Simplifying The Expression (3x^2)/(x+2) * (x^2+3x+2)/(6x^2+6x)
In the realm of mathematics, simplifying complex expressions is a fundamental skill. This article delves into the process of simplifying a specific rational expression: . We will break down each step, providing a clear and concise explanation to aid understanding and mastery. Simplifying rational expressions involves techniques such as factoring polynomials, canceling common factors, and applying basic algebraic principles. By understanding these methods, you will be able to solve mathematical problems effectively and accurately. In simplifying the given expression, the initial step involves factoring each polynomial to identify common factors. Factoring breaks down the complex expressions into simpler components, making them easier to manipulate. For instance, the quadratic expression $x^2 + 3x + 2$ can be factored into $(x+1)(x+2)$, while $6x^2 + 6x$ can be factored into $6x(x+1)$. Recognizing and performing these factorizations correctly is crucial for the subsequent steps. After factoring, the expression becomes $\frac{3x^2}{x+2} \cdot \frac{(x+1)(x+2)}{6x(x+1)}$. The next crucial step is to identify and cancel out common factors present in both the numerator and the denominator. This process is akin to simplifying fractions by dividing both the numerator and the denominator by their greatest common divisor. In our expression, we observe that $(x+2)$ and $(x+1)$ appear in both the numerator and the denominator, allowing us to cancel them out. Additionally, we can simplify $\frac{3x^2}{6x}$ by dividing both terms by $3x$, resulting in $\frac{x}{2}$. This cancellation of common factors is a fundamental technique in simplifying rational expressions. After canceling the common factors, the expression is significantly simplified. The resulting expression is $\frac{3x^2}{x+2} \cdot \frac{(x+1)(x+2)}{6x(x+1)} = \frac{x}{2}$. This final form is much easier to understand and work with. The ability to simplify complex rational expressions to their simplest forms is a crucial skill in algebra and calculus. It allows for easier manipulation and solution of equations and inequalities, as well as simplifying more complex mathematical concepts. By simplifying the expression step-by-step, we have transformed a potentially intimidating problem into a manageable one. The technique of factoring, canceling common factors, and simplifying fractions are all essential tools in mathematics. These skills enable the efficient handling of complex problems and are crucial for achieving precise results. The simplification process not only provides the solution but also enhances the understanding of the underlying mathematical concepts. Mastering these techniques can significantly improve problem-solving abilities and build a solid foundation for further studies in mathematics.
Detailed Steps to Simplify the Expression
To effectively simplify the expression , a methodical approach is essential. This section provides a detailed breakdown of each step involved, ensuring clarity and precision. The process begins with identifying opportunities for factorization within the expression. Factoring is a fundamental algebraic technique that involves breaking down a polynomial into simpler terms, typically the product of two or more factors. This process is critical for simplifying rational expressions because it allows for the identification of common factors that can be canceled out. In the given expression, the quadratic polynomial $x^2 + 3x + 2$ and the binomial $6x^2 + 6x$ are prime candidates for factorization. Factoring $x^2 + 3x + 2$ involves finding two numbers that multiply to 2 and add up to 3. These numbers are 1 and 2, so the polynomial can be factored as $(x+1)(x+2)$. This factorization transforms the quadratic expression into a product of two linear expressions, making it easier to work with. Similarly, factoring $6x^2 + 6x$ involves identifying the greatest common factor (GCF) of the terms. In this case, the GCF is $6x$, which can be factored out of the expression, resulting in $6x(x+1)$. This step simplifies the binomial expression into a product of a monomial and a binomial. By factoring these polynomials, the original expression can be rewritten as $\frac{3x^2}{x+2} \cdot \frac{(x+1)(x+2)}{6x(x+1)}$, we can see that $(x+2)$ and $(x+1)$ appear in both the numerator and the denominator. These common factors can be canceled out, simplifying the expression. Additionally, the term $3x^2$ in the numerator and the term $6x$ in the denominator can be simplified further. The greatest common factor of $3x^2$ and $6x$ is $3x$. Dividing both terms by $3x$ simplifies $\frac{3x^2}{6x}$ to $\frac{x}{2}$. After canceling out the common factors $(x+2)$ and $(x+1)$, and simplifying $\frac{3x^2}{6x}$, the expression is reduced to $\frac{x}{2}$. This simplified form is much more manageable and easier to understand than the original expression. By systematically factoring and canceling common factors, we have effectively reduced a complex rational expression to its simplest form. This ability to simplify expressions is crucial in algebra and calculus, as it allows for easier manipulation and solution of equations and inequalities. Furthermore, this detailed step-by-step approach enhances the understanding of the underlying mathematical concepts and builds a solid foundation for more advanced problem-solving.
Factoring Polynomials: The Key to Simplification
Factoring polynomials is a pivotal skill when simplifying rational expressions, particularly in the context of the expression . Mastery of factoring techniques is essential for breaking down complex polynomials into simpler, manageable components. This section will delve into the specific factoring methods applied in this expression and underscore the broader significance of factoring in algebra. At the heart of simplifying rational expressions lies the ability to factor polynomials accurately. Factoring is the process of decomposing a polynomial into the product of two or more simpler polynomials. This technique is crucial because it reveals common factors within the expression, which can then be canceled out, leading to simplification. For instance, the expression $x^2 + 3x + 2$ is a quadratic polynomial that can be factored into two binomials. The binomial expression $6x^2 + 6x$ can also be simplified by factoring out the greatest common factor (GCF). Understanding and applying these factoring techniques are fundamental to the simplification process. When we examine the given expression, $\frac3 x^2}{x+2} \cdot \frac{x^2+3 x+2}{6 x^2+6 x}$, we identify two key polynomials that require factoring{x+2} \cdot \frac{x^2+3 x+2}{6 x^2+6 x}$. This form clearly reveals common factors that were not immediately apparent in the original expression. The importance of factoring extends beyond simplifying rational expressions. It is a cornerstone of algebra, used extensively in solving equations, graphing functions, and simplifying other types of algebraic expressions. Mastery of factoring techniques enhances problem-solving skills and provides a solid foundation for more advanced mathematical concepts. Factoring is not merely a mechanical process; it requires understanding the structure of polynomials and the relationships between their terms. Proficiency in factoring involves recognizing different patterns and applying appropriate techniques, such as factoring by grouping, factoring quadratic trinomials, and identifying special patterns like the difference of squares. In summary, factoring polynomials is a crucial skill in simplifying rational expressions and a fundamental technique in algebra. By breaking down complex polynomials into simpler factors, we can identify common terms, cancel them out, and simplify expressions effectively. This ability is essential for solving mathematical problems and building a strong foundation in algebra.
Canceling Common Factors: Streamlining the Expression
Canceling common factors is a pivotal step in simplifying rational expressions, especially when dealing with expressions such as . This process, akin to simplifying numerical fractions, streamlines the expression by eliminating redundant terms. Understanding how to identify and cancel common factors is essential for achieving the simplest form of a rational expression. After factoring the polynomials in the expression, the stage is set for canceling common factors. This step involves identifying terms that appear in both the numerator and the denominator and then eliminating them. Canceling common factors is based on the principle that dividing both the numerator and the denominator of a fraction by the same non-zero factor does not change the value of the fraction. This is a fundamental concept in simplifying rational expressions. In the expression $\frac{3 x^2}{x+2} \cdot \frac{x^2+3 x+2}{6 x^2+6 x}$. The next step is to identify and cancel common factors. We can see that $(x+2)$ appears in both the numerator and the denominator. Canceling $(x+2)$ simplifies the expression by removing a common term. Similarly, $(x+1)$ also appears in both the numerator and the denominator and can be canceled out. The presence of common factors like $(x+2)$ and $(x+1)$ highlights the importance of factoring polynomials. Without factoring, these common terms would not be apparent, and the expression could not be simplified effectively. In addition to canceling polynomial factors, we can also simplify numerical and variable factors. In this expression, $3x^2$ in the numerator and $6x$ in the denominator share common factors. The greatest common factor (GCF) of $3x^2$ and $6x$ is $3x$. Dividing both $3x^2$ and $6x$ by $3x$ simplifies the fraction $\frac{3x^2}{6x}$ to $\frac{x}{2}$. After canceling all common factors, the expression is significantly simplified. The cancellation of $(x+2)$ and $(x+1)$ and the simplification of $\frac{3x^2}{6x}$ result in a much simpler form. This streamlined expression is easier to work with and understand. Canceling common factors is not just a mechanical step; it requires a clear understanding of the underlying algebraic principles. It is crucial to ensure that only common factors are canceled and that the cancellation is performed correctly. This process leads to the most simplified form of the rational expression, making it easier to analyze and use in further calculations. In summary, canceling common factors is a critical step in simplifying rational expressions. By identifying and eliminating terms that appear in both the numerator and the denominator, the expression is streamlined, and its simplest form is achieved. This process is essential for solving mathematical problems efficiently and accurately.
Final Simplified Form: $\frac{x}{2}$
After systematically factoring, canceling common factors, and simplifying, the expression is reduced to its simplest form: . This final form is not only concise but also reveals the underlying structure of the original expression. Understanding how to arrive at this simplified form is crucial for mastering algebraic manipulation and problem-solving. The journey to the final simplified form involves several key steps. First, the polynomials in the expression are factored to identify common factors. This step transforms the expression into a product of simpler terms, making it easier to manipulate. Next, common factors present in both the numerator and the denominator are canceled out. This process streamlines the expression by eliminating redundant terms. Finally, any remaining terms are simplified to achieve the most concise form. The original expression, $\frac3 x^2}{x+2} \cdot \frac{x^2+3 x+2}{6 x^2+6 x}$. Next, we identify and cancel common factors. The factors $(x+2)$ and $(x+1)$ appear in both the numerator and the denominator and can be canceled out. Additionally, the term $3x^2$ in the numerator and the term $6x$ in the denominator can be simplified by dividing both terms by their greatest common factor, which is $3x$. This simplification results in $\frac{x}{2}$. After canceling the common factors and simplifying the remaining terms, the expression is reduced to its simplest form{2}$ reveals the fundamental relationship between the terms in the expression. It shows that for any value of $x$, the expression will evaluate to half of $x$, provided that the original expression is defined (i.e., the denominator is not zero). The process of simplifying the expression to $\frac{x}{2}$ is $\frac{x}{2}$. This concise form is achieved through the systematic application of factoring, canceling common factors, and simplifying terms. Mastering these techniques is crucial for algebraic manipulation and problem-solving.
