with polynomials instead of numbers. When working with rational expressions, it's essential to remember the concept of the domain. The domain of a rational expression is the set of all possible values of the variable (usually 'x') that make the expression defined. Since division by zero is undefined, any value of x that makes the denominator of the rational expression equal to zero must be excluded from the domain. Identifying these excluded values is a crucial part of working with rational expressions.

1. Finding an Equivalent Expression in Lowest Terms

Our main goal here is to simplify the given rational expression: (x^2 - 14x + 49) / (6x^2 - 42x). This involves a couple of key steps: factoring both the numerator and the denominator, and then canceling out any common factors. Let's break down each step.

Factoring the Numerator

The numerator, x^2 - 14x + 49, is a quadratic expression. Recognizing patterns is a crucial skill in algebra, and in this case, we can see that the numerator is a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the square of a binomial. The general form of a perfect square trinomial is a^2 ± 2ab + b^2, which factors into (a ± b)^2. In our case, x^2 - 14x + 49 fits this pattern perfectly. We can rewrite it as x^2 - 2(x)(7) + 7^2. This corresponds to the form a^2 - 2ab + b^2, where a = x and b = 7. Therefore, we can factor the numerator as (x - 7)^2, which is the same as (x - 7)(x - 7). Factoring the numerator transforms the expression from a seemingly complex quadratic into a more manageable form, setting the stage for simplification.

Factoring the Denominator

The denominator, 6x^2 - 42x, is not a trinomial, but it can be factored using a different technique: finding the greatest common factor (GCF). The greatest common factor is the largest factor that divides into both terms of the expression. In this case, both 6x^2 and -42x are divisible by 6x. Factoring out 6x from the denominator gives us 6x(x - 7). This step is crucial because it reveals a common factor with the numerator, which will allow us to simplify the expression. Factoring out the GCF not only simplifies the expression but also helps in identifying values that might need to be excluded from the domain later on.

Simplifying the Expression

Now that we've factored both the numerator and the denominator, our expression looks like this: ((x - 7)(x - 7)) / (6x(x - 7)). We can see that (x - 7) is a common factor in both the numerator and the denominator. Just like we would cancel common factors in a regular fraction, we can cancel the (x - 7) term from the top and the bottom. This leaves us with (x - 7) / (6x). This is the simplified form of the original rational expression. Simplifying the expression not only makes it easier to work with but also provides a clearer understanding of its behavior. The simplified form allows us to easily evaluate the expression for different values of x (except for those excluded from the domain) and to perform further algebraic manipulations.

Therefore, the equivalent expression in lowest terms is (x - 7) / (6x).

2. Determining Excluded Values from the Domain

Remember, we can't divide by zero. So, any value of x that makes the denominator of our rational expression equal to zero must be excluded from the domain. This is a critical step in working with rational expressions, as it ensures that the expression remains defined. To find these excluded values, we need to look at the denominator before we simplified the expression. This is because simplifying might hide some of the values that make the original denominator zero.

Analyzing the Original Denominator

The original denominator was 6x^2 - 42x. We already factored this as 6x(x - 7). To find the excluded values, we need to set this factored form equal to zero and solve for x: 6x(x - 7) = 0. This equation is satisfied if either 6x = 0 or (x - 7) = 0.

Solving for Excluded Values

Solving 6x = 0, we get x = 0. This means that x = 0 makes the denominator zero, and therefore, it must be excluded from the domain. Solving (x - 7) = 0, we get x = 7. This means that x = 7 also makes the denominator zero and must be excluded from the domain. These two values, x = 0 and x = 7, are the values that would lead to division by zero, making the expression undefined.

Therefore, the values of x that must be excluded from the domain are x = 0 and x = 7.

Conclusion

Simplifying rational expressions involves factoring, canceling common factors, and identifying values excluded from the domain. By following these steps systematically, you can confidently tackle even the most complex rational expressions. Remember to always factor both the numerator and the denominator completely, cancel out common factors carefully, and determine the excluded values by considering the original denominator. These skills are fundamental to success in algebra and calculus, so practice makes perfect!

For more in-depth information and examples, you can explore resources like Khan Academy's Algebra I course, which offers a comprehensive understanding of rational expressions and related topics.