Simplifying Rational Expressions Step-by-Step (2x^2 + 7x - 4) / (2x^2 + 9x + 4)

In the realm of algebra, simplifying rational expressions is a fundamental skill. Rational expressions, essentially fractions with polynomials in the numerator and denominator, often appear complex at first glance. However, by mastering the art of factorization and cancellation, we can reduce these expressions to their simplest forms. This article delves into the process of simplifying the rational expression (2x^2 + 7x - 4) / (2x^2 + 9x + 4), providing a clear, step-by-step guide suitable for students and anyone seeking to enhance their algebraic proficiency. The core concept behind simplifying any rational expression lies in identifying common factors within the numerator and the denominator. Once these common factors are identified, they can be canceled out, thereby reducing the expression to its simplest equivalent form. This process not only makes the expression more manageable but also facilitates further operations such as addition, subtraction, multiplication, and division of rational expressions. To embark on this simplification journey, we need to revisit the techniques of factoring quadratic expressions. Factoring, in essence, is the process of breaking down a polynomial into a product of simpler polynomials. In the case of quadratic expressions, we aim to express them as a product of two linear factors. This involves identifying two numbers that, when multiplied, give the constant term and, when added, yield the coefficient of the linear term. This foundational skill is the key to unlocking the simplification of the given rational expression. Let's begin by examining the numerator and the denominator separately and applying the principles of factoring to each. This initial step sets the stage for identifying the common factors and ultimately simplifying the expression.

Factoring the Numerator: 2x^2 + 7x - 4

Our first task is to factor the numerator, which is the quadratic expression 2x^2 + 7x - 4. To factor this quadratic, we're looking for two binomials that, when multiplied, result in the original quadratic expression. The most common method to accomplish this is by using the trial and error method or the ac method. Here, we will demonstrate the ac method, which is a systematic approach. In this method, we first identify the coefficients a, b, and c from the quadratic expression ax^2 + bx + c. In our case, a = 2, b = 7, and c = -4. We then calculate the product of a and c, which is 2 * (-4) = -8. Now, we need to find two numbers that multiply to -8 and add up to b, which is 7. After some thought, we can identify these two numbers as 8 and -1 because 8 * (-1) = -8 and 8 + (-1) = 7. Next, we rewrite the middle term (7x) using these two numbers: 2x^2 + 7x - 4 becomes 2x^2 + 8x - x - 4. We have effectively split the middle term into two terms, allowing us to factor by grouping. Now, we group the first two terms and the last two terms: (2x^2 + 8x) + (-x - 4). We factor out the greatest common factor (GCF) from each group. From the first group, the GCF is 2x, and from the second group, the GCF is -1. This gives us: 2x(x + 4) - 1(x + 4). Notice that we now have a common binomial factor of (x + 4) in both terms. We factor out this common binomial factor: (x + 4)(2x - 1). Therefore, the factored form of the numerator 2x^2 + 7x - 4 is (x + 4)(2x - 1). This factorization is a crucial step towards simplifying the entire rational expression. By breaking down the quadratic into its linear factors, we unveil the potential for cancellation with factors in the denominator. Now that we have successfully factored the numerator, we turn our attention to the denominator and apply a similar factoring process. The goal is to express both the numerator and the denominator as products of their factors, paving the way for identifying common terms and simplifying the rational expression. The ability to accurately factor quadratic expressions is a cornerstone of algebraic manipulation, and this example illustrates the methodical approach required to achieve the correct factorization.

Factoring the Denominator: 2x^2 + 9x + 4

Following the successful factorization of the numerator, we now shift our focus to the denominator, which is the quadratic expression 2x^2 + 9x + 4. Similar to the previous step, we aim to factor this quadratic into two binomials. We will again employ the ac method, as it provides a structured approach to factoring quadratics. First, we identify the coefficients a, b, and c from the quadratic expression ax^2 + bx + c. In this case, a = 2, b = 9, and c = 4. We then calculate the product of a and c, which is 2 * 4 = 8. Now, we need to find two numbers that multiply to 8 and add up to b, which is 9. These two numbers are readily identifiable as 8 and 1 because 8 * 1 = 8 and 8 + 1 = 9. We proceed to rewrite the middle term (9x) using these two numbers: 2x^2 + 9x + 4 becomes 2x^2 + 8x + x + 4. We have split the middle term into two terms, which sets the stage for factoring by grouping. We group the first two terms and the last two terms: (2x^2 + 8x) + (x + 4). We factor out the greatest common factor (GCF) from each group. From the first group, the GCF is 2x, and from the second group, the GCF is 1. This gives us: 2x(x + 4) + 1(x + 4). Observe that we now have a common binomial factor of (x + 4) in both terms. We factor out this common binomial factor: (x + 4)(2x + 1). Consequently, the factored form of the denominator 2x^2 + 9x + 4 is (x + 4)(2x + 1). This factorization is a critical accomplishment, as it allows us to express the denominator as a product of its linear factors. With both the numerator and the denominator now in their factored forms, we are well-positioned to identify common factors and simplify the rational expression. The ability to factor quadratic expressions accurately is paramount in simplifying rational expressions, and this step further demonstrates the systematic approach required for successful factorization. Now that we have successfully factored both the numerator and the denominator, we are ready to combine these results and proceed with the final step of simplification.

Combining the Factored Expressions and Simplifying

Having meticulously factored both the numerator and the denominator, we now arrive at the crucial step of combining these results and simplifying the rational expression. Recall that the original expression was (2x^2 + 7x - 4) / (2x^2 + 9x + 4). Through our factoring efforts, we have determined that: The numerator, 2x^2 + 7x - 4, factors into (x + 4)(2x - 1). The denominator, 2x^2 + 9x + 4, factors into (x + 4)(2x + 1). Now, we substitute these factored forms back into the original rational expression: [(x + 4)(2x - 1)] / [(x + 4)(2x + 1)]. This representation clearly reveals the factors present in both the numerator and the denominator. The next step is to identify any common factors that can be canceled out. In this case, we observe that the binomial factor (x + 4) appears in both the numerator and the denominator. This common factor can be canceled, as dividing both the numerator and the denominator by (x + 4) does not change the value of the expression (provided that x ≠ -4, as this would make the factor zero). After canceling the common factor (x + 4), we are left with: (2x - 1) / (2x + 1). This is the simplified form of the original rational expression. The expression (2x - 1) / (2x + 1) cannot be simplified further, as there are no more common factors between the numerator and the denominator. This simplified form is an equivalent expression to the original, but it is in its most reduced state. Simplifying rational expressions is not just a matter of algebraic manipulation; it also has practical implications. Simplified expressions are easier to work with in further calculations, such as solving equations or evaluating the expression for specific values of x. This process highlights the importance of mastering factoring techniques and understanding the principles of cancellation in algebraic expressions. In conclusion, by systematically factoring the numerator and the denominator and then canceling common factors, we have successfully simplified the rational expression (2x^2 + 7x - 4) / (2x^2 + 9x + 4) to its simplest form, (2x - 1) / (2x + 1). This result underscores the power of factorization as a tool for simplifying complex algebraic expressions.

Final Answer

Therefore, the simplified form of the rational expression (2x^2 + 7x - 4) / (2x^2 + 9x + 4) is (2x - 1) / (2x + 1). This simplification was achieved through factoring both the numerator and the denominator and then canceling out the common factor of (x + 4). The final expression represents the most reduced form of the original rational expression.