Factor 3b² - 27b + 60: A Step-by-Step Guide

Hey guys! Ever found yourself staring at a quadratic expression and wondering how to break it down into simpler terms? You're not alone! Factoring is a crucial skill in algebra, and it can unlock a whole world of problem-solving abilities. Today, we're going to dive deep into factoring the expression 3b² - 27b + 60 completely. We'll break it down step-by-step, so even if you're new to factoring, you'll be a pro by the end of this guide. Trust me, it's like cracking a secret code – super satisfying when you get it!

Understanding the Basics of Factoring

Before we jump into the specific expression, let's quickly review what factoring actually means. In simple terms, factoring is the reverse of expanding. When we expand, we multiply terms together to get a larger expression. Factoring is the opposite – we're trying to find the smaller expressions that, when multiplied, give us the original expression. Think of it like this: if expanding is like building a house brick by brick, factoring is like taking the house apart to see the individual bricks. For quadratic expressions like 3b² - 27b + 60, we're looking for two binomials (expressions with two terms) that, when multiplied together, give us the original quadratic. But before we dive into binomials, the first and often most crucial step is to look for a greatest common factor (GCF). This is the largest factor that divides into all the terms in the expression. Identifying and factoring out the GCF will simplify the problem significantly and make the subsequent steps much easier. Factoring is not just about finding the right numbers; it's about understanding the underlying structure of algebraic expressions. It's a fundamental skill that you'll use time and time again in higher-level math courses, from calculus to linear algebra. So, mastering factoring now will set you up for success in the future. And remember, practice makes perfect! The more you factor expressions, the more comfortable and confident you'll become.

Step 1: Identifying the Greatest Common Factor (GCF)

The first thing we always want to do when factoring is to look for a greatest common factor (GCF). This is the largest number and/or variable that divides evenly into all the terms in our expression. In the expression 3b² - 27b + 60, let's look at the coefficients first: 3, -27, and 60. What's the largest number that divides evenly into all three of these? If you guessed 3, you're absolutely right! Now, let's look at the variables. We have b² in the first term, b in the second term, and no variable in the third term. This means the only common factor involving variables is 1, since the third term doesn't have a 'b'. So, our GCF is simply 3. Factoring out the GCF is like peeling away the outer layer to reveal the core of the expression. It simplifies things immensely and makes the subsequent factoring steps much easier. It's a critical step that you should always check for before attempting any other factoring techniques. Think of it this way: if you're trying to assemble a puzzle, you wouldn't start by trying to force pieces together randomly. You'd first sort the pieces, maybe group them by color or shape. Factoring out the GCF is like sorting the pieces of our algebraic puzzle. It helps us see the underlying structure and makes the solution much clearer. And remember, factoring out the GCF doesn't change the value of the expression; it just rewrites it in a different form. This is a key concept in algebra, and it's essential for understanding how to manipulate expressions and equations effectively.

Step 2: Factoring out the GCF

Now that we've identified the GCF as 3, let's factor it out of the expression 3b² - 27b + 60. Factoring out the GCF means dividing each term in the expression by 3 and writing the result inside parentheses. So, we have: 3b² / 3 = b² -27b / 3 = -9b 60 / 3 = 20. Therefore, 3b² - 27b + 60 becomes 3(b² - 9b + 20). See how much simpler the expression inside the parentheses is? That's the power of factoring out the GCF! It transforms a seemingly complex expression into something much more manageable. This is a crucial step because it often reduces the coefficients, making the remaining factoring process easier. By factoring out the GCF, we've essentially taken a big problem and broken it down into two smaller, more approachable problems. We've dealt with the '3' outside the parentheses, and now we can focus on factoring the quadratic expression inside: b² - 9b + 20. This is a common strategy in problem-solving – break down a complex problem into smaller, more manageable parts. It's like tackling a giant to-do list by focusing on one task at a time. Each small victory builds momentum and makes the overall goal seem less daunting. And that's exactly what we're doing here. By factoring out the GCF, we've taken a significant step towards completely factoring the original expression. Now, we're ready to move on to the next step: factoring the quadratic expression inside the parentheses.

Step 3: Factoring the Quadratic Expression

Okay, we've got 3(b² - 9b + 20). Now, let's focus on factoring the quadratic expression inside the parentheses: b² - 9b + 20. This is a trinomial (an expression with three terms) in the form of ax² + bx + c, where a = 1, b = -9, and c = 20. When a = 1, factoring the trinomial becomes a bit simpler. We're looking for two numbers that multiply to c (20) and add up to b (-9). Let's think about the factors of 20: 1 and 20, 2 and 10, 4 and 5. Which of these pairs, when added together (or subtracted, considering the signs), could give us -9? Hmmm... 4 and 5 seem promising, but we need a negative 9. Remember that a negative times a negative is a positive, so let's try -4 and -5. -4 multiplied by -5 is indeed 20, and -4 plus -5 is -9. Bingo! We've found our numbers. This step is like being a detective, searching for clues and piecing them together to solve a mystery. You're looking for the specific combination of numbers that satisfies both the multiplication and addition conditions. It might take some trial and error, but don't get discouraged! With practice, you'll develop an intuition for these types of problems. And remember, there are always multiple strategies you can use. If finding the numbers mentally is challenging, you can try listing out all the factors of c and systematically checking each pair. The key is to find a method that works for you and stick with it. Once you've found the magic numbers, you're ready to write the expression in its factored form.

Step 4: Writing the Factored Form

Now that we've found the two numbers, -4 and -5, that multiply to 20 and add up to -9, we can write the quadratic expression b² - 9b + 20 in its factored form. Since our numbers are -4 and -5, the factored form will be (b - 4)(b - 5). It's that simple! We've taken a quadratic expression and broken it down into two binomials. But we're not quite done yet! Remember the 3 we factored out earlier? We need to include that in our final factored expression. So, the completely factored form of 3b² - 27b + 60 is 3(b - 4)(b - 5). This is the grand finale! We've taken the original expression and transformed it into its most simplified form. It's like taking a complex machine and understanding its individual components, how they fit together, and how the whole thing works. Factoring is not just about finding the answer; it's about gaining a deeper understanding of the structure of algebraic expressions. And this understanding is what will empower you to tackle more challenging problems in the future. Now, let's take a moment to appreciate what we've accomplished. We started with a seemingly daunting expression and, by breaking it down step-by-step, we've successfully factored it completely. This is a testament to the power of systematic problem-solving and the importance of mastering fundamental algebraic skills.

Checking Your Work

It's always a good idea to check your work, especially in math! To check our factored form, 3(b - 4)(b - 5), we can expand it and see if we get back our original expression, 3b² - 27b + 60. Let's start by expanding (b - 4)(b - 5). Using the FOIL method (First, Outer, Inner, Last), we get: b * b = b² b * -5 = -5b -4 * b = -4b -4 * -5 = 20. Combining these terms, we have b² - 5b - 4b + 20, which simplifies to b² - 9b + 20. Now, we need to multiply this expression by the 3 we factored out earlier: 3(b² - 9b + 20). Distributing the 3, we get: 3 * b² = 3b² 3 * -9b = -27b 3 * 20 = 60. So, we have 3b² - 27b + 60. And guess what? That's our original expression! This means our factoring is correct. Checking your work is like double-checking your map before setting off on a journey. It ensures you're on the right track and helps you avoid potential mistakes. In factoring, expanding the factored form is a reliable way to verify your answer. It provides a sense of confidence and solidifies your understanding of the process. And remember, even if you don't get the original expression right away, the process of checking can help you identify where you went wrong and learn from your mistakes. This is a valuable skill that extends beyond math. It's about being meticulous, paying attention to detail, and taking the time to ensure accuracy. These are qualities that will serve you well in all aspects of life.

Conclusion

Alright guys, we did it! We successfully factored the expression 3b² - 27b + 60 completely, and we even checked our work to make sure we got it right. We learned the importance of identifying the GCF, factoring out the GCF, and then factoring the remaining quadratic expression. We also saw how checking our work by expanding the factored form can give us confidence in our answer. Factoring is a fundamental skill in algebra, and mastering it will open doors to more advanced mathematical concepts. It's like learning the alphabet before you can read and write – it's a building block for future success. So, keep practicing, keep exploring, and don't be afraid to tackle challenging problems. The more you practice, the more comfortable and confident you'll become with factoring. And remember, math is not just about memorizing formulas and procedures; it's about developing critical thinking skills and problem-solving abilities. These are skills that will benefit you in all areas of your life. So, embrace the challenge, enjoy the process, and keep learning! And if you ever get stuck, don't hesitate to ask for help. There are tons of resources available, from textbooks and online tutorials to teachers and classmates. The key is to keep learning and growing. Now that you've mastered factoring this expression, you're well on your way to becoming a factoring pro! Go forth and conquer more algebraic challenges!