Factorizing Expressions: Step-by-Step Guide

Hey guys! Today, we're diving into the world of factorization, a crucial skill in algebra. We'll break down two expressions step-by-step, making sure you understand the process. So, grab your pencils, and let's get started!

(i) Factorizing a + 2 + 2ab + a^2b

Okay, so our first expression is a + 2 + 2ab + a^2b. At first glance, it might seem a bit intimidating, but don't worry! We can tackle this by looking for common factors and rearranging terms to make the pattern clearer. This is where we put our thinking caps on and really analyze the expression. What do we see? How can we group terms together to reveal those hidden factors?

Step 1: Rearranging the Terms

In factorization, the first key step involves rearranging the terms to identify potential groupings. This allows us to spot common factors more easily. So, let’s start by rearranging our expression: a + 2 + 2ab + a^2b. A good way to approach this is to group terms that share variables or constants. Looking at the expression, we can group the terms as (a^2b + 2ab) + (a + 2). This grouping immediately helps us see some similarities. Think of it like organizing your closet – you group similar items together to make it easier to find what you need!

Step 2: Identifying Common Factors

Now that we've grouped our terms, let’s identify common factors within each group. In the first group, (a^2b + 2ab), we can see that both terms have ab in common. Remember, a common factor is a term that divides evenly into multiple terms. Factoring out ab from the first group gives us ab(a + 2). This is like finding the biggest container that can hold similar items. In the second group, (a + 2), we can see that there's no immediately obvious common factor other than 1. However, notice that (a + 2) is the same expression we have in the first group's factored form. This is a huge clue that we're on the right track! It's like finding puzzle pieces that fit together – a very satisfying moment.

Step 3: Factoring by Grouping

Here’s where the magic happens! We've identified the common factors within each group, and now we can factor by grouping. Our expression now looks like this: ab(a + 2) + 1(a + 2). Notice that (a + 2) is a common factor for the entire expression. We can factor (a + 2) out, which gives us (a + 2)(ab + 1). And there you have it! We’ve successfully factored the expression. Think of this step as putting the final touches on a masterpiece. You’ve taken all the individual elements and combined them into a cohesive whole.

Step 4: The Final Factorized Expression

So, the final factorized form of a + 2 + 2ab + a^2b is (a + 2)(ab + 1). We've broken down the expression into its simplest multiplicative components. It’s like dismantling a complex machine into its basic parts. Each part is crucial, but together they form something greater. Congratulations! You've just conquered your first factorization problem. Remember, practice makes perfect, so keep at it!

(iii) Factorizing 6(x+2y)^3 - 8(x+2y)^2 + 4(x+2y)

Alright, let's move on to the second expression: 6(x+2y)^3 - 8(x+2y)^2 + 4(x+2y). This one looks a bit more complex, but we'll use the same principles of identifying common factors and breaking it down step by step. The key here is to recognize the repeating term (x+2y) and treat it as a single unit. It’s like recognizing a pattern in a complex design – once you see it, the rest becomes much clearer.

Step 1: Identifying the Common Factor

The first thing to notice here is that all three terms have (x+2y) in common. Not just that, but they also have numerical factors that share a common divisor. This is where our number sense comes into play. We need to find the greatest common factor (GCF) of the coefficients 6, 8, and 4. The GCF of these numbers is 2. This means we can factor out 2 as well as (x+2y). So, the greatest common factor for the entire expression is 2(x+2y). Identifying the GCF is like finding the master key that unlocks all the parts of the expression.

Step 2: Factoring out the GCF

Now, let’s factor out the GCF, 2(x+2y), from the expression. This gives us:

2(x+2y)[3(x+2y)^2 - 4(x+2y) + 2]

We’ve successfully pulled out the common factor, making the remaining expression inside the brackets a bit more manageable. This is like decluttering a room – you remove the unnecessary items to reveal the underlying structure. Now, let’s focus on the expression inside the brackets:

3(x+2y)^2 - 4(x+2y) + 2

Step 3: Analyzing the Remaining Quadratic Expression

We have a quadratic expression in terms of (x+2y). To make things simpler, let’s substitute u = (x+2y). This substitution transforms the expression into:

3u^2 - 4u + 2

This looks much more familiar, right? It's a standard quadratic expression that we can try to factorize. Substituting u for (x+2y) is like changing the language of the expression to something we understand better. It simplifies the problem and makes the next steps clearer.

Step 4: Checking for Factorization of the Quadratic Expression

Now, we need to check if the quadratic expression 3u^2 - 4u + 2 can be factored further. We can try to find two numbers that multiply to give 3 * 2 = 6 and add up to -4. However, there are no such integer numbers. This means that the quadratic expression 3u^2 - 4u + 2 cannot be factored using simple integer methods. Sometimes, expressions just don’t factor nicely, and that’s perfectly okay! It’s like trying to fit puzzle pieces that just don’t quite match – you have to accept that they don’t fit.

Step 5: Substituting Back and the Final Answer

Since we couldn't factor the quadratic expression further, we substitute back (x+2y) for u. Our final factorized expression is:

2(x+2y)[3(x+2y)^2 - 4(x+2y) + 2]

And that’s it! We’ve taken a complex expression and broken it down as much as possible. Even though we couldn’t factor the quadratic part, we still managed to simplify the expression significantly. It’s like reaching the end of a challenging journey – you might not have reached the exact destination you envisioned, but you’ve still made progress.

Conclusion: Mastering Factorization

So, guys, we've tackled two factorization problems today, and hopefully, you're feeling more confident about your skills. Remember, the key to mastering factorization is practice and understanding the basic principles. Look for common factors, rearrange terms, and don’t be afraid to try different approaches. Sometimes, the solution isn't immediately obvious, but with persistence, you'll get there! Factorization is a fundamental tool in algebra, and it will serve you well in more advanced topics. Keep practicing, and you’ll become a factorization pro in no time!

If you have any questions or want to try more examples, feel free to ask. Happy factorizing!