Factoring Made Easy: A Guide To $12xy - 9x - 8y + 6$

Hey math enthusiasts! Today, we're diving deep into the world of factoring. We're going to break down how to completely factor the expression $12xy - 9x - 8y + 6$. Don't worry if it looks a bit intimidating at first; we'll go through it step by step, making it as clear as possible. Factoring is a fundamental skill in algebra, and once you get the hang of it, you'll find it incredibly useful for solving equations and simplifying expressions. So, grab your pencils and let's get started. We will find out what is the completely factored form of $12 x y-9 x-8 y+6$.

Understanding the Basics of Factoring

Before we jump into the specific problem, let's quickly recap what factoring is all about. Factoring is essentially the reverse process of multiplying. When we factor an expression, we're trying to rewrite it as a product of simpler expressions (usually in parentheses). Think of it like breaking down a number into its prime factors. For example, the number 12 can be factored into 2 x 2 x 3. Similarly, we aim to break down our algebraic expression into its component factors. The goal here is to find common factors among the terms and group them together. This process helps us simplify the original expression and make it easier to work with. There are different techniques for factoring, and the method we use depends on the form of the expression. In our case, we'll use a technique called factoring by grouping. Factoring is not just about getting the right answer; it's also about understanding the structure of the expression and how different parts relate to each other. This understanding will be crucial as you progress in mathematics, especially when you start dealing with more complex equations and functions. So, let's get our hands dirty and start learning.

Factoring by grouping is particularly useful when we have four terms, like in our example, $12xy - 9x - 8y + 6$. The strategy involves grouping terms together, finding common factors within each group, and then seeing if we can find a common factor across both groups. We'll aim to create identical binomials that can then be factored out. This can feel a little like a puzzle, but with practice, it becomes more straightforward. The idea is to transform the original expression into a product of simpler expressions. The ability to factor is invaluable in mathematics. It is used everywhere from simplifying algebraic expressions to solving equations to simplifying fractions. So, as we dive into this problem, remember that the goal is not just to get the answer, but also to build a deeper understanding of algebraic structures.

Step-by-Step Factoring of $12xy - 9x - 8y + 6$

Now, let's get to the main event: factoring $12xy - 9x - 8y + 6$. We'll follow these steps carefully:

1. Group the Terms: First, group the terms into two pairs: $(12xy - 9x)$ and $(-8y + 6)$. This is the initial step for factoring by grouping. The grouping is strategically done to identify common factors within each pair of terms. Remember that the sign of the third term (-8y) is included in the second group to preserve the original expression's value. Always be careful with the signs during this grouping process.

2. Factor out Common Factors from Each Group: Look for the greatest common factor (GCF) in each group. For the first group, $(12xy - 9x)$, the GCF is $3x$. Factoring this out, we get $3x(4y - 3)$. For the second group, $(-8y + 6)$, the GCF is $-2$. Factoring this out, we get $-2(4y - 3)$. The key here is to find the largest factor that divides evenly into all terms within a group. This step simplifies each group into a more manageable form and prepares them for the final factoring process. Correctly identifying and factoring out the GCF is crucial for successfully simplifying the expression. Remember, the goal is to make the remaining parts inside the parentheses identical so we can factor further.

3. Factor out the Common Binomial: Notice that both resulting terms, $3x(4y - 3)$ and $-2(4y - 3)$, have a common binomial factor: $(4y - 3)$. Now, factor out this common binomial. This means we rewrite the entire expression as a product of $(4y - 3)$ and the remaining factors ($3x$ and $-2$). So, the factored form becomes $(4y - 3)(3x - 2)$. This step brings us closer to the final factored form of the original expression. The success of factoring hinges on recognizing the common factors and correctly extracting them from the expression. This step neatly transforms the initial, somewhat complex, expression into a product of two simpler factors.

4. Final Result: The completely factored form of $12xy - 9x - 8y + 6$ is $(3x - 2)(4y - 3)$.

And there you have it! We've successfully factored the expression by grouping. The factored form, $(3x - 2)(4y - 3)$, is equivalent to the original expression but is now in a much more simplified form. This simplification is useful for solving equations, simplifying fractions, and understanding the behavior of the expression.

Verification and Further Practice

To make sure we got it right, let's multiply the factors back together to verify our answer. Multiplying $(3x - 2)(4y - 3)$, we get:

$3x * 4y - 3x * 3 - 2 * 4y + 2 * 3 = 12xy - 9x - 8y + 6$

This confirms that our factored form is correct. Verification is a crucial part of any mathematical process. Always double-check your work to ensure accuracy. If you want to get more practice with factoring, here are a few additional examples:

1. $6xy - 4x + 9y - 6$
2. $20ab + 15a - 8b - 6$
3. $10m^2 + 25m - 4m - 10$

Try to factor these expressions using the same method. Practice makes perfect, and the more you practice, the more comfortable and confident you'll become with factoring. Remember to carefully identify the common factors and group the terms correctly. Don't be afraid to take your time and double-check your work. These exercises will help you solidify your understanding of the process.

Conclusion: Mastering the Art of Factoring

Factoring is a powerful technique in algebra, and we've just scratched the surface. We've explored how to completely factor $12xy - 9x - 8y + 6$, and we've seen how factoring by grouping can be a great tool to simplify an expression. Keep practicing, and you'll find that factoring becomes second nature. It's a skill that will serve you well in various areas of mathematics. Remember to always look for the common factors, and don't be afraid to break down the problem into smaller steps. With each problem you solve, you'll gain a deeper understanding and appreciation for the beauty and utility of algebra. Keep exploring, and enjoy the journey of learning and discovery. Now that you have this knowledge, you are ready to tackle more complex expressions and solve more advanced mathematical problems with confidence. The ability to factor is one of the pillars of success in algebra.