Fully Factorising 24f² + 30f A Step-by-Step Guide

In the realm of mathematics, factorisation stands as a fundamental technique for simplifying expressions and solving equations. At its core, factorisation involves breaking down a complex expression into simpler components, known as factors, which when multiplied together, yield the original expression. This process is akin to dissecting a sentence into its individual words, allowing for a more granular understanding of the expression's structure and behavior. In this comprehensive guide, we will delve into the intricacies of fully factorising the expression 24f² + 30f, exploring the underlying principles and step-by-step methods involved. Mastering factorisation is crucial for students and professionals alike, as it forms the bedrock for advanced algebraic manipulations and problem-solving strategies. From simplifying complex equations to identifying patterns and relationships within mathematical constructs, factorisation empowers individuals to tackle a wide array of mathematical challenges with confidence and precision. Therefore, a thorough understanding of factorisation techniques is an invaluable asset for anyone seeking to excel in the field of mathematics. Factorisation not only aids in simplifying expressions but also paves the way for solving equations, understanding polynomial behavior, and even tackling real-world problems involving mathematical modeling. Whether you are a student grappling with algebraic concepts or a seasoned professional seeking to refine your mathematical toolkit, this guide will provide you with the knowledge and skills necessary to confidently factorise expressions and unlock their hidden potential. So, let us embark on this journey of mathematical exploration and unravel the secrets of factorisation, starting with the expression 24f² + 30f.

Understanding the Basics: What is Factorisation?

Before diving into the specific example of 24f² + 30f, it's crucial to grasp the fundamental concept of factorisation. In essence, factorisation is the process of expressing a number or an algebraic expression as a product of its factors. Factors are numbers or expressions that divide evenly into the original number or expression. For instance, the factors of 12 are 1, 2, 3, 4, 6, and 12 because each of these numbers divides 12 without leaving a remainder. Similarly, in algebra, we can factorise expressions containing variables. The goal is to rewrite the expression as a product of simpler expressions, often involving constants and variables. This process simplifies complex expressions, making them easier to manipulate and solve. Think of it like this: you're taking a composite whole and breaking it down into its constituent parts. This not only simplifies the expression but also reveals its underlying structure. Consider the number 36; it can be factorised as 2 x 2 x 3 x 3, revealing its prime factors. In algebra, factorisation follows the same principle but with variables and expressions. For example, the expression x² + 5x + 6 can be factorised into (x + 2)(x + 3). Understanding this basic principle is the key to mastering factorisation. It allows us to rewrite expressions in a more manageable form, which is essential for solving equations, simplifying expressions, and understanding mathematical relationships. In the context of our main example, 24f² + 30f, factorisation will help us identify common factors and rewrite the expression in a simpler, more revealing form. By mastering this foundational concept, you'll be well-equipped to tackle more complex algebraic challenges.

Identifying Common Factors

When tackling the task of fully factorising an expression like 24f² + 30f, the initial and most crucial step involves identifying the common factors present in both terms. This process is akin to searching for shared building blocks within the expression, which can then be extracted to simplify it. In our example, we have two terms: 24f² and 30f. To find the common factors, we need to examine both the numerical coefficients (24 and 30) and the variable components (f² and f). Let's start with the numerical coefficients. We need to find the greatest common divisor (GCD) of 24 and 30. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. The greatest common factor between these two sets is 6. This means that 6 is the largest number that divides both 24 and 30 evenly. Now, let's turn our attention to the variable components. We have f² in the first term and f in the second term. The common factor here is f, as it is the highest power of f that is present in both terms. Remember that f² means f multiplied by itself (f * f), and f is simply f. Therefore, f is a common factor. Combining our findings, we see that the greatest common factor (GCF) of 24f² and 30f is 6f. This is the largest expression that divides both terms without leaving a remainder. Identifying the GCF is the cornerstone of factorisation. Once we've identified 6f as the GCF, we can proceed to the next step, which involves factoring it out of the expression. This process of identifying common factors is not only crucial for factorisation but also strengthens your understanding of number theory and algebraic principles. It's a foundational skill that will serve you well in more advanced mathematical concepts.

Factoring Out the Greatest Common Factor (GCF)

Having identified the greatest common factor (GCF) as 6f for the expression 24f² + 30f, the next step involves factoring it out. This process is essentially the reverse of the distributive property. We're taking the GCF and dividing each term in the original expression by it, then writing the expression as a product of the GCF and the resulting quotient. Let's break it down: We start with 24f² + 30f and our GCF is 6f. Now, we divide each term by 6f: 24f² / 6f = 4f (because 24 divided by 6 is 4, and f² divided by f is f) 30f / 6f = 5 (because 30 divided by 6 is 5, and f divided by f is 1) Now, we can rewrite the expression as the product of the GCF and the results of our divisions: 6f(4f + 5) This is the factored form of the expression. We've successfully extracted the GCF and placed it outside the parentheses, leaving the remaining terms inside. To verify our factorisation, we can use the distributive property to expand the factored expression: 6f * 4f = 24f² 6f * 5 = 30f Adding these together, we get 24f² + 30f, which is our original expression. This confirms that our factorisation is correct. Factoring out the GCF is a powerful technique that simplifies expressions and makes them easier to work with. It's a fundamental skill in algebra and is used extensively in solving equations, simplifying fractions, and many other mathematical applications. By mastering this step, you're building a solid foundation for more advanced algebraic manipulations. In our example, we've successfully factored out the GCF, but it's important to always check if the expression can be factored further. In this case, (4f + 5) cannot be factored further, so we have fully factorised the expression.

Verifying the Factorisation

After factoring an expression, it's paramount to verify the result to ensure accuracy. This step acts as a safeguard against potential errors and solidifies understanding of the factorisation process. For the expression 24f² + 30f, which we factored into 6f(4f + 5), verification involves expanding the factored form and comparing it to the original expression. The expansion process utilizes the distributive property, where the term outside the parentheses (6f) is multiplied by each term inside the parentheses (4f and 5). Let's perform the expansion: 6f * (4f + 5) = (6f * 4f) + (6f * 5) Now, we multiply each term: 6f * 4f = 24f² 6f * 5 = 30f Combining these results, we get: 24f² + 30f This is precisely the original expression we started with. This confirms that our factorisation, 6f(4f + 5), is correct. Verification not only ensures accuracy but also reinforces the understanding of the relationship between factorisation and expansion. It highlights that factorisation is essentially the reverse process of expansion. Expanding the factored form should always yield the original expression if the factorisation is done correctly. This verification step is crucial, especially in more complex factorisation problems where errors can easily occur. It instills a sense of confidence in your solution and prevents carrying forward incorrect factorisations in subsequent calculations. Furthermore, the process of verification deepens your grasp of algebraic manipulation and strengthens your problem-solving skills. It encourages a systematic approach to mathematics, emphasizing the importance of checking your work at each step. In summary, verifying the factorisation is an indispensable step in the factorisation process. It guarantees the correctness of the result and enhances your understanding of algebraic concepts. Always make it a habit to verify your factorisations to ensure accuracy and build confidence in your mathematical abilities.

Fully Factorised Form: 6f(4f + 5)

Having meticulously gone through the steps of identifying the greatest common factor (GCF), factoring it out, and verifying the result, we arrive at the fully factorised form of the expression 24f² + 30f, which is 6f(4f + 5). This form represents the most simplified representation of the original expression as a product of its factors. In this factorised form, 6f and (4f + 5) are the factors of 24f² + 30f. This means that when 6f is multiplied by (4f + 5), the result is 24f² + 30f. The fully factorised form is significant for several reasons. Firstly, it simplifies the expression, making it easier to work with in further algebraic manipulations. For instance, if we were to solve an equation involving 24f² + 30f = 0, the factorised form would make the process significantly simpler. We could set each factor equal to zero (6f = 0 and 4f + 5 = 0) and solve for f. Secondly, the fully factorised form reveals the structure and composition of the expression. It shows the basic building blocks that make up the expression and how they relate to each other. This can be particularly useful in understanding the behavior of the expression and its properties. Furthermore, the fully factorised form is often required in various mathematical contexts, such as simplifying algebraic fractions, finding roots of polynomials, and solving quadratic equations. It is a fundamental skill that is essential for success in algebra and beyond. In our example, we have successfully transformed a seemingly complex expression into a concise and manageable form. The process of factorisation has allowed us to break down the expression into its fundamental components, providing a deeper understanding of its nature. This skill is invaluable in mathematics and will serve as a solid foundation for tackling more complex problems in the future.

Applications of Factorisation

Factorisation, as demonstrated with the expression 24f² + 30f, is not merely an academic exercise; it's a powerful tool with wide-ranging applications in mathematics and beyond. Understanding its practical uses highlights its importance and relevance. One of the primary applications of factorisation lies in solving algebraic equations. Consider the equation 24f² + 30f = 0. Directly solving this quadratic equation might seem challenging. However, by using the factorised form, 6f(4f + 5) = 0, the problem becomes significantly simpler. According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we can set each factor equal to zero: 6f = 0 and 4f + 5 = 0. Solving these simple linear equations, we find the solutions f = 0 and f = -5/4. This illustrates how factorisation transforms a complex quadratic equation into simpler linear equations, making the solution process straightforward. Another crucial application of factorisation is in simplifying algebraic fractions. If we have a fraction with a polynomial expression in the numerator or denominator, factorisation can help us cancel out common factors and reduce the fraction to its simplest form. This simplification is essential in calculus and other advanced mathematical fields. Factorisation is also fundamental in understanding the behavior of polynomial functions. The factors of a polynomial reveal its roots or zeros, which are the points where the graph of the function intersects the x-axis. Knowing the roots of a polynomial is vital for sketching its graph and analyzing its properties. Beyond pure mathematics, factorisation finds applications in various fields such as physics, engineering, and computer science. In physics, it can be used to simplify equations describing physical phenomena. In engineering, it's used in circuit analysis and control systems. In computer science, it plays a role in cryptography and data compression. In essence, factorisation is a cornerstone of mathematical problem-solving. It simplifies complex expressions, reveals underlying structures, and provides a pathway to solve equations and understand mathematical relationships. Its applications are vast and diverse, making it an indispensable tool for anyone working with mathematics in any capacity.

Common Mistakes to Avoid

When factorising expressions, it's crucial to be aware of common pitfalls that can lead to errors. Avoiding these mistakes ensures accuracy and a deeper understanding of the factorisation process. One frequent error is failing to identify the greatest common factor (GCF) correctly. For instance, in the expression 24f² + 30f, a student might identify 2f as a common factor but miss the fact that 6f is the GCF. This would lead to an incomplete factorisation, leaving room for further simplification. Always ensure you've identified the largest possible factor common to all terms. Another common mistake is incorrectly applying the distributive property when factoring out the GCF. This often results in errors in the terms remaining inside the parentheses. For example, when factoring 6f out of 24f² + 30f, a student might incorrectly write 6f(4f² + 5) instead of the correct form, 6f(4f + 5). Remember to divide each term in the original expression by the GCF to determine the terms inside the parentheses. Sign errors are also a common source of mistakes, particularly when dealing with negative coefficients. Pay close attention to the signs of the terms when factoring and expanding to verify your result. A simple sign error can completely change the outcome of the factorisation. Another mistake is stopping the factorisation process prematurely. Sometimes, after factoring out the GCF, the expression inside the parentheses can be factored further. Always check if the resulting expression can be factorised again. This ensures that you have fully factorised the expression. Finally, failing to verify the factorisation is a significant oversight. As discussed earlier, verification is essential to catch any errors made during the factorisation process. Expanding the factored form and comparing it to the original expression is a simple yet effective way to ensure accuracy. By being mindful of these common mistakes and adopting a systematic approach to factorisation, you can minimize errors and develop a strong command of this fundamental algebraic skill.

Practice Problems

To solidify your understanding of factorisation, working through practice problems is essential. The more you practice, the more comfortable and proficient you will become in applying the techniques discussed. Here are a few practice problems to help you hone your skills: 1. Factorise 15x² + 25x. This problem is similar to our example, focusing on identifying and factoring out the greatest common factor (GCF). Look for the GCF of the coefficients and the variables, then factor it out and verify your answer. 2. Factorise 36a³ - 18a². This problem involves higher powers of the variable, requiring careful attention to the variable components when identifying the GCF. Remember to extract the highest power of the variable that is common to both terms. 3. Factorise 12y² + 18y. This problem is another exercise in finding the GCF. Pay attention to the numerical coefficients and the variable components, and ensure you factor out the largest possible factor. 4. Factorise 9b² - 27b. This problem introduces a negative coefficient, so be mindful of the signs when factoring out the GCF. Remember to include the negative sign in the factored form if necessary. 5. Factorise 20z³ + 50z². This problem combines larger coefficients with higher powers of the variable, providing a good challenge for applying the factorisation techniques. Take your time to identify the GCF and factor it out carefully. For each problem, follow the steps outlined in this guide: identify the GCF, factor it out, and verify your result by expanding the factored form. If you encounter difficulties, review the concepts and examples discussed earlier. Remember, practice is key to mastering factorisation. Work through these problems diligently, and you will develop the skills and confidence needed to tackle more complex algebraic challenges. Additionally, seek out more practice problems from textbooks, online resources, or your teacher to further enhance your understanding and proficiency in factorisation.

Conclusion

In conclusion, the process of fully factorising 24f² + 30f exemplifies the core principles of factorisation in mathematics. We successfully broke down the expression into its fundamental components, 6f and (4f + 5), by identifying and extracting the greatest common factor (GCF). This methodical approach not only simplifies the expression but also unveils its underlying structure and relationships. Throughout this guide, we've emphasized the importance of understanding the basics of factorisation, identifying common factors, factoring out the GCF, verifying the results, and recognizing the practical applications of factorisation. We also highlighted common mistakes to avoid, ensuring a more accurate and efficient problem-solving process. The ability to factorise expressions is a cornerstone of algebraic manipulation and is essential for solving equations, simplifying fractions, and understanding polynomial behavior. It is a skill that extends far beyond the classroom, finding applications in various fields such as physics, engineering, and computer science. By mastering factorisation, you equip yourself with a powerful tool for tackling a wide range of mathematical challenges. Furthermore, the practice problems provided offer an opportunity to reinforce your understanding and develop your skills further. Consistent practice is key to becoming proficient in factorisation and building confidence in your mathematical abilities. As you continue your mathematical journey, remember that factorisation is not just a technique but a way of thinking. It encourages a systematic approach to problem-solving, emphasizing the importance of breaking down complex problems into simpler components. This mindset will serve you well in all areas of mathematics and beyond. So, embrace the power of factorisation, practice diligently, and unlock the full potential of your mathematical abilities.