Factoring: -21x + 14y + 56 Explained Simply

Let's dive into the world of factoring and break down the expression -21x + 14y + 56. Factoring, at its core, is like reverse multiplication. It involves finding the common elements within an expression and rewriting it in a simpler, more manageable form. This skill is crucial in algebra and beyond, forming the bedrock for solving equations and simplifying complex expressions. In this comprehensive guide, we’ll walk you through each step, ensuring you grasp the underlying concepts and can confidently tackle similar problems. Our focus will be on identifying the greatest common factor (GCF) and using it to simplify the given expression, making it easier to work with and understand. So, whether you're a student grappling with algebra or just looking to brush up your math skills, this guide is here to help you master the art of factoring. We'll start with the basics, gradually building up to the solution, and provide clear explanations every step of the way. By the end of this article, you'll not only know how to factor this specific expression but also have a solid understanding of the principles behind factoring in general.

Understanding the Basics of Factoring

Before we jump into the specifics of our expression, let's solidify our understanding of factoring itself. Factoring is the process of breaking down a number or an expression into its constituent parts, known as factors. Think of it like dismantling a machine to see the individual components that make it work. In mathematical terms, when we factor, we are looking for the numbers or expressions that, when multiplied together, give us the original number or expression. This is particularly useful when dealing with algebraic expressions, as it allows us to simplify them and make them easier to work with. For instance, consider the number 12. It can be factored in several ways: 1 x 12, 2 x 6, or 3 x 4. Each of these pairs represents a different way of expressing 12 as a product of its factors. Similarly, in algebra, we might encounter expressions like x² + 2x + 1, which can be factored into (x + 1)(x + 1). Understanding these fundamental concepts is key to successfully factoring more complex expressions. The goal is always to find the simplest and most reduced form, which often involves identifying the greatest common factor. This is the largest number or expression that divides evenly into all terms of the original expression. By factoring out the GCF, we effectively streamline the expression, making it easier to analyze and manipulate in further calculations. So, remember, factoring isn't just about finding any factors; it's about finding the most significant ones that help us simplify and understand the underlying structure of the expression.

Identifying the Greatest Common Factor (GCF)

The cornerstone of successful factoring is identifying the Greatest Common Factor, often abbreviated as GCF. The GCF is the largest number or expression that divides evenly into all the terms of a given expression. Finding the GCF is like searching for the common thread that ties all the elements together. It’s the key to unlocking a simplified form of the expression. To illustrate, let’s consider the numbers 12, 18, and 24. To find their GCF, we first list the factors of each number: Factors of 12: 1, 2, 3, 4, 6, 12 Factors of 18: 1, 2, 3, 6, 9, 18 Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. The common factors are 1, 2, 3, and 6. Among these, 6 is the largest, making it the GCF of 12, 18, and 24. In algebraic expressions, the process is similar but involves both numerical and variable factors. For example, in the expression 6x² + 9x, we look for the largest number that divides both 6 and 9, which is 3, and the highest power of x that is common to both terms, which is x. Thus, the GCF of 6x² and 9x is 3x. Mastering the identification of the GCF is paramount because it not only simplifies the factoring process but also reduces the expression to its most basic form. This skill is not just useful in algebra; it's a fundamental concept that applies across various branches of mathematics. So, take the time to practice finding the GCF, and you'll find that factoring becomes a much smoother and more intuitive process. Remember, the GCF is the key to simplifying complex expressions and making them easier to work with.

Applying GCF to the Expression -21x + 14y + 56

Now, let's apply our understanding of the Greatest Common Factor (GCF) to the expression -21x + 14y + 56. Our goal is to identify the largest factor that divides evenly into all three terms: -21x, 14y, and 56. To begin, we'll look at the numerical coefficients: -21, 14, and 56. We need to find the largest number that divides all three of these evenly. The factors of 21 are 1, 3, 7, and 21. The factors of 14 are 1, 2, 7, and 14. The factors of 56 are 1, 2, 4, 7, 8, 14, 28, and 56. The common factors are 1 and 7. Therefore, the greatest common factor of the numerical coefficients is 7. However, since the first term has a negative coefficient (-21), it's often beneficial to factor out a negative number. This doesn't change the mathematical value of the expression but can simplify it and make it easier to work with in subsequent steps. So, in this case, we'll consider -7 as our numerical GCF. Now, let's look at the variable parts of the terms. We have x in the first term, y in the second term, and no variable in the third term. Since there are no variables common to all three terms, we don't have a variable component in our GCF. Thus, the GCF for the entire expression -21x + 14y + 56 is -7. This means that -7 is the largest factor that can be divided out of each term, allowing us to rewrite the expression in a more simplified, factored form. By identifying and applying the GCF, we're taking the first crucial step towards completely factoring the given expression. This methodical approach ensures that we're not just simplifying, but also maintaining the integrity and equivalence of the expression.

Step-by-Step Factoring of -21x + 14y + 56

Having identified -7 as the Greatest Common Factor (GCF) of the expression -21x + 14y + 56, we can now proceed with the factoring process step-by-step. Factoring out the GCF involves dividing each term in the expression by the GCF and then rewriting the expression in a factored form. This is akin to reverse distribution, where we're pulling out a common element rather than multiplying it in. Step 1: Divide each term by the GCF: -21x / -7 = 3x 14y / -7 = -2y 56 / -7 = -8. Notice that dividing a negative term by a negative GCF results in a positive term, and vice versa. This is a crucial point to keep in mind to ensure accuracy. Step 2: Rewrite the expression in factored form. We take the GCF, which is -7, and write it outside a set of parentheses. Inside the parentheses, we place the results of the divisions we performed in Step 1. So, the factored form of the expression becomes: -7(3x - 2y - 8). This expression is mathematically equivalent to the original expression, -21x + 14y + 56, but it's written in a more simplified, factored form. This form is often more useful for solving equations, simplifying expressions, and other algebraic manipulations. To check our work, we can distribute the -7 back into the parentheses: -7 * 3x = -21x -7 * -2y = 14y -7 * -8 = 56. This gives us back our original expression, confirming that our factoring is correct. This step-by-step process highlights the elegance and efficiency of factoring. By identifying and factoring out the GCF, we've transformed a complex expression into a simpler, more manageable form. This is a fundamental skill in algebra and a cornerstone for more advanced mathematical concepts.

Final Factored Form and Verification

After meticulously applying the Greatest Common Factor (GCF), we've arrived at the final factored form of the expression -21x + 14y + 56. The factored form is -7(3x - 2y - 8). This represents the simplified version of our original expression, where we've essentially