Simplifying Rational Expressions How To Simplify (3x^2 + 4x - 4) / (3x^2 - 14x + 8)

In the realm of algebra, simplifying expressions is a fundamental skill. Often, we encounter complex rational expressions that seem daunting at first glance. However, with a systematic approach, we can break them down into simpler forms, making them easier to understand and manipulate. This guide will walk you through the process of simplifying the rational expression $\frac{3 x^2+4 x-4}{3 x^2-14 x+8}$, providing a detailed, step-by-step explanation to ensure clarity. Understanding how to simplify these expressions is crucial for various mathematical applications, from solving equations to graphing functions. The process primarily involves factoring the numerator and the denominator, and then canceling out any common factors. This not only simplifies the expression but also reveals any restrictions on the variable x that would make the expression undefined. Let's dive into the specifics of simplifying this particular expression, highlighting the key techniques and concepts involved. Factoring quadratic expressions is a critical component of this process, and we will explore different methods to achieve this efficiently. We'll also discuss how to identify and handle any potential pitfalls, ensuring you can confidently tackle similar problems in the future. So, grab your pencil and paper, and let's embark on this algebraic journey together. Remember, the goal is not just to find the answer, but to understand the underlying principles that make it possible. With practice, you'll become adept at simplifying rational expressions and appreciate the elegance and power of algebraic manipulation.

Step-by-Step Simplification of the Expression

To simplify the expression $\frac{3 x^2+4 x-4}{3 x^2-14 x+8}$, we need to factor both the numerator and the denominator. Factoring is the process of breaking down a polynomial into a product of simpler polynomials, which is essential for identifying common factors that can be canceled. This simplification technique is crucial in algebra as it helps reduce complex expressions into more manageable forms, making them easier to work with in subsequent calculations or analyses. The first step in factoring is to look for common factors in each term of the polynomial. If there are no common factors across all terms, we proceed to other methods, such as the quadratic formula or factoring by grouping. Factoring not only simplifies expressions but also reveals important information about the roots and behavior of polynomial functions. It is a foundational skill that underpins many algebraic concepts and is widely used in solving equations, graphing functions, and other mathematical applications. Understanding the different factoring techniques and when to apply them is key to mastering algebra. Let's begin by meticulously factoring each part of our expression, laying the groundwork for the simplification that follows. This process will not only help us solve the problem at hand but also reinforce our understanding of factoring, a cornerstone of algebraic manipulation.

Factoring the Numerator: $3x^2 + 4x - 4$

The numerator is a quadratic expression: $3x^2 + 4x - 4$. To factor this, we need to find two numbers that multiply to the product of the leading coefficient (3) and the constant term (-4), which is -12, and add up to the middle coefficient (4). These numbers are 6 and -2. This process of factoring quadratic expressions is a fundamental skill in algebra, often used to solve quadratic equations, simplify rational expressions, and analyze polynomial functions. The key is to correctly identify the factors that satisfy both the multiplication and addition conditions. There are several methods to approach this, including trial and error, the AC method (which we are using here), and the quadratic formula. The AC method involves finding two numbers that multiply to AC (where A is the leading coefficient and C is the constant term) and add up to B (the middle coefficient). Once these numbers are found, the middle term is split, and the expression is factored by grouping. Mastering this technique is essential for efficiently simplifying algebraic expressions and solving a wide range of mathematical problems. Let's proceed with splitting the middle term and factoring by grouping to complete the factorization of the numerator. This step is crucial for simplifying our rational expression, as it allows us to identify and cancel out common factors with the denominator.

We can rewrite the middle term using these numbers:

$3x^2 + 6x - 2x - 4$

Now, we factor by grouping:

$3x(x + 2) - 2(x + 2)$

We can see that $(x + 2)$ is a common factor:

$(3x - 2)(x + 2)$

Thus, the factored form of the numerator is $(3x - 2)(x + 2)$. Factoring by grouping is a powerful technique that allows us to handle quadratic expressions that are not easily factorable by simple observation. It involves splitting the middle term in a way that allows us to group terms and factor out common factors. This method is particularly useful when the leading coefficient is not 1, as it provides a systematic approach to finding the correct factors. The ability to factor by grouping is an essential skill for simplifying algebraic expressions and solving equations. It allows us to break down complex polynomials into simpler factors, making them easier to work with. Now that we have factored the numerator, we will apply the same process to the denominator. Factoring the denominator will allow us to identify common factors between the numerator and denominator, which we can then cancel out to simplify the expression. This is the core of simplifying rational expressions, and the next step is crucial for arriving at the final simplified form.

Factoring the Denominator: $3x^2 - 14x + 8$

Next, we factor the denominator: $3x^2 - 14x + 8$. We need to find two numbers that multiply to (3)(8) = 24 and add up to -14. These numbers are -12 and -2. The process of factoring the denominator mirrors the process we used for the numerator, but with different coefficients. It's important to pay close attention to the signs of the numbers we're looking for, as they play a crucial role in determining the correct factors. The ability to factor quadratic expressions accurately and efficiently is essential for simplifying rational expressions and solving algebraic equations. There are various strategies we can use, including trial and error, the AC method (as we're using here), and recognizing special patterns like perfect square trinomials or differences of squares. Each method has its strengths, and the best approach depends on the specific expression we're dealing with. In this case, the AC method provides a systematic way to find the correct factors, ensuring we don't overlook any possibilities. Once we have the factors, we can rewrite the middle term and factor by grouping, just as we did with the numerator. This step is critical for simplifying the expression, as it allows us to identify common factors between the numerator and the denominator. Let's proceed with factoring the denominator, laying the groundwork for canceling out common factors and arriving at the simplified expression.

Rewrite the middle term:

$3x^2 - 12x - 2x + 8$

Factor by grouping:

$3x(x - 4) - 2(x - 4)$

We can see that $(x - 4)$ is a common factor:

$(3x - 2)(x - 4)$

Thus, the factored form of the denominator is $(3x - 2)(x - 4)$. Factoring by grouping, as we've demonstrated here, is a powerful technique for handling quadratic expressions, particularly those where the leading coefficient is not equal to one. It allows us to systematically break down the expression into simpler factors, making it easier to identify common terms that can be canceled out when simplifying rational expressions. This method involves rewriting the middle term using the factors we found, then grouping the terms and factoring out common factors from each group. The ability to factor by grouping is an essential skill for anyone working with algebraic expressions and equations, as it provides a reliable method for simplifying complex expressions. Now that we have factored both the numerator and the denominator, we are ready to simplify the original expression by canceling out common factors. This step is the heart of simplifying rational expressions, as it allows us to reduce the expression to its simplest form. Let's move on to the next step and see how this works in practice.

Simplifying the Expression

Now we have the factored forms of both the numerator and the denominator. We can rewrite the original expression:

$\frac{(3x - 2)(x + 2)}{(3x - 2)(x - 4)}$

We can see that $(3x - 2)$ is a common factor in both the numerator and the denominator. Identifying common factors in the numerator and denominator is the key to simplifying rational expressions. Once we've factored both parts of the expression, we can look for any factors that appear in both the numerator and the denominator. These common factors can be canceled out, effectively reducing the expression to a simpler form. This process is similar to simplifying fractions with numerical values, where we divide both the numerator and denominator by their greatest common divisor. In the case of rational expressions, we are dividing by common factors that are polynomials. The ability to identify and cancel common factors is essential for simplifying rational expressions and solving equations involving rational expressions. It allows us to reduce complex expressions to their simplest form, making them easier to work with in subsequent calculations or analyses. In this case, we can see that the factor $(3x - 2)$ appears in both the numerator and the denominator, so we can cancel it out. This will leave us with a simplified expression that is much easier to work with. Let's proceed with canceling out the common factor and see what we're left with.

Cancel the common factor $(3x - 2)$:

$\frac{(3x - 2)(x + 2)}{(3x - 2)(x - 4)} = \frac{x + 2}{x - 4}$

Therefore, the simplified form of the expression is $\frac{x + 2}{x - 4}$. Canceling common factors is a critical step in simplifying rational expressions. It allows us to reduce the expression to its simplest form, making it easier to understand and work with. When we cancel a common factor, we are essentially dividing both the numerator and the denominator by that factor, which doesn't change the value of the expression as long as the factor is not equal to zero. In this case, we canceled the common factor $(3x - 2)$, which means we are assuming that $3x - 2 \neq 0$. This is an important consideration, as it can lead to restrictions on the variable x. The simplified expression $\frac{x + 2}{x - 4}$ is equivalent to the original expression for all values of x except those that make the canceled factor equal to zero or the denominator equal to zero. Now that we have simplified the expression, it's important to consider any restrictions on the variable x. These restrictions arise from the original expression and ensure that we are not dividing by zero. Let's take a look at the restrictions on x for this simplified expression.

Restrictions on the Variable x

In the original expression, the denominator was $3x^2 - 14x + 8$, which factors to $(3x - 2)(x - 4)$. The denominator cannot be equal to zero, as division by zero is undefined. Identifying restrictions on the variable x is a crucial step in simplifying rational expressions. These restrictions arise from the original expression's denominator, as division by zero is undefined in mathematics. We need to find any values of x that would make the denominator equal to zero and exclude them from the domain of the expression. This ensures that the simplified expression is equivalent to the original expression for all valid values of x. There are two potential sources of restrictions: factors that were canceled during simplification and factors that remain in the simplified denominator. In this case, we canceled the factor $(3x - 2)$, and we still have the factor $(x - 4)$ in the simplified denominator. We need to set each of these factors equal to zero and solve for x to find the values that must be excluded. Understanding and stating these restrictions is essential for a complete solution, as it clarifies the conditions under which the simplified expression is valid. Let's proceed with finding the restrictions on x for this expression, ensuring we have a complete and accurate solution.

Therefore, we have two restrictions:

1. $3x - 2 \neq 0 \Rightarrow x \neq \frac{2}{3}$
2. $x - 4 \neq 0 \Rightarrow x \neq 4$

These restrictions on x are crucial to state alongside the simplified expression. These restrictions on x are essential to consider because they define the domain of the rational expression. The domain is the set of all possible values of x for which the expression is defined. In this case, the restrictions $x \neq \frac{2}{3}$ and $x \neq 4$ tell us that the expression is defined for all real numbers except $\frac{2}{3}$ and 4. These values must be excluded from the domain because they would make the original denominator equal to zero, resulting in an undefined expression. When simplifying rational expressions, it's important to identify and state these restrictions to ensure that the simplified expression is equivalent to the original expression for all valid values of x. Ignoring these restrictions can lead to incorrect results or misunderstandings about the behavior of the expression. Now that we have identified the restrictions on x, we have a complete solution to the problem. We have simplified the expression and stated the values of x that must be excluded from the domain. This ensures that our solution is mathematically sound and accurate.

Final Answer

The simplified expression is $\frac{x + 2}{x - 4}$, with the restrictions $x \neq \frac{2}{3}$ and $x \neq 4$. This final answer encapsulates the entire process of simplifying the rational expression, from factoring the numerator and denominator to canceling common factors and identifying restrictions on the variable x. It's a comprehensive solution that provides both the simplified form of the expression and the conditions under which it is valid. The ability to arrive at such a complete answer is a testament to a solid understanding of algebraic principles and techniques. Simplifying rational expressions is a fundamental skill in algebra, and mastering it opens doors to more advanced topics, such as solving rational equations, graphing rational functions, and calculus. This process involves several key steps, each of which requires careful attention and precision. From factoring quadratic expressions to identifying and canceling common factors, each step builds upon the previous one to arrive at the final simplified form. The final step of stating the restrictions on the variable x is equally important, as it ensures that the simplified expression is equivalent to the original expression for all valid values of x. With this final answer, we have successfully simplified the given rational expression and provided a complete and accurate solution. This process demonstrates the power of algebraic manipulation and the importance of a systematic approach to problem-solving.