How To Fully Factorise 18m² - 6m A Step By Step Guide

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Factoring expressions is a fundamental skill in algebra, guys. It's like unlocking a secret code that reveals the inner workings of an equation. When you fully factorise an expression, you break it down into its simplest multiplicative components, making it easier to solve equations, simplify fractions, and understand the behavior of functions. Today, we're going to dive deep into how to fully factorise the expression 18m² - 6m. Trust me, by the end of this article, you'll be a factoring pro!

Understanding Factorisation

Before we jump into the specifics of our problem, let's get a solid grasp of what factoring actually means. In simple terms, factoring is the reverse process of expansion. When we expand, we multiply terms together, like distributing a number across a parenthesis. Factoring, on the other hand, involves finding the common factors within an expression and pulling them out to rewrite the expression as a product of simpler terms. Think of it like dismantling a machine into its individual parts – you're breaking down the expression into its fundamental building blocks.

Why is factoring so important, you ask? Well, imagine trying to solve a complex equation without factoring. It would be like trying to navigate a maze blindfolded! Factoring allows us to simplify equations, making them easier to solve. It also helps us identify key characteristics of functions, such as their roots (where they cross the x-axis). Plus, it's a crucial skill for more advanced mathematical concepts like calculus and linear algebra. So, mastering factoring is like equipping yourself with a powerful tool that will serve you well throughout your mathematical journey.

Now, when we talk about fully factorising, we mean breaking down the expression until it can't be factored any further. It's like disassembling that machine until you have only the smallest, indivisible components. This ensures that we have the most simplified representation of the expression, making it easier to work with in any context. There are several techniques for factoring, including finding the greatest common factor (GCF), difference of squares, and quadratic trinomial factoring. We'll be focusing on the GCF method in this article, as it's the key to fully factorising 18m² - 6m.

Identifying the Greatest Common Factor (GCF)

The first step in fully factorising 18m² - 6m is to identify the greatest common factor (GCF). The GCF is the largest factor that divides evenly into all terms in the expression. It's like finding the biggest piece that all the parts of your machine have in common. To find the GCF, we need to consider both the coefficients (the numbers in front of the variables) and the variables themselves.

Let's start with the coefficients: 18 and -6. What's the largest number that divides evenly into both 18 and 6? Well, the factors of 18 are 1, 2, 3, 6, 9, and 18, while the factors of 6 are 1, 2, 3, and 6. The largest number they have in common is 6. So, 6 is part of our GCF. But remember, we need to consider the variables as well.

Now, let's look at the variables. We have m² (which is m * m) and m. What's the largest variable factor they have in common? They both have at least one 'm'. So, 'm' is also part of our GCF. If we had m³ and m², the GCF would be m² because that's the highest power of 'm' that divides evenly into both terms. In this case, since both terms have at least 'm' to the power of 1, our variable GCF is simply 'm'.

Combining the coefficient GCF (6) and the variable GCF (m), we find that the greatest common factor of 18m² and -6m is 6m. This means that 6m is the largest term that can be factored out of both parts of our expression. Identifying the GCF is a crucial step because it ensures that we factor the expression completely, leaving no common factors behind. It's like making sure you've removed the biggest shared component before moving on to the smaller parts.

Factoring Out the GCF: Step-by-Step

Now that we've identified the GCF as 6m, the next step is to factor it out of the expression 18m² - 6m. This involves dividing each term in the expression by the GCF and writing the result in parentheses. Think of it like separating the common piece from each part of your machine and putting the remaining parts into a neatly organized box. Let's break it down step-by-step:

  1. Write down the GCF: Start by writing down the GCF, which we found to be 6m. This will be the term that sits outside the parentheses in our factored expression.
  2. Divide the first term by the GCF: Divide 18m² by 6m. Remember the rules of exponents: when dividing terms with the same base, you subtract the exponents. So, 18m² / 6m = (18/6) * (m²/m) = 3 * m^(2-1) = 3m. This term will be inside the parentheses.
  3. Divide the second term by the GCF: Divide -6m by 6m. This is straightforward: -6m / 6m = -1. This term will also be inside the parentheses.
  4. Write the factored expression: Now, we combine the GCF with the results of our divisions. We write the GCF outside the parentheses, followed by the terms we obtained inside the parentheses, separated by the appropriate sign. So, the factored expression is 6m(3m - 1).

Let's recap what we've done. We took the expression 18m² - 6m, identified the GCF as 6m, and then divided each term by the GCF. This gave us the terms 3m and -1, which we placed inside the parentheses. The GCF, 6m, sits outside the parentheses, multiplying the entire expression inside. This is the essence of factoring: rewriting an expression as a product of its factors. This process might seem tricky at first, but with practice, it becomes second nature. It's like learning to ride a bike – once you get the hang of it, you'll be zipping through factoring problems like a pro!

Verifying the Factorisation

Okay, we've factored the expression, but how do we know we did it correctly? This is where verification comes in. It's like double-checking your work to make sure everything is in order. The easiest way to verify our factorisation is to expand the factored expression and see if it matches the original expression. Remember, expanding is the reverse of factoring, so it's a perfect way to check our work.

Our factored expression is 6m(3m - 1). To expand this, we need to distribute the 6m across the terms inside the parentheses. This means multiplying 6m by both 3m and -1.

  1. Multiply 6m by 3m: 6m * 3m = (6 * 3) * (m * m) = 18m².
  2. Multiply 6m by -1: 6m * -1 = -6m.
  3. Combine the terms: Now, we add the results together: 18m² + (-6m) = 18m² - 6m.

Hey, look at that! The expanded expression, 18m² - 6m, is exactly the same as our original expression. This means our factorisation is correct! We successfully factored 18m² - 6m into 6m(3m - 1). Verification is a crucial step because it gives you confidence in your answer and helps you catch any mistakes. It's like proofreading a document before submitting it – you want to make sure everything is perfect.

If the expanded expression didn't match the original expression, it would mean we made a mistake somewhere in our factoring process. We would then need to go back and carefully review each step, looking for errors in our GCF identification or division. But in this case, we nailed it! We've not only factored the expression but also verified our answer, ensuring we're on the right track. So, always remember to verify your factorisation – it's the key to mastering this important algebraic skill.

Final Factored Form: 6m(3m - 1)

So, there you have it! We've taken the expression 18m² - 6m and fully factorised it into 6m(3m - 1). This is the final, simplified form of the expression, where it's expressed as a product of its factors. We started by understanding the concept of factorisation, then identified the greatest common factor (GCF), factored it out, and finally verified our result. This comprehensive approach ensures that we not only get the correct answer but also understand the underlying principles of factoring.

Remember, the key to successful factoring is practice. The more you work with different expressions, the more comfortable you'll become with identifying common factors and applying the appropriate techniques. Don't be afraid to make mistakes – they're a natural part of the learning process. Just be sure to review your work, verify your answers, and learn from your errors. With dedication and perseverance, you'll become a factoring master in no time!

Fully factorising expressions like 18m² - 6m is a fundamental skill in algebra, guys, and mastering it will open doors to more advanced mathematical concepts. By breaking down expressions into their simplest multiplicative components, we gain a deeper understanding of their structure and behavior. So, keep practicing, keep exploring, and keep pushing your mathematical boundaries. You've got this!

In summary, the fully factorised form of 18m² - 6m is 6m(3m - 1). We achieved this by identifying the GCF (6m) and factoring it out of the original expression. Always remember to verify your work to ensure accuracy and build confidence in your factoring skills. Now, go forth and conquer those factoring challenges!