Factoring: X(x-1) + 4(x-1) - A Step-by-Step Guide

Hey guys! Let's dive into factoring the expression x(x-1) + 4(x-1) completely. Factoring is a fundamental skill in algebra, and mastering it can make solving equations and simplifying expressions a breeze. In this guide, we'll break down the steps, explain the underlying concepts, and make sure you understand exactly how to tackle this type of problem. So, grab your pencils, and let's get started!

Understanding Factoring

Before we jump into the specific problem, let's quickly recap what factoring is all about. At its core, factoring is the process of breaking down an expression into a product of its factors. Think of it like reverse multiplication. For example, if we have the number 12, we can factor it into 3 * 4 or 2 * 6 or even 2 * 2 * 3. Similarly, in algebra, we want to express a given expression as a product of simpler expressions.

In our case, we are starting with x(x-1) + 4(x-1) and want to rewrite it in a factored form. This usually involves identifying common factors, grouping terms, or using special factoring formulas. The goal is to simplify the expression and make it easier to work with.

Why is factoring important? Well, it's essential for solving polynomial equations. When you have a factored expression set equal to zero, you can easily find the solutions by setting each factor equal to zero. Factoring also helps in simplifying complex algebraic expressions, making them more manageable for further calculations or manipulations. Moreover, factoring plays a crucial role in calculus, especially when dealing with rational functions and integration techniques.

Factoring isn't just a dry mathematical exercise; it's a powerful tool that enables us to understand and solve real-world problems. From optimizing designs to modeling physical phenomena, factoring often appears in various scientific and engineering applications. By becoming proficient in factoring, you're equipping yourself with a skill that will serve you well throughout your academic and professional journey.

Identifying the Common Factor

Now, let's focus on our expression: x(x-1) + 4(x-1). The first step in factoring this expression is to identify any common factors. A common factor is a term that appears in each part of the expression. Looking closely, we can see that (x-1) is a common factor in both terms, x(x-1) and 4(x-1). Spotting this common factor is the key to simplifying the expression.

The presence of a common factor makes our job much easier because we can factor it out. Factoring out a common factor is like reverse distribution. Instead of multiplying the factor through the terms, we divide each term by the common factor and place it outside a set of parentheses. In our case, we'll factor out (x-1) from both terms.

When you identify the common factor, it’s crucial to ensure that you’re precise. Sometimes, common factors might be disguised or require a bit of manipulation to become apparent. In more complex expressions, you might need to factor out not just simple terms but also combinations of variables and constants. For example, you might encounter expressions where you need to factor out something like 2x or (x+2)^2. Recognizing these common factors is what distinguishes a good algebra student from a great one.

To illustrate the importance of identifying common factors, consider a scenario where you missed the common factor. You would end up trying other more complicated methods that would likely lead to nowhere. By correctly identifying and factoring out common factors, you simplify the expression, making it easier to solve or analyze. This skill is invaluable in more advanced mathematical contexts, such as calculus and differential equations, where recognizing patterns and simplifying expressions is essential for problem-solving.

Factoring Out the Common Factor

Now that we've identified (x-1) as the common factor in x(x-1) + 4(x-1), let's factor it out. To do this, we rewrite the expression as (x-1)(...). We need to fill in the parentheses with the remaining terms after dividing each original term by (x-1).

So, when we divide x(x-1) by (x-1), we are left with x. And when we divide 4(x-1) by (x-1), we are left with 4. Therefore, we can rewrite the expression as (x-1)(x + 4). This is the factored form of the original expression.

The process of factoring out the common factor involves careful attention to detail. You must ensure that you correctly divide each term by the common factor and accurately represent the remaining terms inside the parentheses. A common mistake is to forget to include a term or to incorrectly divide, leading to an incorrect factored form. Always double-check your work to ensure accuracy.

Factoring out the common factor is a versatile technique that can be applied to various algebraic expressions. Whether you're dealing with binomials, trinomials, or polynomials with multiple terms, the principle remains the same: identify the common factor and factor it out. With practice, you'll become adept at spotting common factors and efficiently factoring them out, simplifying complex expressions with ease.

Checking Your Work

After factoring, it's always a good idea to check your work to ensure that you haven't made any mistakes. The easiest way to do this is by expanding the factored expression and verifying that it matches the original expression. In our case, we factored x(x-1) + 4(x-1) into (x-1)(x+4). Let's expand (x-1)(x+4) to see if we get back the original expression.

To expand (x-1)(x+4), we use the distributive property (also known as the FOIL method). This means we multiply each term in the first set of parentheses by each term in the second set of parentheses:

x * x = x^2 x * 4 = 4x -1 * x = -x -1 * 4 = -4

Now, we add these terms together: x^2 + 4x - x - 4. Simplifying this, we get x^2 + 3x - 4. This is not the same as our original expression. Let's re-examine the original expression x(x-1) + 4(x-1) and expand that.

Expanding x(x-1) + 4(x-1) gives us x^2 - x + 4x - 4. Combining like terms, we get x^2 + 3x - 4. Aha! It matches after expansion. Our factored form (x-1)(x+4) is indeed correct!

By checking your work through expansion, you can catch any errors and gain confidence in your factoring skills. This practice reinforces your understanding of both factoring and expanding, helping you to become more proficient in algebra. Checking your work is not just a formality; it's an essential step in problem-solving that ensures accuracy and builds a solid foundation for more advanced mathematical concepts.

Final Answer

So, after factoring the expression x(x-1) + 4(x-1) completely, we arrive at the factored form (x-1)(x+4). This is our final answer. Factoring this expression involved identifying the common factor (x-1), factoring it out, and then simplifying.

Mastering factoring is a key skill in algebra, and it's something that you'll use over and over again in your mathematical journey. Keep practicing, and don't be afraid to tackle more complex factoring problems. With persistence and a solid understanding of the basic principles, you'll become a factoring pro in no time! Remember to always double-check your work, and happy factoring!

Hope this helps, and happy studying, everyone!